τ w \\tau >\\tau _wcorrespond to the so-called normal Kovacs response.","Timeτ\\tau measures the average number of collisions per particle.Note that, in order to visualize the Kovacs effect, the relativedeviations of T(τ)T(\\tau ) from T st T_{\\mathrm {st}} in the response curves havebeen magnified by a factor r=5r=5 for all the protocols, except forthe transition one, for which r=100r=100.","All the curves correspondto normal and tangential restitution coefficients α=0.8\\alpha =0.8 andβ=0\\beta =0, respectively."],["Results and Discussion","Two data sets from molecular dynamics (MD) simulations (see ) of the granular gas for both the HP and the CP, together with their corresponding theoretical predictions, are represented in Fig.","REFA, which clearly shows the appearance of very large memory effects.","The temperature humps displayed here, of approximately $100\\%$ for the CP and $10\\%$ for the HP, are larger by at least two orders of magnitude than previously observed memory effects in athermal systems, which at most range from a few thousandths to a few hundredths of the stationary value of the relevant variable [17], [19].","The theoretical curves displayed in Fig.","REFA have been obtained by means of a bi-variate Maxwellian approximation, in which all the cumulants are assumed to be zero (see ).","Thus, the essential property driving the giant memory effect here is the existence of two independent temperature scales, translational and rotational, i.e., the breakdown of equipartition as given by the fact that $\\theta \\ne 1$ .","This is further illustrated in Fig.","REFB, which shows $\\theta (\\tau )$ for the same cases as in Fig.","REFA.","Again, the agreement between theory and simulation is excellent, even at the level of the two contributions to the total temperature.","Note that the relaxation time in the CP case is much longer than in the HP one.","Let us denote the earliest minimum and maximum in the temperature evolution as $T_{m}$ and $T_{M}$ , respectively.","We also define $\\mathcal {H}_{m}\\equiv T_{m}/T_{\\mathrm {st}}-1<0$ , $\\mathcal {H}_{M}\\equiv T_{M}/T_{\\mathrm {st}}-1>0$ , accordingly.","In Fig.","REF we present contour plots highlighting the regions with large $|\\mathcal {H}_m|$ (HP normal response, CP anomalous response) and $\\mathcal {H}_M$ (HP anomalous response, CP normal response).","We also plot the transition line $\\mathcal {H}_M=|\\mathcal {H}_m|$ from normal to anomalous response.","Huge Kovacs humps appear, especially in the normal region for the CP, in which the size of reported humps can be as large as $100\\%$ , relative to the steady temperature.","In order to characterize and quantify complexity in the thermal response we define the parameter $\\mathcal {S}$ , $\\mathcal {S} =\\mathrm {sgn} ({\\mathcal {H}_{1}})\\frac{\\min (|\\mathcal {H}_{m}|,\\mathcal {H}_{M})}{\\max (|\\mathcal {H}_{m}|,\\mathcal {H}_{M})},$ where $\\mathcal {H}_{1}$ (equal to either $\\mathcal {H}_{m}$ or $\\mathcal {H}_{M}$ ) is the magnitude of the earliest extremum.","Note that $\\mathcal {S}=0$ if there is only one extremum.","Thus, $\\mathcal {S}\\ne 0$ is the signature of the emergence of more complex response, i.e., with more than one extremum, in the normal-to-anomalous transition.","In the transition region, $|\\mathcal {S}|$ attains its maximum value, $|\\mathcal {S}|=1$ , when both extrema are of the same size and neither dominates.","The sign of $\\mathcal {S}\\in [-1,1]$ is equal to that of the earliest extremum, providing further information on the detailed structure of the response.","Figure REF represents $\\mathcal {S}$ as a function of the coefficients of restitution $\\alpha , \\beta $ , by solving the system of Eqs.","(REF ) and (REF ).","Panels A, C, and D correspond to the HP, whereas panels B and E correspond to the CP.","We have highlighted in blue (red) regions with $\\mathcal {S}<0$ ($\\mathcal {S}>0$ ), whereas all points with “simple” memory behavior, i.e., $\\mathcal {S}=0$ , remain white.","The complex regions are thin but still occupy noticeable sections of the parameter space, especially taking into account that they fall into ranges of experimental values of $\\alpha $ and $\\beta $ commonly present in a variety of materials [23].","In Fig.","REFA (HP), we clearly observe two zones rich in complex memory effects.","In panel C, the first complex zone is zoomed in.","This region is attached to the smooth limit, $\\beta \\sim -1$ , and only displays $\\mathcal {S}>0$ for high inelasticities, up to $\\alpha =1/\\sqrt{2}$ .","In panel D, the second complex region is zoomed in.","Within this region, which is close to the quasielastic limit $\\alpha \\sim 1$ , the system displays both $\\mathcal {S}> 0$ and $\\mathcal {S}< 0$ behavior.","In Fig.","REFB (CP), only one complex Kovacs region next to the quasielastic limit, inside which $\\mathcal {S}> 0$ , has been identified.","Panel E shows a close-up thereof.","It is important to mention that we have found that all the details of the complex regions emerge in the theoretical solution only when the cumulants are taken into account.","This indicates that the temperatures $T_t$ and $T_r$ do not explain in full detail by themselves the complexity of memory effects found in the rough granular gas.","Let us also point out that we have found for the HP a critical narrow region with a discontinuous transition from $\\mathcal {S}> 0$ to $\\mathcal {S}<0$ , which is signaled in panel D in Fig.","REF and represented in time evolution curves in Fig.","REF .","In this critical region, the system displays several different mechanisms for the transition from complex to simple—only one extremum—behavior.","The latter can be either the normal behavior of molecular systems [1], [9], [15] (also present in nonequilibrium systems) or the anomalous behavior exclusive of nonequilibrium systems [17], [19].","This is appropriately tagged in panels A and B of Fig.","REF , in which we have labeled the corresponding normal and anomalous regions.","In the narrow critical region, $\\mathcal {S}$ discontinuously jumps from (small) negative to positive values and three consecutive temperature extrema appear before stabilization in the stationary value is attained.","Otherwise, $\\mathcal {S}$ has a well-defined sign and the transition from complex to simple is continuous.","Figure: Contour plots of large extrema in the Kovacs response.Large minima (|ℋ m ||\\mathcal {H}_{m}|) are represented by bluishcontours and large maxima (ℋ M \\mathcal {H}_{M}) by reddishcontours.","Dashed lines indicate the ℋ M =|ℋ m |\\mathcal {H}_M=|\\mathcal {H}_m|transition curves, for which the predominant extremum changes sign,from maximum to minimum and vice versa.","Above and below these curveswe find ℋ M >|ℋ m |\\mathcal {H}_M> |\\mathcal {H}_m|(ℋ M <|ℋ m |\\mathcal {H}_M< |\\mathcal {H}_m|) and ℋ M <ℋ m \\mathcal {H}_M<\\mathcal {H}_m(ℋ M >|ℋ m |\\mathcal {H}_M>|\\mathcal {H}_m|) behaviors, respectively, in the HP(CP).","(A) Heating protocol (HP).","(B) Cooling protocol(CP).Figure: Kovacs complexity (𝒮\\mathcal {S}) phase diagrams.Density plot of 𝒮\\mathcal {S} vs. the α,β\\alpha , \\beta completespace parameter: (A) for the HP; (B) for the CP.","(C, D) Insets of complex memory regions next to the smoothand the quasielastic limits, respectively, for the HP.","(E)Inset of the complex memory region next to the quasielastic limitfor the CP.","In the HP, as seen in panels A and D,three different types of transition exist: 𝒮<0\\mathcal {S}<0(bluish, β<-0.65\\beta <-0.65), 𝒮>0\\mathcal {S}>0 (reddish,β>-0.65\\beta > -0.65), and the intermediate mechanism(𝒮≈0\\mathcal {S}\\approx 0) for β≈-0.65\\beta \\approx -0.65, as depictedbelow in Fig. B.","However, in the CP, seepanels B and E, only a 𝒮>0\\mathcal {S}>0-typetransition has been observed.Figure: Mechanisms for the transition from normal to anomalous inthe HP.","There are three of these mechanisms, which are shownhere by DSMC simulations (symbols) and our theoretical approach(lines).","Specifically, through (A) 𝒮>0\\mathcal {S}>0 atβ=0\\beta = 0, (B) 𝒮≈0\\mathcal {S} \\approx 0 (actually a tripleKovacs hump transition mechanism; see inset, where the linejoining the simulation points is a guide to the eye) atβ=-0.65\\beta = -0.65, and (C) 𝒮<0\\mathcal {S} < 0 atβ=-0.8\\beta = -0.8.","In order to assist in locating these transitionsin the parameter plane (α,β)(\\alpha ,\\beta ), their positions have beenannotated in Fig.","D.Figure REF displays the evolution curves of the temperature for the three different Kovacs transitions that we have found, in all cases depicted here for the HP: the $\\mathcal {S}>0$ transition in panel A, the $\\mathcal {S} \\approx 0$ transition in panel B (in its inset we show the three consecutive humps), and finally the $\\mathcal {S} < 0$ case in panel C. All theoretical curves are compared against the numerical solution of the kinetic equation, obtained by means of the direct simulation Monte Carlo (DSMC) method (see ).","The agreement is in general excellent, which once more shows the accuracy of our theoretical approach.","Although the size of the humps in the transition regions appear smaller than those in Fig.","REFA with simple memory behavior, yet they are of the same order of magnitude as those previously reported in the smooth granular gas [17]."],["Conclusions","Our work puts forward a general mechanism for the emergence of significantly large memory effects.","Enormous humps can be expected if the time evolution of the system under scrutiny is controlled by at least two independent and comparable in magnitude physical variables (here the translational and rotational temperatures) but with only one (here the total temperature) being relevant for the macroscopic or hydrodynamic description.","In addition, complex Kovacs response, with more than one extremum, can be expected if the time evolution of the system depends on several additional relevant variables.","Here, these additional variables are the fourth-order cumulants, whose sometimes nonmonotonic relaxation [25] probably enhances memory effect complexity.","So far, and despite the large number of previous works on analogous phenomena, only one extremum in the Kovacs response has been reported.","In thermal systems, in which the usual fluctuation-dissipation theorem holds and the stationary (equilibrium) distribution has the canonical shape, this is consistent with linear response results that predict normal behavior with only one maximum [16].","In athermal systems, the Kovacs response also includes anomalous behavior, but once more only one extremum has been observed [17], [19].","Therefore, an interesting prospect is elucidating whether or not the nonlinear theoretical framework developed in Ref.","[19] allows for complex response with more than one extremum.","Memory effects of the size and complexity we have observed here can potentially be present in other athermal or molecular systems.","Several variables of comparable magnitude must be coupled in their time evolution in nonlinear form, even if only a subset thereof is relevant in the macroscopic description.","This may be relevant, for instance, in active matter systems, where nonlinear effects are important in general [19], [29].","We think our results are also especially significant for future experimental work, since we expect these large memory effects to be measurable in granular dynamics experiments; a thermally homogeneous system may be achieved by means of homogeneous turbulent air fluidization [28]."],["Acknowledgements","The authors thank Prof. J. S. Urbach for fruitful discussions.","This work has been supported by the Spanish Agencia Estatal de Investigación Grants (partially financed by the ERDF) No.","MTM2017-84446-C2-2-R, MTM2014-56948-C2-2-P (A.L.","), and No.","FIS2016-76359-P (F.V.R.","and A.S.), and also by Universidad de Sevilla's VI Plan Propio de Investigación Grant PP2018/494 (A.P.).","Use of computing facilities from Extremadura Research Centre for Advanced Technologies (CETA-CIEMAT), funded by the ERDF is also acknowledged."],["Theory","The stochastic force ($\\mathbf {F}^{\\mathrm {wn}}$ ) has the form of a white noise: $\\langle {\\bf F}_i^{\\mathrm {wn}} (t) \\rangle = {\\bf 0}$ , $\\langle {\\bf F}_i ^{\\mathrm {wn}} (t) {\\bf F}_j ^{\\mathrm {wn}} (t^{\\prime }) \\rangle = \\mathsf {I}\\,m^2 \\chi _0^2\\delta _{ij}\\delta (t-t^{\\prime })$ , where indexes $i,j$ refer to particles, $\\mathsf {I}$ is the $3\\times 3$ unit matrix, and $\\chi _0^2$ is the white noise intensity.","In homogeneous states, the Boltzmann–Fokker–Planck equation characterizing the evolution of a granular gas submitted to the stochastic external force $\\mathbf {F}^{\\mathrm {wn}}$ is written as [25] $\\left(\\partial _t -\\frac{\\chi _0^2}{2}\\nabla _{\\mathbf {v}}^2\\right) f(\\mathbf {v} , ;t)={J[\\mathbf {v}, |f(t)]}.$ Above, $f(\\mathbf {v},;t)$ is the velocity distribution function ($\\mathbf {v}$ and $$ being the translational and angular velocities, respectively) and $J[\\mathbf {v}, |f]$ is the collision integral in the (inelastic) Boltzmann equation for rough spheres, which accounts for the collision rules [25] $ \\widehat{}\\cdot \\mathbf {u}^{\\prime }=-\\alpha \\,\\widehat{{}}\\cdot \\mathbf {u},\\quad \\widehat{{}}\\times \\mathbf {u}^{\\prime }=-\\beta \\,\\widehat{{}}\\times \\mathbf {u}.$ Here, the primes denote postcollisional values, $\\widehat{{}}$ is the unit collision vector joining the centers of the two colliding spheres (from the center of particle 1 to the center of particle 2) and $\\mathbf {u}=\\mathbf {v}_1-\\mathbf {v}_2-{\\frac{\\sigma }{2}}\\widehat{{}}\\times ({}_1+{}_2)$ is the relative velocity of the spheres at their contact point.","The coefficient of normal restitution $\\alpha $ takes values between 0 (completely inelastic collision) and 1 (completely elastic collision), while the coefficient of tangential restitution $\\beta $ takes values between $-1$ (completely smooth collision, unchanged angular velocities) and 1 (completely rough collision) [24].","Given any one-particle function $A(\\mathbf {v},)$ , its average is defined as $\\langle A(t)\\rangle =n^{-1}\\int d \\mathbf {v}\\int d \\,A( \\mathbf {v}, )f(\\mathbf {v}, ;t)$ , where the number density is given by $n=\\int d \\mathbf {v}\\int d \\, f( \\mathbf {v}, ;t)$ .","The basic physical properties are the translational ($T_t$ ), rotational ($T_r$ ), and total ($T$ ) granular temperatures, i.e., ${T_t={\\frac{m}{3}}\\langle v^2\\rangle ,\\quad T_r={\\frac{I}{3}}\\langle \\omega ^2\\rangle ,} \\quad { T=\\frac{T_t+T_r}{2}=T_t\\frac{1+\\theta }{2}},$ where $I$ is the moment of inertia.","We have introduced the temperature ratio $\\theta \\equiv T_r/ T_t$ , which is relevant for the analysis that follows and whose steady-state value is independent of the driving amplitude $\\chi _0^2$ .","The evolution equations for $T_t$ , $T_r$ , and $T$ are $\\partial _t T_t-{m\\chi _0^2}=-\\xi _t T_t,\\quad {\\partial _t T_r=-\\xi _r T_r},$ $\\partial _t T-\\frac{m\\chi _0^2}{2}=-\\zeta T.$ The equations for $T_t$ and $T_r$ have been obtained by multiplying both sides of Eq.","(REF ) by the translational and rotational kinetic energies, respectively, and integrating over all particle velocity values.","The parameters $\\xi _t$ and $\\xi _r$ are $\\xi _t=-{\\frac{m}{3n T_t}}\\int d\\mathbf {v}\\int d {}\\,v^2{J[\\mathbf {v}, {}| f]},$ $\\xi _r=-{\\frac{I}{3n T_r}}\\int d\\mathbf {v}\\int d {}\\,\\omega ^2{J[\\mathbf {v},{}| f]},$ respectively.","In general, neither $\\xi _t$ nor $\\xi _r$ does have a definite sign, whereas the cooling rate, $ \\zeta =\\frac{\\xi _t T_t+\\xi _rT_r}{2T}=\\frac{\\xi _t+\\xi _r{\\theta }}{1+\\theta },$ is always positive because energy is dissipated in collisions.","To proceed further, it is convenient to go to dimensionless variables.","Time is measured in a scale $\\tau $ , $\\tau =\\frac{1}{2}\\int _0^t dt^{\\prime }\\,\\nu (t^{\\prime }),\\quad \\nu (t)=4n\\sigma ^2 \\sqrt{\\pi T_t(t)/m},$ which is roughly the accumulated number of collisions per particle, because $\\nu (t)$ is the collision frequency.","Dimensionless velocities are introduced as $\\mathbf {c}(t)\\equiv \\frac{\\mathbf {v}}{\\sqrt{2 T_t(t)/m}}, \\quad \\mathbf {w}(t)\\equiv \\frac{{}}{\\sqrt{2 T_r(t)/I}},$ a reduced velocity distribution function as $\\phi (\\mathbf {c},\\mathbf {w};\\tau )\\equiv \\frac{1}{n}\\left[\\frac{4 T_t(t)T_r(t)}{m I}\\right]^{3/2}f(\\mathbf {v}, {};t),$ and the dimensionless collision kernel as $\\mathcal {J}[\\mathbf {c},\\mathbf {w}|\\phi (\\tau )]=\\frac{2}{n\\nu (t)}\\left[\\frac{4 T_t(t) T_r(t)}{m I}\\right]^{3/2} J[\\mathbf {v}, {}|f(t)].$ In dimensionless variables, the evolution equations for the temperatures can be written as Eqs.","(REF ) and (REF ) in the main text.","Therein, there appear the reduced collisional moments $\\mu _{20}\\equiv \\mu _{20}^{(0)}$ and $\\mu _{02}\\equiv \\mu _{02}^{(0)}$ , where $\\mu _{pq}^{(r)}(\\tau )\\equiv -\\int d\\mathbf {c}\\int d\\mathbf {w}\\,c^pw^q(\\mathbf {c}\\cdot \\mathbf {w})^r{\\mathcal {J}[\\mathbf {c},\\mathbf {w}|\\phi (\\tau )]}.$ Note that, aside from the nondimensionalizing factors, the production rates $\\xi _t$ and $\\xi _r$ are basically identical to $\\mu _{20}$ and $\\mu _{02}$ , respectively.","These are functionals of the whole distribution function and thus the evolution equations for the temperatures are not closed.","In order to close the dynamical equations, a formally exact expansion in orthogonal polynomials can be performed [25].","For isotropic states, we can expand the velocity distribution around the Maxwellian $\\phi _M(c,w) = \\pi ^{-3}e^{-c^2-w^2}$ , $\\phi (\\mathbf {c},\\mathbf {w};\\tau )=\\phi _M(c,w)\\sum _{j=0}^\\infty \\sum _{k=0}^\\infty \\sum _{\\ell =0}^\\infty a_{jk}^{(\\ell )}(\\tau )\\Psi _{jk}^{(\\ell )}(\\mathbf {c},\\mathbf {w}),$ where $\\Psi _{jk}^{(\\ell )}(\\mathbf {c},\\mathbf {w})$ are certain products of Laguerre and Legendre polynomials.","By normalization, $a_{00}^{(0)}=1$ , $a_{10}^{(0)}=a_{01}^{(0)}=0$ , and the lowest nontrivial coefficients are those associated with moments of degree four, namely $a_{20}^{(0)}=\\frac{4}{15}\\langle c^4\\rangle -1,\\quad a_{02}^{(0)}=\\frac{4}{15}\\langle w^4\\rangle -1,$ $a_{11}^{(0)}=\\frac{4}{9}\\langle c^2w^2\\rangle -1,\\quad a_{00}^{(1)}=\\frac{8}{15}\\left[\\langle (\\mathbf {c}\\cdot \\mathbf {w})^2\\rangle -\\frac{1}{3}\\langle c^2w^2\\rangle \\right] ,$ which we call the fourth-order cumulants henceforth.","Maxwellian approximation.- The simplest description is obtained by substituting the Maxwellian velocity distribution into the collision integrals (REF ).","Equivalently, one may consider that all the nontrivial cumulant vanish in this approach, which yields $ \\mu _{20,M}&=1-\\alpha ^2+\\frac{\\kappa (1+\\beta )}{(1+\\kappa )^2}\\left[2+\\kappa (1-\\beta )-\\theta (1+\\beta )\\right], \\\\\\mu _{02,M}&=\\frac{\\kappa (1+\\beta )}{(1+\\kappa )^2}\\left[2+\\kappa ^{-1}(1-\\beta )-\\theta ^{-1}(1+\\beta )\\right].","$ where $\\kappa \\equiv 4I/m\\sigma ^2$ is the dimensionless moment of inertia.","Insertion of Eq.","(REF ) into the evolution equations (REF ) and (REF ) in the main text gives rise to the Maxwellian approximation.","First Sonine approximation.- A more elaborate approximation can be done by incorporating the lowest order cumulants, which we defined in Eqs.","(REF ) and (REF ), as the first corrections to the Maxwellian.","A closed set of six coupled differential equations can be obtained for $\\theta (\\tau )$ , $\\gamma (\\tau )$ , $a_{20}^{(0)}(\\tau )$ , $a_{02}^{(0)}(\\tau )$ , $a_{11}^{(0)}(\\tau )$ , and $a_{00}^{(1)}(\\tau )$ .","To do so, explicit—yet not exact—expressions for the collision integrals $\\mu _{pq}^{(r)}$ with $p+q+2r=2$ and 4 are derived in terms of $\\theta $ and those lowest order cumulants.","These rather involved expressions can be found in the Supplemental Material of Ref.","[25], and are thus omitted here.","The resulting set of six differential equations can be numerically solved with appropriate initial conditions for each physical situation, as discussed in the main text.","In this way, we obtain the time evolution of the temperatures in the so-called first Sonine approximation, to which we refer throughout this work."],["Computer simulations","We use in this work data sets obtained from computer simulations from two independent and different methods: direct simulation Monte Carlo (DSMC) method, which obtains an exact numerical solution of the relevant kinetic equation [in our case Eq.","(REF )] and molecular dynamics (MD) simulation, which solves particles trajectories.","A detailed description of the DSMC method may be found elsewhere [30].","In our DSMC simulations, and in order to reduce statistical noise in the temperature time evolution curves, we have used an average of 100 statistical replicas of a system with $2\\times 10^6$ particles.","In the MD case, we have simulated 1000 inelastic hard spheres at a density $n\\sigma ^{3}=0.01$ and averaged over 500 trajectories."]],"string":"[\n [\n \"On the emergence of large and complex memory effects in nonequilibrium\\n fluids\"\n ],\n [\n \"Abstract Control of cooling and heating processes is essential in many industrial and biological processes.\",\n \"In fact, the time evolution of an observable quantity may differ according to the previous history of the system.\",\n \"For example, a system that is being subject to cooling and then, at a given time $t_{w}$ for which the instantaneous temperature is $T(t_w)=T_{\\\\mathrm{st}}$, is suddenly put in contact with a temperature source at $T_{\\\\mathrm{st}}$ may continue cooling down temporarily or, on the contrary, undergo a temperature rebound.\",\n \"According to current knowledge, there can be only one \\\"spurious\\\" and small peak/low.\",\n \"However, our results prove that, under certain conditions, more than one extremum may appear.\",\n \"Specifically, we have observed regions with two extrema and a critical point with three extrema.\",\n \"We have also detected cases where extraordinarily large extrema are observed, as large as the order of magnitude of the stationary value of the variable of interest.\",\n \"We show this by studying the thermal evolution of a low density set of macroscopic particles that do not preserve kinetic energy upon collision, i.e., a granular gas.\",\n \"We describe the mechanism that signals in this system the emergence of these complex and large memory effects, and explain why similar observations can be expected in a variety of systems.\"\n ],\n [\n \"Introduction\",\n \"Experimental observations reveal that the response to an excitation of complex condensed matter systems may depend on the entire system's history, and not just on the instantaneous value of the state variables [1], [2], [3], [4], [5], [6], [7], [8].\",\n \"This is usually called memory effect.\",\n \"Memory effects signal the breakdown of the thermodynamic (or hydrodynamic or macroscopic, depending on the physical context) description.\",\n \"Some typical memory effects include shape memory in polymers [4], aging and rejuvenation in spin glasses [9], active matter [10], and polymers [11], and the counterintuitive Mpemba effect [12], [13], [14].\",\n \"One of the most relevant memory effects related to thermal processes was originally observed by Kovacs and collaborators [1] in a polymer system, which was subject to quenching to a low temperature $T_{1}$ from an equilibrium state at temperature $T_{0}>T_1$ .\",\n \"After a long enough waiting time $t_{w}$ , but still relaxing towards equilibrium at $T_{1}$ , the temperature was suddenly increased back to an intermediate value $T_{\\\\mathrm {st}}$ , $T_{1}< T_{\\\\mathrm {st}} < T_{0}$ , such that the instantaneous value of the volume $\\\\mathcal {V}(t=t_w)$ equalled the equilibrium value $\\\\mathcal {V}_{\\\\mathrm {st}}$ corresponding to $T_{\\\\mathrm {st}}$ .\",\n \"Subsequently, the volume $\\\\mathcal {V}(t)$ did not remain flat but followed a nonmonotonic evolution.\",\n \"This nonmonotonic behavior, denominated later as Kovacs hump, consists in reaching one maximum before returning to its equilibrium value $\\\\mathcal {V}_{\\\\mathrm {st}}$ .\",\n \"We have described above the typical cooling procedure, but also a heating protocol can be considered ($T_{0}t_w$ .\",\n \"Quite recently, Kovacs-like memory effects have been thoroughly investigated in glassy systems [15], [16], granular fluids [17], [18], active matter [19], and disordered mechanical systems [20].\",\n \"The memory effect is typically quite small: the maximum deviation of $\\\\mathcal {V}(t)$ from the stationary value $\\\\mathcal {V}_{\\\\mathrm {st}}$ is several orders of magnitude smaller than $\\\\mathcal {V}_{\\\\mathrm {st}}$ [1], [17], [18], [15], [19].\",\n \"One of the main aims of our work is to show that the actual memory effects landscape is in general far more complex than expected.\",\n \"First, we show that several extrema—instead of only one—may appear in a single heating/cooling protocol à la Kovacs, contrary to what has been previously observed [15], [16], [17], [18], [19], [20].\",\n \"Second, very large memory humps, of the order of magnitude of the stationary value of the quantity of interest, can be observed.\",\n \"To the best of our knowledge, both features have not yet been reported in the literature.\",\n \"It must be noted that humps much larger than those predicted by linear response theory have recently been found in a nonlinear active matter model [19], but the relative deviation from the steady state is still of a few hundredths therein.\",\n \"Our results are found in a granular fluid but the mechanism presented for these features is quite general.\",\n \"Thus, giant and complex memory effects—not necessarily of the Kovacs-type—may be expected to appear in many natural and artificial systems.\",\n \"These memory effects have obviously important implications in problems like, for instance, system stabilization.\"\n ],\n [\n \"Description of the system and theoretical solution\",\n \"We consider a collection of identical solid spheres at low particle density so that collisions are always instantaneous and binary but inelastic, i.e., energy is not conserved and we deal with a granular gas [21], [22].\",\n \"In this case, particles have homogeneous mass density and we employ the rough hard sphere collisional model with constant coefficients of normal and tangential restitution, $\\\\alpha $ and $\\\\beta $ , respectively, which is quite realistic for a variety of materials at low particle density [23].\",\n \"Let us discuss first why the granular gas of rough spheres is a good candidate for eventually finding complex memory effects.\",\n \"Memory effects appear always in complex systems that consist of many structural units, for which a continuum description seems in principle appropriate.\",\n \"Within this kind of description, the instantaneous value of the complete set of macroscopic variables completely characterizes the system's time evolution [24].\",\n \"However, there are states that cannot be completely described only with the system macroscopic variables, and it is precisely for these states where a memory effect can be observed.\",\n \"As a matter of fact, this kind of distinct states for which the macroscopic description fails are theoretically very well understood in the context of the kinetic theory of gases [24].\",\n \"Furthermore, the granular gas of rough spheres can have extremely long relaxation times before it falls into a state where the macroscopic description is valid [25], [26], giving room to the emergence of eventual long lasting memory effects.\",\n \"And, most importantly, in this kind of system there are always two intrinsic, independent, and potentially large temperature scales—the translational and rotational granular temperatures—with a highly nonlinear coupling.\",\n \"All these facts open new spaces in the search of novel important features in complex memory effects, including eventually multiple extrema.\",\n \"To keep things simple, we consider the granular gas to be in a spatially homogeneous state at all times.\",\n \"The translational velocities are denoted by $\\\\mathbf {v}$ , while the angular (or rotational) velocities are denoted by ${}$ .\",\n \"The system is thermalized by a stochastic but homogeneous volume force [27], [28] characterized by a noise intensity $\\\\chi _0^2$ (see ).\",\n \"The kinetic description of our system starts from the corresponding Boltzmann–Fokker–Planck equation for the granular gas under this kind of forcing [26] (see ).\",\n \"The exact solution to this kinetic equation can be formally expressed by means of an expansion around a Maxwellian distribution with variances $T_t$ (translational temperature) and $T_r$ (rotational temperature) in the translational and angular velocities, respectively.\",\n \"The total granular temperature is given by $T=(T_t+T_r)/2$ , which is proportional to the mean kinetic (translational plus rotational) energy per particle.\",\n \"By adopting a dimensionless time scale $\\\\tau $ , proportional to the number of collisions per particle (see ), the evolution equations for the temperatures can be written as $\\\\frac{\\\\partial \\\\ln \\\\theta (\\\\tau )}{\\\\partial \\\\tau }=\\\\frac{2}{3}\\\\left[\\\\mu _{20}(\\\\tau )-\\\\mu _{02}(\\\\tau )-\\\\gamma (\\\\tau )\\\\right],$ $ \\\\frac{\\\\partial \\\\ln \\\\gamma (\\\\tau )}{\\\\partial \\\\tau }=\\\\mu _{20}(\\\\tau )-\\\\gamma (\\\\tau ).$ Above, $\\\\theta \\\\equiv T_r/T_t$ is the temperature ratio and $\\\\gamma \\\\equiv \\\\left(\\\\frac{T_{\\\\mathrm {noise}}}{T}\\\\frac{1+\\\\theta }{2}\\\\right)^{\\\\frac{3}{2}}$ is a dimensionless measure of the noise intensity, where $T_{\\\\mathrm {noise}}\\\\equiv m(3\\\\chi _0^2/4\\\\sqrt{\\\\pi }n\\\\sigma ^2)^{\\\\frac{2}{3}}$ with $n$ being the particle density, and $m$ and $\\\\sigma $ being the mass and diameter of a sphere, respectively.\",\n \"The reduced collisional moments $\\\\mu _{20}$ and $\\\\mu _{02}$ (see for more reference) are functionals of the whole velocity distribution and therefore the above system of equations is not closed.\",\n \"In order to solve it, we use the first Sonine approximation, which refers to the first nontrivial truncation of the aforementioned exact infinite expansion [25].\",\n \"For this, together with Eqs.\",\n \"(REF ) and (REF ), we need to incorporate the evolution equations for the fourth-order cumulants and the initial values of $\\\\gamma $ , $\\\\theta $ , and these cumulants (see ).\",\n \"We generate a common initial state for all the temperature evolution curves we subsequently analyze.\",\n \"At an arbitrary time, which we choose to be the time origin $\\\\tau =0$ , and over an arbitrary previous microscopic state, we apply an instantaneous thermal pulse to the granular gas.\",\n \"In this way, the rotational modes ($T_r$ ) of the granular gas are quenched, whereas the translational modes ($T_t$ ) are subject to a large heating.\",\n \"As a result, most of the initial kinetic energy is in the translational modes, so that the total initial temperature is $T(0)=T_t(0)/2$ and the temperature ratio is $\\\\theta (0)=0$ .\",\n \"Moreover, all the fourth-order cumulants vanish because the initial distribution that results from the heat pulse is a bi-variate ($T_r, T_t$ ) Maxwellian.\",\n \"By this procedure, the system forgets all the previous thermal history of the system, assuring always the same nonequilibrium initial state.\",\n \"From the initial state we have just characterized, the granular gas is left to cool freely, due to the intrinsically inelastic particle collisions [21], for a waiting time $\\\\tau _w$ .\",\n \"At $\\\\tau =\\\\tau _{w}$ , we suddenly apply the stochastic force, with an intensity such that the corresponding steady temperature $T_{\\\\mathrm {st}}$ to be reached equals the instantaneous temperature value at the moment of turning the noise on, i.e., $T_{\\\\mathrm {st}} = T(\\\\tau _w)$ .\",\n \"If $T(\\\\tau >\\\\tau _w)$ further departs from $T_{\\\\mathrm {st}}$ , then a Kovacs-like memory effect is observed.\",\n \"What we call protocol is the thermal procedure that we have just described.\",\n \"Depending on the waiting time $\\\\tau _w$ for turning the stochastic heating on, the system spans different classes of temperature evolution curves.\",\n \"This is depicted and explained in Fig.\",\n \"REF .\",\n \"For the sake of simplicity, we investigate the two limiting cases in Fig.\",\n \"REF ; i.e., $\\\\tau _w=0$ (heating protocol, HP) and $\\\\tau _w\\\\rightarrow \\\\infty $ (cooling protocol, CP).\",\n \"In the CP, since the system is left cooling down for a long time, the system is already in the homogeneous cooling state (HCS) [21] at $\\\\tau _w$ .\",\n \"In the HCS, the temperature $T(t)$ is the only relevant variable and decays in time following Haff's law [21], whereas the temperature ratio and the fourth-order cumulants are time independent, and their values depend only on the parameters $\\\\alpha $ and $\\\\beta $ [25].\",\n \"Therefore, the conditions for this protocol at $\\\\tau _w$ are $\\\\gamma (\\\\tau _w)=\\\\gamma _{\\\\mathrm {st}}[(1+\\\\theta _{\\\\mathrm {st}})/(1+\\\\theta _{\\\\mathrm {HCS}})]^{\\\\frac{3}{2}}$ , $\\\\theta (\\\\tau _w)=\\\\theta _{\\\\mathrm {HCS}}$ , and the fourth-order cumulants at $\\\\tau =\\\\tau _w$ also equal their HCS values.\",\n \"In the HP, the initial conditions for the Kovacs experiment are different.\",\n \"Since we turn on the stochastic force right after the thermal pulse, the initial conditions are those of a bi-variate Maxwellian.\",\n \"Therefore, we have that $\\\\gamma (\\\\tau _w)=\\\\gamma _{\\\\mathrm {st}}(1+\\\\theta _{\\\\mathrm {st}})^{\\\\frac{3}{2}}$ , $\\\\theta (\\\\tau _w)=0$ , and, in addition, all the cumulants vanish at $\\\\tau =\\\\tau _w$ .\",\n \"Figure: Illustration of the protocols considered in this work.The granular gas is prepared in an initial state (τ=0\\\\tau =0) forwhich all the energy is concentrated in the translational degreesof freedom, as described in the text.\",\n \"In a first stage,0<τ<τ w 0<\\\\tau <\\\\tau _{w}, the granular gas freely cools.\",\n \"Then, at thewaiting time τ=τ w \\\\tau =\\\\tau _{w}, the noise intensity is suddenlyincreased from zero to a value such that the instantaneoustemperature T(τ w )T(\\\\tau _{w}) coincides with the corresponding steadytemperature T st T_{\\\\mathrm {st}}.\",\n \"The curves shown for τ>τ w \\\\tau >\\\\tau _wcorrespond to the so-called normal Kovacs response.\",\n \"Timeτ\\\\tau measures the average number of collisions per particle.Note that, in order to visualize the Kovacs effect, the relativedeviations of T(τ)T(\\\\tau ) from T st T_{\\\\mathrm {st}} in the response curves havebeen magnified by a factor r=5r=5 for all the protocols, except forthe transition one, for which r=100r=100.\",\n \"All the curves correspondto normal and tangential restitution coefficients α=0.8\\\\alpha =0.8 andβ=0\\\\beta =0, respectively.\"\n ],\n [\n \"Results and Discussion\",\n \"Two data sets from molecular dynamics (MD) simulations (see ) of the granular gas for both the HP and the CP, together with their corresponding theoretical predictions, are represented in Fig.\",\n \"REFA, which clearly shows the appearance of very large memory effects.\",\n \"The temperature humps displayed here, of approximately $100\\\\%$ for the CP and $10\\\\%$ for the HP, are larger by at least two orders of magnitude than previously observed memory effects in athermal systems, which at most range from a few thousandths to a few hundredths of the stationary value of the relevant variable [17], [19].\",\n \"The theoretical curves displayed in Fig.\",\n \"REFA have been obtained by means of a bi-variate Maxwellian approximation, in which all the cumulants are assumed to be zero (see ).\",\n \"Thus, the essential property driving the giant memory effect here is the existence of two independent temperature scales, translational and rotational, i.e., the breakdown of equipartition as given by the fact that $\\\\theta \\\\ne 1$ .\",\n \"This is further illustrated in Fig.\",\n \"REFB, which shows $\\\\theta (\\\\tau )$ for the same cases as in Fig.\",\n \"REFA.\",\n \"Again, the agreement between theory and simulation is excellent, even at the level of the two contributions to the total temperature.\",\n \"Note that the relaxation time in the CP case is much longer than in the HP one.\",\n \"Let us denote the earliest minimum and maximum in the temperature evolution as $T_{m}$ and $T_{M}$ , respectively.\",\n \"We also define $\\\\mathcal {H}_{m}\\\\equiv T_{m}/T_{\\\\mathrm {st}}-1<0$ , $\\\\mathcal {H}_{M}\\\\equiv T_{M}/T_{\\\\mathrm {st}}-1>0$ , accordingly.\",\n \"In Fig.\",\n \"REF we present contour plots highlighting the regions with large $|\\\\mathcal {H}_m|$ (HP normal response, CP anomalous response) and $\\\\mathcal {H}_M$ (HP anomalous response, CP normal response).\",\n \"We also plot the transition line $\\\\mathcal {H}_M=|\\\\mathcal {H}_m|$ from normal to anomalous response.\",\n \"Huge Kovacs humps appear, especially in the normal region for the CP, in which the size of reported humps can be as large as $100\\\\%$ , relative to the steady temperature.\",\n \"In order to characterize and quantify complexity in the thermal response we define the parameter $\\\\mathcal {S}$ , $\\\\mathcal {S} =\\\\mathrm {sgn} ({\\\\mathcal {H}_{1}})\\\\frac{\\\\min (|\\\\mathcal {H}_{m}|,\\\\mathcal {H}_{M})}{\\\\max (|\\\\mathcal {H}_{m}|,\\\\mathcal {H}_{M})},$ where $\\\\mathcal {H}_{1}$ (equal to either $\\\\mathcal {H}_{m}$ or $\\\\mathcal {H}_{M}$ ) is the magnitude of the earliest extremum.\",\n \"Note that $\\\\mathcal {S}=0$ if there is only one extremum.\",\n \"Thus, $\\\\mathcal {S}\\\\ne 0$ is the signature of the emergence of more complex response, i.e., with more than one extremum, in the normal-to-anomalous transition.\",\n \"In the transition region, $|\\\\mathcal {S}|$ attains its maximum value, $|\\\\mathcal {S}|=1$ , when both extrema are of the same size and neither dominates.\",\n \"The sign of $\\\\mathcal {S}\\\\in [-1,1]$ is equal to that of the earliest extremum, providing further information on the detailed structure of the response.\",\n \"Figure REF represents $\\\\mathcal {S}$ as a function of the coefficients of restitution $\\\\alpha , \\\\beta $ , by solving the system of Eqs.\",\n \"(REF ) and (REF ).\",\n \"Panels A, C, and D correspond to the HP, whereas panels B and E correspond to the CP.\",\n \"We have highlighted in blue (red) regions with $\\\\mathcal {S}<0$ ($\\\\mathcal {S}>0$ ), whereas all points with “simple” memory behavior, i.e., $\\\\mathcal {S}=0$ , remain white.\",\n \"The complex regions are thin but still occupy noticeable sections of the parameter space, especially taking into account that they fall into ranges of experimental values of $\\\\alpha $ and $\\\\beta $ commonly present in a variety of materials [23].\",\n \"In Fig.\",\n \"REFA (HP), we clearly observe two zones rich in complex memory effects.\",\n \"In panel C, the first complex zone is zoomed in.\",\n \"This region is attached to the smooth limit, $\\\\beta \\\\sim -1$ , and only displays $\\\\mathcal {S}>0$ for high inelasticities, up to $\\\\alpha =1/\\\\sqrt{2}$ .\",\n \"In panel D, the second complex region is zoomed in.\",\n \"Within this region, which is close to the quasielastic limit $\\\\alpha \\\\sim 1$ , the system displays both $\\\\mathcal {S}> 0$ and $\\\\mathcal {S}< 0$ behavior.\",\n \"In Fig.\",\n \"REFB (CP), only one complex Kovacs region next to the quasielastic limit, inside which $\\\\mathcal {S}> 0$ , has been identified.\",\n \"Panel E shows a close-up thereof.\",\n \"It is important to mention that we have found that all the details of the complex regions emerge in the theoretical solution only when the cumulants are taken into account.\",\n \"This indicates that the temperatures $T_t$ and $T_r$ do not explain in full detail by themselves the complexity of memory effects found in the rough granular gas.\",\n \"Let us also point out that we have found for the HP a critical narrow region with a discontinuous transition from $\\\\mathcal {S}> 0$ to $\\\\mathcal {S}<0$ , which is signaled in panel D in Fig.\",\n \"REF and represented in time evolution curves in Fig.\",\n \"REF .\",\n \"In this critical region, the system displays several different mechanisms for the transition from complex to simple—only one extremum—behavior.\",\n \"The latter can be either the normal behavior of molecular systems [1], [9], [15] (also present in nonequilibrium systems) or the anomalous behavior exclusive of nonequilibrium systems [17], [19].\",\n \"This is appropriately tagged in panels A and B of Fig.\",\n \"REF , in which we have labeled the corresponding normal and anomalous regions.\",\n \"In the narrow critical region, $\\\\mathcal {S}$ discontinuously jumps from (small) negative to positive values and three consecutive temperature extrema appear before stabilization in the stationary value is attained.\",\n \"Otherwise, $\\\\mathcal {S}$ has a well-defined sign and the transition from complex to simple is continuous.\",\n \"Figure: Contour plots of large extrema in the Kovacs response.Large minima (|ℋ m ||\\\\mathcal {H}_{m}|) are represented by bluishcontours and large maxima (ℋ M \\\\mathcal {H}_{M}) by reddishcontours.\",\n \"Dashed lines indicate the ℋ M =|ℋ m |\\\\mathcal {H}_M=|\\\\mathcal {H}_m|transition curves, for which the predominant extremum changes sign,from maximum to minimum and vice versa.\",\n \"Above and below these curveswe find ℋ M >|ℋ m |\\\\mathcal {H}_M> |\\\\mathcal {H}_m|(ℋ M <|ℋ m |\\\\mathcal {H}_M< |\\\\mathcal {H}_m|) and ℋ M <ℋ m \\\\mathcal {H}_M<\\\\mathcal {H}_m(ℋ M >|ℋ m |\\\\mathcal {H}_M>|\\\\mathcal {H}_m|) behaviors, respectively, in the HP(CP).\",\n \"(A) Heating protocol (HP).\",\n \"(B) Cooling protocol(CP).Figure: Kovacs complexity (𝒮\\\\mathcal {S}) phase diagrams.Density plot of 𝒮\\\\mathcal {S} vs. the α,β\\\\alpha , \\\\beta completespace parameter: (A) for the HP; (B) for the CP.\",\n \"(C, D) Insets of complex memory regions next to the smoothand the quasielastic limits, respectively, for the HP.\",\n \"(E)Inset of the complex memory region next to the quasielastic limitfor the CP.\",\n \"In the HP, as seen in panels A and D,three different types of transition exist: 𝒮<0\\\\mathcal {S}<0(bluish, β<-0.65\\\\beta <-0.65), 𝒮>0\\\\mathcal {S}>0 (reddish,β>-0.65\\\\beta > -0.65), and the intermediate mechanism(𝒮≈0\\\\mathcal {S}\\\\approx 0) for β≈-0.65\\\\beta \\\\approx -0.65, as depictedbelow in Fig. B.\",\n \"However, in the CP, seepanels B and E, only a 𝒮>0\\\\mathcal {S}>0-typetransition has been observed.Figure: Mechanisms for the transition from normal to anomalous inthe HP.\",\n \"There are three of these mechanisms, which are shownhere by DSMC simulations (symbols) and our theoretical approach(lines).\",\n \"Specifically, through (A) 𝒮>0\\\\mathcal {S}>0 atβ=0\\\\beta = 0, (B) 𝒮≈0\\\\mathcal {S} \\\\approx 0 (actually a tripleKovacs hump transition mechanism; see inset, where the linejoining the simulation points is a guide to the eye) atβ=-0.65\\\\beta = -0.65, and (C) 𝒮<0\\\\mathcal {S} < 0 atβ=-0.8\\\\beta = -0.8.\",\n \"In order to assist in locating these transitionsin the parameter plane (α,β)(\\\\alpha ,\\\\beta ), their positions have beenannotated in Fig.\",\n \"D.Figure REF displays the evolution curves of the temperature for the three different Kovacs transitions that we have found, in all cases depicted here for the HP: the $\\\\mathcal {S}>0$ transition in panel A, the $\\\\mathcal {S} \\\\approx 0$ transition in panel B (in its inset we show the three consecutive humps), and finally the $\\\\mathcal {S} < 0$ case in panel C. All theoretical curves are compared against the numerical solution of the kinetic equation, obtained by means of the direct simulation Monte Carlo (DSMC) method (see ).\",\n \"The agreement is in general excellent, which once more shows the accuracy of our theoretical approach.\",\n \"Although the size of the humps in the transition regions appear smaller than those in Fig.\",\n \"REFA with simple memory behavior, yet they are of the same order of magnitude as those previously reported in the smooth granular gas [17].\"\n ],\n [\n \"Conclusions\",\n \"Our work puts forward a general mechanism for the emergence of significantly large memory effects.\",\n \"Enormous humps can be expected if the time evolution of the system under scrutiny is controlled by at least two independent and comparable in magnitude physical variables (here the translational and rotational temperatures) but with only one (here the total temperature) being relevant for the macroscopic or hydrodynamic description.\",\n \"In addition, complex Kovacs response, with more than one extremum, can be expected if the time evolution of the system depends on several additional relevant variables.\",\n \"Here, these additional variables are the fourth-order cumulants, whose sometimes nonmonotonic relaxation [25] probably enhances memory effect complexity.\",\n \"So far, and despite the large number of previous works on analogous phenomena, only one extremum in the Kovacs response has been reported.\",\n \"In thermal systems, in which the usual fluctuation-dissipation theorem holds and the stationary (equilibrium) distribution has the canonical shape, this is consistent with linear response results that predict normal behavior with only one maximum [16].\",\n \"In athermal systems, the Kovacs response also includes anomalous behavior, but once more only one extremum has been observed [17], [19].\",\n \"Therefore, an interesting prospect is elucidating whether or not the nonlinear theoretical framework developed in Ref.\",\n \"[19] allows for complex response with more than one extremum.\",\n \"Memory effects of the size and complexity we have observed here can potentially be present in other athermal or molecular systems.\",\n \"Several variables of comparable magnitude must be coupled in their time evolution in nonlinear form, even if only a subset thereof is relevant in the macroscopic description.\",\n \"This may be relevant, for instance, in active matter systems, where nonlinear effects are important in general [19], [29].\",\n \"We think our results are also especially significant for future experimental work, since we expect these large memory effects to be measurable in granular dynamics experiments; a thermally homogeneous system may be achieved by means of homogeneous turbulent air fluidization [28].\"\n ],\n [\n \"Acknowledgements\",\n \"The authors thank Prof. J. S. Urbach for fruitful discussions.\",\n \"This work has been supported by the Spanish Agencia Estatal de Investigación Grants (partially financed by the ERDF) No.\",\n \"MTM2017-84446-C2-2-R, MTM2014-56948-C2-2-P (A.L.\",\n \"), and No.\",\n \"FIS2016-76359-P (F.V.R.\",\n \"and A.S.), and also by Universidad de Sevilla's VI Plan Propio de Investigación Grant PP2018/494 (A.P.).\",\n \"Use of computing facilities from Extremadura Research Centre for Advanced Technologies (CETA-CIEMAT), funded by the ERDF is also acknowledged.\"\n ],\n [\n \"Theory\",\n \"The stochastic force ($\\\\mathbf {F}^{\\\\mathrm {wn}}$ ) has the form of a white noise: $\\\\langle {\\\\bf F}_i^{\\\\mathrm {wn}} (t) \\\\rangle = {\\\\bf 0}$ , $\\\\langle {\\\\bf F}_i ^{\\\\mathrm {wn}} (t) {\\\\bf F}_j ^{\\\\mathrm {wn}} (t^{\\\\prime }) \\\\rangle = \\\\mathsf {I}\\\\,m^2 \\\\chi _0^2\\\\delta _{ij}\\\\delta (t-t^{\\\\prime })$ , where indexes $i,j$ refer to particles, $\\\\mathsf {I}$ is the $3\\\\times 3$ unit matrix, and $\\\\chi _0^2$ is the white noise intensity.\",\n \"In homogeneous states, the Boltzmann–Fokker–Planck equation characterizing the evolution of a granular gas submitted to the stochastic external force $\\\\mathbf {F}^{\\\\mathrm {wn}}$ is written as [25] $\\\\left(\\\\partial _t -\\\\frac{\\\\chi _0^2}{2}\\\\nabla _{\\\\mathbf {v}}^2\\\\right) f(\\\\mathbf {v} , ;t)={J[\\\\mathbf {v}, |f(t)]}.$ Above, $f(\\\\mathbf {v},;t)$ is the velocity distribution function ($\\\\mathbf {v}$ and $$ being the translational and angular velocities, respectively) and $J[\\\\mathbf {v}, |f]$ is the collision integral in the (inelastic) Boltzmann equation for rough spheres, which accounts for the collision rules [25] $ \\\\widehat{}\\\\cdot \\\\mathbf {u}^{\\\\prime }=-\\\\alpha \\\\,\\\\widehat{{}}\\\\cdot \\\\mathbf {u},\\\\quad \\\\widehat{{}}\\\\times \\\\mathbf {u}^{\\\\prime }=-\\\\beta \\\\,\\\\widehat{{}}\\\\times \\\\mathbf {u}.$ Here, the primes denote postcollisional values, $\\\\widehat{{}}$ is the unit collision vector joining the centers of the two colliding spheres (from the center of particle 1 to the center of particle 2) and $\\\\mathbf {u}=\\\\mathbf {v}_1-\\\\mathbf {v}_2-{\\\\frac{\\\\sigma }{2}}\\\\widehat{{}}\\\\times ({}_1+{}_2)$ is the relative velocity of the spheres at their contact point.\",\n \"The coefficient of normal restitution $\\\\alpha $ takes values between 0 (completely inelastic collision) and 1 (completely elastic collision), while the coefficient of tangential restitution $\\\\beta $ takes values between $-1$ (completely smooth collision, unchanged angular velocities) and 1 (completely rough collision) [24].\",\n \"Given any one-particle function $A(\\\\mathbf {v},)$ , its average is defined as $\\\\langle A(t)\\\\rangle =n^{-1}\\\\int d \\\\mathbf {v}\\\\int d \\\\,A( \\\\mathbf {v}, )f(\\\\mathbf {v}, ;t)$ , where the number density is given by $n=\\\\int d \\\\mathbf {v}\\\\int d \\\\, f( \\\\mathbf {v}, ;t)$ .\",\n \"The basic physical properties are the translational ($T_t$ ), rotational ($T_r$ ), and total ($T$ ) granular temperatures, i.e., ${T_t={\\\\frac{m}{3}}\\\\langle v^2\\\\rangle ,\\\\quad T_r={\\\\frac{I}{3}}\\\\langle \\\\omega ^2\\\\rangle ,} \\\\quad { T=\\\\frac{T_t+T_r}{2}=T_t\\\\frac{1+\\\\theta }{2}},$ where $I$ is the moment of inertia.\",\n \"We have introduced the temperature ratio $\\\\theta \\\\equiv T_r/ T_t$ , which is relevant for the analysis that follows and whose steady-state value is independent of the driving amplitude $\\\\chi _0^2$ .\",\n \"The evolution equations for $T_t$ , $T_r$ , and $T$ are $\\\\partial _t T_t-{m\\\\chi _0^2}=-\\\\xi _t T_t,\\\\quad {\\\\partial _t T_r=-\\\\xi _r T_r},$ $\\\\partial _t T-\\\\frac{m\\\\chi _0^2}{2}=-\\\\zeta T.$ The equations for $T_t$ and $T_r$ have been obtained by multiplying both sides of Eq.\",\n \"(REF ) by the translational and rotational kinetic energies, respectively, and integrating over all particle velocity values.\",\n \"The parameters $\\\\xi _t$ and $\\\\xi _r$ are $\\\\xi _t=-{\\\\frac{m}{3n T_t}}\\\\int d\\\\mathbf {v}\\\\int d {}\\\\,v^2{J[\\\\mathbf {v}, {}| f]},$ $\\\\xi _r=-{\\\\frac{I}{3n T_r}}\\\\int d\\\\mathbf {v}\\\\int d {}\\\\,\\\\omega ^2{J[\\\\mathbf {v},{}| f]},$ respectively.\",\n \"In general, neither $\\\\xi _t$ nor $\\\\xi _r$ does have a definite sign, whereas the cooling rate, $ \\\\zeta =\\\\frac{\\\\xi _t T_t+\\\\xi _rT_r}{2T}=\\\\frac{\\\\xi _t+\\\\xi _r{\\\\theta }}{1+\\\\theta },$ is always positive because energy is dissipated in collisions.\",\n \"To proceed further, it is convenient to go to dimensionless variables.\",\n \"Time is measured in a scale $\\\\tau $ , $\\\\tau =\\\\frac{1}{2}\\\\int _0^t dt^{\\\\prime }\\\\,\\\\nu (t^{\\\\prime }),\\\\quad \\\\nu (t)=4n\\\\sigma ^2 \\\\sqrt{\\\\pi T_t(t)/m},$ which is roughly the accumulated number of collisions per particle, because $\\\\nu (t)$ is the collision frequency.\",\n \"Dimensionless velocities are introduced as $\\\\mathbf {c}(t)\\\\equiv \\\\frac{\\\\mathbf {v}}{\\\\sqrt{2 T_t(t)/m}}, \\\\quad \\\\mathbf {w}(t)\\\\equiv \\\\frac{{}}{\\\\sqrt{2 T_r(t)/I}},$ a reduced velocity distribution function as $\\\\phi (\\\\mathbf {c},\\\\mathbf {w};\\\\tau )\\\\equiv \\\\frac{1}{n}\\\\left[\\\\frac{4 T_t(t)T_r(t)}{m I}\\\\right]^{3/2}f(\\\\mathbf {v}, {};t),$ and the dimensionless collision kernel as $\\\\mathcal {J}[\\\\mathbf {c},\\\\mathbf {w}|\\\\phi (\\\\tau )]=\\\\frac{2}{n\\\\nu (t)}\\\\left[\\\\frac{4 T_t(t) T_r(t)}{m I}\\\\right]^{3/2} J[\\\\mathbf {v}, {}|f(t)].$ In dimensionless variables, the evolution equations for the temperatures can be written as Eqs.\",\n \"(REF ) and (REF ) in the main text.\",\n \"Therein, there appear the reduced collisional moments $\\\\mu _{20}\\\\equiv \\\\mu _{20}^{(0)}$ and $\\\\mu _{02}\\\\equiv \\\\mu _{02}^{(0)}$ , where $\\\\mu _{pq}^{(r)}(\\\\tau )\\\\equiv -\\\\int d\\\\mathbf {c}\\\\int d\\\\mathbf {w}\\\\,c^pw^q(\\\\mathbf {c}\\\\cdot \\\\mathbf {w})^r{\\\\mathcal {J}[\\\\mathbf {c},\\\\mathbf {w}|\\\\phi (\\\\tau )]}.$ Note that, aside from the nondimensionalizing factors, the production rates $\\\\xi _t$ and $\\\\xi _r$ are basically identical to $\\\\mu _{20}$ and $\\\\mu _{02}$ , respectively.\",\n \"These are functionals of the whole distribution function and thus the evolution equations for the temperatures are not closed.\",\n \"In order to close the dynamical equations, a formally exact expansion in orthogonal polynomials can be performed [25].\",\n \"For isotropic states, we can expand the velocity distribution around the Maxwellian $\\\\phi _M(c,w) = \\\\pi ^{-3}e^{-c^2-w^2}$ , $\\\\phi (\\\\mathbf {c},\\\\mathbf {w};\\\\tau )=\\\\phi _M(c,w)\\\\sum _{j=0}^\\\\infty \\\\sum _{k=0}^\\\\infty \\\\sum _{\\\\ell =0}^\\\\infty a_{jk}^{(\\\\ell )}(\\\\tau )\\\\Psi _{jk}^{(\\\\ell )}(\\\\mathbf {c},\\\\mathbf {w}),$ where $\\\\Psi _{jk}^{(\\\\ell )}(\\\\mathbf {c},\\\\mathbf {w})$ are certain products of Laguerre and Legendre polynomials.\",\n \"By normalization, $a_{00}^{(0)}=1$ , $a_{10}^{(0)}=a_{01}^{(0)}=0$ , and the lowest nontrivial coefficients are those associated with moments of degree four, namely $a_{20}^{(0)}=\\\\frac{4}{15}\\\\langle c^4\\\\rangle -1,\\\\quad a_{02}^{(0)}=\\\\frac{4}{15}\\\\langle w^4\\\\rangle -1,$ $a_{11}^{(0)}=\\\\frac{4}{9}\\\\langle c^2w^2\\\\rangle -1,\\\\quad a_{00}^{(1)}=\\\\frac{8}{15}\\\\left[\\\\langle (\\\\mathbf {c}\\\\cdot \\\\mathbf {w})^2\\\\rangle -\\\\frac{1}{3}\\\\langle c^2w^2\\\\rangle \\\\right] ,$ which we call the fourth-order cumulants henceforth.\",\n \"Maxwellian approximation.- The simplest description is obtained by substituting the Maxwellian velocity distribution into the collision integrals (REF ).\",\n \"Equivalently, one may consider that all the nontrivial cumulant vanish in this approach, which yields $ \\\\mu _{20,M}&=1-\\\\alpha ^2+\\\\frac{\\\\kappa (1+\\\\beta )}{(1+\\\\kappa )^2}\\\\left[2+\\\\kappa (1-\\\\beta )-\\\\theta (1+\\\\beta )\\\\right], \\\\\\\\\\\\mu _{02,M}&=\\\\frac{\\\\kappa (1+\\\\beta )}{(1+\\\\kappa )^2}\\\\left[2+\\\\kappa ^{-1}(1-\\\\beta )-\\\\theta ^{-1}(1+\\\\beta )\\\\right].\",\n \"$ where $\\\\kappa \\\\equiv 4I/m\\\\sigma ^2$ is the dimensionless moment of inertia.\",\n \"Insertion of Eq.\",\n \"(REF ) into the evolution equations (REF ) and (REF ) in the main text gives rise to the Maxwellian approximation.\",\n \"First Sonine approximation.- A more elaborate approximation can be done by incorporating the lowest order cumulants, which we defined in Eqs.\",\n \"(REF ) and (REF ), as the first corrections to the Maxwellian.\",\n \"A closed set of six coupled differential equations can be obtained for $\\\\theta (\\\\tau )$ , $\\\\gamma (\\\\tau )$ , $a_{20}^{(0)}(\\\\tau )$ , $a_{02}^{(0)}(\\\\tau )$ , $a_{11}^{(0)}(\\\\tau )$ , and $a_{00}^{(1)}(\\\\tau )$ .\",\n \"To do so, explicit—yet not exact—expressions for the collision integrals $\\\\mu _{pq}^{(r)}$ with $p+q+2r=2$ and 4 are derived in terms of $\\\\theta $ and those lowest order cumulants.\",\n \"These rather involved expressions can be found in the Supplemental Material of Ref.\",\n \"[25], and are thus omitted here.\",\n \"The resulting set of six differential equations can be numerically solved with appropriate initial conditions for each physical situation, as discussed in the main text.\",\n \"In this way, we obtain the time evolution of the temperatures in the so-called first Sonine approximation, to which we refer throughout this work.\"\n ],\n [\n \"Computer simulations\",\n \"We use in this work data sets obtained from computer simulations from two independent and different methods: direct simulation Monte Carlo (DSMC) method, which obtains an exact numerical solution of the relevant kinetic equation [in our case Eq.\",\n \"(REF )] and molecular dynamics (MD) simulation, which solves particles trajectories.\",\n \"A detailed description of the DSMC method may be found elsewhere [30].\",\n \"In our DSMC simulations, and in order to reduce statistical noise in the temperature time evolution curves, we have used an average of 100 statistical replicas of a system with $2\\\\times 10^6$ particles.\",\n \"In the MD case, we have simulated 1000 inelastic hard spheres at a density $n\\\\sigma ^{3}=0.01$ and averaged over 500 trajectories.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08401"}}},{"rowIdx":391,"cells":{"text":{"kind":"list like","value":[["Flux Transport Dynamo: From Modelling Irregularities to Making\n Predictions"],["Abstract The flux transport dynamo, in which the poloidal magnetic field is generated by the Babcock--Leighton mechanism and the meridional circulation plays a crucial role, has emerged as an attractive model for the solar cycle.","Based on theoretical calculations done with this model, we argue that the fluctuations in the Babcock--Leighton mechanism and the fluctuations in the meridional circulation are the most likely causes of the irregularities of the solar cycle.","With our increased theoretical understanding of how these irregularities arise, it can be possible to predict a future solar cycle by feeding the appropriate observational data in a theoretical dynamo model."],["Introduction","The flux transport dynamo model, which started being developed about a quarter century ago [50], [19], [22], has emerged as an attractive theoretical model for the solar cycle.","There are several modern reviews [11], [12], [4], [34] surveying the current status of the field.","The present paper is not a comprehensive review, but is based on a talk in a Workshop at the International Space Science Institute (ISSI) highlighting the works done by the author and his coworkers.","Readers are assumed to be familiar with the phenomenology of the solar cycle and the basic concepts of MHD (such as flux freezing and magnetic buoyancy).","Readers not having this background are advised to look at the earlier reviews by the author [11], [12], which were written for wider readership.","The initial effort in this field of flux transport dynamo was directed towards developing periodic models to explain the various periodic aspects of the solar cycle.","Once sufficiently sophisticated periodic models were available, the next question was whether these theoretical models can be used to understand how the irregularities of the solar cycle arise.","There is also a related question: if we understand what causes the irregularities of the cycle, then will that enable us to predict future cycles?","We discuss the basic periodic model of the flux transport dynamo in the next Section.","Then § 3 discusses the possible causes of the irregularities of the solar cycle.","Afterwards in § 4 we address the question whether we are now in a position to predict future cycles.","Finally, in § 5 we summarize the limitations of the 2D kinematic dynamo models and the recent efforts of going beyond such simple models."],["The Basic Periodic Model","One completely non-controversial aspect of solar dynamo models is the generation of the toroidal field from the poloidal field by differential rotation.","Since differential rotation has now been mapped by helioseismology, this process can now be included in theoretical dynamo models quite realistically.","The toroidal field is primarily produced in the tachocline at the bottom of the convection zone and rises from there due to magnetic buoyancy to create the sunspots.","Although some authors have argued that the near-surface shear layer discovered by helioseismology can also be important for the generation of the toroidal field [2], the general view is that magnetic buoyancy would limit the growth of magnetic field in this region of strong super-adiabatic gradient.","To this generally accepted view that the toroidal field is primarily produced in the tachocline, the flux transport dynamo model adds the following assumptions.","The generation of the poloidal field from the toroidal field takes place due to the Babcock–Leighton mechanism.","The meridional circulation of the Sun plays a crucial role in the dynamo process.","We now comment on these two assumptions.","Bipolar sunspots on the solar surface appear with a tilt statistically increasing with latitude, in accordance with the so-called Joy's law.","This tilt is produced by the Coriolis force acting on the rising flux tubes [21].","[1] and [36] suggested that the poloidal field of the Sun is produced from the decay of such tilted bipolar sunspot pairs.","There is now enough evidence from observations of the solar surface that the poloidal field does get built up in this way.","The meridional circulation is observed to be poleward at the solar surface and advects the poloidal field generated there, in conformity with observational data of surface magnetic fields.","The material which is advected to the poles has to flow back equatorward through deeper layers within the solar convection zone.","Since this circulation is driven by the turbulent stresses in the convection zone, we expect the meridional circulation not to penetrate much below the bottom of the convection zone, although a slight penetration helps in explaining several aspects of observational data [42], [3].","The early models of the flux transport dynamo assumed the return flow of the meridional circulation to take place at the bottom of the convection zone, where the toroidal field generated by the differential rotation is advected equatorward with this flow, giving a natural explanation of the butterfly diagram representing the equatorward shift of the sunspot belt [19].","Such dynamo models have been remarkably successful in explaining many aspects of the observational data [8].","Figure: A complicated meridional circulation used by in a dynamocalculation—red corresponding to streamlines of clockwise circulation and blueto anti-clockwise circulation.","Note that the flow near the bottom at low latitudesis equatorward.","The butterfly diagram obtained with this circulation is solar-like(sunspot activity drifting to lower latitudes with time).While we still do not have unambiguous measurements of the return flow of the meridional circulation, some groups claim to have found evidence for the return flow well above the bottom of the convection zone [24], [54], [45].","However, [44] argue that the available helioseismology data still cannot rule out a one-cell meridional circulation spanning the whole of the convection zone in each hemisphere.","[27] showed that, even with a meridional circulation much more complicated than the one-cell pattern assumed in the earlier flux transport dynamo papers, it is still possible to match the relevant observational data as long as there is an equatorward flow at the bottom of the convection zone (see Figure 1)."],["The Origin of the Irregularities of the Solar Cycle","The earliest attempts of explaining irregularities of the solar cycle were by regarding them as nonlinear chaos arising out of the nonlinearities of the dynamo equations [51].","[5] argued that the Gnevyshev-Ohl rule in solar cycles (i.e.","the tendency of alternate cycles to lie above and below the running mean of cycle amplitudes) arises out of period doubling due to nonlinearities.","However, the simplest kinds of nonlinearities expected in dynamo equations tend to make the cycles more stable rather than producing irregularities and it has been suggested that stochastic fluctuations are more likely to be the primary reason behind producing irregularities [9].","The Babcock–Leighton mechanism for the generation of the poloidal field depends on the tilts of bipolar sunspots.","While the average tilt is given by Joy's law, we see considerable scatter around this average tilt, presumably caused by the action of turbulence in the convection zone on the rising flux tubes [37].","This scatter in the tilt angles is expected to introduce fluctuations in the Babcock–Leighton mechanism [14].","By including this fluctuation in the dynamo models, it is possible to explain many aspects of the irregularities of the cycles including the grand minima [17].","One other source of irregularities is the fluctuations in the meridional circulation.","A faster meridional circulation will make the solar cycles shorter and vice versa.","While we have actual data of meridional circulation variations for not more than a couple of decades, we have data for durations of solar cycles for more than a century, providing indications that the meridional circulation had fluctuations in the past with correlation times of the order of 30–40 years [32].","When the meridional circulation is slow and the cycles longer, diffusion has more time to act on the magnetic fields, making the cycles weaker.","On such ground, we expect longer cycles to be weaker and shorter cycles to be stronger, leading to what is called the Waldmeier effect [32].","Also, when the meridional circulation is sufficiently weak, theoretical dynamo models show that even grand minima can be induced [31].","To get these results, the correlation time of the meridional circulation fluctuations was taken to be considerably longer than the cycle period.","If the correlation time is taken too short, then one may not get these results [40].","We also emphasize that the effect of diffusion in making longer cycles weaker is vital for getting these results.","We need to take the value of diffusivity sufficiently high such that the diffusion time scale is shorter than or of the order of the cycle period.","This is not the case in the model of [20] in which diffusivity is very low.","A longer cycle in such a low-diffusivity model tends to be stronger because differential rotation has time to generate more toroidal field during a cycle, giving the opposite of the Waldmeier effect.","Differences between high- and low-diffusivity dynamos were studied by [53].","Clearly the high-diffusivity model yields results more in conformity with observational data.","Figure: According to the calculations of , the poloidal fieldstrength (γ\\gamma is the value of the poloidal field compared to its averagevalue) and the amplitude of the meridional circulation at the surface have tolie in the shaded region at the beginning of a cycle in order to force the dynamointo a grand minimum.They estimated the probability of this to be about 1.3%, corresponding toabout 13 grand minima in 11,000 years.By analyzing the contents of C-14 in old tree trunks and Be-10 in polar ice cores, it has now been possible to reconstruct the history of solar activity over a few millenia [48].","It has been estimated that there have been about 27 grand minima in the last 11,000 years [49].","Since grand minima can be caused both by fluctuations in the Babcock-Leighton mechanism and fluctuations in the meridional circulation, a full theoretical model of grand minima should combine both types of fluctuations.","If, at the beginning of a cycle, the poloidal field is too weak due to the fluctuations in the Babcock–Leighton mechanism or the meridional circulation is too weak, then the Sun may be forced into a grand minimum.","Assuming a Gaussian distribution for fluctuations in both the Babcock–Leighton mechanism and the meridional circulation, [18] developed a comprehensive theory of grand minima that agreed remarkably well with the statistical data of grand minima (see Figure 2 and its caption).","However, if there are no sunspots during grand minima, then the Babcock–Leighton mechanism which depends on sunspots may not be operational and how the Sun comes out of the grand minima is still rather poorly understood [33], [28].","While discussing irregularities of the solar cycle, it may be mentioned that these irregularities are correlated reasonably well in the two hemispheres of the Sun.","Strong cycles are usually strong in both the hemispheres and weak cycles are weak in both.","This requires a coupling between the two hemispheres, implying that the turbulent diffusion time over the convection zone could not be more than a few years [6], [23]."],["Predicting solar cycles","The first attempts of predicting solar cycles were based on using observational precursors of solar cycles.","There is considerable evidence that the polar field at the end of a solar cycle is correlated with the next cycle.","Since the polar field at the end of cycle 23 was rather weak, several authors [47], [46] predicted that the next cycle 24 would be weak.","The sunspot minimum between the cycles 23 and 24 (around 2005–2008) was the first sunspot minimum when sufficiently sophisticated models of the flux transport dynamo were available.","Whether these models could be used to predict the next cycle became an important question.","When a kinematic mean field dynamo code is run without introducing any fluctuations, one finds that the code settles down to a periodic solution if the various dynamo parameters are in the correct range.","In order to model actual solar cycles, one has to feed some observational data to the theoretical model in an appropriate manner and then run the code for a future cycle to generate a prediction.","The crucial issue here is to figure out what kind of observational data to feed into the theoretical model and how.","An understanding of what causes the irregularities of the solar cycle is of utmost interest in deciding this.","An attempt by [20] produced the prediction that the cycle 24 would be very strong, in contradiction to what was predicted on the basis of the weak polar field at the end of the cycle 23.","Figure: The sunspot number in the last few years.","The upper star indicatesthe predicted amplitude of cycle 24 according to , while the lower starindicates the predicted amplitude according to .","The circle on thehorizontal axis indicates the time when these predictions were made.Assuming that the fluctuation in the Babcock–Leighton mechanism is the main cause of irregularities in the solar cycle, [14] devised a methodology of feeding observational data of the polar magnetic field into the theoretical model to account for the random kick received by the dynamo due to fluctuations in the Babcock–Leighton mechanism.","The dynamo model of [14] predicted that the cycle 24 would be weak, in conformity with the weakness of the polar field at the end of cycle 23.","[30] explained the physical basis of what causes the correlation between the polar field at the end of a cycle and the strength of the next cycle.","Suppose the fluctuations in the Babcock–Leighton mechanism produced a poloidal field stronger than the usual.","This strong poloidal field will be advected to the poles to produce a strong polar field at the end of the cycle and, if the turbulent diffusion time across the convection zone is not more than a few years, this poloidal field will also diffuse to the bottom of the convection zone to provide a strong seed for the next cycle, making the next cycle strong.","On the other hand, if the poloidal field produced in a cycle is weaker than the average, then we shall get a weak polar field at the end of the cycle and a weak subsequent cycle.","This will give rise to a correlation between the polar field at the end of a cycle and the strength of the next cycle.","If the diffusion is assumed to be weak—as in the model of [20]—then different regions of the convection zone may not be able to communicate through diffusion in a few years and we shall not get this correlation.","The prediction of [14] that the cycle 24 would be weak was a robust prediction in their model because the polar field at the end of a cycle is correlated to the next cycle in their model and they had fed the data of the weak polar field at the end of cycle 23 into their theoretical model in order to generate their prediction.","As can be seen in Figure 3, the actual amplitude of cycle 24 turned out to be very close to what was predicted by [14], making this to be the first successful prediction of a solar cycle from a theoretical dynamo model in the history of our subject.","As we have pointed out in the previous Section, the fluctuations of the meridional circulation also can cause irregularities in the solar cycle.","This was not realized when the various predictions for cycle 24 were made during 2005–2007.","It is observationally found that there is a correlation between the decay rate of a cycle and the strength of the next cycle [26].","Now, a faster meridional circulation, which would make a cycle shorter, surely will make the decay rate faster and also the cycle stronger, as pointed out already (a slower meridional circulation would do the opposite).","If the effect of the fluctuating meridional circulation on the decay rate is immediate, but on the cycle strength is delayed by a few years, then we can explain the observed correlation.","This is confirmed by theoretical dynamo calculations [26].","This shows that it may be possible to use the decay rate at the end of a cycle to predict the effect of the fluctuating meridional circulation on the next cycle.","This issue needs to be looked at carefully."],["Conclusion","We have pointed out that over the years we have acquired an understanding of how the irregularities of the solar cycle arise and that this understanding helps us in predicting future solar activity.","Our point of view is that the fluctuations in the Babcock–Leighton mechanism and the fluctuations in the meridional circulation are the two primary sources of irregularities in the solar cycles.","These fluctuations have to be modelled realistically and fed into a theoretical dynamo model to generate predictions.","It may be noted that we now have a huge amount of data on the magnetic activity of solar-like stars [13].","Some solar-like stars display grand minima and we have evidence for the Waldmeier effect in some of them—see the concluding paragraph of [35].","This suggests that dynamos similar to the solar dynamo may be operational in the interiors of solar-type stars and the irregularities of their cycles also may be produced the same way as the irregularities of solar cycles.","Work on constructing flux transport dynamo models for solar-like stars has just begun [35].","Our ability to model stellar dynamos may throw more light on how the solar dynamo works.","Figure: A study of magnetic field evolution on the solar surface from the 3D kinematicdynamo model of , showing how the polar field builds up from a singletilted bipolar sunspot pair due to the Babcock–Leighton mechanism.","The differentpanels show the distribution of magnetic field at the following epochs after theemergence of the bipolar sunspots: (a) 0.025 yr, (b) 0.25 yr, (c) 1.02 yr,(d) 2.03 yr, (e) 3.05 yr, (f) 4.06 yr.All the theoretical results we discussed are based on axisymmetric 2D kinematic dynamo models.","One inherent limitation of such models is that the rise of a magnetic loop due to magnetic buoyancy and the Babcock–Leighton process of generating poloidal flux from it are intrinsically 3D processes and can be included in 2D models only through very crude averaging procedures [41], [40], [16].","As we have discussed, the fluctuations in the Babcock–Leighton process play a crucial role in producing the irregularities of the solar cycle.","In order to model these fluctuations realistically, it is essential to treat the Babcock–Leighton process itself more realistically than what is possible in 2D models.","The next step should be the construction of 3D kinematic dynamo models for which efforts have begun [52], [38], [39], [25].","Such 3D kinematic dynamo models can treat the Babcock–Leighton mechanism more realistically (see Figure 4) and should provide a better understanding of how fluctuations in the Babcock–Leighton mechanism cause irregularities in the dynamo.","The magnetic field presumably exists in the form of flux tubes throughout the convection zone and one limitation of a mean field model is that flux tubes are not handled properly [10].","A 3D kinematic model allows one to model flux tubes in a more realistic fashion.","A proper inclusion of flux tubes in a dynamo model is essential for explaining such interesting observations as the predominance of negative helicity in the norther hemisphere [43], which is presumably caused by the wrapping of the poloidal field around the rising flux tubes [10], [7], [29].","This process can be modelled in 2D mean field dynamo models only through drastic simplifications [15].","It should be possible to model this better through 3D kinematic dynamo models.","In other words, after the tremendous advances made by the 2D kinematic flux transport model during the last quarter century, it appears that that 3D kinematic dynamo models are likely to occupy the centre stage in the coming years."],["Acknowledgements","This work is partly supported by DST through a J.C. Bose Fellowship.","I thank VarSITI for travel support for attending the workshop at ISSI and thank ISSI for local hospitality during the workshop."]],"string":"[\n [\n \"Flux Transport Dynamo: From Modelling Irregularities to Making\\n Predictions\"\n ],\n [\n \"Abstract The flux transport dynamo, in which the poloidal magnetic field is generated by the Babcock--Leighton mechanism and the meridional circulation plays a crucial role, has emerged as an attractive model for the solar cycle.\",\n \"Based on theoretical calculations done with this model, we argue that the fluctuations in the Babcock--Leighton mechanism and the fluctuations in the meridional circulation are the most likely causes of the irregularities of the solar cycle.\",\n \"With our increased theoretical understanding of how these irregularities arise, it can be possible to predict a future solar cycle by feeding the appropriate observational data in a theoretical dynamo model.\"\n ],\n [\n \"Introduction\",\n \"The flux transport dynamo model, which started being developed about a quarter century ago [50], [19], [22], has emerged as an attractive theoretical model for the solar cycle.\",\n \"There are several modern reviews [11], [12], [4], [34] surveying the current status of the field.\",\n \"The present paper is not a comprehensive review, but is based on a talk in a Workshop at the International Space Science Institute (ISSI) highlighting the works done by the author and his coworkers.\",\n \"Readers are assumed to be familiar with the phenomenology of the solar cycle and the basic concepts of MHD (such as flux freezing and magnetic buoyancy).\",\n \"Readers not having this background are advised to look at the earlier reviews by the author [11], [12], which were written for wider readership.\",\n \"The initial effort in this field of flux transport dynamo was directed towards developing periodic models to explain the various periodic aspects of the solar cycle.\",\n \"Once sufficiently sophisticated periodic models were available, the next question was whether these theoretical models can be used to understand how the irregularities of the solar cycle arise.\",\n \"There is also a related question: if we understand what causes the irregularities of the cycle, then will that enable us to predict future cycles?\",\n \"We discuss the basic periodic model of the flux transport dynamo in the next Section.\",\n \"Then § 3 discusses the possible causes of the irregularities of the solar cycle.\",\n \"Afterwards in § 4 we address the question whether we are now in a position to predict future cycles.\",\n \"Finally, in § 5 we summarize the limitations of the 2D kinematic dynamo models and the recent efforts of going beyond such simple models.\"\n ],\n [\n \"The Basic Periodic Model\",\n \"One completely non-controversial aspect of solar dynamo models is the generation of the toroidal field from the poloidal field by differential rotation.\",\n \"Since differential rotation has now been mapped by helioseismology, this process can now be included in theoretical dynamo models quite realistically.\",\n \"The toroidal field is primarily produced in the tachocline at the bottom of the convection zone and rises from there due to magnetic buoyancy to create the sunspots.\",\n \"Although some authors have argued that the near-surface shear layer discovered by helioseismology can also be important for the generation of the toroidal field [2], the general view is that magnetic buoyancy would limit the growth of magnetic field in this region of strong super-adiabatic gradient.\",\n \"To this generally accepted view that the toroidal field is primarily produced in the tachocline, the flux transport dynamo model adds the following assumptions.\",\n \"The generation of the poloidal field from the toroidal field takes place due to the Babcock–Leighton mechanism.\",\n \"The meridional circulation of the Sun plays a crucial role in the dynamo process.\",\n \"We now comment on these two assumptions.\",\n \"Bipolar sunspots on the solar surface appear with a tilt statistically increasing with latitude, in accordance with the so-called Joy's law.\",\n \"This tilt is produced by the Coriolis force acting on the rising flux tubes [21].\",\n \"[1] and [36] suggested that the poloidal field of the Sun is produced from the decay of such tilted bipolar sunspot pairs.\",\n \"There is now enough evidence from observations of the solar surface that the poloidal field does get built up in this way.\",\n \"The meridional circulation is observed to be poleward at the solar surface and advects the poloidal field generated there, in conformity with observational data of surface magnetic fields.\",\n \"The material which is advected to the poles has to flow back equatorward through deeper layers within the solar convection zone.\",\n \"Since this circulation is driven by the turbulent stresses in the convection zone, we expect the meridional circulation not to penetrate much below the bottom of the convection zone, although a slight penetration helps in explaining several aspects of observational data [42], [3].\",\n \"The early models of the flux transport dynamo assumed the return flow of the meridional circulation to take place at the bottom of the convection zone, where the toroidal field generated by the differential rotation is advected equatorward with this flow, giving a natural explanation of the butterfly diagram representing the equatorward shift of the sunspot belt [19].\",\n \"Such dynamo models have been remarkably successful in explaining many aspects of the observational data [8].\",\n \"Figure: A complicated meridional circulation used by in a dynamocalculation—red corresponding to streamlines of clockwise circulation and blueto anti-clockwise circulation.\",\n \"Note that the flow near the bottom at low latitudesis equatorward.\",\n \"The butterfly diagram obtained with this circulation is solar-like(sunspot activity drifting to lower latitudes with time).While we still do not have unambiguous measurements of the return flow of the meridional circulation, some groups claim to have found evidence for the return flow well above the bottom of the convection zone [24], [54], [45].\",\n \"However, [44] argue that the available helioseismology data still cannot rule out a one-cell meridional circulation spanning the whole of the convection zone in each hemisphere.\",\n \"[27] showed that, even with a meridional circulation much more complicated than the one-cell pattern assumed in the earlier flux transport dynamo papers, it is still possible to match the relevant observational data as long as there is an equatorward flow at the bottom of the convection zone (see Figure 1).\"\n ],\n [\n \"The Origin of the Irregularities of the Solar Cycle\",\n \"The earliest attempts of explaining irregularities of the solar cycle were by regarding them as nonlinear chaos arising out of the nonlinearities of the dynamo equations [51].\",\n \"[5] argued that the Gnevyshev-Ohl rule in solar cycles (i.e.\",\n \"the tendency of alternate cycles to lie above and below the running mean of cycle amplitudes) arises out of period doubling due to nonlinearities.\",\n \"However, the simplest kinds of nonlinearities expected in dynamo equations tend to make the cycles more stable rather than producing irregularities and it has been suggested that stochastic fluctuations are more likely to be the primary reason behind producing irregularities [9].\",\n \"The Babcock–Leighton mechanism for the generation of the poloidal field depends on the tilts of bipolar sunspots.\",\n \"While the average tilt is given by Joy's law, we see considerable scatter around this average tilt, presumably caused by the action of turbulence in the convection zone on the rising flux tubes [37].\",\n \"This scatter in the tilt angles is expected to introduce fluctuations in the Babcock–Leighton mechanism [14].\",\n \"By including this fluctuation in the dynamo models, it is possible to explain many aspects of the irregularities of the cycles including the grand minima [17].\",\n \"One other source of irregularities is the fluctuations in the meridional circulation.\",\n \"A faster meridional circulation will make the solar cycles shorter and vice versa.\",\n \"While we have actual data of meridional circulation variations for not more than a couple of decades, we have data for durations of solar cycles for more than a century, providing indications that the meridional circulation had fluctuations in the past with correlation times of the order of 30–40 years [32].\",\n \"When the meridional circulation is slow and the cycles longer, diffusion has more time to act on the magnetic fields, making the cycles weaker.\",\n \"On such ground, we expect longer cycles to be weaker and shorter cycles to be stronger, leading to what is called the Waldmeier effect [32].\",\n \"Also, when the meridional circulation is sufficiently weak, theoretical dynamo models show that even grand minima can be induced [31].\",\n \"To get these results, the correlation time of the meridional circulation fluctuations was taken to be considerably longer than the cycle period.\",\n \"If the correlation time is taken too short, then one may not get these results [40].\",\n \"We also emphasize that the effect of diffusion in making longer cycles weaker is vital for getting these results.\",\n \"We need to take the value of diffusivity sufficiently high such that the diffusion time scale is shorter than or of the order of the cycle period.\",\n \"This is not the case in the model of [20] in which diffusivity is very low.\",\n \"A longer cycle in such a low-diffusivity model tends to be stronger because differential rotation has time to generate more toroidal field during a cycle, giving the opposite of the Waldmeier effect.\",\n \"Differences between high- and low-diffusivity dynamos were studied by [53].\",\n \"Clearly the high-diffusivity model yields results more in conformity with observational data.\",\n \"Figure: According to the calculations of , the poloidal fieldstrength (γ\\\\gamma is the value of the poloidal field compared to its averagevalue) and the amplitude of the meridional circulation at the surface have tolie in the shaded region at the beginning of a cycle in order to force the dynamointo a grand minimum.They estimated the probability of this to be about 1.3%, corresponding toabout 13 grand minima in 11,000 years.By analyzing the contents of C-14 in old tree trunks and Be-10 in polar ice cores, it has now been possible to reconstruct the history of solar activity over a few millenia [48].\",\n \"It has been estimated that there have been about 27 grand minima in the last 11,000 years [49].\",\n \"Since grand minima can be caused both by fluctuations in the Babcock-Leighton mechanism and fluctuations in the meridional circulation, a full theoretical model of grand minima should combine both types of fluctuations.\",\n \"If, at the beginning of a cycle, the poloidal field is too weak due to the fluctuations in the Babcock–Leighton mechanism or the meridional circulation is too weak, then the Sun may be forced into a grand minimum.\",\n \"Assuming a Gaussian distribution for fluctuations in both the Babcock–Leighton mechanism and the meridional circulation, [18] developed a comprehensive theory of grand minima that agreed remarkably well with the statistical data of grand minima (see Figure 2 and its caption).\",\n \"However, if there are no sunspots during grand minima, then the Babcock–Leighton mechanism which depends on sunspots may not be operational and how the Sun comes out of the grand minima is still rather poorly understood [33], [28].\",\n \"While discussing irregularities of the solar cycle, it may be mentioned that these irregularities are correlated reasonably well in the two hemispheres of the Sun.\",\n \"Strong cycles are usually strong in both the hemispheres and weak cycles are weak in both.\",\n \"This requires a coupling between the two hemispheres, implying that the turbulent diffusion time over the convection zone could not be more than a few years [6], [23].\"\n ],\n [\n \"Predicting solar cycles\",\n \"The first attempts of predicting solar cycles were based on using observational precursors of solar cycles.\",\n \"There is considerable evidence that the polar field at the end of a solar cycle is correlated with the next cycle.\",\n \"Since the polar field at the end of cycle 23 was rather weak, several authors [47], [46] predicted that the next cycle 24 would be weak.\",\n \"The sunspot minimum between the cycles 23 and 24 (around 2005–2008) was the first sunspot minimum when sufficiently sophisticated models of the flux transport dynamo were available.\",\n \"Whether these models could be used to predict the next cycle became an important question.\",\n \"When a kinematic mean field dynamo code is run without introducing any fluctuations, one finds that the code settles down to a periodic solution if the various dynamo parameters are in the correct range.\",\n \"In order to model actual solar cycles, one has to feed some observational data to the theoretical model in an appropriate manner and then run the code for a future cycle to generate a prediction.\",\n \"The crucial issue here is to figure out what kind of observational data to feed into the theoretical model and how.\",\n \"An understanding of what causes the irregularities of the solar cycle is of utmost interest in deciding this.\",\n \"An attempt by [20] produced the prediction that the cycle 24 would be very strong, in contradiction to what was predicted on the basis of the weak polar field at the end of the cycle 23.\",\n \"Figure: The sunspot number in the last few years.\",\n \"The upper star indicatesthe predicted amplitude of cycle 24 according to , while the lower starindicates the predicted amplitude according to .\",\n \"The circle on thehorizontal axis indicates the time when these predictions were made.Assuming that the fluctuation in the Babcock–Leighton mechanism is the main cause of irregularities in the solar cycle, [14] devised a methodology of feeding observational data of the polar magnetic field into the theoretical model to account for the random kick received by the dynamo due to fluctuations in the Babcock–Leighton mechanism.\",\n \"The dynamo model of [14] predicted that the cycle 24 would be weak, in conformity with the weakness of the polar field at the end of cycle 23.\",\n \"[30] explained the physical basis of what causes the correlation between the polar field at the end of a cycle and the strength of the next cycle.\",\n \"Suppose the fluctuations in the Babcock–Leighton mechanism produced a poloidal field stronger than the usual.\",\n \"This strong poloidal field will be advected to the poles to produce a strong polar field at the end of the cycle and, if the turbulent diffusion time across the convection zone is not more than a few years, this poloidal field will also diffuse to the bottom of the convection zone to provide a strong seed for the next cycle, making the next cycle strong.\",\n \"On the other hand, if the poloidal field produced in a cycle is weaker than the average, then we shall get a weak polar field at the end of the cycle and a weak subsequent cycle.\",\n \"This will give rise to a correlation between the polar field at the end of a cycle and the strength of the next cycle.\",\n \"If the diffusion is assumed to be weak—as in the model of [20]—then different regions of the convection zone may not be able to communicate through diffusion in a few years and we shall not get this correlation.\",\n \"The prediction of [14] that the cycle 24 would be weak was a robust prediction in their model because the polar field at the end of a cycle is correlated to the next cycle in their model and they had fed the data of the weak polar field at the end of cycle 23 into their theoretical model in order to generate their prediction.\",\n \"As can be seen in Figure 3, the actual amplitude of cycle 24 turned out to be very close to what was predicted by [14], making this to be the first successful prediction of a solar cycle from a theoretical dynamo model in the history of our subject.\",\n \"As we have pointed out in the previous Section, the fluctuations of the meridional circulation also can cause irregularities in the solar cycle.\",\n \"This was not realized when the various predictions for cycle 24 were made during 2005–2007.\",\n \"It is observationally found that there is a correlation between the decay rate of a cycle and the strength of the next cycle [26].\",\n \"Now, a faster meridional circulation, which would make a cycle shorter, surely will make the decay rate faster and also the cycle stronger, as pointed out already (a slower meridional circulation would do the opposite).\",\n \"If the effect of the fluctuating meridional circulation on the decay rate is immediate, but on the cycle strength is delayed by a few years, then we can explain the observed correlation.\",\n \"This is confirmed by theoretical dynamo calculations [26].\",\n \"This shows that it may be possible to use the decay rate at the end of a cycle to predict the effect of the fluctuating meridional circulation on the next cycle.\",\n \"This issue needs to be looked at carefully.\"\n ],\n [\n \"Conclusion\",\n \"We have pointed out that over the years we have acquired an understanding of how the irregularities of the solar cycle arise and that this understanding helps us in predicting future solar activity.\",\n \"Our point of view is that the fluctuations in the Babcock–Leighton mechanism and the fluctuations in the meridional circulation are the two primary sources of irregularities in the solar cycles.\",\n \"These fluctuations have to be modelled realistically and fed into a theoretical dynamo model to generate predictions.\",\n \"It may be noted that we now have a huge amount of data on the magnetic activity of solar-like stars [13].\",\n \"Some solar-like stars display grand minima and we have evidence for the Waldmeier effect in some of them—see the concluding paragraph of [35].\",\n \"This suggests that dynamos similar to the solar dynamo may be operational in the interiors of solar-type stars and the irregularities of their cycles also may be produced the same way as the irregularities of solar cycles.\",\n \"Work on constructing flux transport dynamo models for solar-like stars has just begun [35].\",\n \"Our ability to model stellar dynamos may throw more light on how the solar dynamo works.\",\n \"Figure: A study of magnetic field evolution on the solar surface from the 3D kinematicdynamo model of , showing how the polar field builds up from a singletilted bipolar sunspot pair due to the Babcock–Leighton mechanism.\",\n \"The differentpanels show the distribution of magnetic field at the following epochs after theemergence of the bipolar sunspots: (a) 0.025 yr, (b) 0.25 yr, (c) 1.02 yr,(d) 2.03 yr, (e) 3.05 yr, (f) 4.06 yr.All the theoretical results we discussed are based on axisymmetric 2D kinematic dynamo models.\",\n \"One inherent limitation of such models is that the rise of a magnetic loop due to magnetic buoyancy and the Babcock–Leighton process of generating poloidal flux from it are intrinsically 3D processes and can be included in 2D models only through very crude averaging procedures [41], [40], [16].\",\n \"As we have discussed, the fluctuations in the Babcock–Leighton process play a crucial role in producing the irregularities of the solar cycle.\",\n \"In order to model these fluctuations realistically, it is essential to treat the Babcock–Leighton process itself more realistically than what is possible in 2D models.\",\n \"The next step should be the construction of 3D kinematic dynamo models for which efforts have begun [52], [38], [39], [25].\",\n \"Such 3D kinematic dynamo models can treat the Babcock–Leighton mechanism more realistically (see Figure 4) and should provide a better understanding of how fluctuations in the Babcock–Leighton mechanism cause irregularities in the dynamo.\",\n \"The magnetic field presumably exists in the form of flux tubes throughout the convection zone and one limitation of a mean field model is that flux tubes are not handled properly [10].\",\n \"A 3D kinematic model allows one to model flux tubes in a more realistic fashion.\",\n \"A proper inclusion of flux tubes in a dynamo model is essential for explaining such interesting observations as the predominance of negative helicity in the norther hemisphere [43], which is presumably caused by the wrapping of the poloidal field around the rising flux tubes [10], [7], [29].\",\n \"This process can be modelled in 2D mean field dynamo models only through drastic simplifications [15].\",\n \"It should be possible to model this better through 3D kinematic dynamo models.\",\n \"In other words, after the tremendous advances made by the 2D kinematic flux transport model during the last quarter century, it appears that that 3D kinematic dynamo models are likely to occupy the centre stage in the coming years.\"\n ],\n [\n \"Acknowledgements\",\n \"This work is partly supported by DST through a J.C. Bose Fellowship.\",\n \"I thank VarSITI for travel support for attending the workshop at ISSI and thank ISSI for local hospitality during the workshop.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08550"}}},{"rowIdx":392,"cells":{"text":{"kind":"list like","value":[["Proton Mass Decomposition from the QCD Energy Momentum Tensor"],["Abstract We report results on the proton mass decomposition and also on related quark and glue momentum fractions.","The results are based on overlap valence fermions on four ensembles of $N_f = 2+1$ DWF configurations with three lattice spacings and three volumes, and several pion masses including the physical pion mass.","With fully non-perturbative renormalization (and universal normalization on both quark and gluon), we find that the quark energy and glue field energy contribute 33(4)(4)\\% and 37(5)(4)\\% respectively in the $\\overline{MS}$ scheme at $\\mu = 2$ GeV.","A quarter of the trace anomaly gives a 23(1)(1)\\% contribution to the proton mass based on the sum rule, given 9(2)(1)\\% contribution from the $u, d,$ and $s$ quark scalar condensates.","The $u,d,s$ and glue momentum fractions in the $\\overline{MS}$ scheme are in good agreement with global analyses at $\\mu = 2$ GeV."],["Acknowledgments","We thank the RBC and UKQCD collaborations for providing us their DWF gauge configurations.","YY is supported by the US National Science Foundation under grant PHY 1653405 “CAREER: Constraining Parton Distribution Functions for New-Physics Searches.\"","Y.C.","and Z.L.","acknowledge the support of the National Science Foundation of China under Grants No.","11575196, No.","11575197, No.","11335001.","This work is partially supported by DOE grant DE-SC0013065 and by the DOE TMD topical collaboration.","This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No.$~$ DE-AC05-00OR22725.","This work used Stampede time under the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1053575.","We also thank the National Energy Research Scientific Computing Center (NERSC) for providing HPC resources that have contributed to the research results reported within this paper.","We acknowledge the facilities of the USQCD Collaboration used for this research in part, which are funded by the Office of Science of the U.S. Department of Energy."],["The non-perturbative renormalization of the quark and gauge EMT","The renormalized momentum fractions $\\langle x \\rangle ^R$ in the $\\overline{\\textrm {MS}}$ scheme at scale $\\mu $ are $\\langle x \\rangle ^R_{u,d,s}=Z^{\\overline{\\textrm {MS}}}_{QQ}(\\mu )\\langle x \\rangle _{u,d,s}+\\delta Z^{\\overline{\\textrm {MS}}}_{QQ}(\\mu )\\sum _{q=u,d,s} \\langle x \\rangle _{q}+Z^{\\overline{\\textrm {MS}}}_{QG}(\\mu )\\langle x \\rangle _g,\\ \\langle x \\rangle ^R_g=Z^{\\overline{\\textrm {MS}}}_{GQ}(\\mu )\\sum _{q=u,d,s} \\langle x \\rangle _{q}+Z^{\\overline{\\textrm {MS}}}_{GG}\\langle x \\rangle _g,$ where $\\langle x \\rangle _{u,d,s,g}$ is the bare momentum fraction under the lattice regularization, and the renormalization constants at $\\overline{\\textrm {MS}}$ scale $\\mu $ are defined through the RI/MOM scheme at scale $\\mu _R$ , $&&\\left(\\begin{array}{cc}Z^{\\overline{\\textrm {MS}}}_{QQ}(\\mu )+N_f\\delta Z^{\\overline{\\textrm {MS}}}_{QQ} (\\mu )&N_fZ^{\\overline{\\textrm {MS}}}_{QG}(\\mu ) \\\\Z^{\\overline{\\textrm {MS}}}_{GQ}(\\mu ) &Z^{\\overline{\\textrm {MS}}}_{GG}(\\mu )\\end{array}\\right)\\equiv \\left\\lbrace \\left[\\left(\\begin{array}{cc}Z_{QQ}(\\mu _R)+N_f\\delta Z_{QQ}&N_fZ_{QG}(\\mu _R)\\\\Z_{GQ}(\\mu _R)&Z_{GG}(\\mu _R)\\end{array}\\right)\\right.\\right.\\nonumber \\\\&&\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\left.\\left.\\left(\\begin{array}{cc}R_{QQ}(\\frac{\\mu }{\\mu _{R}})+{\\cal O}(N_f\\alpha _s^2) &N_fR_{QG}(\\frac{\\mu }{\\mu _{R}})\\\\R_{GQ}(\\frac{\\mu }{\\mu _{R}}) &R_{GG}(\\frac{\\mu }{\\mu _{R}})\\end{array}\\right)\\right]|_{a^2\\mu _R^2\\rightarrow 0}\\right\\rbrace ^{-1}\\\\&&=\\left\\lbrace \\left(\\begin{array}{cc}\\left(Z_{QQ}R_{QQ}\\right)+N_f\\left(\\delta Z_{QQ}R_{QQ}+Z_{QG}R_{GQ}\\right)&N_f\\left((Z_{QQ}+N_f\\delta R_{QQ}) R_{QG}+Z_{QG}R_{GG}\\right)\\\\\\left(Z_{GQ}R_{QQ}+Z_{GG}R_{GQ}\\right) &\\left(N_fZ_{GQ} R_{QG}+Z_{GG}R_{GG}\\right)\\end{array}\\right)(\\mu _R,\\frac{\\mu }{\\mu _R})|_{a^2\\mu _R^2\\rightarrow 0}\\right\\rbrace ^{-1}$ and $Z_{QQ}(\\mu )=\\left[\\left(Z_{QQ}R_{QQ}\\right)(\\mu _R,\\frac{\\mu }{\\mu _R})|_{a^2\\mu _R^2\\rightarrow 0}\\right]^{-1}$ .","In the above equations, the RI/MOM renormalization constants $Z$ of $\\overline{T}^{q,g}_{44}$ are defined with the following conditions suggested by Ref.","[20] for cases with quark external legs: $Z_{QQ}(\\mu _R)&=&\\frac{V\\textrm {Tr}\\left[\\Gamma ^{q}_{\\mu \\mu }\\bar{S}_q^{-1}(p)\\left\\langle \\sum _w \\gamma _5 S_q^{\\dagger }(p,w) \\gamma _5 \\gamma _{\\mu }\\overleftrightarrow{D}_{\\mu } S_q(p,w) \\right\\rangle \\bar{S}_q^{-1}(p)\\right]}{\\left[-i\\Gamma ^{q}_{\\mu \\mu }(\\gamma _{\\mu }\\tilde{p}_{\\mu }-\\frac{1}{4}\\tilde{p}\\!\\!\\!/)\\right]Z_q}|_{p^2=\\mu _R^2},\\\\\\delta Z_{QQ}(\\mu _R)&=&\\frac{V\\textrm {Tr}\\left[\\Gamma ^{q}_{\\mu \\mu }\\bar{S}_q^{-1}(p)\\left\\langle \\overline{T}^q_{\\mu \\mu } S_q(p) \\right\\rangle \\bar{S}_q^{-1}(p)\\right]}{\\left[-i\\Gamma ^{q}_{\\mu \\mu }(\\gamma _{\\mu }\\tilde{p}_{\\mu }-\\frac{1}{4}\\tilde{p}\\!\\!\\!/)\\right]Z_q}|_{p^2=\\mu _R^2},\\\\Z_{GQ}(\\mu _R)&=&\\frac{V\\textrm {Tr}\\left[\\Gamma ^{q}_{\\mu \\mu }\\bar{S}_q^{-1}(p)\\left\\langle \\overline{T}^g_{\\mu \\mu } S_q(p) \\right\\rangle \\bar{S}_q^{-1}(p)\\right]}{\\left[-i\\Gamma ^{q}_{\\mu \\mu }(\\gamma _{\\mu }\\tilde{p}_{\\mu }-\\frac{1}{4}\\tilde{p}\\!\\!\\!/)\\right]Z_q}|_{p^2=\\mu _R^2},$ where $V$ is the lattice volume, $p$ is the momentum of the external quark/gluon state, $\\tilde{p}_{\\mu }=\\textrm {sin}p_{\\mu }$ , $\\Gamma ^q_{\\mu \\mu }=i\\gamma _{\\mu }\\tilde{p}_{\\mu }-i\\frac{\\tilde{p}^2_{\\mu }}{\\tilde{p}^2}\\tilde{p}\\!\\!\\!/$ as suggested by Ref.","[30], the quark propagators are $\\bar{S}_q(p)\\equiv \\langle S_q(p)\\rangle \\equiv \\langle \\sum _{x} e^{ipx}S_q(p,x)\\rangle $ with $S_q(p,x)=\\sum _{y} e^{-ipy}\\psi (x)\\bar{\\psi }(y) $ , and $Z_q$ is defined through the axial-vector vertex correction and Ward identity [31].","Note that the index $\\mu $ in Eqs.","(REF )-() is not summed while the results with different values of $\\mu $ can be averaged.","For the case of gluon external legs, the definitions are the following, as inspired by Refs.","[21], [19], $Z_{QG}(\\mu _R)&=&\\xi ^{-1}Z_{b}(\\mu _R, \\overline{T}^q)-(\\xi ^{-1}-1)Z_{a}(\\mu _R, \\overline{T}^q),\\\\Z_{GG}(\\mu _R)&=&\\xi ^{-1}Z_{b}(\\mu _R, \\overline{T}^g)-(\\xi ^{-1}-1)Z_{a}(\\mu _R, \\overline{T}^g),$ where $\\xi \\equiv \\frac{\\sum _{\\mu }p_{\\mu }^4}{(\\sum _{\\mu }p_{\\mu }^2)^2}$ , and $Z_{a/b}$ are defined by $Z_{a}(\\mu _R, \\overline{T})&=&\\frac{p^2\\langle (k_{\\mu }\\overline{T}_{\\mu \\nu }q_{\\nu })\\mathrm {Tr}[A_{\\rho }(p) A_{\\tau }(-p)\\Gamma _{\\rho \\tau }]\\rangle }{2k^2q^2\\langle \\mathrm {Tr}[A_{\\rho }(p) A_{\\tau }(-p)\\Gamma _{\\rho \\tau }]\\rangle }|_{\\tiny {\\begin{array}{c}p^2=\\mu _R^2, k+q=p, k\\cdot q=0, \\\\ \\Gamma _{\\rho \\tau }=\\delta _{\\rho \\tau }-\\frac{k_{\\rho }k_{\\tau }}{k^2}-\\frac{q_{\\rho }q_{\\tau }}{q^2} \\end{array}}}\\\\Z_{b}(\\mu _R, \\overline{T})&=&\\frac{\\langle (p_{\\mu }\\overline{T}_{\\mu \\nu }p_{\\nu }-l_{\\mu }\\overline{T}_{\\mu \\nu }l_{\\nu })\\mathrm {Tr}[A_{\\rho }(p) A_{\\tau }(-p)\\tilde{\\Gamma }_{\\rho \\tau }]\\rangle }{2p^2\\langle \\mathrm {Tr}[A_{\\rho }(p) A_{\\tau }(-p)\\tilde{\\Gamma }_{\\rho \\tau }]\\rangle }|_{\\tiny {\\begin{array}{c}p^2=\\mu _R^2, l^2=p^2, l\\cdot p=0, \\\\ \\tilde{\\Gamma }_{\\rho \\tau }=\\delta _{\\rho \\tau }-\\frac{p_{\\rho }p_{\\tau }}{p^2}-\\frac{l_{\\rho }l_{\\tau }}{l^2} \\end{array}}},$ with all the repeated indices summed.","The 3-loop level result of the matching coefficient $R_{QQ}(\\frac{\\mu }{\\mu _{R}})=1+\\frac{g^2}{16\\pi ^2}C_F[\\frac{8}{3}\\textrm {log}(\\mu ^2/\\mu _R^2)+\\frac{31}{9}]+{\\cal O}(\\alpha _s^2)$ has been obtained in Ref.","[20], while just the 1-loop level results of the other $R$ 's are available [21]: $R_{QG}&=&-\\frac{g^2}{16\\pi ^2}[\\frac{2}{3}\\textrm {log}(\\mu ^2/\\mu _R^2)+\\frac{4}{9}]+{\\cal O}(\\alpha _s^2),\\ R_{GQ}=-\\frac{g^2C_F}{16\\pi ^2}[\\frac{8}{3}\\textrm {log}(\\mu ^2/\\mu _R^2)+\\frac{22}{9}]+{\\cal O}(\\alpha _s^2), \\nonumber \\\\R_{GG}&=&1+\\frac{g^2N_f}{16\\pi ^2}[\\frac{2}{3}\\textrm {log}(\\mu ^2/\\mu _R^2)+\\frac{10}{9}]-\\frac{g^2N_c}{16\\pi ^2}\\frac{5}{12}+{\\cal O}(\\alpha _s^2).$ Thus we will use the 3-loop matching for the renormalization of the quark momentum fraction and keep all the matching at 1-loop level in the rest of the renormalization calculation.","In the practical lattice calculation, all the $Z$ 's suffer from discretization errors and we need to repeat the calculation of $Z$ at different $p^2$ , match them to the $\\overline{\\textrm {MS}}$ scheme at the $\\mu _R$ scale, evolve them from $\\mu _R$ to a fixed scale such as 2 GeV, and then apply the $a^2\\mu _R^2=a^2p^2$ extrapolation to get the final result of $Z$ .","To apply this extrapolation properly, we should calculate the vertex corrections in the quark and gluon states with exactly the same momenta, and combine them first.","But it is not necessary for most of the cases except for the quark to gluon mixing, as we will discuss case by case.","Figure: The quark EMT renormalization constant Z QQ R QQ Z_{QQ}R_{QQ} (left panel) and the disconnected pieces (δZ QQ R QQ \\delta Z_{QQ}R_{QQ} and Z QG R GQ Z_{QG}R_{GQ}, in the right panel) at different lattice spacings at MS ¯\\overline{\\textrm {MS}} scheme 2 GeV, as a function of a 2 p 2 a^2p^2.","With the momenta along the body-diagonal direction, the discretization errors are small and the results have a mild a 2 p 2 a^2p^2 dependence."],["Quark EMT renormalization","For the quark external state, we choose 12 momenta with $\\frac{\\sum _{\\mu }p_{\\mu }^4}{(\\sum _{\\mu }p_{\\mu }^2)^2}<0.27$ for 30 configurations and 6 valence quark masses at each lattice spacing, and use Landau gauge fixed momentum volume sources to improve the signal.","With the given momenta, we extrapolated the result to the chiral limit and then applied the continuum matching to get the value at $\\overline{\\textrm {MS}}$ 2 GeV.","The values of $Z_{QQ}R_{QQ}$ for three lattice spacings are plotted in the left panel of Fig.","REF , and the values at $a^2p^2=0$ limit are obtained based on linear extrapolation of the data in the range $a^2p^2\\in [4,8]$ .","It turns out that the $a^2p^2$ dependence is small by using $\\tilde{p}_{\\mu }$ in the definition of the tree level operator and projection, when the chosen momenta are along the body-diagonal direction.","The calculation on the 48I ensemble is skipped as its lattice spacing is almost the same as that of the 24I ensemble.","We followed the same strategy used in Ref.","[31], [32] to analyze the systematic uncertainties and the error budgets are collected in Tab.","REF .","The systematic uncertainty of $m_s^{sea}\\ne 0$ are estimated as $\\sim $ 1% on the 24I and 32I ensembles and $\\sim $ 0.25% on 32ID ensemble, based on our previous study with multiple sea quark masses [31].","The values for the off-diagonal parts of the quark EMT cases at $a$ =0.143, 0.111 and 0.083 fm (at $a^2p^2=0$ limit) are slightly different.","They are 0.786(2)(9), 0.796(1)(11) and 0.793(1)(11) respectively.","In the right panel of Fig.","REF , the other two terms needed by the singlet quark EMT renormalization, $\\delta Z_{QQ}R_{QQ}$ and $Z_{QG}R_{GQ}$ for the disconnected contribution, are plotted.","Regardless of the simulation details of the $Z_{QG}$ which will be addressed later, the contribution of second term is much smaller than the first term, and thus the $a^2p^2$ extrapolation can be carried out with the $\\delta Z_{QQ}R_{QQ}$ term alone in the $a^2p^2\\in [4,8]$ range, considering the value of the other term as a systematic uncertainty which is $\\sim $ 0.001.","Note that the CDER technique [18] is applied to the correlation function of $\\delta Z_{QQ}$ with the cutoff $r_0\\sim $ 1 fm, $\\left\\langle \\textrm {Tr}[\\overline{T}^{q}_{\\mu \\mu }] S(p) \\right\\rangle =\\left\\langle \\int \\textrm {d}^4x \\textrm {d}^{4}y \\textrm {Tr}[\\overline{T}^{q,g}_{\\mu \\mu }](x) S(p,y) \\right\\rangle \\simeq \\left\\langle \\int \\textrm {d}^4x \\int _{r\\le r_0}\\textrm {d}^{4}r \\textrm {Tr}[\\overline{T}^{q}_{\\mu \\mu }](x) S(p,x+r) \\right\\rangle $"],["Gauge EMT renormalization","In our previous investigation of the glue EMT renormalization [19], we chose the momenta with two transverse components to calculate $Z_{GG}$ : $Z_{GG}(\\mu _R)&=&\\frac{p^2\\langle (\\overline{T}^{g}_{\\mu \\mu }-\\overline{T}^{g}_{\\nu \\nu })\\mathrm {Tr}[A_{\\rho }(p) A_{\\rho }(-p)]\\rangle }{2p^2_{\\mu }\\langle \\mathrm {Tr}[A_{\\rho }(p) A_{\\rho }(-p)]\\rangle }|_{\\tiny {\\begin{array}{c}p^2=\\mu _R^2,\\\\ \\rho \\ne \\mu \\ne \\nu , \\\\p_\\rho =0,\\\\p_\\nu =0\\end{array}}},$ But the discretization errors at the range $a^2p^2\\in [4,8]$ used in the quark external legs are too large to carry out a reliable fit, when we want to combine it with $Z_{GQ}$ .","Thus in this work, we introduce two improvements to suppress the $a^2p^2$ corrections: 1.","The $a^2p^2$ correction from the HYP smearing on the glue operator: We found that most of the $a^2p^2$ correction in $Z_{GG}$ with the HYP smeared gauge EMT can be removed by the following ratio $f(a^2p^2)$ , $\\tilde{Z}_{GG}(a^2p^2)=Z_{GG}(a^2p^2)f(a^2p^2\\rightarrow 0)/f(a^2p^2),\\ f(a^2p^2)=\\frac{\\langle \\mathrm {Tr}[A^{HYP}_{\\rho }(p) A^{HYP}_{\\rho }(-p)]\\rangle }{\\langle \\mathrm {Tr}[A_{\\rho }(p) A_{\\rho }(-p)]\\rangle },$ where $A_{\\rho }^{HYP}=\\sum _{x} e^{ip\\cdot (x+\\frac{1}{2}\\hat{\\rho })}\\left[\\frac{{\\cal U}_\\rho (x)-{\\cal U}^{\\dagger }_\\rho (x)}{2ig_0a}\\right]_\\text{traceless}$ is the HYP-smeared gauge potential defined from the HYP-smeared gauge link ${\\cal U}_\\rho $ , and $f(a^2p^2)$ is the ratio of two propagators $S(p)\\equiv \\langle \\mathrm {Tr}[A_{\\rho }(p) A_{\\rho }(-p)]\\rangle $ with and without HYP smearing.","Since the scale dependence of $f(a^2p^2)$ is cancelled, only the $a^2p^2$ discretization errors exist up to the normalization $f(a^2p^2\\rightarrow 0)$ .","Note that such a normalization corresponds to the tadpole effect of the HYP smearing which should be included in the renormalization constant $Z_{GG}$ .","Since the value of $S(p)$ doesn't exist at $p^2=0$ , we will have to fit $f(a^2p^2)$ with a polynomial of $a^2p^2$ to get the value of $f(a^2p^2\\rightarrow 0)$ .","2.","We also extend the calculation of the $Z_{GG}$ to the momenta along the body-diagonal direction, by considering the projected renormalization constants defined in Eq. .","They can be rewritten into the combination of the renormalization constant of the traceless diagonal gauge EMT $Z_{GG}$ , and the off-diagonal one $Z_{GG}^{off}$ by $Z_{a}(\\mu _R, \\overline{T}_g)=Z^{off}_{GG}(\\mu _R), Z_{b}(\\mu _R, \\overline{T}_g)=Z^{off}_{GG}(\\mu _R)+\\xi (Z_{GG}-Z^{off}_{GG})(\\mu _R).$ Figure: The gauge EMT renormalization constant, with (left panel) and without (right panel) the a 2 p 2 a^2p^2 improvements described in Eq.","() and ().","The contribution from Z ˜ GG RI R GG \\tilde{Z}^{RI}_{GG}R_{GG} is dominant and that from N f Z GQ R QG N_fZ_{GQ} R_{QG} is negligible.","The a 2 p 2 →0a^2p^2\\rightarrow 0 results with and without improvement can provide consistent predictions.With 949/356/290 configurations respectively, the values of $\\tilde{Z}^{RI}_{GG}R_{GG}$ on 32ID/48I/64I ensembles with 1 step of HYP smearing are illustrated in the left panel of Fig.","REF .","Note that the ensemble 64I, which has almost the same lattice spacing as 32I (0.0837(2) vs. 0.0828(3)) but with a larger volume and physical pion mass, is used for the calculation, as a similar calculation on 32I would require $\\sim $ 5,000 configurations, which are not available, to reach the similar accuracy.","After both improvements above are applied, the $a^2p^2$ dependences of the $\\tilde{Z}^{RI}_{GG}R_{GG}$ are mild.","The contributions from the other term $N_fZ_{GQ} R_{QG}$ are also illustrated on the same figures.","Both the values and their uncertainties ($\\sim $ 0.005) are much smaller than the statistical uncertainty of $\\tilde{Z}^{RI}_{GG}$ and thus can be dropped safely.","With linear extrapolation of the data in the range $a^2p^2\\in [2,7]$ and the polynomial fit of the $f(a^2p^2)$ in the same range, we obtain the renormalization constants at $a$ =0.143, 0.114 and 0.084 fm to be 0.79(3)(6), 0.94(3)(3), and 0.91(3)(4) respectively, with the second error determined from varying the starting/ending points of the $a^2p^2$ range by 1.","For comparison, we also illustrate the results obtained from the previous strategy used in Ref.","[19] in the right panel of Fig.","REF , with the fit of the data in the range of $a^2\\hat{p}^2\\equiv (\\textrm {sin}\\frac{pa}{2})^2\\in [1.5, 5]$ .","The $a^2p^2\\rightarrow 0$ results of based on the polynomial fit are consistent with the present linear-fit ones."],["Mixings","For the mixing from glue to quark, both the contributions from $Z_{QQ} R_{QG}$ and $Z_{QG}R_{GG}$ are small, as illustrated in the left panel of Fig.","REF .","The total contribution can be estimated as $-$ 0.010(10), $-$ 0.005(10), $-$ 0.000(10) per flavor at $a$ =0.143, 0.111 and 0.083 fm respectively, when the linear extrapolation is applied in the range $a^2p^2\\in [4,8]$ .","The last piece of the jigsaw is the mixing from quark to glue.","The CDER technique used for $\\delta Z_{QQ}$ as in Eq.","REF can be applied for the calculation here with the gauge EMT operator.","As in the right panel of Fig.","REF , the contributions from $\\tilde{Z}^{RI}_{GG}R_{GQ}$ are just a few percent, while those from $Z_{GQ}R_{QQ}$ are sizable.","Since the $a^2p^2$ dependence of $\\tilde{Z}^{RI}_{GG}$ is small as in the left panel of Fig.","REF , we can approximate the $\\tilde{Z}^{RI}_{GG}(a^2p^2)$ by its $a^2p^2$ extrapolated value and assign $\\sim $ 3% systematic uncertainty from the difference between the values at $a^2p^2$ equal to 4 and 8.","After combining the correction and applying the linear extrapolation in the range $a^2p^2\\in [4,8]$ , we obtain the total mixing at $a$ =0.143, 0.111 and 0.083 fm as $-$ 0.34(2)(4), $-$ 0.26(2)(3), and $-$ 0.13(2)(1) respectively."]],"string":"[\n [\n \"Proton Mass Decomposition from the QCD Energy Momentum Tensor\"\n ],\n [\n \"Abstract We report results on the proton mass decomposition and also on related quark and glue momentum fractions.\",\n \"The results are based on overlap valence fermions on four ensembles of $N_f = 2+1$ DWF configurations with three lattice spacings and three volumes, and several pion masses including the physical pion mass.\",\n \"With fully non-perturbative renormalization (and universal normalization on both quark and gluon), we find that the quark energy and glue field energy contribute 33(4)(4)\\\\% and 37(5)(4)\\\\% respectively in the $\\\\overline{MS}$ scheme at $\\\\mu = 2$ GeV.\",\n \"A quarter of the trace anomaly gives a 23(1)(1)\\\\% contribution to the proton mass based on the sum rule, given 9(2)(1)\\\\% contribution from the $u, d,$ and $s$ quark scalar condensates.\",\n \"The $u,d,s$ and glue momentum fractions in the $\\\\overline{MS}$ scheme are in good agreement with global analyses at $\\\\mu = 2$ GeV.\"\n ],\n [\n \"Acknowledgments\",\n \"We thank the RBC and UKQCD collaborations for providing us their DWF gauge configurations.\",\n \"YY is supported by the US National Science Foundation under grant PHY 1653405 “CAREER: Constraining Parton Distribution Functions for New-Physics Searches.\\\"\",\n \"Y.C.\",\n \"and Z.L.\",\n \"acknowledge the support of the National Science Foundation of China under Grants No.\",\n \"11575196, No.\",\n \"11575197, No.\",\n \"11335001.\",\n \"This work is partially supported by DOE grant DE-SC0013065 and by the DOE TMD topical collaboration.\",\n \"This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No.$~$ DE-AC05-00OR22725.\",\n \"This work used Stampede time under the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1053575.\",\n \"We also thank the National Energy Research Scientific Computing Center (NERSC) for providing HPC resources that have contributed to the research results reported within this paper.\",\n \"We acknowledge the facilities of the USQCD Collaboration used for this research in part, which are funded by the Office of Science of the U.S. Department of Energy.\"\n ],\n [\n \"The non-perturbative renormalization of the quark and gauge EMT\",\n \"The renormalized momentum fractions $\\\\langle x \\\\rangle ^R$ in the $\\\\overline{\\\\textrm {MS}}$ scheme at scale $\\\\mu $ are $\\\\langle x \\\\rangle ^R_{u,d,s}=Z^{\\\\overline{\\\\textrm {MS}}}_{QQ}(\\\\mu )\\\\langle x \\\\rangle _{u,d,s}+\\\\delta Z^{\\\\overline{\\\\textrm {MS}}}_{QQ}(\\\\mu )\\\\sum _{q=u,d,s} \\\\langle x \\\\rangle _{q}+Z^{\\\\overline{\\\\textrm {MS}}}_{QG}(\\\\mu )\\\\langle x \\\\rangle _g,\\\\ \\\\langle x \\\\rangle ^R_g=Z^{\\\\overline{\\\\textrm {MS}}}_{GQ}(\\\\mu )\\\\sum _{q=u,d,s} \\\\langle x \\\\rangle _{q}+Z^{\\\\overline{\\\\textrm {MS}}}_{GG}\\\\langle x \\\\rangle _g,$ where $\\\\langle x \\\\rangle _{u,d,s,g}$ is the bare momentum fraction under the lattice regularization, and the renormalization constants at $\\\\overline{\\\\textrm {MS}}$ scale $\\\\mu $ are defined through the RI/MOM scheme at scale $\\\\mu _R$ , $&&\\\\left(\\\\begin{array}{cc}Z^{\\\\overline{\\\\textrm {MS}}}_{QQ}(\\\\mu )+N_f\\\\delta Z^{\\\\overline{\\\\textrm {MS}}}_{QQ} (\\\\mu )&N_fZ^{\\\\overline{\\\\textrm {MS}}}_{QG}(\\\\mu ) \\\\\\\\Z^{\\\\overline{\\\\textrm {MS}}}_{GQ}(\\\\mu ) &Z^{\\\\overline{\\\\textrm {MS}}}_{GG}(\\\\mu )\\\\end{array}\\\\right)\\\\equiv \\\\left\\\\lbrace \\\\left[\\\\left(\\\\begin{array}{cc}Z_{QQ}(\\\\mu _R)+N_f\\\\delta Z_{QQ}&N_fZ_{QG}(\\\\mu _R)\\\\\\\\Z_{GQ}(\\\\mu _R)&Z_{GG}(\\\\mu _R)\\\\end{array}\\\\right)\\\\right.\\\\right.\\\\nonumber \\\\\\\\&&\\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\left.\\\\left.\\\\left(\\\\begin{array}{cc}R_{QQ}(\\\\frac{\\\\mu }{\\\\mu _{R}})+{\\\\cal O}(N_f\\\\alpha _s^2) &N_fR_{QG}(\\\\frac{\\\\mu }{\\\\mu _{R}})\\\\\\\\R_{GQ}(\\\\frac{\\\\mu }{\\\\mu _{R}}) &R_{GG}(\\\\frac{\\\\mu }{\\\\mu _{R}})\\\\end{array}\\\\right)\\\\right]|_{a^2\\\\mu _R^2\\\\rightarrow 0}\\\\right\\\\rbrace ^{-1}\\\\\\\\&&=\\\\left\\\\lbrace \\\\left(\\\\begin{array}{cc}\\\\left(Z_{QQ}R_{QQ}\\\\right)+N_f\\\\left(\\\\delta Z_{QQ}R_{QQ}+Z_{QG}R_{GQ}\\\\right)&N_f\\\\left((Z_{QQ}+N_f\\\\delta R_{QQ}) R_{QG}+Z_{QG}R_{GG}\\\\right)\\\\\\\\\\\\left(Z_{GQ}R_{QQ}+Z_{GG}R_{GQ}\\\\right) &\\\\left(N_fZ_{GQ} R_{QG}+Z_{GG}R_{GG}\\\\right)\\\\end{array}\\\\right)(\\\\mu _R,\\\\frac{\\\\mu }{\\\\mu _R})|_{a^2\\\\mu _R^2\\\\rightarrow 0}\\\\right\\\\rbrace ^{-1}$ and $Z_{QQ}(\\\\mu )=\\\\left[\\\\left(Z_{QQ}R_{QQ}\\\\right)(\\\\mu _R,\\\\frac{\\\\mu }{\\\\mu _R})|_{a^2\\\\mu _R^2\\\\rightarrow 0}\\\\right]^{-1}$ .\",\n \"In the above equations, the RI/MOM renormalization constants $Z$ of $\\\\overline{T}^{q,g}_{44}$ are defined with the following conditions suggested by Ref.\",\n \"[20] for cases with quark external legs: $Z_{QQ}(\\\\mu _R)&=&\\\\frac{V\\\\textrm {Tr}\\\\left[\\\\Gamma ^{q}_{\\\\mu \\\\mu }\\\\bar{S}_q^{-1}(p)\\\\left\\\\langle \\\\sum _w \\\\gamma _5 S_q^{\\\\dagger }(p,w) \\\\gamma _5 \\\\gamma _{\\\\mu }\\\\overleftrightarrow{D}_{\\\\mu } S_q(p,w) \\\\right\\\\rangle \\\\bar{S}_q^{-1}(p)\\\\right]}{\\\\left[-i\\\\Gamma ^{q}_{\\\\mu \\\\mu }(\\\\gamma _{\\\\mu }\\\\tilde{p}_{\\\\mu }-\\\\frac{1}{4}\\\\tilde{p}\\\\!\\\\!\\\\!/)\\\\right]Z_q}|_{p^2=\\\\mu _R^2},\\\\\\\\\\\\delta Z_{QQ}(\\\\mu _R)&=&\\\\frac{V\\\\textrm {Tr}\\\\left[\\\\Gamma ^{q}_{\\\\mu \\\\mu }\\\\bar{S}_q^{-1}(p)\\\\left\\\\langle \\\\overline{T}^q_{\\\\mu \\\\mu } S_q(p) \\\\right\\\\rangle \\\\bar{S}_q^{-1}(p)\\\\right]}{\\\\left[-i\\\\Gamma ^{q}_{\\\\mu \\\\mu }(\\\\gamma _{\\\\mu }\\\\tilde{p}_{\\\\mu }-\\\\frac{1}{4}\\\\tilde{p}\\\\!\\\\!\\\\!/)\\\\right]Z_q}|_{p^2=\\\\mu _R^2},\\\\\\\\Z_{GQ}(\\\\mu _R)&=&\\\\frac{V\\\\textrm {Tr}\\\\left[\\\\Gamma ^{q}_{\\\\mu \\\\mu }\\\\bar{S}_q^{-1}(p)\\\\left\\\\langle \\\\overline{T}^g_{\\\\mu \\\\mu } S_q(p) \\\\right\\\\rangle \\\\bar{S}_q^{-1}(p)\\\\right]}{\\\\left[-i\\\\Gamma ^{q}_{\\\\mu \\\\mu }(\\\\gamma _{\\\\mu }\\\\tilde{p}_{\\\\mu }-\\\\frac{1}{4}\\\\tilde{p}\\\\!\\\\!\\\\!/)\\\\right]Z_q}|_{p^2=\\\\mu _R^2},$ where $V$ is the lattice volume, $p$ is the momentum of the external quark/gluon state, $\\\\tilde{p}_{\\\\mu }=\\\\textrm {sin}p_{\\\\mu }$ , $\\\\Gamma ^q_{\\\\mu \\\\mu }=i\\\\gamma _{\\\\mu }\\\\tilde{p}_{\\\\mu }-i\\\\frac{\\\\tilde{p}^2_{\\\\mu }}{\\\\tilde{p}^2}\\\\tilde{p}\\\\!\\\\!\\\\!/$ as suggested by Ref.\",\n \"[30], the quark propagators are $\\\\bar{S}_q(p)\\\\equiv \\\\langle S_q(p)\\\\rangle \\\\equiv \\\\langle \\\\sum _{x} e^{ipx}S_q(p,x)\\\\rangle $ with $S_q(p,x)=\\\\sum _{y} e^{-ipy}\\\\psi (x)\\\\bar{\\\\psi }(y) $ , and $Z_q$ is defined through the axial-vector vertex correction and Ward identity [31].\",\n \"Note that the index $\\\\mu $ in Eqs.\",\n \"(REF )-() is not summed while the results with different values of $\\\\mu $ can be averaged.\",\n \"For the case of gluon external legs, the definitions are the following, as inspired by Refs.\",\n \"[21], [19], $Z_{QG}(\\\\mu _R)&=&\\\\xi ^{-1}Z_{b}(\\\\mu _R, \\\\overline{T}^q)-(\\\\xi ^{-1}-1)Z_{a}(\\\\mu _R, \\\\overline{T}^q),\\\\\\\\Z_{GG}(\\\\mu _R)&=&\\\\xi ^{-1}Z_{b}(\\\\mu _R, \\\\overline{T}^g)-(\\\\xi ^{-1}-1)Z_{a}(\\\\mu _R, \\\\overline{T}^g),$ where $\\\\xi \\\\equiv \\\\frac{\\\\sum _{\\\\mu }p_{\\\\mu }^4}{(\\\\sum _{\\\\mu }p_{\\\\mu }^2)^2}$ , and $Z_{a/b}$ are defined by $Z_{a}(\\\\mu _R, \\\\overline{T})&=&\\\\frac{p^2\\\\langle (k_{\\\\mu }\\\\overline{T}_{\\\\mu \\\\nu }q_{\\\\nu })\\\\mathrm {Tr}[A_{\\\\rho }(p) A_{\\\\tau }(-p)\\\\Gamma _{\\\\rho \\\\tau }]\\\\rangle }{2k^2q^2\\\\langle \\\\mathrm {Tr}[A_{\\\\rho }(p) A_{\\\\tau }(-p)\\\\Gamma _{\\\\rho \\\\tau }]\\\\rangle }|_{\\\\tiny {\\\\begin{array}{c}p^2=\\\\mu _R^2, k+q=p, k\\\\cdot q=0, \\\\\\\\ \\\\Gamma _{\\\\rho \\\\tau }=\\\\delta _{\\\\rho \\\\tau }-\\\\frac{k_{\\\\rho }k_{\\\\tau }}{k^2}-\\\\frac{q_{\\\\rho }q_{\\\\tau }}{q^2} \\\\end{array}}}\\\\\\\\Z_{b}(\\\\mu _R, \\\\overline{T})&=&\\\\frac{\\\\langle (p_{\\\\mu }\\\\overline{T}_{\\\\mu \\\\nu }p_{\\\\nu }-l_{\\\\mu }\\\\overline{T}_{\\\\mu \\\\nu }l_{\\\\nu })\\\\mathrm {Tr}[A_{\\\\rho }(p) A_{\\\\tau }(-p)\\\\tilde{\\\\Gamma }_{\\\\rho \\\\tau }]\\\\rangle }{2p^2\\\\langle \\\\mathrm {Tr}[A_{\\\\rho }(p) A_{\\\\tau }(-p)\\\\tilde{\\\\Gamma }_{\\\\rho \\\\tau }]\\\\rangle }|_{\\\\tiny {\\\\begin{array}{c}p^2=\\\\mu _R^2, l^2=p^2, l\\\\cdot p=0, \\\\\\\\ \\\\tilde{\\\\Gamma }_{\\\\rho \\\\tau }=\\\\delta _{\\\\rho \\\\tau }-\\\\frac{p_{\\\\rho }p_{\\\\tau }}{p^2}-\\\\frac{l_{\\\\rho }l_{\\\\tau }}{l^2} \\\\end{array}}},$ with all the repeated indices summed.\",\n \"The 3-loop level result of the matching coefficient $R_{QQ}(\\\\frac{\\\\mu }{\\\\mu _{R}})=1+\\\\frac{g^2}{16\\\\pi ^2}C_F[\\\\frac{8}{3}\\\\textrm {log}(\\\\mu ^2/\\\\mu _R^2)+\\\\frac{31}{9}]+{\\\\cal O}(\\\\alpha _s^2)$ has been obtained in Ref.\",\n \"[20], while just the 1-loop level results of the other $R$ 's are available [21]: $R_{QG}&=&-\\\\frac{g^2}{16\\\\pi ^2}[\\\\frac{2}{3}\\\\textrm {log}(\\\\mu ^2/\\\\mu _R^2)+\\\\frac{4}{9}]+{\\\\cal O}(\\\\alpha _s^2),\\\\ R_{GQ}=-\\\\frac{g^2C_F}{16\\\\pi ^2}[\\\\frac{8}{3}\\\\textrm {log}(\\\\mu ^2/\\\\mu _R^2)+\\\\frac{22}{9}]+{\\\\cal O}(\\\\alpha _s^2), \\\\nonumber \\\\\\\\R_{GG}&=&1+\\\\frac{g^2N_f}{16\\\\pi ^2}[\\\\frac{2}{3}\\\\textrm {log}(\\\\mu ^2/\\\\mu _R^2)+\\\\frac{10}{9}]-\\\\frac{g^2N_c}{16\\\\pi ^2}\\\\frac{5}{12}+{\\\\cal O}(\\\\alpha _s^2).$ Thus we will use the 3-loop matching for the renormalization of the quark momentum fraction and keep all the matching at 1-loop level in the rest of the renormalization calculation.\",\n \"In the practical lattice calculation, all the $Z$ 's suffer from discretization errors and we need to repeat the calculation of $Z$ at different $p^2$ , match them to the $\\\\overline{\\\\textrm {MS}}$ scheme at the $\\\\mu _R$ scale, evolve them from $\\\\mu _R$ to a fixed scale such as 2 GeV, and then apply the $a^2\\\\mu _R^2=a^2p^2$ extrapolation to get the final result of $Z$ .\",\n \"To apply this extrapolation properly, we should calculate the vertex corrections in the quark and gluon states with exactly the same momenta, and combine them first.\",\n \"But it is not necessary for most of the cases except for the quark to gluon mixing, as we will discuss case by case.\",\n \"Figure: The quark EMT renormalization constant Z QQ R QQ Z_{QQ}R_{QQ} (left panel) and the disconnected pieces (δZ QQ R QQ \\\\delta Z_{QQ}R_{QQ} and Z QG R GQ Z_{QG}R_{GQ}, in the right panel) at different lattice spacings at MS ¯\\\\overline{\\\\textrm {MS}} scheme 2 GeV, as a function of a 2 p 2 a^2p^2.\",\n \"With the momenta along the body-diagonal direction, the discretization errors are small and the results have a mild a 2 p 2 a^2p^2 dependence.\"\n ],\n [\n \"Quark EMT renormalization\",\n \"For the quark external state, we choose 12 momenta with $\\\\frac{\\\\sum _{\\\\mu }p_{\\\\mu }^4}{(\\\\sum _{\\\\mu }p_{\\\\mu }^2)^2}<0.27$ for 30 configurations and 6 valence quark masses at each lattice spacing, and use Landau gauge fixed momentum volume sources to improve the signal.\",\n \"With the given momenta, we extrapolated the result to the chiral limit and then applied the continuum matching to get the value at $\\\\overline{\\\\textrm {MS}}$ 2 GeV.\",\n \"The values of $Z_{QQ}R_{QQ}$ for three lattice spacings are plotted in the left panel of Fig.\",\n \"REF , and the values at $a^2p^2=0$ limit are obtained based on linear extrapolation of the data in the range $a^2p^2\\\\in [4,8]$ .\",\n \"It turns out that the $a^2p^2$ dependence is small by using $\\\\tilde{p}_{\\\\mu }$ in the definition of the tree level operator and projection, when the chosen momenta are along the body-diagonal direction.\",\n \"The calculation on the 48I ensemble is skipped as its lattice spacing is almost the same as that of the 24I ensemble.\",\n \"We followed the same strategy used in Ref.\",\n \"[31], [32] to analyze the systematic uncertainties and the error budgets are collected in Tab.\",\n \"REF .\",\n \"The systematic uncertainty of $m_s^{sea}\\\\ne 0$ are estimated as $\\\\sim $ 1% on the 24I and 32I ensembles and $\\\\sim $ 0.25% on 32ID ensemble, based on our previous study with multiple sea quark masses [31].\",\n \"The values for the off-diagonal parts of the quark EMT cases at $a$ =0.143, 0.111 and 0.083 fm (at $a^2p^2=0$ limit) are slightly different.\",\n \"They are 0.786(2)(9), 0.796(1)(11) and 0.793(1)(11) respectively.\",\n \"In the right panel of Fig.\",\n \"REF , the other two terms needed by the singlet quark EMT renormalization, $\\\\delta Z_{QQ}R_{QQ}$ and $Z_{QG}R_{GQ}$ for the disconnected contribution, are plotted.\",\n \"Regardless of the simulation details of the $Z_{QG}$ which will be addressed later, the contribution of second term is much smaller than the first term, and thus the $a^2p^2$ extrapolation can be carried out with the $\\\\delta Z_{QQ}R_{QQ}$ term alone in the $a^2p^2\\\\in [4,8]$ range, considering the value of the other term as a systematic uncertainty which is $\\\\sim $ 0.001.\",\n \"Note that the CDER technique [18] is applied to the correlation function of $\\\\delta Z_{QQ}$ with the cutoff $r_0\\\\sim $ 1 fm, $\\\\left\\\\langle \\\\textrm {Tr}[\\\\overline{T}^{q}_{\\\\mu \\\\mu }] S(p) \\\\right\\\\rangle =\\\\left\\\\langle \\\\int \\\\textrm {d}^4x \\\\textrm {d}^{4}y \\\\textrm {Tr}[\\\\overline{T}^{q,g}_{\\\\mu \\\\mu }](x) S(p,y) \\\\right\\\\rangle \\\\simeq \\\\left\\\\langle \\\\int \\\\textrm {d}^4x \\\\int _{r\\\\le r_0}\\\\textrm {d}^{4}r \\\\textrm {Tr}[\\\\overline{T}^{q}_{\\\\mu \\\\mu }](x) S(p,x+r) \\\\right\\\\rangle $\"\n ],\n [\n \"Gauge EMT renormalization\",\n \"In our previous investigation of the glue EMT renormalization [19], we chose the momenta with two transverse components to calculate $Z_{GG}$ : $Z_{GG}(\\\\mu _R)&=&\\\\frac{p^2\\\\langle (\\\\overline{T}^{g}_{\\\\mu \\\\mu }-\\\\overline{T}^{g}_{\\\\nu \\\\nu })\\\\mathrm {Tr}[A_{\\\\rho }(p) A_{\\\\rho }(-p)]\\\\rangle }{2p^2_{\\\\mu }\\\\langle \\\\mathrm {Tr}[A_{\\\\rho }(p) A_{\\\\rho }(-p)]\\\\rangle }|_{\\\\tiny {\\\\begin{array}{c}p^2=\\\\mu _R^2,\\\\\\\\ \\\\rho \\\\ne \\\\mu \\\\ne \\\\nu , \\\\\\\\p_\\\\rho =0,\\\\\\\\p_\\\\nu =0\\\\end{array}}},$ But the discretization errors at the range $a^2p^2\\\\in [4,8]$ used in the quark external legs are too large to carry out a reliable fit, when we want to combine it with $Z_{GQ}$ .\",\n \"Thus in this work, we introduce two improvements to suppress the $a^2p^2$ corrections: 1.\",\n \"The $a^2p^2$ correction from the HYP smearing on the glue operator: We found that most of the $a^2p^2$ correction in $Z_{GG}$ with the HYP smeared gauge EMT can be removed by the following ratio $f(a^2p^2)$ , $\\\\tilde{Z}_{GG}(a^2p^2)=Z_{GG}(a^2p^2)f(a^2p^2\\\\rightarrow 0)/f(a^2p^2),\\\\ f(a^2p^2)=\\\\frac{\\\\langle \\\\mathrm {Tr}[A^{HYP}_{\\\\rho }(p) A^{HYP}_{\\\\rho }(-p)]\\\\rangle }{\\\\langle \\\\mathrm {Tr}[A_{\\\\rho }(p) A_{\\\\rho }(-p)]\\\\rangle },$ where $A_{\\\\rho }^{HYP}=\\\\sum _{x} e^{ip\\\\cdot (x+\\\\frac{1}{2}\\\\hat{\\\\rho })}\\\\left[\\\\frac{{\\\\cal U}_\\\\rho (x)-{\\\\cal U}^{\\\\dagger }_\\\\rho (x)}{2ig_0a}\\\\right]_\\\\text{traceless}$ is the HYP-smeared gauge potential defined from the HYP-smeared gauge link ${\\\\cal U}_\\\\rho $ , and $f(a^2p^2)$ is the ratio of two propagators $S(p)\\\\equiv \\\\langle \\\\mathrm {Tr}[A_{\\\\rho }(p) A_{\\\\rho }(-p)]\\\\rangle $ with and without HYP smearing.\",\n \"Since the scale dependence of $f(a^2p^2)$ is cancelled, only the $a^2p^2$ discretization errors exist up to the normalization $f(a^2p^2\\\\rightarrow 0)$ .\",\n \"Note that such a normalization corresponds to the tadpole effect of the HYP smearing which should be included in the renormalization constant $Z_{GG}$ .\",\n \"Since the value of $S(p)$ doesn't exist at $p^2=0$ , we will have to fit $f(a^2p^2)$ with a polynomial of $a^2p^2$ to get the value of $f(a^2p^2\\\\rightarrow 0)$ .\",\n \"2.\",\n \"We also extend the calculation of the $Z_{GG}$ to the momenta along the body-diagonal direction, by considering the projected renormalization constants defined in Eq. .\",\n \"They can be rewritten into the combination of the renormalization constant of the traceless diagonal gauge EMT $Z_{GG}$ , and the off-diagonal one $Z_{GG}^{off}$ by $Z_{a}(\\\\mu _R, \\\\overline{T}_g)=Z^{off}_{GG}(\\\\mu _R), Z_{b}(\\\\mu _R, \\\\overline{T}_g)=Z^{off}_{GG}(\\\\mu _R)+\\\\xi (Z_{GG}-Z^{off}_{GG})(\\\\mu _R).$ Figure: The gauge EMT renormalization constant, with (left panel) and without (right panel) the a 2 p 2 a^2p^2 improvements described in Eq.\",\n \"() and ().\",\n \"The contribution from Z ˜ GG RI R GG \\\\tilde{Z}^{RI}_{GG}R_{GG} is dominant and that from N f Z GQ R QG N_fZ_{GQ} R_{QG} is negligible.\",\n \"The a 2 p 2 →0a^2p^2\\\\rightarrow 0 results with and without improvement can provide consistent predictions.With 949/356/290 configurations respectively, the values of $\\\\tilde{Z}^{RI}_{GG}R_{GG}$ on 32ID/48I/64I ensembles with 1 step of HYP smearing are illustrated in the left panel of Fig.\",\n \"REF .\",\n \"Note that the ensemble 64I, which has almost the same lattice spacing as 32I (0.0837(2) vs. 0.0828(3)) but with a larger volume and physical pion mass, is used for the calculation, as a similar calculation on 32I would require $\\\\sim $ 5,000 configurations, which are not available, to reach the similar accuracy.\",\n \"After both improvements above are applied, the $a^2p^2$ dependences of the $\\\\tilde{Z}^{RI}_{GG}R_{GG}$ are mild.\",\n \"The contributions from the other term $N_fZ_{GQ} R_{QG}$ are also illustrated on the same figures.\",\n \"Both the values and their uncertainties ($\\\\sim $ 0.005) are much smaller than the statistical uncertainty of $\\\\tilde{Z}^{RI}_{GG}$ and thus can be dropped safely.\",\n \"With linear extrapolation of the data in the range $a^2p^2\\\\in [2,7]$ and the polynomial fit of the $f(a^2p^2)$ in the same range, we obtain the renormalization constants at $a$ =0.143, 0.114 and 0.084 fm to be 0.79(3)(6), 0.94(3)(3), and 0.91(3)(4) respectively, with the second error determined from varying the starting/ending points of the $a^2p^2$ range by 1.\",\n \"For comparison, we also illustrate the results obtained from the previous strategy used in Ref.\",\n \"[19] in the right panel of Fig.\",\n \"REF , with the fit of the data in the range of $a^2\\\\hat{p}^2\\\\equiv (\\\\textrm {sin}\\\\frac{pa}{2})^2\\\\in [1.5, 5]$ .\",\n \"The $a^2p^2\\\\rightarrow 0$ results of based on the polynomial fit are consistent with the present linear-fit ones.\"\n ],\n [\n \"Mixings\",\n \"For the mixing from glue to quark, both the contributions from $Z_{QQ} R_{QG}$ and $Z_{QG}R_{GG}$ are small, as illustrated in the left panel of Fig.\",\n \"REF .\",\n \"The total contribution can be estimated as $-$ 0.010(10), $-$ 0.005(10), $-$ 0.000(10) per flavor at $a$ =0.143, 0.111 and 0.083 fm respectively, when the linear extrapolation is applied in the range $a^2p^2\\\\in [4,8]$ .\",\n \"The last piece of the jigsaw is the mixing from quark to glue.\",\n \"The CDER technique used for $\\\\delta Z_{QQ}$ as in Eq.\",\n \"REF can be applied for the calculation here with the gauge EMT operator.\",\n \"As in the right panel of Fig.\",\n \"REF , the contributions from $\\\\tilde{Z}^{RI}_{GG}R_{GQ}$ are just a few percent, while those from $Z_{GQ}R_{QQ}$ are sizable.\",\n \"Since the $a^2p^2$ dependence of $\\\\tilde{Z}^{RI}_{GG}$ is small as in the left panel of Fig.\",\n \"REF , we can approximate the $\\\\tilde{Z}^{RI}_{GG}(a^2p^2)$ by its $a^2p^2$ extrapolated value and assign $\\\\sim $ 3% systematic uncertainty from the difference between the values at $a^2p^2$ equal to 4 and 8.\",\n \"After combining the correction and applying the linear extrapolation in the range $a^2p^2\\\\in [4,8]$ , we obtain the total mixing at $a$ =0.143, 0.111 and 0.083 fm as $-$ 0.34(2)(4), $-$ 0.26(2)(3), and $-$ 0.13(2)(1) respectively.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08677"}}},{"rowIdx":393,"cells":{"text":{"kind":"list like","value":[["Theoretical approach to the ductile fracture of polycrystalline solids"],["Abstract It is shown here that fracture after a brief plastic strain, typically of a few percents, is a necessary consequence of the polycrystalline nature of the materials.","The polycrystal undergoing plastic deformation is modeled as a flowing continuum of random deformable polyhedra, representing the grains, which fill the space without leaving voids.","Adjacent grains slide with a relative velocity proportional to the local shear stress resolved on the plane of the shared grain boundary, when greater than a finite threshold.","The polyhedral grains reshape continuously to preserve matter continuity, being the forces causing grain sliding dominant over those reshaping the grains.","It has been shown in the past that this model does not conserve volume, causing a monotonic hydrostatic pressure variation with strain.","This effect introduces a novel concept in the theory of plasticity because determines that any fine grained polycrystalline material will fail after a finite plastic strain.","Here the hydrostatic pressure dependence on strain is explicitly calculated and shown that has a logarithmic divergence which determines the strain to fracture.","Comparison of theoretical results with strains to fracture given by mechanical tests of commercial alloys show very good agreement."],["Introduction","Asking why things break when subjected to strong enough forces may sound superfluous because breaking objects is one of the most early experiences of every person.","In reality, explaining why solids undergoing plastic deformation are unable of achieving a steady flow regime and collapse past a finite plastic flow, or with almost no flow at all, is a most important scientific and technical problem yet unsolved.","In technical grounds the point is quite serious because of the high expenses associated to fatigue and failure of functional articles.","As well, the design of machine parts and structures is always restricted by the strength of the materials they will be made of, which puts limits to their efficiency and bounds costs from below.","Since the early investigations of Griffith [1], who claimed that the tensile strength of glass is lowered by the presence of very small pre–existent cracks that concentrate stresses when the material is loaded, and Irwin [2] and Orowan [3], who extended the idea to ductile solids, a great amount of effort has been expended in elucidating why solid materials break from the atomic point of view.","Nowadays the question has turned to how solids fail, instead of why they break.","Certainly, the two issues are closely related and answering the former question may clarify the latter, but not necessarily.","Most of the contemporary research on this subject relies on the hypothesis of cracks, and ascribes brittle behavior to the ability of stressed crack tips to propagate conserving their atomically sharp edges.","In ductile solids the tip of the crack blunts, broadens and flows, demanding increasing effort to make it progress [4], [5], [6], [7], [8], [9], [10].","Unfortunately, the problem of stress induced crack propagation has proven to be exceedingly complex, and neither theory nor computer simulations [11], [12], [13] have produced conclusive answers on the fracture process and the origin of brittle or ductile fracture.","The complex evolution of crack growth has been accurately measured [14], [15], [16], confirming atomic scale model predictions [17], [18] that the dynamics of a crack tip is highly unstable, and steady motion in a given direction is in most situations impossible.","We show here that the reason why continued deformation inevitably makes solids to break, undergoing either brittle, ductile or superplastic fracture, is much more basic and simpler than how fracture proceeds.","Resorting to a very general model for the structure of the solid, we demonstrate in what follows that fine grained polycrystalline materials are not able of a steady flow, no matter the strength of the forces involved, and should collapse after reaching a finite plastic strain.","At a scale much larger than the grain size, polycrystalline matter lacks symmetry constrictions and periodicity, and displays same average packing and properties in all directions, and over its whole extention.","Despite this, assimilating an even very fine grained polycrystal to an homogeneous and isotropic continuum may lead to gross errors, no matter the scale, when dealing with it as a dynamical medium.","The faceted nature of the structural constituents of a polycrystal determines that the force fields governing their plastic flow yield $\\nabla \\cdot \\vec{v}\\ne 0$ , where $\\vec{v}$ is the velocity field of the material continuum.","This means that flow makes the specific volume to vary.","Grain elasticity in polycrystalline solids allows for some density variation, and hence the medium can flow up to some limit, yielding ductile behaviour.","However, the consequent pressure build up influences strongly the ongoing deformation, which cannot be steady, and finally produces fracture.","Thus ductility is closely related to compressibility.","The model for the plastic flow of a polycrystalline solid has been extensively studied, principally in the context of superplasticity, but is expected to equally hold for normal ductile solids.","However, a brief account of its physical basis and the resulting general theoretical scheme is in order here.","The plastic deformation of a fine grained polycrystalline solid is modelled as a flowing continuum of random irregular polyhedra of different shapes and sizes, representing grains, which share faces.","The model is essentially the same as the one of Ref. [19].","Grains can move over long paths by sliding along the shared surfaces, or grain boundaries, accommodating effortlessly their shapes to preserve matter continuity.","Certainly, grain shape accommodation demands some effort, but it is assumed much smaller than the one required for grain sliding.","In other words, the shear stress between two sliding grains is greater than the critical resolved shear stress (CRSS) demanded by slip deformation of the crystallites.","This way, grain boundary sliding is the rate limiting process in the plastic strain.","In the present scheme grains always retain their individuality and mass, and are the dynamical entities.","The flow is driven by a field of tensor forces between the grains, determined by the stress tensor.","Figure: Local reference system (x ' y ' z ' )(x^\\prime y^\\prime z^\\prime ), with the z ' z^\\prime axis normal to the plane of the commonboundary of two adjacent grains.","The relative velocity Δv →\\Delta \\vec{v}of the two grains is in the x ' y ' x^{\\prime }y^{\\prime } plane.","The axes of the (xyz)(xyz)frame of reference are in the principal directions of the stresstensor.Fig.","REF shows a local frame of reference $(x^{\\prime }y^{\\prime }z^{\\prime })$ with the $x^{\\prime }y^{\\prime }$ plane coincident with the boundary between two adjacent grains.","The total shear stress in the shared boundary plane then reads $\\tau _{z^{\\prime }}=(\\sigma _{x^{\\prime }z^{\\prime }}^2+\\sigma _{y^{\\prime }z^{\\prime }}^2)^{1/2}$ , where $\\sigma _{i^{\\prime }j^{\\prime }}$ , $i^{\\prime },j^{\\prime }=x^{\\prime },y^{\\prime },z^{\\prime }$ , stands for the components of the stress tensor in this local coordinate system.","There is strong evidence that the sliding relative speed $|\\Delta \\vec{v}|$ of two adjacent grains obeys a linear law of the general form $|\\Delta \\vec{v}|=\\mathcal {Q}(\\tau _{z^{\\prime }}-\\tau _c)$ for $\\tau _{z^{\\prime }}>\\tau _c$ in plastic deformation [19], [20], [21], [22], [23], [24].","Here $\\mathcal {Q}$ is a proportionality coefficient and $\\tau _c$ is a critical shear stress such that $|\\Delta \\vec{v}|=0$ when $\\tau _{z^{\\prime }}\\le \\tau _c$ .","As $\\Delta \\vec{v}$ is parallel to the shear force in the plane of the interface, its components are given by $\\Delta v_{i^{\\prime }}=\\mathcal {Q}(\\tau _{z^{\\prime }}-\\tau _c)(\\sigma _{i^{\\prime }z^{\\prime }}/\\tau _{z^{\\prime }})$ , $i^{\\prime }=x^{\\prime },y^{\\prime }$ , for $\\tau _{z^{\\prime }}\\ge \\tau _c$ .","This expresion for $\\Delta \\vec{v}$ has proven to hold with great accuracy for several aluminium, titanium and magnesium alloys [24], [25], [26].","Hence the force law at the grain scale reads $\\begin{aligned}&\\Delta v_{i^{\\prime }}={\\left\\lbrace \\begin{array}{ll}\\mathcal {Q}\\,\\left( 1-\\dfrac{\\tau _c}{\\tau _{z^{\\prime }}}\\right) \\sigma _{i^{\\prime }z^{\\prime }},\\,&i^{\\prime }=x^{\\prime }, y^{\\prime }, \\, \\text{if}\\, \\tau _{z^{\\prime }}>\\tau _c \\\\0, &\\text{ otherwise},\\end{array}\\right.}","\\\\&\\Delta v_{z^{\\prime }}\\equiv 0.","\\\\\\end{aligned}$ The coefficient $\\mathcal {Q}=\\mathcal {Q}(p,T)$ does not depend on the shear stresses and neither on the orientation of the grain boundary, therefore its dependence on the normal stresses is only via the hydrostatic pressure invariant $p=-(\\sigma _{x^{\\prime }x^{\\prime }}+\\sigma _{y^{\\prime }y^{\\prime }}+\\sigma _{z^{\\prime }z^{\\prime }})/3 .$ The next step is to express the force law (REF ) in the frame of reference $(xyz)$ , common to all grain surfaces, instead of the local ones $(x^\\prime y^\\prime z^\\prime )$ .","Given the rotation matrix $R(\\theta ,\\phi )=(R_{ij}(\\theta ,\\phi ))$ connecting the two frames one can put the local stress tensor $(\\sigma _{i^\\prime j^\\prime })=R(\\theta ,\\phi )(\\sigma _{ij})R^T(\\theta ,\\phi )$ in terms of the stress tensor $(\\sigma _{i j})$ of the externally applied forces and the Euler angles $(\\theta ,\\phi )$ of the grain boundary plane.","The macroscopic force law is obtained from replacing in Eq.","(REF ) and averaging over the Euler angles.","Invoking also Hooke's law one has that $\\nabla \\cdot \\vec{v}=\\frac{\\dot{V}}{V}=-\\frac{\\dot{p}}{B},$ where $B$ is the bulk elastic modulus, $\\dot{p}$ the pressure variation rate, and $\\dot{V}/V$ the volume variation rate per unit volume.","A detailed account of the procedure would be in excess here because can be found in the literature [19], [23], [24], [25], [26]."],["The equations of motion","After a rather tedious set of mathematical steps [19], [24], [25], [26], [27] the procedure outlined above for the special case of an externally applied unidirectional normal stress $\\sigma $ on a polycrystalline solid, isotropic in the scale much larger than the mean grain size $d$ , yields the complete set of macroscopic equations of motion $\\dot{\\varepsilon }=s\\,\\frac{\\tau _c\\mathcal {Q}(p)}{2d}\\left[ \\cot (2\\theta _c)+2\\theta _c-\\frac{\\pi }{2}\\right] ,$ $\\begin{aligned}\\dot{p}= &sB\\frac{\\tau _c\\mathcal {Q}(p)}{2d}\\bigg [ \\frac{1 -\\cos (2\\theta _c)}{\\sin (2\\theta _c)}-2\\theta _c\\bigg (1+\\frac{2}{\\pi \\sin (2\\theta _c)}\\bigg )\\\\&-\\frac{2}{\\pi }\\cos (2\\theta _c)+\\frac{\\pi }{2}\\bigg ],\\end{aligned}$ where $\\dot{\\varepsilon }$ is the strain rate in the direction of the applied stress $\\sigma $ , $s=\\pm 1$ assumes the positive and negative values for tension and compression, respectively, and the auxiliary variable $\\theta _c$ is given by $\\sin (2\\theta _c)=\\frac{4\\tau _c}{3|\\sigma +p|}.$ The properties of the specific material enters the theoretical formulation through the coefficient $\\mathcal {Q}(p,T)$ , governing grain boundary sliding.","It has been studied in detail for fine grained polycrystalline solids and has been shown to be of the general form $\\frac{\\mathcal {Q}(p,T)}{4d}=C_0\\frac{\\Omega ^*}{k_BT}\\exp \\left( -\\frac{\\epsilon _0+\\Omega ^*p}{k_BT}\\right),$ where $k_B$ is the Boltzmann constant, $T$ the absolute temperature, the coefficient $C_0$ depends only on the grain size $d$ , the constant $\\epsilon _0$ is the energy necessary for evaporating a crystal vacancy from the grain boundary, and $\\Omega ^*$ is the excitation volume for the same process.","Eqs.","(REF ), (REF ) and (REF ) show that plastic flow is essentially a time dependent problem.","They govern the coupled time evolution of the three variables, $\\sigma $ , $\\varepsilon $ and $p$ , relevant for the cylindrically symmetric deformation of a polycrystalline continuous medium.","The actual behaviour of these variables in specific circumstances depends also on the initial conditions and deformation path ($\\sigma =\\text{constant}$ , $\\dot{\\varepsilon }=\\text{constant}$ , or any other imposed condition between the variables and their time derivatives).","The observed dependence on history of the plastic properties of ductile solids is usually attributed to structural variations or deformation induced damage.","In the present scheme, history enters through the initial condition for the variable $p$ , which is omitted in the traditional theoretical approaches to plasticity.","Here, the system described by Eqs.","(REF ), (REF ) and (REF ) behaves always the same way, but its evolution depends on the initial conditions for the variables, which include $p$ [27].","In opposition to the classical theory of plasticity, it exists a nontrivial transversal stress $\\sigma _\\perp =-(\\sigma +3p)/2$ which is not an independent variable, but evolves in time as dictated by the equations of motion.","One can set $\\sigma _\\perp =0$ as a natural initial condition if the material has been previously annealed, but $\\sigma _\\perp $ is expected to take finite values on the subsequent deformation.","As the magnitude inside the square brackets in the right hand side of Eq.","(REF ) is positive for any $\\theta _c$ , $\\dot{p}$ has the sign of $s$ .","The transversal stress $\\sigma _\\perp $ then decreases monotonically to negative values for positive $\\sigma +p$ .","Physically, this means that the plastic stretching in one direction is always accompanied by a finite compression in the plane normal to the deformation axis, which increases monotonically with strain.","This explains why necking always precedes ductile fracture [27]."],["The equations for constant strain rate","Replacing Eq.","(REF ) in the identity $d\\varepsilon =(\\dot{\\varepsilon }/\\dot{p})\\, dp$ one has that $\\begin{aligned}d\\varepsilon =&s\\frac{2\\dot{\\varepsilon }d}{B\\tau _c}\\bigg [\\frac{1-\\cos (2\\theta _c)}{\\sin (2\\theta _c)}-2\\theta _c\\bigg (1+\\frac{2}{\\pi \\sin (2\\theta _c)}\\bigg )\\\\&-\\frac{2}{\\pi }\\cos (2\\theta _c)+\\frac{\\pi }{2}\\bigg ]^{-1}\\frac{dp}{\\mathcal {Q}(p,T)},\\end{aligned}$ where $\\dot{\\varepsilon }$ is considered as a given constant.","As long as $\\dot{\\varepsilon }=\\text{constant}$ , Eq.","(REF ) shows that $\\theta _c=\\theta _c(p)$ .","Combining the derivatives of Eqs.","(REF ) and (REF ) with respect to $p$ it can be shown that $\\frac{dp}{\\mathcal {Q}}=-s\\frac{\\tau _ck_BT}{\\Omega ^*\\dot{\\varepsilon }d}\\cot ^2 (2\\theta _c)\\, d\\theta _c .$ Replacing now Eq.","(REF ) in (REF ) and integrating, it is finally obtained $\\begin{aligned}\\varepsilon =&-\\frac{2k_BT}{B\\Omega ^*}\\int _{\\theta _0}^{\\theta _c}d\\theta \\,\\cot ^2(2\\theta _c)\\bigg [\\frac{1-\\cos (2\\theta )}{\\sin (2\\theta )}\\\\&-2\\theta \\bigg (1+\\frac{2}{\\pi \\sin (2\\theta )}\\bigg )-\\frac{2}{\\pi }\\cos (2\\theta )+\\frac{\\pi }{2}\\bigg ]^{-1},\\end{aligned}$ where the limits $\\theta _0$ and $\\theta _c$ correspond to the critical angles for the initial and final values of the strain, $\\varepsilon =0$ and $\\varepsilon $ , respectively.","This way, $\\varepsilon $ is related with the auxiliary variable $\\theta _c$ by an expression of the form $\\varepsilon =\\frac{k_BT}{B\\Omega ^*}[F(\\theta _c)-F(\\theta _0)]\\quad (\\dot{\\varepsilon }=\\text{ constant}),$ where $F(\\theta )$ is the universal function $\\begin{aligned}F(\\theta )&=-2\\int _{\\pi /8}^{\\theta } d\\theta \\,\\cot ^2(2\\theta _c)\\bigg [\\frac{1-\\cos (2\\theta )}{\\sin (2\\theta )}\\\\&-2\\theta \\bigg (1+\\frac{2}{\\pi \\sin (2\\theta )}\\bigg )-\\frac{2}{\\pi }\\cos (2\\theta )+\\frac{\\pi }{2}\\bigg ]^{-1},\\end{aligned}$ which is monotonically decreasing in its whole range $(0,\\pi /4)$ and has two singularities, at $\\theta =0$ and $\\theta =\\pi /4$ .","If the material has been thoroughly annealed prior to the plastic deformation, it holds the initial condition $p=-\\sigma _0/3$ at $\\varepsilon =0$ , where $\\sigma _0$ is the stress at the beginning of the plastic deformation.","The magnitude of $\\varepsilon $ is controlled by the adimensional coefficient appearing in Eqs.","(REF ) and (REF ), which is a very small quantity.","The bulk modulus $B$ for metals is of the order of $10^{11}\\,\\text{Pa}$ .","Previous literature on aluminium and titanium alloys shows that $\\Omega ^*$ is $2.6\\times 10^{-27}\\,\\text{m}^3$ for Al–8090 and $5.9\\times 10^{-28}\\,\\text{m}^3$ for titanium Ti–6Al–4V at rather high temperatures [24].","Assuming $\\Omega ^*$ does not vary too much with $T$ one can take these figures to estimate that, at $T=300\\,\\text{K}$ , $\\frac{k_BT}{B\\Omega ^*}\\sim 2.3\\times 10^{-5}\\, -\\, 7.0\\times 10^{-5}.$ Because of the small value of the coefficient (REF ), any significant strain $\\varepsilon $ demands that the function $F(\\theta )$ be large, of the order of $10^3$ to have a strain of a few percents.","Hence $\\theta $ , or $\\theta _c$ , or both, must be in one of the two asymptotic regions $\\theta \\gtrsim 0$ or $\\theta \\lesssim \\pi /4$ .","The threshold stress $\\tau _c$ for grain sliding is generally in the range $0.5-5\\, \\text{MPa}$ , i. e. much smaller than the applied stresses $\\sigma $ that are customary in mechanical tests.","Hence the divergence at $\\theta =0$ should be the right one and appreciable strains occur for $\\theta _c(\\varepsilon ,\\dot{\\varepsilon },T)\\approx 0.$ The other pole of function $F(\\theta )$ corresponds to very slow flux, as occurring in superplastic deformation."],["Theory in the first order in $\\theta _c$","Up to the first order in $\\theta $ the expression in between the square brackets in Eqs.","(REF ) and (REF ) reduces to $\\begin{aligned}\\bigg [\\frac{1-\\cos (2\\theta )}{\\sin (2\\theta )}-2\\theta \\bigg ( 1&+\\frac{2}{\\pi \\sin (2\\theta )}\\bigg ) \\\\-\\frac{2\\cos (2\\theta )}{\\pi }&+\\frac{\\pi }{2}\\bigg ]\\approx \\frac{\\pi }{2}-\\frac{4}{\\pi }-\\theta .\\end{aligned}$ The constant $\\pi /2-4/\\pi =0.29756$ is not small enough and we can neglect $\\theta $ when compared with it.","Thus, with no significant lost of precision the exact equation $\\begin{aligned}&\\frac{dp}{d\\varepsilon }=sB\\frac{\\tau _c\\mathcal {Q}(p,T)}{2\\dot{\\varepsilon }d}\\bigg [ \\frac{1 -\\cos (2\\theta _c)}{\\sin (2\\theta _c)}\\\\&-2\\theta _c\\bigg (1+\\frac{2}{\\pi \\sin (2\\theta _c)}\\bigg )-\\frac{2}{\\pi }\\cos (2\\theta _c)+\\frac{\\pi }{2}\\bigg ],\\end{aligned}$ can be reduced to the much simpler first order differential equation $\\frac{dp}{d\\varepsilon }=s\\left(\\pi -\\frac{8}{\\pi }\\right)\\frac{C_0B\\tau _c\\Omega ^*}{k_BT\\dot{\\varepsilon }}\\exp \\left( -\\frac{\\epsilon _0+\\Omega ^*p}{k_BT}\\right),$ whose solution can be written as $\\begin{aligned}p-p_0=\\frac{k_BT}{\\Omega ^*}\\ln &\\bigg [ 1-C_0\\frac{\\pi ^2-8}{\\pi }\\frac{\\tau _c B}{\\dot{\\varepsilon }}\\bigg (\\frac{\\Omega ^*}{k_B T}\\bigg )^2 \\\\&\\times \\exp \\bigg ( -\\frac{\\epsilon _0+\\Omega ^*p_0}{k_B T}\\bigg )|\\varepsilon |\\bigg ]\\, .\\end{aligned}$ where it was substituted $s\\varepsilon =|\\varepsilon |$ .","Eq.","(REF ) expresses the main finding of this work: when the modulus $|\\varepsilon |$ of the strain approaches from below the value $\\varepsilon _{\\text{frac}}=\\frac{\\pi \\dot{\\varepsilon }}{(\\pi ^2-8)C_0\\tau _c B}\\bigg (\\frac{k_B T}{\\Omega ^*}\\bigg )^2\\exp \\bigg ( \\frac{\\epsilon _0+\\Omega ^*p_0}{k_B T}\\bigg )$ the hydrostatic pressure $p$ diverges logarithmically.","According to the definition (REF ) positive stresses (tension) contribute negatively to the hydrostatic pressure $p$ .","If the sample is conveniently annealed prior to the tensile test then $p_0=-\\sigma _0 /3$ , where $\\sigma _0$ is the applied initial tensile stress.","As the test proceeds, $p=-(\\sigma +2\\sigma _\\perp )/3$ increases monotonically with $\\varepsilon $ , and the transversal stress $\\sigma _\\perp $ increases from zero to negative (compressive) values.","When $\\varepsilon $ approaches the critical value $\\varepsilon _{\\text{frac}}$ the transversal stress $\\sigma _\\perp $ increases very rapidly, producing the characteristic neck and fracture.","Therefore, Eq.","(REF ) for $\\varepsilon _{\\text{frac}}$ expresses the strain to fracture of the material."],["Necking and strain to fracture","Eq.","(REF ) gives the strain to fracture in terms of the constants of the theory.","However one can express it in terms of more standard coefficients and easily measurable quantities.","Combining Eqs.","(REF ), (REF ), and taking into account the asymptotic approximation (REF ) to write $\\cot (2\\theta _0) +2\\theta _0 -\\frac{\\pi }{2}\\approx \\frac{1}{2\\theta _0}\\approx \\frac{\\sigma _0}{2\\tau _c}\\, ,$ Eq.","(REF ) can be written as $\\varepsilon _{\\text{frac}}=\\frac{\\pi }{(\\pi ^2-8)}\\frac{k_B T}{B\\Omega ^*}\\frac{\\sigma _0}{\\tau _c}.$ We recall that $\\sigma _0$ is the stress registered when the plastic deformation at the chosen constant strain rate $\\dot{\\varepsilon }$ begins.","The bulk modulus $B$ is in tables and the only undetermined parameter is the product $\\Omega ^*\\tau _c$ .","However, $\\Omega ^*\\tau _c$ can be determined independently from other features of the plastic deformation of the sample in order to have a parameter free test of Eq.","(REF ).","To show how well this expression compares with experiment, we include next a study of a representative commercial steel.","Figure: Circles represent the stress–strain experimentaldata for a copper–alloyed high–strength interstitial free steel atthe three strain rates shown in the inset .","The continuouslines represent the predictions of Eq.","() with the parametersoptimizing the fit to the experimental points, shown in Table.Figure REF shows the results of a mechanical test of a copper–alloyed high–strength interstitial free steel at strain rates 1, 20 and $200\\,\\text{s}^{-1}$ [28], together with the fits of Eq.","(REF ) with the asymptotic approximation (REF ).","The high quality of the agreement between theory and experiment is apparent in the figure, and the very little dispersion of the fitting parameters $\\Omega ^* B$ and $\\tau _c$ shown in Table REF reinforces this perception.","The last column of Table REF displays the strain to failure $\\varepsilon _{\\text{frac}}$ for the three strain rates, as given by Eq.","(REF ) where the parameters appearing in the left side of Table REF were substituted.","The values are very close to those measured in the mechanical testings.","Comparisons between predicted strains to fracture with published results of experimental tests for many other commercial alloys exhibit same agreement as the one shown in Fig.","REF and Table REF .","Table: Values for the parameters giving the fitsof Fig.","and calculated strains to failure."],["Conclusions","Although the existence of cracks and imperfections inside a stressed solid may contribute to accelerate fracture, the general cause of ductile fracture is not in them.","An ideal fine–grained polycrystalline material, free of voids and cracks, whose grains are prone to slide, readily accommodating each other's shapes, inevitably should fail after a finite plastic strain.","The reason is an elementary condition that was advanced some years ago [19] but omitted in other studies: whatever the mechanisms for stress–dependent grain boundary sliding and grain shape accommodation may be, they must be consistent with density conservation to produce a steady flow.","However, it is shown here that if the local shear stresses resolved in the planes of grain interfases have a finite threshold for causing grain sliding, density is not conserved in the overall plastic flow.","The grains are increasingly compressed as the sample is being stretched, and hence grain sliding can only proceed at the expenses of elastic volume variations of the crystallites.","Fracture after a brief plastic strain, typically of a few percents, is a necessary consequence of the polycrystalline nature of the materials.","The model gives a simple and precise closed–form equation for the strain to fracture, which is the strain at which the internal hydrostatic pressure diverges."]],"string":"[\n [\n \"Theoretical approach to the ductile fracture of polycrystalline solids\"\n ],\n [\n \"Abstract It is shown here that fracture after a brief plastic strain, typically of a few percents, is a necessary consequence of the polycrystalline nature of the materials.\",\n \"The polycrystal undergoing plastic deformation is modeled as a flowing continuum of random deformable polyhedra, representing the grains, which fill the space without leaving voids.\",\n \"Adjacent grains slide with a relative velocity proportional to the local shear stress resolved on the plane of the shared grain boundary, when greater than a finite threshold.\",\n \"The polyhedral grains reshape continuously to preserve matter continuity, being the forces causing grain sliding dominant over those reshaping the grains.\",\n \"It has been shown in the past that this model does not conserve volume, causing a monotonic hydrostatic pressure variation with strain.\",\n \"This effect introduces a novel concept in the theory of plasticity because determines that any fine grained polycrystalline material will fail after a finite plastic strain.\",\n \"Here the hydrostatic pressure dependence on strain is explicitly calculated and shown that has a logarithmic divergence which determines the strain to fracture.\",\n \"Comparison of theoretical results with strains to fracture given by mechanical tests of commercial alloys show very good agreement.\"\n ],\n [\n \"Introduction\",\n \"Asking why things break when subjected to strong enough forces may sound superfluous because breaking objects is one of the most early experiences of every person.\",\n \"In reality, explaining why solids undergoing plastic deformation are unable of achieving a steady flow regime and collapse past a finite plastic flow, or with almost no flow at all, is a most important scientific and technical problem yet unsolved.\",\n \"In technical grounds the point is quite serious because of the high expenses associated to fatigue and failure of functional articles.\",\n \"As well, the design of machine parts and structures is always restricted by the strength of the materials they will be made of, which puts limits to their efficiency and bounds costs from below.\",\n \"Since the early investigations of Griffith [1], who claimed that the tensile strength of glass is lowered by the presence of very small pre–existent cracks that concentrate stresses when the material is loaded, and Irwin [2] and Orowan [3], who extended the idea to ductile solids, a great amount of effort has been expended in elucidating why solid materials break from the atomic point of view.\",\n \"Nowadays the question has turned to how solids fail, instead of why they break.\",\n \"Certainly, the two issues are closely related and answering the former question may clarify the latter, but not necessarily.\",\n \"Most of the contemporary research on this subject relies on the hypothesis of cracks, and ascribes brittle behavior to the ability of stressed crack tips to propagate conserving their atomically sharp edges.\",\n \"In ductile solids the tip of the crack blunts, broadens and flows, demanding increasing effort to make it progress [4], [5], [6], [7], [8], [9], [10].\",\n \"Unfortunately, the problem of stress induced crack propagation has proven to be exceedingly complex, and neither theory nor computer simulations [11], [12], [13] have produced conclusive answers on the fracture process and the origin of brittle or ductile fracture.\",\n \"The complex evolution of crack growth has been accurately measured [14], [15], [16], confirming atomic scale model predictions [17], [18] that the dynamics of a crack tip is highly unstable, and steady motion in a given direction is in most situations impossible.\",\n \"We show here that the reason why continued deformation inevitably makes solids to break, undergoing either brittle, ductile or superplastic fracture, is much more basic and simpler than how fracture proceeds.\",\n \"Resorting to a very general model for the structure of the solid, we demonstrate in what follows that fine grained polycrystalline materials are not able of a steady flow, no matter the strength of the forces involved, and should collapse after reaching a finite plastic strain.\",\n \"At a scale much larger than the grain size, polycrystalline matter lacks symmetry constrictions and periodicity, and displays same average packing and properties in all directions, and over its whole extention.\",\n \"Despite this, assimilating an even very fine grained polycrystal to an homogeneous and isotropic continuum may lead to gross errors, no matter the scale, when dealing with it as a dynamical medium.\",\n \"The faceted nature of the structural constituents of a polycrystal determines that the force fields governing their plastic flow yield $\\\\nabla \\\\cdot \\\\vec{v}\\\\ne 0$ , where $\\\\vec{v}$ is the velocity field of the material continuum.\",\n \"This means that flow makes the specific volume to vary.\",\n \"Grain elasticity in polycrystalline solids allows for some density variation, and hence the medium can flow up to some limit, yielding ductile behaviour.\",\n \"However, the consequent pressure build up influences strongly the ongoing deformation, which cannot be steady, and finally produces fracture.\",\n \"Thus ductility is closely related to compressibility.\",\n \"The model for the plastic flow of a polycrystalline solid has been extensively studied, principally in the context of superplasticity, but is expected to equally hold for normal ductile solids.\",\n \"However, a brief account of its physical basis and the resulting general theoretical scheme is in order here.\",\n \"The plastic deformation of a fine grained polycrystalline solid is modelled as a flowing continuum of random irregular polyhedra of different shapes and sizes, representing grains, which share faces.\",\n \"The model is essentially the same as the one of Ref. [19].\",\n \"Grains can move over long paths by sliding along the shared surfaces, or grain boundaries, accommodating effortlessly their shapes to preserve matter continuity.\",\n \"Certainly, grain shape accommodation demands some effort, but it is assumed much smaller than the one required for grain sliding.\",\n \"In other words, the shear stress between two sliding grains is greater than the critical resolved shear stress (CRSS) demanded by slip deformation of the crystallites.\",\n \"This way, grain boundary sliding is the rate limiting process in the plastic strain.\",\n \"In the present scheme grains always retain their individuality and mass, and are the dynamical entities.\",\n \"The flow is driven by a field of tensor forces between the grains, determined by the stress tensor.\",\n \"Figure: Local reference system (x ' y ' z ' )(x^\\\\prime y^\\\\prime z^\\\\prime ), with the z ' z^\\\\prime axis normal to the plane of the commonboundary of two adjacent grains.\",\n \"The relative velocity Δv →\\\\Delta \\\\vec{v}of the two grains is in the x ' y ' x^{\\\\prime }y^{\\\\prime } plane.\",\n \"The axes of the (xyz)(xyz)frame of reference are in the principal directions of the stresstensor.Fig.\",\n \"REF shows a local frame of reference $(x^{\\\\prime }y^{\\\\prime }z^{\\\\prime })$ with the $x^{\\\\prime }y^{\\\\prime }$ plane coincident with the boundary between two adjacent grains.\",\n \"The total shear stress in the shared boundary plane then reads $\\\\tau _{z^{\\\\prime }}=(\\\\sigma _{x^{\\\\prime }z^{\\\\prime }}^2+\\\\sigma _{y^{\\\\prime }z^{\\\\prime }}^2)^{1/2}$ , where $\\\\sigma _{i^{\\\\prime }j^{\\\\prime }}$ , $i^{\\\\prime },j^{\\\\prime }=x^{\\\\prime },y^{\\\\prime },z^{\\\\prime }$ , stands for the components of the stress tensor in this local coordinate system.\",\n \"There is strong evidence that the sliding relative speed $|\\\\Delta \\\\vec{v}|$ of two adjacent grains obeys a linear law of the general form $|\\\\Delta \\\\vec{v}|=\\\\mathcal {Q}(\\\\tau _{z^{\\\\prime }}-\\\\tau _c)$ for $\\\\tau _{z^{\\\\prime }}>\\\\tau _c$ in plastic deformation [19], [20], [21], [22], [23], [24].\",\n \"Here $\\\\mathcal {Q}$ is a proportionality coefficient and $\\\\tau _c$ is a critical shear stress such that $|\\\\Delta \\\\vec{v}|=0$ when $\\\\tau _{z^{\\\\prime }}\\\\le \\\\tau _c$ .\",\n \"As $\\\\Delta \\\\vec{v}$ is parallel to the shear force in the plane of the interface, its components are given by $\\\\Delta v_{i^{\\\\prime }}=\\\\mathcal {Q}(\\\\tau _{z^{\\\\prime }}-\\\\tau _c)(\\\\sigma _{i^{\\\\prime }z^{\\\\prime }}/\\\\tau _{z^{\\\\prime }})$ , $i^{\\\\prime }=x^{\\\\prime },y^{\\\\prime }$ , for $\\\\tau _{z^{\\\\prime }}\\\\ge \\\\tau _c$ .\",\n \"This expresion for $\\\\Delta \\\\vec{v}$ has proven to hold with great accuracy for several aluminium, titanium and magnesium alloys [24], [25], [26].\",\n \"Hence the force law at the grain scale reads $\\\\begin{aligned}&\\\\Delta v_{i^{\\\\prime }}={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\mathcal {Q}\\\\,\\\\left( 1-\\\\dfrac{\\\\tau _c}{\\\\tau _{z^{\\\\prime }}}\\\\right) \\\\sigma _{i^{\\\\prime }z^{\\\\prime }},\\\\,&i^{\\\\prime }=x^{\\\\prime }, y^{\\\\prime }, \\\\, \\\\text{if}\\\\, \\\\tau _{z^{\\\\prime }}>\\\\tau _c \\\\\\\\0, &\\\\text{ otherwise},\\\\end{array}\\\\right.}\",\n \"\\\\\\\\&\\\\Delta v_{z^{\\\\prime }}\\\\equiv 0.\",\n \"\\\\\\\\\\\\end{aligned}$ The coefficient $\\\\mathcal {Q}=\\\\mathcal {Q}(p,T)$ does not depend on the shear stresses and neither on the orientation of the grain boundary, therefore its dependence on the normal stresses is only via the hydrostatic pressure invariant $p=-(\\\\sigma _{x^{\\\\prime }x^{\\\\prime }}+\\\\sigma _{y^{\\\\prime }y^{\\\\prime }}+\\\\sigma _{z^{\\\\prime }z^{\\\\prime }})/3 .$ The next step is to express the force law (REF ) in the frame of reference $(xyz)$ , common to all grain surfaces, instead of the local ones $(x^\\\\prime y^\\\\prime z^\\\\prime )$ .\",\n \"Given the rotation matrix $R(\\\\theta ,\\\\phi )=(R_{ij}(\\\\theta ,\\\\phi ))$ connecting the two frames one can put the local stress tensor $(\\\\sigma _{i^\\\\prime j^\\\\prime })=R(\\\\theta ,\\\\phi )(\\\\sigma _{ij})R^T(\\\\theta ,\\\\phi )$ in terms of the stress tensor $(\\\\sigma _{i j})$ of the externally applied forces and the Euler angles $(\\\\theta ,\\\\phi )$ of the grain boundary plane.\",\n \"The macroscopic force law is obtained from replacing in Eq.\",\n \"(REF ) and averaging over the Euler angles.\",\n \"Invoking also Hooke's law one has that $\\\\nabla \\\\cdot \\\\vec{v}=\\\\frac{\\\\dot{V}}{V}=-\\\\frac{\\\\dot{p}}{B},$ where $B$ is the bulk elastic modulus, $\\\\dot{p}$ the pressure variation rate, and $\\\\dot{V}/V$ the volume variation rate per unit volume.\",\n \"A detailed account of the procedure would be in excess here because can be found in the literature [19], [23], [24], [25], [26].\"\n ],\n [\n \"The equations of motion\",\n \"After a rather tedious set of mathematical steps [19], [24], [25], [26], [27] the procedure outlined above for the special case of an externally applied unidirectional normal stress $\\\\sigma $ on a polycrystalline solid, isotropic in the scale much larger than the mean grain size $d$ , yields the complete set of macroscopic equations of motion $\\\\dot{\\\\varepsilon }=s\\\\,\\\\frac{\\\\tau _c\\\\mathcal {Q}(p)}{2d}\\\\left[ \\\\cot (2\\\\theta _c)+2\\\\theta _c-\\\\frac{\\\\pi }{2}\\\\right] ,$ $\\\\begin{aligned}\\\\dot{p}= &sB\\\\frac{\\\\tau _c\\\\mathcal {Q}(p)}{2d}\\\\bigg [ \\\\frac{1 -\\\\cos (2\\\\theta _c)}{\\\\sin (2\\\\theta _c)}-2\\\\theta _c\\\\bigg (1+\\\\frac{2}{\\\\pi \\\\sin (2\\\\theta _c)}\\\\bigg )\\\\\\\\&-\\\\frac{2}{\\\\pi }\\\\cos (2\\\\theta _c)+\\\\frac{\\\\pi }{2}\\\\bigg ],\\\\end{aligned}$ where $\\\\dot{\\\\varepsilon }$ is the strain rate in the direction of the applied stress $\\\\sigma $ , $s=\\\\pm 1$ assumes the positive and negative values for tension and compression, respectively, and the auxiliary variable $\\\\theta _c$ is given by $\\\\sin (2\\\\theta _c)=\\\\frac{4\\\\tau _c}{3|\\\\sigma +p|}.$ The properties of the specific material enters the theoretical formulation through the coefficient $\\\\mathcal {Q}(p,T)$ , governing grain boundary sliding.\",\n \"It has been studied in detail for fine grained polycrystalline solids and has been shown to be of the general form $\\\\frac{\\\\mathcal {Q}(p,T)}{4d}=C_0\\\\frac{\\\\Omega ^*}{k_BT}\\\\exp \\\\left( -\\\\frac{\\\\epsilon _0+\\\\Omega ^*p}{k_BT}\\\\right),$ where $k_B$ is the Boltzmann constant, $T$ the absolute temperature, the coefficient $C_0$ depends only on the grain size $d$ , the constant $\\\\epsilon _0$ is the energy necessary for evaporating a crystal vacancy from the grain boundary, and $\\\\Omega ^*$ is the excitation volume for the same process.\",\n \"Eqs.\",\n \"(REF ), (REF ) and (REF ) show that plastic flow is essentially a time dependent problem.\",\n \"They govern the coupled time evolution of the three variables, $\\\\sigma $ , $\\\\varepsilon $ and $p$ , relevant for the cylindrically symmetric deformation of a polycrystalline continuous medium.\",\n \"The actual behaviour of these variables in specific circumstances depends also on the initial conditions and deformation path ($\\\\sigma =\\\\text{constant}$ , $\\\\dot{\\\\varepsilon }=\\\\text{constant}$ , or any other imposed condition between the variables and their time derivatives).\",\n \"The observed dependence on history of the plastic properties of ductile solids is usually attributed to structural variations or deformation induced damage.\",\n \"In the present scheme, history enters through the initial condition for the variable $p$ , which is omitted in the traditional theoretical approaches to plasticity.\",\n \"Here, the system described by Eqs.\",\n \"(REF ), (REF ) and (REF ) behaves always the same way, but its evolution depends on the initial conditions for the variables, which include $p$ [27].\",\n \"In opposition to the classical theory of plasticity, it exists a nontrivial transversal stress $\\\\sigma _\\\\perp =-(\\\\sigma +3p)/2$ which is not an independent variable, but evolves in time as dictated by the equations of motion.\",\n \"One can set $\\\\sigma _\\\\perp =0$ as a natural initial condition if the material has been previously annealed, but $\\\\sigma _\\\\perp $ is expected to take finite values on the subsequent deformation.\",\n \"As the magnitude inside the square brackets in the right hand side of Eq.\",\n \"(REF ) is positive for any $\\\\theta _c$ , $\\\\dot{p}$ has the sign of $s$ .\",\n \"The transversal stress $\\\\sigma _\\\\perp $ then decreases monotonically to negative values for positive $\\\\sigma +p$ .\",\n \"Physically, this means that the plastic stretching in one direction is always accompanied by a finite compression in the plane normal to the deformation axis, which increases monotonically with strain.\",\n \"This explains why necking always precedes ductile fracture [27].\"\n ],\n [\n \"The equations for constant strain rate\",\n \"Replacing Eq.\",\n \"(REF ) in the identity $d\\\\varepsilon =(\\\\dot{\\\\varepsilon }/\\\\dot{p})\\\\, dp$ one has that $\\\\begin{aligned}d\\\\varepsilon =&s\\\\frac{2\\\\dot{\\\\varepsilon }d}{B\\\\tau _c}\\\\bigg [\\\\frac{1-\\\\cos (2\\\\theta _c)}{\\\\sin (2\\\\theta _c)}-2\\\\theta _c\\\\bigg (1+\\\\frac{2}{\\\\pi \\\\sin (2\\\\theta _c)}\\\\bigg )\\\\\\\\&-\\\\frac{2}{\\\\pi }\\\\cos (2\\\\theta _c)+\\\\frac{\\\\pi }{2}\\\\bigg ]^{-1}\\\\frac{dp}{\\\\mathcal {Q}(p,T)},\\\\end{aligned}$ where $\\\\dot{\\\\varepsilon }$ is considered as a given constant.\",\n \"As long as $\\\\dot{\\\\varepsilon }=\\\\text{constant}$ , Eq.\",\n \"(REF ) shows that $\\\\theta _c=\\\\theta _c(p)$ .\",\n \"Combining the derivatives of Eqs.\",\n \"(REF ) and (REF ) with respect to $p$ it can be shown that $\\\\frac{dp}{\\\\mathcal {Q}}=-s\\\\frac{\\\\tau _ck_BT}{\\\\Omega ^*\\\\dot{\\\\varepsilon }d}\\\\cot ^2 (2\\\\theta _c)\\\\, d\\\\theta _c .$ Replacing now Eq.\",\n \"(REF ) in (REF ) and integrating, it is finally obtained $\\\\begin{aligned}\\\\varepsilon =&-\\\\frac{2k_BT}{B\\\\Omega ^*}\\\\int _{\\\\theta _0}^{\\\\theta _c}d\\\\theta \\\\,\\\\cot ^2(2\\\\theta _c)\\\\bigg [\\\\frac{1-\\\\cos (2\\\\theta )}{\\\\sin (2\\\\theta )}\\\\\\\\&-2\\\\theta \\\\bigg (1+\\\\frac{2}{\\\\pi \\\\sin (2\\\\theta )}\\\\bigg )-\\\\frac{2}{\\\\pi }\\\\cos (2\\\\theta )+\\\\frac{\\\\pi }{2}\\\\bigg ]^{-1},\\\\end{aligned}$ where the limits $\\\\theta _0$ and $\\\\theta _c$ correspond to the critical angles for the initial and final values of the strain, $\\\\varepsilon =0$ and $\\\\varepsilon $ , respectively.\",\n \"This way, $\\\\varepsilon $ is related with the auxiliary variable $\\\\theta _c$ by an expression of the form $\\\\varepsilon =\\\\frac{k_BT}{B\\\\Omega ^*}[F(\\\\theta _c)-F(\\\\theta _0)]\\\\quad (\\\\dot{\\\\varepsilon }=\\\\text{ constant}),$ where $F(\\\\theta )$ is the universal function $\\\\begin{aligned}F(\\\\theta )&=-2\\\\int _{\\\\pi /8}^{\\\\theta } d\\\\theta \\\\,\\\\cot ^2(2\\\\theta _c)\\\\bigg [\\\\frac{1-\\\\cos (2\\\\theta )}{\\\\sin (2\\\\theta )}\\\\\\\\&-2\\\\theta \\\\bigg (1+\\\\frac{2}{\\\\pi \\\\sin (2\\\\theta )}\\\\bigg )-\\\\frac{2}{\\\\pi }\\\\cos (2\\\\theta )+\\\\frac{\\\\pi }{2}\\\\bigg ]^{-1},\\\\end{aligned}$ which is monotonically decreasing in its whole range $(0,\\\\pi /4)$ and has two singularities, at $\\\\theta =0$ and $\\\\theta =\\\\pi /4$ .\",\n \"If the material has been thoroughly annealed prior to the plastic deformation, it holds the initial condition $p=-\\\\sigma _0/3$ at $\\\\varepsilon =0$ , where $\\\\sigma _0$ is the stress at the beginning of the plastic deformation.\",\n \"The magnitude of $\\\\varepsilon $ is controlled by the adimensional coefficient appearing in Eqs.\",\n \"(REF ) and (REF ), which is a very small quantity.\",\n \"The bulk modulus $B$ for metals is of the order of $10^{11}\\\\,\\\\text{Pa}$ .\",\n \"Previous literature on aluminium and titanium alloys shows that $\\\\Omega ^*$ is $2.6\\\\times 10^{-27}\\\\,\\\\text{m}^3$ for Al–8090 and $5.9\\\\times 10^{-28}\\\\,\\\\text{m}^3$ for titanium Ti–6Al–4V at rather high temperatures [24].\",\n \"Assuming $\\\\Omega ^*$ does not vary too much with $T$ one can take these figures to estimate that, at $T=300\\\\,\\\\text{K}$ , $\\\\frac{k_BT}{B\\\\Omega ^*}\\\\sim 2.3\\\\times 10^{-5}\\\\, -\\\\, 7.0\\\\times 10^{-5}.$ Because of the small value of the coefficient (REF ), any significant strain $\\\\varepsilon $ demands that the function $F(\\\\theta )$ be large, of the order of $10^3$ to have a strain of a few percents.\",\n \"Hence $\\\\theta $ , or $\\\\theta _c$ , or both, must be in one of the two asymptotic regions $\\\\theta \\\\gtrsim 0$ or $\\\\theta \\\\lesssim \\\\pi /4$ .\",\n \"The threshold stress $\\\\tau _c$ for grain sliding is generally in the range $0.5-5\\\\, \\\\text{MPa}$ , i. e. much smaller than the applied stresses $\\\\sigma $ that are customary in mechanical tests.\",\n \"Hence the divergence at $\\\\theta =0$ should be the right one and appreciable strains occur for $\\\\theta _c(\\\\varepsilon ,\\\\dot{\\\\varepsilon },T)\\\\approx 0.$ The other pole of function $F(\\\\theta )$ corresponds to very slow flux, as occurring in superplastic deformation.\"\n ],\n [\n \"Theory in the first order in $\\\\theta _c$\",\n \"Up to the first order in $\\\\theta $ the expression in between the square brackets in Eqs.\",\n \"(REF ) and (REF ) reduces to $\\\\begin{aligned}\\\\bigg [\\\\frac{1-\\\\cos (2\\\\theta )}{\\\\sin (2\\\\theta )}-2\\\\theta \\\\bigg ( 1&+\\\\frac{2}{\\\\pi \\\\sin (2\\\\theta )}\\\\bigg ) \\\\\\\\-\\\\frac{2\\\\cos (2\\\\theta )}{\\\\pi }&+\\\\frac{\\\\pi }{2}\\\\bigg ]\\\\approx \\\\frac{\\\\pi }{2}-\\\\frac{4}{\\\\pi }-\\\\theta .\\\\end{aligned}$ The constant $\\\\pi /2-4/\\\\pi =0.29756$ is not small enough and we can neglect $\\\\theta $ when compared with it.\",\n \"Thus, with no significant lost of precision the exact equation $\\\\begin{aligned}&\\\\frac{dp}{d\\\\varepsilon }=sB\\\\frac{\\\\tau _c\\\\mathcal {Q}(p,T)}{2\\\\dot{\\\\varepsilon }d}\\\\bigg [ \\\\frac{1 -\\\\cos (2\\\\theta _c)}{\\\\sin (2\\\\theta _c)}\\\\\\\\&-2\\\\theta _c\\\\bigg (1+\\\\frac{2}{\\\\pi \\\\sin (2\\\\theta _c)}\\\\bigg )-\\\\frac{2}{\\\\pi }\\\\cos (2\\\\theta _c)+\\\\frac{\\\\pi }{2}\\\\bigg ],\\\\end{aligned}$ can be reduced to the much simpler first order differential equation $\\\\frac{dp}{d\\\\varepsilon }=s\\\\left(\\\\pi -\\\\frac{8}{\\\\pi }\\\\right)\\\\frac{C_0B\\\\tau _c\\\\Omega ^*}{k_BT\\\\dot{\\\\varepsilon }}\\\\exp \\\\left( -\\\\frac{\\\\epsilon _0+\\\\Omega ^*p}{k_BT}\\\\right),$ whose solution can be written as $\\\\begin{aligned}p-p_0=\\\\frac{k_BT}{\\\\Omega ^*}\\\\ln &\\\\bigg [ 1-C_0\\\\frac{\\\\pi ^2-8}{\\\\pi }\\\\frac{\\\\tau _c B}{\\\\dot{\\\\varepsilon }}\\\\bigg (\\\\frac{\\\\Omega ^*}{k_B T}\\\\bigg )^2 \\\\\\\\&\\\\times \\\\exp \\\\bigg ( -\\\\frac{\\\\epsilon _0+\\\\Omega ^*p_0}{k_B T}\\\\bigg )|\\\\varepsilon |\\\\bigg ]\\\\, .\\\\end{aligned}$ where it was substituted $s\\\\varepsilon =|\\\\varepsilon |$ .\",\n \"Eq.\",\n \"(REF ) expresses the main finding of this work: when the modulus $|\\\\varepsilon |$ of the strain approaches from below the value $\\\\varepsilon _{\\\\text{frac}}=\\\\frac{\\\\pi \\\\dot{\\\\varepsilon }}{(\\\\pi ^2-8)C_0\\\\tau _c B}\\\\bigg (\\\\frac{k_B T}{\\\\Omega ^*}\\\\bigg )^2\\\\exp \\\\bigg ( \\\\frac{\\\\epsilon _0+\\\\Omega ^*p_0}{k_B T}\\\\bigg )$ the hydrostatic pressure $p$ diverges logarithmically.\",\n \"According to the definition (REF ) positive stresses (tension) contribute negatively to the hydrostatic pressure $p$ .\",\n \"If the sample is conveniently annealed prior to the tensile test then $p_0=-\\\\sigma _0 /3$ , where $\\\\sigma _0$ is the applied initial tensile stress.\",\n \"As the test proceeds, $p=-(\\\\sigma +2\\\\sigma _\\\\perp )/3$ increases monotonically with $\\\\varepsilon $ , and the transversal stress $\\\\sigma _\\\\perp $ increases from zero to negative (compressive) values.\",\n \"When $\\\\varepsilon $ approaches the critical value $\\\\varepsilon _{\\\\text{frac}}$ the transversal stress $\\\\sigma _\\\\perp $ increases very rapidly, producing the characteristic neck and fracture.\",\n \"Therefore, Eq.\",\n \"(REF ) for $\\\\varepsilon _{\\\\text{frac}}$ expresses the strain to fracture of the material.\"\n ],\n [\n \"Necking and strain to fracture\",\n \"Eq.\",\n \"(REF ) gives the strain to fracture in terms of the constants of the theory.\",\n \"However one can express it in terms of more standard coefficients and easily measurable quantities.\",\n \"Combining Eqs.\",\n \"(REF ), (REF ), and taking into account the asymptotic approximation (REF ) to write $\\\\cot (2\\\\theta _0) +2\\\\theta _0 -\\\\frac{\\\\pi }{2}\\\\approx \\\\frac{1}{2\\\\theta _0}\\\\approx \\\\frac{\\\\sigma _0}{2\\\\tau _c}\\\\, ,$ Eq.\",\n \"(REF ) can be written as $\\\\varepsilon _{\\\\text{frac}}=\\\\frac{\\\\pi }{(\\\\pi ^2-8)}\\\\frac{k_B T}{B\\\\Omega ^*}\\\\frac{\\\\sigma _0}{\\\\tau _c}.$ We recall that $\\\\sigma _0$ is the stress registered when the plastic deformation at the chosen constant strain rate $\\\\dot{\\\\varepsilon }$ begins.\",\n \"The bulk modulus $B$ is in tables and the only undetermined parameter is the product $\\\\Omega ^*\\\\tau _c$ .\",\n \"However, $\\\\Omega ^*\\\\tau _c$ can be determined independently from other features of the plastic deformation of the sample in order to have a parameter free test of Eq.\",\n \"(REF ).\",\n \"To show how well this expression compares with experiment, we include next a study of a representative commercial steel.\",\n \"Figure: Circles represent the stress–strain experimentaldata for a copper–alloyed high–strength interstitial free steel atthe three strain rates shown in the inset .\",\n \"The continuouslines represent the predictions of Eq.\",\n \"() with the parametersoptimizing the fit to the experimental points, shown in Table.Figure REF shows the results of a mechanical test of a copper–alloyed high–strength interstitial free steel at strain rates 1, 20 and $200\\\\,\\\\text{s}^{-1}$ [28], together with the fits of Eq.\",\n \"(REF ) with the asymptotic approximation (REF ).\",\n \"The high quality of the agreement between theory and experiment is apparent in the figure, and the very little dispersion of the fitting parameters $\\\\Omega ^* B$ and $\\\\tau _c$ shown in Table REF reinforces this perception.\",\n \"The last column of Table REF displays the strain to failure $\\\\varepsilon _{\\\\text{frac}}$ for the three strain rates, as given by Eq.\",\n \"(REF ) where the parameters appearing in the left side of Table REF were substituted.\",\n \"The values are very close to those measured in the mechanical testings.\",\n \"Comparisons between predicted strains to fracture with published results of experimental tests for many other commercial alloys exhibit same agreement as the one shown in Fig.\",\n \"REF and Table REF .\",\n \"Table: Values for the parameters giving the fitsof Fig.\",\n \"and calculated strains to failure.\"\n ],\n [\n \"Conclusions\",\n \"Although the existence of cracks and imperfections inside a stressed solid may contribute to accelerate fracture, the general cause of ductile fracture is not in them.\",\n \"An ideal fine–grained polycrystalline material, free of voids and cracks, whose grains are prone to slide, readily accommodating each other's shapes, inevitably should fail after a finite plastic strain.\",\n \"The reason is an elementary condition that was advanced some years ago [19] but omitted in other studies: whatever the mechanisms for stress–dependent grain boundary sliding and grain shape accommodation may be, they must be consistent with density conservation to produce a steady flow.\",\n \"However, it is shown here that if the local shear stresses resolved in the planes of grain interfases have a finite threshold for causing grain sliding, density is not conserved in the overall plastic flow.\",\n \"The grains are increasingly compressed as the sample is being stretched, and hence grain sliding can only proceed at the expenses of elastic volume variations of the crystallites.\",\n \"Fracture after a brief plastic strain, typically of a few percents, is a necessary consequence of the polycrystalline nature of the materials.\",\n \"The model gives a simple and precise closed–form equation for the strain to fracture, which is the strain at which the internal hydrostatic pressure diverges.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08670"}}},{"rowIdx":394,"cells":{"text":{"kind":"list like","value":[["Electron dynamics method using a locally projected group diabatic Fock\n matrix for molecules and aggregates"],["Abstract We propose a method using reduced size of Hilbert space to describe an electron dynamics in molecule and aggregate based on our previous theoretical scheme [ T. Yonehara and T. Nakajima, J. Chem.","Phys.","\\textbf{147}, 074110 (2017) ].","The real-time time-dependent density functional theory is combined with newly introduced projected group diabatic Fock matrix.","First, this projection method is applied to a test donor--acceptor dimer, namely, a naphthalene--tetracyanoethylene with and without initial local excitations and light fields.","Secondly, we calculate an absorption spectrum of five-unit-polythiophene monomer.","The importance of feedback of instantaneous density to Fock matrix is also clarified.","In all cases, half of the orbitals were safely reduced without loss of accuracy in descriptions of properties.","The present scheme provides one possible way to investigate and analyze a complex excited electron dynamics in molecular aggregates within a moderate computational cost."],["Introduction","Recently, the authors proposed a concise method for describing the quantum dynamics of excited electrons traveling over constituent molecules in a molecular aggregate system by utilizing a group diabatic Fock (GDF) matrix [1].","The construction scheme for this GDF matrix using atom centered Gaussian basis set was practically demonstrated by Thoss et al.","[2] based on the idea of diabatization of block diagonalization of Hamitonian that was originally introduced by Köppel et al.","[3] Before that, the applications of this block diabatization concept for a practical calculation can be found in the works on electron conduction problems.","[4], [5] This block diabatic scheme for the electron dynamics provides an intuitive understanding of charge and exciton migration as the quantum mechanical transport of electronic energies in a molecular assembly subject to the inherent electronic propensities of the site molecules, starting from any prepared type of initial local excitations considering the light-electron couplings.","In the present article, on the basis of the characteristics of a nearly block structured diabatic representation in the GDF method, we introduce a compact representation within an extracted Hilbert subspace.","The aim of this paper is two-fold: (1) to introduce a scheme for reducing the computational cost in keeping with an accurate description of the dynamics, and (2) to investigate how many local orbitals are required for a description of dynamics and absorption spectrum toward an interpretation of electron dynamics.","Note that the group diabatic representation treated here does not lead by itself to a reduction of computational cost by describing the dynamics with a small subset of the full orbital space.","In fact, the GDF matrix has the same dimension as the original Fock matrix before the block diagonalization.","(1) and (2) are obtained by using a projection scheme introduced here as an extraction of local group diabatic orbitals associated with a size of the Hilbert space needed for a description of electron dynamics.","The reduction of calculation cost is associated with the reduction of the number of extracted local group orbitals as a basis set for constructing matrix representations of electronic operators utilized in a calculation of electron dynamics.","This article improves upon our previous work by providing and assessing a procedure for reducing the computational cost of electron dynamics calculation using the GDF method, which is combined with the real-time time-dependent density functional theory (RT-TDDFT) [6], [7], [8], [9] under the adiabatic approximation for electron functionals.","The point is that electron migration of chemical interest over a molecular aggregate under moderate light conditions occurs in a sub Hilbert space consisting of a low number of excited states described by molecular orbitals within a relevant but not very large energy range around the Fermi energy of a system.","It is instructive to compare other studies with our present method.","There are many studies intensively investigating molecular interactions and their effects on excited energy transfer proceeding in excited molecular aggregates.","[10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] For example, Futera et al.","[23] successfully utilized and assessed the GDF method originally named a projection operator diabatization scheme[2] to evaluate electronic coupling matrix elements with high level ab initio calculations in the electron-transfer process of a molecular-metal/semiconducter interface.","However, a study on a systematic variable description of the excited electron dynamics in a bottom-up approach is rare.","Compared to previous studies on excited electron transfer in molecular aggregates, the scheme introduced in the present article has the advantages of a compact description of excited electron migration with ab initio electronic structure calculations.","In addition our new scheme allows a systematic improvement of the results by enlarging the projection space.","The most prominent feature is that our scheme is intended for a real-time dynamics of excited electrons in molecular aggregates in an external field starting from any pattern of initial local excitations prepared as a perturbation for the electronic state.","In this article, we detail the procedures for constructing the projected local orbital space within the group diabatic (GD) representation, and then demonstrate numerical applications.","We examine the size of a local orbital space related to dynamics in a systematic way by increasing an energy range for projecting a diabatic local orbital space.","A naphthalene(NPTL)$-$ tetracyanoethylene(TCNE) dimer is treated as a test donor$-$ acceptor system.","Additionally, we also demonstrate a convergence of absorption spectrum with respect to a size of orbital space using a five-unit polythiophene molecule(5UT) as a typical electron donor species in solar cell materials.","To ensure the energy balance between local projected orbitals, we employ an energy width parameter for extracting a relevant subspace with the mean value of the highest occupied molecular orbital (HOMO) and lowest occupied molecular orbital (LUMO) energies in the whole system as a reference energy, and we do not use a scheme which requires direct orbital selection.","In Sect.","II, we explain the theoretical method for describing the electron dynamics based on the GDF representation within projected local diabatic orbitals, which is followed by numerical examples in the section III."],["Theoretical method","In this section, after a brief summary of the electron dynamics method [1] using a GD representation,[1], [11], [12], [13], [10], [17] we introduce a scheme for constructing a concise matrix form with use of projected diabatic local orbitals having a dominant contribution to the underlying dynamics.","The determination of diabatic local projection orbitals playing a main role in the present work requires only the parameters of energy ranges for monomers covering the important orbitals around HOMO and LUMO playing in an excited electron dynamics."],["Overview of the locally projected-space group diabatic representation","The transformation from an atomic orbital (AO) representation of physical operators to a GD form consists of the following two transformations being constructed sequentially: (i) a transformation to a representation using the Löwdin orthogonalized atomic basis functions and (ii) a unitary transformation made from the local orbital sets obtained by the diagonalizations of block sub-matrices corresponding to the predetermined monomer groups in the Fock matrix prepared in step (i).","The overview of the process of extracting the diabatic local projection orbitals as the primary topic of the present article is as follows: (a) calculate the mean of the HOMO and LUMO orbital energies of the whole system; (b) set an energy range covering the local orbitals for each monomer in which the value obtained in (a) is placed at the middle; (c) obtain the diabatic local projection orbitals of the monomers in the energy range prepared in (b).","We refer to the matrix representation within these projection orbitals as the locally projected-space group diabatic Fock (LP-GDF) representation, of which the details are explained in later subsections.","The multiplications of matrices associated with the electronic properties and the analyses of the time-dependent electron density using the newly introduced LP-GDF representation are carried out within the projected orbital space.","The information related to the size of the orbital space required for a description of the dynamics without loss of accuracy provides us with insight into the sub-Hilbert space relevant to it.","In the following subsection, for a self-contained form of the present article, we first summarize the electron dynamics scheme using a GD representation[1] and then proceed to describe how to obtain the diabatic local projection orbitals and how to construct compact matrix representations by using them as a subset of the basis functions.","The first step is to prepare a Fock matrix represented by the Löwdin orthogonalized atomic basis function [24] $\\widetilde{F}_{mn} \\equiv \\langle \\widetilde{\\chi }_m | \\widehat{F} | \\widetilde{\\chi }_n \\rangle ,$ where the orthogonalized Löwdin atomic orbitals(AOs) are expressed by $| \\widetilde{\\chi }_n \\rangle = \\sum _j^{\\textrm {AO}} | \\chi _j \\rangle (\\underline{\\underline{S}}^{-1/2})_{jn}$ with $S_{jn} = \\langle \\chi _j | \\chi _n \\rangle $ being the AO overlap matrix element.","Here $\\lbrace \\chi _n \\rbrace $ is the original basis set consisting of AOs."],["Localized orbitals of a subgroup","After the classification of $\\lbrace \\widetilde{\\chi }_n \\rbrace $ into subgroups, e.g., monomers, the block structure of the Fock matrix within the Löwdin basis set is determined with its diagonal blocks $\\lbrace \\underline{\\underline{\\widetilde{F}}}_{G_i G_i}\\rbrace $ and off-diagonal ones $\\lbrace \\underline{\\underline{\\widetilde{F}}}_{G_i G_j}\\rbrace _{i\\ne j}$ with i and j ranging from 1 to $N_g$ .","$G_i$ denotes the i-th subgroup.","$N_g$ is the number of subgroups in the system.","Note that we can employ arbitrary divisions of the component atoms in the whole system.","The diagonalization of diagonal blocks corresponding to subgroup $G$ , $\\underline{\\underline{\\widetilde{F}}}_{GG}=\\underline{\\underline{D}}_{G}\\,\\underline{\\underline{\\overline{F}}}_{GG}\\,\\underline{\\underline{D}}_{G}^{\\dagger },$ gives rise to the unitary transformation matrix $\\underline{\\underline{D}}_{G}$ , whose column vectors are the linear coefficient vectors of the localized eigenstates expanded in terms of the Löwdin orthogonalized atomic basis functions for the group $G$ .","The dagger symbol attached to a matrix indicates its adjoint form.","$ \\underline{\\underline{\\overline{F}}}_{GG} $ is the diagonal matrix having the eigen energies $\\lbrace \\epsilon _{j,{G}}\\rbrace _{j = 1 \\sim M_{G} }$ of the corresponding subgroup $G$ as its elements, and $M_{G}$ is the number of local basis functions spanned at group $G$ .","Here, $ {G} \\in \\lbrace {G}_1,...,{G}_{N_g} \\rbrace $ .","The elements in off-diagonal blocks of the Fock matrix represented by these localized orbitals, associated with different groups, can take non-zero values, which provide a diabatic character in the representation with the use of the collection of these orbital sets.","These group localized orbital sets provide a transformation matrix from a Löwdin representation to the GD one to be explained later.","Here, we provide a short comment on the ambiguity of group division in a Löwdin representation of Fock operator.","In case using diffuse AO orbitals having far larger distribution than a distance between monomers, a center of Löwdin orbital created by the mixture of original AOs can become close to a nuclei belonging to a different group.","Though that is one of the possible problem in the group assignment of Löwdin orbitals associated with a natural group sectoring, it does not cause any problem in case with distances of monomers sufficiently larger than AOs and also Löwdin orbitals.","[17] In this article, we do not use diffuse orbitals and treat a case with distance of monomers being larger than a spatial range in orbital distribution."],["Group diabatic Fock matrix","The GD representation of the Fock operator $\\underline{\\underline{\\overline{F}}}$ , as one of the main ingredients in the GDF electron dynamics scheme is constructed via the transformation of the Löwdin representation matrix of the Fock operator $ \\underline{\\underline{\\widetilde{F}}} $ .","By using the already obtained unitary matrices with the dimensions of the local basis functions associated with groups, $\\lbrace \\underline{\\underline{D}}_{G_i}\\rbrace _{i=1 \\sim N_g}$ , this transformation is expressed by [1], [11], [12], [10] $\\underline{\\underline{\\overline{F}}}_{G_i\\,G_j}=\\underline{\\underline{D}}_{G_i}^{\\dagger }\\,\\underline{\\underline{\\widetilde{F}}}_{G_i\\,G_j}\\,\\underline{\\underline{D}}_{G_j},$ where i and j range from 1 to the number of groups $N_g$ .","The assembly of these sub-matrices $ \\left\\lbrace \\underline{\\underline{\\overline{F}}}_{G_i\\,G_j} \\right\\rbrace _{i,j=1\\sim N_g} $ constructs the GD representation which is called GDF matrix and expressed by $ \\underline{\\underline{F}}^{\\textrm {GD}} $ .","The physical meaning of the components in this final form is as follows.","This sub-matrices $ \\left\\lbrace \\underline{\\underline{\\overline{F}}}_{G_i G_i} \\right\\rbrace _{i=1 \\sim N_g}$ in the diagonal blocks correspond to the local group eigen energies, while $ \\left\\lbrace \\underline{\\underline{\\overline{F}}}_{G_i G_j} \\right\\rbrace _{i \\ne j}$ , placed at the off-diagonal blocks, describes the interactions between different groups.","Note that, in this transformation, the information included in the AO, Löwdin, and GDF representations are the same and no approximation is applied."],["Transformation from the AO representation to the GD representation","A matrix representation of any observable operator $\\hat{O}$ in terms of the constructed GD basis set, $\\underline{\\underline{O}}^{\\mathrm {GD}}$ , is related to that of the original AO basis set, $\\underline{\\underline{O}}^{\\textrm {AO}}$ , as $\\underline{\\underline{O}}^{\\mathrm {GD}}=\\underline{\\underline{U}}^{\\dagger } \\,\\underline{\\underline{O}}^{\\textrm {AO}} \\,\\underline{\\underline{U}},$ where $\\underline{\\underline{U}} \\equiv \\underline{\\underline{S}}^{-1/2} \\underline{\\underline{W}}$ .","Here, the diagonal block in the sub-transformation matrix $ \\underline{\\underline{W}}$ is given by $\\underline{\\underline{W}}_{G_i G_i} \\equiv \\underline{\\underline{D}}_{G_i}$ for $i$ , while the off-diagonal one is defined as $\\underline{\\underline{W}}_{G_i G_j} \\equiv \\underline{\\underline{0}}$ .","The Fock matrix obeys the same transformation rule and is obtained by setting $\\hat{O}=\\hat{F}$ in the above equations, where we know that $ \\underline{\\underline{F}}^{\\textrm {AO}} = \\underline{\\underline{F}} $ and $ \\underline{\\underline{F}}^{\\mathrm {GD}} = \\underline{\\underline{\\overline{F}}} $ .","On the other hand, the transformation of the density matrix from the original AO basis set to the GD one is written as $\\underline{\\underline{\\rho }}^\\mathrm {GD}=\\underline{\\underline{U}} \\,\\underline{\\underline{\\rho }}^\\mathrm {AO} \\,\\underline{\\underline{U}}^{\\dagger }.$ Note that the unitarity of $\\underline{\\underline{W}}$ assures total electron conservation with respect to this transformation, $\\textrm {Tr}\\left[ \\underline{\\underline{\\rho }}^{\\mathrm {AO}} \\underline{\\underline{S}} \\right]= \\mathrm {Tr} \\left[ \\underline{\\underline{\\rho }}^\\mathrm {GD} \\right].$"],["State coupling","We can obtain the essential elements needed for the construction of light–electron coupling by setting $ \\hat{O} = \\hat{\\bf r} $ , $ \\partial _{\\bf r} $ in the previous subsection.","Here, boldface denotes a vector in a three-dimensional Cartesian space, and ${\\bf r}$ denotes a composite variable of the electron position in three-dimensional space.","The first and second operators are responsible for the light–electron coupling in length and velocity forms.","[25], [26] Here, we neglect the nonadiabatic coupling and molecular motion to allow us to focus on an examination of the electron dynamics scheme using projected local diabatic group orbitals.","Light-electron couplings within GD representation are expressed by $\\underline{\\underline{L}}^\\mathrm {GD}=+ e \\underline{\\underline{{\\bf r}}}^\\mathrm {GD} {\\bf E}$ for the length gauge and $\\underline{\\underline{L}}^\\mathrm {GD}=- i \\hbar \\dfrac{e}{c} {\\bf A}\\underline{\\underline{\\partial _{\\bf r}}}^\\mathrm {GD}$ for the velocity gauge.","[25], [26] ${\\bf E}$ and ${\\bf A}$ denote respectively the three-dimensional electric field vector and electromagnetic field vector potential, which generally depend on a point in a three-dimensional space.","$\\underline{\\underline{{\\bf r}}}^{\\textrm {G}D}$ and $\\underline{\\underline{\\partial _{\\bf r}}}^{\\textrm {G}D}$ denote the GD presentation matrices respectively for $\\hat{\\bf O}={\\bf r}$ and $\\partial _{\\bf r}$ in Eq.","(REF ).","In this study, we employ the length gauge.","The details on the treatment for these coupling matrices with further approximations applied in dynamics calculation are given in Subsect.","REF ."],["Locally projected space made from subsets of local site orbitals\n","This subsection is an essential part of the present article proposing the LP-GDF scheme.","Here, we provide the concrete procedures used in it and the corresponding mathematical expressions.","Let us consider the diagonal block of the Fock matrix within the GD representation, $\\left\\lbrace \\underline{\\underline{\\widetilde{F}}}_{G_i G_i}\\right\\rbrace $ and the associated local orbital energies $\\lbrace \\epsilon _{j,G_i} \\rbrace _{j=1\\sim N_{G_i}}$ corresponding to the i–th group site.","In a situation where the interactions between group monomers are weak compared to those among the atoms in each monomer, we can safely employ the referential local ground state, where all of the electrons assigned to a local site are filled in ascending order from the lowest-energy local orbital.","The excited electron dynamics in a molecular aggregate system under moderate sunlight conditions is expected to proceed in low-excited-state manifolds constructed from the local molecular orbitals in a relevant but not very large energy range around the Fermi energy of the system.","In a situation involving rather strong light field or molecular interaction, the Hilbert subspace required for a description of the dynamics will become large.","It is worthwhile to examine the size of the subspace relevant to the excited electron dynamics by varying the type of molecular interactions such as light-matter coupling and initial local excitations.","Thus, in order to examine the size of the orbital space needed for a description of the electron dynamics at a sufficient accuracy, we establish a procedure for projecting a Hilbert subspace around the Fermi energy of the whole molecular aggregate by using a width parameter for the energy range as follows: (i) Set an energy width $\\Delta \\epsilon _i$ covering the sub-space consisting of local orbitals orbitals for the $i$ -th monomer, $G_{i}$ .","Here $i$ ranges from 1 to $N_g$ .","(ii) Calculate the mean $\\eta $ of the HOMO and LUMO energies of the whole system.","(iii) Extract a subset of group diabatic local orbitals in the energy range $ [ \\eta - \\Delta \\epsilon _i/2, \\eta + \\Delta \\epsilon _i/2 ] $ from the whole orbital space.","We refer to this as a projection of the diabatic local orbitals.","(iv) Construct minor matrices as representations of the electronic operators within the projected orbital basis set obtained in (iii).","(v) Follow the same procedure as that in the GDF electron dynamics scheme other than the projection introduced here.","This means that time propagation and property analysis are carried out using the obtained small size matrices.","The mathematical form of LP-GDF procedure is compactly given in Appendix .","In this article, we employ a version of projection independent on the orbitals during a time propagation but constructed from the initial orbitals in dynamics.","The orbital projection scheme using an energy width allows for the systematic and natural treatment for general situations involving the unknown energy orders and spatial localities of the GD orbitals in molecular aggregates.","Note that to ensure the energy balance between local projection orbitals, we employ an energy width parameter for extracting a relevant subspace with the mean value of the highest occupied molecular orbital (HOMO) and lowest occupied molecular orbital (LUMO) energies in the whole system as a reference energy, and we do not use a scheme which requires direct orbital selection.","In the following numerical demonstrations, for focusing on this essential point in the proposed scheme, we employ a parameter for the energy range independent of the monomers, namely, $\\Delta \\epsilon _i$ = $\\Delta \\epsilon $ , which is expressed by $E_{\\rm bw}$ in the figures.","The resultant total number of projection orbitals of the monomers is denoted by $N_{\\rm proj}$ throughout the present article."],["Initial density matrix: local excitation and electron filling","The initial density matrices in the GDF representation are prepared so that the diagonal elements in each diagonal block of the corresponding monomer should be occupied up to the number of electrons assigned to this monomer.","A simple example can be found in our previous article.","[1] Though an off-diagonal filling responsible for the initial coherence is also possible, for simplicity, we consider only the diagonal part in the preparation of the initial state of the density matrix.","This issue will be reported in our future article.","In the article, we treat the case of the spin-restricted model within the GDF representation as explained below.","This is not rigorously the same as that within the canonical (KS or HF) orbital representation.","In a restricted case with the same spatial orbitals for different alpha and beta spins in this model, the occupancy of the GDF local orbitals of each monomer is up to half of the number of electrons assigned to this group from the lowest energy orbital.","Therefore, in a strict sense, the initial density matrix mentioned above differs from that of the true ground state of the whole system.","In fact, this does not cause any problem for examining the migration dynamics of charge and electronic excitations over the constituent monomers in an assembly.","[1] Generally, we can make any type of excitation configuration starting from the reference occupations of the GD orbitals.","If we want an initial density associated with an excess or a deficiency of electrons in each monomer for treating the case accompanied with a charge moiety, we merely need to set the occupations to the corresponding number of electrons in each monomer.","[1] Throughout this article, we treat cases with an overall singlet spin state in a spin-restricted manner.","Appendix provides a practical example for preparing an initial density matrix within GD representation."],["Time propagation of the density matrix in a GD representation","In this study, the calculation of the time propagation of an electronic state is performed in terms of the Liouville–von Neumann equation associated with one particle density matrix as follows: $\\dfrac{\\partial }{\\partial t}\\underline{\\underline{\\rho }}^\\mathrm {GD}=-\\dfrac{i}{\\hbar }\\left[\\underline{\\underline{\\mathcal {F}}}^\\mathrm {GD}\\left\\lbrace \\underline{\\underline{\\rho }}^\\mathrm {GD}(t),t \\right\\rbrace ,\\underline{\\underline{\\rho }}^\\mathrm {GD}\\right],$ where $\\underline{\\underline{\\mathcal {F}}}^\\mathrm {GD}\\equiv \\underline{\\underline{F}}^\\mathrm {GD}+\\underline{\\underline{L}}^\\mathrm {GD}.$ $\\underline{\\underline{L}}^\\mathrm {GD}$ is a light–electron coupling matrix.","For the length gauge, their corresponding matrix elements have forms of $\\underline{\\underline{L}}^\\mathrm {GD}=+ e \\underline{\\underline{{\\bf r}}}^\\mathrm {GD} {\\bf E}$ , [25], [26] where ${\\bf E}$ is the three-dimensional electric field vector, which generally depend on a point in a three-dimensional space.","We used the dipole approximation, namely, long wavelength approximation, [25], [26] since the wavelength of light treated here is sufficiently large compared to the size of the molecular system treated.","In the RT-TDDFT, the Fock matrix depends on the time-dependent density matrix.","For convenience of a discussion on the numerical demonstration to be presented in the later section, we label the dynamics described by Eq.","(REF ) including this dependency as 'RT' meaning 'real time Fock matrix' while the term 'FF' denoting 'frozen Fock matrix' is used to refer to the approximated dynamics with the replacement of the time dependent Fock by that at the initial simulation time.","For a technical simplicity and focusing on the projection scheme of electron dynamics, we employ the pure density functional for including an electronic exchange-correlation throughout this article.","For obtaining the time dependent electron density matrix, we solved the non-linear Liouville–von Neumann equation associated with the RT-TDDFT and introduced LP-GDF matrix by using the following two types of numerically stable time integrators, namely (1) the predictor-corrector second order Magnus scheme with linear Fock extrapolation (PC2M-LF) [27] and (2) exponential propagation with predictor-corrector SCF scheme using final corrector as a resultant density (EPPC1).","[28] PC2M-LF needs one time of update of Fock matrix for each step while an iteration scheme is applied for EPPC1 until a corrector density is converged.","Their details are summarized in Appendix .","We followed a dynamics with time steps of 8 and 20 attosecond using a PC2M-LF and EPPC1 time integrator, respectively.","Hereinafter we abbreviate femtosecond to fs and attosecond to as.","In many cases with a moderate dynamics of density the use of former scheme is sufficient, while the latter was needed for a stable calculation of absorption spectrum using short laser pulses.","The Fock, electron dipole transition matrices within the AO representation required for the dynamics calculation were evaluated by using the NTChem2013 software package.","[29] We should comment on conservations of properties during dynamics.","As a special case, we examined a case starting from a locally excited state without external light field.","While within the LP-GD scheme a trace of density matrix is perfectly conserved both for RT-TDDFT and Frozen Fock approximation, an expectation value of Fock operator is conserved only for the latter one.","Despite this, in a practical sense, this LP-GD RT-TDDFT scheme provides a good result with respect to a convergence to reference data as $\\Delta _\\epsilon $ increases, which will be shown in Sect.","with demonstrations of a userbility of the method through calculations of charge migration dynamics and absorption spectrum."],["Numerical demonstration","Here, the size effect of the projected space on the electron dynamics and absorption spectrum are examined by using the LP-GDF electron dynamics method introduced in the previous section.","We compare the time-dependent behaviors of the Mulliken charge of electron donor molecules for NPTL–TCNE with different initial excitation and continuum light field by varying $E_{\\rm bw}$ .","The convergence of absorption spectrum for 5UT is also examined in the same manner after showing explicitly the importance of the self-consistency between time-dependent density and the time-dependent Fock matrix during electron dynamics.","With respect to the NPTL–TCNE, readers can find further information in our previous article.","[1] We validate the efficiency of the present projection method by showing a convergence of result with respect to a reduced size of local orbital space.","The schematics of the molecular configurations used in this article as shown in Fig.","REF .","We also provide information related to the dependence of the computational cost in the electron dynamics calculations with and without Fock build on the size of the projection space, which are detailed in Appendix .","Figure: Schematics of the geometries of the molecular aggregates treated in this article:(a) NPTL–TCNE dimer (b) 5UTThe centroids of aggregates were set to origin for all the systems.See the supplementary material for detailed information of their Cartesian coordinates."],["NPTL–TCNE dimer","As a first test system, we treat a dimer system consisting of NPTL and TCNE.","In this combination of monomers, NPTL serves as an electron donor molecule, while TCNE plays role as an electron acceptor.","The molecular geometries and relative orientations of the monomers used here are the same as those in Panel (l) in Fig.","3 in our previous article [1] on the original GDF electron dynamics scheme where readers can find further information including the literature of experimental data.","The geometry of each monomer was optimized at the DFT/6-31G(d) level with the use of the PBE exchange correlation functional.","[30] The Fock matrix associated with initial optimized KS orbitals and its following time-dependent density matrix for the construction of the GDF matrix was also calculated at the same ab initio level.","Both molecules have planar geometries in the optimized geometry in their ground electronic states.","Here, as shown in Fig.","REF , the principal axis of NPTL was set to be parallel to the X axis, while we set TCNE to be parallel to the Y axis.","The molecular planes of these flat molecules are parallel to the X–Y plane.","They were placed in a parallel orientation with a slide of 1.24 $\\textrm {Å}$ along the Y axis.","The distance between molecular planes was fixed at 4 $\\textrm {Å}$ .","Although a dimer is treated here, we considered the crystal data[31] reported in the literature with respect to the relative orientation.","This selection of molecular configuration yields a non-vanishing overlap between the frontier orbitals, i.e., the HOMO of NPTL and the LUMO of TCNE.","[1] See the supplementary material for a further information on the geometrical coordinate of this system as well as GD and canonical orbital energies.","Figure: Convergence of the results for the charge separation dynamics in an NPTL–TCNE dimerwith an increase in the energy range determining the projected local orbital space.The energy width used for orbital projection is expressed byE bw E_{\\rm bw} = Δϵ\\Delta \\epsilon in Hartree.The numbers of projected orbitals, N proj N_{\\rm proj}, are 13, 46, 74 and 120respectively for E bw _{bw} = 0.2, 0.5, 1.0 and 1.6.The exact results obtained by using the full 296 orbitals are indicated by solid red line.See the main text for the details of situations with respect to the treatment of Fock matrix,initial local excitation and light field employed in the panels.Fig.","REF provides the results of the excited electron dynamics involving initial local excitations and external light fields.","The time dependent behaviors of the Mulliken charge of the donor molecule, NPTL, are displayed with the variation of the energy range covering the projection orbital space $\\Delta \\epsilon $ .","In the panels of the figure, $\\Delta \\epsilon $ is expressed as $E_{\\textrm {bw}}$ .","In panels, we plotted the Mulliken charge of NPTL as an electron donor system for each case corresponding to the vertical axis.","The horizontal axis denotes the time in femtoseconds.","The results of RT-TDDFT LP-GDF scheme are displayed in panels of (a–d) while corresponding results of frozen Fock approximation are presented in (e–h).","(a/e) and (b/f) show the results starting from the initial local excitation respectively in the donor NPTL and acceptor TCNE, while the other moieties are initially in the ground states within the GDF representation.","In these four panels, no light field is irradiated to the system.","In panels (c/g) and (d/h), a continuum light field is applied, and its field parameters as a wave length and unit vector of polarization are 700 nm corresponding to $\\omega =0.065$ au and $\\left( \\frac{1}{\\sqrt{3}}, \\frac{1}{\\sqrt{3}}, \\frac{1}{\\sqrt{3}} \\right)$ in the XYZ Cartesian coordinate, respectively.","The field strength $Es$ used in the cases of panels (c/g) is 0.015 a.u., while a larger strength of 0.02 is applied in (d/h).","The function form of light field is given in Appendix .","The continuum light field was replaced by a pulse with a sufficiently large time width $t_w=2.4\\times 10^{4}$ fs and peak time $t_c=0$ .","As seen in the RT cases (a) and (b) accompanied with initial local excitations without light fields show the fast convergence with respect to the increase of $E_{\\textrm {bw}}$ .","In fact, $E_{\\rm bw}=1.0$ associated with 74 projected local orbitals provides a quantitative convergence to the result obtained by the full 296 orbitals.","On the contrary, in (c) and (d) from initial local ground states with continuum light field having moderate strength, the quantitative convergence are achieved by $E_{\\textrm {bw}}$ =1.6 providing 120 projected orbitals.","As found in the panels (c/g) and (d/h), the FF approximation provides faster convergence compared to RT results, corresponding to the cases involved with light fields.","On the other hand, the differences of the converged results of FF from those of RT means the importance of a consideration of feedback from time dependent density matrix to the Fock matrix.","In (c/d/g/h), we employed the light stronger than that of usual sun light for a severe assessment of the present scheme in cases accompanied with hard excitations.","In fact, in this strength, the non-linear effect about the density matrix result in the difference in convergence rates between frozen Fock type and RT-TDDFT calculations with respect to an increase of number of projected orbitals.","Figure: N proj N_{\\rm proj} dependency of max deviations for donnor moleculeduring dynamics from the results by full orbital calculation in Fig.",".Deviation is expressed as an absolute value but not relative one.The panels of (a), (b), (c) and (d) in this figure correspond to those of(a/e), (b/f), (c/g) and (d/h) in Fig.",", respectively.“RT” and “FF” in panels correspond to the results obtained by electron dynamics calculationusing RT-TDDFT scheme considering the self-consistency and frozen Fock approximation.To clarify the convergence of time dependent behaviors of charge migration dynamics in Fig.","REF with respect to the number of projected orbitals, we displayed the max deviations of the approximation results measured from the reference data obtained by full orbitals in Fig.","REF as a function of $N_{\\rm proj}$ .","As found in the figure, in all cases we can reduce a half the number of the orbitals keeping with the accuracy matched to the result obtained by using full orbitals.","The associated CPU times spent for the calculations and the ratios of them to those obtained by the full orbitals are summarized in Appendix .","In the last part of this subsection, as a supplementary information, we provide an evidence of the efficiency of the orbital extraction scheme for the standard RT-TDDFT and Frozen Fock approximation by guiding to the corresponding results.","The electron dynamics calculation using the orbital extraction (projection) in standard KS orbital sequence is equivalent to the GDF electron dynamics calculation with $N_g=1$ , namely no group division.","This is because a diagonalization of Löwdin AO representation in case of $N_g=1$ gives rise to exactly the same eigen states for the original Kohn-Sham equation at an initial simulation time.","In the other words, in the case of $N_g=1$ , total eigen states of Fock operator do not change through a transformation from a representation with original AO to that with Löwdin orthonormalized AOs.","We calculated charge migration of this system by treating the whole system as a monomer, namely, by setting $N_g=1$ .","The energy threshold value for orbital extraction and number of selected orbitals were varied within LP scheme.","The data were compared with those obtained by the methods without any orbital projection nor any group division, namely full orbital type of standard RT-TDDFT or Frozen Fock approach.","Because the setting of $N_g=1$ does not allow local excitation of each monomers at initial simulation time in the same sense as $N_g \\ge 2$ , we calculated only the light field cases starting from initial ground state of whole system, namely, no excitation of that.","The results and detailed discussion are included in the Supplementary Information.","We should also comment that the Frozen Fock approach without GD representation can provide a better convergence as seen in the comparison between the panel (g) in Fig.","2 and (c) in Fig.","S2 with respect to the $E_{\\rm bw}=0.2$ and full orbital cases.","In the same sense, RT-TDDFT without GD approach can be superior in convergence as found in the comparison between the panel (d) in Fig.","2 and (b) in Fig.","S2 for $E_{\\rm bw}=1.0$ and full orbital cases."],["5UT","In the second demonstration of the present work, we here apply the method to the calculation of absorption spectrum of 5UT.","This molecule plays an important role not only as an electron donor but also a hole transfer material in an organic solar cell.","[32] The existence of sulfur atoms in $\\pi $ conjugate system gives rise to the stability of molecular structure as well as the increase of charge conduction through the electron overlap contributed by their d-orbitals.","[33] Here we do not discuss a chemical aspect of this system, and just focus on the performance of the method by looking the dependency of results on the size of projected local orbital space.","The molecular geometry of 5UT treated here was optimized in the PBE/6-31G(d) level calculation, which is commonly used in the electron dynamics calculation.","The geometry information is summarized in the supplementary materials.","The light pulse was applied and the time-dependent information of induced electronic dipole moments was converted to absorption spectrum, of which details on functional forms are given in Appendix .","We used EPPC1 as the time integrator with the time step of 20as.","The total time of simulation carried out was 30 fs.","The functional form of light field is given in Appendix .","The field strength, $E_s$ , central frequency, $\\omega $ , pulse peak time, $t_c$ , pulse width time, $t_w$ , and angular frequency, $\\omega $ , of the applied light pulse were 0.001au, 100(=2.42 fs), 40(=0.97 fs) and 0.0285, respectively.","The polarization vector of light field was set to be parallel to the vector of (1,1,1) in the Cartesian coordinate.","We set the group number $N_g$ =1.","The initial density used in the electron dynamics was set to that of the DFT ground state.","Figure: The importance of feedback of time-dependent information to the Fock matix.We presented absorption spectrums of 5UT with the variation of the damping factor,γ\\gamma , and time cut from the initial in the absorption calculation, T CUT T_{\\rm CUT}.See also the supplementary materials for the mathematical form used for the absorption spectrum.Left panels (a–c) and right ones (d–f) correspond to the results of RT and FF, respectively.Note that in all cases here, the full 394 orbitals are used in the dynamics calculationsfor preparing the light pulse induced electronic dipole moments for the spectrum calculations.Triangles denotes the positions of optically allowed excitationsassociated with the absorption energy obtained by linear response TDDFT calculations.The absorption spectrum evaluated from the formula given in Appendix numerically depends on the damping parameter, $\\gamma $ , and the initial cut off time, $T_{\\rm CUT}$ .","We checked that the simulation time was long enough for the examined energy range from 0 to 4 eV.","In order to select a set of reliable parameters for absorption spectrum, we compared the obtained spectrum by varying $\\gamma $ and $T_{\\rm CUT}$ , which is presented in Fig.","REF including the information of positions of optically-allowed excitations in energy space obtained by LR-TDDFT.","The transition properties evaluated by LR-TDDFT including excitation energies, oscillator strength, and dipole moments are summarized in the supplementary material.","We also compared the results with and without the feedback of time-dependent density to the Fock matrix in order to show that the instantaneous back-reaction of the density to effective Hamiltonian in the simulation is the key factor for the reproduction of the LR-TDDFT results, which are correspondingly labeled with RT and FF defined in the section of theoretical method, Subsection REF .","With respect to the parameters, we found the best parameters $\\gamma =0.004$ both for RT and FF cases in the aspects of the balance of proper smoothness and spectrum peak widths.","In turn, the best parameters of $T_{\\rm CUT}$ is 5.3 fs and 6.2 for RT and FF based on the observation of the appearance of maximum peaks.","Again, we can see that RT calculations presented in the panels (a–c) correctly reproduce the first peak position given by LR-TDDFT while FF ones fail.","Then, we chose the RT type calculation for spectrum calculation with $\\gamma =0.004$ and $T_{\\rm CUT}$ =5.3 fs and proceed to check the dependency on the size of projected local orbital space.","Figure: Projected size dependency and convergence of absorption spectrum.we compared the results by projected local orbitals, N proj N_{\\rm proj}=64, 89, and 164correspondingly associated with E bw E_{\\rm bw}=0.5, 0.75, and 1.5to that obtained by using N orbs N_{\\rm orbs}=394 full orbitals for 5UT molecule.Here we employed the damping factor of γ=0.004\\gamma =0.004 and T CUT T_{\\rm CUT}=5.3 fsbased on the observation in Fig.",".Triangles denotes the positions of optically allowed excitations associated with theabsorption energy obtained by linear response TDDFT calculations,which are labeled with the symbol 'LR.OA'.Fig.","REF presents the convergence of the absorption spectrum using RT-TDDFT calculation combined with LP-GDF scheme with respect to the size of projected local orbital space.","We can observe the monotonic convergence of the results by projection scheme to that obtained by the 394 full orbitals.","In fact, the spectrum shapes by $E_{\\rm bw}$ =0.5, 0.75 and 1.5 corresponding to $N_{\\rm proj}$ =64, 89 and 164 projected orbitals monotonically approaches to that of the full orbital calculation.","More than half of the total orbitals were reduced in this spectrum calculation.","The information of CPU time is also included in the supplementary material.","The success in the reduction of diabatic local orbital space required for the sufficient description suggests the efficiency of the introduced projection method for RT-TDDFT."],["Concluding remarks","In this study, we introduced and assessed the electron dynamics method as a combination of the local orbital projection and GDF electron dynamics scheme within a framework of RT-TDDFT.","Through the examination of complex charge migration induced by local excitation and light field for the NPTL–TCNE dimer as a test donor-acceptor system, we showed that the present method allows us to investigate the size of the Hilbert subspace associated with the excited electron dynamics of molecular aggregates.","In the application of the projection method to absorption spectrum, we numerically demonstrated the reproduction of the result with use of the reduced number of orbitals.","We can expect that this method paves a way to a practical investigation of electron dynamics of molecular aggregates with a further combination of the techniques for reducing the cost in Fock build.","We conclude by providing a perspective.","The molecular motion, non-adiabatic mixing, and transition between electronic states can affect charge and exciton migration dynamics.","There, a nuclear quantum effect may play an important role in electron dynamics in molecular aggregate systems having high density of states.","The GD representation scheme can afford a realistic and practical model Hamiltonian for treating these issues.","In our future work, we will report on a coarse grained quantum dynamics in molecular aggregates having a nano-size for describing a coupled dynamics of charge and exciton migration associated with their birth, extinction and transport."],["Supplementary material","See supplementary material for the issues of (I) validity check of the time interval employed for describing the charge migration dynamics and (II) Efficiency of orbital projection method for standard RT-TDDFT and Frozen Fock method without a group division in charge migration dynamics for NPTL–TCNE system.","(III) group diabatic and canonical orbital energies of NPTL–TCNE dimer (IV) optical transition properties of 5UT obtained by LR-TDDFT calculation (V) geometry data of molecular aggregates of a NPTL–TCNE dimer and 5UT.","This research was supported by MEXT, Japan, “Next-Generation Supercomputer Project” (the K computer project) and “Priority Issue on Post-K Computer” (Development of new fundamental technologies for high-efficiency energy creation, conversion/storage and use).","Some of the computations in the present study were performed using the Research Center for Computational Science, Okazaki, Japan, and also HOKUSAI system in RIKEN, Wako, Japan."],["Initial density matrix within GD representation associated with local excitations","Essential points in preparing initial density matrix are summarized in a following example.","Consider the spin-restricted case with three monomers each of which has 2 electrons and 3 orbitals.","Here set $N_g=3$ .","See the main text about the approximate treatment associated with this term of 'spin-restricted'.","If we want to prepare the initial condition such that the first and third monomers are initially excited from local HOMO to LUMO, the initial density matrix in GDF scheme is constructed as follows: [1] set reference density matrix, namely GDF ground state, $\\underline{\\underline{\\rho }}^{\\textrm {GD:ground}}$ and [2] carry out local HOMO LUMO excitation and obtain the aimed GDF density, $\\underline{\\underline{\\rho }}^{\\textrm {GD:1,3-HL}}$ namely, $& \\underline{\\underline{\\rho }}^{\\textrm {GD:ground}} \\equiv \\left(\\begin{array}{ccccccccc}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right) \\\\\\quad & \\Longrightarrow \\\\\\quad & \\underline{\\underline{\\rho }}^{\\textrm {GD:1,3-HL}} \\equiv \\left(\\begin{array}{ccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right).$"],["Time integrator","We detail the time integrators used for solving non-linear Liouville von Neumann equation including time-dependent Hamiltonian depending on the density matrix employed in this study, namely, (1) the predictor-corrector second order Magnus scheme with linear Fock extrapolation (PC2M-LF) and (2) exponential propagation with predictor-corrector SCF scheme using final corrector as a resultant density (EPPC1).","Note again that PC2M-LF needs one time of update of Fock matrix for each step while EPPC1 includes an iteration scheme with respect to a convergence of corrector density.","See the main text for the original articles of these schemes."],["PC2M-LF","This scheme consists of the following five processes for each time step, $[1] &\\, F_{3} := - \\frac{3}{4}F_{1a} + \\frac{7}{4} F_{1b} \\\\[2] &\\, D_{4} :=e^{- \\frac{i}{\\hbar } \\frac{dt}{2} F_{3} } D_{2} e^{ \\frac{i}{\\hbar } \\frac{dt}{2} F_{3} }\\\\[3] &\\, F_{5} := F[D_{4}] \\\\[4] &\\, D_{6} :=e^{- \\frac{i}{\\hbar } dt F_{5} } D_{2} e^{ \\frac{i}{\\hbar } dt F_{5} } \\\\[5] &\\,( F_{1a}, F_{1b} ) := ( F_{1b}, F_{5} ) \\,\\, \\Rightarrow \\,\\, \\textrm {End of this step}$ If we consider a time propagation from $t$ to $t+dt$ , $F_3$ and $F_5$ respectively denote the Fock matrices at $t+\\frac{1}{4}dt$ and $t+\\frac{1}{2}dt$ while $D_2$ , $D_4$ and $D_6$ correspond to the density matrices at $t$ , $t+\\frac{1}{2}dt$ and $t+dt$ .","$F_3$ in the first step [1] is constructed from the Fock matrices, $F_{1a}$ and $F_{1b}$ , respectively denote the Fock matrices in the previous two times respectively at $t-\\frac{1}{2}dt$ and $t-\\frac{3}{2}dt$ , by taking a linear extrapolation of them.","It is known that owing to the time dividing points set properly this scheme has the same accuracy of second order Magnus expansion.","The time propagation matrix having exponential form is treated exactly using a spectrum representation obtained by the diagonalization of Fock matrix, which is also applied in the EPPC1 scheme explained in the next subsection."],["EPPC1","This integrator involved with a micro iteration consists of the following procedures: $[1] & \\, F_{N} := F[D_{N}] \\\\[2] & \\, D_{N+1}^{\\rm pred} :=e^{- \\frac{i}{\\hbar } dt F_{N} } \\, D_{N} \\, e^{ \\frac{i}{\\hbar } dt F_{N} } \\\\[3] & \\, F^{\\rm pred}_{N+1} := F[D_{N+1}^{\\rm pred}] \\\\[4] & \\, F^{\\rm mid}:= \\frac{1}{2} \\left( F_{N} + F_{N+1}^{\\rm pred} \\right) \\\\[5] & \\, D_{N+1}^{\\rm corr} :=e^{- \\frac{i}{\\hbar } dt F^{\\rm mid} } \\, D_{N} \\, e^{ \\frac{i}{\\hbar } dt F^{\\rm mid} } \\\\[6] & \\,\\,\\left\\lbrace \\begin{array}{l}\\textrm {If} \\,\\, || D_{N+1}^{\\rm corr} - D_{N+1}^{\\rm pred} || < \\epsilon \\,\\,\\, \\\\\\quad \\textrm {then} \\,\\,\\,D_{N+1} := D_{N+1}^{\\rm corr}\\,\\,\\, \\Rightarrow \\,\\,\\textrm {End of this step} \\\\\\textrm {If} \\,\\, || D_{N+1}^{\\rm corr} - D_{N+1}^{\\rm pred} || \\ge \\epsilon \\,\\,\\, \\\\\\quad \\textrm {then} \\,\\,\\,D_{N+1}^{\\rm pred} := D_{N+1}^{\\rm corr}\\,\\,\\, \\Rightarrow \\,\\,\\textrm {Return to [3]}\\end{array}\\right.$ Here, a time propagation is carried out from $t$ to $t+dt$ , which are correspondingly labeled with $N$ and $N+1$ .","$F$ and $D$ are the Fock and density matrices.","'mid' denotes a mid point of $t$ and $t+dt$ , namely, $t+\\frac{1}{2}dt$ .","'pred' and 'corr' are abbreviations of 'predictor' and 'corrector', respectively.","The double '$||$ ' symbol means taking the Frobenius norm of a matrix placed between these two symbols.","The threshold is expressed by $\\epsilon =n\\alpha \\xi $ with $n$ being the dimension number of a corresponding matrix.","$\\alpha $ is set to be a value proportional to the absolute maximum eigen value of the matrix under consideration.","Here, we used 1 as $\\alpha $ for simplicity.","Thus, $\\xi $ determines the strictness of the self-consistency between the instantaneous density and the Fock matrix."],["Mathematical form of LP-GDF procedure","Here we provide the formal mathematical expression of LP-GDF scheme.","At first, one knows that within the GD representation the identity operator is written as $\\hat{1}\\simeq \\sum _{i=1}^{N_g} \\sum _{j=1}^{N_{G_i}}\\mid \\phi _j^{G_i}\\rangle \\langle \\phi _{j}^{G_{i}}\\mid ,$ where $\\left\\lbrace \\mid \\phi _j^{G_i} \\rangle \\equiv \\sum _k\\mid \\widetilde{\\chi }_{k,G_i} \\rangle \\left[ \\underline{\\underline{D}}_{G_i} \\right]_{kj}\\right\\rbrace $ with $ \\mid \\widetilde{\\chi }_{k,G_i} \\rangle $ being the k-th Löwdin orthogonalized atomic orbital basis function spanned in the G$_i$ -th group is the set of GD localized orbitals for the group labeled by $G_i$ and $j$ ranges from 1 to $N_{G_i}$ , which denotes the number of basis functions spanned at the site, $G_i$ .","$i$ ranges from 1 to $N_g$ , i.e., the number of monomer group sites.","We used the symbol for approximation in the equation because of the practical use of a finite basis set during computation.","Note that the GDF orbitals created from Löwdin orthonormal basis remain to be orthogonal under unitary transformations even if the transformations are carried out in each group.","In turn, with respect to the structure of matrix representation of the Fock operator, only the GDF orbital pairs between different monomers are Fock non-orthogonal while the pairs within the same group are Fock orthogonal .","Next, we introduce projection operators in order to realize steps (i)–(v) in Subsect.REF as follows: $\\hat{P} \\equiv \\sum _{i=1}^{N_g} \\sum _{j \\in \\Omega _i}^{N_{G_i}}\\mid \\phi _j^{G_i}\\rangle \\langle \\phi _j^{G_i}\\mid $ where $\\Omega _i \\equiv \\left\\lbrace j;| \\epsilon _{j,G_i} - \\bar{\\epsilon } |\\le \\Delta \\epsilon _i/2\\right\\rbrace \\quad \\textrm {and}\\quad \\bar{\\epsilon } \\equiv (\\epsilon _{\\textrm {H}}+\\epsilon _{\\textrm {L}})/2.$ Here, $\\epsilon _{j,G_i}$ is the local orbital energy associated with the GD orbital $\\phi _j^{G_i}$ , while $\\epsilon _{\\textrm {H}}$ and $\\epsilon _{\\textrm {L}}$ are the HOMO and LUMO energies of the whole system.","By using these mathematical tools within LP-GDF formulation, we approximate a one electron operator $\\hat{O}$ as follows: $\\hat{O}\\approx \\hat{O}_P \\equiv \\hat{P}\\hat{O}\\hat{P}.$"],["Function form of Laser pulse field","The vector potential of light field as a function of time in the long wave length approximation employed takes a form of ${\\bf A}(t) = \\sum _j^{N_\\textrm {p}}{\\bf A}_{j} \\, f(t;t_{c_j},t_{w_j}) \\, \\text{cos}(\\omega _j (t-t_{c_j})+\\delta _j),$ where bold font means the three dimensional vector in the Cartesian coordinate space.","Here, the envelope function is defined by $ f(t;t_c,t_w) \\equiv \\textrm {exp} \\left( - \\left( \\dfrac{t-t_c}{t_w} \\right)^2 \\right) $ .","The physical meanings of parameters appeared above are as follows; $t$ denotes time, $t_c$ is a field peak time, $t_w$ stands for a typical gaussian decay time, $\\omega $ means a central angular frequency of field, and $\\delta $ is a carrier envelope phase.","$N_\\textrm {p}$ denotes a number of pulses.","The electric field vector of external light corresponding to ${\\bf A}(t)$ is given by $ {\\bf E}(t) = -\\dfrac{1}{c} \\dfrac{\\partial {\\bf A}(t) }{\\partial t} $ .","In the present article, we employed $N_\\textrm {p}=1$ and $\\delta =0$ ."],["Absorption spectrum","The absorption spectrum was evaluated using the following function, $S(\\omega )\\equiv \\dfrac{1}{3}\\textrm {Tr}[\\underline{\\underline{\\widetilde{\\sigma }}}(\\omega )],$ where $\\underline{\\widetilde{\\underline{\\sigma }}}(\\omega )\\equiv \\dfrac{4\\pi \\omega }{c}\\textrm {Im}[\\underline{\\underline{\\widetilde{\\alpha }}}(\\omega )]$ is an absorption cross section with $[\\underline{\\underline{\\widetilde{\\alpha }}}(\\omega )]_{jk}\\equiv \\dfrac{\\widetilde{\\mu }^{\\rm {ind}}_j(\\omega )}{\\widetilde{E}_k(\\omega )}=\\dfrac{ \\int _{T_{\\rm CUT}}^{T_{\\rm FIN}} e^{ i \\omega t } e^{ - \\gamma t} \\mu _{j}(t) }{ \\int _{T_{\\rm CUT}}^{T_{\\rm FIN}} e^{ i \\omega t } E_k(t) }$ .","$\\gamma $ is a damping factor.","Here, $ \\mu ^{\\rm {ind}}_j(t) \\equiv \\mu _j(t) - \\mu _j(0) $ and ${E_k(t)}$ are induced dipole moment and external electric field, respectively.","The symbol of tilde means taking the Fourier transformation.","$T_{\\rm FIN}$ is a final simulation time.","$T_{\\rm CUT}$ is the initial time used for a transformation from time dependent induced dipole moment of electrons and external field to absorption spectrum."],["Computational time","Here, we provide information about the computational cost of electron dynamics including analysis of the time-dependent properties.","Note that the overhead of parallel computation is also included in time.","The aim here is to show examples of the performance of the method in cases with use of moderate computer facilities.","The specifications of the computer and compiler utilized in this article are as follows: [For NPTL–TCNE] { Intel(R) Xeon(R) Gold 6148 0 @ 2.40GHz with a cache size of 27.5MB, The Intel FORTRAN compiler in Version 18.0.2.199 with level three optimization. }","[For 5UT] { Intel(R) Xeon(R) Gold 6152 0 @ 2.10GHz with a cache size of 30.9MB, The Intel FORTRAN compiler in Version 17.0.4.196 with level three optimization. }","In all the calculations 40 cpu cores were used in a parallel manner using openMP for matrix-matrix product and MPI for Fock build if needed."],["NPTL–TCNE","Tab.","REF and Tab.","REF present the comparison of the computation times using different number of projected local orbitals in cases of PC2M-LF and EPPC1 methods, respectively.","The system treated is the NPTL–TCNE dimer.","A ratio to the time spent in cases with full orbitals expresses an acceleration in computation by using projected space.","The tables include the comparisons between RT and FF.","Note that the time intervals of one time step employed here are different between these two integrators.","We checked the reproduction of the same result by using these two integrators, which is presented in the supplementary material.","In Tab.","REF for PC2M-LF, though we can see that a moderate acceleration is attained with a help of projected space and acceleration in FF is superior to RT one including Fock build, the use of RT recommended for the reliable calculation and provides still moderate acceleration by projections of local orbital space.","In fact, the computation time in RT with $E_{\\rm bw}=1.6$ corresponding to $N_{\\rm proj}=120$ which provide converged results is reduced to one fourth of that spent in the cases using the full orbitals.","As seen in Tab.","REF , the acceleration in the calculation by projection in case of EPPC1 including micro iterations accompanied with Fock build for a convergence of corrector density is less than those of PC2M-LF in Tab.","REF .","Despite of this, the case of $E_{\\rm bw}=1.6$ with $N_{\\rm proj}=120$ need approximately the half of the computational time in the full orbital case.","Table: CPU time spent for the dynamics calculations plus the time-dependent analysisfor NPTL–TCNE with PC2M-LF scheme with 1875 time steps using 8 as time interval.The unit of time is second.Numeric value in [] denotes the ratio to the cost in case with use of full orbitalswhile that in {}\\lbrace \\rbrace mean the number of Fock build per one time step.Note the single Fock build per one time step in the PC2M-LF scheme.The spent time needed in the cases of Fig.","are the samein each group of the panels (a-d) for RT and (e-h) for FF, namely,in the above RT and FF columns correspond to the group of (a-d) and (e-h) in Fig.",".See the text for the meanings of E bw =ΔϵE_{\\rm bw}=\\Delta \\epsilon , N proj N_{\\rm proj}, RT and FF.The wall time means the wall clock time associated withopenMP for dynamics and MPI for Fock build.Table: CPU time spent for the dynamics calculations plus the time-dependent analysisfor NPTL–TCNE with EPPC1 scheme with 750 time steps using 20 as time interval.The unit of time is second.Numeric value in [] denotes the ratio to the cost in case with use of full orbitalswhile that presented in {}\\lbrace \\rbrace correspond to the average number of cycles inmicro-iterations needed for the density convergence with ξ=10 -8 \\xi =10^{-8}.See the text for the meanings of E bw =ΔϵE_{\\rm bw}=\\Delta \\epsilon , N proj N_{\\rm proj}, RT and FF.","(a-d) and (e-h) correspond to RT and FF, respectively,and all the symbols are the same as Fig.","in the main text.The definition of the wall time is the same as that in Tab.","."],["5UT","Tab.","REF presents the comparison of the computation times using different number of projected local orbitals in cases of EPPC1 methods for 5UT system.","The way of presentation is the same as the table for NPTL–TCNE.","Here also we show the comparisons between RT and FF.","As seen in Tab.","REF , the case of $E_{\\rm bw}=1.5$ with $N_{\\rm proj}=164$ associated with the converged spectrum presented in Fig.","5 in the main text needs merely less than half of the computational time in the case with the full 394 orbitals.","Table: CPU time spent for the dynamics calculations plus the time-dependent analysisfor 5UT with EPPC1 scheme with 1500 time steps using 20as time steps.The unit of time is second.Numeric value in [] denotes the ratio to the cost in case with use of full orbitalswhile that presented in {}\\lbrace \\rbrace correspond to the average number of cycles inmicro-iterations needed for the density convergence with ξ=10 -8 \\xi =10^{-8}.See the text for the meanings of E bw =ΔϵE_{\\rm bw}=\\Delta \\epsilon and N proj N_{\\rm proj}.The wall time means the wall clock time, and the values in parenthesesdenote the CPU efficiency in percent associated openMP for dynamics and MPI for Fock build.Supplementary material for “Electron dynamics method using a locally projected group diabatic Fock matrix for molecule and aggregate”"],["Convergence of charge migration dynamics with respect\nto the time increment using different integrator for NPTL–TCNE system.","Fig.","SREF shows the convergence of charge migration dynamics for NPTL–TCNE system with respect to time increments.","The panels correspond to those in Fig.","2 in the main article.","For a stringent check, we used different integrators, namely, PC2M-LF and EPPC1, with the time increments of 8 and 20 as in each step, respectively.","The perfect agreement with each other means the convergence of the result.","Figure: Check of the convergence of charge migration dynamics with respectto the time increment using different integrator for NPTL–TCNE system.See the main text in this supplementary material about the panels."],["\nEfficiency of orbital projection method for standard RT-TDDFT and Frozen Fock method\nwithout a group division in charge migration dynamics for NPTL–TCNE system.","We demonstrate an efficiency of the present local projection (LP) scheme for a standard RT-TDDFT.","Note that the standard RT-TDDFT is equivalent to the case with $N_g$ =1 and a ground state of whole system as an initial state within LP-GDF electron dynamics scheme.","This is also verified for the calculation of absorption spectrum for 5UT with $N_g$ =1, which is presented in Fig.","5 in the main text.","Here, for charge migration dynamics in NPTL–TCNE system, we show the efficiency of LP scheme for the standard RT-TDDFT without any group division in the Fock matrix.","We also show the results for the Frozen Fock approach.","In this context, panels (a/b) and (c/d) in Fig.","SREF with $N_g$ =1 in LP-GDF electron dynamics correspond to (c/d) and (g/h) of Fig.","2 in the main text with $N_g$ =2, respectively.","The same properties are plotted.","Time integrator employed here is EPPC1.","Strictly speaking, the initial electronic condition is different for both schemes since the former standard case starts from the initial density matrix made from the ground state of whole system while the initial density matrix was prepared by using the combination of local ground states of monomers in the latter LP-GDF case.","Despite this, the time dependent behavior is similar for both cases, which inversely supports validity of group division in Fig.","2.","Fig.","SREF is the maximum error analysis for the LP results compared from the full orbital calculations.","Panels (A) and (B) correspond to the cases of (a/b) and (c/d) in Fig.","SREF The plots of the data were carried out in the same way as Fig.","3 in the main text.","These results indicate that the LP scheme can safely reduce the computational cost with respect to orbital numbers required for the description also for charge migration dynamics under a light field with using standard RT-TDDFT equivalent to LP-GDF-electron dynamics scheme with $N_g$ =1.","Figure: Light induced charge migration dynamics with N g N_g=1 for NPTL–TCNE.Panels (a), (b), (c) and (d) in this figurerespectively correspond to (c), (d), (g) and (h) in Fig.","2 in the main text.Figure: N proj N_{\\textrm {proj}} dependency of max deviations for donormolecule during dynamics from the results by full orbital calculationin Fig.","S.RT and FF in panel (A) were evaluated from the resultsin (a) and (c) of Fig.","S.while RT and FF in panel (B) were evaluated from the results in (b) and (d)."],["Group diabatic and canonical orbital energies of NPTL–TCNE dimer","Tab.","REF summarizes the energy differences of LUMO+1, LUMO and HOMO-1 measured from HOMO, which are presented for group diabatic and canonical representations associated with monomers and dimer system, respectively.","Table: (Upper table)Energy differences of group diabatized HOMO-1-1, LUMO, and LUMO+1+1 measuredfrom group diabatized HOMO of the electron donor NPTL and acceptor TCNE moietiesin a molecular complex, which are labeled with the symbols ofΔϵ H-1,H G \\Delta \\epsilon ^{\\rm G}_{{\\rm H}-1,{\\rm H}},Δϵ L,H G \\Delta \\epsilon ^{\\rm G}_{{\\rm L},{\\rm H}} , andΔϵ L+1,H G \\Delta \\epsilon ^{\\rm G}_{{\\rm L}+1,{\\rm H}} , respectively.Note that the monomer interactions are includedin the evaluations of the local orbital energies of monomers.","(Lower table)Energy differences of canonical HOMO-1-1, LUMO and LUMO+1+1 measured from HOMOfor the whole NPTL–TCNE dimer system, which are denoted by the symbols ofΔϵ H-1,H W \\Delta \\epsilon ^{\\rm W}_{{\\rm H}-1,{\\rm H}},Δϵ L,H W \\Delta \\epsilon ^{\\rm W}_{{\\rm L},{\\rm H}} , andΔϵ L+1,H W \\Delta \\epsilon ^{\\rm W}_{{\\rm L}+1,{\\rm H}} , respectively.In both tables, the unit is atomic unit.GD and CA mean 'group diabatic' and 'canonical', respectively."],["Excitation energies, transition dipoles and oscillator strength of 5UT","Tab.SREF summarizes the information on lower excitations of 5UT at the geometry given in the later section in this supplementary material.","We showed the excitation energies, transition dipole moment vectors, and oscillator strength.","Here, $(i \\rightarrow j)$ , $\\Delta E_{ij}\\equiv |E_i-E_j|$ , $\\vec{L}_{ij}$ and $f_{ij} \\equiv \\frac{2}{3}(\\Delta E_{ij}) |\\vec{L}_{ij}|^2 $ denote a state pairs of transition, energy difference of them, transition dipole moment vector for them, and its corresponding oscillator strength, respectively.","$E_i$ means $i$ -th adiabatic state with 0 being ground electronic state.","The geometry of this molecular system is given later in this supplementary material.","According to magnitudes of oscillator strength, we considered that 0$\\rightarrow $ 1, 0$\\rightarrow $ 3 and 0$\\rightarrow $ 5 are optically allowed transitions, and used as reference excitation energies in the presentation of Fig.","4 and 5 in the main text.","Though it is not the aim to reproduce the experimental data in the present article since the main focus is the projection scheme in electron dynamics calculation, we provide information near to the situation treated here with respect to the target system.","The experimental data of the position of the first peak in absorption spectrum of optical electronic transition for this 5UT system, however, in the CHCl$_3$ solvent but not in a gas phase, is reported in the former works, [34] and its value is 2.98 eV.","Though the absorption spectrum for electronic transition in a molecular system can be changed by an existence of a polar solvent associated with a complex modification of electric transition and structure relaxation, this is not significant in this system.","On the other hand, as seen in Fig.","4 of the main article as the comparison part of RT and FF results, the value given by RT-TDDFT, 2.2 eV, being consistent with LR-TDDFT result here within the present limited ab initio level is close to the experimental value mentioned above, compared to the peak value, 1.6 eV, given by FF approximation as found in Fig.","4.","This observation also supports the basic assertion of the importance of the self-consistent treatment for the Fock matrix and time-dependent density matrix in a calculation of absorption spectrum.","The remaining discrepancy in the LR-TDDFT result here is remedied by using more appropriate higher level basis set[34] and exchange-correlation functionals [32].","In fact, at the same structure, the use of the representative hybrid functionals with a long-range correction, namely, CAM-B3LYP[36] and LC-BOP[37], provide the optically allowed first electronic transition energies in gas phase, 2.80 eV and 2.99, respectively, while their corresponding values for the polarizable continuum model(PCM)[38] with CHCl$_3$ solvent at the same structure are 2.68 and 2.88.","In the calculations using the PCM combined with LR-TDDFT here, we used the ab initio program package for electronic structure calculation, GAMESS.","[39] Table: Transition properties of 5UT obtained by LR-TDDFT calculation with PBE/6-31G(d).See the main text in this supplementary material for the details."],["Geometry data of molecular systems","We summarize the data of Cartesian coordinates in Angstrom for the molecular systems used in the present article.","The information of atoms are also included below."],["NPTL–TCNE","C 1.824214 0.710474 -1.999379 C 1.824214 -0.710473 -1.999379 C 0.628132 -1.408746 -1.999379 C -0.619806 -0.721697 -1.999379 C -0.619809 0.721696 -1.999379 C 0.628130 1.408742 -1.999379 C -1.867746 1.408746 -1.999379 C -3.063829 0.710474 -1.999379 C -3.063829 -0.710473 -1.999379 C -1.867746 -1.408742 -1.999379 H 2.775875 1.251891 -1.999379 H 2.775876 -1.251890 -1.999379 H 0.624772 -2.504937 -1.999379 H 0.624771 2.504934 -1.999379 H -1.864388 2.504937 -1.999379 H -4.015491 1.251889 -1.999379 H -4.015490 -1.251891 -1.999379 H -1.864388 -2.504934 -1.999379 N 2.832029 2.070410 2.000621 N -1.591624 2.070452 2.000621 C 1.839916 1.437817 2.000621 C -0.599565 1.437776 2.000621 C 0.620188 0.694253 2.000621 C 0.620194 -0.694285 2.000621 C 1.839957 -1.437800 2.000621 C -0.599550 -1.437827 2.000621 N 2.832015 -2.070408 2.000621 N -1.591638 -2.070398 2.000621"],["5UT","C -8.572000 1.432812 0.000000 C -9.102000 0.159612 0.000000 S -7.858600 -1.051488 0.000000 C -6.587100 0.165912 0.000000 C -7.149500 1.439512 0.000000 H -9.185200 2.337012 0.000000 H -10.148900 -0.142688 0.000000 H -6.546000 2.351012 0.000000 C -4.632800 -1.495088 0.000000 C -5.196000 -0.220788 0.000000 S -3.927900 0.998512 0.000000 C -2.656500 -0.219888 0.000000 C -3.219600 -1.495488 0.000000 H -5.236800 -2.406288 0.000000 H -2.615500 -2.406588 0.000000 C -0.706100 1.441412 0.000000 C -1.270100 0.165512 0.000000 S 0.000000 -1.054088 0.000000 C 1.270100 0.165512 0.000000 C 0.706100 1.441412 0.000000 H -1.310300 2.352412 0.000000 H 1.310300 2.352412 0.000000 C 3.219600 -1.495488 0.000000 C 2.656500 -0.219888 0.000000 S 3.927900 0.998512 0.000000 C 5.196000 -0.220788 0.000000 C 4.632800 -1.495088 0.000000 H 2.615500 -2.406588 0.000000 H 5.236800 -2.406288 0.000000 C 7.149500 1.439512 0.000000 C 6.587100 0.165912 0.000000 S 7.858600 -1.051488 0.000000 C 9.102000 0.159612 0.000000 C 8.572000 1.432812 0.000000 H 6.546000 2.351012 0.000000 H 10.148900 -0.142688 0.000000 H 9.185200 2.337012 0.000000"]],"string":"[\n [\n \"Electron dynamics method using a locally projected group diabatic Fock\\n matrix for molecules and aggregates\"\n ],\n [\n \"Abstract We propose a method using reduced size of Hilbert space to describe an electron dynamics in molecule and aggregate based on our previous theoretical scheme [ T. Yonehara and T. Nakajima, J. Chem.\",\n \"Phys.\",\n \"\\\\textbf{147}, 074110 (2017) ].\",\n \"The real-time time-dependent density functional theory is combined with newly introduced projected group diabatic Fock matrix.\",\n \"First, this projection method is applied to a test donor--acceptor dimer, namely, a naphthalene--tetracyanoethylene with and without initial local excitations and light fields.\",\n \"Secondly, we calculate an absorption spectrum of five-unit-polythiophene monomer.\",\n \"The importance of feedback of instantaneous density to Fock matrix is also clarified.\",\n \"In all cases, half of the orbitals were safely reduced without loss of accuracy in descriptions of properties.\",\n \"The present scheme provides one possible way to investigate and analyze a complex excited electron dynamics in molecular aggregates within a moderate computational cost.\"\n ],\n [\n \"Introduction\",\n \"Recently, the authors proposed a concise method for describing the quantum dynamics of excited electrons traveling over constituent molecules in a molecular aggregate system by utilizing a group diabatic Fock (GDF) matrix [1].\",\n \"The construction scheme for this GDF matrix using atom centered Gaussian basis set was practically demonstrated by Thoss et al.\",\n \"[2] based on the idea of diabatization of block diagonalization of Hamitonian that was originally introduced by Köppel et al.\",\n \"[3] Before that, the applications of this block diabatization concept for a practical calculation can be found in the works on electron conduction problems.\",\n \"[4], [5] This block diabatic scheme for the electron dynamics provides an intuitive understanding of charge and exciton migration as the quantum mechanical transport of electronic energies in a molecular assembly subject to the inherent electronic propensities of the site molecules, starting from any prepared type of initial local excitations considering the light-electron couplings.\",\n \"In the present article, on the basis of the characteristics of a nearly block structured diabatic representation in the GDF method, we introduce a compact representation within an extracted Hilbert subspace.\",\n \"The aim of this paper is two-fold: (1) to introduce a scheme for reducing the computational cost in keeping with an accurate description of the dynamics, and (2) to investigate how many local orbitals are required for a description of dynamics and absorption spectrum toward an interpretation of electron dynamics.\",\n \"Note that the group diabatic representation treated here does not lead by itself to a reduction of computational cost by describing the dynamics with a small subset of the full orbital space.\",\n \"In fact, the GDF matrix has the same dimension as the original Fock matrix before the block diagonalization.\",\n \"(1) and (2) are obtained by using a projection scheme introduced here as an extraction of local group diabatic orbitals associated with a size of the Hilbert space needed for a description of electron dynamics.\",\n \"The reduction of calculation cost is associated with the reduction of the number of extracted local group orbitals as a basis set for constructing matrix representations of electronic operators utilized in a calculation of electron dynamics.\",\n \"This article improves upon our previous work by providing and assessing a procedure for reducing the computational cost of electron dynamics calculation using the GDF method, which is combined with the real-time time-dependent density functional theory (RT-TDDFT) [6], [7], [8], [9] under the adiabatic approximation for electron functionals.\",\n \"The point is that electron migration of chemical interest over a molecular aggregate under moderate light conditions occurs in a sub Hilbert space consisting of a low number of excited states described by molecular orbitals within a relevant but not very large energy range around the Fermi energy of a system.\",\n \"It is instructive to compare other studies with our present method.\",\n \"There are many studies intensively investigating molecular interactions and their effects on excited energy transfer proceeding in excited molecular aggregates.\",\n \"[10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] For example, Futera et al.\",\n \"[23] successfully utilized and assessed the GDF method originally named a projection operator diabatization scheme[2] to evaluate electronic coupling matrix elements with high level ab initio calculations in the electron-transfer process of a molecular-metal/semiconducter interface.\",\n \"However, a study on a systematic variable description of the excited electron dynamics in a bottom-up approach is rare.\",\n \"Compared to previous studies on excited electron transfer in molecular aggregates, the scheme introduced in the present article has the advantages of a compact description of excited electron migration with ab initio electronic structure calculations.\",\n \"In addition our new scheme allows a systematic improvement of the results by enlarging the projection space.\",\n \"The most prominent feature is that our scheme is intended for a real-time dynamics of excited electrons in molecular aggregates in an external field starting from any pattern of initial local excitations prepared as a perturbation for the electronic state.\",\n \"In this article, we detail the procedures for constructing the projected local orbital space within the group diabatic (GD) representation, and then demonstrate numerical applications.\",\n \"We examine the size of a local orbital space related to dynamics in a systematic way by increasing an energy range for projecting a diabatic local orbital space.\",\n \"A naphthalene(NPTL)$-$ tetracyanoethylene(TCNE) dimer is treated as a test donor$-$ acceptor system.\",\n \"Additionally, we also demonstrate a convergence of absorption spectrum with respect to a size of orbital space using a five-unit polythiophene molecule(5UT) as a typical electron donor species in solar cell materials.\",\n \"To ensure the energy balance between local projected orbitals, we employ an energy width parameter for extracting a relevant subspace with the mean value of the highest occupied molecular orbital (HOMO) and lowest occupied molecular orbital (LUMO) energies in the whole system as a reference energy, and we do not use a scheme which requires direct orbital selection.\",\n \"In Sect.\",\n \"II, we explain the theoretical method for describing the electron dynamics based on the GDF representation within projected local diabatic orbitals, which is followed by numerical examples in the section III.\"\n ],\n [\n \"Theoretical method\",\n \"In this section, after a brief summary of the electron dynamics method [1] using a GD representation,[1], [11], [12], [13], [10], [17] we introduce a scheme for constructing a concise matrix form with use of projected diabatic local orbitals having a dominant contribution to the underlying dynamics.\",\n \"The determination of diabatic local projection orbitals playing a main role in the present work requires only the parameters of energy ranges for monomers covering the important orbitals around HOMO and LUMO playing in an excited electron dynamics.\"\n ],\n [\n \"Overview of the locally projected-space group diabatic representation\",\n \"The transformation from an atomic orbital (AO) representation of physical operators to a GD form consists of the following two transformations being constructed sequentially: (i) a transformation to a representation using the Löwdin orthogonalized atomic basis functions and (ii) a unitary transformation made from the local orbital sets obtained by the diagonalizations of block sub-matrices corresponding to the predetermined monomer groups in the Fock matrix prepared in step (i).\",\n \"The overview of the process of extracting the diabatic local projection orbitals as the primary topic of the present article is as follows: (a) calculate the mean of the HOMO and LUMO orbital energies of the whole system; (b) set an energy range covering the local orbitals for each monomer in which the value obtained in (a) is placed at the middle; (c) obtain the diabatic local projection orbitals of the monomers in the energy range prepared in (b).\",\n \"We refer to the matrix representation within these projection orbitals as the locally projected-space group diabatic Fock (LP-GDF) representation, of which the details are explained in later subsections.\",\n \"The multiplications of matrices associated with the electronic properties and the analyses of the time-dependent electron density using the newly introduced LP-GDF representation are carried out within the projected orbital space.\",\n \"The information related to the size of the orbital space required for a description of the dynamics without loss of accuracy provides us with insight into the sub-Hilbert space relevant to it.\",\n \"In the following subsection, for a self-contained form of the present article, we first summarize the electron dynamics scheme using a GD representation[1] and then proceed to describe how to obtain the diabatic local projection orbitals and how to construct compact matrix representations by using them as a subset of the basis functions.\",\n \"The first step is to prepare a Fock matrix represented by the Löwdin orthogonalized atomic basis function [24] $\\\\widetilde{F}_{mn} \\\\equiv \\\\langle \\\\widetilde{\\\\chi }_m | \\\\widehat{F} | \\\\widetilde{\\\\chi }_n \\\\rangle ,$ where the orthogonalized Löwdin atomic orbitals(AOs) are expressed by $| \\\\widetilde{\\\\chi }_n \\\\rangle = \\\\sum _j^{\\\\textrm {AO}} | \\\\chi _j \\\\rangle (\\\\underline{\\\\underline{S}}^{-1/2})_{jn}$ with $S_{jn} = \\\\langle \\\\chi _j | \\\\chi _n \\\\rangle $ being the AO overlap matrix element.\",\n \"Here $\\\\lbrace \\\\chi _n \\\\rbrace $ is the original basis set consisting of AOs.\"\n ],\n [\n \"Localized orbitals of a subgroup\",\n \"After the classification of $\\\\lbrace \\\\widetilde{\\\\chi }_n \\\\rbrace $ into subgroups, e.g., monomers, the block structure of the Fock matrix within the Löwdin basis set is determined with its diagonal blocks $\\\\lbrace \\\\underline{\\\\underline{\\\\widetilde{F}}}_{G_i G_i}\\\\rbrace $ and off-diagonal ones $\\\\lbrace \\\\underline{\\\\underline{\\\\widetilde{F}}}_{G_i G_j}\\\\rbrace _{i\\\\ne j}$ with i and j ranging from 1 to $N_g$ .\",\n \"$G_i$ denotes the i-th subgroup.\",\n \"$N_g$ is the number of subgroups in the system.\",\n \"Note that we can employ arbitrary divisions of the component atoms in the whole system.\",\n \"The diagonalization of diagonal blocks corresponding to subgroup $G$ , $\\\\underline{\\\\underline{\\\\widetilde{F}}}_{GG}=\\\\underline{\\\\underline{D}}_{G}\\\\,\\\\underline{\\\\underline{\\\\overline{F}}}_{GG}\\\\,\\\\underline{\\\\underline{D}}_{G}^{\\\\dagger },$ gives rise to the unitary transformation matrix $\\\\underline{\\\\underline{D}}_{G}$ , whose column vectors are the linear coefficient vectors of the localized eigenstates expanded in terms of the Löwdin orthogonalized atomic basis functions for the group $G$ .\",\n \"The dagger symbol attached to a matrix indicates its adjoint form.\",\n \"$ \\\\underline{\\\\underline{\\\\overline{F}}}_{GG} $ is the diagonal matrix having the eigen energies $\\\\lbrace \\\\epsilon _{j,{G}}\\\\rbrace _{j = 1 \\\\sim M_{G} }$ of the corresponding subgroup $G$ as its elements, and $M_{G}$ is the number of local basis functions spanned at group $G$ .\",\n \"Here, $ {G} \\\\in \\\\lbrace {G}_1,...,{G}_{N_g} \\\\rbrace $ .\",\n \"The elements in off-diagonal blocks of the Fock matrix represented by these localized orbitals, associated with different groups, can take non-zero values, which provide a diabatic character in the representation with the use of the collection of these orbital sets.\",\n \"These group localized orbital sets provide a transformation matrix from a Löwdin representation to the GD one to be explained later.\",\n \"Here, we provide a short comment on the ambiguity of group division in a Löwdin representation of Fock operator.\",\n \"In case using diffuse AO orbitals having far larger distribution than a distance between monomers, a center of Löwdin orbital created by the mixture of original AOs can become close to a nuclei belonging to a different group.\",\n \"Though that is one of the possible problem in the group assignment of Löwdin orbitals associated with a natural group sectoring, it does not cause any problem in case with distances of monomers sufficiently larger than AOs and also Löwdin orbitals.\",\n \"[17] In this article, we do not use diffuse orbitals and treat a case with distance of monomers being larger than a spatial range in orbital distribution.\"\n ],\n [\n \"Group diabatic Fock matrix\",\n \"The GD representation of the Fock operator $\\\\underline{\\\\underline{\\\\overline{F}}}$ , as one of the main ingredients in the GDF electron dynamics scheme is constructed via the transformation of the Löwdin representation matrix of the Fock operator $ \\\\underline{\\\\underline{\\\\widetilde{F}}} $ .\",\n \"By using the already obtained unitary matrices with the dimensions of the local basis functions associated with groups, $\\\\lbrace \\\\underline{\\\\underline{D}}_{G_i}\\\\rbrace _{i=1 \\\\sim N_g}$ , this transformation is expressed by [1], [11], [12], [10] $\\\\underline{\\\\underline{\\\\overline{F}}}_{G_i\\\\,G_j}=\\\\underline{\\\\underline{D}}_{G_i}^{\\\\dagger }\\\\,\\\\underline{\\\\underline{\\\\widetilde{F}}}_{G_i\\\\,G_j}\\\\,\\\\underline{\\\\underline{D}}_{G_j},$ where i and j range from 1 to the number of groups $N_g$ .\",\n \"The assembly of these sub-matrices $ \\\\left\\\\lbrace \\\\underline{\\\\underline{\\\\overline{F}}}_{G_i\\\\,G_j} \\\\right\\\\rbrace _{i,j=1\\\\sim N_g} $ constructs the GD representation which is called GDF matrix and expressed by $ \\\\underline{\\\\underline{F}}^{\\\\textrm {GD}} $ .\",\n \"The physical meaning of the components in this final form is as follows.\",\n \"This sub-matrices $ \\\\left\\\\lbrace \\\\underline{\\\\underline{\\\\overline{F}}}_{G_i G_i} \\\\right\\\\rbrace _{i=1 \\\\sim N_g}$ in the diagonal blocks correspond to the local group eigen energies, while $ \\\\left\\\\lbrace \\\\underline{\\\\underline{\\\\overline{F}}}_{G_i G_j} \\\\right\\\\rbrace _{i \\\\ne j}$ , placed at the off-diagonal blocks, describes the interactions between different groups.\",\n \"Note that, in this transformation, the information included in the AO, Löwdin, and GDF representations are the same and no approximation is applied.\"\n ],\n [\n \"Transformation from the AO representation to the GD representation\",\n \"A matrix representation of any observable operator $\\\\hat{O}$ in terms of the constructed GD basis set, $\\\\underline{\\\\underline{O}}^{\\\\mathrm {GD}}$ , is related to that of the original AO basis set, $\\\\underline{\\\\underline{O}}^{\\\\textrm {AO}}$ , as $\\\\underline{\\\\underline{O}}^{\\\\mathrm {GD}}=\\\\underline{\\\\underline{U}}^{\\\\dagger } \\\\,\\\\underline{\\\\underline{O}}^{\\\\textrm {AO}} \\\\,\\\\underline{\\\\underline{U}},$ where $\\\\underline{\\\\underline{U}} \\\\equiv \\\\underline{\\\\underline{S}}^{-1/2} \\\\underline{\\\\underline{W}}$ .\",\n \"Here, the diagonal block in the sub-transformation matrix $ \\\\underline{\\\\underline{W}}$ is given by $\\\\underline{\\\\underline{W}}_{G_i G_i} \\\\equiv \\\\underline{\\\\underline{D}}_{G_i}$ for $i$ , while the off-diagonal one is defined as $\\\\underline{\\\\underline{W}}_{G_i G_j} \\\\equiv \\\\underline{\\\\underline{0}}$ .\",\n \"The Fock matrix obeys the same transformation rule and is obtained by setting $\\\\hat{O}=\\\\hat{F}$ in the above equations, where we know that $ \\\\underline{\\\\underline{F}}^{\\\\textrm {AO}} = \\\\underline{\\\\underline{F}} $ and $ \\\\underline{\\\\underline{F}}^{\\\\mathrm {GD}} = \\\\underline{\\\\underline{\\\\overline{F}}} $ .\",\n \"On the other hand, the transformation of the density matrix from the original AO basis set to the GD one is written as $\\\\underline{\\\\underline{\\\\rho }}^\\\\mathrm {GD}=\\\\underline{\\\\underline{U}} \\\\,\\\\underline{\\\\underline{\\\\rho }}^\\\\mathrm {AO} \\\\,\\\\underline{\\\\underline{U}}^{\\\\dagger }.$ Note that the unitarity of $\\\\underline{\\\\underline{W}}$ assures total electron conservation with respect to this transformation, $\\\\textrm {Tr}\\\\left[ \\\\underline{\\\\underline{\\\\rho }}^{\\\\mathrm {AO}} \\\\underline{\\\\underline{S}} \\\\right]= \\\\mathrm {Tr} \\\\left[ \\\\underline{\\\\underline{\\\\rho }}^\\\\mathrm {GD} \\\\right].$\"\n ],\n [\n \"State coupling\",\n \"We can obtain the essential elements needed for the construction of light–electron coupling by setting $ \\\\hat{O} = \\\\hat{\\\\bf r} $ , $ \\\\partial _{\\\\bf r} $ in the previous subsection.\",\n \"Here, boldface denotes a vector in a three-dimensional Cartesian space, and ${\\\\bf r}$ denotes a composite variable of the electron position in three-dimensional space.\",\n \"The first and second operators are responsible for the light–electron coupling in length and velocity forms.\",\n \"[25], [26] Here, we neglect the nonadiabatic coupling and molecular motion to allow us to focus on an examination of the electron dynamics scheme using projected local diabatic group orbitals.\",\n \"Light-electron couplings within GD representation are expressed by $\\\\underline{\\\\underline{L}}^\\\\mathrm {GD}=+ e \\\\underline{\\\\underline{{\\\\bf r}}}^\\\\mathrm {GD} {\\\\bf E}$ for the length gauge and $\\\\underline{\\\\underline{L}}^\\\\mathrm {GD}=- i \\\\hbar \\\\dfrac{e}{c} {\\\\bf A}\\\\underline{\\\\underline{\\\\partial _{\\\\bf r}}}^\\\\mathrm {GD}$ for the velocity gauge.\",\n \"[25], [26] ${\\\\bf E}$ and ${\\\\bf A}$ denote respectively the three-dimensional electric field vector and electromagnetic field vector potential, which generally depend on a point in a three-dimensional space.\",\n \"$\\\\underline{\\\\underline{{\\\\bf r}}}^{\\\\textrm {G}D}$ and $\\\\underline{\\\\underline{\\\\partial _{\\\\bf r}}}^{\\\\textrm {G}D}$ denote the GD presentation matrices respectively for $\\\\hat{\\\\bf O}={\\\\bf r}$ and $\\\\partial _{\\\\bf r}$ in Eq.\",\n \"(REF ).\",\n \"In this study, we employ the length gauge.\",\n \"The details on the treatment for these coupling matrices with further approximations applied in dynamics calculation are given in Subsect.\",\n \"REF .\"\n ],\n [\n \"Locally projected space made from subsets of local site orbitals\\n\",\n \"This subsection is an essential part of the present article proposing the LP-GDF scheme.\",\n \"Here, we provide the concrete procedures used in it and the corresponding mathematical expressions.\",\n \"Let us consider the diagonal block of the Fock matrix within the GD representation, $\\\\left\\\\lbrace \\\\underline{\\\\underline{\\\\widetilde{F}}}_{G_i G_i}\\\\right\\\\rbrace $ and the associated local orbital energies $\\\\lbrace \\\\epsilon _{j,G_i} \\\\rbrace _{j=1\\\\sim N_{G_i}}$ corresponding to the i–th group site.\",\n \"In a situation where the interactions between group monomers are weak compared to those among the atoms in each monomer, we can safely employ the referential local ground state, where all of the electrons assigned to a local site are filled in ascending order from the lowest-energy local orbital.\",\n \"The excited electron dynamics in a molecular aggregate system under moderate sunlight conditions is expected to proceed in low-excited-state manifolds constructed from the local molecular orbitals in a relevant but not very large energy range around the Fermi energy of the system.\",\n \"In a situation involving rather strong light field or molecular interaction, the Hilbert subspace required for a description of the dynamics will become large.\",\n \"It is worthwhile to examine the size of the subspace relevant to the excited electron dynamics by varying the type of molecular interactions such as light-matter coupling and initial local excitations.\",\n \"Thus, in order to examine the size of the orbital space needed for a description of the electron dynamics at a sufficient accuracy, we establish a procedure for projecting a Hilbert subspace around the Fermi energy of the whole molecular aggregate by using a width parameter for the energy range as follows: (i) Set an energy width $\\\\Delta \\\\epsilon _i$ covering the sub-space consisting of local orbitals orbitals for the $i$ -th monomer, $G_{i}$ .\",\n \"Here $i$ ranges from 1 to $N_g$ .\",\n \"(ii) Calculate the mean $\\\\eta $ of the HOMO and LUMO energies of the whole system.\",\n \"(iii) Extract a subset of group diabatic local orbitals in the energy range $ [ \\\\eta - \\\\Delta \\\\epsilon _i/2, \\\\eta + \\\\Delta \\\\epsilon _i/2 ] $ from the whole orbital space.\",\n \"We refer to this as a projection of the diabatic local orbitals.\",\n \"(iv) Construct minor matrices as representations of the electronic operators within the projected orbital basis set obtained in (iii).\",\n \"(v) Follow the same procedure as that in the GDF electron dynamics scheme other than the projection introduced here.\",\n \"This means that time propagation and property analysis are carried out using the obtained small size matrices.\",\n \"The mathematical form of LP-GDF procedure is compactly given in Appendix .\",\n \"In this article, we employ a version of projection independent on the orbitals during a time propagation but constructed from the initial orbitals in dynamics.\",\n \"The orbital projection scheme using an energy width allows for the systematic and natural treatment for general situations involving the unknown energy orders and spatial localities of the GD orbitals in molecular aggregates.\",\n \"Note that to ensure the energy balance between local projection orbitals, we employ an energy width parameter for extracting a relevant subspace with the mean value of the highest occupied molecular orbital (HOMO) and lowest occupied molecular orbital (LUMO) energies in the whole system as a reference energy, and we do not use a scheme which requires direct orbital selection.\",\n \"In the following numerical demonstrations, for focusing on this essential point in the proposed scheme, we employ a parameter for the energy range independent of the monomers, namely, $\\\\Delta \\\\epsilon _i$ = $\\\\Delta \\\\epsilon $ , which is expressed by $E_{\\\\rm bw}$ in the figures.\",\n \"The resultant total number of projection orbitals of the monomers is denoted by $N_{\\\\rm proj}$ throughout the present article.\"\n ],\n [\n \"Initial density matrix: local excitation and electron filling\",\n \"The initial density matrices in the GDF representation are prepared so that the diagonal elements in each diagonal block of the corresponding monomer should be occupied up to the number of electrons assigned to this monomer.\",\n \"A simple example can be found in our previous article.\",\n \"[1] Though an off-diagonal filling responsible for the initial coherence is also possible, for simplicity, we consider only the diagonal part in the preparation of the initial state of the density matrix.\",\n \"This issue will be reported in our future article.\",\n \"In the article, we treat the case of the spin-restricted model within the GDF representation as explained below.\",\n \"This is not rigorously the same as that within the canonical (KS or HF) orbital representation.\",\n \"In a restricted case with the same spatial orbitals for different alpha and beta spins in this model, the occupancy of the GDF local orbitals of each monomer is up to half of the number of electrons assigned to this group from the lowest energy orbital.\",\n \"Therefore, in a strict sense, the initial density matrix mentioned above differs from that of the true ground state of the whole system.\",\n \"In fact, this does not cause any problem for examining the migration dynamics of charge and electronic excitations over the constituent monomers in an assembly.\",\n \"[1] Generally, we can make any type of excitation configuration starting from the reference occupations of the GD orbitals.\",\n \"If we want an initial density associated with an excess or a deficiency of electrons in each monomer for treating the case accompanied with a charge moiety, we merely need to set the occupations to the corresponding number of electrons in each monomer.\",\n \"[1] Throughout this article, we treat cases with an overall singlet spin state in a spin-restricted manner.\",\n \"Appendix provides a practical example for preparing an initial density matrix within GD representation.\"\n ],\n [\n \"Time propagation of the density matrix in a GD representation\",\n \"In this study, the calculation of the time propagation of an electronic state is performed in terms of the Liouville–von Neumann equation associated with one particle density matrix as follows: $\\\\dfrac{\\\\partial }{\\\\partial t}\\\\underline{\\\\underline{\\\\rho }}^\\\\mathrm {GD}=-\\\\dfrac{i}{\\\\hbar }\\\\left[\\\\underline{\\\\underline{\\\\mathcal {F}}}^\\\\mathrm {GD}\\\\left\\\\lbrace \\\\underline{\\\\underline{\\\\rho }}^\\\\mathrm {GD}(t),t \\\\right\\\\rbrace ,\\\\underline{\\\\underline{\\\\rho }}^\\\\mathrm {GD}\\\\right],$ where $\\\\underline{\\\\underline{\\\\mathcal {F}}}^\\\\mathrm {GD}\\\\equiv \\\\underline{\\\\underline{F}}^\\\\mathrm {GD}+\\\\underline{\\\\underline{L}}^\\\\mathrm {GD}.$ $\\\\underline{\\\\underline{L}}^\\\\mathrm {GD}$ is a light–electron coupling matrix.\",\n \"For the length gauge, their corresponding matrix elements have forms of $\\\\underline{\\\\underline{L}}^\\\\mathrm {GD}=+ e \\\\underline{\\\\underline{{\\\\bf r}}}^\\\\mathrm {GD} {\\\\bf E}$ , [25], [26] where ${\\\\bf E}$ is the three-dimensional electric field vector, which generally depend on a point in a three-dimensional space.\",\n \"We used the dipole approximation, namely, long wavelength approximation, [25], [26] since the wavelength of light treated here is sufficiently large compared to the size of the molecular system treated.\",\n \"In the RT-TDDFT, the Fock matrix depends on the time-dependent density matrix.\",\n \"For convenience of a discussion on the numerical demonstration to be presented in the later section, we label the dynamics described by Eq.\",\n \"(REF ) including this dependency as 'RT' meaning 'real time Fock matrix' while the term 'FF' denoting 'frozen Fock matrix' is used to refer to the approximated dynamics with the replacement of the time dependent Fock by that at the initial simulation time.\",\n \"For a technical simplicity and focusing on the projection scheme of electron dynamics, we employ the pure density functional for including an electronic exchange-correlation throughout this article.\",\n \"For obtaining the time dependent electron density matrix, we solved the non-linear Liouville–von Neumann equation associated with the RT-TDDFT and introduced LP-GDF matrix by using the following two types of numerically stable time integrators, namely (1) the predictor-corrector second order Magnus scheme with linear Fock extrapolation (PC2M-LF) [27] and (2) exponential propagation with predictor-corrector SCF scheme using final corrector as a resultant density (EPPC1).\",\n \"[28] PC2M-LF needs one time of update of Fock matrix for each step while an iteration scheme is applied for EPPC1 until a corrector density is converged.\",\n \"Their details are summarized in Appendix .\",\n \"We followed a dynamics with time steps of 8 and 20 attosecond using a PC2M-LF and EPPC1 time integrator, respectively.\",\n \"Hereinafter we abbreviate femtosecond to fs and attosecond to as.\",\n \"In many cases with a moderate dynamics of density the use of former scheme is sufficient, while the latter was needed for a stable calculation of absorption spectrum using short laser pulses.\",\n \"The Fock, electron dipole transition matrices within the AO representation required for the dynamics calculation were evaluated by using the NTChem2013 software package.\",\n \"[29] We should comment on conservations of properties during dynamics.\",\n \"As a special case, we examined a case starting from a locally excited state without external light field.\",\n \"While within the LP-GD scheme a trace of density matrix is perfectly conserved both for RT-TDDFT and Frozen Fock approximation, an expectation value of Fock operator is conserved only for the latter one.\",\n \"Despite this, in a practical sense, this LP-GD RT-TDDFT scheme provides a good result with respect to a convergence to reference data as $\\\\Delta _\\\\epsilon $ increases, which will be shown in Sect.\",\n \"with demonstrations of a userbility of the method through calculations of charge migration dynamics and absorption spectrum.\"\n ],\n [\n \"Numerical demonstration\",\n \"Here, the size effect of the projected space on the electron dynamics and absorption spectrum are examined by using the LP-GDF electron dynamics method introduced in the previous section.\",\n \"We compare the time-dependent behaviors of the Mulliken charge of electron donor molecules for NPTL–TCNE with different initial excitation and continuum light field by varying $E_{\\\\rm bw}$ .\",\n \"The convergence of absorption spectrum for 5UT is also examined in the same manner after showing explicitly the importance of the self-consistency between time-dependent density and the time-dependent Fock matrix during electron dynamics.\",\n \"With respect to the NPTL–TCNE, readers can find further information in our previous article.\",\n \"[1] We validate the efficiency of the present projection method by showing a convergence of result with respect to a reduced size of local orbital space.\",\n \"The schematics of the molecular configurations used in this article as shown in Fig.\",\n \"REF .\",\n \"We also provide information related to the dependence of the computational cost in the electron dynamics calculations with and without Fock build on the size of the projection space, which are detailed in Appendix .\",\n \"Figure: Schematics of the geometries of the molecular aggregates treated in this article:(a) NPTL–TCNE dimer (b) 5UTThe centroids of aggregates were set to origin for all the systems.See the supplementary material for detailed information of their Cartesian coordinates.\"\n ],\n [\n \"NPTL–TCNE dimer\",\n \"As a first test system, we treat a dimer system consisting of NPTL and TCNE.\",\n \"In this combination of monomers, NPTL serves as an electron donor molecule, while TCNE plays role as an electron acceptor.\",\n \"The molecular geometries and relative orientations of the monomers used here are the same as those in Panel (l) in Fig.\",\n \"3 in our previous article [1] on the original GDF electron dynamics scheme where readers can find further information including the literature of experimental data.\",\n \"The geometry of each monomer was optimized at the DFT/6-31G(d) level with the use of the PBE exchange correlation functional.\",\n \"[30] The Fock matrix associated with initial optimized KS orbitals and its following time-dependent density matrix for the construction of the GDF matrix was also calculated at the same ab initio level.\",\n \"Both molecules have planar geometries in the optimized geometry in their ground electronic states.\",\n \"Here, as shown in Fig.\",\n \"REF , the principal axis of NPTL was set to be parallel to the X axis, while we set TCNE to be parallel to the Y axis.\",\n \"The molecular planes of these flat molecules are parallel to the X–Y plane.\",\n \"They were placed in a parallel orientation with a slide of 1.24 $\\\\textrm {Å}$ along the Y axis.\",\n \"The distance between molecular planes was fixed at 4 $\\\\textrm {Å}$ .\",\n \"Although a dimer is treated here, we considered the crystal data[31] reported in the literature with respect to the relative orientation.\",\n \"This selection of molecular configuration yields a non-vanishing overlap between the frontier orbitals, i.e., the HOMO of NPTL and the LUMO of TCNE.\",\n \"[1] See the supplementary material for a further information on the geometrical coordinate of this system as well as GD and canonical orbital energies.\",\n \"Figure: Convergence of the results for the charge separation dynamics in an NPTL–TCNE dimerwith an increase in the energy range determining the projected local orbital space.The energy width used for orbital projection is expressed byE bw E_{\\\\rm bw} = Δϵ\\\\Delta \\\\epsilon in Hartree.The numbers of projected orbitals, N proj N_{\\\\rm proj}, are 13, 46, 74 and 120respectively for E bw _{bw} = 0.2, 0.5, 1.0 and 1.6.The exact results obtained by using the full 296 orbitals are indicated by solid red line.See the main text for the details of situations with respect to the treatment of Fock matrix,initial local excitation and light field employed in the panels.Fig.\",\n \"REF provides the results of the excited electron dynamics involving initial local excitations and external light fields.\",\n \"The time dependent behaviors of the Mulliken charge of the donor molecule, NPTL, are displayed with the variation of the energy range covering the projection orbital space $\\\\Delta \\\\epsilon $ .\",\n \"In the panels of the figure, $\\\\Delta \\\\epsilon $ is expressed as $E_{\\\\textrm {bw}}$ .\",\n \"In panels, we plotted the Mulliken charge of NPTL as an electron donor system for each case corresponding to the vertical axis.\",\n \"The horizontal axis denotes the time in femtoseconds.\",\n \"The results of RT-TDDFT LP-GDF scheme are displayed in panels of (a–d) while corresponding results of frozen Fock approximation are presented in (e–h).\",\n \"(a/e) and (b/f) show the results starting from the initial local excitation respectively in the donor NPTL and acceptor TCNE, while the other moieties are initially in the ground states within the GDF representation.\",\n \"In these four panels, no light field is irradiated to the system.\",\n \"In panels (c/g) and (d/h), a continuum light field is applied, and its field parameters as a wave length and unit vector of polarization are 700 nm corresponding to $\\\\omega =0.065$ au and $\\\\left( \\\\frac{1}{\\\\sqrt{3}}, \\\\frac{1}{\\\\sqrt{3}}, \\\\frac{1}{\\\\sqrt{3}} \\\\right)$ in the XYZ Cartesian coordinate, respectively.\",\n \"The field strength $Es$ used in the cases of panels (c/g) is 0.015 a.u., while a larger strength of 0.02 is applied in (d/h).\",\n \"The function form of light field is given in Appendix .\",\n \"The continuum light field was replaced by a pulse with a sufficiently large time width $t_w=2.4\\\\times 10^{4}$ fs and peak time $t_c=0$ .\",\n \"As seen in the RT cases (a) and (b) accompanied with initial local excitations without light fields show the fast convergence with respect to the increase of $E_{\\\\textrm {bw}}$ .\",\n \"In fact, $E_{\\\\rm bw}=1.0$ associated with 74 projected local orbitals provides a quantitative convergence to the result obtained by the full 296 orbitals.\",\n \"On the contrary, in (c) and (d) from initial local ground states with continuum light field having moderate strength, the quantitative convergence are achieved by $E_{\\\\textrm {bw}}$ =1.6 providing 120 projected orbitals.\",\n \"As found in the panels (c/g) and (d/h), the FF approximation provides faster convergence compared to RT results, corresponding to the cases involved with light fields.\",\n \"On the other hand, the differences of the converged results of FF from those of RT means the importance of a consideration of feedback from time dependent density matrix to the Fock matrix.\",\n \"In (c/d/g/h), we employed the light stronger than that of usual sun light for a severe assessment of the present scheme in cases accompanied with hard excitations.\",\n \"In fact, in this strength, the non-linear effect about the density matrix result in the difference in convergence rates between frozen Fock type and RT-TDDFT calculations with respect to an increase of number of projected orbitals.\",\n \"Figure: N proj N_{\\\\rm proj} dependency of max deviations for donnor moleculeduring dynamics from the results by full orbital calculation in Fig.\",\n \".Deviation is expressed as an absolute value but not relative one.The panels of (a), (b), (c) and (d) in this figure correspond to those of(a/e), (b/f), (c/g) and (d/h) in Fig.\",\n \", respectively.“RT” and “FF” in panels correspond to the results obtained by electron dynamics calculationusing RT-TDDFT scheme considering the self-consistency and frozen Fock approximation.To clarify the convergence of time dependent behaviors of charge migration dynamics in Fig.\",\n \"REF with respect to the number of projected orbitals, we displayed the max deviations of the approximation results measured from the reference data obtained by full orbitals in Fig.\",\n \"REF as a function of $N_{\\\\rm proj}$ .\",\n \"As found in the figure, in all cases we can reduce a half the number of the orbitals keeping with the accuracy matched to the result obtained by using full orbitals.\",\n \"The associated CPU times spent for the calculations and the ratios of them to those obtained by the full orbitals are summarized in Appendix .\",\n \"In the last part of this subsection, as a supplementary information, we provide an evidence of the efficiency of the orbital extraction scheme for the standard RT-TDDFT and Frozen Fock approximation by guiding to the corresponding results.\",\n \"The electron dynamics calculation using the orbital extraction (projection) in standard KS orbital sequence is equivalent to the GDF electron dynamics calculation with $N_g=1$ , namely no group division.\",\n \"This is because a diagonalization of Löwdin AO representation in case of $N_g=1$ gives rise to exactly the same eigen states for the original Kohn-Sham equation at an initial simulation time.\",\n \"In the other words, in the case of $N_g=1$ , total eigen states of Fock operator do not change through a transformation from a representation with original AO to that with Löwdin orthonormalized AOs.\",\n \"We calculated charge migration of this system by treating the whole system as a monomer, namely, by setting $N_g=1$ .\",\n \"The energy threshold value for orbital extraction and number of selected orbitals were varied within LP scheme.\",\n \"The data were compared with those obtained by the methods without any orbital projection nor any group division, namely full orbital type of standard RT-TDDFT or Frozen Fock approach.\",\n \"Because the setting of $N_g=1$ does not allow local excitation of each monomers at initial simulation time in the same sense as $N_g \\\\ge 2$ , we calculated only the light field cases starting from initial ground state of whole system, namely, no excitation of that.\",\n \"The results and detailed discussion are included in the Supplementary Information.\",\n \"We should also comment that the Frozen Fock approach without GD representation can provide a better convergence as seen in the comparison between the panel (g) in Fig.\",\n \"2 and (c) in Fig.\",\n \"S2 with respect to the $E_{\\\\rm bw}=0.2$ and full orbital cases.\",\n \"In the same sense, RT-TDDFT without GD approach can be superior in convergence as found in the comparison between the panel (d) in Fig.\",\n \"2 and (b) in Fig.\",\n \"S2 for $E_{\\\\rm bw}=1.0$ and full orbital cases.\"\n ],\n [\n \"5UT\",\n \"In the second demonstration of the present work, we here apply the method to the calculation of absorption spectrum of 5UT.\",\n \"This molecule plays an important role not only as an electron donor but also a hole transfer material in an organic solar cell.\",\n \"[32] The existence of sulfur atoms in $\\\\pi $ conjugate system gives rise to the stability of molecular structure as well as the increase of charge conduction through the electron overlap contributed by their d-orbitals.\",\n \"[33] Here we do not discuss a chemical aspect of this system, and just focus on the performance of the method by looking the dependency of results on the size of projected local orbital space.\",\n \"The molecular geometry of 5UT treated here was optimized in the PBE/6-31G(d) level calculation, which is commonly used in the electron dynamics calculation.\",\n \"The geometry information is summarized in the supplementary materials.\",\n \"The light pulse was applied and the time-dependent information of induced electronic dipole moments was converted to absorption spectrum, of which details on functional forms are given in Appendix .\",\n \"We used EPPC1 as the time integrator with the time step of 20as.\",\n \"The total time of simulation carried out was 30 fs.\",\n \"The functional form of light field is given in Appendix .\",\n \"The field strength, $E_s$ , central frequency, $\\\\omega $ , pulse peak time, $t_c$ , pulse width time, $t_w$ , and angular frequency, $\\\\omega $ , of the applied light pulse were 0.001au, 100(=2.42 fs), 40(=0.97 fs) and 0.0285, respectively.\",\n \"The polarization vector of light field was set to be parallel to the vector of (1,1,1) in the Cartesian coordinate.\",\n \"We set the group number $N_g$ =1.\",\n \"The initial density used in the electron dynamics was set to that of the DFT ground state.\",\n \"Figure: The importance of feedback of time-dependent information to the Fock matix.We presented absorption spectrums of 5UT with the variation of the damping factor,γ\\\\gamma , and time cut from the initial in the absorption calculation, T CUT T_{\\\\rm CUT}.See also the supplementary materials for the mathematical form used for the absorption spectrum.Left panels (a–c) and right ones (d–f) correspond to the results of RT and FF, respectively.Note that in all cases here, the full 394 orbitals are used in the dynamics calculationsfor preparing the light pulse induced electronic dipole moments for the spectrum calculations.Triangles denotes the positions of optically allowed excitationsassociated with the absorption energy obtained by linear response TDDFT calculations.The absorption spectrum evaluated from the formula given in Appendix numerically depends on the damping parameter, $\\\\gamma $ , and the initial cut off time, $T_{\\\\rm CUT}$ .\",\n \"We checked that the simulation time was long enough for the examined energy range from 0 to 4 eV.\",\n \"In order to select a set of reliable parameters for absorption spectrum, we compared the obtained spectrum by varying $\\\\gamma $ and $T_{\\\\rm CUT}$ , which is presented in Fig.\",\n \"REF including the information of positions of optically-allowed excitations in energy space obtained by LR-TDDFT.\",\n \"The transition properties evaluated by LR-TDDFT including excitation energies, oscillator strength, and dipole moments are summarized in the supplementary material.\",\n \"We also compared the results with and without the feedback of time-dependent density to the Fock matrix in order to show that the instantaneous back-reaction of the density to effective Hamiltonian in the simulation is the key factor for the reproduction of the LR-TDDFT results, which are correspondingly labeled with RT and FF defined in the section of theoretical method, Subsection REF .\",\n \"With respect to the parameters, we found the best parameters $\\\\gamma =0.004$ both for RT and FF cases in the aspects of the balance of proper smoothness and spectrum peak widths.\",\n \"In turn, the best parameters of $T_{\\\\rm CUT}$ is 5.3 fs and 6.2 for RT and FF based on the observation of the appearance of maximum peaks.\",\n \"Again, we can see that RT calculations presented in the panels (a–c) correctly reproduce the first peak position given by LR-TDDFT while FF ones fail.\",\n \"Then, we chose the RT type calculation for spectrum calculation with $\\\\gamma =0.004$ and $T_{\\\\rm CUT}$ =5.3 fs and proceed to check the dependency on the size of projected local orbital space.\",\n \"Figure: Projected size dependency and convergence of absorption spectrum.we compared the results by projected local orbitals, N proj N_{\\\\rm proj}=64, 89, and 164correspondingly associated with E bw E_{\\\\rm bw}=0.5, 0.75, and 1.5to that obtained by using N orbs N_{\\\\rm orbs}=394 full orbitals for 5UT molecule.Here we employed the damping factor of γ=0.004\\\\gamma =0.004 and T CUT T_{\\\\rm CUT}=5.3 fsbased on the observation in Fig.\",\n \".Triangles denotes the positions of optically allowed excitations associated with theabsorption energy obtained by linear response TDDFT calculations,which are labeled with the symbol 'LR.OA'.Fig.\",\n \"REF presents the convergence of the absorption spectrum using RT-TDDFT calculation combined with LP-GDF scheme with respect to the size of projected local orbital space.\",\n \"We can observe the monotonic convergence of the results by projection scheme to that obtained by the 394 full orbitals.\",\n \"In fact, the spectrum shapes by $E_{\\\\rm bw}$ =0.5, 0.75 and 1.5 corresponding to $N_{\\\\rm proj}$ =64, 89 and 164 projected orbitals monotonically approaches to that of the full orbital calculation.\",\n \"More than half of the total orbitals were reduced in this spectrum calculation.\",\n \"The information of CPU time is also included in the supplementary material.\",\n \"The success in the reduction of diabatic local orbital space required for the sufficient description suggests the efficiency of the introduced projection method for RT-TDDFT.\"\n ],\n [\n \"Concluding remarks\",\n \"In this study, we introduced and assessed the electron dynamics method as a combination of the local orbital projection and GDF electron dynamics scheme within a framework of RT-TDDFT.\",\n \"Through the examination of complex charge migration induced by local excitation and light field for the NPTL–TCNE dimer as a test donor-acceptor system, we showed that the present method allows us to investigate the size of the Hilbert subspace associated with the excited electron dynamics of molecular aggregates.\",\n \"In the application of the projection method to absorption spectrum, we numerically demonstrated the reproduction of the result with use of the reduced number of orbitals.\",\n \"We can expect that this method paves a way to a practical investigation of electron dynamics of molecular aggregates with a further combination of the techniques for reducing the cost in Fock build.\",\n \"We conclude by providing a perspective.\",\n \"The molecular motion, non-adiabatic mixing, and transition between electronic states can affect charge and exciton migration dynamics.\",\n \"There, a nuclear quantum effect may play an important role in electron dynamics in molecular aggregate systems having high density of states.\",\n \"The GD representation scheme can afford a realistic and practical model Hamiltonian for treating these issues.\",\n \"In our future work, we will report on a coarse grained quantum dynamics in molecular aggregates having a nano-size for describing a coupled dynamics of charge and exciton migration associated with their birth, extinction and transport.\"\n ],\n [\n \"Supplementary material\",\n \"See supplementary material for the issues of (I) validity check of the time interval employed for describing the charge migration dynamics and (II) Efficiency of orbital projection method for standard RT-TDDFT and Frozen Fock method without a group division in charge migration dynamics for NPTL–TCNE system.\",\n \"(III) group diabatic and canonical orbital energies of NPTL–TCNE dimer (IV) optical transition properties of 5UT obtained by LR-TDDFT calculation (V) geometry data of molecular aggregates of a NPTL–TCNE dimer and 5UT.\",\n \"This research was supported by MEXT, Japan, “Next-Generation Supercomputer Project” (the K computer project) and “Priority Issue on Post-K Computer” (Development of new fundamental technologies for high-efficiency energy creation, conversion/storage and use).\",\n \"Some of the computations in the present study were performed using the Research Center for Computational Science, Okazaki, Japan, and also HOKUSAI system in RIKEN, Wako, Japan.\"\n ],\n [\n \"Initial density matrix within GD representation associated with local excitations\",\n \"Essential points in preparing initial density matrix are summarized in a following example.\",\n \"Consider the spin-restricted case with three monomers each of which has 2 electrons and 3 orbitals.\",\n \"Here set $N_g=3$ .\",\n \"See the main text about the approximate treatment associated with this term of 'spin-restricted'.\",\n \"If we want to prepare the initial condition such that the first and third monomers are initially excited from local HOMO to LUMO, the initial density matrix in GDF scheme is constructed as follows: [1] set reference density matrix, namely GDF ground state, $\\\\underline{\\\\underline{\\\\rho }}^{\\\\textrm {GD:ground}}$ and [2] carry out local HOMO LUMO excitation and obtain the aimed GDF density, $\\\\underline{\\\\underline{\\\\rho }}^{\\\\textrm {GD:1,3-HL}}$ namely, $& \\\\underline{\\\\underline{\\\\rho }}^{\\\\textrm {GD:ground}} \\\\equiv \\\\left(\\\\begin{array}{ccccccccc}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\\\0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\\\\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\\\0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\\\\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\end{array}\\\\right) \\\\\\\\\\\\quad & \\\\Longrightarrow \\\\\\\\\\\\quad & \\\\underline{\\\\underline{\\\\rho }}^{\\\\textrm {GD:1,3-HL}} \\\\equiv \\\\left(\\\\begin{array}{ccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\\\0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\\\\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\end{array}\\\\right).$\"\n ],\n [\n \"Time integrator\",\n \"We detail the time integrators used for solving non-linear Liouville von Neumann equation including time-dependent Hamiltonian depending on the density matrix employed in this study, namely, (1) the predictor-corrector second order Magnus scheme with linear Fock extrapolation (PC2M-LF) and (2) exponential propagation with predictor-corrector SCF scheme using final corrector as a resultant density (EPPC1).\",\n \"Note again that PC2M-LF needs one time of update of Fock matrix for each step while EPPC1 includes an iteration scheme with respect to a convergence of corrector density.\",\n \"See the main text for the original articles of these schemes.\"\n ],\n [\n \"PC2M-LF\",\n \"This scheme consists of the following five processes for each time step, $[1] &\\\\, F_{3} := - \\\\frac{3}{4}F_{1a} + \\\\frac{7}{4} F_{1b} \\\\\\\\[2] &\\\\, D_{4} :=e^{- \\\\frac{i}{\\\\hbar } \\\\frac{dt}{2} F_{3} } D_{2} e^{ \\\\frac{i}{\\\\hbar } \\\\frac{dt}{2} F_{3} }\\\\\\\\[3] &\\\\, F_{5} := F[D_{4}] \\\\\\\\[4] &\\\\, D_{6} :=e^{- \\\\frac{i}{\\\\hbar } dt F_{5} } D_{2} e^{ \\\\frac{i}{\\\\hbar } dt F_{5} } \\\\\\\\[5] &\\\\,( F_{1a}, F_{1b} ) := ( F_{1b}, F_{5} ) \\\\,\\\\, \\\\Rightarrow \\\\,\\\\, \\\\textrm {End of this step}$ If we consider a time propagation from $t$ to $t+dt$ , $F_3$ and $F_5$ respectively denote the Fock matrices at $t+\\\\frac{1}{4}dt$ and $t+\\\\frac{1}{2}dt$ while $D_2$ , $D_4$ and $D_6$ correspond to the density matrices at $t$ , $t+\\\\frac{1}{2}dt$ and $t+dt$ .\",\n \"$F_3$ in the first step [1] is constructed from the Fock matrices, $F_{1a}$ and $F_{1b}$ , respectively denote the Fock matrices in the previous two times respectively at $t-\\\\frac{1}{2}dt$ and $t-\\\\frac{3}{2}dt$ , by taking a linear extrapolation of them.\",\n \"It is known that owing to the time dividing points set properly this scheme has the same accuracy of second order Magnus expansion.\",\n \"The time propagation matrix having exponential form is treated exactly using a spectrum representation obtained by the diagonalization of Fock matrix, which is also applied in the EPPC1 scheme explained in the next subsection.\"\n ],\n [\n \"EPPC1\",\n \"This integrator involved with a micro iteration consists of the following procedures: $[1] & \\\\, F_{N} := F[D_{N}] \\\\\\\\[2] & \\\\, D_{N+1}^{\\\\rm pred} :=e^{- \\\\frac{i}{\\\\hbar } dt F_{N} } \\\\, D_{N} \\\\, e^{ \\\\frac{i}{\\\\hbar } dt F_{N} } \\\\\\\\[3] & \\\\, F^{\\\\rm pred}_{N+1} := F[D_{N+1}^{\\\\rm pred}] \\\\\\\\[4] & \\\\, F^{\\\\rm mid}:= \\\\frac{1}{2} \\\\left( F_{N} + F_{N+1}^{\\\\rm pred} \\\\right) \\\\\\\\[5] & \\\\, D_{N+1}^{\\\\rm corr} :=e^{- \\\\frac{i}{\\\\hbar } dt F^{\\\\rm mid} } \\\\, D_{N} \\\\, e^{ \\\\frac{i}{\\\\hbar } dt F^{\\\\rm mid} } \\\\\\\\[6] & \\\\,\\\\,\\\\left\\\\lbrace \\\\begin{array}{l}\\\\textrm {If} \\\\,\\\\, || D_{N+1}^{\\\\rm corr} - D_{N+1}^{\\\\rm pred} || < \\\\epsilon \\\\,\\\\,\\\\, \\\\\\\\\\\\quad \\\\textrm {then} \\\\,\\\\,\\\\,D_{N+1} := D_{N+1}^{\\\\rm corr}\\\\,\\\\,\\\\, \\\\Rightarrow \\\\,\\\\,\\\\textrm {End of this step} \\\\\\\\\\\\textrm {If} \\\\,\\\\, || D_{N+1}^{\\\\rm corr} - D_{N+1}^{\\\\rm pred} || \\\\ge \\\\epsilon \\\\,\\\\,\\\\, \\\\\\\\\\\\quad \\\\textrm {then} \\\\,\\\\,\\\\,D_{N+1}^{\\\\rm pred} := D_{N+1}^{\\\\rm corr}\\\\,\\\\,\\\\, \\\\Rightarrow \\\\,\\\\,\\\\textrm {Return to [3]}\\\\end{array}\\\\right.$ Here, a time propagation is carried out from $t$ to $t+dt$ , which are correspondingly labeled with $N$ and $N+1$ .\",\n \"$F$ and $D$ are the Fock and density matrices.\",\n \"'mid' denotes a mid point of $t$ and $t+dt$ , namely, $t+\\\\frac{1}{2}dt$ .\",\n \"'pred' and 'corr' are abbreviations of 'predictor' and 'corrector', respectively.\",\n \"The double '$||$ ' symbol means taking the Frobenius norm of a matrix placed between these two symbols.\",\n \"The threshold is expressed by $\\\\epsilon =n\\\\alpha \\\\xi $ with $n$ being the dimension number of a corresponding matrix.\",\n \"$\\\\alpha $ is set to be a value proportional to the absolute maximum eigen value of the matrix under consideration.\",\n \"Here, we used 1 as $\\\\alpha $ for simplicity.\",\n \"Thus, $\\\\xi $ determines the strictness of the self-consistency between the instantaneous density and the Fock matrix.\"\n ],\n [\n \"Mathematical form of LP-GDF procedure\",\n \"Here we provide the formal mathematical expression of LP-GDF scheme.\",\n \"At first, one knows that within the GD representation the identity operator is written as $\\\\hat{1}\\\\simeq \\\\sum _{i=1}^{N_g} \\\\sum _{j=1}^{N_{G_i}}\\\\mid \\\\phi _j^{G_i}\\\\rangle \\\\langle \\\\phi _{j}^{G_{i}}\\\\mid ,$ where $\\\\left\\\\lbrace \\\\mid \\\\phi _j^{G_i} \\\\rangle \\\\equiv \\\\sum _k\\\\mid \\\\widetilde{\\\\chi }_{k,G_i} \\\\rangle \\\\left[ \\\\underline{\\\\underline{D}}_{G_i} \\\\right]_{kj}\\\\right\\\\rbrace $ with $ \\\\mid \\\\widetilde{\\\\chi }_{k,G_i} \\\\rangle $ being the k-th Löwdin orthogonalized atomic orbital basis function spanned in the G$_i$ -th group is the set of GD localized orbitals for the group labeled by $G_i$ and $j$ ranges from 1 to $N_{G_i}$ , which denotes the number of basis functions spanned at the site, $G_i$ .\",\n \"$i$ ranges from 1 to $N_g$ , i.e., the number of monomer group sites.\",\n \"We used the symbol for approximation in the equation because of the practical use of a finite basis set during computation.\",\n \"Note that the GDF orbitals created from Löwdin orthonormal basis remain to be orthogonal under unitary transformations even if the transformations are carried out in each group.\",\n \"In turn, with respect to the structure of matrix representation of the Fock operator, only the GDF orbital pairs between different monomers are Fock non-orthogonal while the pairs within the same group are Fock orthogonal .\",\n \"Next, we introduce projection operators in order to realize steps (i)–(v) in Subsect.REF as follows: $\\\\hat{P} \\\\equiv \\\\sum _{i=1}^{N_g} \\\\sum _{j \\\\in \\\\Omega _i}^{N_{G_i}}\\\\mid \\\\phi _j^{G_i}\\\\rangle \\\\langle \\\\phi _j^{G_i}\\\\mid $ where $\\\\Omega _i \\\\equiv \\\\left\\\\lbrace j;| \\\\epsilon _{j,G_i} - \\\\bar{\\\\epsilon } |\\\\le \\\\Delta \\\\epsilon _i/2\\\\right\\\\rbrace \\\\quad \\\\textrm {and}\\\\quad \\\\bar{\\\\epsilon } \\\\equiv (\\\\epsilon _{\\\\textrm {H}}+\\\\epsilon _{\\\\textrm {L}})/2.$ Here, $\\\\epsilon _{j,G_i}$ is the local orbital energy associated with the GD orbital $\\\\phi _j^{G_i}$ , while $\\\\epsilon _{\\\\textrm {H}}$ and $\\\\epsilon _{\\\\textrm {L}}$ are the HOMO and LUMO energies of the whole system.\",\n \"By using these mathematical tools within LP-GDF formulation, we approximate a one electron operator $\\\\hat{O}$ as follows: $\\\\hat{O}\\\\approx \\\\hat{O}_P \\\\equiv \\\\hat{P}\\\\hat{O}\\\\hat{P}.$\"\n ],\n [\n \"Function form of Laser pulse field\",\n \"The vector potential of light field as a function of time in the long wave length approximation employed takes a form of ${\\\\bf A}(t) = \\\\sum _j^{N_\\\\textrm {p}}{\\\\bf A}_{j} \\\\, f(t;t_{c_j},t_{w_j}) \\\\, \\\\text{cos}(\\\\omega _j (t-t_{c_j})+\\\\delta _j),$ where bold font means the three dimensional vector in the Cartesian coordinate space.\",\n \"Here, the envelope function is defined by $ f(t;t_c,t_w) \\\\equiv \\\\textrm {exp} \\\\left( - \\\\left( \\\\dfrac{t-t_c}{t_w} \\\\right)^2 \\\\right) $ .\",\n \"The physical meanings of parameters appeared above are as follows; $t$ denotes time, $t_c$ is a field peak time, $t_w$ stands for a typical gaussian decay time, $\\\\omega $ means a central angular frequency of field, and $\\\\delta $ is a carrier envelope phase.\",\n \"$N_\\\\textrm {p}$ denotes a number of pulses.\",\n \"The electric field vector of external light corresponding to ${\\\\bf A}(t)$ is given by $ {\\\\bf E}(t) = -\\\\dfrac{1}{c} \\\\dfrac{\\\\partial {\\\\bf A}(t) }{\\\\partial t} $ .\",\n \"In the present article, we employed $N_\\\\textrm {p}=1$ and $\\\\delta =0$ .\"\n ],\n [\n \"Absorption spectrum\",\n \"The absorption spectrum was evaluated using the following function, $S(\\\\omega )\\\\equiv \\\\dfrac{1}{3}\\\\textrm {Tr}[\\\\underline{\\\\underline{\\\\widetilde{\\\\sigma }}}(\\\\omega )],$ where $\\\\underline{\\\\widetilde{\\\\underline{\\\\sigma }}}(\\\\omega )\\\\equiv \\\\dfrac{4\\\\pi \\\\omega }{c}\\\\textrm {Im}[\\\\underline{\\\\underline{\\\\widetilde{\\\\alpha }}}(\\\\omega )]$ is an absorption cross section with $[\\\\underline{\\\\underline{\\\\widetilde{\\\\alpha }}}(\\\\omega )]_{jk}\\\\equiv \\\\dfrac{\\\\widetilde{\\\\mu }^{\\\\rm {ind}}_j(\\\\omega )}{\\\\widetilde{E}_k(\\\\omega )}=\\\\dfrac{ \\\\int _{T_{\\\\rm CUT}}^{T_{\\\\rm FIN}} e^{ i \\\\omega t } e^{ - \\\\gamma t} \\\\mu _{j}(t) }{ \\\\int _{T_{\\\\rm CUT}}^{T_{\\\\rm FIN}} e^{ i \\\\omega t } E_k(t) }$ .\",\n \"$\\\\gamma $ is a damping factor.\",\n \"Here, $ \\\\mu ^{\\\\rm {ind}}_j(t) \\\\equiv \\\\mu _j(t) - \\\\mu _j(0) $ and ${E_k(t)}$ are induced dipole moment and external electric field, respectively.\",\n \"The symbol of tilde means taking the Fourier transformation.\",\n \"$T_{\\\\rm FIN}$ is a final simulation time.\",\n \"$T_{\\\\rm CUT}$ is the initial time used for a transformation from time dependent induced dipole moment of electrons and external field to absorption spectrum.\"\n ],\n [\n \"Computational time\",\n \"Here, we provide information about the computational cost of electron dynamics including analysis of the time-dependent properties.\",\n \"Note that the overhead of parallel computation is also included in time.\",\n \"The aim here is to show examples of the performance of the method in cases with use of moderate computer facilities.\",\n \"The specifications of the computer and compiler utilized in this article are as follows: [For NPTL–TCNE] { Intel(R) Xeon(R) Gold 6148 0 @ 2.40GHz with a cache size of 27.5MB, The Intel FORTRAN compiler in Version 18.0.2.199 with level three optimization. }\",\n \"[For 5UT] { Intel(R) Xeon(R) Gold 6152 0 @ 2.10GHz with a cache size of 30.9MB, The Intel FORTRAN compiler in Version 17.0.4.196 with level three optimization. }\",\n \"In all the calculations 40 cpu cores were used in a parallel manner using openMP for matrix-matrix product and MPI for Fock build if needed.\"\n ],\n [\n \"NPTL–TCNE\",\n \"Tab.\",\n \"REF and Tab.\",\n \"REF present the comparison of the computation times using different number of projected local orbitals in cases of PC2M-LF and EPPC1 methods, respectively.\",\n \"The system treated is the NPTL–TCNE dimer.\",\n \"A ratio to the time spent in cases with full orbitals expresses an acceleration in computation by using projected space.\",\n \"The tables include the comparisons between RT and FF.\",\n \"Note that the time intervals of one time step employed here are different between these two integrators.\",\n \"We checked the reproduction of the same result by using these two integrators, which is presented in the supplementary material.\",\n \"In Tab.\",\n \"REF for PC2M-LF, though we can see that a moderate acceleration is attained with a help of projected space and acceleration in FF is superior to RT one including Fock build, the use of RT recommended for the reliable calculation and provides still moderate acceleration by projections of local orbital space.\",\n \"In fact, the computation time in RT with $E_{\\\\rm bw}=1.6$ corresponding to $N_{\\\\rm proj}=120$ which provide converged results is reduced to one fourth of that spent in the cases using the full orbitals.\",\n \"As seen in Tab.\",\n \"REF , the acceleration in the calculation by projection in case of EPPC1 including micro iterations accompanied with Fock build for a convergence of corrector density is less than those of PC2M-LF in Tab.\",\n \"REF .\",\n \"Despite of this, the case of $E_{\\\\rm bw}=1.6$ with $N_{\\\\rm proj}=120$ need approximately the half of the computational time in the full orbital case.\",\n \"Table: CPU time spent for the dynamics calculations plus the time-dependent analysisfor NPTL–TCNE with PC2M-LF scheme with 1875 time steps using 8 as time interval.The unit of time is second.Numeric value in [] denotes the ratio to the cost in case with use of full orbitalswhile that in {}\\\\lbrace \\\\rbrace mean the number of Fock build per one time step.Note the single Fock build per one time step in the PC2M-LF scheme.The spent time needed in the cases of Fig.\",\n \"are the samein each group of the panels (a-d) for RT and (e-h) for FF, namely,in the above RT and FF columns correspond to the group of (a-d) and (e-h) in Fig.\",\n \".See the text for the meanings of E bw =ΔϵE_{\\\\rm bw}=\\\\Delta \\\\epsilon , N proj N_{\\\\rm proj}, RT and FF.The wall time means the wall clock time associated withopenMP for dynamics and MPI for Fock build.Table: CPU time spent for the dynamics calculations plus the time-dependent analysisfor NPTL–TCNE with EPPC1 scheme with 750 time steps using 20 as time interval.The unit of time is second.Numeric value in [] denotes the ratio to the cost in case with use of full orbitalswhile that presented in {}\\\\lbrace \\\\rbrace correspond to the average number of cycles inmicro-iterations needed for the density convergence with ξ=10 -8 \\\\xi =10^{-8}.See the text for the meanings of E bw =ΔϵE_{\\\\rm bw}=\\\\Delta \\\\epsilon , N proj N_{\\\\rm proj}, RT and FF.\",\n \"(a-d) and (e-h) correspond to RT and FF, respectively,and all the symbols are the same as Fig.\",\n \"in the main text.The definition of the wall time is the same as that in Tab.\",\n \".\"\n ],\n [\n \"5UT\",\n \"Tab.\",\n \"REF presents the comparison of the computation times using different number of projected local orbitals in cases of EPPC1 methods for 5UT system.\",\n \"The way of presentation is the same as the table for NPTL–TCNE.\",\n \"Here also we show the comparisons between RT and FF.\",\n \"As seen in Tab.\",\n \"REF , the case of $E_{\\\\rm bw}=1.5$ with $N_{\\\\rm proj}=164$ associated with the converged spectrum presented in Fig.\",\n \"5 in the main text needs merely less than half of the computational time in the case with the full 394 orbitals.\",\n \"Table: CPU time spent for the dynamics calculations plus the time-dependent analysisfor 5UT with EPPC1 scheme with 1500 time steps using 20as time steps.The unit of time is second.Numeric value in [] denotes the ratio to the cost in case with use of full orbitalswhile that presented in {}\\\\lbrace \\\\rbrace correspond to the average number of cycles inmicro-iterations needed for the density convergence with ξ=10 -8 \\\\xi =10^{-8}.See the text for the meanings of E bw =ΔϵE_{\\\\rm bw}=\\\\Delta \\\\epsilon and N proj N_{\\\\rm proj}.The wall time means the wall clock time, and the values in parenthesesdenote the CPU efficiency in percent associated openMP for dynamics and MPI for Fock build.Supplementary material for “Electron dynamics method using a locally projected group diabatic Fock matrix for molecule and aggregate”\"\n ],\n [\n \"Convergence of charge migration dynamics with respect\\nto the time increment using different integrator for NPTL–TCNE system.\",\n \"Fig.\",\n \"SREF shows the convergence of charge migration dynamics for NPTL–TCNE system with respect to time increments.\",\n \"The panels correspond to those in Fig.\",\n \"2 in the main article.\",\n \"For a stringent check, we used different integrators, namely, PC2M-LF and EPPC1, with the time increments of 8 and 20 as in each step, respectively.\",\n \"The perfect agreement with each other means the convergence of the result.\",\n \"Figure: Check of the convergence of charge migration dynamics with respectto the time increment using different integrator for NPTL–TCNE system.See the main text in this supplementary material about the panels.\"\n ],\n [\n \"\\nEfficiency of orbital projection method for standard RT-TDDFT and Frozen Fock method\\nwithout a group division in charge migration dynamics for NPTL–TCNE system.\",\n \"We demonstrate an efficiency of the present local projection (LP) scheme for a standard RT-TDDFT.\",\n \"Note that the standard RT-TDDFT is equivalent to the case with $N_g$ =1 and a ground state of whole system as an initial state within LP-GDF electron dynamics scheme.\",\n \"This is also verified for the calculation of absorption spectrum for 5UT with $N_g$ =1, which is presented in Fig.\",\n \"5 in the main text.\",\n \"Here, for charge migration dynamics in NPTL–TCNE system, we show the efficiency of LP scheme for the standard RT-TDDFT without any group division in the Fock matrix.\",\n \"We also show the results for the Frozen Fock approach.\",\n \"In this context, panels (a/b) and (c/d) in Fig.\",\n \"SREF with $N_g$ =1 in LP-GDF electron dynamics correspond to (c/d) and (g/h) of Fig.\",\n \"2 in the main text with $N_g$ =2, respectively.\",\n \"The same properties are plotted.\",\n \"Time integrator employed here is EPPC1.\",\n \"Strictly speaking, the initial electronic condition is different for both schemes since the former standard case starts from the initial density matrix made from the ground state of whole system while the initial density matrix was prepared by using the combination of local ground states of monomers in the latter LP-GDF case.\",\n \"Despite this, the time dependent behavior is similar for both cases, which inversely supports validity of group division in Fig.\",\n \"2.\",\n \"Fig.\",\n \"SREF is the maximum error analysis for the LP results compared from the full orbital calculations.\",\n \"Panels (A) and (B) correspond to the cases of (a/b) and (c/d) in Fig.\",\n \"SREF The plots of the data were carried out in the same way as Fig.\",\n \"3 in the main text.\",\n \"These results indicate that the LP scheme can safely reduce the computational cost with respect to orbital numbers required for the description also for charge migration dynamics under a light field with using standard RT-TDDFT equivalent to LP-GDF-electron dynamics scheme with $N_g$ =1.\",\n \"Figure: Light induced charge migration dynamics with N g N_g=1 for NPTL–TCNE.Panels (a), (b), (c) and (d) in this figurerespectively correspond to (c), (d), (g) and (h) in Fig.\",\n \"2 in the main text.Figure: N proj N_{\\\\textrm {proj}} dependency of max deviations for donormolecule during dynamics from the results by full orbital calculationin Fig.\",\n \"S.RT and FF in panel (A) were evaluated from the resultsin (a) and (c) of Fig.\",\n \"S.while RT and FF in panel (B) were evaluated from the results in (b) and (d).\"\n ],\n [\n \"Group diabatic and canonical orbital energies of NPTL–TCNE dimer\",\n \"Tab.\",\n \"REF summarizes the energy differences of LUMO+1, LUMO and HOMO-1 measured from HOMO, which are presented for group diabatic and canonical representations associated with monomers and dimer system, respectively.\",\n \"Table: (Upper table)Energy differences of group diabatized HOMO-1-1, LUMO, and LUMO+1+1 measuredfrom group diabatized HOMO of the electron donor NPTL and acceptor TCNE moietiesin a molecular complex, which are labeled with the symbols ofΔϵ H-1,H G \\\\Delta \\\\epsilon ^{\\\\rm G}_{{\\\\rm H}-1,{\\\\rm H}},Δϵ L,H G \\\\Delta \\\\epsilon ^{\\\\rm G}_{{\\\\rm L},{\\\\rm H}} , andΔϵ L+1,H G \\\\Delta \\\\epsilon ^{\\\\rm G}_{{\\\\rm L}+1,{\\\\rm H}} , respectively.Note that the monomer interactions are includedin the evaluations of the local orbital energies of monomers.\",\n \"(Lower table)Energy differences of canonical HOMO-1-1, LUMO and LUMO+1+1 measured from HOMOfor the whole NPTL–TCNE dimer system, which are denoted by the symbols ofΔϵ H-1,H W \\\\Delta \\\\epsilon ^{\\\\rm W}_{{\\\\rm H}-1,{\\\\rm H}},Δϵ L,H W \\\\Delta \\\\epsilon ^{\\\\rm W}_{{\\\\rm L},{\\\\rm H}} , andΔϵ L+1,H W \\\\Delta \\\\epsilon ^{\\\\rm W}_{{\\\\rm L}+1,{\\\\rm H}} , respectively.In both tables, the unit is atomic unit.GD and CA mean 'group diabatic' and 'canonical', respectively.\"\n ],\n [\n \"Excitation energies, transition dipoles and oscillator strength of 5UT\",\n \"Tab.SREF summarizes the information on lower excitations of 5UT at the geometry given in the later section in this supplementary material.\",\n \"We showed the excitation energies, transition dipole moment vectors, and oscillator strength.\",\n \"Here, $(i \\\\rightarrow j)$ , $\\\\Delta E_{ij}\\\\equiv |E_i-E_j|$ , $\\\\vec{L}_{ij}$ and $f_{ij} \\\\equiv \\\\frac{2}{3}(\\\\Delta E_{ij}) |\\\\vec{L}_{ij}|^2 $ denote a state pairs of transition, energy difference of them, transition dipole moment vector for them, and its corresponding oscillator strength, respectively.\",\n \"$E_i$ means $i$ -th adiabatic state with 0 being ground electronic state.\",\n \"The geometry of this molecular system is given later in this supplementary material.\",\n \"According to magnitudes of oscillator strength, we considered that 0$\\\\rightarrow $ 1, 0$\\\\rightarrow $ 3 and 0$\\\\rightarrow $ 5 are optically allowed transitions, and used as reference excitation energies in the presentation of Fig.\",\n \"4 and 5 in the main text.\",\n \"Though it is not the aim to reproduce the experimental data in the present article since the main focus is the projection scheme in electron dynamics calculation, we provide information near to the situation treated here with respect to the target system.\",\n \"The experimental data of the position of the first peak in absorption spectrum of optical electronic transition for this 5UT system, however, in the CHCl$_3$ solvent but not in a gas phase, is reported in the former works, [34] and its value is 2.98 eV.\",\n \"Though the absorption spectrum for electronic transition in a molecular system can be changed by an existence of a polar solvent associated with a complex modification of electric transition and structure relaxation, this is not significant in this system.\",\n \"On the other hand, as seen in Fig.\",\n \"4 of the main article as the comparison part of RT and FF results, the value given by RT-TDDFT, 2.2 eV, being consistent with LR-TDDFT result here within the present limited ab initio level is close to the experimental value mentioned above, compared to the peak value, 1.6 eV, given by FF approximation as found in Fig.\",\n \"4.\",\n \"This observation also supports the basic assertion of the importance of the self-consistent treatment for the Fock matrix and time-dependent density matrix in a calculation of absorption spectrum.\",\n \"The remaining discrepancy in the LR-TDDFT result here is remedied by using more appropriate higher level basis set[34] and exchange-correlation functionals [32].\",\n \"In fact, at the same structure, the use of the representative hybrid functionals with a long-range correction, namely, CAM-B3LYP[36] and LC-BOP[37], provide the optically allowed first electronic transition energies in gas phase, 2.80 eV and 2.99, respectively, while their corresponding values for the polarizable continuum model(PCM)[38] with CHCl$_3$ solvent at the same structure are 2.68 and 2.88.\",\n \"In the calculations using the PCM combined with LR-TDDFT here, we used the ab initio program package for electronic structure calculation, GAMESS.\",\n \"[39] Table: Transition properties of 5UT obtained by LR-TDDFT calculation with PBE/6-31G(d).See the main text in this supplementary material for the details.\"\n ],\n [\n \"Geometry data of molecular systems\",\n \"We summarize the data of Cartesian coordinates in Angstrom for the molecular systems used in the present article.\",\n \"The information of atoms are also included below.\"\n ],\n [\n \"NPTL–TCNE\",\n \"C 1.824214 0.710474 -1.999379 C 1.824214 -0.710473 -1.999379 C 0.628132 -1.408746 -1.999379 C -0.619806 -0.721697 -1.999379 C -0.619809 0.721696 -1.999379 C 0.628130 1.408742 -1.999379 C -1.867746 1.408746 -1.999379 C -3.063829 0.710474 -1.999379 C -3.063829 -0.710473 -1.999379 C -1.867746 -1.408742 -1.999379 H 2.775875 1.251891 -1.999379 H 2.775876 -1.251890 -1.999379 H 0.624772 -2.504937 -1.999379 H 0.624771 2.504934 -1.999379 H -1.864388 2.504937 -1.999379 H -4.015491 1.251889 -1.999379 H -4.015490 -1.251891 -1.999379 H -1.864388 -2.504934 -1.999379 N 2.832029 2.070410 2.000621 N -1.591624 2.070452 2.000621 C 1.839916 1.437817 2.000621 C -0.599565 1.437776 2.000621 C 0.620188 0.694253 2.000621 C 0.620194 -0.694285 2.000621 C 1.839957 -1.437800 2.000621 C -0.599550 -1.437827 2.000621 N 2.832015 -2.070408 2.000621 N -1.591638 -2.070398 2.000621\"\n ],\n [\n \"5UT\",\n \"C -8.572000 1.432812 0.000000 C -9.102000 0.159612 0.000000 S -7.858600 -1.051488 0.000000 C -6.587100 0.165912 0.000000 C -7.149500 1.439512 0.000000 H -9.185200 2.337012 0.000000 H -10.148900 -0.142688 0.000000 H -6.546000 2.351012 0.000000 C -4.632800 -1.495088 0.000000 C -5.196000 -0.220788 0.000000 S -3.927900 0.998512 0.000000 C -2.656500 -0.219888 0.000000 C -3.219600 -1.495488 0.000000 H -5.236800 -2.406288 0.000000 H -2.615500 -2.406588 0.000000 C -0.706100 1.441412 0.000000 C -1.270100 0.165512 0.000000 S 0.000000 -1.054088 0.000000 C 1.270100 0.165512 0.000000 C 0.706100 1.441412 0.000000 H -1.310300 2.352412 0.000000 H 1.310300 2.352412 0.000000 C 3.219600 -1.495488 0.000000 C 2.656500 -0.219888 0.000000 S 3.927900 0.998512 0.000000 C 5.196000 -0.220788 0.000000 C 4.632800 -1.495088 0.000000 H 2.615500 -2.406588 0.000000 H 5.236800 -2.406288 0.000000 C 7.149500 1.439512 0.000000 C 6.587100 0.165912 0.000000 S 7.858600 -1.051488 0.000000 C 9.102000 0.159612 0.000000 C 8.572000 1.432812 0.000000 H 6.546000 2.351012 0.000000 H 10.148900 -0.142688 0.000000 H 9.185200 2.337012 0.000000\"\n ]\n]"},"id":{"kind":"string","value":"1808.08362"}}},{"rowIdx":395,"cells":{"text":{"kind":"list like","value":[["Collective Mode Interferences in Light--Matter Interactions"],["Abstract We present a theoretical and experimental analysis of transient optical properties of a dense cold atomic gas.","After the rapid extinction of a weak coherent driving field (mean photon number $\\sim 1.5$), a transient `flash' is observed.","Surprisingly the decay of the `flash' is faster than the decay of the fastest superradiant mode of the system.","We show that this `faster than superradiance decay' is expected due to the interference between collective eigenmodes that exhibit a range of frequency shifts away from the bare atomic transition.","Experimental results confirm that the initial decay rate of the superradiant flash increases with optical depth, in agreement with the numerical simulations for the experimental conditions."],["Eigenmode distribution","In Fig.","REF we plot the eigenmodes for an ensemble of $N=400$ atoms as in Fig.","1b in the main text, except now showing 100 random realizations rather than a single realization.","This gives a more complete representation of the overall eigenmode distribution.","Importantly, it shows that barely any eigenmode decay rates exceed $2\\Gamma _0$ even for 100 repetitions while the decay rate measured for the equivalent optical depth in Fig.","4a in the main text exceeds $\\sim 2.5\\Gamma _0$ .","This excludes the possibility that the enhanced decay rate could be due to spurious large eigenmode decay rates.","Figure: Eigenvalue distribution.a Eigenmode shifts Δ p \\Delta _p and widths Γ p \\Gamma _p for an ensemble of N=400N=400 interacting atoms using the same parameters as Fig.","1b from the main text, repeated 100 times.","b,c Eigenvalue probability density PP."]],"string":"[\n [\n \"Collective Mode Interferences in Light--Matter Interactions\"\n ],\n [\n \"Abstract We present a theoretical and experimental analysis of transient optical properties of a dense cold atomic gas.\",\n \"After the rapid extinction of a weak coherent driving field (mean photon number $\\\\sim 1.5$), a transient `flash' is observed.\",\n \"Surprisingly the decay of the `flash' is faster than the decay of the fastest superradiant mode of the system.\",\n \"We show that this `faster than superradiance decay' is expected due to the interference between collective eigenmodes that exhibit a range of frequency shifts away from the bare atomic transition.\",\n \"Experimental results confirm that the initial decay rate of the superradiant flash increases with optical depth, in agreement with the numerical simulations for the experimental conditions.\"\n ],\n [\n \"Eigenmode distribution\",\n \"In Fig.\",\n \"REF we plot the eigenmodes for an ensemble of $N=400$ atoms as in Fig.\",\n \"1b in the main text, except now showing 100 random realizations rather than a single realization.\",\n \"This gives a more complete representation of the overall eigenmode distribution.\",\n \"Importantly, it shows that barely any eigenmode decay rates exceed $2\\\\Gamma _0$ even for 100 repetitions while the decay rate measured for the equivalent optical depth in Fig.\",\n \"4a in the main text exceeds $\\\\sim 2.5\\\\Gamma _0$ .\",\n \"This excludes the possibility that the enhanced decay rate could be due to spurious large eigenmode decay rates.\",\n \"Figure: Eigenvalue distribution.a Eigenmode shifts Δ p \\\\Delta _p and widths Γ p \\\\Gamma _p for an ensemble of N=400N=400 interacting atoms using the same parameters as Fig.\",\n \"1b from the main text, repeated 100 times.\",\n \"b,c Eigenvalue probability density PP.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08415"}}},{"rowIdx":396,"cells":{"text":{"kind":"list like","value":[["Tree-based Particle Smoothing Algorithms in a Hidden Markov Model"],["Abstract We provide a new strategy built on the divide-and-conquer approach by Lindsten et al.","(2017) to investigate the smoothing problem in a hidden Markov model.","We employ this approach to decompose a hidden Markov model into sub-models with intermediate target distributions based on an auxiliary tree structure and produce independent samples from the sub-models at the leaf nodes towards the original model of interest at the root.","We review the target distribution in the sub-models suggested by Lindsten et al.","and propose two new classes of target distributions, which are the estimates of the (joint) filtering distributions and the (joint) smoothing distributions.","The first proposed type is straightforwardly constructible by running a filtering algorithm in advance.","The algorithm using the second type of target distributions has an advantage of roughly retaining the marginals of all random variables invariant at all levels of the tree at the cost of approximating the marginal smoothing distributions in advance.","We further propose the constructions of these target distributions using pre-generated Monte Carlo samples.","We show empirically the algorithms with the proposed intermediate target distributions give stable and comparable results as the conventional smoothing methods in a linear Gaussian model and a non-linear model."],["Introduction","A hidden Markov model (HMM) is a discrete-time stochastic process $\\lbrace X_{t}, Y_{t}\\rbrace _{t \\ge 0}$ where $\\lbrace X_{t}\\rbrace _{t \\ge 0}$ is an unobserved Markov chain.","We only have access to $\\lbrace Y_{t}\\rbrace $ whose distribution depends on $\\lbrace X_{t}\\rbrace $ .","We make the following assumptions in the entire article: The densities of the initial state $X_{0}$ , the transition density $X_{t+1}$ given $X_{t} = x_{t}$ and the emission density $Y_{t}$ given $X_{t} = x_{t}$ taken with respect to some dominating measure exist and are denoted as follows: $X_{0} &\\sim p_{0}(x_{0}) &&\\\\X_{t+1}| \\lbrace X_{t} = x_{t} \\rbrace &\\sim p(x_{t+1} | x_{t}) && \\text{~~for } t = 0, \\ldots , T-1,\\\\Y_{t} | \\lbrace X_{t} = x_{t}\\rbrace &\\sim p(y_{t}|x_{t}) && \\text{~~for } t = 0, \\ldots , T,$ where $T$ is the final time step of the process.","We are interested in the (marginal) smoothing distributions $\\lbrace p(x_{t}|y_{0:T})\\rbrace _{t = 0, \\ldots , T}$ or the joint smoothing distribution $p(x_{0:T}|y_{0:T})$ where $x_{0:T}$ and $y_{0:T}$ are abbreviations of $(x_{0}, \\ldots , x_{T})$ and $(y_{0}, \\ldots , y_{T})$ , respectively.","Exact solutions are available for linear Gaussian HMM using a Rauch–Tung–Striebel smoother (RTSs) [21] and in a HMM with finite-space Markov chains [3].","In most other cases, the smoothing distributions are not analytically tractable.","A large body of work uses Monte Carlo methods to approximate the smoothing distributions $\\lbrace p(x_{t}|y_{0:T})\\rbrace _{t = 0, \\ldots , T}$ or the joint smoothing distribution $p(x_{0:T}|y_{0:T})$ .","Sequential Monte Carlo (SMC) methods [7] are commonly used to sequentially update the filtering distributions $\\lbrace p(x_{t}|y_{0:t})\\rbrace _{t = 0, \\ldots , T}$ .","SMC can in principle be used to estimate the joint smoothing density $p(x_{0:T}|y_{0:T})$ by updating the entire history of the random samples in each resampling step.","However, the performance can be poor, as path degeneracy will occur in many settings [2].","Advanced sequential Monte Carlo methods with desirable theoretical and practical results have been developed in recent years including sequential Quasi-Monte Carlo (SQMC) [11], divide-and-conquer sequential Monte Carlo (D&C SMC) [17], multilevel sequential Monte Carlo (MSMC) [4] and variational sequential Monte Carlo (VSMC) [20].","Other smoothing algorithms have been suggested previously.","[8] develop the forward filtering backward smoothing algorithm (FFBSm) for sampling from $\\lbrace p(x_{t}|y_{0:T})\\rbrace _{t = 0, \\ldots , T}$ based on the formula proposed by [14].","[12] propose the forward filtering backward simulation algorithm (FFBSi) which generates samples from the joint smoothing distribution $p(x_{0:T}|y_{0:T})$ .","[5] propose a two-filter smoother (TFS) which employs a standard forward particle filter and a backward information filter to sample from $\\lbrace p(x_{t}|y_{0:T})\\rbrace _{t = 0, \\ldots , T}$ .","Typically, these algorithms have quadratic complexities in $N$ for generating $N$ samples.","[9] and [16] propose two smoothing algorithms with lower computational complexity, but their methods do not provide unbiased estimates.","In this article, we suggest using the divide-and-conquer sequential Monte Carlo (D&C SMC) [17] approach to address the smoothing problem.","The D&C SMC algorithm performs statistical inferences in probabilistic graphical models.","It splits the random variables of the target distribution into multiple levels of disjoint sets based upon an auxiliary tree $\\mathcal {T}$ .","An intermediate target distribution needs to be assigned to each set of random variables yielding sub-models for each non-leaf node.","The choice of these intermediate target distributions is key for a good overall performance of the algorithm.","By sampling independently from the leaf nodes and gradually propagating, merging and resampling from the leaf nodes to the root, the D&C SMC algorithm eventually produces samples from the target distribution.","The merging step involves importance sampling.","Using the idea of D&C SMC, we aim to estimate the joint smoothing distribution $p(x_{0:T}|y_{0:T})$ and thus call the algorithm: `tree-based particle smoothing algorithm' (TPS).","The key differences between TPS and other smoothing algorithms lie in its non-sequential and more adaptive merging step of the samples.","Our main contribution is the proposition and investigation of three classes of intermediate target distributions to be used in the algorithm.","We denote a leaf node corresponding to a single random variable $X_{j}$ by $\\mathcal {T}_{j} \\in \\mathcal {T}$ and a non-leaf node corresponding to the random variables $X_{j:l}$ by $\\mathcal {T}_{j:l} \\in \\mathcal {T} (j < l)$ .","The first class advised by [17] has the density proportional to the product of all transition and emission densities associated to the target variable $X_{j}$ (resp.","$X_{j:l}$ ) in the sub-model.","This is equivalent to the unnormalised likelihood of a new HMM starting at time $j$ (resp.","from time $j$ to $l$ ) given the observations of the same time interval with an uninformative prior of $X_{j}$ if $j \\ne 0$ .","The second class uses an estimate of the filtering distribution $p(x_{j} | y_{0:j})$ at $\\mathcal {T}_{j} \\in \\mathcal {T}$ and an estimate of the joint filtering distribution $p(x_{j:l} | y_{0:l})$ at $\\mathcal {T}_{j:l} \\in \\mathcal {T}$ .","Working with this estimate involves tuning a preliminary particle filter.","The third class uses estimates of the marginal smoothing distribution $p(x_{j} | y_{0:T})$ at $\\mathcal {T}_{j} \\in \\mathcal {T}$ and of the joint smoothing distribution $p(x_{j:l} | y_{0:T})$ at $\\mathcal {T}_{j:l} \\in \\mathcal {T}$ .","We will see that this class of immediate distributions is optimal in a certain sense.","Furthermore, under this construction, we approximately retain the marginal distribution of all single random variables $\\lbrace X_{j}\\rbrace _{j = 0}^{T}$ invariant as the marginal smoothing distributions $\\lbrace p(x_{j} | y_{0:T}) \\rbrace _{j = 0}^{T}$ at every level of the tree.","The price of implementing TPS using the second class of intermediate target distributions relies on both the estimates of the filtering and the (marginal) smoothing distributions, but not necessarily the joint smoothing distribution.","We then propose some parametric and non-parametric approaches to construct these intermediate distributions based on the pre-generated Monte Carlo samples considering both efficiency and accuracy.","The article is structured as follows.","We first describe the divide-and-conquer approach for particle smoothing in Section .","We discuss the intermediate target distributions and the constructions of the initial sampling distributions at the leaf nodes in Section .","In Section , we conduct simulation studies in a linear Gaussian and non-linear non-Gaussian HMM to compare TPS with other smoothing algorithms.","The article finishes with a discussion in Section ."],["Tree-based Particle Smoothing Algorithm (TPS)","This section outlines an algorithm we call `tree-based particle smoothing algorithm' (TPS).","[17] describe the construction of an auxiliary tree for general probabilistic graphical models.","We demonstrate a unique construction of an auxiliary binary tree from a HMM bearing intermediate target distributions specified at each node.","We then illustrate the sampling procedure for the target distributions at the nodes.","We present an algorithm which can be applied recursively from the leaf nodes towards the root and yet generate the target samples."],["Construction of an auxiliary tree","TPS splits a HMM into sub-models based upon a binary tree decomposition.","It first divides the random variables $X_{0:T}$ into two disjoint subsets and recursively apply binary splits to the resulting two subsets until the resulting subset consists of only a single random variable.","Each generated subset corresponds to a tree node and is assigned an intermediate target distribution.","The root characterises the complete model with the target distribution $p(x_{0:T}|y_{0:T})$ .","Initial samples are generated at the leaf nodes, independent between leaves.","Theses samples are recursively merged using importance sampling until the root of the tree is reached.","We propose one intuitive way of implementing the binary splits which ensures that the left subtree is always a complete binary tree and contains at least as many nodes as the right subtree.","We split a non-leaf node with the variables $X_{j:l}$ where $0 \\le j < l \\le T$ , into two children $\\mathcal {T}_{j:k-1}$ and $\\mathcal {T}_{k:l}$ with the random variables $X_{j:k-1}$ and $X_{k:l}$ , where $k = j + 2^{p},$ and $p = \\lceil \\frac{\\log (l-j+1)}{\\log 2}\\rceil - 1.$ The auxiliary tree when $T = 5$ is shown in Figure REF .","This construction has several advantages: The random variables within each node have consecutive time indices.","The left subtree is also a complete binary tree of $2^{\\lceil \\frac{\\log (T+1)}{\\log 2}\\rceil }$ leave nodes.","$\\lbrace y_{T+1}, y_{T+2}, \\ldots \\rbrace $ become available, as samples from the complete subtree would not need to be updated.","Moreover, the tree has a height of $\\big (\\lceil \\frac{\\log (T+1)}{\\log 2}\\rceil + 1\\big )$ levels, which implies a maximum number of $\\lceil \\frac{\\log (T+1)}{\\log 2}\\rceil $ updates of the samples corresponding to a single random variable with different target distributions at different levels of the tree.","Usually, more updates potentially indicate more resampling steps, which may cause more serious degeneracy problems.","In Figure REF , the samples corresponding to $X_{0}, \\ldots , X_{3}$ need to be updated three times from the leave nodes and those of $X_{4}, X_{5}$ need to be updated twice.","When running a bootstrap particle filter to solve the smoothing problem, the samples at time step $t = 0$ need to be updated $T$ times and thus the maximum number of the updates become $T$ , which is no less than $ \\lceil \\frac{\\log (T+1)}{\\log 2}\\rceil $ .","[17] also propose a general way of constructing the auxiliary tree in a self-similar model family, where a HMM belongs to.","Their construction in the context of a HMM may not be identical to ours with no restriction on the choice of the cutting point.","Figure: An auxiliary binary tree consisting of random variables when T=5T = 5"],["Sampling procedure in the sub-models of tree","We describe the sampling approach from the target distribution at a leaf and non-leaf node of the constructed binary tree $\\mathcal {T}$ described in Section REF .","We denote a target density by $f_{j}$ which can be straightforwardly sampled from at a leaf node $\\mathcal {T}_{j} \\in \\mathcal {T}$ , a proper importance density by $h_{j:l}$ and a target density by $f_{j:l}$ respectively at a non-root tree node $\\mathcal {T}_{j:l} \\in \\mathcal {T}$ where $0 < l - j 0$ whenever $f(\\mathbf {x_{1}}, \\mathbf {x_{2}}) > 0$ , where we assume $h_{1}$ and $h_{2}$ are the probability densities of two independent (joint) random variables $\\mathbf {X_{1}}$ and $\\mathbf {X_{2}}$ .","We claim that proposal $f_{1}({\\mathbf {x_{1}}}) f_{2}({ \\mathbf {x_{2}}})$ has the smallest KL divergence among all proposals of the form $h_{1}(\\mathbf {x_{1}})h_{2}({\\mathbf {x_{2}}})$ where $f_{1}({\\mathbf {x_{1}}})$ and $f_{2}({ \\mathbf {x_{2}}})$ are the marginal densities of $f(\\mathbf {x_{1}, x_{2}})$ with respect to $\\mathbf {X_{1}}$ and $\\mathbf {X_{2}}$ , respectively.","Theorem 1 Let $f$ be a probability density function defined on $\\mathbb {R}^{n_{1} + n_{2}}$ , let $h_{1}$ and $h_{2}$ be probability density functions on $\\mathbb {R}^{n_{1}}$ and $\\mathbb {R}^{n_{2}}$ , respectively.","If $h_{1}(\\mathbf {x_{1}}) h_{2} (\\mathbf {x_{2}}) > 0$ whenever $f(\\mathbf {x_{1}, x_{2}}) >0,$ then $ \\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_{1}}} f( \\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\bigg (\\frac{f(\\mathbf {x_{1}}, \\mathbf {x_{2}})}{h_{1}(\\mathbf {x_{1}}) h_{2}(\\mathbf {x_{2}}) } \\bigg ) \\mathrm {d} \\mathbf {x_{1}} \\mathrm {d} \\mathbf {x_{2}} \\ge \\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_{1}}} f(\\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\bigg (\\frac{f(\\mathbf {x_{1}}, \\mathbf {x_{2}})}{f_{1}(\\mathbf {x_{1}}) f_{2}(\\mathbf {x_{2}}) } \\bigg ) \\mathrm {d} \\mathbf {x_{1}} \\mathrm {d} \\mathbf {x_{2}},$ where $f_{1}({ \\mathbf {x_{1}}}) = \\int _{\\mathbb {R}^{n_{2}}} f({\\mathbf {x_{1}}}, \\mathbf {{x_{2}}}) \\mathrm {d} { \\mathbf {x_{2}}}$ and $f_{2}({ \\mathbf {x_{2}}}) = \\int _{\\mathbb {R}^{n_{1}}} f(\\mathbf {{x_{1}}}, \\mathbf {{x_{2}}}) \\mathrm {d} {\\mathbf {x_{1}}}$ are the densities of the marginal distributions of $f(\\mathbf {x_{1}}, \\mathbf {x_{2}})$ .","The proof of Theorem REF is in the Appendix."],["Estimates of smoothing distributions as target","We provide an alternative way of constructing the intermediate target distributions using the marginal smoothing distributions motivated by Theorem REF .","Since the closed-form solutions to the marginal smoothing distributions are not available in general, we employ the estimates of the distributions at the nodes.","At the root, we still use $f_{0:T} = p(x_{0:T}|y_{0:T})$ .","At a leaf node $\\mathcal {T}_{j} \\in \\mathcal {T}$ , we define $f_{j}(x_{j}) = \\hat{p}(x_{j} | y_{0:T}) \\approx p(x_{j} | y_{0:T})$ , which requires estimating the marginal smoothing distribution.","We thus name the algorithm TPS-ES.","At a non-leaf and non-root node $\\mathcal {T}_{j:l}$ , we define the target distribution $f_{j:l}$ : $f_{j:l}(x_{j:l}) &\\propto & \\hat{p}(x_{j} | y_{0:j}) \\frac{ \\hat{p}(x_{l} | y_{0:T})}{ \\hat{p}(x_{l} | y_{0:l})} \\prod ^{l-1}_{i = j} \\bigg \\lbrace p(x_{i+1} | x_{i}) p(y_{i+1} | x_{i+1}) \\bigg \\rbrace \\\\ &\\approx & p(x_{j} | y_{0:j}) \\frac{p(x_{l} | y_{0:T}) }{p(x_{l} | y_{0:l}) } \\prod ^{l-1}_{i = j} \\bigg \\lbrace p(x_{i+1} | x_{i}) p(y_{i+1} | x_{i+1}) \\bigg \\rbrace \\\\& = & p(x_{j:l} | y_{0:T}),$ where $\\hat{p}(x_{j} | y_{0:j})$ denotes a probability density approximating the filtering density at the $j$ th time step.","Hence, given the estimate smoothing densities $\\lbrace \\hat{p}(x_{j} | y_{0:T}) \\rbrace _{j = 0, \\ldots , T}$ and the estimating filtering densities $\\lbrace \\hat{p}(x_{j} | y_{0:j}) \\rbrace _{j = 0, \\ldots , T}$ , we build an estimator of the distribution $p(x_{j:l} | y_{0:T})$ at $\\mathcal {T}_{j:l} \\in \\mathcal {T}$ .","Merging the particles at $\\mathcal {T}_{j:l}$ from its children at $\\mathcal {T}_{j:k-1} \\in \\mathcal {T}$ and $\\mathcal {T}_{k:l} \\in \\mathcal {T}$ amounts to correlating the two sets of samples while roughly preserving their marginal distributions.","The weight of the merged sample $\\tilde{x}^{(i)}_{j:l} = (\\tilde{x}_{j:k-1}^{({i})}, \\tilde{x}_{k:l}^{({i})})$ in Equation (REF ) becomes: $\\hat{w}^{(i)}_{j:l} = \\tilde{w}^{(i)}_{j:l} \\frac{\\hat{p}(\\tilde{x}^{({i})}_{k-1}|y_{0:k-1})}{ \\hat{p}(\\tilde{x}^{({i})}_{k-1}|y_{0:T}) \\hat{p}(\\tilde{x}^{({i})}_{k}|y_{0:k})} p(\\tilde{x}^{({i})}_{k} | \\tilde{x}^{({i})}_{k-1}) p(y_{k}|\\tilde{x}^{({i})}_{k}).$ Applying TPS-ES demands the constructions of $\\lbrace \\hat{p}(x_{j} | y_{0:j}) \\rbrace _{j = 0, \\ldots , T}$ and $\\lbrace \\hat{p}(x_{j} | y_{0:T}) \\rbrace _{j = 0, \\ldots , T}$ in advance.","The new weight formula in Equation (REF ) additionally incorporates the ratio between the estimated filtering and smoothing densities of $x_{k-1}$ compared with Equation (REF ).","TPS-ES exhibits a sound property regarding the Kullback–Leibler divergence discussed in Section REF .","Given the target distribution $f_{j:l}(x_{j:l}) = \\hat{p}(x_{j:l}|y_{0:T}) $ estimating $p(x_{j:l}|y_{0:T}) $ at $\\mathcal {T}_{j:l}$ , the proposal $h_{j:l}(x_{j:l}) = f_{j:k-1}(x_{j:k-1}) f_{k:l}(x_{k:l})$ estimates $p(x_{j:k-1} | y_{0:T}) p(x_{k:l} | y_{0:T})$ .","We notice $p(x_{j:k-1} | y_{0:T})$ and $p(x_{k:l} | y_{0:T})$ are the marginal distributions, and their product forms a proposal attaining the minimum KL divergence from $p(x_{j:l}|y_{0:T})$ .","Hence, what the proposal density $h_{j:l}(x_{j:l})$ estimates has a minimum KL divergence from the smoothing density that our target distribution $f_{j:l}(x_{j:l})$ estimates.","Moreover, TPS-ES can be practically useful in some extreme models whereas the empirical marginal densities from other Monte Carlo smoothing algorithms may miss modes caused by the poor proposals.","Since TPS-ES leaves the marginal distributions of all random variables roughly invariant at all levels of the tree, we can diagnose each importance sampling step by inspecting the empirical marginals of the corresponding variables.","If there is a substantial difference between the empirical marginal distributions, we need to examine the combination step."],["Initial sampling distribution at leaf nodes","We illustrate the constructions of the univariate distributions $\\lbrace \\hat{p}(x_{j} | y_{1:j})\\rbrace _{j = 0, \\ldots , T}$ and $\\lbrace \\hat{p}(x_{j} | y_{0:T})_{j = 0, \\ldots ,T}\\rbrace $ mentioned in Section REF and Section REF , which are used in the initial sampling distributions at the leaf nodes.","In general, the solutions of the filtering and smoothing distribution of a HMM are analytically intractable and need to be estimated from Monte Carlo samples with some exceptions including linear Gaussian and discrete HMMs.","We aim to generate a probability density $\\hat{f}$ estimating a target density $f$ given the weighted samples $\\lbrace x_{i}, w_{i}\\rbrace _{i = 1}^{n}$ from $f$ .","In the context of $f$ being a filtering or smoothing distribution, we can obtain the weighted samples by running a filtering algorithm or a smoothing algorithm.","We are not interested in the empirical distribution since it is discrete and generally does not cover the full support of the random variable of interest.","We first consider some parametric approaches.","We can fit the data with some common probability distributions including a normal distribution and Student's $t$ -distribution.","We can also accommodate a mixture model to fit multiple modes of the target densities.","The parameters of the distributions can be estimated in various ways including moment matching, maximum likelihood method and EM algorithm.","The parametric approaches are reasonably quick and simple.","For instance, assuming a Gaussian distribution requires the evaluation of the mean and variance and can be easily obtained from the samples using moment matching.","The generation and evaluation of densities of the new particles are straightforward and fast to implement.","Nevertheless, the target distribution may not be well approximated under the parametric assumption.","Alternatively, we can employ some non-parametric approaches for instance, a kernel density estimator (KDS).","We need to select the type of kernels and bandwidth in advance.","The complexity of generating $N$ new samples is $O\\big (\\log (n)N\\big )$ and the evaluation of the densities is more computationally expensive with complexity $O(nN)$ .","We propose another non-parametric approximation method using piecewise constant functions with a lower computational effort than a KDS.","We first build a uniform grid consisting of the points $x_{1} < x_{2} < \\ldots < x_{n}$ with densities $d_{1}, \\ldots , d_{n}$ estimated by a KDS such that $x_{i+1} - x_{i} = \\Delta > 0$ for $i = 1, \\ldots , n$ .","The resulting probability density function formed by these grid points using piecewise constant functions is: $f(x) = \\sum _{i = 1}^{n} \\mathbb {1}_{x \\in [x_{i} - \\Delta /2, x_{i} + \\Delta /2)} d_{i}.$ The evaluation of the sample densities reduces significantly from $O(nN)$ to $O\\big (N \\big )$ compared to a KDS.","Such probability density functions using piecewise constant functions have several disadvantages though enjoy a fast computation of estimated densities.","Firstly, the estimator is biased since the proposal density generally does not cover the full support of the target density.","Moreover, in TPS-ES, if the estimated filtering and smoothing distributions are both generated using the piecewise constant functions with different samples, there is no guarantee their densities have the same support, which may cause zero or infinite weight in Equation (REF ).","To avoid this, we consider the mixture probability distributions using the piecewise constant functions accommodating the samples from both the filtering and smoothing distributions.","Assume at time step $j$ , the first uniform grid consists of the points $x^{f}_{1} < x^{f}_{2} < \\ldots < x^{f}_{n^{f}}$ such that $x^{f}_{i+1} - x^{f}_{i} = \\Delta ^{f}$ for $i = 1, \\ldots , n^{f}$ with estimated filtering densities $d^{f}_{1}, \\ldots , d^{f}_{n}$ from a KDS and assume the second uniform grid consists of the points $x^{s}_{1} < x^{s}_{2} < \\ldots < x^{s}_{n^{s}}$ such that $x^{s}_{i+1} - x^{s}_{i} = \\Delta ^{s}$ for $i = 1, \\ldots , n^{s}$ with estimated smoothing densities $d^{s}_{1}, \\ldots , d^{s}_{n}$ from another KDS.","Then the resulting estimated filtering density $\\hat{p}(x | y_{0:j})$ is given by $\\hat{p}(x | y_{0:j}) = \\alpha ^{f} \\sum _{i = 1}^{n^{f}} \\mathbb {1}_{x \\in [x^{f}_{i} - \\Delta ^{f}/2, x^{f}_{i} + \\Delta ^{f}/2)} d^{f}_{i} + (1-\\alpha ^{f}) \\sum _{i = 1}^{n^{s}} \\mathbb {1}_{x \\in [x^{s}_{i} - \\Delta ^{s}/2, x^{s}_{i} + \\Delta ^{s}/2)} d^{s}_{i},$ where $0 < \\alpha ^{f} < 1$ .","Similarly, the estimated smoothing density $\\hat{p}(x | y_{0:T})$ is given by $\\hat{p}(x | y_{0:T}) = \\alpha ^{s} \\sum _{i = 1}^{n^{s}} \\mathbb {1}_{x \\in [x^{s}_{i} - \\Delta ^{s}/2, x^{s}_{i} + \\Delta ^{s}/2)} d^{s}_{i} + (1-\\alpha ^{s}) \\sum _{i = 1}^{n^{f}} \\mathbb {1}_{x \\in [x^{f}_{i} - \\Delta ^{f}/2, x^{f}_{i} + \\Delta ^{f}/2)} d^{f}_{i},$ where $0 < \\alpha ^{s} < 1$ .","We have no conclusion of the values of $\\alpha ^{f}$ and $\\alpha ^{s}$ so far and choose them with values close to 1.","The resulting grid with the set of points $\\lbrace x^{f}_{1}, x^{f}_{2}, \\ldots , x^{f}_{n^{f}}, x^{s}_{1}, x^{s}_{2}, \\ldots , x^{s}_{n^{s}}\\rbrace $ is generally not uniform, but we ensure the estimated filtering and smoothing densities have the same support, though still finite."],["Simulations","We conduct simulations in a linear Gaussian HMM and a non-linear non-Gaussian HMM in this section.","We implement TPS-EF and other smoothing algorithms with roughly the same computational effort.","In the second example, we further compare TPS-EF and TPS-ES."],["Gaussian Linear Model","We consider a simple linear Gaussian HMM similar to [8].","$X_{t} &= 0.8 X_{t-1} + V_{t} ~~~&& t = 1, \\ldots , T,\\\\Y_{t} &= X_{t} + W_{t} ~~~&& t = 0, \\ldots , T.$ where $T = 127$ , where $X_0,V_1,\\dots ,V_T,W_0,\\dots ,W_T$ are independent with $X_{0} \\sim \\mathcal {N}(0,1)$ , $V_{t}\\sim N(0,1)$ , $W_{t}\\sim N(0,1)$ .","We implement the following smoothing algorithms.","We run TPS using normal distributions as the initial sampling distributions (TPS-N) whose means and variances are estimated using moment matching from the samples of a bootstrap particle filter.","The choice of a normal distribution is motivated by the fact that in this case the true smoothing distribution is a normal distribution.","We also implement the tree-based particle smoothing algorithm suggest by [17] (TPS-L), the Rauch–Tung–Striebel smoother (RTSs) [21] yielding the closed-form solutions, the bootstrap particle filter (BPF) which updates the entire history of the particles in each step, the forward filtering backward smoothing algorithm (FFBSm) [8] and the forward filtering backward simulation (FFBSi) [12].","We have implemented the above methods in R ourselves.","We set the required sample size $N = 10000$ in TPS-N as a benchmark and denote $n = 10000$ the number of samples pre-generated from a bootstrap particle filter in FFBSm, FFBSi, TPS-N and TPS-L. We adjust the number of particles in other algorithms to roughly keep the same running time.","As the implementations are not deterministic, we allow a 10% error regarding the running time for the rest of the algorithms compared to TPS-N. We run each algorithm $M = 500$ times with the same set of observations $\\lbrace y_{t}\\rbrace _{t = 0}^{127}$ .","As a criterion for comparison, we define the mean square error of means (MSEm) and variances (MSEv) in the $m$ th simulation: $\\text{MSEm}_{m} &=& \\frac{1}{T+1} \\sum ^{T}_{t = 0} \\big ( \\widehat{\\mathbb {E}}^{m}[X_{t}| Y_{0:T}] - \\mathbb {E}[X_{t}| Y_{0:T}]\\big )^{2}, \\\\\\text{MSEv}_{m} &=& \\frac{1}{T+1} \\sum ^{T}_{t = 0} \\big ( \\widehat{\\text{Var}}^{m}[X_{t}| Y_{0:T}] - \\text{Var}[X_{t}| Y_{0:T}]\\big )^{2}, \\\\$ where $ \\widehat{\\mathbb {E}}[X^{m}_{t}| Y_{0:T}]$ and $\\widehat{\\text{Var}}[x^{m}| y_{0:T}] $ are the Monte Carlo estimates of the mean and variance of the smoothing distribution at time step $t$ in the $m$ th simulation.","$\\mathbb {E}[X_{t}| Y_{0:T}]$ and $\\text{Var}[X_{t}| Y_{0:T}]$ are the true smoothing means and variances from a Rauch–Tung–Striebel smoother [21].","The simulation results are shown in Table REF .","When $N = n$ , the two tree-based sampling algorithms: TPS-L and TPS-N enjoy the same complexity $O(N)$ as BPF, and generate far more particles than FFBSm and FFBSi with quadratic complexities.","TPS-L has the smallest mean of MSEm and MSEv followed by TPS-N, which outperform FFBSm and FFBSi significantly in terms of MSEm.","Table: Simulation errors in the linear model"],["Non-linear Model","We consider a well-known non-linear model [13], [1]: $X_{t} &= \\frac{1}{2} X_{t-1} + 25 \\frac{X_{t-1}}{1 + X_{t-1}^2} + 8 \\cos (1.2 t) + V_{t},~~&&t = 1,2, \\ldots , T,\\\\Y_{t} &= \\frac{X^{2}_{t}}{20} + W_{t},~~&&t = 0,2, \\ldots , T,$ where $T = 511$ , where $X_0, V_1,...,V_T, W_0,...W_T$ are independent with $X_{0} \\sim \\mathcal {N}(0, 1)$ , $V_{t} \\sim \\mathcal {N}(0, \\tau ^2)$ and $W_{t} \\sim \\mathcal {N}(0,\\sigma ^2).$ We run the same algorithms BPF, FFBSm, FFBSi and TPS-L as in Section REF .","In TPS-EF, we use piecewise constant functions defined in Equation (REF ) for the approximation of the initial sampling distributions.","We call the algorithm TPS-EFP and set $N = n = 10000$ as a benchmark.","As before, we correspondingly adjust the sample sizes in other algorithms to achieve roughly the same computational effort.","We calculate the mean and standard deviation of the MSE of means (MSEm) in $M = 500$ simulations with the same set of observations.","Given no closed-form solutions to the true smoothing distributions, we apply a discrete analogue to the distributions of the initial hidden state $p_{0}(x_{0})$ and the transition distributions $\\lbrace p(x_{t+1}|x_{t})\\rbrace _{t = 0, \\ldots , 126}$ .","We then approximate the smoothing distributions of the original HMM using the solutions of the discrete-space HMM.","The MSEm of the $m$ th simulation in the non-linear model is defined as: $ \\text{MSEm}_{m} = \\frac{1}{T+1} \\sum ^{T}_{t = 0} \\big (\\widehat{\\mathbb {E}}^{m}[X_{t}| Y_{0:T}] - \\mathbb {E} (\\hat{X}_{t}\\big | y_{0:T}) )^{2}, $ where $\\mathbb {E} (\\hat{X}_{t}\\big | y_{0:T})$ is the mean of the smoothing distribution at time step $t$ of the discrete-space HMM.","We additionally perform Kolmogorov–Smirnov test [19] which measures a distance between the empirical distribution and the target probability distribution.","In the context of the smoothing problem in a non-linear hidden Markov model, the Kolmogorov–Smirnov statistic can be defined as $ D = \\sup _{x} | F^{(t)}_{1,N}(x) - F^{(t)}_{2}(x) |, $ where $F^{(t)}_{1,N}$ is the empirical cumulative function generated by $N$ samples at the time step $t$ from a smoothing algorithm and $F^{(t)}_{2}$ is the cumulative distribution function at time step $t$ of the smoothing distribution from a discrete-space HMM derived from the true model.","We denote KS$_{m}$ to be the sum of the KS statistic of all time steps in the $m$ th simulation.","Table: Simulation errors in the non-linear modelThe simulation results with different values of $\\tau $ and $\\sigma $ are shown in Table REF .","In the first two situations, TPS-L shows the largest error and KS statistic, especially when $\\tau = 1$ and $\\sigma = 5$ .","This can be explained by the poor proposal from the initial sampling distribution constructed by the algorithm.","We examine this by plotting the cumulative distribution function (CDF) of the initial sampling distribution $f_{j}$ in TPS-L, the filtering distribution $p(x_{j}| y_{0:j})$ and the marginal smoothing distribution $p(x_{j}| y_{0:T})$ at a particular time step when $j = 271$ .","In Figure REF , the CDF of the initial sampling are far more dissimilar to the marginal smoothing distribution than the filtering one, which contributes to very ineffective importance sampling steps during the built-up of the tree.","Figure: CDF of the smoothing, filtering and initial sampling distribution at time step j=271j = 271 of TPS-L in the non-linear model when τ=1,σ=5\\tau = 1, \\sigma = 5.Other algorithms provide different results in the three parameter settings.","When $\\tau = 1, \\sigma = 1$ , TPS-EFP shows much smaller MSEm and KS followed by BPF.","BPF however has the largest mean of KS.","When $\\tau = 1, \\sigma = 5$ , TPS-EFP has a larger mean of MSEm than BPF.","In terms of the KS statistic, TPS-EFP outperforms other smoothing algorithms.","When $\\tau = 5, \\sigma = 1$ , TPS-EFP and TPS-L produce dominant results with vastly smaller MSEm.","They also exhibit the smallest mean of KS among the smoothing algorithms whereas the BPF gives the largest result though generating the most samples.","To conclude, TPS-EFP and TPS-L perform well when the ratio between the standard deviation in the transition and emission density, i.e.","when $\\tau / \\sigma $ is large.","TPS-EFP has a more stable and appreciable performance, which provides low MSEm and consistently the smallest KS among the five smoothing algorithms.","In contrast, the result of TPS-L may be misleading due to its instability.","BPF works well regarding MSEm in some situations, but poorly in terms of Kolmogorov–Smirnov statistic.","FFBSm and FFBSi produces less accurate results due to higher computational complexity."],["Comparing TPS-EF and TPS-ES in the non-linear model","In this section, we conduct simulations in the same non-linear model using tree-based particle smoothing algorithm with estimated filtering (TPS-EF) and smoothing (TPS-ES) distributions as the intermediate target distributions.","As TPS-ES is not a good competitor given a relatively small sample size in Section REF , we compare its performance with TPS-EF with more computational budget.","We demonstrate the implementations of the two algorithms.","We apply the same smoothing algorithm TPS-EFP as described in Section REF which utilises the piecewise constant functions to estimate the filtering distributions.","As TPS-ES requires the estimated smoothing distributions as the initial sampling distributions, we achieve this by using piecewise constant functions for the estimation based on the samples from an initial run of TPS-EFP and thus call the algorithm TPS-ESP.","We specify the parameters in the simulations of TPS-EFP and TPS-ESP.","We denote the sample size by $N$ .","We set the parameters $\\alpha ^{s} = \\alpha ^{f} = 0.95$ appeared in Equation (REF ) and (REF ).","In TPS-EFP and TPS-ESP, the estimated filtering distributions are both constructed from $n$ samples from the particle filters.","Additionally, in TPS-ESP, the estimated smoothing distributions are constructed from TPS-EFP with $n^{\\prime }$ samples.","We run TPS-ESP in two different situations: The first one has the same sample size $N$ as TPS-EFP and requires more computational effort to estimate the initial sampling distributions based on $n^{\\prime }$ Monte Carlo samples.","The second one has roughly the same computational effort as TPS-EFP which generates fewer Monte Carlo samples for the estimation of the initial sampling distributions and the target samples.","We compare TPS-EFP and TPS-ESP with respect to the mean square error and Kolmogorov–Smirnov statistic defined in Section REF .","We run TPS-EFP and TPS-ESP for $M = 200$ times with different values of $\\tau $ and $\\sigma $ whose results are shown in Table REF .","TPS-ESP has an evident improvement of the Kolmogorov–Smirnov (KS) statistic in most situations and the comparisons between MSEm vary.","The MSEm of TPS-ESP always decreases when generating the same number of samples as TPS-EFP.","However, TPS-ESP does not provide convinced results under roughly the same computational effort.","Overall, the performance of TPS-ESP depends on the computational budget.","Given the same sample size as in TPS-EFP, TPS-ESP can potentially decrease both MSEm and KS statistic.","This may not be true when the algorithm is kept the same overall effort as TPS-EFP.","Table: Simulation errors between TPS-EF and TPS-ES in the non-linear model"],["Conclusion","This article introduces a Monte Carlo sampling method we call TPS built on the D&C SMC [17] to estimate the joint smoothing distribution $p(x_{0:T}|y_{0:T})$ in a hidden Markov model.","The method decomposes the model into sub-models with intermediate target distributions using a binary tree structure.","TPS samples independently from the leaves of the tree and gradually merges and resamples to target the new distributions upon the auxiliary tree.","We propose one generic way of constructing a binary tree which sequentially splits the joint random variables $X_{0:T}$ .","Furthermore, we discuss the sampling procedure of the target samples at a non-leave node by combining the samples from its children using importance sampling.","The computational effort is adjustable with a possible reduction to a linear effort with respect to the sample size.","Using the above settings, we investigate three algorithms with different types of intermediate target distributions at the non-root nodes.","TPS-L [17] constructs intermediate target distributions conditional on the observations from the same time interval as the target variables and imposes an uninformative prior.","TPS-L is very simple to implement with no additional tuning algorithms.","The algorithm is at the risk of providing very poor initial sampling distribution based on little information from the observations.","TPS-EF employs intermediate target distributions estimating of the (joint) filtering distributions which conditions on the observations up to the last time step in the target variable.","It is straightforward for implementation with an initial run of a filtering algorithm.","Nevertheless, the proposal in the importance sampling step may still not be satisfactory when the marginal filtering and smoothing distributions are vastly different.","TPS-ES builds the distributions estimating of the (joint) smoothing distributions which conditions on all the observations.","It roughly retains the marginal smoothing distributions from the intermediate target distributions at all levels of the auxiliary tree despite its more intensive computations.","We further propose the constructions of the estimated filtering and smoothing distributions based on the Monte Carlo samples.","Considering both accuracy and computational effort, we recommend parametric approaches such as normal assumptions in a linear Gaussian model and non-parametric approaches such as using piecewise constant functions in a non-linear model.","In the simulation studies, TPS-L has the smallest error in the linear model, but very unstable results in the different settings of the non-linear model.","TPS-EF exhibits more desirable simulation outcomes.","It is computationally less expensive than the most smoothing algorithms with quadratic complexity.","It also produces the smallest mean square errors in the linear Gaussian model and consistently the smallest average Kolmogorov–Smirnov statistic in different situations under the non-linear model.","In particular, it outperforms other algorithms substantially when the variance of the transition density is much larger than the emission density.","TPS-ES, however, has a better approximation of the smoothing distribution with respect to the Kolmogorov–Smirnov statistic compared with TPS-EF at the cost of an additional run of a smoothing algorithm.","To conclude, TPS with two proposed choices of the intermediate target distribution presents a new approach of addressing the smoothing problem which shows the following advantages: We have flexibilities of choosing and constructing the intermediate target distributions, which can potentially produce better proposals in the importance sampling steps.","TPS can escape from the quadratic complexity with respect to the sample size computationally, and produce more particles and accurate simulation results than some smoothing algorithms.","Nevertheless, its performance depends on the implementation of other filtering or smoothing algorithms and the estimation of the target distributions.","Due to its flexible and relatively fast implementations with stable and comparable simulation results, we regard it as a competitor with other smoothing algorithms."],["Proof of Theorem ","By Jensen's inequality, $&& \\int _{\\mathbb {R}^{n_{1}}} f_{1}(\\mathbf {x_{1}}) \\log \\big ( f_{1}(\\mathbf {x_{1}}) \\big ) \\mathrm {d} \\mathbf {x_{1}} - \\int _{\\mathbb {R}^{n_{1}}} f_{1}(\\mathbf {x_{1}}) \\log \\big ( h_{1}(\\mathbf {x_{1}}) \\big ) \\mathrm {d}\\mathbf {x_{1}}\\\\ &=& \\int _{\\mathbb {R}^{n_{1}}} f_{1} (\\mathbf {x_{1}}) \\log \\bigg ( \\frac{f_{1}(\\mathbf {x_{1}})}{h_{1} (\\mathbf {x_{1}})} \\bigg ) \\mathrm {d} \\mathbf {x_{1}}= \\mathbb {E} \\bigg [ \\log \\bigg ( \\frac{f_{1}(\\mathbf {X_{1}})}{ h_{1}(\\mathbf {X_{1}}) } \\bigg ) \\bigg ] = \\mathbb {E} \\bigg [ - \\log \\bigg ( \\frac{h_{1}(\\mathbf {X_{1}})}{ f_{1}(\\mathbf {X_{1}}) } \\bigg ) \\bigg ] \\\\&\\ge & - \\log \\bigg \\lbrace \\mathbb {E} \\bigg [ \\frac{h_{1}(\\mathbf {X_{1}})}{f_{1}(\\mathbf {X_{1}})} \\bigg ] \\bigg \\rbrace = 0.","$ Using this and the definition of marginal distribution, $&& \\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_1}} f(\\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\big ( f_{1}(\\mathbf {x_{1}}) \\big ) \\mathrm {d} \\mathbf {x_{1}} \\mathrm {d} \\mathbf {x_{2}}= \\int _{\\mathbb {R}^{n_{1}}} f_{1}(\\mathbf {x_{1}}) \\log \\big ( f_{1}(\\mathbf {x_{1}}) \\big ) \\mathrm {d} \\mathbf {x_{1}} \\nonumber \\\\&\\ge & \\int _{\\mathbb {R}^{n_{1}}} f_{1}(\\mathbf {x_{1}}) \\log \\big ( h_{1}(\\mathbf {x_{1}}) \\big ) \\mathrm {d} \\mathbf {x_{1}} = \\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_1}} f(\\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\big ( h_{1}(\\mathbf {x_{1}}) \\big ) \\mathrm {d} \\mathbf {x_{1}}\\mathrm {d} \\mathbf {x_{2}}.","\\nonumber \\\\$ Similarly, $\\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_{1}}} f(\\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\big ( f_{2}(\\mathbf {x_{2}}) \\big ) \\mathrm {d} \\mathbf {x_{1}} \\mathrm {d} \\mathbf {x_{2}} \\ge \\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_{1}}} f(\\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\big ( h_{2}(\\mathbf {x_{2}}) \\big ) \\mathrm {d} \\mathbf {x_{1}}\\mathrm {d} \\mathbf {x_{2}}.$ Multiplying (REF ) and (REF ) by -1 and adding them, we have $\\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_{1}}} f( \\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\bigg ( \\frac{1}{f_{1}(\\mathbf {x_{1}}) f_{2}(\\mathbf {x_{2}})} \\bigg ) \\mathrm {d} \\mathbf {x_{1}} \\mathrm {d}\\mathbf {x_{2}} \\le \\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_{1}}} f( \\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\bigg (\\frac{1}{h_{1}(\\mathbf {x_{1}}) h_{2}(\\mathbf {x_{2}})} \\bigg ) \\mathrm {d} \\mathbf {x_{1}} \\mathrm {d} \\mathbf {x_{2}}.$ Adding $ \\displaystyle \\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_{1}}} f( \\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\big ( f(\\mathbf {x_{1}}, \\mathbf {x_{2}}) \\big ) \\mathrm {d} \\mathbf {x_{1}} \\mathrm {d} \\mathbf {x_{2}} $ to both sides yields the result."]],"string":"[\n [\n \"Tree-based Particle Smoothing Algorithms in a Hidden Markov Model\"\n ],\n [\n \"Abstract We provide a new strategy built on the divide-and-conquer approach by Lindsten et al.\",\n \"(2017) to investigate the smoothing problem in a hidden Markov model.\",\n \"We employ this approach to decompose a hidden Markov model into sub-models with intermediate target distributions based on an auxiliary tree structure and produce independent samples from the sub-models at the leaf nodes towards the original model of interest at the root.\",\n \"We review the target distribution in the sub-models suggested by Lindsten et al.\",\n \"and propose two new classes of target distributions, which are the estimates of the (joint) filtering distributions and the (joint) smoothing distributions.\",\n \"The first proposed type is straightforwardly constructible by running a filtering algorithm in advance.\",\n \"The algorithm using the second type of target distributions has an advantage of roughly retaining the marginals of all random variables invariant at all levels of the tree at the cost of approximating the marginal smoothing distributions in advance.\",\n \"We further propose the constructions of these target distributions using pre-generated Monte Carlo samples.\",\n \"We show empirically the algorithms with the proposed intermediate target distributions give stable and comparable results as the conventional smoothing methods in a linear Gaussian model and a non-linear model.\"\n ],\n [\n \"Introduction\",\n \"A hidden Markov model (HMM) is a discrete-time stochastic process $\\\\lbrace X_{t}, Y_{t}\\\\rbrace _{t \\\\ge 0}$ where $\\\\lbrace X_{t}\\\\rbrace _{t \\\\ge 0}$ is an unobserved Markov chain.\",\n \"We only have access to $\\\\lbrace Y_{t}\\\\rbrace $ whose distribution depends on $\\\\lbrace X_{t}\\\\rbrace $ .\",\n \"We make the following assumptions in the entire article: The densities of the initial state $X_{0}$ , the transition density $X_{t+1}$ given $X_{t} = x_{t}$ and the emission density $Y_{t}$ given $X_{t} = x_{t}$ taken with respect to some dominating measure exist and are denoted as follows: $X_{0} &\\\\sim p_{0}(x_{0}) &&\\\\\\\\X_{t+1}| \\\\lbrace X_{t} = x_{t} \\\\rbrace &\\\\sim p(x_{t+1} | x_{t}) && \\\\text{~~for } t = 0, \\\\ldots , T-1,\\\\\\\\Y_{t} | \\\\lbrace X_{t} = x_{t}\\\\rbrace &\\\\sim p(y_{t}|x_{t}) && \\\\text{~~for } t = 0, \\\\ldots , T,$ where $T$ is the final time step of the process.\",\n \"We are interested in the (marginal) smoothing distributions $\\\\lbrace p(x_{t}|y_{0:T})\\\\rbrace _{t = 0, \\\\ldots , T}$ or the joint smoothing distribution $p(x_{0:T}|y_{0:T})$ where $x_{0:T}$ and $y_{0:T}$ are abbreviations of $(x_{0}, \\\\ldots , x_{T})$ and $(y_{0}, \\\\ldots , y_{T})$ , respectively.\",\n \"Exact solutions are available for linear Gaussian HMM using a Rauch–Tung–Striebel smoother (RTSs) [21] and in a HMM with finite-space Markov chains [3].\",\n \"In most other cases, the smoothing distributions are not analytically tractable.\",\n \"A large body of work uses Monte Carlo methods to approximate the smoothing distributions $\\\\lbrace p(x_{t}|y_{0:T})\\\\rbrace _{t = 0, \\\\ldots , T}$ or the joint smoothing distribution $p(x_{0:T}|y_{0:T})$ .\",\n \"Sequential Monte Carlo (SMC) methods [7] are commonly used to sequentially update the filtering distributions $\\\\lbrace p(x_{t}|y_{0:t})\\\\rbrace _{t = 0, \\\\ldots , T}$ .\",\n \"SMC can in principle be used to estimate the joint smoothing density $p(x_{0:T}|y_{0:T})$ by updating the entire history of the random samples in each resampling step.\",\n \"However, the performance can be poor, as path degeneracy will occur in many settings [2].\",\n \"Advanced sequential Monte Carlo methods with desirable theoretical and practical results have been developed in recent years including sequential Quasi-Monte Carlo (SQMC) [11], divide-and-conquer sequential Monte Carlo (D&C SMC) [17], multilevel sequential Monte Carlo (MSMC) [4] and variational sequential Monte Carlo (VSMC) [20].\",\n \"Other smoothing algorithms have been suggested previously.\",\n \"[8] develop the forward filtering backward smoothing algorithm (FFBSm) for sampling from $\\\\lbrace p(x_{t}|y_{0:T})\\\\rbrace _{t = 0, \\\\ldots , T}$ based on the formula proposed by [14].\",\n \"[12] propose the forward filtering backward simulation algorithm (FFBSi) which generates samples from the joint smoothing distribution $p(x_{0:T}|y_{0:T})$ .\",\n \"[5] propose a two-filter smoother (TFS) which employs a standard forward particle filter and a backward information filter to sample from $\\\\lbrace p(x_{t}|y_{0:T})\\\\rbrace _{t = 0, \\\\ldots , T}$ .\",\n \"Typically, these algorithms have quadratic complexities in $N$ for generating $N$ samples.\",\n \"[9] and [16] propose two smoothing algorithms with lower computational complexity, but their methods do not provide unbiased estimates.\",\n \"In this article, we suggest using the divide-and-conquer sequential Monte Carlo (D&C SMC) [17] approach to address the smoothing problem.\",\n \"The D&C SMC algorithm performs statistical inferences in probabilistic graphical models.\",\n \"It splits the random variables of the target distribution into multiple levels of disjoint sets based upon an auxiliary tree $\\\\mathcal {T}$ .\",\n \"An intermediate target distribution needs to be assigned to each set of random variables yielding sub-models for each non-leaf node.\",\n \"The choice of these intermediate target distributions is key for a good overall performance of the algorithm.\",\n \"By sampling independently from the leaf nodes and gradually propagating, merging and resampling from the leaf nodes to the root, the D&C SMC algorithm eventually produces samples from the target distribution.\",\n \"The merging step involves importance sampling.\",\n \"Using the idea of D&C SMC, we aim to estimate the joint smoothing distribution $p(x_{0:T}|y_{0:T})$ and thus call the algorithm: `tree-based particle smoothing algorithm' (TPS).\",\n \"The key differences between TPS and other smoothing algorithms lie in its non-sequential and more adaptive merging step of the samples.\",\n \"Our main contribution is the proposition and investigation of three classes of intermediate target distributions to be used in the algorithm.\",\n \"We denote a leaf node corresponding to a single random variable $X_{j}$ by $\\\\mathcal {T}_{j} \\\\in \\\\mathcal {T}$ and a non-leaf node corresponding to the random variables $X_{j:l}$ by $\\\\mathcal {T}_{j:l} \\\\in \\\\mathcal {T} (j < l)$ .\",\n \"The first class advised by [17] has the density proportional to the product of all transition and emission densities associated to the target variable $X_{j}$ (resp.\",\n \"$X_{j:l}$ ) in the sub-model.\",\n \"This is equivalent to the unnormalised likelihood of a new HMM starting at time $j$ (resp.\",\n \"from time $j$ to $l$ ) given the observations of the same time interval with an uninformative prior of $X_{j}$ if $j \\\\ne 0$ .\",\n \"The second class uses an estimate of the filtering distribution $p(x_{j} | y_{0:j})$ at $\\\\mathcal {T}_{j} \\\\in \\\\mathcal {T}$ and an estimate of the joint filtering distribution $p(x_{j:l} | y_{0:l})$ at $\\\\mathcal {T}_{j:l} \\\\in \\\\mathcal {T}$ .\",\n \"Working with this estimate involves tuning a preliminary particle filter.\",\n \"The third class uses estimates of the marginal smoothing distribution $p(x_{j} | y_{0:T})$ at $\\\\mathcal {T}_{j} \\\\in \\\\mathcal {T}$ and of the joint smoothing distribution $p(x_{j:l} | y_{0:T})$ at $\\\\mathcal {T}_{j:l} \\\\in \\\\mathcal {T}$ .\",\n \"We will see that this class of immediate distributions is optimal in a certain sense.\",\n \"Furthermore, under this construction, we approximately retain the marginal distribution of all single random variables $\\\\lbrace X_{j}\\\\rbrace _{j = 0}^{T}$ invariant as the marginal smoothing distributions $\\\\lbrace p(x_{j} | y_{0:T}) \\\\rbrace _{j = 0}^{T}$ at every level of the tree.\",\n \"The price of implementing TPS using the second class of intermediate target distributions relies on both the estimates of the filtering and the (marginal) smoothing distributions, but not necessarily the joint smoothing distribution.\",\n \"We then propose some parametric and non-parametric approaches to construct these intermediate distributions based on the pre-generated Monte Carlo samples considering both efficiency and accuracy.\",\n \"The article is structured as follows.\",\n \"We first describe the divide-and-conquer approach for particle smoothing in Section .\",\n \"We discuss the intermediate target distributions and the constructions of the initial sampling distributions at the leaf nodes in Section .\",\n \"In Section , we conduct simulation studies in a linear Gaussian and non-linear non-Gaussian HMM to compare TPS with other smoothing algorithms.\",\n \"The article finishes with a discussion in Section .\"\n ],\n [\n \"Tree-based Particle Smoothing Algorithm (TPS)\",\n \"This section outlines an algorithm we call `tree-based particle smoothing algorithm' (TPS).\",\n \"[17] describe the construction of an auxiliary tree for general probabilistic graphical models.\",\n \"We demonstrate a unique construction of an auxiliary binary tree from a HMM bearing intermediate target distributions specified at each node.\",\n \"We then illustrate the sampling procedure for the target distributions at the nodes.\",\n \"We present an algorithm which can be applied recursively from the leaf nodes towards the root and yet generate the target samples.\"\n ],\n [\n \"Construction of an auxiliary tree\",\n \"TPS splits a HMM into sub-models based upon a binary tree decomposition.\",\n \"It first divides the random variables $X_{0:T}$ into two disjoint subsets and recursively apply binary splits to the resulting two subsets until the resulting subset consists of only a single random variable.\",\n \"Each generated subset corresponds to a tree node and is assigned an intermediate target distribution.\",\n \"The root characterises the complete model with the target distribution $p(x_{0:T}|y_{0:T})$ .\",\n \"Initial samples are generated at the leaf nodes, independent between leaves.\",\n \"Theses samples are recursively merged using importance sampling until the root of the tree is reached.\",\n \"We propose one intuitive way of implementing the binary splits which ensures that the left subtree is always a complete binary tree and contains at least as many nodes as the right subtree.\",\n \"We split a non-leaf node with the variables $X_{j:l}$ where $0 \\\\le j < l \\\\le T$ , into two children $\\\\mathcal {T}_{j:k-1}$ and $\\\\mathcal {T}_{k:l}$ with the random variables $X_{j:k-1}$ and $X_{k:l}$ , where $k = j + 2^{p},$ and $p = \\\\lceil \\\\frac{\\\\log (l-j+1)}{\\\\log 2}\\\\rceil - 1.$ The auxiliary tree when $T = 5$ is shown in Figure REF .\",\n \"This construction has several advantages: The random variables within each node have consecutive time indices.\",\n \"The left subtree is also a complete binary tree of $2^{\\\\lceil \\\\frac{\\\\log (T+1)}{\\\\log 2}\\\\rceil }$ leave nodes.\",\n \"$\\\\lbrace y_{T+1}, y_{T+2}, \\\\ldots \\\\rbrace $ become available, as samples from the complete subtree would not need to be updated.\",\n \"Moreover, the tree has a height of $\\\\big (\\\\lceil \\\\frac{\\\\log (T+1)}{\\\\log 2}\\\\rceil + 1\\\\big )$ levels, which implies a maximum number of $\\\\lceil \\\\frac{\\\\log (T+1)}{\\\\log 2}\\\\rceil $ updates of the samples corresponding to a single random variable with different target distributions at different levels of the tree.\",\n \"Usually, more updates potentially indicate more resampling steps, which may cause more serious degeneracy problems.\",\n \"In Figure REF , the samples corresponding to $X_{0}, \\\\ldots , X_{3}$ need to be updated three times from the leave nodes and those of $X_{4}, X_{5}$ need to be updated twice.\",\n \"When running a bootstrap particle filter to solve the smoothing problem, the samples at time step $t = 0$ need to be updated $T$ times and thus the maximum number of the updates become $T$ , which is no less than $ \\\\lceil \\\\frac{\\\\log (T+1)}{\\\\log 2}\\\\rceil $ .\",\n \"[17] also propose a general way of constructing the auxiliary tree in a self-similar model family, where a HMM belongs to.\",\n \"Their construction in the context of a HMM may not be identical to ours with no restriction on the choice of the cutting point.\",\n \"Figure: An auxiliary binary tree consisting of random variables when T=5T = 5\"\n ],\n [\n \"Sampling procedure in the sub-models of tree\",\n \"We describe the sampling approach from the target distribution at a leaf and non-leaf node of the constructed binary tree $\\\\mathcal {T}$ described in Section REF .\",\n \"We denote a target density by $f_{j}$ which can be straightforwardly sampled from at a leaf node $\\\\mathcal {T}_{j} \\\\in \\\\mathcal {T}$ , a proper importance density by $h_{j:l}$ and a target density by $f_{j:l}$ respectively at a non-root tree node $\\\\mathcal {T}_{j:l} \\\\in \\\\mathcal {T}$ where $0 < l - j 0$ whenever $f(\\\\mathbf {x_{1}}, \\\\mathbf {x_{2}}) > 0$ , where we assume $h_{1}$ and $h_{2}$ are the probability densities of two independent (joint) random variables $\\\\mathbf {X_{1}}$ and $\\\\mathbf {X_{2}}$ .\",\n \"We claim that proposal $f_{1}({\\\\mathbf {x_{1}}}) f_{2}({ \\\\mathbf {x_{2}}})$ has the smallest KL divergence among all proposals of the form $h_{1}(\\\\mathbf {x_{1}})h_{2}({\\\\mathbf {x_{2}}})$ where $f_{1}({\\\\mathbf {x_{1}}})$ and $f_{2}({ \\\\mathbf {x_{2}}})$ are the marginal densities of $f(\\\\mathbf {x_{1}, x_{2}})$ with respect to $\\\\mathbf {X_{1}}$ and $\\\\mathbf {X_{2}}$ , respectively.\",\n \"Theorem 1 Let $f$ be a probability density function defined on $\\\\mathbb {R}^{n_{1} + n_{2}}$ , let $h_{1}$ and $h_{2}$ be probability density functions on $\\\\mathbb {R}^{n_{1}}$ and $\\\\mathbb {R}^{n_{2}}$ , respectively.\",\n \"If $h_{1}(\\\\mathbf {x_{1}}) h_{2} (\\\\mathbf {x_{2}}) > 0$ whenever $f(\\\\mathbf {x_{1}, x_{2}}) >0,$ then $ \\\\int _{\\\\mathbb {R}^{n_{2}}} \\\\int _{\\\\mathbb {R}^{n_{1}}} f( \\\\mathbf {x_{1}}, \\\\mathbf {x_{2}}) \\\\log \\\\bigg (\\\\frac{f(\\\\mathbf {x_{1}}, \\\\mathbf {x_{2}})}{h_{1}(\\\\mathbf {x_{1}}) h_{2}(\\\\mathbf {x_{2}}) } \\\\bigg ) \\\\mathrm {d} \\\\mathbf {x_{1}} \\\\mathrm {d} \\\\mathbf {x_{2}} \\\\ge \\\\int _{\\\\mathbb {R}^{n_{2}}} \\\\int _{\\\\mathbb {R}^{n_{1}}} f(\\\\mathbf {x_{1}}, \\\\mathbf {x_{2}}) \\\\log \\\\bigg (\\\\frac{f(\\\\mathbf {x_{1}}, \\\\mathbf {x_{2}})}{f_{1}(\\\\mathbf {x_{1}}) f_{2}(\\\\mathbf {x_{2}}) } \\\\bigg ) \\\\mathrm {d} \\\\mathbf {x_{1}} \\\\mathrm {d} \\\\mathbf {x_{2}},$ where $f_{1}({ \\\\mathbf {x_{1}}}) = \\\\int _{\\\\mathbb {R}^{n_{2}}} f({\\\\mathbf {x_{1}}}, \\\\mathbf {{x_{2}}}) \\\\mathrm {d} { \\\\mathbf {x_{2}}}$ and $f_{2}({ \\\\mathbf {x_{2}}}) = \\\\int _{\\\\mathbb {R}^{n_{1}}} f(\\\\mathbf {{x_{1}}}, \\\\mathbf {{x_{2}}}) \\\\mathrm {d} {\\\\mathbf {x_{1}}}$ are the densities of the marginal distributions of $f(\\\\mathbf {x_{1}}, \\\\mathbf {x_{2}})$ .\",\n \"The proof of Theorem REF is in the Appendix.\"\n ],\n [\n \"Estimates of smoothing distributions as target\",\n \"We provide an alternative way of constructing the intermediate target distributions using the marginal smoothing distributions motivated by Theorem REF .\",\n \"Since the closed-form solutions to the marginal smoothing distributions are not available in general, we employ the estimates of the distributions at the nodes.\",\n \"At the root, we still use $f_{0:T} = p(x_{0:T}|y_{0:T})$ .\",\n \"At a leaf node $\\\\mathcal {T}_{j} \\\\in \\\\mathcal {T}$ , we define $f_{j}(x_{j}) = \\\\hat{p}(x_{j} | y_{0:T}) \\\\approx p(x_{j} | y_{0:T})$ , which requires estimating the marginal smoothing distribution.\",\n \"We thus name the algorithm TPS-ES.\",\n \"At a non-leaf and non-root node $\\\\mathcal {T}_{j:l}$ , we define the target distribution $f_{j:l}$ : $f_{j:l}(x_{j:l}) &\\\\propto & \\\\hat{p}(x_{j} | y_{0:j}) \\\\frac{ \\\\hat{p}(x_{l} | y_{0:T})}{ \\\\hat{p}(x_{l} | y_{0:l})} \\\\prod ^{l-1}_{i = j} \\\\bigg \\\\lbrace p(x_{i+1} | x_{i}) p(y_{i+1} | x_{i+1}) \\\\bigg \\\\rbrace \\\\\\\\ &\\\\approx & p(x_{j} | y_{0:j}) \\\\frac{p(x_{l} | y_{0:T}) }{p(x_{l} | y_{0:l}) } \\\\prod ^{l-1}_{i = j} \\\\bigg \\\\lbrace p(x_{i+1} | x_{i}) p(y_{i+1} | x_{i+1}) \\\\bigg \\\\rbrace \\\\\\\\& = & p(x_{j:l} | y_{0:T}),$ where $\\\\hat{p}(x_{j} | y_{0:j})$ denotes a probability density approximating the filtering density at the $j$ th time step.\",\n \"Hence, given the estimate smoothing densities $\\\\lbrace \\\\hat{p}(x_{j} | y_{0:T}) \\\\rbrace _{j = 0, \\\\ldots , T}$ and the estimating filtering densities $\\\\lbrace \\\\hat{p}(x_{j} | y_{0:j}) \\\\rbrace _{j = 0, \\\\ldots , T}$ , we build an estimator of the distribution $p(x_{j:l} | y_{0:T})$ at $\\\\mathcal {T}_{j:l} \\\\in \\\\mathcal {T}$ .\",\n \"Merging the particles at $\\\\mathcal {T}_{j:l}$ from its children at $\\\\mathcal {T}_{j:k-1} \\\\in \\\\mathcal {T}$ and $\\\\mathcal {T}_{k:l} \\\\in \\\\mathcal {T}$ amounts to correlating the two sets of samples while roughly preserving their marginal distributions.\",\n \"The weight of the merged sample $\\\\tilde{x}^{(i)}_{j:l} = (\\\\tilde{x}_{j:k-1}^{({i})}, \\\\tilde{x}_{k:l}^{({i})})$ in Equation (REF ) becomes: $\\\\hat{w}^{(i)}_{j:l} = \\\\tilde{w}^{(i)}_{j:l} \\\\frac{\\\\hat{p}(\\\\tilde{x}^{({i})}_{k-1}|y_{0:k-1})}{ \\\\hat{p}(\\\\tilde{x}^{({i})}_{k-1}|y_{0:T}) \\\\hat{p}(\\\\tilde{x}^{({i})}_{k}|y_{0:k})} p(\\\\tilde{x}^{({i})}_{k} | \\\\tilde{x}^{({i})}_{k-1}) p(y_{k}|\\\\tilde{x}^{({i})}_{k}).$ Applying TPS-ES demands the constructions of $\\\\lbrace \\\\hat{p}(x_{j} | y_{0:j}) \\\\rbrace _{j = 0, \\\\ldots , T}$ and $\\\\lbrace \\\\hat{p}(x_{j} | y_{0:T}) \\\\rbrace _{j = 0, \\\\ldots , T}$ in advance.\",\n \"The new weight formula in Equation (REF ) additionally incorporates the ratio between the estimated filtering and smoothing densities of $x_{k-1}$ compared with Equation (REF ).\",\n \"TPS-ES exhibits a sound property regarding the Kullback–Leibler divergence discussed in Section REF .\",\n \"Given the target distribution $f_{j:l}(x_{j:l}) = \\\\hat{p}(x_{j:l}|y_{0:T}) $ estimating $p(x_{j:l}|y_{0:T}) $ at $\\\\mathcal {T}_{j:l}$ , the proposal $h_{j:l}(x_{j:l}) = f_{j:k-1}(x_{j:k-1}) f_{k:l}(x_{k:l})$ estimates $p(x_{j:k-1} | y_{0:T}) p(x_{k:l} | y_{0:T})$ .\",\n \"We notice $p(x_{j:k-1} | y_{0:T})$ and $p(x_{k:l} | y_{0:T})$ are the marginal distributions, and their product forms a proposal attaining the minimum KL divergence from $p(x_{j:l}|y_{0:T})$ .\",\n \"Hence, what the proposal density $h_{j:l}(x_{j:l})$ estimates has a minimum KL divergence from the smoothing density that our target distribution $f_{j:l}(x_{j:l})$ estimates.\",\n \"Moreover, TPS-ES can be practically useful in some extreme models whereas the empirical marginal densities from other Monte Carlo smoothing algorithms may miss modes caused by the poor proposals.\",\n \"Since TPS-ES leaves the marginal distributions of all random variables roughly invariant at all levels of the tree, we can diagnose each importance sampling step by inspecting the empirical marginals of the corresponding variables.\",\n \"If there is a substantial difference between the empirical marginal distributions, we need to examine the combination step.\"\n ],\n [\n \"Initial sampling distribution at leaf nodes\",\n \"We illustrate the constructions of the univariate distributions $\\\\lbrace \\\\hat{p}(x_{j} | y_{1:j})\\\\rbrace _{j = 0, \\\\ldots , T}$ and $\\\\lbrace \\\\hat{p}(x_{j} | y_{0:T})_{j = 0, \\\\ldots ,T}\\\\rbrace $ mentioned in Section REF and Section REF , which are used in the initial sampling distributions at the leaf nodes.\",\n \"In general, the solutions of the filtering and smoothing distribution of a HMM are analytically intractable and need to be estimated from Monte Carlo samples with some exceptions including linear Gaussian and discrete HMMs.\",\n \"We aim to generate a probability density $\\\\hat{f}$ estimating a target density $f$ given the weighted samples $\\\\lbrace x_{i}, w_{i}\\\\rbrace _{i = 1}^{n}$ from $f$ .\",\n \"In the context of $f$ being a filtering or smoothing distribution, we can obtain the weighted samples by running a filtering algorithm or a smoothing algorithm.\",\n \"We are not interested in the empirical distribution since it is discrete and generally does not cover the full support of the random variable of interest.\",\n \"We first consider some parametric approaches.\",\n \"We can fit the data with some common probability distributions including a normal distribution and Student's $t$ -distribution.\",\n \"We can also accommodate a mixture model to fit multiple modes of the target densities.\",\n \"The parameters of the distributions can be estimated in various ways including moment matching, maximum likelihood method and EM algorithm.\",\n \"The parametric approaches are reasonably quick and simple.\",\n \"For instance, assuming a Gaussian distribution requires the evaluation of the mean and variance and can be easily obtained from the samples using moment matching.\",\n \"The generation and evaluation of densities of the new particles are straightforward and fast to implement.\",\n \"Nevertheless, the target distribution may not be well approximated under the parametric assumption.\",\n \"Alternatively, we can employ some non-parametric approaches for instance, a kernel density estimator (KDS).\",\n \"We need to select the type of kernels and bandwidth in advance.\",\n \"The complexity of generating $N$ new samples is $O\\\\big (\\\\log (n)N\\\\big )$ and the evaluation of the densities is more computationally expensive with complexity $O(nN)$ .\",\n \"We propose another non-parametric approximation method using piecewise constant functions with a lower computational effort than a KDS.\",\n \"We first build a uniform grid consisting of the points $x_{1} < x_{2} < \\\\ldots < x_{n}$ with densities $d_{1}, \\\\ldots , d_{n}$ estimated by a KDS such that $x_{i+1} - x_{i} = \\\\Delta > 0$ for $i = 1, \\\\ldots , n$ .\",\n \"The resulting probability density function formed by these grid points using piecewise constant functions is: $f(x) = \\\\sum _{i = 1}^{n} \\\\mathbb {1}_{x \\\\in [x_{i} - \\\\Delta /2, x_{i} + \\\\Delta /2)} d_{i}.$ The evaluation of the sample densities reduces significantly from $O(nN)$ to $O\\\\big (N \\\\big )$ compared to a KDS.\",\n \"Such probability density functions using piecewise constant functions have several disadvantages though enjoy a fast computation of estimated densities.\",\n \"Firstly, the estimator is biased since the proposal density generally does not cover the full support of the target density.\",\n \"Moreover, in TPS-ES, if the estimated filtering and smoothing distributions are both generated using the piecewise constant functions with different samples, there is no guarantee their densities have the same support, which may cause zero or infinite weight in Equation (REF ).\",\n \"To avoid this, we consider the mixture probability distributions using the piecewise constant functions accommodating the samples from both the filtering and smoothing distributions.\",\n \"Assume at time step $j$ , the first uniform grid consists of the points $x^{f}_{1} < x^{f}_{2} < \\\\ldots < x^{f}_{n^{f}}$ such that $x^{f}_{i+1} - x^{f}_{i} = \\\\Delta ^{f}$ for $i = 1, \\\\ldots , n^{f}$ with estimated filtering densities $d^{f}_{1}, \\\\ldots , d^{f}_{n}$ from a KDS and assume the second uniform grid consists of the points $x^{s}_{1} < x^{s}_{2} < \\\\ldots < x^{s}_{n^{s}}$ such that $x^{s}_{i+1} - x^{s}_{i} = \\\\Delta ^{s}$ for $i = 1, \\\\ldots , n^{s}$ with estimated smoothing densities $d^{s}_{1}, \\\\ldots , d^{s}_{n}$ from another KDS.\",\n \"Then the resulting estimated filtering density $\\\\hat{p}(x | y_{0:j})$ is given by $\\\\hat{p}(x | y_{0:j}) = \\\\alpha ^{f} \\\\sum _{i = 1}^{n^{f}} \\\\mathbb {1}_{x \\\\in [x^{f}_{i} - \\\\Delta ^{f}/2, x^{f}_{i} + \\\\Delta ^{f}/2)} d^{f}_{i} + (1-\\\\alpha ^{f}) \\\\sum _{i = 1}^{n^{s}} \\\\mathbb {1}_{x \\\\in [x^{s}_{i} - \\\\Delta ^{s}/2, x^{s}_{i} + \\\\Delta ^{s}/2)} d^{s}_{i},$ where $0 < \\\\alpha ^{f} < 1$ .\",\n \"Similarly, the estimated smoothing density $\\\\hat{p}(x | y_{0:T})$ is given by $\\\\hat{p}(x | y_{0:T}) = \\\\alpha ^{s} \\\\sum _{i = 1}^{n^{s}} \\\\mathbb {1}_{x \\\\in [x^{s}_{i} - \\\\Delta ^{s}/2, x^{s}_{i} + \\\\Delta ^{s}/2)} d^{s}_{i} + (1-\\\\alpha ^{s}) \\\\sum _{i = 1}^{n^{f}} \\\\mathbb {1}_{x \\\\in [x^{f}_{i} - \\\\Delta ^{f}/2, x^{f}_{i} + \\\\Delta ^{f}/2)} d^{f}_{i},$ where $0 < \\\\alpha ^{s} < 1$ .\",\n \"We have no conclusion of the values of $\\\\alpha ^{f}$ and $\\\\alpha ^{s}$ so far and choose them with values close to 1.\",\n \"The resulting grid with the set of points $\\\\lbrace x^{f}_{1}, x^{f}_{2}, \\\\ldots , x^{f}_{n^{f}}, x^{s}_{1}, x^{s}_{2}, \\\\ldots , x^{s}_{n^{s}}\\\\rbrace $ is generally not uniform, but we ensure the estimated filtering and smoothing densities have the same support, though still finite.\"\n ],\n [\n \"Simulations\",\n \"We conduct simulations in a linear Gaussian HMM and a non-linear non-Gaussian HMM in this section.\",\n \"We implement TPS-EF and other smoothing algorithms with roughly the same computational effort.\",\n \"In the second example, we further compare TPS-EF and TPS-ES.\"\n ],\n [\n \"Gaussian Linear Model\",\n \"We consider a simple linear Gaussian HMM similar to [8].\",\n \"$X_{t} &= 0.8 X_{t-1} + V_{t} ~~~&& t = 1, \\\\ldots , T,\\\\\\\\Y_{t} &= X_{t} + W_{t} ~~~&& t = 0, \\\\ldots , T.$ where $T = 127$ , where $X_0,V_1,\\\\dots ,V_T,W_0,\\\\dots ,W_T$ are independent with $X_{0} \\\\sim \\\\mathcal {N}(0,1)$ , $V_{t}\\\\sim N(0,1)$ , $W_{t}\\\\sim N(0,1)$ .\",\n \"We implement the following smoothing algorithms.\",\n \"We run TPS using normal distributions as the initial sampling distributions (TPS-N) whose means and variances are estimated using moment matching from the samples of a bootstrap particle filter.\",\n \"The choice of a normal distribution is motivated by the fact that in this case the true smoothing distribution is a normal distribution.\",\n \"We also implement the tree-based particle smoothing algorithm suggest by [17] (TPS-L), the Rauch–Tung–Striebel smoother (RTSs) [21] yielding the closed-form solutions, the bootstrap particle filter (BPF) which updates the entire history of the particles in each step, the forward filtering backward smoothing algorithm (FFBSm) [8] and the forward filtering backward simulation (FFBSi) [12].\",\n \"We have implemented the above methods in R ourselves.\",\n \"We set the required sample size $N = 10000$ in TPS-N as a benchmark and denote $n = 10000$ the number of samples pre-generated from a bootstrap particle filter in FFBSm, FFBSi, TPS-N and TPS-L. We adjust the number of particles in other algorithms to roughly keep the same running time.\",\n \"As the implementations are not deterministic, we allow a 10% error regarding the running time for the rest of the algorithms compared to TPS-N. We run each algorithm $M = 500$ times with the same set of observations $\\\\lbrace y_{t}\\\\rbrace _{t = 0}^{127}$ .\",\n \"As a criterion for comparison, we define the mean square error of means (MSEm) and variances (MSEv) in the $m$ th simulation: $\\\\text{MSEm}_{m} &=& \\\\frac{1}{T+1} \\\\sum ^{T}_{t = 0} \\\\big ( \\\\widehat{\\\\mathbb {E}}^{m}[X_{t}| Y_{0:T}] - \\\\mathbb {E}[X_{t}| Y_{0:T}]\\\\big )^{2}, \\\\\\\\\\\\text{MSEv}_{m} &=& \\\\frac{1}{T+1} \\\\sum ^{T}_{t = 0} \\\\big ( \\\\widehat{\\\\text{Var}}^{m}[X_{t}| Y_{0:T}] - \\\\text{Var}[X_{t}| Y_{0:T}]\\\\big )^{2}, \\\\\\\\$ where $ \\\\widehat{\\\\mathbb {E}}[X^{m}_{t}| Y_{0:T}]$ and $\\\\widehat{\\\\text{Var}}[x^{m}| y_{0:T}] $ are the Monte Carlo estimates of the mean and variance of the smoothing distribution at time step $t$ in the $m$ th simulation.\",\n \"$\\\\mathbb {E}[X_{t}| Y_{0:T}]$ and $\\\\text{Var}[X_{t}| Y_{0:T}]$ are the true smoothing means and variances from a Rauch–Tung–Striebel smoother [21].\",\n \"The simulation results are shown in Table REF .\",\n \"When $N = n$ , the two tree-based sampling algorithms: TPS-L and TPS-N enjoy the same complexity $O(N)$ as BPF, and generate far more particles than FFBSm and FFBSi with quadratic complexities.\",\n \"TPS-L has the smallest mean of MSEm and MSEv followed by TPS-N, which outperform FFBSm and FFBSi significantly in terms of MSEm.\",\n \"Table: Simulation errors in the linear model\"\n ],\n [\n \"Non-linear Model\",\n \"We consider a well-known non-linear model [13], [1]: $X_{t} &= \\\\frac{1}{2} X_{t-1} + 25 \\\\frac{X_{t-1}}{1 + X_{t-1}^2} + 8 \\\\cos (1.2 t) + V_{t},~~&&t = 1,2, \\\\ldots , T,\\\\\\\\Y_{t} &= \\\\frac{X^{2}_{t}}{20} + W_{t},~~&&t = 0,2, \\\\ldots , T,$ where $T = 511$ , where $X_0, V_1,...,V_T, W_0,...W_T$ are independent with $X_{0} \\\\sim \\\\mathcal {N}(0, 1)$ , $V_{t} \\\\sim \\\\mathcal {N}(0, \\\\tau ^2)$ and $W_{t} \\\\sim \\\\mathcal {N}(0,\\\\sigma ^2).$ We run the same algorithms BPF, FFBSm, FFBSi and TPS-L as in Section REF .\",\n \"In TPS-EF, we use piecewise constant functions defined in Equation (REF ) for the approximation of the initial sampling distributions.\",\n \"We call the algorithm TPS-EFP and set $N = n = 10000$ as a benchmark.\",\n \"As before, we correspondingly adjust the sample sizes in other algorithms to achieve roughly the same computational effort.\",\n \"We calculate the mean and standard deviation of the MSE of means (MSEm) in $M = 500$ simulations with the same set of observations.\",\n \"Given no closed-form solutions to the true smoothing distributions, we apply a discrete analogue to the distributions of the initial hidden state $p_{0}(x_{0})$ and the transition distributions $\\\\lbrace p(x_{t+1}|x_{t})\\\\rbrace _{t = 0, \\\\ldots , 126}$ .\",\n \"We then approximate the smoothing distributions of the original HMM using the solutions of the discrete-space HMM.\",\n \"The MSEm of the $m$ th simulation in the non-linear model is defined as: $ \\\\text{MSEm}_{m} = \\\\frac{1}{T+1} \\\\sum ^{T}_{t = 0} \\\\big (\\\\widehat{\\\\mathbb {E}}^{m}[X_{t}| Y_{0:T}] - \\\\mathbb {E} (\\\\hat{X}_{t}\\\\big | y_{0:T}) )^{2}, $ where $\\\\mathbb {E} (\\\\hat{X}_{t}\\\\big | y_{0:T})$ is the mean of the smoothing distribution at time step $t$ of the discrete-space HMM.\",\n \"We additionally perform Kolmogorov–Smirnov test [19] which measures a distance between the empirical distribution and the target probability distribution.\",\n \"In the context of the smoothing problem in a non-linear hidden Markov model, the Kolmogorov–Smirnov statistic can be defined as $ D = \\\\sup _{x} | F^{(t)}_{1,N}(x) - F^{(t)}_{2}(x) |, $ where $F^{(t)}_{1,N}$ is the empirical cumulative function generated by $N$ samples at the time step $t$ from a smoothing algorithm and $F^{(t)}_{2}$ is the cumulative distribution function at time step $t$ of the smoothing distribution from a discrete-space HMM derived from the true model.\",\n \"We denote KS$_{m}$ to be the sum of the KS statistic of all time steps in the $m$ th simulation.\",\n \"Table: Simulation errors in the non-linear modelThe simulation results with different values of $\\\\tau $ and $\\\\sigma $ are shown in Table REF .\",\n \"In the first two situations, TPS-L shows the largest error and KS statistic, especially when $\\\\tau = 1$ and $\\\\sigma = 5$ .\",\n \"This can be explained by the poor proposal from the initial sampling distribution constructed by the algorithm.\",\n \"We examine this by plotting the cumulative distribution function (CDF) of the initial sampling distribution $f_{j}$ in TPS-L, the filtering distribution $p(x_{j}| y_{0:j})$ and the marginal smoothing distribution $p(x_{j}| y_{0:T})$ at a particular time step when $j = 271$ .\",\n \"In Figure REF , the CDF of the initial sampling are far more dissimilar to the marginal smoothing distribution than the filtering one, which contributes to very ineffective importance sampling steps during the built-up of the tree.\",\n \"Figure: CDF of the smoothing, filtering and initial sampling distribution at time step j=271j = 271 of TPS-L in the non-linear model when τ=1,σ=5\\\\tau = 1, \\\\sigma = 5.Other algorithms provide different results in the three parameter settings.\",\n \"When $\\\\tau = 1, \\\\sigma = 1$ , TPS-EFP shows much smaller MSEm and KS followed by BPF.\",\n \"BPF however has the largest mean of KS.\",\n \"When $\\\\tau = 1, \\\\sigma = 5$ , TPS-EFP has a larger mean of MSEm than BPF.\",\n \"In terms of the KS statistic, TPS-EFP outperforms other smoothing algorithms.\",\n \"When $\\\\tau = 5, \\\\sigma = 1$ , TPS-EFP and TPS-L produce dominant results with vastly smaller MSEm.\",\n \"They also exhibit the smallest mean of KS among the smoothing algorithms whereas the BPF gives the largest result though generating the most samples.\",\n \"To conclude, TPS-EFP and TPS-L perform well when the ratio between the standard deviation in the transition and emission density, i.e.\",\n \"when $\\\\tau / \\\\sigma $ is large.\",\n \"TPS-EFP has a more stable and appreciable performance, which provides low MSEm and consistently the smallest KS among the five smoothing algorithms.\",\n \"In contrast, the result of TPS-L may be misleading due to its instability.\",\n \"BPF works well regarding MSEm in some situations, but poorly in terms of Kolmogorov–Smirnov statistic.\",\n \"FFBSm and FFBSi produces less accurate results due to higher computational complexity.\"\n ],\n [\n \"Comparing TPS-EF and TPS-ES in the non-linear model\",\n \"In this section, we conduct simulations in the same non-linear model using tree-based particle smoothing algorithm with estimated filtering (TPS-EF) and smoothing (TPS-ES) distributions as the intermediate target distributions.\",\n \"As TPS-ES is not a good competitor given a relatively small sample size in Section REF , we compare its performance with TPS-EF with more computational budget.\",\n \"We demonstrate the implementations of the two algorithms.\",\n \"We apply the same smoothing algorithm TPS-EFP as described in Section REF which utilises the piecewise constant functions to estimate the filtering distributions.\",\n \"As TPS-ES requires the estimated smoothing distributions as the initial sampling distributions, we achieve this by using piecewise constant functions for the estimation based on the samples from an initial run of TPS-EFP and thus call the algorithm TPS-ESP.\",\n \"We specify the parameters in the simulations of TPS-EFP and TPS-ESP.\",\n \"We denote the sample size by $N$ .\",\n \"We set the parameters $\\\\alpha ^{s} = \\\\alpha ^{f} = 0.95$ appeared in Equation (REF ) and (REF ).\",\n \"In TPS-EFP and TPS-ESP, the estimated filtering distributions are both constructed from $n$ samples from the particle filters.\",\n \"Additionally, in TPS-ESP, the estimated smoothing distributions are constructed from TPS-EFP with $n^{\\\\prime }$ samples.\",\n \"We run TPS-ESP in two different situations: The first one has the same sample size $N$ as TPS-EFP and requires more computational effort to estimate the initial sampling distributions based on $n^{\\\\prime }$ Monte Carlo samples.\",\n \"The second one has roughly the same computational effort as TPS-EFP which generates fewer Monte Carlo samples for the estimation of the initial sampling distributions and the target samples.\",\n \"We compare TPS-EFP and TPS-ESP with respect to the mean square error and Kolmogorov–Smirnov statistic defined in Section REF .\",\n \"We run TPS-EFP and TPS-ESP for $M = 200$ times with different values of $\\\\tau $ and $\\\\sigma $ whose results are shown in Table REF .\",\n \"TPS-ESP has an evident improvement of the Kolmogorov–Smirnov (KS) statistic in most situations and the comparisons between MSEm vary.\",\n \"The MSEm of TPS-ESP always decreases when generating the same number of samples as TPS-EFP.\",\n \"However, TPS-ESP does not provide convinced results under roughly the same computational effort.\",\n \"Overall, the performance of TPS-ESP depends on the computational budget.\",\n \"Given the same sample size as in TPS-EFP, TPS-ESP can potentially decrease both MSEm and KS statistic.\",\n \"This may not be true when the algorithm is kept the same overall effort as TPS-EFP.\",\n \"Table: Simulation errors between TPS-EF and TPS-ES in the non-linear model\"\n ],\n [\n \"Conclusion\",\n \"This article introduces a Monte Carlo sampling method we call TPS built on the D&C SMC [17] to estimate the joint smoothing distribution $p(x_{0:T}|y_{0:T})$ in a hidden Markov model.\",\n \"The method decomposes the model into sub-models with intermediate target distributions using a binary tree structure.\",\n \"TPS samples independently from the leaves of the tree and gradually merges and resamples to target the new distributions upon the auxiliary tree.\",\n \"We propose one generic way of constructing a binary tree which sequentially splits the joint random variables $X_{0:T}$ .\",\n \"Furthermore, we discuss the sampling procedure of the target samples at a non-leave node by combining the samples from its children using importance sampling.\",\n \"The computational effort is adjustable with a possible reduction to a linear effort with respect to the sample size.\",\n \"Using the above settings, we investigate three algorithms with different types of intermediate target distributions at the non-root nodes.\",\n \"TPS-L [17] constructs intermediate target distributions conditional on the observations from the same time interval as the target variables and imposes an uninformative prior.\",\n \"TPS-L is very simple to implement with no additional tuning algorithms.\",\n \"The algorithm is at the risk of providing very poor initial sampling distribution based on little information from the observations.\",\n \"TPS-EF employs intermediate target distributions estimating of the (joint) filtering distributions which conditions on the observations up to the last time step in the target variable.\",\n \"It is straightforward for implementation with an initial run of a filtering algorithm.\",\n \"Nevertheless, the proposal in the importance sampling step may still not be satisfactory when the marginal filtering and smoothing distributions are vastly different.\",\n \"TPS-ES builds the distributions estimating of the (joint) smoothing distributions which conditions on all the observations.\",\n \"It roughly retains the marginal smoothing distributions from the intermediate target distributions at all levels of the auxiliary tree despite its more intensive computations.\",\n \"We further propose the constructions of the estimated filtering and smoothing distributions based on the Monte Carlo samples.\",\n \"Considering both accuracy and computational effort, we recommend parametric approaches such as normal assumptions in a linear Gaussian model and non-parametric approaches such as using piecewise constant functions in a non-linear model.\",\n \"In the simulation studies, TPS-L has the smallest error in the linear model, but very unstable results in the different settings of the non-linear model.\",\n \"TPS-EF exhibits more desirable simulation outcomes.\",\n \"It is computationally less expensive than the most smoothing algorithms with quadratic complexity.\",\n \"It also produces the smallest mean square errors in the linear Gaussian model and consistently the smallest average Kolmogorov–Smirnov statistic in different situations under the non-linear model.\",\n \"In particular, it outperforms other algorithms substantially when the variance of the transition density is much larger than the emission density.\",\n \"TPS-ES, however, has a better approximation of the smoothing distribution with respect to the Kolmogorov–Smirnov statistic compared with TPS-EF at the cost of an additional run of a smoothing algorithm.\",\n \"To conclude, TPS with two proposed choices of the intermediate target distribution presents a new approach of addressing the smoothing problem which shows the following advantages: We have flexibilities of choosing and constructing the intermediate target distributions, which can potentially produce better proposals in the importance sampling steps.\",\n \"TPS can escape from the quadratic complexity with respect to the sample size computationally, and produce more particles and accurate simulation results than some smoothing algorithms.\",\n \"Nevertheless, its performance depends on the implementation of other filtering or smoothing algorithms and the estimation of the target distributions.\",\n \"Due to its flexible and relatively fast implementations with stable and comparable simulation results, we regard it as a competitor with other smoothing algorithms.\"\n ],\n [\n \"Proof of Theorem \",\n \"By Jensen's inequality, $&& \\\\int _{\\\\mathbb {R}^{n_{1}}} f_{1}(\\\\mathbf {x_{1}}) \\\\log \\\\big ( f_{1}(\\\\mathbf {x_{1}}) \\\\big ) \\\\mathrm {d} \\\\mathbf {x_{1}} - \\\\int _{\\\\mathbb {R}^{n_{1}}} f_{1}(\\\\mathbf {x_{1}}) \\\\log \\\\big ( h_{1}(\\\\mathbf {x_{1}}) \\\\big ) \\\\mathrm {d}\\\\mathbf {x_{1}}\\\\\\\\ &=& \\\\int _{\\\\mathbb {R}^{n_{1}}} f_{1} (\\\\mathbf {x_{1}}) \\\\log \\\\bigg ( \\\\frac{f_{1}(\\\\mathbf {x_{1}})}{h_{1} (\\\\mathbf {x_{1}})} \\\\bigg ) \\\\mathrm {d} \\\\mathbf {x_{1}}= \\\\mathbb {E} \\\\bigg [ \\\\log \\\\bigg ( \\\\frac{f_{1}(\\\\mathbf {X_{1}})}{ h_{1}(\\\\mathbf {X_{1}}) } \\\\bigg ) \\\\bigg ] = \\\\mathbb {E} \\\\bigg [ - \\\\log \\\\bigg ( \\\\frac{h_{1}(\\\\mathbf {X_{1}})}{ f_{1}(\\\\mathbf {X_{1}}) } \\\\bigg ) \\\\bigg ] \\\\\\\\&\\\\ge & - \\\\log \\\\bigg \\\\lbrace \\\\mathbb {E} \\\\bigg [ \\\\frac{h_{1}(\\\\mathbf {X_{1}})}{f_{1}(\\\\mathbf {X_{1}})} \\\\bigg ] \\\\bigg \\\\rbrace = 0.\",\n \"$ Using this and the definition of marginal distribution, $&& \\\\int _{\\\\mathbb {R}^{n_{2}}} \\\\int _{\\\\mathbb {R}^{n_1}} f(\\\\mathbf {x_{1}}, \\\\mathbf {x_{2}}) \\\\log \\\\big ( f_{1}(\\\\mathbf {x_{1}}) \\\\big ) \\\\mathrm {d} \\\\mathbf {x_{1}} \\\\mathrm {d} \\\\mathbf {x_{2}}= \\\\int _{\\\\mathbb {R}^{n_{1}}} f_{1}(\\\\mathbf {x_{1}}) \\\\log \\\\big ( f_{1}(\\\\mathbf {x_{1}}) \\\\big ) \\\\mathrm {d} \\\\mathbf {x_{1}} \\\\nonumber \\\\\\\\&\\\\ge & \\\\int _{\\\\mathbb {R}^{n_{1}}} f_{1}(\\\\mathbf {x_{1}}) \\\\log \\\\big ( h_{1}(\\\\mathbf {x_{1}}) \\\\big ) \\\\mathrm {d} \\\\mathbf {x_{1}} = \\\\int _{\\\\mathbb {R}^{n_{2}}} \\\\int _{\\\\mathbb {R}^{n_1}} f(\\\\mathbf {x_{1}}, \\\\mathbf {x_{2}}) \\\\log \\\\big ( h_{1}(\\\\mathbf {x_{1}}) \\\\big ) \\\\mathrm {d} \\\\mathbf {x_{1}}\\\\mathrm {d} \\\\mathbf {x_{2}}.\",\n \"\\\\nonumber \\\\\\\\$ Similarly, $\\\\int _{\\\\mathbb {R}^{n_{2}}} \\\\int _{\\\\mathbb {R}^{n_{1}}} f(\\\\mathbf {x_{1}}, \\\\mathbf {x_{2}}) \\\\log \\\\big ( f_{2}(\\\\mathbf {x_{2}}) \\\\big ) \\\\mathrm {d} \\\\mathbf {x_{1}} \\\\mathrm {d} \\\\mathbf {x_{2}} \\\\ge \\\\int _{\\\\mathbb {R}^{n_{2}}} \\\\int _{\\\\mathbb {R}^{n_{1}}} f(\\\\mathbf {x_{1}}, \\\\mathbf {x_{2}}) \\\\log \\\\big ( h_{2}(\\\\mathbf {x_{2}}) \\\\big ) \\\\mathrm {d} \\\\mathbf {x_{1}}\\\\mathrm {d} \\\\mathbf {x_{2}}.$ Multiplying (REF ) and (REF ) by -1 and adding them, we have $\\\\int _{\\\\mathbb {R}^{n_{2}}} \\\\int _{\\\\mathbb {R}^{n_{1}}} f( \\\\mathbf {x_{1}}, \\\\mathbf {x_{2}}) \\\\log \\\\bigg ( \\\\frac{1}{f_{1}(\\\\mathbf {x_{1}}) f_{2}(\\\\mathbf {x_{2}})} \\\\bigg ) \\\\mathrm {d} \\\\mathbf {x_{1}} \\\\mathrm {d}\\\\mathbf {x_{2}} \\\\le \\\\int _{\\\\mathbb {R}^{n_{2}}} \\\\int _{\\\\mathbb {R}^{n_{1}}} f( \\\\mathbf {x_{1}}, \\\\mathbf {x_{2}}) \\\\log \\\\bigg (\\\\frac{1}{h_{1}(\\\\mathbf {x_{1}}) h_{2}(\\\\mathbf {x_{2}})} \\\\bigg ) \\\\mathrm {d} \\\\mathbf {x_{1}} \\\\mathrm {d} \\\\mathbf {x_{2}}.$ Adding $ \\\\displaystyle \\\\int _{\\\\mathbb {R}^{n_{2}}} \\\\int _{\\\\mathbb {R}^{n_{1}}} f( \\\\mathbf {x_{1}}, \\\\mathbf {x_{2}}) \\\\log \\\\big ( f(\\\\mathbf {x_{1}}, \\\\mathbf {x_{2}}) \\\\big ) \\\\mathrm {d} \\\\mathbf {x_{1}} \\\\mathrm {d} \\\\mathbf {x_{2}} $ to both sides yields the result.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08400"}}},{"rowIdx":397,"cells":{"text":{"kind":"list like","value":[["Integrated Server for Measurement-Device-Independent Quantum Key\n Distribution Network"],["Abstract Quantum key distribution (QKD), harnessing quantum physics and optoelectronics, may promise unconditionally secure information exchange in theory.","Recently, theoretical and experimental advances in measurement-device-independent (MDI-) QKD have successfully closed the physical backdoor in detection terminals.","However, the issues of scalability, stability, cost and loss prevent QKD systems from widespread application in practice.","Here, we propose and experimentally demonstrate a solution to build a star-topology quantum access network with an integrated server.","By using femtosecond laser direct writing technique, we construct integrated circuits for all the elements of Bell state analyzer together and are able to integrate several such analyzers on single photonic chip.","The measured high-visibility Bell state analysis suggests integrated server a promising platform for the practical application of MDI-QKD network."],["Integrated Server for Measurement-Device-Independent Quantum Key Distribution Network Ci-Yu Wang State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Jun Gao State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Zhi-Qiang Jiao State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Lu-Feng Qiao State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Ruo-Jing Ren State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Zhen Feng State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Yuan Chen State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Zeng-Quan Yan State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Yao Wang State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Hao Tang State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Xian-Min Jin xianmin.jin@sjtu.edu.cn State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Quantum key distribution (QKD), harnessing quantum physics and optoelectronics, may promise unconditionally secure information exchange in theory.","Recently, theoretical and experimental advances in measurement-device-independent (MDI-) QKD have successfully closed the physical backdoor in detection terminals.","However, the issues of scalability, stability, cost and loss prevent QKD systems from widespread application in practice.","Here, we propose and experimentally demonstrate a solution to build a star-topology quantum access network with an integrated server.","By using femtosecond laser direct writing technique, we construct integrated circuits for all the elements of Bell state analyzer together and are able to integrate several such analyzers on single photonic chip.","The measured high-visibility Bell state analysis suggests integrated server a promising platform for the practical application of MDI-QKD network.","Living in a new era of `Internet Of Everything', the ability to build a secure communication network is essential more than ever.","Since the emergence of QKD[1] – an indispensable part of quantum communication, many proof-of-principle demonstrations of quantum internet[2], [3], [4], [5], [6], [7], [8] have been done in increasingly complex systems.","Experiments have boomed in bulk components implementations[9], [10], [11], [12], leaving connecting multiple users an expensive as well as a massive construction project.","Integrated quantum device, featuring its cheap, compact, stable and flexible performance, becomes a revolutionary solution to establish quantum communication links.","Much efforts have been put forward on versatile integrated QKD platforms[13], [14], [15].","Chip-based client[16], [17], [18], [19], [20], [21] has been developed to investigate the practical possibilities for miniaturization of the quantum communication terminal.","Recently, Bunandar et al.","[22] achieved a 43-km-long link with silicon integrated devices, paving the way for one-way long-distance and high-speed chip-based QKD.","However, these point-to-point schemes have been threatened by many attacks through physical loopholes in detection terminals.","MDI-QKD[23], [24], [25], [26] introduces an independent untrusted third party (Charlie) to announce the measurement results, and the end users (Alice and Bob) only need to encode and send their photons (FIG.","1a).","It is immune to all attacks on detection and has a high tolerance of finite basis-dependent flaws.","In this letter, we propose a solution to build a star-topology[27], [28], [29], [30] quantum access network with an integrated server.","The star topology is found perfectly matching the features of MDI-QKD scheme.","All clients only need to build one quantum channel with the server, where an all-optical router can switch and connect any two clients on an integrated Bell state analyzer (FIG.","1b).","The keys are not touched in the routing process.","Furthermore, this solution is very resource-efficient for multi-user networks.","The proposed star network needs $N$ quantum channels to fully connect any two clients, which is far below the requirement of mesh network (FIG.","1c), $C^2_N$ quantum channels, constructed with point-to-point schemes.","Meanwhile, this solution needs plenty of Bell state analyzers for simultaneously supporting QKD among many pairs of clients, which, methodologically, can be well solved by constructing integrated circuits for all the elements of Bell state analyzer and assembling an array of such integrated analyzers on single photonic chip in server, as is shown in FIG.","1d.","Figure: Schematic of integrated server for MDI-QKD a. Overview of MDI-QKD protocol.","Two clients (Alice and Bob) encode their photons in BB84 bases by polarization modulator (Pol-M) and send them to an untrusted relay Charlie to conduct Bell state analysis.","Intensity modulators (IM) are used to generate decoy states.","The Bell state analyzer is composed of one 50/50 beam splitter (BS) and a polarization beam splitter(PBS) on each of its output ports for projection.","b. Star-topology network.","c. Mesh-topology network.","d. Integrated server for proof-of-principle test.","Two single photons from Alice and Bob are employed to qualify on-chip Bell state analyzer.","Signals from off-chip avalanche photodiode detectors (APD) are processed by a field-programmable gate array (FPGA).Figure: Design of integrated elements of on-chip Bell state analyzer.","a.","Top view geometry of integrated analyzer consisting of one polarization insensitive coupler (PIC) and two identical polarization directional couplers (PDC).","r: radius of fabrication circular arc.","CL: coupling length.","CW: coupling width.","b. Contour map of index simulation of PIC with designed two crossings.","c. Power distribution in waveguides with a Gaussian beam injected from In1.","d. Measured transmission of couplers by scanning coupling lengths when coupling width is fixed at 8m, for horizontal-(red square) and vertical-(blue circle) polarization input light.","Solid lines are the best-fitting curves according to coupling mode theory.","Dashed line marks the right coupling length for realizing a PDC.We employ femtosecond laser direct writing[31] to realize all the elements of Bell state analyzer, since the written chip can uniquely be single-mode-fiber compatible for low insertion loss, and can be fabricated in a single-step, mask-free and cost-effective way.","By controlling the nonlinear absorption, we are able to tune the mode profile and birefringence of the waveguide for polarization manipulating integrated devices[32], [33], [34], [35], [36], [37].","The waveguides used in this work are fabricated in boro-silicate glass via femtosecond laser direct writing.","Waveguides are formed by laser-induced permanent refractive index change, locating at 170 $$ m under the substrate surface by focusing femtosecond laser (wavelength 513nm, repetition rate 1 MHz, pulse energy 210 nJ, writing speed 20 mm/s) through a 50$\\times $ objective (numerical aperture 0.55).","Profiting from the mode control capacity of femtosecond laser direct writing, the written chip can be linked to a standard V-groove fiber array packaging with facet coupling as low as 1dB.","FIG.","2a shows the detailed design of the Bell state analyzer.","The special configuration of polarization insensitve coupler (PIC) is found not very sensitive to fabrication parameters in terms of splitting ratio.","The simulation results with commercially available softwares (RSoft BeamPROP and CAD suite) are shown in FIG.","2b and 2c.","Polarization insensitiveness is made possible by geometrically controlling interaction length with different coupling width to compensate the birefringence-induced difference of coupling coefficients between horizontal (H) and vertical (V) polarized light[38], [39].","The characterized performance shown in TABLE 1 confirms the high repeatability and uniformity of our PIC.","According to coupling mode theory, photons transferring from one waveguide to the other follows a sinusoidal law as shown in FIG.","2d.","Slight difference in coupling coefficients for H and V polarized light leads to different oscillation periods, and finally an opposite output behaves as a polarization directional coupler (PDC).","Two waveguides are prototyped with a fixed coupling width 8 $$ m, small enough but not overlapped, to ensure that the two propagating modes become coupled during evanescent field overlap.","Figure: Characterized polarization performance of on-chip Bell state analyzer.","a.","Measured polarization projection with a probe laser at 780 nm.","b.","Measured polarization projection with a probe laser at 786 nm.","c. and d. Measured projection coincidence counts per 100 seconds with heralded single photon at 780 nm injected from In1 and In2.Table: 50/50 PIC performance.","Output splitting ratios under different input conditions.Figure: Experimental results of Bell state analysis with integrated server.","a. Hong-Ou-Mandel Interference for all the coincidence counts.","An average visibility over 90%\\% is reached.","b.-e.","Relative conditional probability distribution with different input states in rectilinear and diagonal bases.","b. and d. show the theoretical distributions and c. and e. show the experimental coincidence distributions.The main task of MDI-QKD is to conduct Bell state analysis, thus we first use a probe laser to characterize our integrated analyzer.","The results obtained by injecting photons from In1 and In2 are presented with hatched and colored histograms respectively, and are piled together to clearly reveal their uniformity for different inputs and for different wavelengths of light at 780 nm (FIG.","3a) and 786 nm (FIG.","3b).","Our probe laser can only be tuned in a limited range, and however, the good performance at two different wavelengths implies the ability to fabricate our integrated analyzer in telecom band through similar parameter optimization method.","To test the performance of polarization projection on two PDCs, we prepare four typical input states for BB84 protocol bases, polarized at 0$^\\circ $ (H), 90$^\\circ $ (V), 45$^\\circ $ (D) and 135$^\\circ $ (A).","The normalized projection results agree with the expected relative ratio for H/V and A/D bases (red dashed lines).","More importantly, the average polarization extinction ratio for H/V bases is about 20:1, which can well identify $|\\psi ^+\\rangle $ out of $|\\phi ^\\pm \\rangle $ for Bell state analysis.","However, the response and noise level of detection system for laser and single photons are different.","Thus, we repeat the experiment with input states of heralded single photons.","We generate photon pairs via Type-II spontaneous parametric down-conversion by pumping a beta-barium borate (BBO) crystal with a Ti:sapphire laser.","Heralding efficiency of conditionally detecting one single photon by the other one is over 25$\\%$ .","Apparently, the detection and coupling efficiencies for different outputs can not be identical in practice, which may modify the projection results.","As we can see, the projection coincidence counts per 100 seconds shown in FIG.","3c and 3d slightly deviate from the results shown in FIG.","3a, but, are all around the expected relative ratio (red dashed lines).","Interestingly, the average polarization extinction ratio for H/V bases is as high as 59:1, which of course reflects the real performance of the device at single-photon level.","In order to qualify the integrated server suitable for MDI-QKD, we couple the pair of photons into one of on-chip Bell state analyzers, as is shown in FIG.","1d, both prepared in D, and scan their relative delay to find the zone of Hong-Ou-Mandel Interference[40].","Photons from Alice and Bob are encoded independently in BB84 protocol bases by rotating half wave plates.","Since our two PDCs can project incoming states H and V at the output ports 1, 4 and 2, 3 respectively, coincidence measurements could deterministically discriminate the states $|\\psi ^+\\rangle $ and $|\\psi ^-\\rangle $ by two-fold coincidences C12, C34 and C13, C24 via bunching or anti-bunching effect[41].","Meanwhile, two-fold coincidences C14 and C23 can not distinguish the states $|\\phi ^+\\rangle $ and $|\\phi ^-\\rangle $ , and therefore are not used as successful events in MDI-QKD protocols[42].","The Visibilities for all combinations are $89.8\\%, 92.4\\%, 93.0\\%, 92.3\\%, 92.2\\%$ and $83.7\\%$ for C12, C13, C14, C23, C24 and C34 (FIG.","4a).","Two-photon data are collected for eight linearly independent input conditions FIG.","4b and 4c (HH, HV, VH, VV), and FIG.","4d and 4e (DD, DA, AD, AA), with half wave plates and polarization compensation.","The gain values at each rectilinear ($Q_{i,j}^{r}$ ) and diagonal ($Q_{i,j}^{d}$ ) basis can be caculated from the coincidence measurements by[24]: $Q_{i,j}^{r}=\\frac{1}{4}(C_{Sum}^{HH}+C_{Sum}^{VV}+C_{Sum}^{HV}+C_{Sum}^{VH})$ $Q_{i,j}^{d}=\\frac{1}{4}(C_{Sum}^{++}+C_{Sum}^{--}+C_{Sum}^{+-}+C_{Sum}^{-+})$ i and j represent the different average coincidence counts used by Alice and Bob while we adopt the successful events by: $C_{Sum}^{AB}=C_{12}^{AB}+C_{34}^{AB}+C_{13}^{AB}+C_{24}^{AB}$ We get the final gain values: $Q_{i,j}^{r}=1.37\\times 10^{-6}$ bits/pulse, $Q_{i,j}^{d}=1.75\\times 10^{-6}$ bits/pulse.","We estimate the quantum bit error rate (QBER) by: $E_{i,j}^{r}=\\frac{C_{Sum}^{HH}+C_{Sum}^{VV}}{C_{Sum}^{HH}+C_{Sum}^{VV}+C_{Sum}^{HV}+C_{Sum}^{VH}}$ $\\begin{aligned}E_{i,j}^{d}=&\\frac{C_{13}^{++}+C_{24}^{++}+C_{13}^{--}+C_{24}^{--}}{C_{Sum}^{++}+C_{Sum}^{--}+C_{Sum}^{+-}+C_{Sum}^{-+}}\\\\+&\\frac{C_{12}^{+-}+C_{34}^{+-}+C_{12}^{-+}+C_{34}^{-+}}{C_{Sum}^{++}+C_{Sum}^{--}+C_{Sum}^{+-}+C_{Sum}^{-+}}\\end{aligned}$ From our experimental data, we estimate $E_{i,j}^{r}=0.058, E_{i,j}^{d}=0.081$ .","The nonideal QBER can be attributed to multi-photon events of the testing photon source, residual birefringence in PIC and polarization-dependent loss.","The last two points are being improved to meet the stringent requirement for low QBER and therefore for high final bit rate.","In summary, we propose and experimentally demonstrate an integrated server for MDI-QKD network and its compatibility to a star-topology quantum access network.","Every client sends their encoded photons to server through their sole channel.","Server can link any two clients simply by switching their photons to one of Bell state analyzers on a photonic chip.","All the combination between two clients is the same as a standard MDI-QKD and therefore is unconditionally secure.","The number of required channels is much less than mesh-topology with point-to-point schemes for realizing fully connected MDI-QKD networks, which is particularly significant when the networks go to large scales.","Acknowledgements: The authors thank Jian-Wei Pan, Hang Li, Xiao-Ling Pang, and Jian-Peng Dou for helpful discussions.","This work was supported by National Key R&D Program of China (2017YFA0303700); National Natural Science Foundation of China (NSFC) (11374211, 61734005, 11690033); Shanghai Municipal Education Commission (SMEC)(16SG09, 2017-01-07-00-02-E00049); Science and Technology Commission of Shanghai Municipality (STCSM) (15QA1402200, 16JC1400405).","X.-M.J. acknowledges support from the National Young 1000 Talents Plan."]],"string":"[\n [\n \"Integrated Server for Measurement-Device-Independent Quantum Key\\n Distribution Network\"\n ],\n [\n \"Abstract Quantum key distribution (QKD), harnessing quantum physics and optoelectronics, may promise unconditionally secure information exchange in theory.\",\n \"Recently, theoretical and experimental advances in measurement-device-independent (MDI-) QKD have successfully closed the physical backdoor in detection terminals.\",\n \"However, the issues of scalability, stability, cost and loss prevent QKD systems from widespread application in practice.\",\n \"Here, we propose and experimentally demonstrate a solution to build a star-topology quantum access network with an integrated server.\",\n \"By using femtosecond laser direct writing technique, we construct integrated circuits for all the elements of Bell state analyzer together and are able to integrate several such analyzers on single photonic chip.\",\n \"The measured high-visibility Bell state analysis suggests integrated server a promising platform for the practical application of MDI-QKD network.\"\n ],\n [\n \"Integrated Server for Measurement-Device-Independent Quantum Key Distribution Network Ci-Yu Wang State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Jun Gao State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Zhi-Qiang Jiao State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Lu-Feng Qiao State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Ruo-Jing Ren State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Zhen Feng State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Yuan Chen State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Zeng-Quan Yan State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Yao Wang State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Hao Tang State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Xian-Min Jin xianmin.jin@sjtu.edu.cn State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Quantum key distribution (QKD), harnessing quantum physics and optoelectronics, may promise unconditionally secure information exchange in theory.\",\n \"Recently, theoretical and experimental advances in measurement-device-independent (MDI-) QKD have successfully closed the physical backdoor in detection terminals.\",\n \"However, the issues of scalability, stability, cost and loss prevent QKD systems from widespread application in practice.\",\n \"Here, we propose and experimentally demonstrate a solution to build a star-topology quantum access network with an integrated server.\",\n \"By using femtosecond laser direct writing technique, we construct integrated circuits for all the elements of Bell state analyzer together and are able to integrate several such analyzers on single photonic chip.\",\n \"The measured high-visibility Bell state analysis suggests integrated server a promising platform for the practical application of MDI-QKD network.\",\n \"Living in a new era of `Internet Of Everything', the ability to build a secure communication network is essential more than ever.\",\n \"Since the emergence of QKD[1] – an indispensable part of quantum communication, many proof-of-principle demonstrations of quantum internet[2], [3], [4], [5], [6], [7], [8] have been done in increasingly complex systems.\",\n \"Experiments have boomed in bulk components implementations[9], [10], [11], [12], leaving connecting multiple users an expensive as well as a massive construction project.\",\n \"Integrated quantum device, featuring its cheap, compact, stable and flexible performance, becomes a revolutionary solution to establish quantum communication links.\",\n \"Much efforts have been put forward on versatile integrated QKD platforms[13], [14], [15].\",\n \"Chip-based client[16], [17], [18], [19], [20], [21] has been developed to investigate the practical possibilities for miniaturization of the quantum communication terminal.\",\n \"Recently, Bunandar et al.\",\n \"[22] achieved a 43-km-long link with silicon integrated devices, paving the way for one-way long-distance and high-speed chip-based QKD.\",\n \"However, these point-to-point schemes have been threatened by many attacks through physical loopholes in detection terminals.\",\n \"MDI-QKD[23], [24], [25], [26] introduces an independent untrusted third party (Charlie) to announce the measurement results, and the end users (Alice and Bob) only need to encode and send their photons (FIG.\",\n \"1a).\",\n \"It is immune to all attacks on detection and has a high tolerance of finite basis-dependent flaws.\",\n \"In this letter, we propose a solution to build a star-topology[27], [28], [29], [30] quantum access network with an integrated server.\",\n \"The star topology is found perfectly matching the features of MDI-QKD scheme.\",\n \"All clients only need to build one quantum channel with the server, where an all-optical router can switch and connect any two clients on an integrated Bell state analyzer (FIG.\",\n \"1b).\",\n \"The keys are not touched in the routing process.\",\n \"Furthermore, this solution is very resource-efficient for multi-user networks.\",\n \"The proposed star network needs $N$ quantum channels to fully connect any two clients, which is far below the requirement of mesh network (FIG.\",\n \"1c), $C^2_N$ quantum channels, constructed with point-to-point schemes.\",\n \"Meanwhile, this solution needs plenty of Bell state analyzers for simultaneously supporting QKD among many pairs of clients, which, methodologically, can be well solved by constructing integrated circuits for all the elements of Bell state analyzer and assembling an array of such integrated analyzers on single photonic chip in server, as is shown in FIG.\",\n \"1d.\",\n \"Figure: Schematic of integrated server for MDI-QKD a. Overview of MDI-QKD protocol.\",\n \"Two clients (Alice and Bob) encode their photons in BB84 bases by polarization modulator (Pol-M) and send them to an untrusted relay Charlie to conduct Bell state analysis.\",\n \"Intensity modulators (IM) are used to generate decoy states.\",\n \"The Bell state analyzer is composed of one 50/50 beam splitter (BS) and a polarization beam splitter(PBS) on each of its output ports for projection.\",\n \"b. Star-topology network.\",\n \"c. Mesh-topology network.\",\n \"d. Integrated server for proof-of-principle test.\",\n \"Two single photons from Alice and Bob are employed to qualify on-chip Bell state analyzer.\",\n \"Signals from off-chip avalanche photodiode detectors (APD) are processed by a field-programmable gate array (FPGA).Figure: Design of integrated elements of on-chip Bell state analyzer.\",\n \"a.\",\n \"Top view geometry of integrated analyzer consisting of one polarization insensitive coupler (PIC) and two identical polarization directional couplers (PDC).\",\n \"r: radius of fabrication circular arc.\",\n \"CL: coupling length.\",\n \"CW: coupling width.\",\n \"b. Contour map of index simulation of PIC with designed two crossings.\",\n \"c. Power distribution in waveguides with a Gaussian beam injected from In1.\",\n \"d. Measured transmission of couplers by scanning coupling lengths when coupling width is fixed at 8m, for horizontal-(red square) and vertical-(blue circle) polarization input light.\",\n \"Solid lines are the best-fitting curves according to coupling mode theory.\",\n \"Dashed line marks the right coupling length for realizing a PDC.We employ femtosecond laser direct writing[31] to realize all the elements of Bell state analyzer, since the written chip can uniquely be single-mode-fiber compatible for low insertion loss, and can be fabricated in a single-step, mask-free and cost-effective way.\",\n \"By controlling the nonlinear absorption, we are able to tune the mode profile and birefringence of the waveguide for polarization manipulating integrated devices[32], [33], [34], [35], [36], [37].\",\n \"The waveguides used in this work are fabricated in boro-silicate glass via femtosecond laser direct writing.\",\n \"Waveguides are formed by laser-induced permanent refractive index change, locating at 170 $$ m under the substrate surface by focusing femtosecond laser (wavelength 513nm, repetition rate 1 MHz, pulse energy 210 nJ, writing speed 20 mm/s) through a 50$\\\\times $ objective (numerical aperture 0.55).\",\n \"Profiting from the mode control capacity of femtosecond laser direct writing, the written chip can be linked to a standard V-groove fiber array packaging with facet coupling as low as 1dB.\",\n \"FIG.\",\n \"2a shows the detailed design of the Bell state analyzer.\",\n \"The special configuration of polarization insensitve coupler (PIC) is found not very sensitive to fabrication parameters in terms of splitting ratio.\",\n \"The simulation results with commercially available softwares (RSoft BeamPROP and CAD suite) are shown in FIG.\",\n \"2b and 2c.\",\n \"Polarization insensitiveness is made possible by geometrically controlling interaction length with different coupling width to compensate the birefringence-induced difference of coupling coefficients between horizontal (H) and vertical (V) polarized light[38], [39].\",\n \"The characterized performance shown in TABLE 1 confirms the high repeatability and uniformity of our PIC.\",\n \"According to coupling mode theory, photons transferring from one waveguide to the other follows a sinusoidal law as shown in FIG.\",\n \"2d.\",\n \"Slight difference in coupling coefficients for H and V polarized light leads to different oscillation periods, and finally an opposite output behaves as a polarization directional coupler (PDC).\",\n \"Two waveguides are prototyped with a fixed coupling width 8 $$ m, small enough but not overlapped, to ensure that the two propagating modes become coupled during evanescent field overlap.\",\n \"Figure: Characterized polarization performance of on-chip Bell state analyzer.\",\n \"a.\",\n \"Measured polarization projection with a probe laser at 780 nm.\",\n \"b.\",\n \"Measured polarization projection with a probe laser at 786 nm.\",\n \"c. and d. Measured projection coincidence counts per 100 seconds with heralded single photon at 780 nm injected from In1 and In2.Table: 50/50 PIC performance.\",\n \"Output splitting ratios under different input conditions.Figure: Experimental results of Bell state analysis with integrated server.\",\n \"a. Hong-Ou-Mandel Interference for all the coincidence counts.\",\n \"An average visibility over 90%\\\\% is reached.\",\n \"b.-e.\",\n \"Relative conditional probability distribution with different input states in rectilinear and diagonal bases.\",\n \"b. and d. show the theoretical distributions and c. and e. show the experimental coincidence distributions.The main task of MDI-QKD is to conduct Bell state analysis, thus we first use a probe laser to characterize our integrated analyzer.\",\n \"The results obtained by injecting photons from In1 and In2 are presented with hatched and colored histograms respectively, and are piled together to clearly reveal their uniformity for different inputs and for different wavelengths of light at 780 nm (FIG.\",\n \"3a) and 786 nm (FIG.\",\n \"3b).\",\n \"Our probe laser can only be tuned in a limited range, and however, the good performance at two different wavelengths implies the ability to fabricate our integrated analyzer in telecom band through similar parameter optimization method.\",\n \"To test the performance of polarization projection on two PDCs, we prepare four typical input states for BB84 protocol bases, polarized at 0$^\\\\circ $ (H), 90$^\\\\circ $ (V), 45$^\\\\circ $ (D) and 135$^\\\\circ $ (A).\",\n \"The normalized projection results agree with the expected relative ratio for H/V and A/D bases (red dashed lines).\",\n \"More importantly, the average polarization extinction ratio for H/V bases is about 20:1, which can well identify $|\\\\psi ^+\\\\rangle $ out of $|\\\\phi ^\\\\pm \\\\rangle $ for Bell state analysis.\",\n \"However, the response and noise level of detection system for laser and single photons are different.\",\n \"Thus, we repeat the experiment with input states of heralded single photons.\",\n \"We generate photon pairs via Type-II spontaneous parametric down-conversion by pumping a beta-barium borate (BBO) crystal with a Ti:sapphire laser.\",\n \"Heralding efficiency of conditionally detecting one single photon by the other one is over 25$\\\\%$ .\",\n \"Apparently, the detection and coupling efficiencies for different outputs can not be identical in practice, which may modify the projection results.\",\n \"As we can see, the projection coincidence counts per 100 seconds shown in FIG.\",\n \"3c and 3d slightly deviate from the results shown in FIG.\",\n \"3a, but, are all around the expected relative ratio (red dashed lines).\",\n \"Interestingly, the average polarization extinction ratio for H/V bases is as high as 59:1, which of course reflects the real performance of the device at single-photon level.\",\n \"In order to qualify the integrated server suitable for MDI-QKD, we couple the pair of photons into one of on-chip Bell state analyzers, as is shown in FIG.\",\n \"1d, both prepared in D, and scan their relative delay to find the zone of Hong-Ou-Mandel Interference[40].\",\n \"Photons from Alice and Bob are encoded independently in BB84 protocol bases by rotating half wave plates.\",\n \"Since our two PDCs can project incoming states H and V at the output ports 1, 4 and 2, 3 respectively, coincidence measurements could deterministically discriminate the states $|\\\\psi ^+\\\\rangle $ and $|\\\\psi ^-\\\\rangle $ by two-fold coincidences C12, C34 and C13, C24 via bunching or anti-bunching effect[41].\",\n \"Meanwhile, two-fold coincidences C14 and C23 can not distinguish the states $|\\\\phi ^+\\\\rangle $ and $|\\\\phi ^-\\\\rangle $ , and therefore are not used as successful events in MDI-QKD protocols[42].\",\n \"The Visibilities for all combinations are $89.8\\\\%, 92.4\\\\%, 93.0\\\\%, 92.3\\\\%, 92.2\\\\%$ and $83.7\\\\%$ for C12, C13, C14, C23, C24 and C34 (FIG.\",\n \"4a).\",\n \"Two-photon data are collected for eight linearly independent input conditions FIG.\",\n \"4b and 4c (HH, HV, VH, VV), and FIG.\",\n \"4d and 4e (DD, DA, AD, AA), with half wave plates and polarization compensation.\",\n \"The gain values at each rectilinear ($Q_{i,j}^{r}$ ) and diagonal ($Q_{i,j}^{d}$ ) basis can be caculated from the coincidence measurements by[24]: $Q_{i,j}^{r}=\\\\frac{1}{4}(C_{Sum}^{HH}+C_{Sum}^{VV}+C_{Sum}^{HV}+C_{Sum}^{VH})$ $Q_{i,j}^{d}=\\\\frac{1}{4}(C_{Sum}^{++}+C_{Sum}^{--}+C_{Sum}^{+-}+C_{Sum}^{-+})$ i and j represent the different average coincidence counts used by Alice and Bob while we adopt the successful events by: $C_{Sum}^{AB}=C_{12}^{AB}+C_{34}^{AB}+C_{13}^{AB}+C_{24}^{AB}$ We get the final gain values: $Q_{i,j}^{r}=1.37\\\\times 10^{-6}$ bits/pulse, $Q_{i,j}^{d}=1.75\\\\times 10^{-6}$ bits/pulse.\",\n \"We estimate the quantum bit error rate (QBER) by: $E_{i,j}^{r}=\\\\frac{C_{Sum}^{HH}+C_{Sum}^{VV}}{C_{Sum}^{HH}+C_{Sum}^{VV}+C_{Sum}^{HV}+C_{Sum}^{VH}}$ $\\\\begin{aligned}E_{i,j}^{d}=&\\\\frac{C_{13}^{++}+C_{24}^{++}+C_{13}^{--}+C_{24}^{--}}{C_{Sum}^{++}+C_{Sum}^{--}+C_{Sum}^{+-}+C_{Sum}^{-+}}\\\\\\\\+&\\\\frac{C_{12}^{+-}+C_{34}^{+-}+C_{12}^{-+}+C_{34}^{-+}}{C_{Sum}^{++}+C_{Sum}^{--}+C_{Sum}^{+-}+C_{Sum}^{-+}}\\\\end{aligned}$ From our experimental data, we estimate $E_{i,j}^{r}=0.058, E_{i,j}^{d}=0.081$ .\",\n \"The nonideal QBER can be attributed to multi-photon events of the testing photon source, residual birefringence in PIC and polarization-dependent loss.\",\n \"The last two points are being improved to meet the stringent requirement for low QBER and therefore for high final bit rate.\",\n \"In summary, we propose and experimentally demonstrate an integrated server for MDI-QKD network and its compatibility to a star-topology quantum access network.\",\n \"Every client sends their encoded photons to server through their sole channel.\",\n \"Server can link any two clients simply by switching their photons to one of Bell state analyzers on a photonic chip.\",\n \"All the combination between two clients is the same as a standard MDI-QKD and therefore is unconditionally secure.\",\n \"The number of required channels is much less than mesh-topology with point-to-point schemes for realizing fully connected MDI-QKD networks, which is particularly significant when the networks go to large scales.\",\n \"Acknowledgements: The authors thank Jian-Wei Pan, Hang Li, Xiao-Ling Pang, and Jian-Peng Dou for helpful discussions.\",\n \"This work was supported by National Key R&D Program of China (2017YFA0303700); National Natural Science Foundation of China (NSFC) (11374211, 61734005, 11690033); Shanghai Municipal Education Commission (SMEC)(16SG09, 2017-01-07-00-02-E00049); Science and Technology Commission of Shanghai Municipality (STCSM) (15QA1402200, 16JC1400405).\",\n \"X.-M.J. acknowledges support from the National Young 1000 Talents Plan.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08586"}}},{"rowIdx":398,"cells":{"text":{"kind":"list like","value":[["An extension of a theorem of Zermelo"],["Abstract We show that if (M,E,E') satisfies the first order Zermelo-Fraenkel axioms of set theory when the membership relation is E and also when the membership relation is E', and in both cases the formulas are allowed to contain both E and E', then (M,E) and (M,E') are isomorphic, and the isomorphism is definable in (M,E,E').","This extends Zermelo's 1930 theorem about second order ZFC."],["We show that if $(M,\\in _1,\\in _2)$ satisfies the first order Zermelo-Fraenkel axioms of set theory when the membership relation is $\\in _1$ and also when the membership relation is $\\in _2$ , and in both cases the formulas are allowed to contain both $\\in _1$ and $\\in _2$ , then $(M,\\in _1)\\cong (M,\\in _2)$ , and the isomorphism is definable in $(M,\\in _1,\\in _2)$ .","This extends Zermelo's 1930 theorem in [6].","Zermelo [6] proved that if $(M,\\in _1)$ and $(M,\\in _2)$ both satisfy the second order Zermelo-Fraenkel axioms in which the Separation Schema and the Replacement Schema of $ZFC$ are replaced by single second order axioms, then $(M,\\in _1)\\cong (M,\\in _2)$ .","We extend this as follows: Let us consider the vocabulary $\\lbrace \\in _1,\\in _2\\rbrace $ , where both $\\in _1$ and $\\in _2$ are binary predicate symbols.","Let $ZFC(\\in _1)$ denote the first order Zermelo-Fraenkel axioms of set theory when $\\in _1$ is the membership relation but formulas are allowed to contain $\\in _2$ too.","Similarly, in $ZFC(\\in _2)$ the membership relation is $\\in _2$ but formulas are allowed to contain $\\in _1$ too.","We prove the following theorem: Theorem 1 If $(M,\\in _1,\\in _2)\\models ZFC(\\in _1)\\cup ZFC(\\in _2)$ , then $(M,\\in _1)\\cong (M,\\in _2)$ via a definable class function.","The result of Zermelo readily follows from our theorem.","The important difference between our result and Zermelo's result is that our theories $ZFC(\\in _1)$ and $ZFC(\\in _2)$ are first order theories.","It is important that we allow in these axiom systems formulas from the extended vocabulary $\\lbrace \\in _1,\\in _2\\rbrace $ .","Without this the result would be blatantly false as there are countable non-isomorphic models of $ZFC$ , assuming there are models of $ZFC$ at all.","Since the isomorphism in Theorem REF is definable, the result can be seen as a provable theorem of the first order theory $ZFC(\\in _1)\\cup ZFC(\\in _2)$ .","Theorem REF resembles the categoricity conclusion for set theory in [2].","There are two main differences: First, the axiomatization of set theory in [2] is informal, based on the Axiom of Extensionality and an informal full Comprehension Axiom, while our result is completely formal and in the context of ZFC.","Secondly, it is assumed in [2] that the two $\\in $ -relations give rise to the same (informal) structure of the ordinals, owing to the uniqueness of the ordinal concept.","We do not make this assumption but rather prove that the two $\\in $ -relations have isomorphic ordinals.","Martin's work has been extended to class theory in [5].","Theorem REF was stated without proof in [3].","We call our theorem an internal categoricity result because it shows that one cannot have in one and the same domain two non-isomorphic membership-relations $\\in _1$ and $\\in _2$ if these relations can “talk” about each other.","Our theorem is a strong robustness result for set theory.","Essentially, the model cannot be changed “internally”.","To get a non-isomorphic model one has to go “outside” the model.","Such robustness is important for set theory because set theory is already the “outside” of mathematics, the framework where mathematics is (or can be) built.","How are the numerous independence results in harmony with this internal categoricity?","Let us take the Continuum Hypothesis CH as an example.","CH is independent of $ZFC$ in the sense that both $ZFC\\cup \\lbrace CH\\rbrace $ and $ZFC\\cup \\lbrace \\lnot CH\\rbrace $ are consistent, if $ZFC$ itself is.","Internal categoricity means simply that if $(M,\\in _1)$ satisfies $CH$ and $(M,\\in _2)$ satisfies $\\lnot CH$ , then either $(M,\\in _1)$ or $(M,\\in _2)$ does not satisfy the Separation Schema or the Replacement Schema if formulas are allowed to mention the other membership-relation.","Such models cannot be internal to each other in the sense discussed.","In the below proof we will work in $ZFC(\\in _1)\\cup ZFC(\\in _2)$ but in fact operate all the time in either $\\in _1$ -set theory or in $\\in _2$ -set theory.","We have to keep the two set theories separate even though they also interact via the Separation and Replacement Schemas in the joint vocabulary $\\lbrace \\in _1,\\in _2\\rbrace $ .","Let $\\mathrm {tr}_i(x)$ be the formula $\\forall t\\in _i x\\forall w\\in _i t(w\\in _1 x)$ .","Let $\\mathrm {TC}_i(x)$ be the unique $u$ such that $\\mathrm {tr}_i(u)\\wedge \\forall v\\in _ix(v\\in _iu)\\wedge \\forall v((\\mathrm {tr}_i(v)\\wedge \\forall w\\in _ix(w\\in _iv))\\rightarrow \\forall w\\in _iu(w\\in _i v))$ (“$u$ is the $\\in _i$ -transitive closure of $x$ ”).","When we write $\\mathrm {TC}_i(\\lbrace x\\rbrace )$ , we mean by $\\lbrace x\\rbrace $ the singleton $\\lbrace x\\rbrace $ in the sense of $\\in _i$ .","Let $\\phi (x,y)$ be the formula $\\exists f\\psi (x,y,f)$ , where $\\psi (x,y,f)$ is the conjunction of the following formulas (where $f(t)$ , $f(w)$ and $f(x)$ are in the sense of $\\in _1$ ): (i) In the sense of $\\in _1$ , the set $f$ is a function with $\\mathrm {TC}_1(\\lbrace x\\rbrace )$ as its domain.","(ii) $\\forall t\\in _1 \\mathrm {TC}_1(x)(f(t)\\in _2 \\mathrm {TC}_2(y))$ (iii) $\\forall t\\in _2 \\mathrm {TC}_2(y)\\exists w\\in _1 \\mathrm {TC}_1(x)(t=f(w))$ (iv) $\\forall t\\in _1 \\mathrm {TC}_1(x)\\forall w\\in _1 \\mathrm {TC}_1(\\lbrace x\\rbrace )(t\\in _1w\\leftrightarrow f(t)\\in _2 f(w))$ (v) $f(x)=y$ In the sense of $\\in _1$ , the set $f$ is a function with $\\mathrm {TC}_1(\\lbrace x\\rbrace )$ as its domain.","$\\forall t\\in _1 \\mathrm {TC}_1(x)(f(t)\\in _2 \\mathrm {TC}_2(y))$ $\\forall t\\in _2 \\mathrm {TC}_2(y)\\exists w\\in _1 \\mathrm {TC}_1(x)(t=f(w))$ $\\forall t\\in _1 \\mathrm {TC}_1(x)\\forall w\\in _1 \\mathrm {TC}_1(\\lbrace x\\rbrace )(t\\in _1w\\leftrightarrow f(t)\\in _2 f(w))$ $f(x)=y$ We prove a sequence of lemmas about the formulas $\\phi (x,y)$ and $\\psi (x,y,f)$ : Lemma 2 If $\\psi (x,y,f)$ and $\\psi (x,y,f^{\\prime })$ , then $f=f^{\\prime }$ .","To prove $f=f^{\\prime }$ assume $w\\in _1 \\mathrm {TC}_1(\\lbrace x\\rbrace )$ .","We show $f(w)=f^{\\prime }(w)$ .","W.l.o.g.","$f(s)=f^{\\prime }(s)$ for $s\\in _1 w$ .","Suppose $t\\in _2 f(w)$ .","Clearly, $t\\in _2 \\mathrm {TC}_2(\\lbrace y\\rbrace )$ .","By (iii), $t=f(s)$ for some $s\\in _1 \\mathrm {TC}_1(\\lbrace x\\rbrace )$ .","By (iv), $s\\in _1 w$ .","By (iv) again, $f^{\\prime }(s)\\in _2 f^{\\prime }(w)$ .","By assumption, $f(s)=f^{\\prime }(s)$ .","Hence $t\\in _2 f^{\\prime }(w)$ .","Thus $\\forall t(t\\in _2 f(w)\\rightarrow t\\in _2 f^{\\prime }(w))$ .","By symmetry, $f(w)=f^{\\prime }(w)$ .","Lemma 3 Suppose $\\psi (x,y,f)$ .","If $x^{\\prime }\\in _1x$ , then $\\phi (x^{\\prime },f(x^{\\prime }))$ .","If $y^{\\prime }\\in _2y$ , then there is $x^{\\prime }\\in _1x$ such that $f(x^{\\prime })=y^{\\prime }$ and $\\phi (x^{\\prime },y^{\\prime })$ .","Let $y^{\\prime }=f(x^{\\prime })$ and $f^{\\prime }=f\\upharpoonright \\mathrm {TC}_1(\\lbrace x^{\\prime }\\rbrace )$ .","Clearly now $\\psi (x^{\\prime },y^{\\prime },f^{\\prime })$ .","Hence $\\phi (x^{\\prime },f(x^{\\prime }))$ .","The other claim is proved similarly.","Lemma 4 If $\\phi (x,y)$ and $\\phi (x,y^{\\prime })$ , then $y=y^{\\prime }$ .","If $\\phi (x,y)$ and $\\phi (x^{\\prime },y)$ , then $x=x^{\\prime }$ .","We may assume the claim holds for all $\\in _1$ -elements of $x$ .","Suppose $\\psi (x,y,f)$ and $\\psi (x,y^{\\prime },f^{\\prime })$ .","We prove $y=y^{\\prime }$ .","Let $s\\in _2 y$ .","By Lemma REF there is $t\\in _1 x$ such that $f(t)=s$ and $\\phi (t,s)$ .","By (iv), $s\\in _1 x$ .","Let $s^{\\prime }=f^{\\prime }(t)$ .","By (iv), $s^{\\prime }\\in _2 y^{\\prime }$ .","By Lemma REF again, $\\phi (t,s^{\\prime })$ .","By the Induction Hypothesis, $s=s^{\\prime }$ .","We have proved $\\forall s(s\\in _2 y\\rightarrow s\\in _2 y^{\\prime })$ .","The converse follows from symmetry.","Now to the second claim.","We may assume the claim holds for all $\\in _2$ -elements of $y$ .","Suppose $\\psi (x,y,f)$ and $\\psi (x^{\\prime },y,f^{\\prime })$ .","We prove $x=x^{\\prime }$ .","Let $s\\in _1 x$ .","Thus $f(s)\\in _2 y$ .","There is $s^{\\prime }\\in _1 \\mathrm {TC}_1(\\lbrace x^{\\prime }\\rbrace )$ such that $f^{\\prime }(s^{\\prime })=f(s)$ .","Now $\\phi (s,f(s))$ and $\\phi (s^{\\prime },f(s))$ by Lemma REF .","Since $f(s)\\in _2 y$ , $s=s^{\\prime }$ .","Hence $s\\in _1 x^{\\prime }$ .","We have proved $\\forall s(s\\in _1 x\\rightarrow s\\in _1 x^{\\prime })$ .","The converse follows from symmetry.","Lemma 5 If $\\phi (x,y)$ and $\\phi (x^{\\prime },y^{\\prime })$ , then $x\\in _1x^{\\prime }\\leftrightarrow y\\in _2y^{\\prime }$ .","Suppose $\\psi (x,y,f)$ and $\\psi (x^{\\prime },y^{\\prime },f^{\\prime })$ .","Suppose $x\\in _1 x^{\\prime }$ .","Then $z=f^{\\prime }(x)\\in _2 y^{\\prime }$ .","By Lemma REF , $\\phi (x,z)$ .","We have $\\phi (x,y)$ and $\\phi (x,z)$ .","By Lemma REF , $y=z$ .","Hence $y\\in _2 y^{\\prime }$ .","The converse is similar.","Let $\\mbox{On}_1(x)$ be the $\\in _1$ -formula saying that $x$ is an ordinal i.e.","a transitive set of transitive sets, and similarly $\\mbox{On}_2(x)$ .","For $\\mbox{On}_1(\\alpha )$ let $V^1_\\alpha $ be the $\\alpha ^{th}$ level of the cumulative hierarchy in the sense of $\\in _1$ , and similarly $V^2_y$ when $\\mbox{On}_2(y)$ .","Lemma 6 If $\\phi (\\alpha ,y)$ , then $\\mbox{On}_1(\\alpha )$ if and only if $\\mbox{On}_2(y)$ .","If $\\alpha $ is a limit ordinal then so is $y$ i.e.","if $\\forall u\\in _1 \\alpha \\exists v\\in _1\\alpha (u\\in _1 v)$ , then $\\forall u\\in _2 y\\exists v\\in _2 y(u\\in _2 v)$ , and vice versa.","Let us fix $y$ .","Suppose $\\psi (\\alpha ,y,f)$ .","We prove that $y$ is a transitive set of transitive sets.","Suppose $w\\in _2s\\in _2y$ .","There are $t\\in _1 \\alpha $ and $u\\in _1 t$ such that $f(t)=s$ and $f(u)=w$ .","Now $w\\in _2 y$ follows from $u\\in _1\\alpha $ .","This shows that $\\mathrm {tr}_2(y)$ .","Similarly one proves that all $s\\in _2y$ satisfy $\\mathrm {tr}_2(y)$ .","This ends the proof of the first claim.","The second claim is proved similarly.","Lemma 7 Suppose $\\psi (\\alpha ,y,f)$ .","If $\\mbox{On}_1(\\alpha )$ (or equivalently $\\mbox{On}_2(y)$ ), then there is $\\bar{f}\\supseteq f$ such that $\\psi (V^1_\\alpha ,V^2_y,\\bar{f})$ .","We use induction on $\\alpha $ .","Suppose the claim holds for $\\alpha $ .","We prove the claim for $\\alpha +1$ .","Suppose to this end $\\psi (\\alpha +1,y+1,f)$ .","We construct $\\bar{f}$ such that $\\psi (V^1_{\\alpha +1},V^2_{y+1},\\bar{f})$ .","From $\\psi (\\alpha +1,y+1,f)$ we obtain $\\psi (\\alpha ,y,f\\upharpoonright \\alpha )$ .","By assumption there is $g\\supseteq f\\upharpoonright \\alpha $ such that $\\psi (V^1_\\alpha ,V^2_y,g)$ .","Let $\\theta (u,v)$ be the formula $\\forall w(w\\in _2 v\\leftrightarrow (w\\in _2 V^2_y\\wedge \\exists t\\in _1 u(g(t)=w))).$ It follows from the Separation Schema of $ZFC(\\in _2)$ that for all $u\\in _1V^1_{\\alpha +1}$ there is $v$ such that $\\theta (u,v)$ .","By the Replacement Schema of $ZFC(\\in _1)$ , we can let $\\bar{f}$ be a function (in the sense of $\\lbrace \\in _1\\rbrace $ ) such that for all $u\\in _1V^1_{\\alpha +1}$ we have $\\theta (u,\\bar{f}(u))$ .","It is easy to see, using the Separation Schema of $ZFC(\\in _1)$ , that $\\psi (V^1_{\\alpha +1},V^2_{y+1},\\bar{f})$ .","Suppose then the claim holds for all $\\beta <\\alpha =\\bigcup \\alpha $ .","For each $\\beta <\\alpha $ there is thus some $g_\\beta $ such that $\\psi (V^1_\\beta ,V^2_{f(\\beta )},g_\\beta )$ .","By the Replacement Schema of $ZFC(\\in _1)$ we can form the $\\in _1$ -set $\\bar{f}=\\bigcup _{\\beta <\\alpha }g_\\beta $ .","It is easy to see that $\\psi (V^1_\\alpha ,V^2_y,\\bar{f})$ .","Lemma 8 $\\forall x\\exists y\\phi (x,y)$ and $\\forall y\\exists x\\phi (x,y)$ .","Let us first assume that both $\\forall \\alpha (\\mbox{On}_1(\\alpha )\\rightarrow \\exists y\\phi (\\alpha ,y))$ and $\\forall y(\\mbox{On}_2(y)\\rightarrow \\exists \\alpha \\phi (\\alpha ,y)).$ hold.","In order to prove $\\forall x\\exists y\\phi (x,y)$ , suppose $x$ is given.","There is $\\alpha $ such that $\\mbox{On}_1(\\alpha )$ and $x\\in _1 V^1_\\alpha $ .","By (REF ) there are $v$ and $f$ such that $\\psi (\\alpha ,v,f)$ .","By Lemma REF there is $\\bar{f}\\supseteq f$ such that $\\psi (V^1_\\alpha , V^2_v,\\bar{f})$ .","By Lemma REF , $\\phi (x,\\bar{f}(x))$ .","Thus $\\exists y\\phi (x,y)$ .","In order to prove $\\forall y\\exists x\\phi (x,y)$ , suppose $y$ is given.","There is $v$ such that $\\mbox{On}_2(v)$ and $y\\in _2 V^2_v$ .","By (REF ) there are $\\alpha $ and $f$ such that $\\psi (\\alpha ,v,f)$ .","By Lemma REF there is $\\bar{f}\\supseteq f$ such that $\\psi (V^1_\\alpha , V^2_v)$ .","By condition (iii) of the definition of $\\psi $ there is $w\\in _1 V^1_\\alpha $ such that $\\bar{f}(w)=y$ .","By Lemma REF , $\\phi (w,\\bar{f}(w))$ .","Thus $\\exists x\\phi (x,y)$ .","Thus it suffices to show that the failure of (REF ) or (REF ) to hold leads to a contradiction.","Case 1: $\\lnot $ (REF )$\\wedge \\lnot $ (REF ).","Let $\\alpha $ be the $\\in _1$ -least $\\alpha $ such that $\\mbox{On}_1(\\alpha )\\wedge \\lnot \\exists y\\phi (\\alpha ,y)$ .","Let $y$ be the $\\in _2$ -least $y$ such that $\\mbox{On}_2(y)\\wedge \\lnot \\exists \\beta \\phi (\\beta ,y)$ .","It is easy to see that $\\phi (\\alpha ,y)$ , a contradiction.","Case 2: (REF )$\\wedge \\lnot $ (REF ).","Let $y$ be the $\\in _2$ -least $y$ such that $\\mbox{On}_2(y)\\wedge \\lnot \\exists \\alpha \\phi (\\alpha ,y)$ .","Now, $\\forall t\\in _2 y\\exists \\alpha (\\mbox{On}_1(\\alpha )\\wedge \\phi (\\alpha ,t))$ .","Clearly, $y$ is an $\\in _2$ -limit ordinal.","Suppose $z\\in _2 V^2_t$ , where $t\\in _2 y$ .","Let $\\alpha $ and $f$ be such that $\\mbox{On}_1(\\alpha )\\wedge \\psi (\\alpha ,t,f)$ .","By Lemma REF there is $\\bar{f}\\supseteq f$ such that $\\psi (V^1_\\alpha , V^2_t,\\bar{f})$ .","There is $x\\in _1 V^1_\\alpha $ such that $\\bar{f}(x)=z$ .","Thus $\\phi (x,z)$ and hence $\\forall z\\in _2 V^2_y\\exists x\\ \\phi (x,z).$ By the Replacement Schema in $ZFC(\\in _2)$ there is $c$ such that $\\forall z\\in _2 V^2_y\\exists x\\in _2 c\\ \\phi (x,z).$ Let $\\alpha $ be such that $c\\in _1 V^1_\\alpha $ .","By (REF ) there are $t$ and $f$ such that $\\phi (\\alpha ,t,f)$ .","Necessarily, $t\\in _2 y$ .","By Lemma REF there is $\\bar{f}\\supseteq f$ such that $\\psi (V^1_\\alpha , V^2_t,\\bar{f})$ .","In particular, $\\bar{f}(c)\\in _2V^2_y$ .","By (REF ) there is $b\\in _2 c$ such that $\\phi (b,\\bar{f}(c))$ .","Since also $\\phi (c,\\bar{f}(c))$ , Lemma REF gives $c=b$ .","Thus $c\\in _2 c$ , a contradiction.","Case 3: $\\lnot $ (REF )$\\wedge $ (REF ).","This case is analogous to Case 2.","Proposition 9 The class defined by $\\phi (x,y)$ is an isomorphism between the $\\in _1$ -reduct and the $\\in _2$ -reduct.","By Lemmas REF , REF and REF .","A similar result holds for first order Peano arithmetic, extending the categoricity result of Dedekind [1] of second order Peano arithmetic.","The proof (see [4]) of this is similar, but somewhat easier."]],"string":"[\n [\n \"An extension of a theorem of Zermelo\"\n ],\n [\n \"Abstract We show that if (M,E,E') satisfies the first order Zermelo-Fraenkel axioms of set theory when the membership relation is E and also when the membership relation is E', and in both cases the formulas are allowed to contain both E and E', then (M,E) and (M,E') are isomorphic, and the isomorphism is definable in (M,E,E').\",\n \"This extends Zermelo's 1930 theorem about second order ZFC.\"\n ],\n [\n \"We show that if $(M,\\\\in _1,\\\\in _2)$ satisfies the first order Zermelo-Fraenkel axioms of set theory when the membership relation is $\\\\in _1$ and also when the membership relation is $\\\\in _2$ , and in both cases the formulas are allowed to contain both $\\\\in _1$ and $\\\\in _2$ , then $(M,\\\\in _1)\\\\cong (M,\\\\in _2)$ , and the isomorphism is definable in $(M,\\\\in _1,\\\\in _2)$ .\",\n \"This extends Zermelo's 1930 theorem in [6].\",\n \"Zermelo [6] proved that if $(M,\\\\in _1)$ and $(M,\\\\in _2)$ both satisfy the second order Zermelo-Fraenkel axioms in which the Separation Schema and the Replacement Schema of $ZFC$ are replaced by single second order axioms, then $(M,\\\\in _1)\\\\cong (M,\\\\in _2)$ .\",\n \"We extend this as follows: Let us consider the vocabulary $\\\\lbrace \\\\in _1,\\\\in _2\\\\rbrace $ , where both $\\\\in _1$ and $\\\\in _2$ are binary predicate symbols.\",\n \"Let $ZFC(\\\\in _1)$ denote the first order Zermelo-Fraenkel axioms of set theory when $\\\\in _1$ is the membership relation but formulas are allowed to contain $\\\\in _2$ too.\",\n \"Similarly, in $ZFC(\\\\in _2)$ the membership relation is $\\\\in _2$ but formulas are allowed to contain $\\\\in _1$ too.\",\n \"We prove the following theorem: Theorem 1 If $(M,\\\\in _1,\\\\in _2)\\\\models ZFC(\\\\in _1)\\\\cup ZFC(\\\\in _2)$ , then $(M,\\\\in _1)\\\\cong (M,\\\\in _2)$ via a definable class function.\",\n \"The result of Zermelo readily follows from our theorem.\",\n \"The important difference between our result and Zermelo's result is that our theories $ZFC(\\\\in _1)$ and $ZFC(\\\\in _2)$ are first order theories.\",\n \"It is important that we allow in these axiom systems formulas from the extended vocabulary $\\\\lbrace \\\\in _1,\\\\in _2\\\\rbrace $ .\",\n \"Without this the result would be blatantly false as there are countable non-isomorphic models of $ZFC$ , assuming there are models of $ZFC$ at all.\",\n \"Since the isomorphism in Theorem REF is definable, the result can be seen as a provable theorem of the first order theory $ZFC(\\\\in _1)\\\\cup ZFC(\\\\in _2)$ .\",\n \"Theorem REF resembles the categoricity conclusion for set theory in [2].\",\n \"There are two main differences: First, the axiomatization of set theory in [2] is informal, based on the Axiom of Extensionality and an informal full Comprehension Axiom, while our result is completely formal and in the context of ZFC.\",\n \"Secondly, it is assumed in [2] that the two $\\\\in $ -relations give rise to the same (informal) structure of the ordinals, owing to the uniqueness of the ordinal concept.\",\n \"We do not make this assumption but rather prove that the two $\\\\in $ -relations have isomorphic ordinals.\",\n \"Martin's work has been extended to class theory in [5].\",\n \"Theorem REF was stated without proof in [3].\",\n \"We call our theorem an internal categoricity result because it shows that one cannot have in one and the same domain two non-isomorphic membership-relations $\\\\in _1$ and $\\\\in _2$ if these relations can “talk” about each other.\",\n \"Our theorem is a strong robustness result for set theory.\",\n \"Essentially, the model cannot be changed “internally”.\",\n \"To get a non-isomorphic model one has to go “outside” the model.\",\n \"Such robustness is important for set theory because set theory is already the “outside” of mathematics, the framework where mathematics is (or can be) built.\",\n \"How are the numerous independence results in harmony with this internal categoricity?\",\n \"Let us take the Continuum Hypothesis CH as an example.\",\n \"CH is independent of $ZFC$ in the sense that both $ZFC\\\\cup \\\\lbrace CH\\\\rbrace $ and $ZFC\\\\cup \\\\lbrace \\\\lnot CH\\\\rbrace $ are consistent, if $ZFC$ itself is.\",\n \"Internal categoricity means simply that if $(M,\\\\in _1)$ satisfies $CH$ and $(M,\\\\in _2)$ satisfies $\\\\lnot CH$ , then either $(M,\\\\in _1)$ or $(M,\\\\in _2)$ does not satisfy the Separation Schema or the Replacement Schema if formulas are allowed to mention the other membership-relation.\",\n \"Such models cannot be internal to each other in the sense discussed.\",\n \"In the below proof we will work in $ZFC(\\\\in _1)\\\\cup ZFC(\\\\in _2)$ but in fact operate all the time in either $\\\\in _1$ -set theory or in $\\\\in _2$ -set theory.\",\n \"We have to keep the two set theories separate even though they also interact via the Separation and Replacement Schemas in the joint vocabulary $\\\\lbrace \\\\in _1,\\\\in _2\\\\rbrace $ .\",\n \"Let $\\\\mathrm {tr}_i(x)$ be the formula $\\\\forall t\\\\in _i x\\\\forall w\\\\in _i t(w\\\\in _1 x)$ .\",\n \"Let $\\\\mathrm {TC}_i(x)$ be the unique $u$ such that $\\\\mathrm {tr}_i(u)\\\\wedge \\\\forall v\\\\in _ix(v\\\\in _iu)\\\\wedge \\\\forall v((\\\\mathrm {tr}_i(v)\\\\wedge \\\\forall w\\\\in _ix(w\\\\in _iv))\\\\rightarrow \\\\forall w\\\\in _iu(w\\\\in _i v))$ (“$u$ is the $\\\\in _i$ -transitive closure of $x$ ”).\",\n \"When we write $\\\\mathrm {TC}_i(\\\\lbrace x\\\\rbrace )$ , we mean by $\\\\lbrace x\\\\rbrace $ the singleton $\\\\lbrace x\\\\rbrace $ in the sense of $\\\\in _i$ .\",\n \"Let $\\\\phi (x,y)$ be the formula $\\\\exists f\\\\psi (x,y,f)$ , where $\\\\psi (x,y,f)$ is the conjunction of the following formulas (where $f(t)$ , $f(w)$ and $f(x)$ are in the sense of $\\\\in _1$ ): (i) In the sense of $\\\\in _1$ , the set $f$ is a function with $\\\\mathrm {TC}_1(\\\\lbrace x\\\\rbrace )$ as its domain.\",\n \"(ii) $\\\\forall t\\\\in _1 \\\\mathrm {TC}_1(x)(f(t)\\\\in _2 \\\\mathrm {TC}_2(y))$ (iii) $\\\\forall t\\\\in _2 \\\\mathrm {TC}_2(y)\\\\exists w\\\\in _1 \\\\mathrm {TC}_1(x)(t=f(w))$ (iv) $\\\\forall t\\\\in _1 \\\\mathrm {TC}_1(x)\\\\forall w\\\\in _1 \\\\mathrm {TC}_1(\\\\lbrace x\\\\rbrace )(t\\\\in _1w\\\\leftrightarrow f(t)\\\\in _2 f(w))$ (v) $f(x)=y$ In the sense of $\\\\in _1$ , the set $f$ is a function with $\\\\mathrm {TC}_1(\\\\lbrace x\\\\rbrace )$ as its domain.\",\n \"$\\\\forall t\\\\in _1 \\\\mathrm {TC}_1(x)(f(t)\\\\in _2 \\\\mathrm {TC}_2(y))$ $\\\\forall t\\\\in _2 \\\\mathrm {TC}_2(y)\\\\exists w\\\\in _1 \\\\mathrm {TC}_1(x)(t=f(w))$ $\\\\forall t\\\\in _1 \\\\mathrm {TC}_1(x)\\\\forall w\\\\in _1 \\\\mathrm {TC}_1(\\\\lbrace x\\\\rbrace )(t\\\\in _1w\\\\leftrightarrow f(t)\\\\in _2 f(w))$ $f(x)=y$ We prove a sequence of lemmas about the formulas $\\\\phi (x,y)$ and $\\\\psi (x,y,f)$ : Lemma 2 If $\\\\psi (x,y,f)$ and $\\\\psi (x,y,f^{\\\\prime })$ , then $f=f^{\\\\prime }$ .\",\n \"To prove $f=f^{\\\\prime }$ assume $w\\\\in _1 \\\\mathrm {TC}_1(\\\\lbrace x\\\\rbrace )$ .\",\n \"We show $f(w)=f^{\\\\prime }(w)$ .\",\n \"W.l.o.g.\",\n \"$f(s)=f^{\\\\prime }(s)$ for $s\\\\in _1 w$ .\",\n \"Suppose $t\\\\in _2 f(w)$ .\",\n \"Clearly, $t\\\\in _2 \\\\mathrm {TC}_2(\\\\lbrace y\\\\rbrace )$ .\",\n \"By (iii), $t=f(s)$ for some $s\\\\in _1 \\\\mathrm {TC}_1(\\\\lbrace x\\\\rbrace )$ .\",\n \"By (iv), $s\\\\in _1 w$ .\",\n \"By (iv) again, $f^{\\\\prime }(s)\\\\in _2 f^{\\\\prime }(w)$ .\",\n \"By assumption, $f(s)=f^{\\\\prime }(s)$ .\",\n \"Hence $t\\\\in _2 f^{\\\\prime }(w)$ .\",\n \"Thus $\\\\forall t(t\\\\in _2 f(w)\\\\rightarrow t\\\\in _2 f^{\\\\prime }(w))$ .\",\n \"By symmetry, $f(w)=f^{\\\\prime }(w)$ .\",\n \"Lemma 3 Suppose $\\\\psi (x,y,f)$ .\",\n \"If $x^{\\\\prime }\\\\in _1x$ , then $\\\\phi (x^{\\\\prime },f(x^{\\\\prime }))$ .\",\n \"If $y^{\\\\prime }\\\\in _2y$ , then there is $x^{\\\\prime }\\\\in _1x$ such that $f(x^{\\\\prime })=y^{\\\\prime }$ and $\\\\phi (x^{\\\\prime },y^{\\\\prime })$ .\",\n \"Let $y^{\\\\prime }=f(x^{\\\\prime })$ and $f^{\\\\prime }=f\\\\upharpoonright \\\\mathrm {TC}_1(\\\\lbrace x^{\\\\prime }\\\\rbrace )$ .\",\n \"Clearly now $\\\\psi (x^{\\\\prime },y^{\\\\prime },f^{\\\\prime })$ .\",\n \"Hence $\\\\phi (x^{\\\\prime },f(x^{\\\\prime }))$ .\",\n \"The other claim is proved similarly.\",\n \"Lemma 4 If $\\\\phi (x,y)$ and $\\\\phi (x,y^{\\\\prime })$ , then $y=y^{\\\\prime }$ .\",\n \"If $\\\\phi (x,y)$ and $\\\\phi (x^{\\\\prime },y)$ , then $x=x^{\\\\prime }$ .\",\n \"We may assume the claim holds for all $\\\\in _1$ -elements of $x$ .\",\n \"Suppose $\\\\psi (x,y,f)$ and $\\\\psi (x,y^{\\\\prime },f^{\\\\prime })$ .\",\n \"We prove $y=y^{\\\\prime }$ .\",\n \"Let $s\\\\in _2 y$ .\",\n \"By Lemma REF there is $t\\\\in _1 x$ such that $f(t)=s$ and $\\\\phi (t,s)$ .\",\n \"By (iv), $s\\\\in _1 x$ .\",\n \"Let $s^{\\\\prime }=f^{\\\\prime }(t)$ .\",\n \"By (iv), $s^{\\\\prime }\\\\in _2 y^{\\\\prime }$ .\",\n \"By Lemma REF again, $\\\\phi (t,s^{\\\\prime })$ .\",\n \"By the Induction Hypothesis, $s=s^{\\\\prime }$ .\",\n \"We have proved $\\\\forall s(s\\\\in _2 y\\\\rightarrow s\\\\in _2 y^{\\\\prime })$ .\",\n \"The converse follows from symmetry.\",\n \"Now to the second claim.\",\n \"We may assume the claim holds for all $\\\\in _2$ -elements of $y$ .\",\n \"Suppose $\\\\psi (x,y,f)$ and $\\\\psi (x^{\\\\prime },y,f^{\\\\prime })$ .\",\n \"We prove $x=x^{\\\\prime }$ .\",\n \"Let $s\\\\in _1 x$ .\",\n \"Thus $f(s)\\\\in _2 y$ .\",\n \"There is $s^{\\\\prime }\\\\in _1 \\\\mathrm {TC}_1(\\\\lbrace x^{\\\\prime }\\\\rbrace )$ such that $f^{\\\\prime }(s^{\\\\prime })=f(s)$ .\",\n \"Now $\\\\phi (s,f(s))$ and $\\\\phi (s^{\\\\prime },f(s))$ by Lemma REF .\",\n \"Since $f(s)\\\\in _2 y$ , $s=s^{\\\\prime }$ .\",\n \"Hence $s\\\\in _1 x^{\\\\prime }$ .\",\n \"We have proved $\\\\forall s(s\\\\in _1 x\\\\rightarrow s\\\\in _1 x^{\\\\prime })$ .\",\n \"The converse follows from symmetry.\",\n \"Lemma 5 If $\\\\phi (x,y)$ and $\\\\phi (x^{\\\\prime },y^{\\\\prime })$ , then $x\\\\in _1x^{\\\\prime }\\\\leftrightarrow y\\\\in _2y^{\\\\prime }$ .\",\n \"Suppose $\\\\psi (x,y,f)$ and $\\\\psi (x^{\\\\prime },y^{\\\\prime },f^{\\\\prime })$ .\",\n \"Suppose $x\\\\in _1 x^{\\\\prime }$ .\",\n \"Then $z=f^{\\\\prime }(x)\\\\in _2 y^{\\\\prime }$ .\",\n \"By Lemma REF , $\\\\phi (x,z)$ .\",\n \"We have $\\\\phi (x,y)$ and $\\\\phi (x,z)$ .\",\n \"By Lemma REF , $y=z$ .\",\n \"Hence $y\\\\in _2 y^{\\\\prime }$ .\",\n \"The converse is similar.\",\n \"Let $\\\\mbox{On}_1(x)$ be the $\\\\in _1$ -formula saying that $x$ is an ordinal i.e.\",\n \"a transitive set of transitive sets, and similarly $\\\\mbox{On}_2(x)$ .\",\n \"For $\\\\mbox{On}_1(\\\\alpha )$ let $V^1_\\\\alpha $ be the $\\\\alpha ^{th}$ level of the cumulative hierarchy in the sense of $\\\\in _1$ , and similarly $V^2_y$ when $\\\\mbox{On}_2(y)$ .\",\n \"Lemma 6 If $\\\\phi (\\\\alpha ,y)$ , then $\\\\mbox{On}_1(\\\\alpha )$ if and only if $\\\\mbox{On}_2(y)$ .\",\n \"If $\\\\alpha $ is a limit ordinal then so is $y$ i.e.\",\n \"if $\\\\forall u\\\\in _1 \\\\alpha \\\\exists v\\\\in _1\\\\alpha (u\\\\in _1 v)$ , then $\\\\forall u\\\\in _2 y\\\\exists v\\\\in _2 y(u\\\\in _2 v)$ , and vice versa.\",\n \"Let us fix $y$ .\",\n \"Suppose $\\\\psi (\\\\alpha ,y,f)$ .\",\n \"We prove that $y$ is a transitive set of transitive sets.\",\n \"Suppose $w\\\\in _2s\\\\in _2y$ .\",\n \"There are $t\\\\in _1 \\\\alpha $ and $u\\\\in _1 t$ such that $f(t)=s$ and $f(u)=w$ .\",\n \"Now $w\\\\in _2 y$ follows from $u\\\\in _1\\\\alpha $ .\",\n \"This shows that $\\\\mathrm {tr}_2(y)$ .\",\n \"Similarly one proves that all $s\\\\in _2y$ satisfy $\\\\mathrm {tr}_2(y)$ .\",\n \"This ends the proof of the first claim.\",\n \"The second claim is proved similarly.\",\n \"Lemma 7 Suppose $\\\\psi (\\\\alpha ,y,f)$ .\",\n \"If $\\\\mbox{On}_1(\\\\alpha )$ (or equivalently $\\\\mbox{On}_2(y)$ ), then there is $\\\\bar{f}\\\\supseteq f$ such that $\\\\psi (V^1_\\\\alpha ,V^2_y,\\\\bar{f})$ .\",\n \"We use induction on $\\\\alpha $ .\",\n \"Suppose the claim holds for $\\\\alpha $ .\",\n \"We prove the claim for $\\\\alpha +1$ .\",\n \"Suppose to this end $\\\\psi (\\\\alpha +1,y+1,f)$ .\",\n \"We construct $\\\\bar{f}$ such that $\\\\psi (V^1_{\\\\alpha +1},V^2_{y+1},\\\\bar{f})$ .\",\n \"From $\\\\psi (\\\\alpha +1,y+1,f)$ we obtain $\\\\psi (\\\\alpha ,y,f\\\\upharpoonright \\\\alpha )$ .\",\n \"By assumption there is $g\\\\supseteq f\\\\upharpoonright \\\\alpha $ such that $\\\\psi (V^1_\\\\alpha ,V^2_y,g)$ .\",\n \"Let $\\\\theta (u,v)$ be the formula $\\\\forall w(w\\\\in _2 v\\\\leftrightarrow (w\\\\in _2 V^2_y\\\\wedge \\\\exists t\\\\in _1 u(g(t)=w))).$ It follows from the Separation Schema of $ZFC(\\\\in _2)$ that for all $u\\\\in _1V^1_{\\\\alpha +1}$ there is $v$ such that $\\\\theta (u,v)$ .\",\n \"By the Replacement Schema of $ZFC(\\\\in _1)$ , we can let $\\\\bar{f}$ be a function (in the sense of $\\\\lbrace \\\\in _1\\\\rbrace $ ) such that for all $u\\\\in _1V^1_{\\\\alpha +1}$ we have $\\\\theta (u,\\\\bar{f}(u))$ .\",\n \"It is easy to see, using the Separation Schema of $ZFC(\\\\in _1)$ , that $\\\\psi (V^1_{\\\\alpha +1},V^2_{y+1},\\\\bar{f})$ .\",\n \"Suppose then the claim holds for all $\\\\beta <\\\\alpha =\\\\bigcup \\\\alpha $ .\",\n \"For each $\\\\beta <\\\\alpha $ there is thus some $g_\\\\beta $ such that $\\\\psi (V^1_\\\\beta ,V^2_{f(\\\\beta )},g_\\\\beta )$ .\",\n \"By the Replacement Schema of $ZFC(\\\\in _1)$ we can form the $\\\\in _1$ -set $\\\\bar{f}=\\\\bigcup _{\\\\beta <\\\\alpha }g_\\\\beta $ .\",\n \"It is easy to see that $\\\\psi (V^1_\\\\alpha ,V^2_y,\\\\bar{f})$ .\",\n \"Lemma 8 $\\\\forall x\\\\exists y\\\\phi (x,y)$ and $\\\\forall y\\\\exists x\\\\phi (x,y)$ .\",\n \"Let us first assume that both $\\\\forall \\\\alpha (\\\\mbox{On}_1(\\\\alpha )\\\\rightarrow \\\\exists y\\\\phi (\\\\alpha ,y))$ and $\\\\forall y(\\\\mbox{On}_2(y)\\\\rightarrow \\\\exists \\\\alpha \\\\phi (\\\\alpha ,y)).$ hold.\",\n \"In order to prove $\\\\forall x\\\\exists y\\\\phi (x,y)$ , suppose $x$ is given.\",\n \"There is $\\\\alpha $ such that $\\\\mbox{On}_1(\\\\alpha )$ and $x\\\\in _1 V^1_\\\\alpha $ .\",\n \"By (REF ) there are $v$ and $f$ such that $\\\\psi (\\\\alpha ,v,f)$ .\",\n \"By Lemma REF there is $\\\\bar{f}\\\\supseteq f$ such that $\\\\psi (V^1_\\\\alpha , V^2_v,\\\\bar{f})$ .\",\n \"By Lemma REF , $\\\\phi (x,\\\\bar{f}(x))$ .\",\n \"Thus $\\\\exists y\\\\phi (x,y)$ .\",\n \"In order to prove $\\\\forall y\\\\exists x\\\\phi (x,y)$ , suppose $y$ is given.\",\n \"There is $v$ such that $\\\\mbox{On}_2(v)$ and $y\\\\in _2 V^2_v$ .\",\n \"By (REF ) there are $\\\\alpha $ and $f$ such that $\\\\psi (\\\\alpha ,v,f)$ .\",\n \"By Lemma REF there is $\\\\bar{f}\\\\supseteq f$ such that $\\\\psi (V^1_\\\\alpha , V^2_v)$ .\",\n \"By condition (iii) of the definition of $\\\\psi $ there is $w\\\\in _1 V^1_\\\\alpha $ such that $\\\\bar{f}(w)=y$ .\",\n \"By Lemma REF , $\\\\phi (w,\\\\bar{f}(w))$ .\",\n \"Thus $\\\\exists x\\\\phi (x,y)$ .\",\n \"Thus it suffices to show that the failure of (REF ) or (REF ) to hold leads to a contradiction.\",\n \"Case 1: $\\\\lnot $ (REF )$\\\\wedge \\\\lnot $ (REF ).\",\n \"Let $\\\\alpha $ be the $\\\\in _1$ -least $\\\\alpha $ such that $\\\\mbox{On}_1(\\\\alpha )\\\\wedge \\\\lnot \\\\exists y\\\\phi (\\\\alpha ,y)$ .\",\n \"Let $y$ be the $\\\\in _2$ -least $y$ such that $\\\\mbox{On}_2(y)\\\\wedge \\\\lnot \\\\exists \\\\beta \\\\phi (\\\\beta ,y)$ .\",\n \"It is easy to see that $\\\\phi (\\\\alpha ,y)$ , a contradiction.\",\n \"Case 2: (REF )$\\\\wedge \\\\lnot $ (REF ).\",\n \"Let $y$ be the $\\\\in _2$ -least $y$ such that $\\\\mbox{On}_2(y)\\\\wedge \\\\lnot \\\\exists \\\\alpha \\\\phi (\\\\alpha ,y)$ .\",\n \"Now, $\\\\forall t\\\\in _2 y\\\\exists \\\\alpha (\\\\mbox{On}_1(\\\\alpha )\\\\wedge \\\\phi (\\\\alpha ,t))$ .\",\n \"Clearly, $y$ is an $\\\\in _2$ -limit ordinal.\",\n \"Suppose $z\\\\in _2 V^2_t$ , where $t\\\\in _2 y$ .\",\n \"Let $\\\\alpha $ and $f$ be such that $\\\\mbox{On}_1(\\\\alpha )\\\\wedge \\\\psi (\\\\alpha ,t,f)$ .\",\n \"By Lemma REF there is $\\\\bar{f}\\\\supseteq f$ such that $\\\\psi (V^1_\\\\alpha , V^2_t,\\\\bar{f})$ .\",\n \"There is $x\\\\in _1 V^1_\\\\alpha $ such that $\\\\bar{f}(x)=z$ .\",\n \"Thus $\\\\phi (x,z)$ and hence $\\\\forall z\\\\in _2 V^2_y\\\\exists x\\\\ \\\\phi (x,z).$ By the Replacement Schema in $ZFC(\\\\in _2)$ there is $c$ such that $\\\\forall z\\\\in _2 V^2_y\\\\exists x\\\\in _2 c\\\\ \\\\phi (x,z).$ Let $\\\\alpha $ be such that $c\\\\in _1 V^1_\\\\alpha $ .\",\n \"By (REF ) there are $t$ and $f$ such that $\\\\phi (\\\\alpha ,t,f)$ .\",\n \"Necessarily, $t\\\\in _2 y$ .\",\n \"By Lemma REF there is $\\\\bar{f}\\\\supseteq f$ such that $\\\\psi (V^1_\\\\alpha , V^2_t,\\\\bar{f})$ .\",\n \"In particular, $\\\\bar{f}(c)\\\\in _2V^2_y$ .\",\n \"By (REF ) there is $b\\\\in _2 c$ such that $\\\\phi (b,\\\\bar{f}(c))$ .\",\n \"Since also $\\\\phi (c,\\\\bar{f}(c))$ , Lemma REF gives $c=b$ .\",\n \"Thus $c\\\\in _2 c$ , a contradiction.\",\n \"Case 3: $\\\\lnot $ (REF )$\\\\wedge $ (REF ).\",\n \"This case is analogous to Case 2.\",\n \"Proposition 9 The class defined by $\\\\phi (x,y)$ is an isomorphism between the $\\\\in _1$ -reduct and the $\\\\in _2$ -reduct.\",\n \"By Lemmas REF , REF and REF .\",\n \"A similar result holds for first order Peano arithmetic, extending the categoricity result of Dedekind [1] of second order Peano arithmetic.\",\n \"The proof (see [4]) of this is similar, but somewhat easier.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08621"}}},{"rowIdx":399,"cells":{"text":{"kind":"list like","value":[["Embedded Pilot-Aided Channel Estimation for OTFS in Delay-Doppler\n Channels"],["Abstract Orthogonal time frequency space (OTFS) modulation was shown to provide significant error performance advantages over orthogonal frequency division multiplexing (OFDM) in delay--Doppler channels.","In order to detect OTFS modulated data, the channel impulse response needs to be known at the receiver.","In this paper, we propose embedded pilot-aided channel estimation schemes for OTFS.","In each OTFS frame, we arrange pilot, guard, and data symbols in the delay--Doppler plane to suitably avoid interference between pilot and data symbols at the receiver.","We develop such symbol arrangements for OTFS over multipath channels with integer and fractional Doppler shifts, respectively.","At the receiver, channel estimation is performed based on a threshold method and the estimated channel information is used for data detection via a message passing (MP) algorithm.","Thanks to our specific embedded symbol arrangements, both channel estimation and data detection are performed within the same OTFS frame with a minimum overhead.","We compare by simulations the error performance of OTFS using the proposed channel estimation and OTFS with ideally known channel information and observe only a marginal performance loss.","We also demonstrate that the proposed channel estimation in OTFS significantly outperforms OFDM with known channel information.","Finally, we present extensions of the proposed schemes to MIMO and multi-user uplink/downlink."],["Introduction","Orthogonal frequency division multiplexing (OFDM) is a popular modulation scheme that are currently deployed in 4G long term evolution (LTE) mobile systems.","OFDM is known to achieve good robustness and high spectral efficiency for time-invariant frequency selective channels.","However, for high-mobility environments such as high-speed railway mobile communications, the channels can be typically time-varying with high Doppler spreads.","Under such high Doppler conditions, OFDM is no longer robust and suffers heavy performance degradations.","Hence, new modulation schemes that are robust to channel time-variations are being extensively explored.","Recently, orthogonal time frequency space (OTFS) modulation was proposed in [1], [2].","OTFS exhibits significant advantages over OFDM in multipath delay–Doppler channels where each path exhibits a different delay and Doppler shift.","In particular, the idea of transmission in the delay-Doppler domain was introduced in [1], [2].","The delay–Doppler domain provides as an alternative representation of a time-varying channel geometry due to moving objects (e.g.","transmitters, receivers, or reflectors) in the scene.","Leveraging on this representation, OTFS multiplexes each information symbol over a two dimensional (2D) orthogonal basis functions, specifically designed to combat the dynamics of time-varying multipath channels.","Then the information symbols placed in the delay-Doppler coordinate system can be converted to the standard time-frequency domain used by traditional modulation schemes such as OFDM.","More recently, in [12], a simplified OTFS structure was proposed by including OFDM for time-frequency signal modulation.","Its extension to the multiple-input multiple-output (MIMO) case was presented in [13].","In general, OTFS uses the delay-Doppler channel response [1], [2], [3] to parameterize the effects of a time-varying channel on any transmitted waveform.","In the delay-Doppler domain, the response captures the dominant scatterers in the channel, with their specific delay and Doppler parameters.","In the time-frequency domain, this corresponds to a standard time-varying impulse response.","Estimating delay-Doppler channel response at the receiver is necessary to perform OTFS detection [4]-[11].","Hence, in [10], [11], [14], [15], pilot-aided channel estimation techniques were investigated.","In [11], an entire OTFS frame was used for pilot transmission and the estimated channel information was used for data detection in next frame.","This method may not be effective if the channel estimation becomes outdated in the following frame.","In [10], [15], OTFS channel estimation was conducted in the time–frequency domain, resulting in higher implementation complexity than that of [11], [14], where the channel estimation was conducted in delay–Doppler domain.","In [14], channel estimation was considered for OTFS with ideal pulse-shaping waveform over channels with integer Doppler shifts only, i.e., when the channel Doppler taps are aligned to integer delay–Doppler grid.","Motivated by [14], in this paper, we consider multipath channels with integer and fractional Doppler shifts, respectivelyFractional Doppler shifts usually occur with a low Doppler resolution..","Under such setting, we propose an embedded OTFS channel estimation scheme for point-to-point single-input single-output (SISO) system with ideal and rectangular pulse-shaping waveforms, respectively.","Specifically, for each OTFS frame, we arrange a single pilot symbol, guard symbols, and data symbols in the delay–Doppler grid to suitably avoid the interferences between pilot and data symbols.","At the receiver, channel estimation is performed based on a threshold method and the estimated channel information is used for data detection via a message passing (MP) algorithm in [4].","Depending on the channel and symbol arrangement, the threshold is chosen to optimize the estimation accuracy.","Thanks to our specific embedded symbol arrangements, both channel estimation and data detection are performed within the same OTFS frame with a minimum overhead (1% for integer Doppler case and 8% for fractional Doppler case).","We compare by simulations the performance of OTFS using the proposed channel estimation schemes and OTFS with perfectly known channel information and observe only a marginal performance degradation.","Further, we show that OTFS with our channel estimation significantly outperforms OFDM, with known channel information.","Finally, we present the extensions of the proposed channel estimation schemes to MIMO and multi-user uplink/downlink.","The rest of the paper is organized as follows.","Section II reviews basic OTFS concepts and results, which lay the foundations for the development of OTFS-based channel estimation schemes in Section III.","Numerical results are presented in Section IV.","Extensions of the proposed channel estimation schemes to other different OTFS systems are presented in Section V followed by the conclusions in Section VI."],["OTFS: Basic concepts and results","In this section, we first review the basic concepts and results of OTFS from [1], [2], [4]."],["Basic OTFS concepts/notations","– The time–frequency signal plane is discretized to a $M\\times N$ grid (for some integers $N, M >0$ ) by sampling time and frequency axes at intervals $T$ (seconds) and $\\Delta f$ (Hz), respectively, i.e., $\\Lambda = \\bigl \\lbrace (nT,m\\Delta f),\\; n=0,\\hdots ,N-1, m=0,\\hdots ,M-1\\bigr \\rbrace \\nonumber $ – The modulated time–frequency samples $X[n,m], n=0,\\hdots ,N-1, m=0,\\hdots ,M-1$ , are transmitted over an OTFS frame with duration $T_f = NT$ and bandwidth $B = M\\Delta f$ .","– The delay–Doppler plane is discretized to a $M\\times N$ information grid $\\Gamma = \\Biggl \\lbrace \\left(\\frac{k}{NT},\\frac{l}{M\\Delta f}\\right),\\; k=0,\\hdots ,N-1, l=0,\\hdots ,M-1\\Biggr \\rbrace , \\nonumber $ where $1/M\\Delta f$ and $1/NT$ represent the quantization steps of the delay and Doppler frequency axes, respectively."],["OTFS mod/demod","The modulator first maps a set of $NM$ information symbols $\\lbrace x[k,l], k=0,\\ldots ,N-1, l=0,\\ldots , M-1\\rbrace $ from a modulation alphabet $\\mathbb {A} = \\lbrace a_1, \\cdots , a_{Q} \\rbrace $ (e.g.","QAM symbols) of size $Q$ , arranged on the delay–Doppler information grid $\\Gamma $ , to $X[n,m]$ in the time–frequency domain grid using the inverse symplectic finite Fourier transform (ISFFT).","Next, the Heisenberg transform is applied to $X[n,m]$ using transmit pulse $g_{\\rm tx}(t)$ to create the time-domain signal $s(t)$ .","The signal $s(t)$ is then transmitted over the wireless channel with complex baseband channel impulse response $h(\\tau ,\\nu )$ , which characterizes the channel response to an impulse with delay $\\tau $ and Doppler $\\nu $ [17].","The received signal $r(t)$ is processed with the Wigner transform (implementing a receiver filter with an impulse response $g_{\\rm rx}(t)$ ) followed by a sampler, yielding $Y[n,m]$ in the time–frequency domain.","We then apply SFFT on $Y[n,m]$ to obtain received symbols $y[k,l]$ in the delay–Doppler domain for symbol detection [1]."],["OTFS input–output analysis","We now look at the relations between received symbols $y[k,l]$ and transmitted symbols $x[k,l]$ .","We assume that $h(\\tau ,\\nu )$ has finite support bounded by $[0,\\tau _{\\rm max}]$ on the delay axis and $[-\\nu _{\\rm max},\\nu _{\\rm max}]$ on the Doppler axis, where $\\tau _{\\rm max}$ and $\\nu _{\\rm max}$ are the maximum delay and the maximum Doppler shift among all channel paths.","Since typically there are only a small number of reflectors in the channel with associated delays and Dopplers, very few parameters are needed to model the channel in the delay-Doppler domain.","The sparse representation of the channel is $h(\\tau ,\\nu ) = \\sum _{i=1}^{P} h_i \\delta (\\tau -\\tau _i) \\delta (\\nu -\\nu _i)$ where $P$ is the number of propagation paths, $h_i$ , $\\tau _i$ , and $\\nu _i$ represent the complex gain, delay, and Doppler shift associated with the $i$ -th path, and $\\delta (\\cdot )$ denotes the Dirac delta function.","We denote by $l_{\\tau _i}, k_{\\nu _i}$ the delay and Doppler taps for the $i$ -th path (relatively to the delay–Doppler grid $\\Gamma $ ) defined as $\\tau _i = \\frac{l_{\\tau _i}}{M\\Delta f},\\;\\;\\nu _i = \\frac{k_{\\nu _i} + \\kappa _{\\nu _i}}{NT}$ where $-\\frac{1}{2}< \\kappa _{\\nu _i} \\le \\frac{1}{2}$ represents the fractional Doppler, i.e., the fractional shift from the nearest Doppler tap $k_{\\nu _i}$ .","We do not need to consider fractional delays, since the resolution $1/M\\Delta f$ of the time axis is sufficient to approximate the path delays to the nearest sampling points in typical wide-band systems [19].","Let us denote $l_{\\tau }$ and $k_{\\nu }$ the delay and Doppler taps corresponding to the largest delay $\\tau _{\\rm max}$ and Doppler $\\nu _{\\rm max}$ .","We also assume that the pulses $g_{\\text{tx}}(t)$ and $g_{\\text{rx}}(t)$ are ideal, meaning that they satisfy the bi-orthogonal property condition [1], i.e., the cross-ambiguity function $A_{g_{\\text{rx}}, g_{\\text{tx}}}(t,f) = 0$ for $t \\in (nT-\\tau _{\\rm max},nT+\\tau _{\\rm max})$ , $f \\in (m\\Delta f-\\nu _{\\rm max},m\\Delta f+\\nu _{\\rm max})$ , $\\forall n,m$ , except for $n=0, m=0$ , where $A_{g_{\\text{rx}}, g_{\\text{tx}}}(t,f) = 1$ with $t \\in (-\\tau _{\\rm max},\\tau _{\\rm max})$ and $f \\in (-\\nu _{\\rm max}, +\\nu _{\\rm max})$ .","The case of non-ideal yet practical rectangular pulses is discussed in Section III."],["Integer Doppler shifts","The relation between $y[k,l]$ and $x[k,l]$ was derived in [4] as $y[k,l]=\\!\\!\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\sum _{l^{\\prime }=0}^{l_\\tau } b[k^{\\prime },l^{\\prime }] \\hat{h}[k^{\\prime },l^{\\prime }] x[[k-k^{\\prime }]_N,&[l-l^{\\prime }]_M] \\nonumber \\\\& + v[k,l] $ where $\\hat{h}[k^{\\prime },l^{\\prime }] = h[k^{\\prime },l^{\\prime }] e^{-j2\\pi \\frac{k^{\\prime }}{N T}\\frac{l^{\\prime }}{M\\Delta f}} $ , $b[k^{\\prime },l^{\\prime }] \\in \\lbrace 0,1\\rbrace $ is the path indicator, i.e., $b[k^{\\prime },l^{\\prime }]=1$ indicates that there is a path with Doppler tap $k^{\\prime }$ and delay tap $l^{\\prime }$ with corresponding path magnitude $\\hat{h}[k^{\\prime },l^{\\prime }]$ , otherwise, there is no such path, i.e., $b[k^{\\prime },l^{\\prime }]=0$ and $\\hat{h}[k^{\\prime },l^{\\prime }]=0$ .","Finally, the term $v[k,l] \\sim \\mathcal {CN} (0, \\sigma ^2)$ is an additive white noise with variance $\\sigma ^2$ , and $[\\cdot ]_N$ , $[\\cdot ]_M$ denote modulo $N$ and $M$ operations, respectively.","We have the total number of paths: $\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\sum _{l^{\\prime }=0}^{l_\\tau } b[k^{\\prime },l^{\\prime }]= P.$ Each path circularly shifts the transmitted symbols by the delay and Doppler taps.","Figure: The integer Doppler case"],["Fractional Doppler shifts","Similarly, the following result was derived in [4] for the fractional Doppler case $\\!\\!\\!\\!\\!\\!\\!\\!","\\!\\!\\!\\!","y[k,l]\\!\\!\\!\\!&=&\\!\\!\\!\\!","\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\sum _{l^{\\prime }=0}^{l_\\tau } b[k^{\\prime },l^{\\prime }] \\sum _{q=0}^{N-1} \\bar{h}[k^{\\prime },l^{\\prime },\\kappa ^{\\prime },q] \\nonumber \\\\&&\\quad x\\left[[k-k^{\\prime }+q]_N, [l - l^{\\prime }]_M\\right] + v[k,l]$ where $\\kappa ^{\\prime }$ denotes the fractional Doppler associated with the $(k^{\\prime },l^{\\prime })$ path, with the path gain $\\bar{h}[k^{\\prime },l^{\\prime },\\kappa ^{\\prime },q] = \\left(\\frac{e^{j {2\\pi } (-q - \\kappa ^{\\prime }) }-1}{N e^{j \\frac{2\\pi }{N} (- q - \\kappa ^{\\prime })}-N}\\right) h[k^{\\prime },l^{\\prime }] e^{-j2\\pi \\frac{k^{\\prime }+\\kappa ^{\\prime }}{N T}\\frac{l^{\\prime }}{M\\Delta f}}.$ It can be seen that with fractional Doppler shifts, each received symbol is affected by more transmitted symbols than in the case of integer Doppler in (REF ).","We can see from (REF ) that when $\\kappa ^{\\prime }=0$ , (REF ) simplifies to (REF ) as expected."],["OTFS data detection via message passing (MP)","From the received symbols $y[k,l]$ , if the channel parameters are known, we can employ the message passing (MP) algorithm in [4] to detect the data symbols $x[k,l]$ using the set of $MN$ linear equations (REF ) or (REF )."],["Embedded channel estimation for point-to-point SISO Case","We first assume that OTFS with ideal waveforms for multipath channel with integer and fractional Doppler cases.","Then we consider the extension to OTFS with practical rectangular waveforms."],["Integer Doppler Case","Let $x_p$ denote the pilot symbol with pilot SNR of ${\\rm SNR}_{p}$ , $x_d[k,l]$ denote the data symbols with data SNR of ${\\rm SNR}_{d}$ located at location $[k,l]$ in the delay–Doppler information grid, and 0 denotes the guard symbol.","Motivated by [14], we place one pilot symbol $x_p$ , $N_n$ of the guard symbols, and $MN-N_n-1$ information symbols in the delay–Doppler grid $\\Gamma $ for each OTFS frame transmission.","The symbols are located in such a way so that at the receiver, we can separate two distinct groups of received symbols: the first group that involves pilot and guard symbols is used for channel estimation, and the second group for data detection.","Moreover, the guard symbols guarantee that the received symbols for channel estimation and data detection are not interfered with each other.","This helps to provide a more accurate channel estimation to be used for data detection within the same frame.","Figure: The fractional Doppler case: Full guard symbolsFor a pilot, we first choose arbitrary grid location $[k_p,l_p]$ such that $0 \\le k_p \\le N-1 $ , and $0 \\le l_p \\le M-1$ .","For ease of representation, we choose $0 \\le l_p-l_{\\tau } \\le l_p \\le l_p+l_{\\tau } \\le M-1$ , and $0 \\le k_p-2k_{\\nu } \\le k_p \\le k_p+2k_{\\nu } \\le N-1$ .","Recall that $l_\\tau $ and $k_\\nu $ denote the taps corresponding to the maximum delay and Doppler values.","We arrange the pilot, guard, and data symbols in the delay–Doppler grid for an OTFS frame transmission as in Fig.","REF : $x[k,l] ={\\left\\lbrace \\begin{array}{ll}x_p & ~~k=k_p, l =l_p,\\\\0 & ~~k_p-2k_{\\nu } \\le k \\le k_p+2k_{\\nu }, \\\\&~~\\quad l_p-l_{\\tau } \\le l \\le l_p+l_{\\tau }, \\\\x_d[k,l] & ~~\\text{otherwise.}\\end{array}\\right.}","$ In this case, we have $N_n=(2l_{\\tau }+1)(4k_{\\nu }+1)-1$ guard symbols.","For example, in Long-Term Evolution (LTE) channels, the overhead for pilot and guard symbols is less than 1% of the data frame [16].","At the receiver, we use the received symbols $y[k,l]$ , $k_p - k_{\\nu }\\le k \\le k_p + k_{\\nu }, l_p \\le l \\le l_p+l_{\\tau }$ for channel estimation.","Then the remaining received symbols $y[k,l]$ on the grid are used for data detection, as shown in Fig.","REF .","Due to the transmit symbol arrangement in (REF ), using (REF ), we can express the received symbols for channel estimation as $y[k,l] = b[k-k_p,l-l_p] \\hat{h}[k-k_p,l-l_p] x_p + v[k,l].","$ for $k \\in [k_p - k_{\\nu }, k_p + k_{\\nu }], l \\in [l_p, l_p+l_{\\tau }]$ .","We can see that if there is a path with Doppler tap $k-k_p$ and delay tap $l-l_p$ , i.e., $b[k-k_p,l-l_p]=1$ , we have $y[k,l] = \\hat{h}[k-k_p,l-l_p] x_p + v[k,l]$ .","Otherwise, $y[k,l] = v[k,l]$ .","Similarly, we can express the received symbols for data detection as in (REF ), demonstrating no interference between the received symbols for channel estimation and data detection.","We propose a simple channel estimation algorithm as follows.","For $k \\in [k_p - k_{\\nu }, k_p + k_{\\nu }], l \\in [l_p, l_p+l_{\\tau }]$ , if the magnitude $|y[k,l]| \\ge \\mathcal {T}$ , where $\\mathcal {T}$ is some positive detection threshold, then we estimate $b[k-k_p,l-l_p]=1$ and $\\hat{h}[k-k_p,l-l_p]=y[k,l]/x_p$ .","Otherwise, we set $b[k-k_p,l-l_p]=\\hat{h}[k-k_p,l-l_p]=0$ .","The proposed threshold-based scheme relies on the fact that if a path exists, the received symbol is the scaled pilot signal with additive white Gaussian noise (see (REF )).","Otherwise, it is only noise.","By varying the threshold $\\mathcal {T}$ , we can alter the miss detection or false alarm probabilities on path detection.","As a result, the error performance of data detection is affected by $\\mathcal {T}$ , as will be shown in Section .","We then use the estimated information for data detection, i.e., the received symbols $y[k,l]$ for data detection are $y[k,l] =\\!\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\sum _{l^{\\prime }=0}^{l_\\tau } b[k^{\\prime },l^{\\prime }] \\hat{h}[k^{\\prime },l^{\\prime }] x_d[[k-k^{\\prime }]_N,&[l-l^{\\prime }]_M] \\nonumber \\\\& + v[k,l] $ for $k \\notin [k_p - k_{\\nu }, k_p + k_{\\nu }]$ or $ l \\notin [l_p,l_p+l_{\\tau }]$ .","Note that we have a total of $MN-(2k_\\nu +1)(l_\\tau +1)$ received symbols to detect a smaller number of $MN-(2l_{\\tau }+1)(4k_{\\nu }+1)$ data symbols via the MP algorithm in [4]."],["The fractional Doppler case","We consider two cases using full guard symbols and reduced guard symbols, respectively.","The former case offers better channel estimation at the expense of the lower spectral efficiency by using more guard symbols and less data symbols, in contrast to the latter case."],["The case with full guard symbols","We arrange the pilot, guard, and data symbols in the delay–Doppler grid, as depicted in Fig.","REF : $x[k,l] ={\\left\\lbrace \\begin{array}{ll}x_p, & k=k_p, l =l_p\\\\0, & 0 \\le k \\le N-1, l_p \\!-\\!l_\\tau \\le l \\le l_p \\!+\\!","l_\\tau \\\\x_d[k,l], & \\text{otherwise.}","\\end{array}\\right.","}$ For simplicity of notation, we choose $0 \\le l_p-l_{\\tau } \\le l_p \\le l_p+l_{\\tau } \\le M-1$ .","We have the number of guard symbols $N_n=(2l_{\\tau }+1)N-1$ , and the overhead for pilot and guard symbols is about 8% in LTE channels [16].","Figure: The fractional Doppler case: Reduced guard symbolsAt the receiver, we use the received symbols $y[k,l], 0 \\le k \\le N-1, l_p \\le l \\le l_p + l_\\tau $ for channel estimation, and the remaining received symbols $y[k,l]$ for data detection (see Fig.","REF ).","Using (REF ), the received symbols $y[k,l]$ for channel estimation are $y[k,l] \\!=\\!\\!\\!\\!\\!\\!\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\!\\!b[k^{\\prime },l\\!-\\!l_p] \\bar{h}[k^{\\prime },l\\!-\\!l_p,\\kappa ^{\\prime },[k_p\\!+\\!k^{\\prime }\\!-\\!k]_N\\!","]\\,x_p \\!\\!+\\!\\!","v[k,l] \\nonumber $ for $k \\in [0, N-1], l \\in [l_p, l_p + l_\\tau ]$ .","We can rewrite $y[k,l]$ as $y[k,l] = \\tilde{b}[l-l_p] \\tilde{h}[[k-k_p]_N,l-l_p]x_p + v[k,l] $ where $\\tilde{b}[l-l_p] ={\\left\\lbrace \\begin{array}{ll}1, & \\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\!\\!\\!b[k^{\\prime },l-l_p] \\ge 1\\\\0, & \\text{otherwise} \\nonumber \\end{array}\\right.","}$ is the path indicator, and $\\tilde{h}[[k-k_p\\!","]_N,l-l_p] = \\!\\!\\!\\!\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\!\\!\\!b[k^{\\prime },l-l_p] \\bar{h}[k^{\\prime },l-l_p,\\kappa ^{\\prime },[k_p\\!+\\!k^{\\prime }-k]_N\\!","]\\nonumber $ is the effective path gain from the pilot symbol $x_p$ at location $[k_p,l_p]$ to the received symbol $y[k,l]$ .","Then $\\tilde{b}[l-l_p]=1$ indicates that there is at least one path with delay tap $l-l_p$ , otherwise, $\\tilde{b}[l-l_p]=0$ .","Based on (REF ), we propose the following threshold-based channel estimation algorithm.","For $k \\in [0, N-1], l \\in [l_p, l_p + l_\\tau ]$ , if $|y[k,l]| \\ge \\mathcal {T}$ , then we have $\\tilde{b}[l-l_p]=1$ , and $\\tilde{h}[[k-k_p]_N,l-l_p] = y[k,l]/x_p.$ Otherwise, we set $ \\tilde{b}[l-l_p]=\\tilde{h}[[k-k_p]_N,l-l_p] = 0$ .","Unlike the integer Doppler case, where we estimate whether an individual path with given delay and Doppler taps exists, in this case, we estimate whether there exists at least one path with a given delay tap.","For data detection, similar to (REF ), we rewrite (REF ) as $y[k,l]= \\sum _{l^{\\prime }=0}^{l_\\tau }\\tilde{b}[l^{\\prime }] \\sum _{k^{\\prime }=0}^{N-1} \\tilde{h}[k^{\\prime },l^{\\prime }] x_d[[k\\!-\\!k_{i^{\\prime }}]_N,&[l-l^{\\prime }]_M] \\nonumber \\\\&+v[k,l] $ for $k \\in [0,N-1]$ and $l \\notin [l_p, l_p + l_\\tau ]$ .","Now we can adapt the MP algorithm in [4] for data detection in (REF ).","Note that, to guarantee no interference between the received symbols for channel estimation and data detection, the guard symbols need to expand over a wider range over the Doppler axis, when compared to the integer Doppler case."],["The case of reduced guard symbols","Employing full guard symbols to avoid interferences provide more accurate channel estimation but with reduced spectral efficiency.","To improve the spectral efficiency, we can reduce the number of guard symbols and thus increase the number of data symbols, as discussed below.","We arrange the symbols as in Fig.","REF $x[k,l] ={\\left\\lbrace \\begin{array}{ll}x_p & ~~k=k_p, l =l_p,\\\\0 & ~~k_p-2k_{\\nu }-2 \\hat{k} \\le k \\le k_p+2k_{\\nu } + 2 \\hat{k}, \\\\&~~\\quad l_p-l_{\\tau } \\le l \\le l_p+l_{\\tau }, \\\\x_d[k,l] & ~~\\text{otherwise}\\end{array}\\right.}","$ for some integer $\\hat{k}$ .","For smaller $\\hat{k}$ , less guard and more data symbols are used, resulting in an increased spectral efficiency.","The received symbols $y[k,l], k_p-k_{\\nu }- \\hat{k} \\le k \\le k_p+k_{\\nu } + \\hat{k}, l_p \\le l \\le l_p + l_\\tau $ are used for channel estimation, while the remaining $y[k,l]$ are used for data detection (see Fig.","REF ) From (REF ), for channel estimation, we have $y[k,l] = \\tilde{b}[l\\!-\\!l_p] \\tilde{h}[[k\\!-\\!k_p]_N,l\\!-\\!l_p]x_p \\!+\\!","\\mathcal {I}[k,l] \\!+\\!","v[k,l] $ for $k_p-k_{\\nu }- \\hat{k} \\le k \\le k_p+k_{\\nu } + \\hat{k}, l_p \\le l \\le l_p + l_\\tau $ .","The second term $\\mathcal {I}[k,l]$ is the interferences from all neighboring data symbols $x_d[k,l]$ , i.e., $\\mathcal {I}[k,l]&=&\\!\\!\\!\\!","\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\sum _{l^{\\prime }=0}^{l_\\tau } b[k^{\\prime },l^{\\prime }]\\hspace{-19.91692pt} \\sum _{q \\notin [k_p-2k_{\\nu }- 2\\hat{k}, k_p+2k_{\\nu } + 2\\hat{k}]} \\hspace{-19.91692pt} \\bar{h}[k^{\\prime },l^{\\prime },\\kappa ^{\\prime },q] \\nonumber \\\\&&\\quad x_d\\left[[k-k^{\\prime }+q]_N, [l - l^{\\prime }]_M\\right]$ We observe that the interference $\\mathcal {I}[k,l]$ gets larger for smaller $\\hat{k}$ , and similarly for the interference from pilot symbols to the received symbols for data detection.","Similar to the case of full guard symbols, we develop a threshold-based algorithm to estimate $\\tilde{b}[l-l_p]$ and $ \\tilde{h}[[k-k_p]_N,l-l_p]$ based on (REF ) by treating $\\mathcal {I}[k,l]$ as additive noise.","Based on the simulation results (see next section), we demonstrate that the performance gap of the full guard symbols case (8% overhead) and reduced guard symbols case (2% overhead) is indeed marginal."],["OTFS with rectangular waveforms","So far, we have assumed ideal transmit $g_{\\text{tx}}(t)$ and receive $g_{\\text{rx}}(t)$ pulses.","Since the ideal pulses cannot be realized in practice, we now investigate OTFS with the more practical rectangular pulses at both transmitter and receiver.","Although these pulses do not satisfy the bi-orthogonality conditions [5], we show that the proposed embedded channel estimation schemes can also be employed for this case.","Consider the integer Doppler case for simplicity.","With rectangular pulses, the input-output symbol relationship in [5] can be rewritten as $y[k,l]=\\!\\!\\!\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\sum _{l^{\\prime }=0}^{l_\\tau } b[k^{\\prime },l^{\\prime }] \\hat{h}[k^{\\prime },l^{\\prime }]\\alpha [k,l] x[[k\\!-\\!k^{\\prime }]_N,&[l\\!-\\!l^{\\prime }]_M\\!]","\\nonumber \\\\& + v[k,l] \\nonumber $ where $\\alpha [k,l] & ={\\left\\lbrace \\begin{array}{ll}e^{j2\\pi \\left(\\frac{l-l^{\\prime }}{M}\\right) \\frac{k^{\\prime }}{N} } & l^{\\prime } \\le l < M\\\\\\frac{N-1}{N} e^{j2\\pi \\left(\\frac{l-l^{\\prime }}{M}\\right) \\frac{k^{\\prime }}{N} } e^{-j 2 \\pi \\left(\\frac{[k-k^{\\prime }]_N}{N} \\right)} & 0\\le l0$ ) by sampling time and frequency axes at intervals $T$ (seconds) and $\\\\Delta f$ (Hz), respectively, i.e., $\\\\Lambda = \\\\bigl \\\\lbrace (nT,m\\\\Delta f),\\\\; n=0,\\\\hdots ,N-1, m=0,\\\\hdots ,M-1\\\\bigr \\\\rbrace \\\\nonumber $ – The modulated time–frequency samples $X[n,m], n=0,\\\\hdots ,N-1, m=0,\\\\hdots ,M-1$ , are transmitted over an OTFS frame with duration $T_f = NT$ and bandwidth $B = M\\\\Delta f$ .\",\n \"– The delay–Doppler plane is discretized to a $M\\\\times N$ information grid $\\\\Gamma = \\\\Biggl \\\\lbrace \\\\left(\\\\frac{k}{NT},\\\\frac{l}{M\\\\Delta f}\\\\right),\\\\; k=0,\\\\hdots ,N-1, l=0,\\\\hdots ,M-1\\\\Biggr \\\\rbrace , \\\\nonumber $ where $1/M\\\\Delta f$ and $1/NT$ represent the quantization steps of the delay and Doppler frequency axes, respectively.\"\n ],\n [\n \"OTFS mod/demod\",\n \"The modulator first maps a set of $NM$ information symbols $\\\\lbrace x[k,l], k=0,\\\\ldots ,N-1, l=0,\\\\ldots , M-1\\\\rbrace $ from a modulation alphabet $\\\\mathbb {A} = \\\\lbrace a_1, \\\\cdots , a_{Q} \\\\rbrace $ (e.g.\",\n \"QAM symbols) of size $Q$ , arranged on the delay–Doppler information grid $\\\\Gamma $ , to $X[n,m]$ in the time–frequency domain grid using the inverse symplectic finite Fourier transform (ISFFT).\",\n \"Next, the Heisenberg transform is applied to $X[n,m]$ using transmit pulse $g_{\\\\rm tx}(t)$ to create the time-domain signal $s(t)$ .\",\n \"The signal $s(t)$ is then transmitted over the wireless channel with complex baseband channel impulse response $h(\\\\tau ,\\\\nu )$ , which characterizes the channel response to an impulse with delay $\\\\tau $ and Doppler $\\\\nu $ [17].\",\n \"The received signal $r(t)$ is processed with the Wigner transform (implementing a receiver filter with an impulse response $g_{\\\\rm rx}(t)$ ) followed by a sampler, yielding $Y[n,m]$ in the time–frequency domain.\",\n \"We then apply SFFT on $Y[n,m]$ to obtain received symbols $y[k,l]$ in the delay–Doppler domain for symbol detection [1].\"\n ],\n [\n \"OTFS input–output analysis\",\n \"We now look at the relations between received symbols $y[k,l]$ and transmitted symbols $x[k,l]$ .\",\n \"We assume that $h(\\\\tau ,\\\\nu )$ has finite support bounded by $[0,\\\\tau _{\\\\rm max}]$ on the delay axis and $[-\\\\nu _{\\\\rm max},\\\\nu _{\\\\rm max}]$ on the Doppler axis, where $\\\\tau _{\\\\rm max}$ and $\\\\nu _{\\\\rm max}$ are the maximum delay and the maximum Doppler shift among all channel paths.\",\n \"Since typically there are only a small number of reflectors in the channel with associated delays and Dopplers, very few parameters are needed to model the channel in the delay-Doppler domain.\",\n \"The sparse representation of the channel is $h(\\\\tau ,\\\\nu ) = \\\\sum _{i=1}^{P} h_i \\\\delta (\\\\tau -\\\\tau _i) \\\\delta (\\\\nu -\\\\nu _i)$ where $P$ is the number of propagation paths, $h_i$ , $\\\\tau _i$ , and $\\\\nu _i$ represent the complex gain, delay, and Doppler shift associated with the $i$ -th path, and $\\\\delta (\\\\cdot )$ denotes the Dirac delta function.\",\n \"We denote by $l_{\\\\tau _i}, k_{\\\\nu _i}$ the delay and Doppler taps for the $i$ -th path (relatively to the delay–Doppler grid $\\\\Gamma $ ) defined as $\\\\tau _i = \\\\frac{l_{\\\\tau _i}}{M\\\\Delta f},\\\\;\\\\;\\\\nu _i = \\\\frac{k_{\\\\nu _i} + \\\\kappa _{\\\\nu _i}}{NT}$ where $-\\\\frac{1}{2}< \\\\kappa _{\\\\nu _i} \\\\le \\\\frac{1}{2}$ represents the fractional Doppler, i.e., the fractional shift from the nearest Doppler tap $k_{\\\\nu _i}$ .\",\n \"We do not need to consider fractional delays, since the resolution $1/M\\\\Delta f$ of the time axis is sufficient to approximate the path delays to the nearest sampling points in typical wide-band systems [19].\",\n \"Let us denote $l_{\\\\tau }$ and $k_{\\\\nu }$ the delay and Doppler taps corresponding to the largest delay $\\\\tau _{\\\\rm max}$ and Doppler $\\\\nu _{\\\\rm max}$ .\",\n \"We also assume that the pulses $g_{\\\\text{tx}}(t)$ and $g_{\\\\text{rx}}(t)$ are ideal, meaning that they satisfy the bi-orthogonal property condition [1], i.e., the cross-ambiguity function $A_{g_{\\\\text{rx}}, g_{\\\\text{tx}}}(t,f) = 0$ for $t \\\\in (nT-\\\\tau _{\\\\rm max},nT+\\\\tau _{\\\\rm max})$ , $f \\\\in (m\\\\Delta f-\\\\nu _{\\\\rm max},m\\\\Delta f+\\\\nu _{\\\\rm max})$ , $\\\\forall n,m$ , except for $n=0, m=0$ , where $A_{g_{\\\\text{rx}}, g_{\\\\text{tx}}}(t,f) = 1$ with $t \\\\in (-\\\\tau _{\\\\rm max},\\\\tau _{\\\\rm max})$ and $f \\\\in (-\\\\nu _{\\\\rm max}, +\\\\nu _{\\\\rm max})$ .\",\n \"The case of non-ideal yet practical rectangular pulses is discussed in Section III.\"\n ],\n [\n \"Integer Doppler shifts\",\n \"The relation between $y[k,l]$ and $x[k,l]$ was derived in [4] as $y[k,l]=\\\\!\\\\!\\\\sum _{k^{\\\\prime }=-k_\\\\nu }^{k_\\\\nu } \\\\sum _{l^{\\\\prime }=0}^{l_\\\\tau } b[k^{\\\\prime },l^{\\\\prime }] \\\\hat{h}[k^{\\\\prime },l^{\\\\prime }] x[[k-k^{\\\\prime }]_N,&[l-l^{\\\\prime }]_M] \\\\nonumber \\\\\\\\& + v[k,l] $ where $\\\\hat{h}[k^{\\\\prime },l^{\\\\prime }] = h[k^{\\\\prime },l^{\\\\prime }] e^{-j2\\\\pi \\\\frac{k^{\\\\prime }}{N T}\\\\frac{l^{\\\\prime }}{M\\\\Delta f}} $ , $b[k^{\\\\prime },l^{\\\\prime }] \\\\in \\\\lbrace 0,1\\\\rbrace $ is the path indicator, i.e., $b[k^{\\\\prime },l^{\\\\prime }]=1$ indicates that there is a path with Doppler tap $k^{\\\\prime }$ and delay tap $l^{\\\\prime }$ with corresponding path magnitude $\\\\hat{h}[k^{\\\\prime },l^{\\\\prime }]$ , otherwise, there is no such path, i.e., $b[k^{\\\\prime },l^{\\\\prime }]=0$ and $\\\\hat{h}[k^{\\\\prime },l^{\\\\prime }]=0$ .\",\n \"Finally, the term $v[k,l] \\\\sim \\\\mathcal {CN} (0, \\\\sigma ^2)$ is an additive white noise with variance $\\\\sigma ^2$ , and $[\\\\cdot ]_N$ , $[\\\\cdot ]_M$ denote modulo $N$ and $M$ operations, respectively.\",\n \"We have the total number of paths: $\\\\sum _{k^{\\\\prime }=-k_\\\\nu }^{k_\\\\nu } \\\\sum _{l^{\\\\prime }=0}^{l_\\\\tau } b[k^{\\\\prime },l^{\\\\prime }]= P.$ Each path circularly shifts the transmitted symbols by the delay and Doppler taps.\",\n \"Figure: The integer Doppler case\"\n ],\n [\n \"Fractional Doppler shifts\",\n \"Similarly, the following result was derived in [4] for the fractional Doppler case $\\\\!\\\\!\\\\!\\\\!\\\\!\\\\!\\\\!\\\\!\",\n \"\\\\!\\\\!\\\\!\\\\!\",\n \"y[k,l]\\\\!\\\\!\\\\!\\\\!&=&\\\\!\\\\!\\\\!\\\\!\",\n \"\\\\sum _{k^{\\\\prime }=-k_\\\\nu }^{k_\\\\nu } \\\\sum _{l^{\\\\prime }=0}^{l_\\\\tau } b[k^{\\\\prime },l^{\\\\prime }] \\\\sum _{q=0}^{N-1} \\\\bar{h}[k^{\\\\prime },l^{\\\\prime },\\\\kappa ^{\\\\prime },q] \\\\nonumber \\\\\\\\&&\\\\quad x\\\\left[[k-k^{\\\\prime }+q]_N, [l - l^{\\\\prime }]_M\\\\right] + v[k,l]$ where $\\\\kappa ^{\\\\prime }$ denotes the fractional Doppler associated with the $(k^{\\\\prime },l^{\\\\prime })$ path, with the path gain $\\\\bar{h}[k^{\\\\prime },l^{\\\\prime },\\\\kappa ^{\\\\prime },q] = \\\\left(\\\\frac{e^{j {2\\\\pi } (-q - \\\\kappa ^{\\\\prime }) }-1}{N e^{j \\\\frac{2\\\\pi }{N} (- q - \\\\kappa ^{\\\\prime })}-N}\\\\right) h[k^{\\\\prime },l^{\\\\prime }] e^{-j2\\\\pi \\\\frac{k^{\\\\prime }+\\\\kappa ^{\\\\prime }}{N T}\\\\frac{l^{\\\\prime }}{M\\\\Delta f}}.$ It can be seen that with fractional Doppler shifts, each received symbol is affected by more transmitted symbols than in the case of integer Doppler in (REF ).\",\n \"We can see from (REF ) that when $\\\\kappa ^{\\\\prime }=0$ , (REF ) simplifies to (REF ) as expected.\"\n ],\n [\n \"OTFS data detection via message passing (MP)\",\n \"From the received symbols $y[k,l]$ , if the channel parameters are known, we can employ the message passing (MP) algorithm in [4] to detect the data symbols $x[k,l]$ using the set of $MN$ linear equations (REF ) or (REF ).\"\n ],\n [\n \"Embedded channel estimation for point-to-point SISO Case\",\n \"We first assume that OTFS with ideal waveforms for multipath channel with integer and fractional Doppler cases.\",\n \"Then we consider the extension to OTFS with practical rectangular waveforms.\"\n ],\n [\n \"Integer Doppler Case\",\n \"Let $x_p$ denote the pilot symbol with pilot SNR of ${\\\\rm SNR}_{p}$ , $x_d[k,l]$ denote the data symbols with data SNR of ${\\\\rm SNR}_{d}$ located at location $[k,l]$ in the delay–Doppler information grid, and 0 denotes the guard symbol.\",\n \"Motivated by [14], we place one pilot symbol $x_p$ , $N_n$ of the guard symbols, and $MN-N_n-1$ information symbols in the delay–Doppler grid $\\\\Gamma $ for each OTFS frame transmission.\",\n \"The symbols are located in such a way so that at the receiver, we can separate two distinct groups of received symbols: the first group that involves pilot and guard symbols is used for channel estimation, and the second group for data detection.\",\n \"Moreover, the guard symbols guarantee that the received symbols for channel estimation and data detection are not interfered with each other.\",\n \"This helps to provide a more accurate channel estimation to be used for data detection within the same frame.\",\n \"Figure: The fractional Doppler case: Full guard symbolsFor a pilot, we first choose arbitrary grid location $[k_p,l_p]$ such that $0 \\\\le k_p \\\\le N-1 $ , and $0 \\\\le l_p \\\\le M-1$ .\",\n \"For ease of representation, we choose $0 \\\\le l_p-l_{\\\\tau } \\\\le l_p \\\\le l_p+l_{\\\\tau } \\\\le M-1$ , and $0 \\\\le k_p-2k_{\\\\nu } \\\\le k_p \\\\le k_p+2k_{\\\\nu } \\\\le N-1$ .\",\n \"Recall that $l_\\\\tau $ and $k_\\\\nu $ denote the taps corresponding to the maximum delay and Doppler values.\",\n \"We arrange the pilot, guard, and data symbols in the delay–Doppler grid for an OTFS frame transmission as in Fig.\",\n \"REF : $x[k,l] ={\\\\left\\\\lbrace \\\\begin{array}{ll}x_p & ~~k=k_p, l =l_p,\\\\\\\\0 & ~~k_p-2k_{\\\\nu } \\\\le k \\\\le k_p+2k_{\\\\nu }, \\\\\\\\&~~\\\\quad l_p-l_{\\\\tau } \\\\le l \\\\le l_p+l_{\\\\tau }, \\\\\\\\x_d[k,l] & ~~\\\\text{otherwise.}\\\\end{array}\\\\right.}\",\n \"$ In this case, we have $N_n=(2l_{\\\\tau }+1)(4k_{\\\\nu }+1)-1$ guard symbols.\",\n \"For example, in Long-Term Evolution (LTE) channels, the overhead for pilot and guard symbols is less than 1% of the data frame [16].\",\n \"At the receiver, we use the received symbols $y[k,l]$ , $k_p - k_{\\\\nu }\\\\le k \\\\le k_p + k_{\\\\nu }, l_p \\\\le l \\\\le l_p+l_{\\\\tau }$ for channel estimation.\",\n \"Then the remaining received symbols $y[k,l]$ on the grid are used for data detection, as shown in Fig.\",\n \"REF .\",\n \"Due to the transmit symbol arrangement in (REF ), using (REF ), we can express the received symbols for channel estimation as $y[k,l] = b[k-k_p,l-l_p] \\\\hat{h}[k-k_p,l-l_p] x_p + v[k,l].\",\n \"$ for $k \\\\in [k_p - k_{\\\\nu }, k_p + k_{\\\\nu }], l \\\\in [l_p, l_p+l_{\\\\tau }]$ .\",\n \"We can see that if there is a path with Doppler tap $k-k_p$ and delay tap $l-l_p$ , i.e., $b[k-k_p,l-l_p]=1$ , we have $y[k,l] = \\\\hat{h}[k-k_p,l-l_p] x_p + v[k,l]$ .\",\n \"Otherwise, $y[k,l] = v[k,l]$ .\",\n \"Similarly, we can express the received symbols for data detection as in (REF ), demonstrating no interference between the received symbols for channel estimation and data detection.\",\n \"We propose a simple channel estimation algorithm as follows.\",\n \"For $k \\\\in [k_p - k_{\\\\nu }, k_p + k_{\\\\nu }], l \\\\in [l_p, l_p+l_{\\\\tau }]$ , if the magnitude $|y[k,l]| \\\\ge \\\\mathcal {T}$ , where $\\\\mathcal {T}$ is some positive detection threshold, then we estimate $b[k-k_p,l-l_p]=1$ and $\\\\hat{h}[k-k_p,l-l_p]=y[k,l]/x_p$ .\",\n \"Otherwise, we set $b[k-k_p,l-l_p]=\\\\hat{h}[k-k_p,l-l_p]=0$ .\",\n \"The proposed threshold-based scheme relies on the fact that if a path exists, the received symbol is the scaled pilot signal with additive white Gaussian noise (see (REF )).\",\n \"Otherwise, it is only noise.\",\n \"By varying the threshold $\\\\mathcal {T}$ , we can alter the miss detection or false alarm probabilities on path detection.\",\n \"As a result, the error performance of data detection is affected by $\\\\mathcal {T}$ , as will be shown in Section .\",\n \"We then use the estimated information for data detection, i.e., the received symbols $y[k,l]$ for data detection are $y[k,l] =\\\\!\\\\sum _{k^{\\\\prime }=-k_\\\\nu }^{k_\\\\nu } \\\\sum _{l^{\\\\prime }=0}^{l_\\\\tau } b[k^{\\\\prime },l^{\\\\prime }] \\\\hat{h}[k^{\\\\prime },l^{\\\\prime }] x_d[[k-k^{\\\\prime }]_N,&[l-l^{\\\\prime }]_M] \\\\nonumber \\\\\\\\& + v[k,l] $ for $k \\\\notin [k_p - k_{\\\\nu }, k_p + k_{\\\\nu }]$ or $ l \\\\notin [l_p,l_p+l_{\\\\tau }]$ .\",\n \"Note that we have a total of $MN-(2k_\\\\nu +1)(l_\\\\tau +1)$ received symbols to detect a smaller number of $MN-(2l_{\\\\tau }+1)(4k_{\\\\nu }+1)$ data symbols via the MP algorithm in [4].\"\n ],\n [\n \"The fractional Doppler case\",\n \"We consider two cases using full guard symbols and reduced guard symbols, respectively.\",\n \"The former case offers better channel estimation at the expense of the lower spectral efficiency by using more guard symbols and less data symbols, in contrast to the latter case.\"\n ],\n [\n \"The case with full guard symbols\",\n \"We arrange the pilot, guard, and data symbols in the delay–Doppler grid, as depicted in Fig.\",\n \"REF : $x[k,l] ={\\\\left\\\\lbrace \\\\begin{array}{ll}x_p, & k=k_p, l =l_p\\\\\\\\0, & 0 \\\\le k \\\\le N-1, l_p \\\\!-\\\\!l_\\\\tau \\\\le l \\\\le l_p \\\\!+\\\\!\",\n \"l_\\\\tau \\\\\\\\x_d[k,l], & \\\\text{otherwise.}\",\n \"\\\\end{array}\\\\right.\",\n \"}$ For simplicity of notation, we choose $0 \\\\le l_p-l_{\\\\tau } \\\\le l_p \\\\le l_p+l_{\\\\tau } \\\\le M-1$ .\",\n \"We have the number of guard symbols $N_n=(2l_{\\\\tau }+1)N-1$ , and the overhead for pilot and guard symbols is about 8% in LTE channels [16].\",\n \"Figure: The fractional Doppler case: Reduced guard symbolsAt the receiver, we use the received symbols $y[k,l], 0 \\\\le k \\\\le N-1, l_p \\\\le l \\\\le l_p + l_\\\\tau $ for channel estimation, and the remaining received symbols $y[k,l]$ for data detection (see Fig.\",\n \"REF ).\",\n \"Using (REF ), the received symbols $y[k,l]$ for channel estimation are $y[k,l] \\\\!=\\\\!\\\\!\\\\!\\\\!\\\\!\\\\!\\\\sum _{k^{\\\\prime }=-k_\\\\nu }^{k_\\\\nu } \\\\!\\\\!b[k^{\\\\prime },l\\\\!-\\\\!l_p] \\\\bar{h}[k^{\\\\prime },l\\\\!-\\\\!l_p,\\\\kappa ^{\\\\prime },[k_p\\\\!+\\\\!k^{\\\\prime }\\\\!-\\\\!k]_N\\\\!\",\n \"]\\\\,x_p \\\\!\\\\!+\\\\!\\\\!\",\n \"v[k,l] \\\\nonumber $ for $k \\\\in [0, N-1], l \\\\in [l_p, l_p + l_\\\\tau ]$ .\",\n \"We can rewrite $y[k,l]$ as $y[k,l] = \\\\tilde{b}[l-l_p] \\\\tilde{h}[[k-k_p]_N,l-l_p]x_p + v[k,l] $ where $\\\\tilde{b}[l-l_p] ={\\\\left\\\\lbrace \\\\begin{array}{ll}1, & \\\\sum _{k^{\\\\prime }=-k_\\\\nu }^{k_\\\\nu } \\\\!\\\\!\\\\!b[k^{\\\\prime },l-l_p] \\\\ge 1\\\\\\\\0, & \\\\text{otherwise} \\\\nonumber \\\\end{array}\\\\right.\",\n \"}$ is the path indicator, and $\\\\tilde{h}[[k-k_p\\\\!\",\n \"]_N,l-l_p] = \\\\!\\\\!\\\\!\\\\!\\\\sum _{k^{\\\\prime }=-k_\\\\nu }^{k_\\\\nu } \\\\!\\\\!\\\\!b[k^{\\\\prime },l-l_p] \\\\bar{h}[k^{\\\\prime },l-l_p,\\\\kappa ^{\\\\prime },[k_p\\\\!+\\\\!k^{\\\\prime }-k]_N\\\\!\",\n \"]\\\\nonumber $ is the effective path gain from the pilot symbol $x_p$ at location $[k_p,l_p]$ to the received symbol $y[k,l]$ .\",\n \"Then $\\\\tilde{b}[l-l_p]=1$ indicates that there is at least one path with delay tap $l-l_p$ , otherwise, $\\\\tilde{b}[l-l_p]=0$ .\",\n \"Based on (REF ), we propose the following threshold-based channel estimation algorithm.\",\n \"For $k \\\\in [0, N-1], l \\\\in [l_p, l_p + l_\\\\tau ]$ , if $|y[k,l]| \\\\ge \\\\mathcal {T}$ , then we have $\\\\tilde{b}[l-l_p]=1$ , and $\\\\tilde{h}[[k-k_p]_N,l-l_p] = y[k,l]/x_p.$ Otherwise, we set $ \\\\tilde{b}[l-l_p]=\\\\tilde{h}[[k-k_p]_N,l-l_p] = 0$ .\",\n \"Unlike the integer Doppler case, where we estimate whether an individual path with given delay and Doppler taps exists, in this case, we estimate whether there exists at least one path with a given delay tap.\",\n \"For data detection, similar to (REF ), we rewrite (REF ) as $y[k,l]= \\\\sum _{l^{\\\\prime }=0}^{l_\\\\tau }\\\\tilde{b}[l^{\\\\prime }] \\\\sum _{k^{\\\\prime }=0}^{N-1} \\\\tilde{h}[k^{\\\\prime },l^{\\\\prime }] x_d[[k\\\\!-\\\\!k_{i^{\\\\prime }}]_N,&[l-l^{\\\\prime }]_M] \\\\nonumber \\\\\\\\&+v[k,l] $ for $k \\\\in [0,N-1]$ and $l \\\\notin [l_p, l_p + l_\\\\tau ]$ .\",\n \"Now we can adapt the MP algorithm in [4] for data detection in (REF ).\",\n \"Note that, to guarantee no interference between the received symbols for channel estimation and data detection, the guard symbols need to expand over a wider range over the Doppler axis, when compared to the integer Doppler case.\"\n ],\n [\n \"The case of reduced guard symbols\",\n \"Employing full guard symbols to avoid interferences provide more accurate channel estimation but with reduced spectral efficiency.\",\n \"To improve the spectral efficiency, we can reduce the number of guard symbols and thus increase the number of data symbols, as discussed below.\",\n \"We arrange the symbols as in Fig.\",\n \"REF $x[k,l] ={\\\\left\\\\lbrace \\\\begin{array}{ll}x_p & ~~k=k_p, l =l_p,\\\\\\\\0 & ~~k_p-2k_{\\\\nu }-2 \\\\hat{k} \\\\le k \\\\le k_p+2k_{\\\\nu } + 2 \\\\hat{k}, \\\\\\\\&~~\\\\quad l_p-l_{\\\\tau } \\\\le l \\\\le l_p+l_{\\\\tau }, \\\\\\\\x_d[k,l] & ~~\\\\text{otherwise}\\\\end{array}\\\\right.}\",\n \"$ for some integer $\\\\hat{k}$ .\",\n \"For smaller $\\\\hat{k}$ , less guard and more data symbols are used, resulting in an increased spectral efficiency.\",\n \"The received symbols $y[k,l], k_p-k_{\\\\nu }- \\\\hat{k} \\\\le k \\\\le k_p+k_{\\\\nu } + \\\\hat{k}, l_p \\\\le l \\\\le l_p + l_\\\\tau $ are used for channel estimation, while the remaining $y[k,l]$ are used for data detection (see Fig.\",\n \"REF ) From (REF ), for channel estimation, we have $y[k,l] = \\\\tilde{b}[l\\\\!-\\\\!l_p] \\\\tilde{h}[[k\\\\!-\\\\!k_p]_N,l\\\\!-\\\\!l_p]x_p \\\\!+\\\\!\",\n \"\\\\mathcal {I}[k,l] \\\\!+\\\\!\",\n \"v[k,l] $ for $k_p-k_{\\\\nu }- \\\\hat{k} \\\\le k \\\\le k_p+k_{\\\\nu } + \\\\hat{k}, l_p \\\\le l \\\\le l_p + l_\\\\tau $ .\",\n \"The second term $\\\\mathcal {I}[k,l]$ is the interferences from all neighboring data symbols $x_d[k,l]$ , i.e., $\\\\mathcal {I}[k,l]&=&\\\\!\\\\!\\\\!\\\\!\",\n \"\\\\sum _{k^{\\\\prime }=-k_\\\\nu }^{k_\\\\nu } \\\\sum _{l^{\\\\prime }=0}^{l_\\\\tau } b[k^{\\\\prime },l^{\\\\prime }]\\\\hspace{-19.91692pt} \\\\sum _{q \\\\notin [k_p-2k_{\\\\nu }- 2\\\\hat{k}, k_p+2k_{\\\\nu } + 2\\\\hat{k}]} \\\\hspace{-19.91692pt} \\\\bar{h}[k^{\\\\prime },l^{\\\\prime },\\\\kappa ^{\\\\prime },q] \\\\nonumber \\\\\\\\&&\\\\quad x_d\\\\left[[k-k^{\\\\prime }+q]_N, [l - l^{\\\\prime }]_M\\\\right]$ We observe that the interference $\\\\mathcal {I}[k,l]$ gets larger for smaller $\\\\hat{k}$ , and similarly for the interference from pilot symbols to the received symbols for data detection.\",\n \"Similar to the case of full guard symbols, we develop a threshold-based algorithm to estimate $\\\\tilde{b}[l-l_p]$ and $ \\\\tilde{h}[[k-k_p]_N,l-l_p]$ based on (REF ) by treating $\\\\mathcal {I}[k,l]$ as additive noise.\",\n \"Based on the simulation results (see next section), we demonstrate that the performance gap of the full guard symbols case (8% overhead) and reduced guard symbols case (2% overhead) is indeed marginal.\"\n ],\n [\n \"OTFS with rectangular waveforms\",\n \"So far, we have assumed ideal transmit $g_{\\\\text{tx}}(t)$ and receive $g_{\\\\text{rx}}(t)$ pulses.\",\n \"Since the ideal pulses cannot be realized in practice, we now investigate OTFS with the more practical rectangular pulses at both transmitter and receiver.\",\n \"Although these pulses do not satisfy the bi-orthogonality conditions [5], we show that the proposed embedded channel estimation schemes can also be employed for this case.\",\n \"Consider the integer Doppler case for simplicity.\",\n \"With rectangular pulses, the input-output symbol relationship in [5] can be rewritten as $y[k,l]=\\\\!\\\\!\\\\!\\\\sum _{k^{\\\\prime }=-k_\\\\nu }^{k_\\\\nu } \\\\sum _{l^{\\\\prime }=0}^{l_\\\\tau } b[k^{\\\\prime },l^{\\\\prime }] \\\\hat{h}[k^{\\\\prime },l^{\\\\prime }]\\\\alpha [k,l] x[[k\\\\!-\\\\!k^{\\\\prime }]_N,&[l\\\\!-\\\\!l^{\\\\prime }]_M\\\\!]\",\n \"\\\\nonumber \\\\\\\\& + v[k,l] \\\\nonumber $ where $\\\\alpha [k,l] & ={\\\\left\\\\lbrace \\\\begin{array}{ll}e^{j2\\\\pi \\\\left(\\\\frac{l-l^{\\\\prime }}{M}\\\\right) \\\\frac{k^{\\\\prime }}{N} } & l^{\\\\prime } \\\\le l < M\\\\\\\\\\\\frac{N-1}{N} e^{j2\\\\pi \\\\left(\\\\frac{l-l^{\\\\prime }}{M}\\\\right) \\\\frac{k^{\\\\prime }}{N} } e^{-j 2 \\\\pi \\\\left(\\\\frac{[k-k^{\\\\prime }]_N}{N} \\\\right)} & 0\\\\le l
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[
"DeepTracker: Visualizing the Training Process of Convolutional Neural\n Networks"
],
[
"Abstract Deep convolutional neural networks (CNNs) have achieved remarkable success in various fields.",
"However, training an excellent CNN is practically a trial-and-error process that consumes a tremendous amount of time and computer resources.",
"To accelerate the training process and reduce the number of trials, experts need to understand what has occurred in the training process and why the resulting CNN behaves as such.",
"However, current popular training platforms, such as TensorFlow, only provide very little and general information, such as training/validation errors, which is far from enough to serve this purpose.",
"To bridge this gap and help domain experts with their training tasks in a practical environment, we propose a visual analytics system, DeepTracker, to facilitate the exploration of the rich dynamics of CNN training processes and to identify the unusual patterns that are hidden behind the huge amount of training log.",
"Specifically,we combine a hierarchical index mechanism and a set of hierarchical small multiples to help experts explore the entire training log from different levels of detail.",
"We also introduce a novel cube-style visualization to reveal the complex correlations among multiple types of heterogeneous training data including neuron weights, validation images, and training iterations.",
"Three case studies are conducted to demonstrate how DeepTracker provides its users with valuable knowledge in an industry-level CNN training process, namely in our case, training ResNet-50 on the ImageNet dataset.",
"We show that our method can be easily applied to other state-of-the-art \"very deep\" CNN models."
],
[
"Introduction",
"Deep convolutional neural networks (CNNs) have achieved huge success in solving problems related to computer vision, such as image classification [23], [35], object detection [13], semantic segmentation [28].",
"However, in practice, training a high-quality CNN is often a complicated, confusing, and tedious trial-and-error procedure [6].",
"For a large CNN, one complete training trial may take a couple of weeks.",
"However, domain experts often have to repeat this process several times with slightly different settings to obtain a satisfying network, which may take several weeks or even months.",
"To accelerate this process, experts have to understand the training processes further to check whether they are on the right track, find latent mistakes, and make proper adjustments in the next trial.",
"[rgb]0,0,0 Visualizing the concealed rich training dynamics (e.g., the changes of loss/accuracy and weights/gradients/activations over time) is of vital importance to the understanding of CNN training process.",
"Unfortunately, CNNs usually contain a large number of interacting and non-linear parts [7] and recently become wider and deeper [35], [38], [17], [37].",
"Both of these bring considerable difficulties for experts in reasoning about CNN training behaviors.",
"[rgb]0,0,0 Many previous studies investigate what features have been learned by a CNN in one (e.g., usually the last one) or several representative snapshots during the training process [12], [41], [36], [11], [30], [43], [27], [32], [4], , [21].",
"However, little research focuses on visualizing the overall training dynamics.",
"One recent work [26] visualizes the training process of deep neural networks, but it is not tailored for CNNs and not scalable enough to analyze the CNNs that are not only wide but also deep.",
"[rgb]0,0,0Besides, tools like TensorBoard, Nvidia Digits, and Deeplearning4jTensorBoard: https://www.tensorflow.org/get_started/summaries_and_tensorboard; Nvidia Digits: https://developer.nvidia.com/digits; Deeplearning4j: https://deeplearning4j.org/visualization are able to show some high-level training dynamics, such as loss and the mean of weights in a layer.",
"However, these tools can neither handle industry-level training (i.e., training a large CNN on a very large dataset) nor answer complex questions.",
"For example, how the model's performance on each class of images changes over time?",
"How the changes of parameters impact the classification results for each class?",
"Given so many layers or image classes, which of them are worth paying more attention to and what is the best manner to support comparative analysis?",
"[rgb]0,0,0With these concerns, we are in urgent need of a scalable visualization solution to conducting more advanced analytical tasks.",
"To this end, we have to deal with two major challenges.",
"First, the system needs to handle the large-scale training log data.",
"Typically, millions of CNN parameters and tens of thousands of validation images are involved in a training process.",
"In addition, the training is an iteration-based process that usually requires a million iterations to complete, which makes things worse, because the parameters and classification results need to be recorded for every few iterations.",
"In our experiments (Sec.",
"), a sampled training log may easily exceed a couple of terabytes per training.",
"To allow an interactive exploration of the data at such scale, it requires not only an effective data storage and index mechanism but also a scalable visualization technique.",
"Second, the log information is heterogeneous.",
"The full log contains structure (e.g., neural network), numeric (e.g., neuron weights), image (e.g., validation dataset), and nominal data (e.g., classification results).",
"Given that significant insights are often hidden underneath the complex relationships among these data, our system also needs to present all these types of data intuitively and assist experts in their analysis tasks.",
"To address these challenges, we use a downsampling method to store the raw data, and then preprocess and organize these data in a hierarchical manner.",
"We also design several efficient index mechanisms to support real-time interactions.",
"To help experts identify the iterations of interest quickly, we propose an application-specific anomaly detection method.",
"[rgb]0,0,0We also integrate many filtering and aggregation approaches to reduce the amount of presenting information and ensure preserving the noteworthy information.",
"For visualization, we design a set of hierarchical small multiples that is combined with a network structure layout to facilitate an effective exploration of the entire training log from different levels of detail.",
"To reveal the complex correlations among neuron weights, validation images, and training iterations, we further introduce a cube-style visualization.",
"The cube integrates the small multiples and a matrix-based correlation view, thereby allowing experts to effectively slice and dice the data from different perspectives.",
"The main contributions of this paper are summarized as follows: A systematic characterization of the problem of visualizing the rich dynamics in CNN training processes, and a thorough discussion and summary of the design requirements and space.",
"A visual analytics system that integrates a tailored large data storage and index mechanism, an anomaly iteration detection algorithm, and a set of well-designed visualization techniques.",
"A couple of new visualization and interaction techniques including hierarchical small multiples, grid-based correlation view, and cube-style visualization.",
"Background Figure: Illustration of a CNN architecture that contains three types of layers (i.e., CONV layer, POOL layer, and FC layer) and transforms an image volume into a class score vector.A typical CNN can be viewed as a sequence of layers (Fig.",
"REF ) that transforms an image volume (e.g., a $224\\times 224$ image with three color channels of R, G, and B) into an output volume (e.g., a vector of size $1,000$ indicating the probability for an input image to belong to $1,000$ predefined classes) [24].",
"There are three main types of layers to build a CNN architecture, namely, convolutional layer (CONV layer), pooling layer (POOL layer), and fully-connected layer (FC layer).",
"A CONV layer comprises numerous neurons that are connected to a local region in the previous layer's output volume through weighted edges, many of which share the same weights through a parameter sharing scheme.",
"The weights in each neuron compose a filter, the basic unit for detecting visual features in the input image, such as a blotch of color or the shape of an area.",
"The output of each neuron is computed via a dot product operation between the weights and inputs from the previous layer, and is then optionally applied via an elementwise activation function (e.g., ReLU, $max(0, x)$ ).",
"A POOL layer is usually inserted between successive CONV layers to reduce the volume of input through a downsampling operation, thereby reducing the amount of parameters and computation in the network.",
"The only difference between FC layer and CONV layer is that, in contrast to the neurons in CONV layer that are locally connected and have shared parameters, the neurons in FC layers have full connections to the previous layer's output volume.",
"In addition, the output of the last FC layer is fed to a classifier (e.g., Softmax), computing the scores for all predefined classes, where each score represents the probability that an input image belongs to the corresponding class.",
"To obtain an effective CNN, the weight parameters in the CONV and FC layers need to be trained using gradient descent methods [9] to ensure the consistency between the predicted class labels and the predefined class for each training image.",
"Specifically, a training process involves two separate datasets, namely, the training set $\\mathit {D_t}$ and the validation set $\\mathit {D_v}$ .",
"To start the training, the parameters of weighted layers are usually initialized via Gaussian distribution [14], and $\\mathit {D_t}$ is partitioned into non-overlapping batches.",
"For each iteration, one batch of images in $\\mathit {D_t}$ is fed to the network.",
"Afterward, the output classification results are compared with the ground truth (i.e., the known labels of the fed images) to compute the gradients with respect to all neurons.",
"These gradients are then used to update the weights of each neuron.",
"When all batches in $\\mathit {D_t}$ are completed (i.e., finish one epoch), $D_t$ is reshuffled and partitioned into new non-overlapping batches.",
"After several epoches, the initially-randomized neural network will be gradually shaped into a specified network targeting at a specific task.",
"Meanwhile, for every given number of iterations, the network is evaluated via $\\mathit {D_v}$ .",
"Similarly, $D_v$ is fed into the network and then the output classification results are collected and compared with the ground truth.",
"However, the results are only used to validate whether a training process goes well and never used to update neuron weights.",
"Related Work CNN Visualization CNNs have recently received considerable attention in the field of visualization [34].",
"Existing approaches can be classified into two categories, namely, feature-oriented and evolution-oriented.",
"Feature-oriented approaches aim to visualize and interpret how a specific CNN behaves on the input images to disclose what features it has learned.",
"Most of the existing studies belong to this category.",
"Some studies [42], [41] modify part of the input and measure the resulting variation in the output or intermediate hidden layers of a CNN.",
"By visualizing the resulting variations (e.g., using a heatmap), users can identify which parts of the input image contributes the most to the classification results.",
"By contrast, other studies [41], [36], [30], [11] attempt to synthesize an image that is most relevant to the activation (i.e., the output of a layer after an activation function) of interest to help experts determine which features of a specified image are important for the relevant neurons.",
"For example, Mahendran and Vedaldi [30] reconstruct the input images that can either activate the neurons of interest or produce the same activations as another input image.",
"Besides, some methods focus on retrieving a set of images that can maximally activate a specific neuron [13], [12].",
"In this manner, users can discover which types of features or images are captured by a specific neuron.",
"Built on this method, Liu et al.",
"[27] develop a visual analytics system that integrates a set of visualizations to explore the features learned by neurons and reveal the relationships among them.",
"In addition, some work [32], [10], [21], utilizes dimension reduction technique to project the high-dimension activation vectors of FC or intermediate hidden layers onto a 2D space to facilitate revealing the relationships among outputs.",
"In contrast to those studies that investigate how a specified network works, only few studies concentrate on visualizing network training processes.",
"One typical method is to pick several snapshots of a CNN over the training process and then leverage feature-oriented approaches to compare how a CNN behaves differently on a given set of inputs at various iterations [41], [10], [43], [4].",
"For example, Zeiler and Fergus [41] use deconvnet approach to visualize a series of synthesized images that are most relevant to one activation of interest at a few manually picked snapshots, and then observe the differences between them.",
"[rgb]0,0,0 Zeng et al.",
"[43] present a matrix visualization to show the weight differences of filters of one layer as well as this layer's input and output in two model snapshots.",
"Their system also supports side by side comparison on the learned features of neurons (the computation method is similar to the work [13]) in different model snapshots.",
"One limitation of these methods is that when there are myriad filters, images, and iterations, it is challenging to select proper ones to observe and compare.",
"By contrast, we attempt to reveal the rich dynamics of the network parameters and the classification results at a larger scale to help experts find the notable filters, images, and iterations.",
"In this point, Alsallakh et al.",
"[4] analyze the same data facets (i.e., input images, network parameters, and classification results).",
"However, the work focuses more on identifying the hierarchical similarity structures between classes and still belongs to the category of multiple snapshots comparison, thereby suffering from the same limitation.",
"To analyze the evolution of CNN parameters, Eliana [29] treats the parameters of the entire network at one iteration as a high-dimension vector, uses PCA to map the vectors at all iterations onto a 3D space, and creates trajectories for these points.",
"However, this visualization is too abstract to extract useful insights for understanding and debugging the training process.",
"To analyze the classification results, Rauber et al.",
"[32] create a 2D trails graph to present an overview of how the CNN classification results evolve by leveraging the projection techniques.",
"However, this method suffers from scalability and visual clutter problems when applied to large-scale datasets (e.g., ImageNet).",
"Besides, this method only provides a very high-level overview and cannot answer those questions that involve individual classes or images.",
"[rgb]0,0,0The work most similar to ours should be the one by Liu et al.",
"[26], whereas the work mainly targets at deep generative models and would have serious scalability issue if applied in an industry-level CNN training.",
"Besides, compared with that work, we specifically provide a series of hierarchical methods that are tailored for CNNs to visualize the training dynamics, including classification results and network parameters.",
"[rgb]0,0,0Furthermore, we design a novel correlation matrix and 2.5D cube-style visualization to help experts examine the complex relationships exist among network parameters, classification results, and iterations.",
"Multiple Time Series Visualization Time series data have been extensively studied due to their ubiquity.",
"Numerous approaches [3], [2], [5] to time series visualization have been proposed.",
"Our system also falls into this field, since training logs are essentially a type of time-based information.",
"Specifically, our system is most related to existing work that visualizes multiple time series.",
"To visualize multiple time series, one method is to place multiple charts (e.g., line charts) in the same graph space to produce overlapping curves.",
"However, this method reduces the legibility of individual time-series [20].",
"In contrast to overlaying multiple series, one popular alternative is to use small multiples [39] that split the space into individual graphs, each showing one time series.",
"Using small multiples also allows for an effective comparison across charts.",
"Several factors may also affect the performance of small multiples [19], [20], such as the types of charts used (e.g., line charts and horizon graphs), the number of series and the vertical space allocated to each series.",
"The improper use of time-series charts, the increased series, and the small vertical space of each series will result in a serious visual clutter problem.",
"In this work, the evolution of the image classification results of each class and the weight parameters of each layer can be viewed as two types of multiple time series data.",
"Given the large number of classes and layers in a practical CNN training, we identify several space-efficient charts that can characterize the training dynamics.",
"Besides, we propose a similarity-based layout and a hierarchical exploration method to support the exploration of relationships among multiple time series to address the visual clutter problem.",
"We also present a novel cube-based visualization, targeting at the exploration of complex relationships among various types of heterogeneous time-series data (e.g., image classification results, neuron weights, and training iterations).",
"Requirement Analysis DeepTracker was developed in approximately nine months, in which we collaborate closely with three experts (denoted by $\\mathrm {E}_a$ , $\\mathrm {E}_b$ , and $\\mathrm {E}_c$ ) who have considerable experience in CNNs.",
"We held regular discussions with these experts once a week.",
"From our regular discussions, we learned that the training process should be carefully babysat.",
"The experts have to tune and fix a number of hyper-parameters (learning rate, batch size, layer number, filter number per layer, weight decay, momentum, etc.)",
"before starting a training trial.",
"These hyper-parameters, which strongly influence how a CNN is trained, are usually selected based on the experiences learned from their previous trials.",
"During a training, there are several useful quantities the experts want to monitor, such as loss function (the residual error between the prediction results and the ground truth), train/validation error rate (the percentage of mislabeled images), weight update ratio (the ratio of the update magnitudes to the value magnitudes), and weight/gradient/activation distributions per layer.",
"Basically, both loss and error rate should decrease over time; the consistent increase or violent fluctuation of loss on $D_t$ may indicate a problem; a big gap between the error rates of $D_t$ and $D_v$ may suggest that the model is over-fitting, while the absence of any gap may indicate that the model has a limited learning capability; the update ratioIn most cases, if the update ratio is lower than 1e-3, the learning rate might be too low; if it is higher than 1e-3, the learning rate is probably too high.",
"is expected to be around 1e-3; the weight distributions per layer in the beginning should overall follow Gaussian distributions with different standard deviation settingsXavier Initialization [14] is applied in our model.",
"Basically, the deeper (close to loss layer) the layer is, the smaller the sd is.",
"and may become diverse after training; and exploding or vanishing gradient values are a bad sign for the training.",
"The aforementioned rules of thumb are all based on the analysis of high level statistical information.",
"However, the experts still strongly desire to examine more details underlying the statistic, so that they can gain more knowledge and suit the remedy to the case when problems occur.",
"For example, the experience tells the experts that it is better to continue training models (e.g., ResNet [17]) with the same learning rate for a while rather than turn it down immediately, when the overall error rate stops to decrease.",
"The experts are very curious about what happens to the model and its performance on each class of images during the period.",
"Also, we may often see a fluctuation of loss or an occasional exploding gradient values, and what brings about this?",
"Are there any layers or filters that behave abnormally?",
"These details are never uncovered before.",
"Thus, the experts strongly desire a tool to enable them to explore the hidden dynamics behind a CNN training.",
"After several rounds of structured interviews with the experts, we finally summarized and compiled the following list of requirements: [R.1] Exploring multiple facets of neuron weight information.",
"A single iteration may have millions of weight updates, but individual values are generally meaningless to the experts.",
"Instead, the experts are more interested in the statistical information of these weights at different levels of detail (e.g., CONV layer level and filter level).",
"All three experts emphasized the importance of intuitively and effectively examining general types of statistics, such as sum and variance, between these levels.",
"Besides, $\\mathrm {E}_a$ and $\\mathrm {E}_b$ strongly desire to see the weight change degree for filters over iterations to identify which filters change dramatically at which iteration or how a filter changes across iterations.",
"Comparing multiple layers.",
"All three experts like to compare layer-level statistical information.",
"For example, they want to know whether a specified measure of different layers show a similar trend or distribution (e.g., whether the sum is positive or negative).",
"Accordingly, our visualization should also help these experts in performing such comparisons.",
"Tracking the classification results of validation classes.",
"Validation is a critical step in the training process that “test drives” the trained CNN and tracks its performance change across iterations [6].",
"Previous tools only measure the global validation loss/error, thereby concealing the rich dynamics of how the image labels of each class change over time.",
"When the error rates do not reduce as expected, the experts find that such highly-aggregated information is useless and cannot help them understand why or when the training runs into a bottleneck.",
"Therefore, the experts want to know how the images of different classes behave differently and identify those classes that are easy or difficult to train.",
"Detecting important iterations.",
"One complete training process usually contains millions of iterations, but it is obvious that not all iterations are equally important.",
"For example, some image classes may suddenly change their error rates after certain weight updates.",
"The experts are interested in these patterns, which may help reveal in-depth relationships between filters and image features.",
"However, the overall error rate trend does not help much, since it generally decreases slowly and steadily.",
"Thus, the experts hope that our system can help them identify the abnormal iterations and the corresponding classes.",
"Examining individual validation classes.",
"Our initial prototype shows that different classes clearly have different error rate evolution patterns.",
"Thus, experts $\\mathrm {E}_b$ and $\\mathrm {E}_c$ are curious about those classes with very poor or excellent performance, thereby desiring to further explore the image label information for these classes.",
"For example, they want to see whether and why some images are always misclassified.",
"Enabling correlation exploration.",
"Apart from analyzing the weight and validation data separately, the experts are also curious about their relational patterns.",
"They are specifically interested in uncovering the relationships between the layers or filters and the validation images, such as how the changes in network parameters respond to the image labeling results for each class.",
"By connecting these two pieces of information together, they hope to gain fundamental insights into the behaviors of CNNs and to improve training results.",
"System Overview Figure: Three components of DeepTracker.",
"The raw data are preprocessed into a hierarchical structure and then stored into five application-specific data indexes to enable real-time interactions.",
"On top of the efficient data storage, several visualizations are combined together to help experts with analysis tasks from different levels and perspectives.DeepTracker is a web-based application developed under the full-stack framework, MEAN.ts (i.e., MongoDB, Express, AngularJs, Node, and Typescript).",
"The back-end part of our application is deployed in a server with 3.10GHz Intel Xeon E5-2687W CPU and 32GB memory.",
"The architecture of our system (Fig.",
"REF ) begins with the data processing part, where the entire training log is hierarchically organized, and several application-specific indexes are built to support real-time interactions.",
"On top of the efficient data storage, we build three coordinated views, namely, the Validation View, the Layer View, and the Correlation View, to support an interactive analysis from different levels and perspectives.",
"The Validation View aims at providing a visual summary of CNN performance on different validation classes.",
"By combining our anomaly detection algorithm and small multiples, experts can easily identify different image class behavior patterns and critical iterations for each image class (R3, R4).",
"Experts may also drill down to the class of interest to explore further image label information (R5).",
"The Layer View aligns the weight information with the CNN structure to help experts explore various statistical information in the network hierarchy.",
"Experts can further drill up or down in the network hierarchy to compare these measures at different levels of detail (R1, R2).",
"The Correlation View presents a novel grid-based visualization method to provide an overview of the correlation between the image classification results and the neuron weights of each filter (R6).",
"[rgb]0,0,0 The three views compose a cube, with which the experts can simultaneously explore the changes of class-level performances, the variations of filter-level weights, and the correlations between them.",
"Data Acquisition and Construction [rgb]0,0,0 The primary motivation of this work is to monitor industry-level CNN training processes.",
"Therefore, we conduct our experiments with ResNet-50 [17] and ImageNet Dataset [33].",
"ResNet-50, containing 50 weighted layers (i.e., CONV and FC layers), is among the most popular CNNs that have been recently used in practice and meanwhile ImageNet 2012 is also among the largest and most challenging publicly available datasets.",
"The dataset includes $1,000$ classes of images, each class containing $1,300$ training images and 50 validation images.",
"Training such a model needs around 120 epoches (nearly 1.2 millions iterations when batch size is 128) to achieve convergence.",
"Simply dumping all the information for every iteration can easily have the size of dumped data exceed several petabytes and take about 4 weeks.",
"Through discussion, we all thought that $1,600$ is a reasonable interval to capture meaningful changes (about 7 times per epoch).",
"This reduces the log to a manageable size (about 1TB).",
"[rgb]0,0,0 For each dump, we recorded two pieces of information, namely, neuron weights/gradients of CONV layer and FC layer and image classification results.",
"The parameters on BN layers were not recorded, as they can be totally recovered given the weights on CONV and FC layers and always need to be updated when applied in a new dataset.",
"Besides, we did not record the activations of each layer/filter for every validation image, as doing so is technically impracticable considering the extremely large models and datasets and the limited disk storage.",
"Further, activation evolution visualization is beyond the research scope of this paper.",
"Sec.",
"discusses activation visualization is indeed a perfect complementary technique to our work.",
"We organized the weight/gradient information according to the natural hierarchial structure of ResNet-50.",
"It consists of four CONV modules (plus the first CONV layer and the final FC layer, there are 50 layers in total).",
"Each module contains several bottleneck blocks [17] that comprise three to four basic CONV layers (data storage in Fig.",
"REF ).",
"Thus, we grouped all neuron weights to align with such hierarchy.",
"[rgb]0,0,0 In a similar manner, we organized the classification results hierarchically from individual level, class level, to model level.",
"We stored all the data into MongoDBMongoDB is a free and open-source cross-platform document-oriented (NoSql) database.",
"www.mongodb.com.",
"In particular, we precomputed all the relevant aggregation values, such as weight means and error rates, for each filter, layer, image, and class.",
"Nevertheless, the distilled data still remain too large to load into memories (about dozens of gigabytes per training).",
"Therefore, we analyzed the frequent needs of the experts and built several indexes to enable real-time interactions, including Layer-Stat index $I_{ls}$ , Layer-Filter index $I_{lf}$ , Iter-Filter index $I_{if}$ , Cls-Stat index $I_{cs}$ , and Cls-Image index $I_{ci}$ .",
"$I_{ls}$ retrieves the statistic values (e.g., mean and sd) at every iterations for any given layer; $I_{lf}$ lists all the filter-level information (e.g., changing degree of each filter) at every iterations for any given layer; $I_{if}$ searches the top changing filters from all layers at any given iteration; $I_{cs}$ extracts class-level information (e.g., class performances, the different types of abnormal images Sec.",
"REF ) over all iterations for any given class; $I_{ci}$ fetches the meta-data of images for any given classes.",
"Visualization In this section, we describe our three coordinate views, namely, the Validation View, the Layer View, and the Correlation View, that help experts accomplish the aforementioned analytical tasks.",
"Validation View Several urgent requirements (R3, R4, R5) from the experts need to examine how the evolving CNN acts differently on the validation images of each class rather than how the overall validation error rate differs over training.",
"Thus, we design the Validation View (Fig.",
"REF & REF ) to present all image classes in $D_v$ .",
"Figure: From top (a) to bottom (b), image classes are more and more difficult to train.Visual Encoding [rgb]0,0,0By default, the view starts with a visualization of cluster-level performance (R3).",
"The classes with similar evolving trends form a cluster and then their error rates at every iteration are averaged.",
"The averaged error rates are then depicted as a colored stripe, where the x-axis encodes the iterations and the error rates are encoded by colors (Fig.",
"REF ).",
"We choose k-means as the clustering algorithm and $k$ can be adjusted according to demands (Fig.",
"REF shows the case when $k=4$ ).",
"Experts can open up one cluster to further examine the performance in class-level (R3).",
"The design is based on the following considerations.",
"First, the experts are more interested in the overall classes than in individual iterations.",
"Thus, small multiples technique (juxtaposed techniques) is chosen for their superior performance in high-level comparison tasks (e.g., global trends for every series) [20].",
"Second, we cannot present all the classes ($1,000$ in ImageNet) at one time, we have to consider a hierarchical and highly space-efficient visualization.",
"However, many traditional charts, such as line charts and horizon graphs, require a larger vertical space [19] than 1D heatmaps.",
"Compared with traditional charts, heatmaps are also more easy to do side by side comparison for their symmetrical space (i.e., no irregular white spaces).",
"As a result, all experts prefer the heatmap-based small multiples.",
"Image-level performances, R5.",
"The class-level color strips can be further unfolded to explore the image-level evolution patterns.",
"Unfolding a heatmap reveals a pixel chart (Fig.",
"REF d), with each row (1px height) representing an image and each column (1px width) representing an dumped iteration (consistent with the class heatmap).",
"We use red and green colors to indicate the incorrect and correct classifications, respectively.",
"Meanwhile, the experts can zoom/pan the pixel chart for a closer inspection.",
"Clicking on a row shows the original corresponding image.",
"Anomaly iterations, R4.",
"[rgb]0,0,0As experts are concerned about the iterations with abnormal behaviors, we particularly propose a algorithm to detect these anomaly iterations (refer to Sec.",
"REF ).",
"Experts can choose to only show the classes with anomaly iterations (Fig.",
"REF ).",
"At this point, for each class-level color stripe, we use triangular glyphs to highlight these detected anomaly iterations.",
"The upside-down triangles ($\\bigtriangledown $ ) and normal triangles ($\\triangle $ ) indicate those anomaly iterations that are detected by the left-rule and right-rule, respectively.",
"The widths of triangles encode the anomaly scores.",
"Experts can set a threshold value to filter the triangles with low anomaly scores.",
"Anomaly Detection In our scenario, the classification results for an image can be represented by a 0/1 sequence ($[a_1, ..., a_n]$ ), where each element represents a correct or incorrect result at the corresponding validation iteration.",
"The experts are curious about the iterations when a significant amount of 1/0 flips (i.e., 0 to 1 or 1 to 0) occur for a class.",
"In general, this problem can be modeled and solved using Markovian-based anomaly detection algorithms [1].",
"Despite the popularity of using Markovian methods to detect outliers in discrete sequence, we decide to employ rule-based models [1] for two reasons.",
"First, Markovian methods are a black box and the resulting outlier values are sometimes difficult to comprehend.",
"Second, the experts have explicitly described two types of iterations they are very interested in, namely, those iterations when many images with values that remain stable for many previous iterations suddenly flip (denoted by the left-rule) and those iterations when many images flip and keep their values stable after many iterations (denoted by the right-rule).",
"Fortunately, these anomalies can be easily modeled using rules.",
"The rule-based models primarily estimate the value $P(a_i|a_{i-k},\\ldots ,a_{i-1})$ , which can be expressed in the following rule form: $ a_{i-k},\\ldots ,a_{i-1} \\Rightarrow a_{i}.$ In our scenario, if an image has the same value (either 0 or 1) in the previous consecutive $k$ iterations ($i-k,\\ldots , i-1$ ), then its value must be the same at iteration $i$ (the left-rule).",
"Otherwise, iteration $i$ is considered an outlier for the specified image.",
"Based on these considerations, we develop an application-specific algorithm to detect anomaly iterations in the validation history.",
"The algorithm includes the following steps: Rule-Judgement: The algorithm computes a vector $[l_{i1}, \\ldots , l_{in}]$ for every image $i$ , where $l_{ij} =1$ if the left-rule is satisfied, otherwise, $l_{ij} =0$ ; Aggregation: For each class that contains $m$ images, the algorithm aggregates all the computed vectors for each image into one $[L_1, ..., L_n]$ , where $L_j = \\sum _{i=1}^{m}l_{ij}$ , [rgb]0,0,0denoting the left anomaly score at iteration $j$ for this class.",
"The approach is a window-based method, and the experts can adjust the window size $k$ to control the sensitivity of the anomalies.",
"In a similar manner, we detect the anomalies from the opposite direction for the right-rule.",
"Layer View The Layer View focuses on weight-related tasks (R1, R2).",
"The view consists of two connected parts, namely, the CNN structure and the hierarchical small multiples (Fig.",
"REF ), so that experts can hierarchically explore and compare various types of statistic in the context of the network structure.",
"Figure: Visual encodings in the Layer View: (a) a CONV layer, (b,d) hierarchical bars, (c) links between the CNN structure and (e) the hierarchical small multiple charts, (f) a pixel chart for one layer.CNN structure.",
"The experts hope our tool can help them explore the statistical information of each layer and meanwhile know their relative positions in the entire network (R1).",
"Thus, we adopt NetscopeNetscope is a web-based tool for visualizing neural network architectures.",
"http://ethereon.github.io/netscope/, a popular neural network visualizer, in our system.",
"[rgb]0,0,0 The green rectangle is data input layer, the red ones are the CONV layers, and the purple ones mean the pooling layers.",
"The links between these rectangles show the network structure.",
"We further add blue level bars (Fig.",
"REF b) to encode the latent hierarchy (from CONV modules, bottlenecks to basic CONV layers, Sec. ).",
"The right most level bars represent CONV modules (Sec.",
"), which are recursively divided into smaller level bars until reaching elementary CONV layers (i.e., red rectangles).",
"Hierarchical small multiples.",
"To assist experts in exploring and comparing layers of a deep CNN (R1, R2), a space-efficient visualization technique is demanded.",
"Thus, we leverage hierarchical small multiples to show layers of interest (Fig.",
"REF e).",
"[rgb]0,0,0By default, experts are presented with the information about CONV modules and then can drill down to see more information about low-level CONV layers with interactions with the network graph (i.e., click on the corresponding level bars).",
"The width of outcropping rectangles (Fig.",
"REF d) encodes the aggregation level of current layer charts.",
"For example, the top second layer chart in (Fig.",
"REF e) shows the bottleneck-level aggregation information and the following three layer charts show the basic CONV layer level information.",
"Besides, the links (Fig.",
"REF c) mark the real positions of the layer charts in the network structure.",
"The small multiples support multiple types of charts including line chart, horizon graphs [18] and box plots to emphasize the different aspects of the statistical data.",
"The experts use box-plots to see the rough distribution of statistical values and use basic line charts to examine individual values.",
"Besides, the experts prefer to use horizon graphs when performing tasks in regard to trend tracking and comparison (R2), because of its effectiveness in visualizing divergent weight values [20].",
"Similar to unfolding the class heatmap to a pixel chart, the experts are also allowed to open the layers of interest as a pixel chart (Fig.",
"REF f) that presents the filter-level information (R1).",
"Each row (1px height) in the pixel chart represents one filter, and each column (1px width) indicates one iteration.",
"We use sequential colors to encode pixel value (e.g., the cosine similarity between two subsequent dumped iterations).",
"Correlation View This view helps experts establish connections between the filters and images.",
"In particular, the experts want to understand further how the changes in network parameters are related to class performances (R6).",
"For example, several anomaly iterations may be detected for a single class.",
"For each detected anomaly iteration, we can identify a set of anomaly filters (i.e., the top $k$ filters with largest changes at that iteration).",
"Since different classes may share anomaly iterations and different anomaly iterations may share anomaly filters, are there any filters that are commonly seen in these iterations?",
"Do any of the anomaly classes or filters strongly co-occur?",
"We designed the Correlation View to answer these questions.",
"[t] Target set $S_{target}$ .",
"Minimum partition for $S_{target}$ .",
"$S_{result}$ = $\\emptyset $ each target set $s_t$ in $S_{target}$ $S_{new}$ = $\\emptyset $ each mini set $s_r$ in $S_{result}$ $itersection$ = $s_t \\bigcap s_r$ $S_{new}$ = $S_{new} \\bigcup intersection \\bigcup (s_r - s_t)$ $s_t$ = $s_t - s_r$ $s_t$ is not empty $S_{new} \\bigcup s_t$ remove all empty set in $S_{new}$ $S_{result}$ = $S_{new}$ return $S_{result}$ Minimum Set Partition Filter set partition.",
"We first introduce the mini-set concept to organize anomaly filters that are shared by multiple anomaly iterations and different classes.",
"For each class $C_i\\in \\lbrace C_i|{1\\le i \\le n}\\rbrace $ , we denote its anomaly iterations by $T_i = \\lbrace t_{i,k}|{1\\le k\\le n_i}\\rbrace $ .",
"Thus, all anomaly iterations are $\\cup _{1\\le i \\le n}T_i$ , denoted by $T$ .",
"For each anomaly iteration $t\\in T$ , we denote its anomaly filters at CNN layer $L_j\\in \\lbrace L_j|{1\\le j \\le m}\\rbrace $ by $s_{j,t}$ .",
"Thus, for each layer $L_j$ , we can collect all anomaly filter sets $\\lbrace s_{j,t}|t\\in T\\rbrace $ (denoted by $S_j$ ) and all anomaly filters $\\cup _{t\\in T}s_{j,t}$ (denoted by $s_{j}$ ).",
"Thus, mini-set aims to find a minimum set partitions of $s_{j}$ (denoted by $s_j^\\ast $ ) that each $s_{j,t}$ can be assembled from some elements (i.e., mini-sets) in $s_j^\\ast $ .",
"We specifically propose a Set Partition Algorithm (Alg.",
"REF ) to find $s_j^\\ast $ .",
"The algorithm accepts a target set as input (i.e., $S_j$ ).",
"$S_{result}$ is initially empty and a new anomaly filter set is used at each time to partition the mini-sets contained in $S_{result}$ (cf.",
"lines 4 to 7).",
"If the new anomaly filter set is not empty after partitioning, then it is added as a new mini-set (cf.",
"line 8).",
"Finally, the partitions contained in $S_{result}$ will be returned.",
"Figure: (a) The abstract version of correlation view, where rows and columns represent layers and image classes, respectively.",
"A sequential color scheme is used to encode the number of anomaly filters.",
"(b) The complex version of correlation view, where the detailed information of individual anomaly filters are shown.",
"(c) A layout solution for coordinated analysis without using skewed axes.Visual encoding.",
"To intuitively represent these relationships, we introduce a grid-style visualization (Fig.",
"REF ), where rows and columns represent layers and image classes, respectively.",
"The number of rows and columns equal to the number of layers with anomaly filters and classes with anomaly iterations, respectively.",
"[rgb]0,0,0 We start from a abstract version.",
"In this version (Fig.",
"REF a), for $\\mathrm {Cell}_{i,j}$ , a sequential color scheme is used to encode the number of anomaly filters ($\\cup _{t\\in T_j}s_{i,t}$ ).",
"The darker the color it is, the more anomaly filters appear in layer $L_i$ that are related to class $C_i$ .",
"From this visualization, we can easily observe the correlations between layers and classes, while it also hides much detailed information.",
"We cannot answer questions like whether the filters in $\\mathrm {Cell}_{i,j}$ are the same with $\\mathrm {Cell}_{i,k}$ (this kind of information shows how many classes this filter can impact and help examine the relationships among classes), or whether there are filters in $\\mathrm {Cell}_{i,j}$ appearing in more than one anomaly iteration (this shows the importance of these filters for class $C_j$ ).",
"To solve these problems, we provide an advanced version (Fig.",
"REF b).",
"For $\\mathrm {Cell}_{i,j}$ , the width and height encode the number of anomaly iterations and the number of anomaly filters of the corresponding class and layer (i.e., $|T_j|$ and $|s_{i}|$ ), respectively.",
"Based on these numbers, the columns and rows are further divided with vertical/horizontal lines.",
"For a class (e.g., $\\mathrm {Column}_{j}$ ), $|T_j|$ vertical lines are drawn to represent all related anomaly iterations (i.e., $[t_{j,1}, t_{j,2},\\ldots ,t_{j,n_j}]$ ).",
"For each row of layer $L_i$ , there are $|s_i^\\ast |$ horizontal lines representing all mini-sets in that layer.",
"The intersections between these horizontal and vertical lines are highlighted with blue rectangles if the corresponding mini-set is part of the anomaly filters of the corresponding anomaly iteration.",
"The height of the rectangles represents the number of filters of the corresponding mini-set.",
"[rgb]0,0,0Obviously the introduction of mini-sets dramatically cuts down the number of horizontal lines and blue rectangles, otherwise each anomaly filter require one horizontal line and one rectangles, which may cause serious visual clutter problem.",
"In fact, mini-sets can be viewed as a partially aggregation version instead of representing all the anomaly filters as horizontal lines and rectangles.",
"Users can set the minimum appearing number of mini-sets to filter the sets with lower importance.",
"Cube Visualization The log data contain three main aspects of information, namely, iterations, validation information, and weight information.",
"The three aforementioned views are designed to show all possible 2-combinations of these three types of information, respectively.",
"[rgb]0,0,0 Although these views can be used individually, they need to be combined together to form a complete picture.",
"Thus, we propose a novel and intuitive visualization technique, which naturally and seamlessly stitches the three views together, based on their shared axis (inspired from Binx [8]), into a “cube” shape (Fig.",
"REF ).",
"When experts find or highlight a pattern of interest, they can easily track the pattern over the edges to find the related information in the other two views easily.",
"[rgb]0,0,0 The use of skewed axes may bring about a possible perspective distortion problem.",
"Nevertheless, the advantages of the cube-style design far outweigh the disadvantages.",
"Given the limited pixels in a computer screen and each view requiring a large display space, the experts all agree that the cube-style design is the most space-efficient and intuitive manner to show all the information.",
"Furthermore, with such design, it prevents the experts from switching from multiple views, reducing the cognitive burden and the load of memory.",
"This allows the experts to conduct correlation analysis more effectively.",
"We also provide a compromised solution to handle the distortion problem, that is, laying out the three views in the form like Fig.",
"REF c. So the experts can firstly examine the layer view (horizontally) or validation view (vertically) together with the correlation view, and then switch to cube mode to explore the three views together.",
"[rgb]0,0,0 Notice that there are some different settings for several views in the cube.",
"For the layer view (front), only the layers with anomaly filters are activated (see the activated blue bars in the front view of Fig.",
"REF ), the weight variation of each anomaly filter is represented as a horizontal color strip.",
"For the validation view (top), only the classes with anomaly iterations are preserved.",
"The following lists several common exploration pipelines: P1: from the layer view (front), we can quickly check the distribution of activated layers in the overall network and pick some anomaly filters of interests.",
"Then, by tracking along the horizon axis to the correlation view (right), we can examine which classes these filters impact and how important these filters are to the classes.",
"Finally, we can observe the evolving patterns of these classes and the corresponding anomaly iterations in validation view (top).",
"P2: from the validation view (top), we can firstly mark several anomaly iterations of some classes.",
"Then, we can check the corresponding columns in the correlation view, finding the rows that contain anomaly filters and exploring the importance of these filters to these classes and how these filters impact the other classes.",
"Finally, by highlighting these corresponding rows, we can observe them in the layer view to see how these filters behave around the picked anomaly iterations.",
"P3: from the correlation view (right), we can search the horizontal lines across many rectangles (it means these filters impact many classes at the same time) or the rectangles that appear more than one time in the same cell (these filters are judged as anomalies many times for a class and may have great impact on this class).",
"With these selected horizontal lines or rectangles, we can simultaneously track their corresponding weight variation information in the layer view (front) and class performance information in the validation view (top).",
"Use Examples We derived these examples through the assistance of our collaborating experts, who were familiar with our designs and data.",
"As a remark, the following results are from the experiment with 8 times larger batch size and learning rate setting than the basic setting introduced in Sec. .",
"Exploring Validation Results Figure: Overview of validation classes.",
"(a) Two curves show the overall training/validation error rate.",
"(b) Two turning points align well the boundary of stage s2s2.",
"(c) Three peaks appear in the stage s1s1 and align well with (f) the detected anomaly iterations.",
"(d) Two types of mushroom images have different behaviors in the class.",
"(e) Most images in the class flip at the anomaly iteration.The first scenario demonstrates how the experts use DeepTracker to explore the image classification results (R3, R4, and R5).",
"Performance evolution patterns.",
"Fig.",
"REF a shows a typical visualization of training/validation errors that may appear on any popular training platform.",
"The timeline at the top shows a total of 1.2 million iterations.",
"Beneath the timeline, four line segments represent four stages ($s1$ , $s2$ , $s3$ , and $s4$ in Fig.",
"REF ) in the training process, [rgb]0,0,0where the later stage has one-tenth of the learning rate of the previous stage.",
"We can observe that two sudden drops in the curves match well with the boundaries of the training stages (Fig.",
"REF b).",
"However, this is a well-known pattern to the experts.",
"On the other hand, although the overall error rate continues to decrease, the class-level error rates show a more complicated story, that is new to the experts.",
"By quickly scanning the small multiples in cluster-level, the experts identify there are generally four types of class evolving patterns (Fig.",
"REF ).",
"From top to bottom, the four types are more and more difficult to train.",
"[rgb]0,0,0For example, for the type at the top, these classes are recognized correctly after a few iterations.",
"By contrast, the classes at the bottom always have high error rates in the entire training process, which means that the resulting network fails to recognize the related images.",
"From this, the experts learn that the model has spent most of time to improve its performance on the classes of middle-level classification, since the model has already performed well on the easy-trained classes at a very early stage and is always performing miserably on the hard-trained classes over the entire training.",
"From these patterns, the experts consider it promising to accelerate the training process and improve the overall performance by treating classes differently during the training process.",
"That is, stop feeding the easy-trained classes in an appropriate early stage, put more efforts on training the classes of middle difficulties for classification, and figure out why some classes always have extremely high error rates.",
"One similar attempt has been made in a recent work [25].",
"Anomaly iterations.",
"The experts are curious about the three sudden peaks in stage $s1$ , and then mark these three iterations with dotted lines (Fig.",
"REF c), which look like anomaly iterations (R4).",
"However, the colors in the small multiples do not have clear patterns related to these iterations.",
"Then, the experts turn on the anomaly detection and immediately find that many triangles are aligned well with the dotted lines (Fig.",
"REF f), thereby confirming our suspicion.",
"Then, the experts can click on the corresponding image icons to see the detailed images that contribute to the three peaks.",
"In addition, there are more anomaly iterations in stage $s1$ then in the later stages.",
"This interesting pattern can be explained by the reduction of learning rate and the convergence of the model in the later stages.",
"At the same time, it also implies that the learning rate in stage $s1$ is slightly too high, leading to the instability in the model (the case in Sec.",
"REF indicates the same finding for the discovery of potential “dead” filters).",
"Details in classes.",
"To further examine what happens at the anomaly iterations for a class, the experts can further check the image-level information of the class (R5).",
"For example, the experts are curious about the abnormally large anomaly iteration in the class of “mushroom” (Fig.",
"REF d) that are captured by both the left-rule and the right-rule.",
"Then, they click and expand color stripe to see the pixel chart of images.",
"First, they confirm that this iteration is indeed special for this class, because nearly all images flip during particular that [rgb]0,0,0dumped interval (Fig.",
"REF e).",
"Thus, they may further investigate to find the layers or filters that cause such flips based on the filter updates around that iteration.",
"[rgb]0,0,0In particular, the experts comment that, it seems that after the iteration, the CNN model has jumped to a better local optimal for the class, because the green color is more stable after the iteration.",
"This may result from the reduction of the learning rate (from $s1$ to $s2$ ).",
"This kind of patterns appear frequently in many classes during the whole training process, many of them occurring not in the learning rate transition point.",
"The experts wonder that the model should be trying to jump from one local optimal to another better local for these classes continuously, so as to reduce the overall error rate gradually.",
"This insight has never been obtained because the experts initially thought that the error rate for one class should decrease steadily.",
"Besides, the experts also find that, at the bottom of the pixel chart, several images are mislabeled in the entire training process, although the class is easy to train overall(Fig.",
"REF d).",
"To understand why, the experts click on these images to examine them, and find that the contents in the mislabeled images have a clear color pattern different from that of the rest of the mushroom images.",
"The correctly labeled mushrooms are all red, while the mislabeled ones are white or orange.",
"This finding indicates color is a critical feature that the CNN has learned to classify this class of images.",
"Exploring Weight-Relevant Information Figure: (a, b, c) The sd values of each layer decrease slowly and have different scales in different CONV modules.",
"(d) The weight changes in filters are large at the beginning.",
"(e) One outlier filter is detected, whose weights never change during the entire training process.This scenario shows how to discover patterns in neuron weights via the Layer View (R1, R2).",
"First, the experts choose to show the sd (standard deviation) of the weights at the layer-level using horizontal graphs (Fig.",
"REF ).",
"As the experts expected, all the trends show a similar pattern of slow decrease, indicating the weights in the entire model is converging over iterations.",
"Besides, the experts also find that deeper layers (closer to the loss layer) tend to have smaller sd values.",
"[rgb]0,0,0In particular, by tuning the band number (finally to 3) of the horizon graphs, they found the sd values of a CONV module are usually twice as large as those of the one below it (a, b, and c in Fig.",
"REF ).",
"Given that we apply Xavier initializationREF and for ResNet-50, the input sizes of layers in a CONV module are twice as large as the ones in the layers of its previous CONV module, the observed result is not beyond the experts' expectation.",
"This suggests that there is no problem exist on the initialization approach.",
"Analogously, the experts find that the weight means of each layer become negative quickly (from green to blue instantly, Fig.",
"REF a) except for the FC layer (Fig.",
"REF b).",
"At first, the pattern looks strange to the experts.",
"Then, the experts realize that it is reasonable to have more negative weights than positive ones, since negative values are often used to filter out trivial features owing to the use of ReLU activations.",
"The increase of negative weights suggests that the network is trained to extract useful information from the original image layer by layer, and then finally remain the most relevant features.",
"[rgb]0,0,0As for the FC layer, it plays a function of shaping the extracted useful features into feature vectors of given dimension for further classification.",
"One strange phenomenon intrigues the experts, that is, the FC layer weight means are always positive in many-times training ResNet-50 (with different batch sizes and learning rates) on ImageNet Dataset, whereas becoming negative when training ResNet-164 on Cifar Dataset [22].",
"This finding is worth a further investigation.",
"Apart from layer-level values, the experts also explore the filter-level information (R1).",
"In our system, two different ways (i.e., filter-based or iteration-based) are used to normalize weight changes at the filter-level.",
"For filter-based normalization, changes are grouped and normalized by filters, which aims to help experts see the change distribution over iterations for individual filters.",
"Similarly, iteration-based normalization allows experts to examine the distribution over filters for individual iterations.",
"For example, Fig.",
"REF d visualizes the filter changes in one of the CONV layer belonging to the second CONV module using filter-based normalization.",
"The experts find that the changes are drastic in stage $s1$ and become relatively small in the later stages because of the decrease in learning rate and the convergence of the model.",
"However, the experts also identify two strange filters among 64 filters in the first CONV layer that have a constant deep blue color (Fig.",
"REF e).",
"By further checking, the experts find that the weights of these two filters never change during the entire training process.",
"Figure: (a, b) The means of weights in each CONV layer become negative quickly (from green to blue) except for the FC layer.",
"(c) Three filters are always more actively changed than the other filters in the later part of training progress.",
"[rgb]0,0,0This is a total surprise to the experts.",
"Excluding programming bugs, the most likely reason should be due to the dying-ReLU problem, namely, these two filters are inactive for essentially all inputs and no gradients flow backward through the neurons of the two filters.",
"The experts suspect the dying-ReLU problem results from the high learning rate in the beginning of train.",
"In fact, the experts usually follow a rule of thumb to set the hyper-parameter learning rate, that is, multiply the learning rate by k if the batch size is multiplied by k. This rule is currently formally introduced in a recent work [15].",
"In this experiment, we use 32 times larger batch size GPUs (32 GPUs) than the mini-batch size 32 for one GPU to train the model with the corresponding size of learning rate, dying-ReLU problem still occurs.",
"This reminds the experts that the rule may not so accurate for extremely large batch size sometimes, but the problem can be solved by carrying out a warmup strategy (i.e., using lower learning rate at the start of training [17]), which the experts haven't done in previous trainings.",
"One further interesting finding is that by inactivating these two \"dead\" filters (i.e., set their weights as 0 so that they are inactive for any inputs), the experts find the overall performance not affected at all, whereas if we inactivate other random-picked filters in the first CONV layer of the model, the number of mislabeled images in $D_v$ would increase few thousands.",
"Thus, the experts finally modified the network configure and eliminated these two filters, so that the model can run faster while costing less memory.",
"Fig.",
"REF c visualizes the weight changes in one middle layer using iteration-based normalization.",
"The experts find that a small number of filters are always more actively changed than the other filters (long deep blue lines in Fig.",
"REF c) in the later part of iterations.",
"This pattern implies that the updates inside a layer may be highly divergent.",
"[rgb]0,0,0In the later part of the training, where the learning rate is getting smaller and the model is converging, only a couple of filters are still continually actively updated for every iteration.",
"We have tried to inactivate these identified filters, the result showing that the overall performance is not affected.",
"It is not beyond the experts' expectation due to the ResNet's ensemble-like behavior [40] (even several entire layers can be removed without impacting performance).",
"In the end, the experts still cannot fully explain the behavior of these continually updating filters.",
"One possible reason could be that these special filters are not trained well (not converge) to extract some specific features, thus reacting violently for every iteration even in the later stages of the training.",
"Exploring Filter-Image Correlations In this scenario, we demonstrate how the experts use the Correlation View to explore correlations between images and filters.",
"Figure: A cube-style visualization that fuses three coordinated views together to reveal the rich dynamics in a CNN training process:(top) the Validation View shows the error rate changes of validation classes; (front) the Layer View shows the weight changes in CNNfilters; (right) the Correlation View shows the potential relationships between filters and validation classes.Shallow-layer filters vs. deep-layer filters.",
"[rgb]0,0,0At first, the experts choose to only show the top $k$ (100 in this case) changing filters in the layer view.",
"By checking the network structure visualization, the experts find that the activated shallow layers (the layers close to data input layer) are more than the activated deep layers, and most activated layers are the last basic CONV layers of bottlenecks for deep CONV modules.",
"Besides, Fig.",
"REF a shows that deep CONV modules tend to contain more anomaly filters (especially for the CONV modules 4).",
"The experts think that this kind of knowledge is of great importance for network compression [16].",
"Then, the experts go to the complicated version to examine the more detailed correlation information.",
"They filter the mini-sets with very few appearing times, finding that anomaly filters in shallow layers are generally shared by more anomaly classes (columns) and iterations (vertical lines in one column) than those in deep layers (Fig.",
"REF a).",
"The experts think that this pattern may relate to the fact [40] that shallow-layer filters are more likely to capture basic visual features than deep ones, thereby the huge change of these filters affecting more classes of images (e.g., the long and opaque lines marked by b in Fig.",
"REF ).",
"By contrast, a deep filter tends to learn higher-level features, thus only relating to specific classes of images.",
"To further explore the correlations, the experts select two mini-sets (b1 and b2 in Fig.",
"REF ), for comparison.",
"Both the horizontal lines of b1 and b2 are opaque and thick.",
"By tracking them in the Layer View and the Validation View, the experts can see that b1 is in the first CONV layer, and related to many classes.",
"The experts open these classes and discover that many images in them have a common feature, i.e., a large background of blue sky or ocean (b1 in Fig.",
"REF ).",
"This discovery suggests that these filters in E1 may target the basic pattern to help identify images that contain large blue areas.",
"By contrast, b2 is located at the fifth [rgb]0,0,0CONV module and related to only three classes.",
"Interestingly, the images in the three classes also share a more concrete feature, i.e., objects in a bush (b2 in Fig.",
"REF ).",
"[rgb]0,0,0In short, this case confirms that we are on the right track to reveal how the model weight changes relate to the classification result changes.",
"Important filters for a class.",
"To find stronger correlations between filters and classes, the expert focus on anomaly filters that appear more than once in a cell for a specific class.",
"For example, the experts find two appearances of the same mini-set (containing two anomaly filters) for the class of “gong” (c1 in Fig.",
"REF ).",
"Tracking horizontally (along with the pink-highlighted area), the experts find that the mini-set does not appear in other anomaly iterations, which also implies a strong correlation between filters in the mini-set and the class.",
"Then, the experts click on these two rectangular glyphs to highlight the corresponding iterations on the timeline (c2 in Fig.",
"REF ) and the filter locations in the Layer View (c3 in Fig.",
"REF ).",
"It is clear that the gong class is not a well trained class as it has a very large yellow area (indicate a relatively high error rate) in the Validation View.",
"However, the experts also find a period in the middle when this class has a relatively good performance (c4 in Fig.",
"REF ), which happens to contain the highlighted anomaly iterations.",
"Meanwhile, the Layer View shows that the highlighted filters are also updated dramatically during the period of good performance (c3 in Fig.",
"REF ).",
"Considering these patterns, the experts speculate that the filters in the mini-set have a strong impact on the classification of gong images.",
"As expected, we conduct experiments to inactivate these two filters, finding the overall performance and the performance on “gong” class are not impacted (see the reason in the last paragraph in Sec.",
"REF ).",
"Nevertheless, it provides the experts with a new manner to investigate the functions of filters co-working together for classifying one class of images.",
"That is, increase the threshold to find anomaly filters as many as possible, find the mini-sets containing many filters to some classes from multiple layers, and then inactivate them all to validate corresponding impacts.",
"Abnormal anomaly filters.",
"The experts are also attracted by two mini-sets (d1 and e1 in Fig.",
"REF ), because of their abnormal color patterns.",
"The filters in these two mini-sets exhibit large changes all the time in the latter part of the training, which are very different from the other anomaly filters.",
"Thus, the experts are interested in these filters and further check their correlated classes in the Correlation View (right).",
"Interestingly, each abnormal mini-set only appears with two classes (d2 and e2 in Fig.",
"REF ), and each pair of classes have very similar performances displayed in the Validation View (d3 and e3 in Fig.",
"REF ).",
"By checking the detailed images of these classes, the experts discover some common patterns.",
"For example, for mini-set e1, the corresponding classes are about mushrooms growing on grass and dogs playing on grass (e3 in Fig.",
"REF ).",
"For mini-set d1, the corresponding classes are related to curved shapes, such as parachutes and round textures (d3 in Fig.",
"REF ).",
"Although the experts are still unclear about why these two mini-sets have such a special behavior, they believe that these filters are likely to play important roles in identifying middle-level features such as grass and curved shapes.",
"We also conduct further experiments to validate the impact of inactivating these filters and the results are similar to the previous case (i.e., important filters for a class).",
"Expert Feedback Usability [rgb]0,0,0DeepTracker is built upon a close collaboration with three domain experts, who constantly underscore their requirements and provide suggestions during the implementation process.",
"After several iterations of refinement, the experts were happy with the current version.",
"They all praised our way of effectively exploring such extreme large-scale training log via a hierarchical manner.",
"$\\mathrm {E}_a$ and $\\mathrm {E}_b$ mentioned that the well-designed validation and layer views were very intuitive and helped them greatly.",
"For example, the layer view allowing the experts to effectively observe and compare layer-related information (e.g., weight/gradient distribution) can help them diagnose network structures.",
"The detecting of dying-ReLU problem in the early stage of a training is useful for tuning the hyper-parameters (e.g., learning rate).",
"This kind of knowledge can also be leveraged to conduct model compression [16], so as to improve the model in respect to computing speed and memory cost.",
"Although the experts still cannot figure out the exact reason that some filters are always more actively updated in the later training stages, they believe the insight that would be obtained from the future investigation will be helpful in diagnosing and improving network structures.",
"Besides, the divergent evolving patterns of classes and the numerous anomaly iterations found in validation view provide the experts with a new promising direction to train a better model.",
"Both $\\mathrm {E}_a$ and $\\mathrm {E}_c$ were particularly fond of the cube-style visualization and deemed it as a new perspective to observe the training of CNNs for them.",
"They both have found many interesting patterns with the cube visualization, some of which were reasonable or could be explained after thinking for a while.",
"However, the experts also failed to figure out some other patterns, notwithstanding they conducted several testing experiments.",
"Nevertheless, they were still excited that our system can help them identify potential subjects for further study.",
"Generality During the implementation, we were concerned about the generality of DeepTracker, that is, whether the design was biased to the specific requirements from these three experts.",
"Therefore, to check how our system is accepted by broader expert communities, we presented our system in a workshop, which involved about 20 experts in the machine learning and visualization fields.",
"In addition, we also interviewed another group of twelve experts, who worked on a large project about using CNNs to improve image search quality.",
"We presented the latest version of DeepTracker to the experts, encouraged them to experiment with the system, and collected their feedback in the process.",
"Exceeding our expectation, DeepTracker was well accepted by these experts.",
"Although they proposed several new requirements, the experts shared many major interests with our three collaborators, such as tracking class level performance, filter-level information, and the relationships between them.",
"After introducing our system, they immediately understood the purposes of each view and all appreciated this novel, intuitive, and expressive way to watch training processes.",
"Although the demo was performed on our experiment datasets, the experts saw its potential in their project and immediately asked for collaboration with us, so that they could plug in and analyze their own datasets.",
"Improvement Apart from this positive feedback, the experts also made several interesting suggestions to further improve DeepTracker.",
"For example, two experts suggested that our current system only differentiates correct or incorrect classifications for validation images (i.e., 1 and 0).",
"However, the exact incorrect labels should also be presented because such information can help identify confusing or similar classes.",
"One expert mentioned that he showed strong interest on what happens before the anomaly iteration and suggested dump data of every iteration at that abnormal interval for fine-grained analysis.",
"Another expert suggested that our system shoud be integrated with online dashboards, as visualizing the training log on the fly can allow them to terminate the training early and save time if the visualization shows anything undesired.",
"Discussion DeepTracker is our first step to open the “black box” of CNN training.",
"Although the experts have high expectations of this tool, we all agree to start with two fundamental pieces of information: neuron weights and validation results.",
"Considering our target users and the large scale of datasets, we try to avoid using sophisticated visual encodings to ensure a fluent exploration experience.",
"Unsurprisingly, our bare-to-metal visualizations are preferred by the experts, and they use it to find many patterns easily, either expected or unexpected.",
"However, we still have several limitations.",
"First and foremost, although our system can effectively help experts identify strange or interesting patterns, there is still a gap between finding patterns and accelerating CNN training.",
"The experts still have to reason about and understand what these patterns mean or how to use them to accelerate model training in future.",
"We think it is not a problem faced just by our system, but by all CNN visualizations in general.",
"Like previous work, DeepTracker may only peel a hole in the box and reveal limited information.",
"But we hope that, by peeling enough holes, all these strange patterns will start to connect and make sense by themselves, thus providing a clear picture of CNNs.",
"[rgb]0,0,0Second, we have adopted many space-efficient visualizations and interaction techniques (e.g., hierarchy, filtering, and aggregation) to address the scalability issue.",
"Our current design can well support showing dozens of layers and classes in the same time.",
"The correlation view shares all the filter strategies with the other two views, and vice versa.",
"Thus, our system can perform well in most cases.",
"Nevertheless, the worst scenario still requires to display hundreds or thousands of small multiples at the same time.",
"A possible solution is to employ task-specific aggregation or filtering methods to show the data of interests.",
"Third, we propose a rule-based anomaly detection method that requires experts to manually pick a reasonable window size $k$ and set the threshold for filtering.",
"The number and patterns of anomalies are sensitive to these settings.",
"One potential solution to this problem is to develop an automatic method to enumerate all potential parameter settings and identify those can detect a reasonable amount of significant anomalies and provide these settings to the experts as guidance.",
"Finally, we only conduct experiments on ResNet-50 [17], but our method can also be applied to other state-of-the-art deep CNN models, which often have similar hierarchical structures (e.g., “inception block” in google-inception-v4 [37]).",
"[rgb]0,0,0 Besides, the cube visualization is a general technique, which can be used to explore multiple heterogeneous time series data and their complex correlations.",
"However, to further generalize it, a strict user study has to be conducted to find the best manner to use it, such as the axis skew degree, and the minimum height/width for each row/column in the three faces.",
"Conclusion We propose a novel visual analytics solution to disclose the rich dynamics of CNN training processes.",
"Knowing such information can help machine learning experts better understand, debug, and optimize CNNs.",
"We develop a comprehensive system that mainly comprises the validation, layer, and correlation views to facilitate an interactive exploration of the evolution of image classification results, network parameters, and the correlation between them.",
"We conduct experiments by training a very deep CNN (ResNet-50) on ImageNet, one of the largest labeled image datasets that is commonly used in practice, to demonstrate the applicability of our system.",
"The positive feedback from our collaborating experts and the experts from an internal workshop validates the usefulness and effectiveness of our system.",
"Future studies may integrate some feature-oriented visualization techniques, which typically require recording the activation information for input instances.",
"Feature visualizations can provide insights on what features a filter in a given snapshot of CNN has learned.",
"Our system can track critical iterations to take snapshots for a CNN over training, and then use feature visualization techniques to analyze the learned features evolving patterns for the detected important filters.",
"The other urgent need is to deploy the system in a real-time environment.",
"To this end, we have to consider some new design and interaction requirements to fill the gap between finding patterns and accelerating CNN training.",
"The authors would like to thank Kai Yan for providing support in editing the relevant media materials.",
"The work is supported by the National Basic Research Program of China (973 program) under Grant No.",
"2014CB340304 and ITC Grant with No.",
"UIT/138."
],
[
"Background",
"A typical CNN can be viewed as a sequence of layers (Fig.",
"REF ) that transforms an image volume (e.g., a $224\\times 224$ image with three color channels of R, G, and B) into an output volume (e.g., a vector of size $1,000$ indicating the probability for an input image to belong to $1,000$ predefined classes) [24].",
"There are three main types of layers to build a CNN architecture, namely, convolutional layer (CONV layer), pooling layer (POOL layer), and fully-connected layer (FC layer).",
"A CONV layer comprises numerous neurons that are connected to a local region in the previous layer's output volume through weighted edges, many of which share the same weights through a parameter sharing scheme.",
"The weights in each neuron compose a filter, the basic unit for detecting visual features in the input image, such as a blotch of color or the shape of an area.",
"The output of each neuron is computed via a dot product operation between the weights and inputs from the previous layer, and is then optionally applied via an elementwise activation function (e.g., ReLU, $max(0, x)$ ).",
"A POOL layer is usually inserted between successive CONV layers to reduce the volume of input through a downsampling operation, thereby reducing the amount of parameters and computation in the network.",
"The only difference between FC layer and CONV layer is that, in contrast to the neurons in CONV layer that are locally connected and have shared parameters, the neurons in FC layers have full connections to the previous layer's output volume.",
"In addition, the output of the last FC layer is fed to a classifier (e.g., Softmax), computing the scores for all predefined classes, where each score represents the probability that an input image belongs to the corresponding class.",
"To obtain an effective CNN, the weight parameters in the CONV and FC layers need to be trained using gradient descent methods [9] to ensure the consistency between the predicted class labels and the predefined class for each training image.",
"Specifically, a training process involves two separate datasets, namely, the training set $\\mathit {D_t}$ and the validation set $\\mathit {D_v}$ .",
"To start the training, the parameters of weighted layers are usually initialized via Gaussian distribution [14], and $\\mathit {D_t}$ is partitioned into non-overlapping batches.",
"For each iteration, one batch of images in $\\mathit {D_t}$ is fed to the network.",
"Afterward, the output classification results are compared with the ground truth (i.e., the known labels of the fed images) to compute the gradients with respect to all neurons.",
"These gradients are then used to update the weights of each neuron.",
"When all batches in $\\mathit {D_t}$ are completed (i.e., finish one epoch), $D_t$ is reshuffled and partitioned into new non-overlapping batches.",
"After several epoches, the initially-randomized neural network will be gradually shaped into a specified network targeting at a specific task.",
"Meanwhile, for every given number of iterations, the network is evaluated via $\\mathit {D_v}$ .",
"Similarly, $D_v$ is fed into the network and then the output classification results are collected and compared with the ground truth.",
"However, the results are only used to validate whether a training process goes well and never used to update neuron weights."
],
[
"CNN Visualization",
"CNNs have recently received considerable attention in the field of visualization [34].",
"Existing approaches can be classified into two categories, namely, feature-oriented and evolution-oriented.",
"Feature-oriented approaches aim to visualize and interpret how a specific CNN behaves on the input images to disclose what features it has learned.",
"Most of the existing studies belong to this category.",
"Some studies [42], [41] modify part of the input and measure the resulting variation in the output or intermediate hidden layers of a CNN.",
"By visualizing the resulting variations (e.g., using a heatmap), users can identify which parts of the input image contributes the most to the classification results.",
"By contrast, other studies [41], [36], [30], [11] attempt to synthesize an image that is most relevant to the activation (i.e., the output of a layer after an activation function) of interest to help experts determine which features of a specified image are important for the relevant neurons.",
"For example, Mahendran and Vedaldi [30] reconstruct the input images that can either activate the neurons of interest or produce the same activations as another input image.",
"Besides, some methods focus on retrieving a set of images that can maximally activate a specific neuron [13], [12].",
"In this manner, users can discover which types of features or images are captured by a specific neuron.",
"Built on this method, Liu et al.",
"[27] develop a visual analytics system that integrates a set of visualizations to explore the features learned by neurons and reveal the relationships among them.",
"In addition, some work [32], [10], [21], utilizes dimension reduction technique to project the high-dimension activation vectors of FC or intermediate hidden layers onto a 2D space to facilitate revealing the relationships among outputs.",
"In contrast to those studies that investigate how a specified network works, only few studies concentrate on visualizing network training processes.",
"One typical method is to pick several snapshots of a CNN over the training process and then leverage feature-oriented approaches to compare how a CNN behaves differently on a given set of inputs at various iterations [41], [10], [43], [4].",
"For example, Zeiler and Fergus [41] use deconvnet approach to visualize a series of synthesized images that are most relevant to one activation of interest at a few manually picked snapshots, and then observe the differences between them.",
"[rgb]0,0,0 Zeng et al.",
"[43] present a matrix visualization to show the weight differences of filters of one layer as well as this layer's input and output in two model snapshots.",
"Their system also supports side by side comparison on the learned features of neurons (the computation method is similar to the work [13]) in different model snapshots.",
"One limitation of these methods is that when there are myriad filters, images, and iterations, it is challenging to select proper ones to observe and compare.",
"By contrast, we attempt to reveal the rich dynamics of the network parameters and the classification results at a larger scale to help experts find the notable filters, images, and iterations.",
"In this point, Alsallakh et al.",
"[4] analyze the same data facets (i.e., input images, network parameters, and classification results).",
"However, the work focuses more on identifying the hierarchical similarity structures between classes and still belongs to the category of multiple snapshots comparison, thereby suffering from the same limitation.",
"To analyze the evolution of CNN parameters, Eliana [29] treats the parameters of the entire network at one iteration as a high-dimension vector, uses PCA to map the vectors at all iterations onto a 3D space, and creates trajectories for these points.",
"However, this visualization is too abstract to extract useful insights for understanding and debugging the training process.",
"To analyze the classification results, Rauber et al.",
"[32] create a 2D trails graph to present an overview of how the CNN classification results evolve by leveraging the projection techniques.",
"However, this method suffers from scalability and visual clutter problems when applied to large-scale datasets (e.g., ImageNet).",
"Besides, this method only provides a very high-level overview and cannot answer those questions that involve individual classes or images.",
"[rgb]0,0,0The work most similar to ours should be the one by Liu et al.",
"[26], whereas the work mainly targets at deep generative models and would have serious scalability issue if applied in an industry-level CNN training.",
"Besides, compared with that work, we specifically provide a series of hierarchical methods that are tailored for CNNs to visualize the training dynamics, including classification results and network parameters.",
"[rgb]0,0,0Furthermore, we design a novel correlation matrix and 2.5D cube-style visualization to help experts examine the complex relationships exist among network parameters, classification results, and iterations."
],
[
"Multiple Time Series Visualization",
"Time series data have been extensively studied due to their ubiquity.",
"Numerous approaches [3], [2], [5] to time series visualization have been proposed.",
"Our system also falls into this field, since training logs are essentially a type of time-based information.",
"Specifically, our system is most related to existing work that visualizes multiple time series.",
"To visualize multiple time series, one method is to place multiple charts (e.g., line charts) in the same graph space to produce overlapping curves.",
"However, this method reduces the legibility of individual time-series [20].",
"In contrast to overlaying multiple series, one popular alternative is to use small multiples [39] that split the space into individual graphs, each showing one time series.",
"Using small multiples also allows for an effective comparison across charts.",
"Several factors may also affect the performance of small multiples [19], [20], such as the types of charts used (e.g., line charts and horizon graphs), the number of series and the vertical space allocated to each series.",
"The improper use of time-series charts, the increased series, and the small vertical space of each series will result in a serious visual clutter problem.",
"In this work, the evolution of the image classification results of each class and the weight parameters of each layer can be viewed as two types of multiple time series data.",
"Given the large number of classes and layers in a practical CNN training, we identify several space-efficient charts that can characterize the training dynamics.",
"Besides, we propose a similarity-based layout and a hierarchical exploration method to support the exploration of relationships among multiple time series to address the visual clutter problem.",
"We also present a novel cube-based visualization, targeting at the exploration of complex relationships among various types of heterogeneous time-series data (e.g., image classification results, neuron weights, and training iterations)."
],
[
"Requirement Analysis",
"DeepTracker was developed in approximately nine months, in which we collaborate closely with three experts (denoted by $\\mathrm {E}_a$ , $\\mathrm {E}_b$ , and $\\mathrm {E}_c$ ) who have considerable experience in CNNs.",
"We held regular discussions with these experts once a week.",
"From our regular discussions, we learned that the training process should be carefully babysat.",
"The experts have to tune and fix a number of hyper-parameters (learning rate, batch size, layer number, filter number per layer, weight decay, momentum, etc.)",
"before starting a training trial.",
"These hyper-parameters, which strongly influence how a CNN is trained, are usually selected based on the experiences learned from their previous trials.",
"During a training, there are several useful quantities the experts want to monitor, such as loss function (the residual error between the prediction results and the ground truth), train/validation error rate (the percentage of mislabeled images), weight update ratio (the ratio of the update magnitudes to the value magnitudes), and weight/gradient/activation distributions per layer.",
"Basically, both loss and error rate should decrease over time; the consistent increase or violent fluctuation of loss on $D_t$ may indicate a problem; a big gap between the error rates of $D_t$ and $D_v$ may suggest that the model is over-fitting, while the absence of any gap may indicate that the model has a limited learning capability; the update ratioIn most cases, if the update ratio is lower than 1e-3, the learning rate might be too low; if it is higher than 1e-3, the learning rate is probably too high.",
"is expected to be around 1e-3; the weight distributions per layer in the beginning should overall follow Gaussian distributions with different standard deviation settingsXavier Initialization [14] is applied in our model.",
"Basically, the deeper (close to loss layer) the layer is, the smaller the sd is.",
"and may become diverse after training; and exploding or vanishing gradient values are a bad sign for the training.",
"The aforementioned rules of thumb are all based on the analysis of high level statistical information.",
"However, the experts still strongly desire to examine more details underlying the statistic, so that they can gain more knowledge and suit the remedy to the case when problems occur.",
"For example, the experience tells the experts that it is better to continue training models (e.g., ResNet [17]) with the same learning rate for a while rather than turn it down immediately, when the overall error rate stops to decrease.",
"The experts are very curious about what happens to the model and its performance on each class of images during the period.",
"Also, we may often see a fluctuation of loss or an occasional exploding gradient values, and what brings about this?",
"Are there any layers or filters that behave abnormally?",
"These details are never uncovered before.",
"Thus, the experts strongly desire a tool to enable them to explore the hidden dynamics behind a CNN training.",
"After several rounds of structured interviews with the experts, we finally summarized and compiled the following list of requirements: [R.1] Exploring multiple facets of neuron weight information.",
"A single iteration may have millions of weight updates, but individual values are generally meaningless to the experts.",
"Instead, the experts are more interested in the statistical information of these weights at different levels of detail (e.g., CONV layer level and filter level).",
"All three experts emphasized the importance of intuitively and effectively examining general types of statistics, such as sum and variance, between these levels.",
"Besides, $\\mathrm {E}_a$ and $\\mathrm {E}_b$ strongly desire to see the weight change degree for filters over iterations to identify which filters change dramatically at which iteration or how a filter changes across iterations.",
"Comparing multiple layers.",
"All three experts like to compare layer-level statistical information.",
"For example, they want to know whether a specified measure of different layers show a similar trend or distribution (e.g., whether the sum is positive or negative).",
"Accordingly, our visualization should also help these experts in performing such comparisons.",
"Tracking the classification results of validation classes.",
"Validation is a critical step in the training process that “test drives” the trained CNN and tracks its performance change across iterations [6].",
"Previous tools only measure the global validation loss/error, thereby concealing the rich dynamics of how the image labels of each class change over time.",
"When the error rates do not reduce as expected, the experts find that such highly-aggregated information is useless and cannot help them understand why or when the training runs into a bottleneck.",
"Therefore, the experts want to know how the images of different classes behave differently and identify those classes that are easy or difficult to train.",
"Detecting important iterations.",
"One complete training process usually contains millions of iterations, but it is obvious that not all iterations are equally important.",
"For example, some image classes may suddenly change their error rates after certain weight updates.",
"The experts are interested in these patterns, which may help reveal in-depth relationships between filters and image features.",
"However, the overall error rate trend does not help much, since it generally decreases slowly and steadily.",
"Thus, the experts hope that our system can help them identify the abnormal iterations and the corresponding classes.",
"Examining individual validation classes.",
"Our initial prototype shows that different classes clearly have different error rate evolution patterns.",
"Thus, experts $\\mathrm {E}_b$ and $\\mathrm {E}_c$ are curious about those classes with very poor or excellent performance, thereby desiring to further explore the image label information for these classes.",
"For example, they want to see whether and why some images are always misclassified.",
"Enabling correlation exploration.",
"Apart from analyzing the weight and validation data separately, the experts are also curious about their relational patterns.",
"They are specifically interested in uncovering the relationships between the layers or filters and the validation images, such as how the changes in network parameters respond to the image labeling results for each class.",
"By connecting these two pieces of information together, they hope to gain fundamental insights into the behaviors of CNNs and to improve training results.",
"System Overview Figure: Three components of DeepTracker.",
"The raw data are preprocessed into a hierarchical structure and then stored into five application-specific data indexes to enable real-time interactions.",
"On top of the efficient data storage, several visualizations are combined together to help experts with analysis tasks from different levels and perspectives.DeepTracker is a web-based application developed under the full-stack framework, MEAN.ts (i.e., MongoDB, Express, AngularJs, Node, and Typescript).",
"The back-end part of our application is deployed in a server with 3.10GHz Intel Xeon E5-2687W CPU and 32GB memory.",
"The architecture of our system (Fig.",
"REF ) begins with the data processing part, where the entire training log is hierarchically organized, and several application-specific indexes are built to support real-time interactions.",
"On top of the efficient data storage, we build three coordinated views, namely, the Validation View, the Layer View, and the Correlation View, to support an interactive analysis from different levels and perspectives.",
"The Validation View aims at providing a visual summary of CNN performance on different validation classes.",
"By combining our anomaly detection algorithm and small multiples, experts can easily identify different image class behavior patterns and critical iterations for each image class (R3, R4).",
"Experts may also drill down to the class of interest to explore further image label information (R5).",
"The Layer View aligns the weight information with the CNN structure to help experts explore various statistical information in the network hierarchy.",
"Experts can further drill up or down in the network hierarchy to compare these measures at different levels of detail (R1, R2).",
"The Correlation View presents a novel grid-based visualization method to provide an overview of the correlation between the image classification results and the neuron weights of each filter (R6).",
"[rgb]0,0,0 The three views compose a cube, with which the experts can simultaneously explore the changes of class-level performances, the variations of filter-level weights, and the correlations between them.",
"Data Acquisition and Construction [rgb]0,0,0 The primary motivation of this work is to monitor industry-level CNN training processes.",
"Therefore, we conduct our experiments with ResNet-50 [17] and ImageNet Dataset [33].",
"ResNet-50, containing 50 weighted layers (i.e., CONV and FC layers), is among the most popular CNNs that have been recently used in practice and meanwhile ImageNet 2012 is also among the largest and most challenging publicly available datasets.",
"The dataset includes $1,000$ classes of images, each class containing $1,300$ training images and 50 validation images.",
"Training such a model needs around 120 epoches (nearly 1.2 millions iterations when batch size is 128) to achieve convergence.",
"Simply dumping all the information for every iteration can easily have the size of dumped data exceed several petabytes and take about 4 weeks.",
"Through discussion, we all thought that $1,600$ is a reasonable interval to capture meaningful changes (about 7 times per epoch).",
"This reduces the log to a manageable size (about 1TB).",
"[rgb]0,0,0 For each dump, we recorded two pieces of information, namely, neuron weights/gradients of CONV layer and FC layer and image classification results.",
"The parameters on BN layers were not recorded, as they can be totally recovered given the weights on CONV and FC layers and always need to be updated when applied in a new dataset.",
"Besides, we did not record the activations of each layer/filter for every validation image, as doing so is technically impracticable considering the extremely large models and datasets and the limited disk storage.",
"Further, activation evolution visualization is beyond the research scope of this paper.",
"Sec.",
"discusses activation visualization is indeed a perfect complementary technique to our work.",
"We organized the weight/gradient information according to the natural hierarchial structure of ResNet-50.",
"It consists of four CONV modules (plus the first CONV layer and the final FC layer, there are 50 layers in total).",
"Each module contains several bottleneck blocks [17] that comprise three to four basic CONV layers (data storage in Fig.",
"REF ).",
"Thus, we grouped all neuron weights to align with such hierarchy.",
"[rgb]0,0,0 In a similar manner, we organized the classification results hierarchically from individual level, class level, to model level.",
"We stored all the data into MongoDBMongoDB is a free and open-source cross-platform document-oriented (NoSql) database.",
"www.mongodb.com.",
"In particular, we precomputed all the relevant aggregation values, such as weight means and error rates, for each filter, layer, image, and class.",
"Nevertheless, the distilled data still remain too large to load into memories (about dozens of gigabytes per training).",
"Therefore, we analyzed the frequent needs of the experts and built several indexes to enable real-time interactions, including Layer-Stat index $I_{ls}$ , Layer-Filter index $I_{lf}$ , Iter-Filter index $I_{if}$ , Cls-Stat index $I_{cs}$ , and Cls-Image index $I_{ci}$ .",
"$I_{ls}$ retrieves the statistic values (e.g., mean and sd) at every iterations for any given layer; $I_{lf}$ lists all the filter-level information (e.g., changing degree of each filter) at every iterations for any given layer; $I_{if}$ searches the top changing filters from all layers at any given iteration; $I_{cs}$ extracts class-level information (e.g., class performances, the different types of abnormal images Sec.",
"REF ) over all iterations for any given class; $I_{ci}$ fetches the meta-data of images for any given classes.",
"Visualization In this section, we describe our three coordinate views, namely, the Validation View, the Layer View, and the Correlation View, that help experts accomplish the aforementioned analytical tasks.",
"Validation View Several urgent requirements (R3, R4, R5) from the experts need to examine how the evolving CNN acts differently on the validation images of each class rather than how the overall validation error rate differs over training.",
"Thus, we design the Validation View (Fig.",
"REF & REF ) to present all image classes in $D_v$ .",
"Figure: From top (a) to bottom (b), image classes are more and more difficult to train.Visual Encoding [rgb]0,0,0By default, the view starts with a visualization of cluster-level performance (R3).",
"The classes with similar evolving trends form a cluster and then their error rates at every iteration are averaged.",
"The averaged error rates are then depicted as a colored stripe, where the x-axis encodes the iterations and the error rates are encoded by colors (Fig.",
"REF ).",
"We choose k-means as the clustering algorithm and $k$ can be adjusted according to demands (Fig.",
"REF shows the case when $k=4$ ).",
"Experts can open up one cluster to further examine the performance in class-level (R3).",
"The design is based on the following considerations.",
"First, the experts are more interested in the overall classes than in individual iterations.",
"Thus, small multiples technique (juxtaposed techniques) is chosen for their superior performance in high-level comparison tasks (e.g., global trends for every series) [20].",
"Second, we cannot present all the classes ($1,000$ in ImageNet) at one time, we have to consider a hierarchical and highly space-efficient visualization.",
"However, many traditional charts, such as line charts and horizon graphs, require a larger vertical space [19] than 1D heatmaps.",
"Compared with traditional charts, heatmaps are also more easy to do side by side comparison for their symmetrical space (i.e., no irregular white spaces).",
"As a result, all experts prefer the heatmap-based small multiples.",
"Image-level performances, R5.",
"The class-level color strips can be further unfolded to explore the image-level evolution patterns.",
"Unfolding a heatmap reveals a pixel chart (Fig.",
"REF d), with each row (1px height) representing an image and each column (1px width) representing an dumped iteration (consistent with the class heatmap).",
"We use red and green colors to indicate the incorrect and correct classifications, respectively.",
"Meanwhile, the experts can zoom/pan the pixel chart for a closer inspection.",
"Clicking on a row shows the original corresponding image.",
"Anomaly iterations, R4.",
"[rgb]0,0,0As experts are concerned about the iterations with abnormal behaviors, we particularly propose a algorithm to detect these anomaly iterations (refer to Sec.",
"REF ).",
"Experts can choose to only show the classes with anomaly iterations (Fig.",
"REF ).",
"At this point, for each class-level color stripe, we use triangular glyphs to highlight these detected anomaly iterations.",
"The upside-down triangles ($\\bigtriangledown $ ) and normal triangles ($\\triangle $ ) indicate those anomaly iterations that are detected by the left-rule and right-rule, respectively.",
"The widths of triangles encode the anomaly scores.",
"Experts can set a threshold value to filter the triangles with low anomaly scores.",
"Anomaly Detection In our scenario, the classification results for an image can be represented by a 0/1 sequence ($[a_1, ..., a_n]$ ), where each element represents a correct or incorrect result at the corresponding validation iteration.",
"The experts are curious about the iterations when a significant amount of 1/0 flips (i.e., 0 to 1 or 1 to 0) occur for a class.",
"In general, this problem can be modeled and solved using Markovian-based anomaly detection algorithms [1].",
"Despite the popularity of using Markovian methods to detect outliers in discrete sequence, we decide to employ rule-based models [1] for two reasons.",
"First, Markovian methods are a black box and the resulting outlier values are sometimes difficult to comprehend.",
"Second, the experts have explicitly described two types of iterations they are very interested in, namely, those iterations when many images with values that remain stable for many previous iterations suddenly flip (denoted by the left-rule) and those iterations when many images flip and keep their values stable after many iterations (denoted by the right-rule).",
"Fortunately, these anomalies can be easily modeled using rules.",
"The rule-based models primarily estimate the value $P(a_i|a_{i-k},\\ldots ,a_{i-1})$ , which can be expressed in the following rule form: $ a_{i-k},\\ldots ,a_{i-1} \\Rightarrow a_{i}.$ In our scenario, if an image has the same value (either 0 or 1) in the previous consecutive $k$ iterations ($i-k,\\ldots , i-1$ ), then its value must be the same at iteration $i$ (the left-rule).",
"Otherwise, iteration $i$ is considered an outlier for the specified image.",
"Based on these considerations, we develop an application-specific algorithm to detect anomaly iterations in the validation history.",
"The algorithm includes the following steps: Rule-Judgement: The algorithm computes a vector $[l_{i1}, \\ldots , l_{in}]$ for every image $i$ , where $l_{ij} =1$ if the left-rule is satisfied, otherwise, $l_{ij} =0$ ; Aggregation: For each class that contains $m$ images, the algorithm aggregates all the computed vectors for each image into one $[L_1, ..., L_n]$ , where $L_j = \\sum _{i=1}^{m}l_{ij}$ , [rgb]0,0,0denoting the left anomaly score at iteration $j$ for this class.",
"The approach is a window-based method, and the experts can adjust the window size $k$ to control the sensitivity of the anomalies.",
"In a similar manner, we detect the anomalies from the opposite direction for the right-rule.",
"Layer View The Layer View focuses on weight-related tasks (R1, R2).",
"The view consists of two connected parts, namely, the CNN structure and the hierarchical small multiples (Fig.",
"REF ), so that experts can hierarchically explore and compare various types of statistic in the context of the network structure.",
"Figure: Visual encodings in the Layer View: (a) a CONV layer, (b,d) hierarchical bars, (c) links between the CNN structure and (e) the hierarchical small multiple charts, (f) a pixel chart for one layer.CNN structure.",
"The experts hope our tool can help them explore the statistical information of each layer and meanwhile know their relative positions in the entire network (R1).",
"Thus, we adopt NetscopeNetscope is a web-based tool for visualizing neural network architectures.",
"http://ethereon.github.io/netscope/, a popular neural network visualizer, in our system.",
"[rgb]0,0,0 The green rectangle is data input layer, the red ones are the CONV layers, and the purple ones mean the pooling layers.",
"The links between these rectangles show the network structure.",
"We further add blue level bars (Fig.",
"REF b) to encode the latent hierarchy (from CONV modules, bottlenecks to basic CONV layers, Sec. ).",
"The right most level bars represent CONV modules (Sec.",
"), which are recursively divided into smaller level bars until reaching elementary CONV layers (i.e., red rectangles).",
"Hierarchical small multiples.",
"To assist experts in exploring and comparing layers of a deep CNN (R1, R2), a space-efficient visualization technique is demanded.",
"Thus, we leverage hierarchical small multiples to show layers of interest (Fig.",
"REF e).",
"[rgb]0,0,0By default, experts are presented with the information about CONV modules and then can drill down to see more information about low-level CONV layers with interactions with the network graph (i.e., click on the corresponding level bars).",
"The width of outcropping rectangles (Fig.",
"REF d) encodes the aggregation level of current layer charts.",
"For example, the top second layer chart in (Fig.",
"REF e) shows the bottleneck-level aggregation information and the following three layer charts show the basic CONV layer level information.",
"Besides, the links (Fig.",
"REF c) mark the real positions of the layer charts in the network structure.",
"The small multiples support multiple types of charts including line chart, horizon graphs [18] and box plots to emphasize the different aspects of the statistical data.",
"The experts use box-plots to see the rough distribution of statistical values and use basic line charts to examine individual values.",
"Besides, the experts prefer to use horizon graphs when performing tasks in regard to trend tracking and comparison (R2), because of its effectiveness in visualizing divergent weight values [20].",
"Similar to unfolding the class heatmap to a pixel chart, the experts are also allowed to open the layers of interest as a pixel chart (Fig.",
"REF f) that presents the filter-level information (R1).",
"Each row (1px height) in the pixel chart represents one filter, and each column (1px width) indicates one iteration.",
"We use sequential colors to encode pixel value (e.g., the cosine similarity between two subsequent dumped iterations).",
"Correlation View This view helps experts establish connections between the filters and images.",
"In particular, the experts want to understand further how the changes in network parameters are related to class performances (R6).",
"For example, several anomaly iterations may be detected for a single class.",
"For each detected anomaly iteration, we can identify a set of anomaly filters (i.e., the top $k$ filters with largest changes at that iteration).",
"Since different classes may share anomaly iterations and different anomaly iterations may share anomaly filters, are there any filters that are commonly seen in these iterations?",
"Do any of the anomaly classes or filters strongly co-occur?",
"We designed the Correlation View to answer these questions.",
"[t] Target set $S_{target}$ .",
"Minimum partition for $S_{target}$ .",
"$S_{result}$ = $\\emptyset $ each target set $s_t$ in $S_{target}$ $S_{new}$ = $\\emptyset $ each mini set $s_r$ in $S_{result}$ $itersection$ = $s_t \\bigcap s_r$ $S_{new}$ = $S_{new} \\bigcup intersection \\bigcup (s_r - s_t)$ $s_t$ = $s_t - s_r$ $s_t$ is not empty $S_{new} \\bigcup s_t$ remove all empty set in $S_{new}$ $S_{result}$ = $S_{new}$ return $S_{result}$ Minimum Set Partition Filter set partition.",
"We first introduce the mini-set concept to organize anomaly filters that are shared by multiple anomaly iterations and different classes.",
"For each class $C_i\\in \\lbrace C_i|{1\\le i \\le n}\\rbrace $ , we denote its anomaly iterations by $T_i = \\lbrace t_{i,k}|{1\\le k\\le n_i}\\rbrace $ .",
"Thus, all anomaly iterations are $\\cup _{1\\le i \\le n}T_i$ , denoted by $T$ .",
"For each anomaly iteration $t\\in T$ , we denote its anomaly filters at CNN layer $L_j\\in \\lbrace L_j|{1\\le j \\le m}\\rbrace $ by $s_{j,t}$ .",
"Thus, for each layer $L_j$ , we can collect all anomaly filter sets $\\lbrace s_{j,t}|t\\in T\\rbrace $ (denoted by $S_j$ ) and all anomaly filters $\\cup _{t\\in T}s_{j,t}$ (denoted by $s_{j}$ ).",
"Thus, mini-set aims to find a minimum set partitions of $s_{j}$ (denoted by $s_j^\\ast $ ) that each $s_{j,t}$ can be assembled from some elements (i.e., mini-sets) in $s_j^\\ast $ .",
"We specifically propose a Set Partition Algorithm (Alg.",
"REF ) to find $s_j^\\ast $ .",
"The algorithm accepts a target set as input (i.e., $S_j$ ).",
"$S_{result}$ is initially empty and a new anomaly filter set is used at each time to partition the mini-sets contained in $S_{result}$ (cf.",
"lines 4 to 7).",
"If the new anomaly filter set is not empty after partitioning, then it is added as a new mini-set (cf.",
"line 8).",
"Finally, the partitions contained in $S_{result}$ will be returned.",
"Figure: (a) The abstract version of correlation view, where rows and columns represent layers and image classes, respectively.",
"A sequential color scheme is used to encode the number of anomaly filters.",
"(b) The complex version of correlation view, where the detailed information of individual anomaly filters are shown.",
"(c) A layout solution for coordinated analysis without using skewed axes.Visual encoding.",
"To intuitively represent these relationships, we introduce a grid-style visualization (Fig.",
"REF ), where rows and columns represent layers and image classes, respectively.",
"The number of rows and columns equal to the number of layers with anomaly filters and classes with anomaly iterations, respectively.",
"[rgb]0,0,0 We start from a abstract version.",
"In this version (Fig.",
"REF a), for $\\mathrm {Cell}_{i,j}$ , a sequential color scheme is used to encode the number of anomaly filters ($\\cup _{t\\in T_j}s_{i,t}$ ).",
"The darker the color it is, the more anomaly filters appear in layer $L_i$ that are related to class $C_i$ .",
"From this visualization, we can easily observe the correlations between layers and classes, while it also hides much detailed information.",
"We cannot answer questions like whether the filters in $\\mathrm {Cell}_{i,j}$ are the same with $\\mathrm {Cell}_{i,k}$ (this kind of information shows how many classes this filter can impact and help examine the relationships among classes), or whether there are filters in $\\mathrm {Cell}_{i,j}$ appearing in more than one anomaly iteration (this shows the importance of these filters for class $C_j$ ).",
"To solve these problems, we provide an advanced version (Fig.",
"REF b).",
"For $\\mathrm {Cell}_{i,j}$ , the width and height encode the number of anomaly iterations and the number of anomaly filters of the corresponding class and layer (i.e., $|T_j|$ and $|s_{i}|$ ), respectively.",
"Based on these numbers, the columns and rows are further divided with vertical/horizontal lines.",
"For a class (e.g., $\\mathrm {Column}_{j}$ ), $|T_j|$ vertical lines are drawn to represent all related anomaly iterations (i.e., $[t_{j,1}, t_{j,2},\\ldots ,t_{j,n_j}]$ ).",
"For each row of layer $L_i$ , there are $|s_i^\\ast |$ horizontal lines representing all mini-sets in that layer.",
"The intersections between these horizontal and vertical lines are highlighted with blue rectangles if the corresponding mini-set is part of the anomaly filters of the corresponding anomaly iteration.",
"The height of the rectangles represents the number of filters of the corresponding mini-set.",
"[rgb]0,0,0Obviously the introduction of mini-sets dramatically cuts down the number of horizontal lines and blue rectangles, otherwise each anomaly filter require one horizontal line and one rectangles, which may cause serious visual clutter problem.",
"In fact, mini-sets can be viewed as a partially aggregation version instead of representing all the anomaly filters as horizontal lines and rectangles.",
"Users can set the minimum appearing number of mini-sets to filter the sets with lower importance.",
"Cube Visualization The log data contain three main aspects of information, namely, iterations, validation information, and weight information.",
"The three aforementioned views are designed to show all possible 2-combinations of these three types of information, respectively.",
"[rgb]0,0,0 Although these views can be used individually, they need to be combined together to form a complete picture.",
"Thus, we propose a novel and intuitive visualization technique, which naturally and seamlessly stitches the three views together, based on their shared axis (inspired from Binx [8]), into a “cube” shape (Fig.",
"REF ).",
"When experts find or highlight a pattern of interest, they can easily track the pattern over the edges to find the related information in the other two views easily.",
"[rgb]0,0,0 The use of skewed axes may bring about a possible perspective distortion problem.",
"Nevertheless, the advantages of the cube-style design far outweigh the disadvantages.",
"Given the limited pixels in a computer screen and each view requiring a large display space, the experts all agree that the cube-style design is the most space-efficient and intuitive manner to show all the information.",
"Furthermore, with such design, it prevents the experts from switching from multiple views, reducing the cognitive burden and the load of memory.",
"This allows the experts to conduct correlation analysis more effectively.",
"We also provide a compromised solution to handle the distortion problem, that is, laying out the three views in the form like Fig.",
"REF c. So the experts can firstly examine the layer view (horizontally) or validation view (vertically) together with the correlation view, and then switch to cube mode to explore the three views together.",
"[rgb]0,0,0 Notice that there are some different settings for several views in the cube.",
"For the layer view (front), only the layers with anomaly filters are activated (see the activated blue bars in the front view of Fig.",
"REF ), the weight variation of each anomaly filter is represented as a horizontal color strip.",
"For the validation view (top), only the classes with anomaly iterations are preserved.",
"The following lists several common exploration pipelines: P1: from the layer view (front), we can quickly check the distribution of activated layers in the overall network and pick some anomaly filters of interests.",
"Then, by tracking along the horizon axis to the correlation view (right), we can examine which classes these filters impact and how important these filters are to the classes.",
"Finally, we can observe the evolving patterns of these classes and the corresponding anomaly iterations in validation view (top).",
"P2: from the validation view (top), we can firstly mark several anomaly iterations of some classes.",
"Then, we can check the corresponding columns in the correlation view, finding the rows that contain anomaly filters and exploring the importance of these filters to these classes and how these filters impact the other classes.",
"Finally, by highlighting these corresponding rows, we can observe them in the layer view to see how these filters behave around the picked anomaly iterations.",
"P3: from the correlation view (right), we can search the horizontal lines across many rectangles (it means these filters impact many classes at the same time) or the rectangles that appear more than one time in the same cell (these filters are judged as anomalies many times for a class and may have great impact on this class).",
"With these selected horizontal lines or rectangles, we can simultaneously track their corresponding weight variation information in the layer view (front) and class performance information in the validation view (top).",
"Use Examples We derived these examples through the assistance of our collaborating experts, who were familiar with our designs and data.",
"As a remark, the following results are from the experiment with 8 times larger batch size and learning rate setting than the basic setting introduced in Sec. .",
"Exploring Validation Results Figure: Overview of validation classes.",
"(a) Two curves show the overall training/validation error rate.",
"(b) Two turning points align well the boundary of stage s2s2.",
"(c) Three peaks appear in the stage s1s1 and align well with (f) the detected anomaly iterations.",
"(d) Two types of mushroom images have different behaviors in the class.",
"(e) Most images in the class flip at the anomaly iteration.The first scenario demonstrates how the experts use DeepTracker to explore the image classification results (R3, R4, and R5).",
"Performance evolution patterns.",
"Fig.",
"REF a shows a typical visualization of training/validation errors that may appear on any popular training platform.",
"The timeline at the top shows a total of 1.2 million iterations.",
"Beneath the timeline, four line segments represent four stages ($s1$ , $s2$ , $s3$ , and $s4$ in Fig.",
"REF ) in the training process, [rgb]0,0,0where the later stage has one-tenth of the learning rate of the previous stage.",
"We can observe that two sudden drops in the curves match well with the boundaries of the training stages (Fig.",
"REF b).",
"However, this is a well-known pattern to the experts.",
"On the other hand, although the overall error rate continues to decrease, the class-level error rates show a more complicated story, that is new to the experts.",
"By quickly scanning the small multiples in cluster-level, the experts identify there are generally four types of class evolving patterns (Fig.",
"REF ).",
"From top to bottom, the four types are more and more difficult to train.",
"[rgb]0,0,0For example, for the type at the top, these classes are recognized correctly after a few iterations.",
"By contrast, the classes at the bottom always have high error rates in the entire training process, which means that the resulting network fails to recognize the related images.",
"From this, the experts learn that the model has spent most of time to improve its performance on the classes of middle-level classification, since the model has already performed well on the easy-trained classes at a very early stage and is always performing miserably on the hard-trained classes over the entire training.",
"From these patterns, the experts consider it promising to accelerate the training process and improve the overall performance by treating classes differently during the training process.",
"That is, stop feeding the easy-trained classes in an appropriate early stage, put more efforts on training the classes of middle difficulties for classification, and figure out why some classes always have extremely high error rates.",
"One similar attempt has been made in a recent work [25].",
"Anomaly iterations.",
"The experts are curious about the three sudden peaks in stage $s1$ , and then mark these three iterations with dotted lines (Fig.",
"REF c), which look like anomaly iterations (R4).",
"However, the colors in the small multiples do not have clear patterns related to these iterations.",
"Then, the experts turn on the anomaly detection and immediately find that many triangles are aligned well with the dotted lines (Fig.",
"REF f), thereby confirming our suspicion.",
"Then, the experts can click on the corresponding image icons to see the detailed images that contribute to the three peaks.",
"In addition, there are more anomaly iterations in stage $s1$ then in the later stages.",
"This interesting pattern can be explained by the reduction of learning rate and the convergence of the model in the later stages.",
"At the same time, it also implies that the learning rate in stage $s1$ is slightly too high, leading to the instability in the model (the case in Sec.",
"REF indicates the same finding for the discovery of potential “dead” filters).",
"Details in classes.",
"To further examine what happens at the anomaly iterations for a class, the experts can further check the image-level information of the class (R5).",
"For example, the experts are curious about the abnormally large anomaly iteration in the class of “mushroom” (Fig.",
"REF d) that are captured by both the left-rule and the right-rule.",
"Then, they click and expand color stripe to see the pixel chart of images.",
"First, they confirm that this iteration is indeed special for this class, because nearly all images flip during particular that [rgb]0,0,0dumped interval (Fig.",
"REF e).",
"Thus, they may further investigate to find the layers or filters that cause such flips based on the filter updates around that iteration.",
"[rgb]0,0,0In particular, the experts comment that, it seems that after the iteration, the CNN model has jumped to a better local optimal for the class, because the green color is more stable after the iteration.",
"This may result from the reduction of the learning rate (from $s1$ to $s2$ ).",
"This kind of patterns appear frequently in many classes during the whole training process, many of them occurring not in the learning rate transition point.",
"The experts wonder that the model should be trying to jump from one local optimal to another better local for these classes continuously, so as to reduce the overall error rate gradually.",
"This insight has never been obtained because the experts initially thought that the error rate for one class should decrease steadily.",
"Besides, the experts also find that, at the bottom of the pixel chart, several images are mislabeled in the entire training process, although the class is easy to train overall(Fig.",
"REF d).",
"To understand why, the experts click on these images to examine them, and find that the contents in the mislabeled images have a clear color pattern different from that of the rest of the mushroom images.",
"The correctly labeled mushrooms are all red, while the mislabeled ones are white or orange.",
"This finding indicates color is a critical feature that the CNN has learned to classify this class of images.",
"Exploring Weight-Relevant Information Figure: (a, b, c) The sd values of each layer decrease slowly and have different scales in different CONV modules.",
"(d) The weight changes in filters are large at the beginning.",
"(e) One outlier filter is detected, whose weights never change during the entire training process.This scenario shows how to discover patterns in neuron weights via the Layer View (R1, R2).",
"First, the experts choose to show the sd (standard deviation) of the weights at the layer-level using horizontal graphs (Fig.",
"REF ).",
"As the experts expected, all the trends show a similar pattern of slow decrease, indicating the weights in the entire model is converging over iterations.",
"Besides, the experts also find that deeper layers (closer to the loss layer) tend to have smaller sd values.",
"[rgb]0,0,0In particular, by tuning the band number (finally to 3) of the horizon graphs, they found the sd values of a CONV module are usually twice as large as those of the one below it (a, b, and c in Fig.",
"REF ).",
"Given that we apply Xavier initializationREF and for ResNet-50, the input sizes of layers in a CONV module are twice as large as the ones in the layers of its previous CONV module, the observed result is not beyond the experts' expectation.",
"This suggests that there is no problem exist on the initialization approach.",
"Analogously, the experts find that the weight means of each layer become negative quickly (from green to blue instantly, Fig.",
"REF a) except for the FC layer (Fig.",
"REF b).",
"At first, the pattern looks strange to the experts.",
"Then, the experts realize that it is reasonable to have more negative weights than positive ones, since negative values are often used to filter out trivial features owing to the use of ReLU activations.",
"The increase of negative weights suggests that the network is trained to extract useful information from the original image layer by layer, and then finally remain the most relevant features.",
"[rgb]0,0,0As for the FC layer, it plays a function of shaping the extracted useful features into feature vectors of given dimension for further classification.",
"One strange phenomenon intrigues the experts, that is, the FC layer weight means are always positive in many-times training ResNet-50 (with different batch sizes and learning rates) on ImageNet Dataset, whereas becoming negative when training ResNet-164 on Cifar Dataset [22].",
"This finding is worth a further investigation.",
"Apart from layer-level values, the experts also explore the filter-level information (R1).",
"In our system, two different ways (i.e., filter-based or iteration-based) are used to normalize weight changes at the filter-level.",
"For filter-based normalization, changes are grouped and normalized by filters, which aims to help experts see the change distribution over iterations for individual filters.",
"Similarly, iteration-based normalization allows experts to examine the distribution over filters for individual iterations.",
"For example, Fig.",
"REF d visualizes the filter changes in one of the CONV layer belonging to the second CONV module using filter-based normalization.",
"The experts find that the changes are drastic in stage $s1$ and become relatively small in the later stages because of the decrease in learning rate and the convergence of the model.",
"However, the experts also identify two strange filters among 64 filters in the first CONV layer that have a constant deep blue color (Fig.",
"REF e).",
"By further checking, the experts find that the weights of these two filters never change during the entire training process.",
"Figure: (a, b) The means of weights in each CONV layer become negative quickly (from green to blue) except for the FC layer.",
"(c) Three filters are always more actively changed than the other filters in the later part of training progress.",
"[rgb]0,0,0This is a total surprise to the experts.",
"Excluding programming bugs, the most likely reason should be due to the dying-ReLU problem, namely, these two filters are inactive for essentially all inputs and no gradients flow backward through the neurons of the two filters.",
"The experts suspect the dying-ReLU problem results from the high learning rate in the beginning of train.",
"In fact, the experts usually follow a rule of thumb to set the hyper-parameter learning rate, that is, multiply the learning rate by k if the batch size is multiplied by k. This rule is currently formally introduced in a recent work [15].",
"In this experiment, we use 32 times larger batch size GPUs (32 GPUs) than the mini-batch size 32 for one GPU to train the model with the corresponding size of learning rate, dying-ReLU problem still occurs.",
"This reminds the experts that the rule may not so accurate for extremely large batch size sometimes, but the problem can be solved by carrying out a warmup strategy (i.e., using lower learning rate at the start of training [17]), which the experts haven't done in previous trainings.",
"One further interesting finding is that by inactivating these two \"dead\" filters (i.e., set their weights as 0 so that they are inactive for any inputs), the experts find the overall performance not affected at all, whereas if we inactivate other random-picked filters in the first CONV layer of the model, the number of mislabeled images in $D_v$ would increase few thousands.",
"Thus, the experts finally modified the network configure and eliminated these two filters, so that the model can run faster while costing less memory.",
"Fig.",
"REF c visualizes the weight changes in one middle layer using iteration-based normalization.",
"The experts find that a small number of filters are always more actively changed than the other filters (long deep blue lines in Fig.",
"REF c) in the later part of iterations.",
"This pattern implies that the updates inside a layer may be highly divergent.",
"[rgb]0,0,0In the later part of the training, where the learning rate is getting smaller and the model is converging, only a couple of filters are still continually actively updated for every iteration.",
"We have tried to inactivate these identified filters, the result showing that the overall performance is not affected.",
"It is not beyond the experts' expectation due to the ResNet's ensemble-like behavior [40] (even several entire layers can be removed without impacting performance).",
"In the end, the experts still cannot fully explain the behavior of these continually updating filters.",
"One possible reason could be that these special filters are not trained well (not converge) to extract some specific features, thus reacting violently for every iteration even in the later stages of the training.",
"Exploring Filter-Image Correlations In this scenario, we demonstrate how the experts use the Correlation View to explore correlations between images and filters.",
"Figure: A cube-style visualization that fuses three coordinated views together to reveal the rich dynamics in a CNN training process:(top) the Validation View shows the error rate changes of validation classes; (front) the Layer View shows the weight changes in CNNfilters; (right) the Correlation View shows the potential relationships between filters and validation classes.Shallow-layer filters vs. deep-layer filters.",
"[rgb]0,0,0At first, the experts choose to only show the top $k$ (100 in this case) changing filters in the layer view.",
"By checking the network structure visualization, the experts find that the activated shallow layers (the layers close to data input layer) are more than the activated deep layers, and most activated layers are the last basic CONV layers of bottlenecks for deep CONV modules.",
"Besides, Fig.",
"REF a shows that deep CONV modules tend to contain more anomaly filters (especially for the CONV modules 4).",
"The experts think that this kind of knowledge is of great importance for network compression [16].",
"Then, the experts go to the complicated version to examine the more detailed correlation information.",
"They filter the mini-sets with very few appearing times, finding that anomaly filters in shallow layers are generally shared by more anomaly classes (columns) and iterations (vertical lines in one column) than those in deep layers (Fig.",
"REF a).",
"The experts think that this pattern may relate to the fact [40] that shallow-layer filters are more likely to capture basic visual features than deep ones, thereby the huge change of these filters affecting more classes of images (e.g., the long and opaque lines marked by b in Fig.",
"REF ).",
"By contrast, a deep filter tends to learn higher-level features, thus only relating to specific classes of images.",
"To further explore the correlations, the experts select two mini-sets (b1 and b2 in Fig.",
"REF ), for comparison.",
"Both the horizontal lines of b1 and b2 are opaque and thick.",
"By tracking them in the Layer View and the Validation View, the experts can see that b1 is in the first CONV layer, and related to many classes.",
"The experts open these classes and discover that many images in them have a common feature, i.e., a large background of blue sky or ocean (b1 in Fig.",
"REF ).",
"This discovery suggests that these filters in E1 may target the basic pattern to help identify images that contain large blue areas.",
"By contrast, b2 is located at the fifth [rgb]0,0,0CONV module and related to only three classes.",
"Interestingly, the images in the three classes also share a more concrete feature, i.e., objects in a bush (b2 in Fig.",
"REF ).",
"[rgb]0,0,0In short, this case confirms that we are on the right track to reveal how the model weight changes relate to the classification result changes.",
"Important filters for a class.",
"To find stronger correlations between filters and classes, the expert focus on anomaly filters that appear more than once in a cell for a specific class.",
"For example, the experts find two appearances of the same mini-set (containing two anomaly filters) for the class of “gong” (c1 in Fig.",
"REF ).",
"Tracking horizontally (along with the pink-highlighted area), the experts find that the mini-set does not appear in other anomaly iterations, which also implies a strong correlation between filters in the mini-set and the class.",
"Then, the experts click on these two rectangular glyphs to highlight the corresponding iterations on the timeline (c2 in Fig.",
"REF ) and the filter locations in the Layer View (c3 in Fig.",
"REF ).",
"It is clear that the gong class is not a well trained class as it has a very large yellow area (indicate a relatively high error rate) in the Validation View.",
"However, the experts also find a period in the middle when this class has a relatively good performance (c4 in Fig.",
"REF ), which happens to contain the highlighted anomaly iterations.",
"Meanwhile, the Layer View shows that the highlighted filters are also updated dramatically during the period of good performance (c3 in Fig.",
"REF ).",
"Considering these patterns, the experts speculate that the filters in the mini-set have a strong impact on the classification of gong images.",
"As expected, we conduct experiments to inactivate these two filters, finding the overall performance and the performance on “gong” class are not impacted (see the reason in the last paragraph in Sec.",
"REF ).",
"Nevertheless, it provides the experts with a new manner to investigate the functions of filters co-working together for classifying one class of images.",
"That is, increase the threshold to find anomaly filters as many as possible, find the mini-sets containing many filters to some classes from multiple layers, and then inactivate them all to validate corresponding impacts.",
"Abnormal anomaly filters.",
"The experts are also attracted by two mini-sets (d1 and e1 in Fig.",
"REF ), because of their abnormal color patterns.",
"The filters in these two mini-sets exhibit large changes all the time in the latter part of the training, which are very different from the other anomaly filters.",
"Thus, the experts are interested in these filters and further check their correlated classes in the Correlation View (right).",
"Interestingly, each abnormal mini-set only appears with two classes (d2 and e2 in Fig.",
"REF ), and each pair of classes have very similar performances displayed in the Validation View (d3 and e3 in Fig.",
"REF ).",
"By checking the detailed images of these classes, the experts discover some common patterns.",
"For example, for mini-set e1, the corresponding classes are about mushrooms growing on grass and dogs playing on grass (e3 in Fig.",
"REF ).",
"For mini-set d1, the corresponding classes are related to curved shapes, such as parachutes and round textures (d3 in Fig.",
"REF ).",
"Although the experts are still unclear about why these two mini-sets have such a special behavior, they believe that these filters are likely to play important roles in identifying middle-level features such as grass and curved shapes.",
"We also conduct further experiments to validate the impact of inactivating these filters and the results are similar to the previous case (i.e., important filters for a class).",
"Expert Feedback Usability [rgb]0,0,0DeepTracker is built upon a close collaboration with three domain experts, who constantly underscore their requirements and provide suggestions during the implementation process.",
"After several iterations of refinement, the experts were happy with the current version.",
"They all praised our way of effectively exploring such extreme large-scale training log via a hierarchical manner.",
"$\\mathrm {E}_a$ and $\\mathrm {E}_b$ mentioned that the well-designed validation and layer views were very intuitive and helped them greatly.",
"For example, the layer view allowing the experts to effectively observe and compare layer-related information (e.g., weight/gradient distribution) can help them diagnose network structures.",
"The detecting of dying-ReLU problem in the early stage of a training is useful for tuning the hyper-parameters (e.g., learning rate).",
"This kind of knowledge can also be leveraged to conduct model compression [16], so as to improve the model in respect to computing speed and memory cost.",
"Although the experts still cannot figure out the exact reason that some filters are always more actively updated in the later training stages, they believe the insight that would be obtained from the future investigation will be helpful in diagnosing and improving network structures.",
"Besides, the divergent evolving patterns of classes and the numerous anomaly iterations found in validation view provide the experts with a new promising direction to train a better model.",
"Both $\\mathrm {E}_a$ and $\\mathrm {E}_c$ were particularly fond of the cube-style visualization and deemed it as a new perspective to observe the training of CNNs for them.",
"They both have found many interesting patterns with the cube visualization, some of which were reasonable or could be explained after thinking for a while.",
"However, the experts also failed to figure out some other patterns, notwithstanding they conducted several testing experiments.",
"Nevertheless, they were still excited that our system can help them identify potential subjects for further study.",
"Generality During the implementation, we were concerned about the generality of DeepTracker, that is, whether the design was biased to the specific requirements from these three experts.",
"Therefore, to check how our system is accepted by broader expert communities, we presented our system in a workshop, which involved about 20 experts in the machine learning and visualization fields.",
"In addition, we also interviewed another group of twelve experts, who worked on a large project about using CNNs to improve image search quality.",
"We presented the latest version of DeepTracker to the experts, encouraged them to experiment with the system, and collected their feedback in the process.",
"Exceeding our expectation, DeepTracker was well accepted by these experts.",
"Although they proposed several new requirements, the experts shared many major interests with our three collaborators, such as tracking class level performance, filter-level information, and the relationships between them.",
"After introducing our system, they immediately understood the purposes of each view and all appreciated this novel, intuitive, and expressive way to watch training processes.",
"Although the demo was performed on our experiment datasets, the experts saw its potential in their project and immediately asked for collaboration with us, so that they could plug in and analyze their own datasets.",
"Improvement Apart from this positive feedback, the experts also made several interesting suggestions to further improve DeepTracker.",
"For example, two experts suggested that our current system only differentiates correct or incorrect classifications for validation images (i.e., 1 and 0).",
"However, the exact incorrect labels should also be presented because such information can help identify confusing or similar classes.",
"One expert mentioned that he showed strong interest on what happens before the anomaly iteration and suggested dump data of every iteration at that abnormal interval for fine-grained analysis.",
"Another expert suggested that our system shoud be integrated with online dashboards, as visualizing the training log on the fly can allow them to terminate the training early and save time if the visualization shows anything undesired.",
"Discussion DeepTracker is our first step to open the “black box” of CNN training.",
"Although the experts have high expectations of this tool, we all agree to start with two fundamental pieces of information: neuron weights and validation results.",
"Considering our target users and the large scale of datasets, we try to avoid using sophisticated visual encodings to ensure a fluent exploration experience.",
"Unsurprisingly, our bare-to-metal visualizations are preferred by the experts, and they use it to find many patterns easily, either expected or unexpected.",
"However, we still have several limitations.",
"First and foremost, although our system can effectively help experts identify strange or interesting patterns, there is still a gap between finding patterns and accelerating CNN training.",
"The experts still have to reason about and understand what these patterns mean or how to use them to accelerate model training in future.",
"We think it is not a problem faced just by our system, but by all CNN visualizations in general.",
"Like previous work, DeepTracker may only peel a hole in the box and reveal limited information.",
"But we hope that, by peeling enough holes, all these strange patterns will start to connect and make sense by themselves, thus providing a clear picture of CNNs.",
"[rgb]0,0,0Second, we have adopted many space-efficient visualizations and interaction techniques (e.g., hierarchy, filtering, and aggregation) to address the scalability issue.",
"Our current design can well support showing dozens of layers and classes in the same time.",
"The correlation view shares all the filter strategies with the other two views, and vice versa.",
"Thus, our system can perform well in most cases.",
"Nevertheless, the worst scenario still requires to display hundreds or thousands of small multiples at the same time.",
"A possible solution is to employ task-specific aggregation or filtering methods to show the data of interests.",
"Third, we propose a rule-based anomaly detection method that requires experts to manually pick a reasonable window size $k$ and set the threshold for filtering.",
"The number and patterns of anomalies are sensitive to these settings.",
"One potential solution to this problem is to develop an automatic method to enumerate all potential parameter settings and identify those can detect a reasonable amount of significant anomalies and provide these settings to the experts as guidance.",
"Finally, we only conduct experiments on ResNet-50 [17], but our method can also be applied to other state-of-the-art deep CNN models, which often have similar hierarchical structures (e.g., “inception block” in google-inception-v4 [37]).",
"[rgb]0,0,0 Besides, the cube visualization is a general technique, which can be used to explore multiple heterogeneous time series data and their complex correlations.",
"However, to further generalize it, a strict user study has to be conducted to find the best manner to use it, such as the axis skew degree, and the minimum height/width for each row/column in the three faces.",
"Conclusion We propose a novel visual analytics solution to disclose the rich dynamics of CNN training processes.",
"Knowing such information can help machine learning experts better understand, debug, and optimize CNNs.",
"We develop a comprehensive system that mainly comprises the validation, layer, and correlation views to facilitate an interactive exploration of the evolution of image classification results, network parameters, and the correlation between them.",
"We conduct experiments by training a very deep CNN (ResNet-50) on ImageNet, one of the largest labeled image datasets that is commonly used in practice, to demonstrate the applicability of our system.",
"The positive feedback from our collaborating experts and the experts from an internal workshop validates the usefulness and effectiveness of our system.",
"Future studies may integrate some feature-oriented visualization techniques, which typically require recording the activation information for input instances.",
"Feature visualizations can provide insights on what features a filter in a given snapshot of CNN has learned.",
"Our system can track critical iterations to take snapshots for a CNN over training, and then use feature visualization techniques to analyze the learned features evolving patterns for the detected important filters.",
"The other urgent need is to deploy the system in a real-time environment.",
"To this end, we have to consider some new design and interaction requirements to fill the gap between finding patterns and accelerating CNN training.",
"The authors would like to thank Kai Yan for providing support in editing the relevant media materials.",
"The work is supported by the National Basic Research Program of China (973 program) under Grant No.",
"2014CB340304 and ITC Grant with No.",
"UIT/138."
],
[
"System Overview",
"DeepTracker is a web-based application developed under the full-stack framework, MEAN.ts (i.e., MongoDB, Express, AngularJs, Node, and Typescript).",
"The back-end part of our application is deployed in a server with 3.10GHz Intel Xeon E5-2687W CPU and 32GB memory.",
"The architecture of our system (Fig.",
"REF ) begins with the data processing part, where the entire training log is hierarchically organized, and several application-specific indexes are built to support real-time interactions.",
"On top of the efficient data storage, we build three coordinated views, namely, the Validation View, the Layer View, and the Correlation View, to support an interactive analysis from different levels and perspectives.",
"The Validation View aims at providing a visual summary of CNN performance on different validation classes.",
"By combining our anomaly detection algorithm and small multiples, experts can easily identify different image class behavior patterns and critical iterations for each image class (R3, R4).",
"Experts may also drill down to the class of interest to explore further image label information (R5).",
"The Layer View aligns the weight information with the CNN structure to help experts explore various statistical information in the network hierarchy.",
"Experts can further drill up or down in the network hierarchy to compare these measures at different levels of detail (R1, R2).",
"The Correlation View presents a novel grid-based visualization method to provide an overview of the correlation between the image classification results and the neuron weights of each filter (R6).",
"[rgb]0,0,0 The three views compose a cube, with which the experts can simultaneously explore the changes of class-level performances, the variations of filter-level weights, and the correlations between them."
],
[
"Data Acquisition and Construction",
"[rgb]0,0,0 The primary motivation of this work is to monitor industry-level CNN training processes.",
"Therefore, we conduct our experiments with ResNet-50 [17] and ImageNet Dataset [33].",
"ResNet-50, containing 50 weighted layers (i.e., CONV and FC layers), is among the most popular CNNs that have been recently used in practice and meanwhile ImageNet 2012 is also among the largest and most challenging publicly available datasets.",
"The dataset includes $1,000$ classes of images, each class containing $1,300$ training images and 50 validation images.",
"Training such a model needs around 120 epoches (nearly 1.2 millions iterations when batch size is 128) to achieve convergence.",
"Simply dumping all the information for every iteration can easily have the size of dumped data exceed several petabytes and take about 4 weeks.",
"Through discussion, we all thought that $1,600$ is a reasonable interval to capture meaningful changes (about 7 times per epoch).",
"This reduces the log to a manageable size (about 1TB).",
"[rgb]0,0,0 For each dump, we recorded two pieces of information, namely, neuron weights/gradients of CONV layer and FC layer and image classification results.",
"The parameters on BN layers were not recorded, as they can be totally recovered given the weights on CONV and FC layers and always need to be updated when applied in a new dataset.",
"Besides, we did not record the activations of each layer/filter for every validation image, as doing so is technically impracticable considering the extremely large models and datasets and the limited disk storage.",
"Further, activation evolution visualization is beyond the research scope of this paper.",
"Sec.",
"discusses activation visualization is indeed a perfect complementary technique to our work.",
"We organized the weight/gradient information according to the natural hierarchial structure of ResNet-50.",
"It consists of four CONV modules (plus the first CONV layer and the final FC layer, there are 50 layers in total).",
"Each module contains several bottleneck blocks [17] that comprise three to four basic CONV layers (data storage in Fig.",
"REF ).",
"Thus, we grouped all neuron weights to align with such hierarchy.",
"[rgb]0,0,0 In a similar manner, we organized the classification results hierarchically from individual level, class level, to model level.",
"We stored all the data into MongoDBMongoDB is a free and open-source cross-platform document-oriented (NoSql) database.",
"www.mongodb.com.",
"In particular, we precomputed all the relevant aggregation values, such as weight means and error rates, for each filter, layer, image, and class.",
"Nevertheless, the distilled data still remain too large to load into memories (about dozens of gigabytes per training).",
"Therefore, we analyzed the frequent needs of the experts and built several indexes to enable real-time interactions, including Layer-Stat index $I_{ls}$ , Layer-Filter index $I_{lf}$ , Iter-Filter index $I_{if}$ , Cls-Stat index $I_{cs}$ , and Cls-Image index $I_{ci}$ .",
"$I_{ls}$ retrieves the statistic values (e.g., mean and sd) at every iterations for any given layer; $I_{lf}$ lists all the filter-level information (e.g., changing degree of each filter) at every iterations for any given layer; $I_{if}$ searches the top changing filters from all layers at any given iteration; $I_{cs}$ extracts class-level information (e.g., class performances, the different types of abnormal images Sec.",
"REF ) over all iterations for any given class; $I_{ci}$ fetches the meta-data of images for any given classes.",
"Visualization In this section, we describe our three coordinate views, namely, the Validation View, the Layer View, and the Correlation View, that help experts accomplish the aforementioned analytical tasks.",
"Validation View Several urgent requirements (R3, R4, R5) from the experts need to examine how the evolving CNN acts differently on the validation images of each class rather than how the overall validation error rate differs over training.",
"Thus, we design the Validation View (Fig.",
"REF & REF ) to present all image classes in $D_v$ .",
"Figure: From top (a) to bottom (b), image classes are more and more difficult to train.Visual Encoding [rgb]0,0,0By default, the view starts with a visualization of cluster-level performance (R3).",
"The classes with similar evolving trends form a cluster and then their error rates at every iteration are averaged.",
"The averaged error rates are then depicted as a colored stripe, where the x-axis encodes the iterations and the error rates are encoded by colors (Fig.",
"REF ).",
"We choose k-means as the clustering algorithm and $k$ can be adjusted according to demands (Fig.",
"REF shows the case when $k=4$ ).",
"Experts can open up one cluster to further examine the performance in class-level (R3).",
"The design is based on the following considerations.",
"First, the experts are more interested in the overall classes than in individual iterations.",
"Thus, small multiples technique (juxtaposed techniques) is chosen for their superior performance in high-level comparison tasks (e.g., global trends for every series) [20].",
"Second, we cannot present all the classes ($1,000$ in ImageNet) at one time, we have to consider a hierarchical and highly space-efficient visualization.",
"However, many traditional charts, such as line charts and horizon graphs, require a larger vertical space [19] than 1D heatmaps.",
"Compared with traditional charts, heatmaps are also more easy to do side by side comparison for their symmetrical space (i.e., no irregular white spaces).",
"As a result, all experts prefer the heatmap-based small multiples.",
"Image-level performances, R5.",
"The class-level color strips can be further unfolded to explore the image-level evolution patterns.",
"Unfolding a heatmap reveals a pixel chart (Fig.",
"REF d), with each row (1px height) representing an image and each column (1px width) representing an dumped iteration (consistent with the class heatmap).",
"We use red and green colors to indicate the incorrect and correct classifications, respectively.",
"Meanwhile, the experts can zoom/pan the pixel chart for a closer inspection.",
"Clicking on a row shows the original corresponding image.",
"Anomaly iterations, R4.",
"[rgb]0,0,0As experts are concerned about the iterations with abnormal behaviors, we particularly propose a algorithm to detect these anomaly iterations (refer to Sec.",
"REF ).",
"Experts can choose to only show the classes with anomaly iterations (Fig.",
"REF ).",
"At this point, for each class-level color stripe, we use triangular glyphs to highlight these detected anomaly iterations.",
"The upside-down triangles ($\\bigtriangledown $ ) and normal triangles ($\\triangle $ ) indicate those anomaly iterations that are detected by the left-rule and right-rule, respectively.",
"The widths of triangles encode the anomaly scores.",
"Experts can set a threshold value to filter the triangles with low anomaly scores.",
"Anomaly Detection In our scenario, the classification results for an image can be represented by a 0/1 sequence ($[a_1, ..., a_n]$ ), where each element represents a correct or incorrect result at the corresponding validation iteration.",
"The experts are curious about the iterations when a significant amount of 1/0 flips (i.e., 0 to 1 or 1 to 0) occur for a class.",
"In general, this problem can be modeled and solved using Markovian-based anomaly detection algorithms [1].",
"Despite the popularity of using Markovian methods to detect outliers in discrete sequence, we decide to employ rule-based models [1] for two reasons.",
"First, Markovian methods are a black box and the resulting outlier values are sometimes difficult to comprehend.",
"Second, the experts have explicitly described two types of iterations they are very interested in, namely, those iterations when many images with values that remain stable for many previous iterations suddenly flip (denoted by the left-rule) and those iterations when many images flip and keep their values stable after many iterations (denoted by the right-rule).",
"Fortunately, these anomalies can be easily modeled using rules.",
"The rule-based models primarily estimate the value $P(a_i|a_{i-k},\\ldots ,a_{i-1})$ , which can be expressed in the following rule form: $ a_{i-k},\\ldots ,a_{i-1} \\Rightarrow a_{i}.$ In our scenario, if an image has the same value (either 0 or 1) in the previous consecutive $k$ iterations ($i-k,\\ldots , i-1$ ), then its value must be the same at iteration $i$ (the left-rule).",
"Otherwise, iteration $i$ is considered an outlier for the specified image.",
"Based on these considerations, we develop an application-specific algorithm to detect anomaly iterations in the validation history.",
"The algorithm includes the following steps: Rule-Judgement: The algorithm computes a vector $[l_{i1}, \\ldots , l_{in}]$ for every image $i$ , where $l_{ij} =1$ if the left-rule is satisfied, otherwise, $l_{ij} =0$ ; Aggregation: For each class that contains $m$ images, the algorithm aggregates all the computed vectors for each image into one $[L_1, ..., L_n]$ , where $L_j = \\sum _{i=1}^{m}l_{ij}$ , [rgb]0,0,0denoting the left anomaly score at iteration $j$ for this class.",
"The approach is a window-based method, and the experts can adjust the window size $k$ to control the sensitivity of the anomalies.",
"In a similar manner, we detect the anomalies from the opposite direction for the right-rule.",
"Layer View The Layer View focuses on weight-related tasks (R1, R2).",
"The view consists of two connected parts, namely, the CNN structure and the hierarchical small multiples (Fig.",
"REF ), so that experts can hierarchically explore and compare various types of statistic in the context of the network structure.",
"Figure: Visual encodings in the Layer View: (a) a CONV layer, (b,d) hierarchical bars, (c) links between the CNN structure and (e) the hierarchical small multiple charts, (f) a pixel chart for one layer.CNN structure.",
"The experts hope our tool can help them explore the statistical information of each layer and meanwhile know their relative positions in the entire network (R1).",
"Thus, we adopt NetscopeNetscope is a web-based tool for visualizing neural network architectures.",
"http://ethereon.github.io/netscope/, a popular neural network visualizer, in our system.",
"[rgb]0,0,0 The green rectangle is data input layer, the red ones are the CONV layers, and the purple ones mean the pooling layers.",
"The links between these rectangles show the network structure.",
"We further add blue level bars (Fig.",
"REF b) to encode the latent hierarchy (from CONV modules, bottlenecks to basic CONV layers, Sec. ).",
"The right most level bars represent CONV modules (Sec.",
"), which are recursively divided into smaller level bars until reaching elementary CONV layers (i.e., red rectangles).",
"Hierarchical small multiples.",
"To assist experts in exploring and comparing layers of a deep CNN (R1, R2), a space-efficient visualization technique is demanded.",
"Thus, we leverage hierarchical small multiples to show layers of interest (Fig.",
"REF e).",
"[rgb]0,0,0By default, experts are presented with the information about CONV modules and then can drill down to see more information about low-level CONV layers with interactions with the network graph (i.e., click on the corresponding level bars).",
"The width of outcropping rectangles (Fig.",
"REF d) encodes the aggregation level of current layer charts.",
"For example, the top second layer chart in (Fig.",
"REF e) shows the bottleneck-level aggregation information and the following three layer charts show the basic CONV layer level information.",
"Besides, the links (Fig.",
"REF c) mark the real positions of the layer charts in the network structure.",
"The small multiples support multiple types of charts including line chart, horizon graphs [18] and box plots to emphasize the different aspects of the statistical data.",
"The experts use box-plots to see the rough distribution of statistical values and use basic line charts to examine individual values.",
"Besides, the experts prefer to use horizon graphs when performing tasks in regard to trend tracking and comparison (R2), because of its effectiveness in visualizing divergent weight values [20].",
"Similar to unfolding the class heatmap to a pixel chart, the experts are also allowed to open the layers of interest as a pixel chart (Fig.",
"REF f) that presents the filter-level information (R1).",
"Each row (1px height) in the pixel chart represents one filter, and each column (1px width) indicates one iteration.",
"We use sequential colors to encode pixel value (e.g., the cosine similarity between two subsequent dumped iterations).",
"Correlation View This view helps experts establish connections between the filters and images.",
"In particular, the experts want to understand further how the changes in network parameters are related to class performances (R6).",
"For example, several anomaly iterations may be detected for a single class.",
"For each detected anomaly iteration, we can identify a set of anomaly filters (i.e., the top $k$ filters with largest changes at that iteration).",
"Since different classes may share anomaly iterations and different anomaly iterations may share anomaly filters, are there any filters that are commonly seen in these iterations?",
"Do any of the anomaly classes or filters strongly co-occur?",
"We designed the Correlation View to answer these questions.",
"[t] Target set $S_{target}$ .",
"Minimum partition for $S_{target}$ .",
"$S_{result}$ = $\\emptyset $ each target set $s_t$ in $S_{target}$ $S_{new}$ = $\\emptyset $ each mini set $s_r$ in $S_{result}$ $itersection$ = $s_t \\bigcap s_r$ $S_{new}$ = $S_{new} \\bigcup intersection \\bigcup (s_r - s_t)$ $s_t$ = $s_t - s_r$ $s_t$ is not empty $S_{new} \\bigcup s_t$ remove all empty set in $S_{new}$ $S_{result}$ = $S_{new}$ return $S_{result}$ Minimum Set Partition Filter set partition.",
"We first introduce the mini-set concept to organize anomaly filters that are shared by multiple anomaly iterations and different classes.",
"For each class $C_i\\in \\lbrace C_i|{1\\le i \\le n}\\rbrace $ , we denote its anomaly iterations by $T_i = \\lbrace t_{i,k}|{1\\le k\\le n_i}\\rbrace $ .",
"Thus, all anomaly iterations are $\\cup _{1\\le i \\le n}T_i$ , denoted by $T$ .",
"For each anomaly iteration $t\\in T$ , we denote its anomaly filters at CNN layer $L_j\\in \\lbrace L_j|{1\\le j \\le m}\\rbrace $ by $s_{j,t}$ .",
"Thus, for each layer $L_j$ , we can collect all anomaly filter sets $\\lbrace s_{j,t}|t\\in T\\rbrace $ (denoted by $S_j$ ) and all anomaly filters $\\cup _{t\\in T}s_{j,t}$ (denoted by $s_{j}$ ).",
"Thus, mini-set aims to find a minimum set partitions of $s_{j}$ (denoted by $s_j^\\ast $ ) that each $s_{j,t}$ can be assembled from some elements (i.e., mini-sets) in $s_j^\\ast $ .",
"We specifically propose a Set Partition Algorithm (Alg.",
"REF ) to find $s_j^\\ast $ .",
"The algorithm accepts a target set as input (i.e., $S_j$ ).",
"$S_{result}$ is initially empty and a new anomaly filter set is used at each time to partition the mini-sets contained in $S_{result}$ (cf.",
"lines 4 to 7).",
"If the new anomaly filter set is not empty after partitioning, then it is added as a new mini-set (cf.",
"line 8).",
"Finally, the partitions contained in $S_{result}$ will be returned.",
"Figure: (a) The abstract version of correlation view, where rows and columns represent layers and image classes, respectively.",
"A sequential color scheme is used to encode the number of anomaly filters.",
"(b) The complex version of correlation view, where the detailed information of individual anomaly filters are shown.",
"(c) A layout solution for coordinated analysis without using skewed axes.Visual encoding.",
"To intuitively represent these relationships, we introduce a grid-style visualization (Fig.",
"REF ), where rows and columns represent layers and image classes, respectively.",
"The number of rows and columns equal to the number of layers with anomaly filters and classes with anomaly iterations, respectively.",
"[rgb]0,0,0 We start from a abstract version.",
"In this version (Fig.",
"REF a), for $\\mathrm {Cell}_{i,j}$ , a sequential color scheme is used to encode the number of anomaly filters ($\\cup _{t\\in T_j}s_{i,t}$ ).",
"The darker the color it is, the more anomaly filters appear in layer $L_i$ that are related to class $C_i$ .",
"From this visualization, we can easily observe the correlations between layers and classes, while it also hides much detailed information.",
"We cannot answer questions like whether the filters in $\\mathrm {Cell}_{i,j}$ are the same with $\\mathrm {Cell}_{i,k}$ (this kind of information shows how many classes this filter can impact and help examine the relationships among classes), or whether there are filters in $\\mathrm {Cell}_{i,j}$ appearing in more than one anomaly iteration (this shows the importance of these filters for class $C_j$ ).",
"To solve these problems, we provide an advanced version (Fig.",
"REF b).",
"For $\\mathrm {Cell}_{i,j}$ , the width and height encode the number of anomaly iterations and the number of anomaly filters of the corresponding class and layer (i.e., $|T_j|$ and $|s_{i}|$ ), respectively.",
"Based on these numbers, the columns and rows are further divided with vertical/horizontal lines.",
"For a class (e.g., $\\mathrm {Column}_{j}$ ), $|T_j|$ vertical lines are drawn to represent all related anomaly iterations (i.e., $[t_{j,1}, t_{j,2},\\ldots ,t_{j,n_j}]$ ).",
"For each row of layer $L_i$ , there are $|s_i^\\ast |$ horizontal lines representing all mini-sets in that layer.",
"The intersections between these horizontal and vertical lines are highlighted with blue rectangles if the corresponding mini-set is part of the anomaly filters of the corresponding anomaly iteration.",
"The height of the rectangles represents the number of filters of the corresponding mini-set.",
"[rgb]0,0,0Obviously the introduction of mini-sets dramatically cuts down the number of horizontal lines and blue rectangles, otherwise each anomaly filter require one horizontal line and one rectangles, which may cause serious visual clutter problem.",
"In fact, mini-sets can be viewed as a partially aggregation version instead of representing all the anomaly filters as horizontal lines and rectangles.",
"Users can set the minimum appearing number of mini-sets to filter the sets with lower importance.",
"Cube Visualization The log data contain three main aspects of information, namely, iterations, validation information, and weight information.",
"The three aforementioned views are designed to show all possible 2-combinations of these three types of information, respectively.",
"[rgb]0,0,0 Although these views can be used individually, they need to be combined together to form a complete picture.",
"Thus, we propose a novel and intuitive visualization technique, which naturally and seamlessly stitches the three views together, based on their shared axis (inspired from Binx [8]), into a “cube” shape (Fig.",
"REF ).",
"When experts find or highlight a pattern of interest, they can easily track the pattern over the edges to find the related information in the other two views easily.",
"[rgb]0,0,0 The use of skewed axes may bring about a possible perspective distortion problem.",
"Nevertheless, the advantages of the cube-style design far outweigh the disadvantages.",
"Given the limited pixels in a computer screen and each view requiring a large display space, the experts all agree that the cube-style design is the most space-efficient and intuitive manner to show all the information.",
"Furthermore, with such design, it prevents the experts from switching from multiple views, reducing the cognitive burden and the load of memory.",
"This allows the experts to conduct correlation analysis more effectively.",
"We also provide a compromised solution to handle the distortion problem, that is, laying out the three views in the form like Fig.",
"REF c. So the experts can firstly examine the layer view (horizontally) or validation view (vertically) together with the correlation view, and then switch to cube mode to explore the three views together.",
"[rgb]0,0,0 Notice that there are some different settings for several views in the cube.",
"For the layer view (front), only the layers with anomaly filters are activated (see the activated blue bars in the front view of Fig.",
"REF ), the weight variation of each anomaly filter is represented as a horizontal color strip.",
"For the validation view (top), only the classes with anomaly iterations are preserved.",
"The following lists several common exploration pipelines: P1: from the layer view (front), we can quickly check the distribution of activated layers in the overall network and pick some anomaly filters of interests.",
"Then, by tracking along the horizon axis to the correlation view (right), we can examine which classes these filters impact and how important these filters are to the classes.",
"Finally, we can observe the evolving patterns of these classes and the corresponding anomaly iterations in validation view (top).",
"P2: from the validation view (top), we can firstly mark several anomaly iterations of some classes.",
"Then, we can check the corresponding columns in the correlation view, finding the rows that contain anomaly filters and exploring the importance of these filters to these classes and how these filters impact the other classes.",
"Finally, by highlighting these corresponding rows, we can observe them in the layer view to see how these filters behave around the picked anomaly iterations.",
"P3: from the correlation view (right), we can search the horizontal lines across many rectangles (it means these filters impact many classes at the same time) or the rectangles that appear more than one time in the same cell (these filters are judged as anomalies many times for a class and may have great impact on this class).",
"With these selected horizontal lines or rectangles, we can simultaneously track their corresponding weight variation information in the layer view (front) and class performance information in the validation view (top).",
"Use Examples We derived these examples through the assistance of our collaborating experts, who were familiar with our designs and data.",
"As a remark, the following results are from the experiment with 8 times larger batch size and learning rate setting than the basic setting introduced in Sec. .",
"Exploring Validation Results Figure: Overview of validation classes.",
"(a) Two curves show the overall training/validation error rate.",
"(b) Two turning points align well the boundary of stage s2s2.",
"(c) Three peaks appear in the stage s1s1 and align well with (f) the detected anomaly iterations.",
"(d) Two types of mushroom images have different behaviors in the class.",
"(e) Most images in the class flip at the anomaly iteration.The first scenario demonstrates how the experts use DeepTracker to explore the image classification results (R3, R4, and R5).",
"Performance evolution patterns.",
"Fig.",
"REF a shows a typical visualization of training/validation errors that may appear on any popular training platform.",
"The timeline at the top shows a total of 1.2 million iterations.",
"Beneath the timeline, four line segments represent four stages ($s1$ , $s2$ , $s3$ , and $s4$ in Fig.",
"REF ) in the training process, [rgb]0,0,0where the later stage has one-tenth of the learning rate of the previous stage.",
"We can observe that two sudden drops in the curves match well with the boundaries of the training stages (Fig.",
"REF b).",
"However, this is a well-known pattern to the experts.",
"On the other hand, although the overall error rate continues to decrease, the class-level error rates show a more complicated story, that is new to the experts.",
"By quickly scanning the small multiples in cluster-level, the experts identify there are generally four types of class evolving patterns (Fig.",
"REF ).",
"From top to bottom, the four types are more and more difficult to train.",
"[rgb]0,0,0For example, for the type at the top, these classes are recognized correctly after a few iterations.",
"By contrast, the classes at the bottom always have high error rates in the entire training process, which means that the resulting network fails to recognize the related images.",
"From this, the experts learn that the model has spent most of time to improve its performance on the classes of middle-level classification, since the model has already performed well on the easy-trained classes at a very early stage and is always performing miserably on the hard-trained classes over the entire training.",
"From these patterns, the experts consider it promising to accelerate the training process and improve the overall performance by treating classes differently during the training process.",
"That is, stop feeding the easy-trained classes in an appropriate early stage, put more efforts on training the classes of middle difficulties for classification, and figure out why some classes always have extremely high error rates.",
"One similar attempt has been made in a recent work [25].",
"Anomaly iterations.",
"The experts are curious about the three sudden peaks in stage $s1$ , and then mark these three iterations with dotted lines (Fig.",
"REF c), which look like anomaly iterations (R4).",
"However, the colors in the small multiples do not have clear patterns related to these iterations.",
"Then, the experts turn on the anomaly detection and immediately find that many triangles are aligned well with the dotted lines (Fig.",
"REF f), thereby confirming our suspicion.",
"Then, the experts can click on the corresponding image icons to see the detailed images that contribute to the three peaks.",
"In addition, there are more anomaly iterations in stage $s1$ then in the later stages.",
"This interesting pattern can be explained by the reduction of learning rate and the convergence of the model in the later stages.",
"At the same time, it also implies that the learning rate in stage $s1$ is slightly too high, leading to the instability in the model (the case in Sec.",
"REF indicates the same finding for the discovery of potential “dead” filters).",
"Details in classes.",
"To further examine what happens at the anomaly iterations for a class, the experts can further check the image-level information of the class (R5).",
"For example, the experts are curious about the abnormally large anomaly iteration in the class of “mushroom” (Fig.",
"REF d) that are captured by both the left-rule and the right-rule.",
"Then, they click and expand color stripe to see the pixel chart of images.",
"First, they confirm that this iteration is indeed special for this class, because nearly all images flip during particular that [rgb]0,0,0dumped interval (Fig.",
"REF e).",
"Thus, they may further investigate to find the layers or filters that cause such flips based on the filter updates around that iteration.",
"[rgb]0,0,0In particular, the experts comment that, it seems that after the iteration, the CNN model has jumped to a better local optimal for the class, because the green color is more stable after the iteration.",
"This may result from the reduction of the learning rate (from $s1$ to $s2$ ).",
"This kind of patterns appear frequently in many classes during the whole training process, many of them occurring not in the learning rate transition point.",
"The experts wonder that the model should be trying to jump from one local optimal to another better local for these classes continuously, so as to reduce the overall error rate gradually.",
"This insight has never been obtained because the experts initially thought that the error rate for one class should decrease steadily.",
"Besides, the experts also find that, at the bottom of the pixel chart, several images are mislabeled in the entire training process, although the class is easy to train overall(Fig.",
"REF d).",
"To understand why, the experts click on these images to examine them, and find that the contents in the mislabeled images have a clear color pattern different from that of the rest of the mushroom images.",
"The correctly labeled mushrooms are all red, while the mislabeled ones are white or orange.",
"This finding indicates color is a critical feature that the CNN has learned to classify this class of images.",
"Exploring Weight-Relevant Information Figure: (a, b, c) The sd values of each layer decrease slowly and have different scales in different CONV modules.",
"(d) The weight changes in filters are large at the beginning.",
"(e) One outlier filter is detected, whose weights never change during the entire training process.This scenario shows how to discover patterns in neuron weights via the Layer View (R1, R2).",
"First, the experts choose to show the sd (standard deviation) of the weights at the layer-level using horizontal graphs (Fig.",
"REF ).",
"As the experts expected, all the trends show a similar pattern of slow decrease, indicating the weights in the entire model is converging over iterations.",
"Besides, the experts also find that deeper layers (closer to the loss layer) tend to have smaller sd values.",
"[rgb]0,0,0In particular, by tuning the band number (finally to 3) of the horizon graphs, they found the sd values of a CONV module are usually twice as large as those of the one below it (a, b, and c in Fig.",
"REF ).",
"Given that we apply Xavier initializationREF and for ResNet-50, the input sizes of layers in a CONV module are twice as large as the ones in the layers of its previous CONV module, the observed result is not beyond the experts' expectation.",
"This suggests that there is no problem exist on the initialization approach.",
"Analogously, the experts find that the weight means of each layer become negative quickly (from green to blue instantly, Fig.",
"REF a) except for the FC layer (Fig.",
"REF b).",
"At first, the pattern looks strange to the experts.",
"Then, the experts realize that it is reasonable to have more negative weights than positive ones, since negative values are often used to filter out trivial features owing to the use of ReLU activations.",
"The increase of negative weights suggests that the network is trained to extract useful information from the original image layer by layer, and then finally remain the most relevant features.",
"[rgb]0,0,0As for the FC layer, it plays a function of shaping the extracted useful features into feature vectors of given dimension for further classification.",
"One strange phenomenon intrigues the experts, that is, the FC layer weight means are always positive in many-times training ResNet-50 (with different batch sizes and learning rates) on ImageNet Dataset, whereas becoming negative when training ResNet-164 on Cifar Dataset [22].",
"This finding is worth a further investigation.",
"Apart from layer-level values, the experts also explore the filter-level information (R1).",
"In our system, two different ways (i.e., filter-based or iteration-based) are used to normalize weight changes at the filter-level.",
"For filter-based normalization, changes are grouped and normalized by filters, which aims to help experts see the change distribution over iterations for individual filters.",
"Similarly, iteration-based normalization allows experts to examine the distribution over filters for individual iterations.",
"For example, Fig.",
"REF d visualizes the filter changes in one of the CONV layer belonging to the second CONV module using filter-based normalization.",
"The experts find that the changes are drastic in stage $s1$ and become relatively small in the later stages because of the decrease in learning rate and the convergence of the model.",
"However, the experts also identify two strange filters among 64 filters in the first CONV layer that have a constant deep blue color (Fig.",
"REF e).",
"By further checking, the experts find that the weights of these two filters never change during the entire training process.",
"Figure: (a, b) The means of weights in each CONV layer become negative quickly (from green to blue) except for the FC layer.",
"(c) Three filters are always more actively changed than the other filters in the later part of training progress.",
"[rgb]0,0,0This is a total surprise to the experts.",
"Excluding programming bugs, the most likely reason should be due to the dying-ReLU problem, namely, these two filters are inactive for essentially all inputs and no gradients flow backward through the neurons of the two filters.",
"The experts suspect the dying-ReLU problem results from the high learning rate in the beginning of train.",
"In fact, the experts usually follow a rule of thumb to set the hyper-parameter learning rate, that is, multiply the learning rate by k if the batch size is multiplied by k. This rule is currently formally introduced in a recent work [15].",
"In this experiment, we use 32 times larger batch size GPUs (32 GPUs) than the mini-batch size 32 for one GPU to train the model with the corresponding size of learning rate, dying-ReLU problem still occurs.",
"This reminds the experts that the rule may not so accurate for extremely large batch size sometimes, but the problem can be solved by carrying out a warmup strategy (i.e., using lower learning rate at the start of training [17]), which the experts haven't done in previous trainings.",
"One further interesting finding is that by inactivating these two \"dead\" filters (i.e., set their weights as 0 so that they are inactive for any inputs), the experts find the overall performance not affected at all, whereas if we inactivate other random-picked filters in the first CONV layer of the model, the number of mislabeled images in $D_v$ would increase few thousands.",
"Thus, the experts finally modified the network configure and eliminated these two filters, so that the model can run faster while costing less memory.",
"Fig.",
"REF c visualizes the weight changes in one middle layer using iteration-based normalization.",
"The experts find that a small number of filters are always more actively changed than the other filters (long deep blue lines in Fig.",
"REF c) in the later part of iterations.",
"This pattern implies that the updates inside a layer may be highly divergent.",
"[rgb]0,0,0In the later part of the training, where the learning rate is getting smaller and the model is converging, only a couple of filters are still continually actively updated for every iteration.",
"We have tried to inactivate these identified filters, the result showing that the overall performance is not affected.",
"It is not beyond the experts' expectation due to the ResNet's ensemble-like behavior [40] (even several entire layers can be removed without impacting performance).",
"In the end, the experts still cannot fully explain the behavior of these continually updating filters.",
"One possible reason could be that these special filters are not trained well (not converge) to extract some specific features, thus reacting violently for every iteration even in the later stages of the training.",
"Exploring Filter-Image Correlations In this scenario, we demonstrate how the experts use the Correlation View to explore correlations between images and filters.",
"Figure: A cube-style visualization that fuses three coordinated views together to reveal the rich dynamics in a CNN training process:(top) the Validation View shows the error rate changes of validation classes; (front) the Layer View shows the weight changes in CNNfilters; (right) the Correlation View shows the potential relationships between filters and validation classes.Shallow-layer filters vs. deep-layer filters.",
"[rgb]0,0,0At first, the experts choose to only show the top $k$ (100 in this case) changing filters in the layer view.",
"By checking the network structure visualization, the experts find that the activated shallow layers (the layers close to data input layer) are more than the activated deep layers, and most activated layers are the last basic CONV layers of bottlenecks for deep CONV modules.",
"Besides, Fig.",
"REF a shows that deep CONV modules tend to contain more anomaly filters (especially for the CONV modules 4).",
"The experts think that this kind of knowledge is of great importance for network compression [16].",
"Then, the experts go to the complicated version to examine the more detailed correlation information.",
"They filter the mini-sets with very few appearing times, finding that anomaly filters in shallow layers are generally shared by more anomaly classes (columns) and iterations (vertical lines in one column) than those in deep layers (Fig.",
"REF a).",
"The experts think that this pattern may relate to the fact [40] that shallow-layer filters are more likely to capture basic visual features than deep ones, thereby the huge change of these filters affecting more classes of images (e.g., the long and opaque lines marked by b in Fig.",
"REF ).",
"By contrast, a deep filter tends to learn higher-level features, thus only relating to specific classes of images.",
"To further explore the correlations, the experts select two mini-sets (b1 and b2 in Fig.",
"REF ), for comparison.",
"Both the horizontal lines of b1 and b2 are opaque and thick.",
"By tracking them in the Layer View and the Validation View, the experts can see that b1 is in the first CONV layer, and related to many classes.",
"The experts open these classes and discover that many images in them have a common feature, i.e., a large background of blue sky or ocean (b1 in Fig.",
"REF ).",
"This discovery suggests that these filters in E1 may target the basic pattern to help identify images that contain large blue areas.",
"By contrast, b2 is located at the fifth [rgb]0,0,0CONV module and related to only three classes.",
"Interestingly, the images in the three classes also share a more concrete feature, i.e., objects in a bush (b2 in Fig.",
"REF ).",
"[rgb]0,0,0In short, this case confirms that we are on the right track to reveal how the model weight changes relate to the classification result changes.",
"Important filters for a class.",
"To find stronger correlations between filters and classes, the expert focus on anomaly filters that appear more than once in a cell for a specific class.",
"For example, the experts find two appearances of the same mini-set (containing two anomaly filters) for the class of “gong” (c1 in Fig.",
"REF ).",
"Tracking horizontally (along with the pink-highlighted area), the experts find that the mini-set does not appear in other anomaly iterations, which also implies a strong correlation between filters in the mini-set and the class.",
"Then, the experts click on these two rectangular glyphs to highlight the corresponding iterations on the timeline (c2 in Fig.",
"REF ) and the filter locations in the Layer View (c3 in Fig.",
"REF ).",
"It is clear that the gong class is not a well trained class as it has a very large yellow area (indicate a relatively high error rate) in the Validation View.",
"However, the experts also find a period in the middle when this class has a relatively good performance (c4 in Fig.",
"REF ), which happens to contain the highlighted anomaly iterations.",
"Meanwhile, the Layer View shows that the highlighted filters are also updated dramatically during the period of good performance (c3 in Fig.",
"REF ).",
"Considering these patterns, the experts speculate that the filters in the mini-set have a strong impact on the classification of gong images.",
"As expected, we conduct experiments to inactivate these two filters, finding the overall performance and the performance on “gong” class are not impacted (see the reason in the last paragraph in Sec.",
"REF ).",
"Nevertheless, it provides the experts with a new manner to investigate the functions of filters co-working together for classifying one class of images.",
"That is, increase the threshold to find anomaly filters as many as possible, find the mini-sets containing many filters to some classes from multiple layers, and then inactivate them all to validate corresponding impacts.",
"Abnormal anomaly filters.",
"The experts are also attracted by two mini-sets (d1 and e1 in Fig.",
"REF ), because of their abnormal color patterns.",
"The filters in these two mini-sets exhibit large changes all the time in the latter part of the training, which are very different from the other anomaly filters.",
"Thus, the experts are interested in these filters and further check their correlated classes in the Correlation View (right).",
"Interestingly, each abnormal mini-set only appears with two classes (d2 and e2 in Fig.",
"REF ), and each pair of classes have very similar performances displayed in the Validation View (d3 and e3 in Fig.",
"REF ).",
"By checking the detailed images of these classes, the experts discover some common patterns.",
"For example, for mini-set e1, the corresponding classes are about mushrooms growing on grass and dogs playing on grass (e3 in Fig.",
"REF ).",
"For mini-set d1, the corresponding classes are related to curved shapes, such as parachutes and round textures (d3 in Fig.",
"REF ).",
"Although the experts are still unclear about why these two mini-sets have such a special behavior, they believe that these filters are likely to play important roles in identifying middle-level features such as grass and curved shapes.",
"We also conduct further experiments to validate the impact of inactivating these filters and the results are similar to the previous case (i.e., important filters for a class).",
"Expert Feedback Usability [rgb]0,0,0DeepTracker is built upon a close collaboration with three domain experts, who constantly underscore their requirements and provide suggestions during the implementation process.",
"After several iterations of refinement, the experts were happy with the current version.",
"They all praised our way of effectively exploring such extreme large-scale training log via a hierarchical manner.",
"$\\mathrm {E}_a$ and $\\mathrm {E}_b$ mentioned that the well-designed validation and layer views were very intuitive and helped them greatly.",
"For example, the layer view allowing the experts to effectively observe and compare layer-related information (e.g., weight/gradient distribution) can help them diagnose network structures.",
"The detecting of dying-ReLU problem in the early stage of a training is useful for tuning the hyper-parameters (e.g., learning rate).",
"This kind of knowledge can also be leveraged to conduct model compression [16], so as to improve the model in respect to computing speed and memory cost.",
"Although the experts still cannot figure out the exact reason that some filters are always more actively updated in the later training stages, they believe the insight that would be obtained from the future investigation will be helpful in diagnosing and improving network structures.",
"Besides, the divergent evolving patterns of classes and the numerous anomaly iterations found in validation view provide the experts with a new promising direction to train a better model.",
"Both $\\mathrm {E}_a$ and $\\mathrm {E}_c$ were particularly fond of the cube-style visualization and deemed it as a new perspective to observe the training of CNNs for them.",
"They both have found many interesting patterns with the cube visualization, some of which were reasonable or could be explained after thinking for a while.",
"However, the experts also failed to figure out some other patterns, notwithstanding they conducted several testing experiments.",
"Nevertheless, they were still excited that our system can help them identify potential subjects for further study.",
"Generality During the implementation, we were concerned about the generality of DeepTracker, that is, whether the design was biased to the specific requirements from these three experts.",
"Therefore, to check how our system is accepted by broader expert communities, we presented our system in a workshop, which involved about 20 experts in the machine learning and visualization fields.",
"In addition, we also interviewed another group of twelve experts, who worked on a large project about using CNNs to improve image search quality.",
"We presented the latest version of DeepTracker to the experts, encouraged them to experiment with the system, and collected their feedback in the process.",
"Exceeding our expectation, DeepTracker was well accepted by these experts.",
"Although they proposed several new requirements, the experts shared many major interests with our three collaborators, such as tracking class level performance, filter-level information, and the relationships between them.",
"After introducing our system, they immediately understood the purposes of each view and all appreciated this novel, intuitive, and expressive way to watch training processes.",
"Although the demo was performed on our experiment datasets, the experts saw its potential in their project and immediately asked for collaboration with us, so that they could plug in and analyze their own datasets.",
"Improvement Apart from this positive feedback, the experts also made several interesting suggestions to further improve DeepTracker.",
"For example, two experts suggested that our current system only differentiates correct or incorrect classifications for validation images (i.e., 1 and 0).",
"However, the exact incorrect labels should also be presented because such information can help identify confusing or similar classes.",
"One expert mentioned that he showed strong interest on what happens before the anomaly iteration and suggested dump data of every iteration at that abnormal interval for fine-grained analysis.",
"Another expert suggested that our system shoud be integrated with online dashboards, as visualizing the training log on the fly can allow them to terminate the training early and save time if the visualization shows anything undesired.",
"Discussion DeepTracker is our first step to open the “black box” of CNN training.",
"Although the experts have high expectations of this tool, we all agree to start with two fundamental pieces of information: neuron weights and validation results.",
"Considering our target users and the large scale of datasets, we try to avoid using sophisticated visual encodings to ensure a fluent exploration experience.",
"Unsurprisingly, our bare-to-metal visualizations are preferred by the experts, and they use it to find many patterns easily, either expected or unexpected.",
"However, we still have several limitations.",
"First and foremost, although our system can effectively help experts identify strange or interesting patterns, there is still a gap between finding patterns and accelerating CNN training.",
"The experts still have to reason about and understand what these patterns mean or how to use them to accelerate model training in future.",
"We think it is not a problem faced just by our system, but by all CNN visualizations in general.",
"Like previous work, DeepTracker may only peel a hole in the box and reveal limited information.",
"But we hope that, by peeling enough holes, all these strange patterns will start to connect and make sense by themselves, thus providing a clear picture of CNNs.",
"[rgb]0,0,0Second, we have adopted many space-efficient visualizations and interaction techniques (e.g., hierarchy, filtering, and aggregation) to address the scalability issue.",
"Our current design can well support showing dozens of layers and classes in the same time.",
"The correlation view shares all the filter strategies with the other two views, and vice versa.",
"Thus, our system can perform well in most cases.",
"Nevertheless, the worst scenario still requires to display hundreds or thousands of small multiples at the same time.",
"A possible solution is to employ task-specific aggregation or filtering methods to show the data of interests.",
"Third, we propose a rule-based anomaly detection method that requires experts to manually pick a reasonable window size $k$ and set the threshold for filtering.",
"The number and patterns of anomalies are sensitive to these settings.",
"One potential solution to this problem is to develop an automatic method to enumerate all potential parameter settings and identify those can detect a reasonable amount of significant anomalies and provide these settings to the experts as guidance.",
"Finally, we only conduct experiments on ResNet-50 [17], but our method can also be applied to other state-of-the-art deep CNN models, which often have similar hierarchical structures (e.g., “inception block” in google-inception-v4 [37]).",
"[rgb]0,0,0 Besides, the cube visualization is a general technique, which can be used to explore multiple heterogeneous time series data and their complex correlations.",
"However, to further generalize it, a strict user study has to be conducted to find the best manner to use it, such as the axis skew degree, and the minimum height/width for each row/column in the three faces.",
"Conclusion We propose a novel visual analytics solution to disclose the rich dynamics of CNN training processes.",
"Knowing such information can help machine learning experts better understand, debug, and optimize CNNs.",
"We develop a comprehensive system that mainly comprises the validation, layer, and correlation views to facilitate an interactive exploration of the evolution of image classification results, network parameters, and the correlation between them.",
"We conduct experiments by training a very deep CNN (ResNet-50) on ImageNet, one of the largest labeled image datasets that is commonly used in practice, to demonstrate the applicability of our system.",
"The positive feedback from our collaborating experts and the experts from an internal workshop validates the usefulness and effectiveness of our system.",
"Future studies may integrate some feature-oriented visualization techniques, which typically require recording the activation information for input instances.",
"Feature visualizations can provide insights on what features a filter in a given snapshot of CNN has learned.",
"Our system can track critical iterations to take snapshots for a CNN over training, and then use feature visualization techniques to analyze the learned features evolving patterns for the detected important filters.",
"The other urgent need is to deploy the system in a real-time environment.",
"To this end, we have to consider some new design and interaction requirements to fill the gap between finding patterns and accelerating CNN training.",
"The authors would like to thank Kai Yan for providing support in editing the relevant media materials.",
"The work is supported by the National Basic Research Program of China (973 program) under Grant No.",
"2014CB340304 and ITC Grant with No.",
"UIT/138."
],
[
"Visualization",
"In this section, we describe our three coordinate views, namely, the Validation View, the Layer View, and the Correlation View, that help experts accomplish the aforementioned analytical tasks."
],
[
"Validation View",
"Several urgent requirements (R3, R4, R5) from the experts need to examine how the evolving CNN acts differently on the validation images of each class rather than how the overall validation error rate differs over training.",
"Thus, we design the Validation View (Fig.",
"REF & REF ) to present all image classes in $D_v$ .",
"Figure: From top (a) to bottom (b), image classes are more and more difficult to train.",
"[rgb]0,0,0By default, the view starts with a visualization of cluster-level performance (R3).",
"The classes with similar evolving trends form a cluster and then their error rates at every iteration are averaged.",
"The averaged error rates are then depicted as a colored stripe, where the x-axis encodes the iterations and the error rates are encoded by colors (Fig.",
"REF ).",
"We choose k-means as the clustering algorithm and $k$ can be adjusted according to demands (Fig.",
"REF shows the case when $k=4$ ).",
"Experts can open up one cluster to further examine the performance in class-level (R3).",
"The design is based on the following considerations.",
"First, the experts are more interested in the overall classes than in individual iterations.",
"Thus, small multiples technique (juxtaposed techniques) is chosen for their superior performance in high-level comparison tasks (e.g., global trends for every series) [20].",
"Second, we cannot present all the classes ($1,000$ in ImageNet) at one time, we have to consider a hierarchical and highly space-efficient visualization.",
"However, many traditional charts, such as line charts and horizon graphs, require a larger vertical space [19] than 1D heatmaps.",
"Compared with traditional charts, heatmaps are also more easy to do side by side comparison for their symmetrical space (i.e., no irregular white spaces).",
"As a result, all experts prefer the heatmap-based small multiples.",
"Image-level performances, R5.",
"The class-level color strips can be further unfolded to explore the image-level evolution patterns.",
"Unfolding a heatmap reveals a pixel chart (Fig.",
"REF d), with each row (1px height) representing an image and each column (1px width) representing an dumped iteration (consistent with the class heatmap).",
"We use red and green colors to indicate the incorrect and correct classifications, respectively.",
"Meanwhile, the experts can zoom/pan the pixel chart for a closer inspection.",
"Clicking on a row shows the original corresponding image.",
"Anomaly iterations, R4.",
"[rgb]0,0,0As experts are concerned about the iterations with abnormal behaviors, we particularly propose a algorithm to detect these anomaly iterations (refer to Sec.",
"REF ).",
"Experts can choose to only show the classes with anomaly iterations (Fig.",
"REF ).",
"At this point, for each class-level color stripe, we use triangular glyphs to highlight these detected anomaly iterations.",
"The upside-down triangles ($\\bigtriangledown $ ) and normal triangles ($\\triangle $ ) indicate those anomaly iterations that are detected by the left-rule and right-rule, respectively.",
"The widths of triangles encode the anomaly scores.",
"Experts can set a threshold value to filter the triangles with low anomaly scores."
],
[
"Anomaly Detection",
"In our scenario, the classification results for an image can be represented by a 0/1 sequence ($[a_1, ..., a_n]$ ), where each element represents a correct or incorrect result at the corresponding validation iteration.",
"The experts are curious about the iterations when a significant amount of 1/0 flips (i.e., 0 to 1 or 1 to 0) occur for a class.",
"In general, this problem can be modeled and solved using Markovian-based anomaly detection algorithms [1].",
"Despite the popularity of using Markovian methods to detect outliers in discrete sequence, we decide to employ rule-based models [1] for two reasons.",
"First, Markovian methods are a black box and the resulting outlier values are sometimes difficult to comprehend.",
"Second, the experts have explicitly described two types of iterations they are very interested in, namely, those iterations when many images with values that remain stable for many previous iterations suddenly flip (denoted by the left-rule) and those iterations when many images flip and keep their values stable after many iterations (denoted by the right-rule).",
"Fortunately, these anomalies can be easily modeled using rules.",
"The rule-based models primarily estimate the value $P(a_i|a_{i-k},\\ldots ,a_{i-1})$ , which can be expressed in the following rule form: $ a_{i-k},\\ldots ,a_{i-1} \\Rightarrow a_{i}.$ In our scenario, if an image has the same value (either 0 or 1) in the previous consecutive $k$ iterations ($i-k,\\ldots , i-1$ ), then its value must be the same at iteration $i$ (the left-rule).",
"Otherwise, iteration $i$ is considered an outlier for the specified image.",
"Based on these considerations, we develop an application-specific algorithm to detect anomaly iterations in the validation history.",
"The algorithm includes the following steps: Rule-Judgement: The algorithm computes a vector $[l_{i1}, \\ldots , l_{in}]$ for every image $i$ , where $l_{ij} =1$ if the left-rule is satisfied, otherwise, $l_{ij} =0$ ; Aggregation: For each class that contains $m$ images, the algorithm aggregates all the computed vectors for each image into one $[L_1, ..., L_n]$ , where $L_j = \\sum _{i=1}^{m}l_{ij}$ , [rgb]0,0,0denoting the left anomaly score at iteration $j$ for this class.",
"The approach is a window-based method, and the experts can adjust the window size $k$ to control the sensitivity of the anomalies.",
"In a similar manner, we detect the anomalies from the opposite direction for the right-rule.",
"Layer View The Layer View focuses on weight-related tasks (R1, R2).",
"The view consists of two connected parts, namely, the CNN structure and the hierarchical small multiples (Fig.",
"REF ), so that experts can hierarchically explore and compare various types of statistic in the context of the network structure.",
"Figure: Visual encodings in the Layer View: (a) a CONV layer, (b,d) hierarchical bars, (c) links between the CNN structure and (e) the hierarchical small multiple charts, (f) a pixel chart for one layer.CNN structure.",
"The experts hope our tool can help them explore the statistical information of each layer and meanwhile know their relative positions in the entire network (R1).",
"Thus, we adopt NetscopeNetscope is a web-based tool for visualizing neural network architectures.",
"http://ethereon.github.io/netscope/, a popular neural network visualizer, in our system.",
"[rgb]0,0,0 The green rectangle is data input layer, the red ones are the CONV layers, and the purple ones mean the pooling layers.",
"The links between these rectangles show the network structure.",
"We further add blue level bars (Fig.",
"REF b) to encode the latent hierarchy (from CONV modules, bottlenecks to basic CONV layers, Sec. ).",
"The right most level bars represent CONV modules (Sec.",
"), which are recursively divided into smaller level bars until reaching elementary CONV layers (i.e., red rectangles).",
"Hierarchical small multiples.",
"To assist experts in exploring and comparing layers of a deep CNN (R1, R2), a space-efficient visualization technique is demanded.",
"Thus, we leverage hierarchical small multiples to show layers of interest (Fig.",
"REF e).",
"[rgb]0,0,0By default, experts are presented with the information about CONV modules and then can drill down to see more information about low-level CONV layers with interactions with the network graph (i.e., click on the corresponding level bars).",
"The width of outcropping rectangles (Fig.",
"REF d) encodes the aggregation level of current layer charts.",
"For example, the top second layer chart in (Fig.",
"REF e) shows the bottleneck-level aggregation information and the following three layer charts show the basic CONV layer level information.",
"Besides, the links (Fig.",
"REF c) mark the real positions of the layer charts in the network structure.",
"The small multiples support multiple types of charts including line chart, horizon graphs [18] and box plots to emphasize the different aspects of the statistical data.",
"The experts use box-plots to see the rough distribution of statistical values and use basic line charts to examine individual values.",
"Besides, the experts prefer to use horizon graphs when performing tasks in regard to trend tracking and comparison (R2), because of its effectiveness in visualizing divergent weight values [20].",
"Similar to unfolding the class heatmap to a pixel chart, the experts are also allowed to open the layers of interest as a pixel chart (Fig.",
"REF f) that presents the filter-level information (R1).",
"Each row (1px height) in the pixel chart represents one filter, and each column (1px width) indicates one iteration.",
"We use sequential colors to encode pixel value (e.g., the cosine similarity between two subsequent dumped iterations).",
"Correlation View This view helps experts establish connections between the filters and images.",
"In particular, the experts want to understand further how the changes in network parameters are related to class performances (R6).",
"For example, several anomaly iterations may be detected for a single class.",
"For each detected anomaly iteration, we can identify a set of anomaly filters (i.e., the top $k$ filters with largest changes at that iteration).",
"Since different classes may share anomaly iterations and different anomaly iterations may share anomaly filters, are there any filters that are commonly seen in these iterations?",
"Do any of the anomaly classes or filters strongly co-occur?",
"We designed the Correlation View to answer these questions.",
"[t] Target set $S_{target}$ .",
"Minimum partition for $S_{target}$ .",
"$S_{result}$ = $\\emptyset $ each target set $s_t$ in $S_{target}$ $S_{new}$ = $\\emptyset $ each mini set $s_r$ in $S_{result}$ $itersection$ = $s_t \\bigcap s_r$ $S_{new}$ = $S_{new} \\bigcup intersection \\bigcup (s_r - s_t)$ $s_t$ = $s_t - s_r$ $s_t$ is not empty $S_{new} \\bigcup s_t$ remove all empty set in $S_{new}$ $S_{result}$ = $S_{new}$ return $S_{result}$ Minimum Set Partition Filter set partition.",
"We first introduce the mini-set concept to organize anomaly filters that are shared by multiple anomaly iterations and different classes.",
"For each class $C_i\\in \\lbrace C_i|{1\\le i \\le n}\\rbrace $ , we denote its anomaly iterations by $T_i = \\lbrace t_{i,k}|{1\\le k\\le n_i}\\rbrace $ .",
"Thus, all anomaly iterations are $\\cup _{1\\le i \\le n}T_i$ , denoted by $T$ .",
"For each anomaly iteration $t\\in T$ , we denote its anomaly filters at CNN layer $L_j\\in \\lbrace L_j|{1\\le j \\le m}\\rbrace $ by $s_{j,t}$ .",
"Thus, for each layer $L_j$ , we can collect all anomaly filter sets $\\lbrace s_{j,t}|t\\in T\\rbrace $ (denoted by $S_j$ ) and all anomaly filters $\\cup _{t\\in T}s_{j,t}$ (denoted by $s_{j}$ ).",
"Thus, mini-set aims to find a minimum set partitions of $s_{j}$ (denoted by $s_j^\\ast $ ) that each $s_{j,t}$ can be assembled from some elements (i.e., mini-sets) in $s_j^\\ast $ .",
"We specifically propose a Set Partition Algorithm (Alg.",
"REF ) to find $s_j^\\ast $ .",
"The algorithm accepts a target set as input (i.e., $S_j$ ).",
"$S_{result}$ is initially empty and a new anomaly filter set is used at each time to partition the mini-sets contained in $S_{result}$ (cf.",
"lines 4 to 7).",
"If the new anomaly filter set is not empty after partitioning, then it is added as a new mini-set (cf.",
"line 8).",
"Finally, the partitions contained in $S_{result}$ will be returned.",
"Figure: (a) The abstract version of correlation view, where rows and columns represent layers and image classes, respectively.",
"A sequential color scheme is used to encode the number of anomaly filters.",
"(b) The complex version of correlation view, where the detailed information of individual anomaly filters are shown.",
"(c) A layout solution for coordinated analysis without using skewed axes.Visual encoding.",
"To intuitively represent these relationships, we introduce a grid-style visualization (Fig.",
"REF ), where rows and columns represent layers and image classes, respectively.",
"The number of rows and columns equal to the number of layers with anomaly filters and classes with anomaly iterations, respectively.",
"[rgb]0,0,0 We start from a abstract version.",
"In this version (Fig.",
"REF a), for $\\mathrm {Cell}_{i,j}$ , a sequential color scheme is used to encode the number of anomaly filters ($\\cup _{t\\in T_j}s_{i,t}$ ).",
"The darker the color it is, the more anomaly filters appear in layer $L_i$ that are related to class $C_i$ .",
"From this visualization, we can easily observe the correlations between layers and classes, while it also hides much detailed information.",
"We cannot answer questions like whether the filters in $\\mathrm {Cell}_{i,j}$ are the same with $\\mathrm {Cell}_{i,k}$ (this kind of information shows how many classes this filter can impact and help examine the relationships among classes), or whether there are filters in $\\mathrm {Cell}_{i,j}$ appearing in more than one anomaly iteration (this shows the importance of these filters for class $C_j$ ).",
"To solve these problems, we provide an advanced version (Fig.",
"REF b).",
"For $\\mathrm {Cell}_{i,j}$ , the width and height encode the number of anomaly iterations and the number of anomaly filters of the corresponding class and layer (i.e., $|T_j|$ and $|s_{i}|$ ), respectively.",
"Based on these numbers, the columns and rows are further divided with vertical/horizontal lines.",
"For a class (e.g., $\\mathrm {Column}_{j}$ ), $|T_j|$ vertical lines are drawn to represent all related anomaly iterations (i.e., $[t_{j,1}, t_{j,2},\\ldots ,t_{j,n_j}]$ ).",
"For each row of layer $L_i$ , there are $|s_i^\\ast |$ horizontal lines representing all mini-sets in that layer.",
"The intersections between these horizontal and vertical lines are highlighted with blue rectangles if the corresponding mini-set is part of the anomaly filters of the corresponding anomaly iteration.",
"The height of the rectangles represents the number of filters of the corresponding mini-set.",
"[rgb]0,0,0Obviously the introduction of mini-sets dramatically cuts down the number of horizontal lines and blue rectangles, otherwise each anomaly filter require one horizontal line and one rectangles, which may cause serious visual clutter problem.",
"In fact, mini-sets can be viewed as a partially aggregation version instead of representing all the anomaly filters as horizontal lines and rectangles.",
"Users can set the minimum appearing number of mini-sets to filter the sets with lower importance.",
"Cube Visualization The log data contain three main aspects of information, namely, iterations, validation information, and weight information.",
"The three aforementioned views are designed to show all possible 2-combinations of these three types of information, respectively.",
"[rgb]0,0,0 Although these views can be used individually, they need to be combined together to form a complete picture.",
"Thus, we propose a novel and intuitive visualization technique, which naturally and seamlessly stitches the three views together, based on their shared axis (inspired from Binx [8]), into a “cube” shape (Fig.",
"REF ).",
"When experts find or highlight a pattern of interest, they can easily track the pattern over the edges to find the related information in the other two views easily.",
"[rgb]0,0,0 The use of skewed axes may bring about a possible perspective distortion problem.",
"Nevertheless, the advantages of the cube-style design far outweigh the disadvantages.",
"Given the limited pixels in a computer screen and each view requiring a large display space, the experts all agree that the cube-style design is the most space-efficient and intuitive manner to show all the information.",
"Furthermore, with such design, it prevents the experts from switching from multiple views, reducing the cognitive burden and the load of memory.",
"This allows the experts to conduct correlation analysis more effectively.",
"We also provide a compromised solution to handle the distortion problem, that is, laying out the three views in the form like Fig.",
"REF c. So the experts can firstly examine the layer view (horizontally) or validation view (vertically) together with the correlation view, and then switch to cube mode to explore the three views together.",
"[rgb]0,0,0 Notice that there are some different settings for several views in the cube.",
"For the layer view (front), only the layers with anomaly filters are activated (see the activated blue bars in the front view of Fig.",
"REF ), the weight variation of each anomaly filter is represented as a horizontal color strip.",
"For the validation view (top), only the classes with anomaly iterations are preserved.",
"The following lists several common exploration pipelines: P1: from the layer view (front), we can quickly check the distribution of activated layers in the overall network and pick some anomaly filters of interests.",
"Then, by tracking along the horizon axis to the correlation view (right), we can examine which classes these filters impact and how important these filters are to the classes.",
"Finally, we can observe the evolving patterns of these classes and the corresponding anomaly iterations in validation view (top).",
"P2: from the validation view (top), we can firstly mark several anomaly iterations of some classes.",
"Then, we can check the corresponding columns in the correlation view, finding the rows that contain anomaly filters and exploring the importance of these filters to these classes and how these filters impact the other classes.",
"Finally, by highlighting these corresponding rows, we can observe them in the layer view to see how these filters behave around the picked anomaly iterations.",
"P3: from the correlation view (right), we can search the horizontal lines across many rectangles (it means these filters impact many classes at the same time) or the rectangles that appear more than one time in the same cell (these filters are judged as anomalies many times for a class and may have great impact on this class).",
"With these selected horizontal lines or rectangles, we can simultaneously track their corresponding weight variation information in the layer view (front) and class performance information in the validation view (top).",
"Use Examples We derived these examples through the assistance of our collaborating experts, who were familiar with our designs and data.",
"As a remark, the following results are from the experiment with 8 times larger batch size and learning rate setting than the basic setting introduced in Sec. .",
"Exploring Validation Results Figure: Overview of validation classes.",
"(a) Two curves show the overall training/validation error rate.",
"(b) Two turning points align well the boundary of stage s2s2.",
"(c) Three peaks appear in the stage s1s1 and align well with (f) the detected anomaly iterations.",
"(d) Two types of mushroom images have different behaviors in the class.",
"(e) Most images in the class flip at the anomaly iteration.The first scenario demonstrates how the experts use DeepTracker to explore the image classification results (R3, R4, and R5).",
"Performance evolution patterns.",
"Fig.",
"REF a shows a typical visualization of training/validation errors that may appear on any popular training platform.",
"The timeline at the top shows a total of 1.2 million iterations.",
"Beneath the timeline, four line segments represent four stages ($s1$ , $s2$ , $s3$ , and $s4$ in Fig.",
"REF ) in the training process, [rgb]0,0,0where the later stage has one-tenth of the learning rate of the previous stage.",
"We can observe that two sudden drops in the curves match well with the boundaries of the training stages (Fig.",
"REF b).",
"However, this is a well-known pattern to the experts.",
"On the other hand, although the overall error rate continues to decrease, the class-level error rates show a more complicated story, that is new to the experts.",
"By quickly scanning the small multiples in cluster-level, the experts identify there are generally four types of class evolving patterns (Fig.",
"REF ).",
"From top to bottom, the four types are more and more difficult to train.",
"[rgb]0,0,0For example, for the type at the top, these classes are recognized correctly after a few iterations.",
"By contrast, the classes at the bottom always have high error rates in the entire training process, which means that the resulting network fails to recognize the related images.",
"From this, the experts learn that the model has spent most of time to improve its performance on the classes of middle-level classification, since the model has already performed well on the easy-trained classes at a very early stage and is always performing miserably on the hard-trained classes over the entire training.",
"From these patterns, the experts consider it promising to accelerate the training process and improve the overall performance by treating classes differently during the training process.",
"That is, stop feeding the easy-trained classes in an appropriate early stage, put more efforts on training the classes of middle difficulties for classification, and figure out why some classes always have extremely high error rates.",
"One similar attempt has been made in a recent work [25].",
"Anomaly iterations.",
"The experts are curious about the three sudden peaks in stage $s1$ , and then mark these three iterations with dotted lines (Fig.",
"REF c), which look like anomaly iterations (R4).",
"However, the colors in the small multiples do not have clear patterns related to these iterations.",
"Then, the experts turn on the anomaly detection and immediately find that many triangles are aligned well with the dotted lines (Fig.",
"REF f), thereby confirming our suspicion.",
"Then, the experts can click on the corresponding image icons to see the detailed images that contribute to the three peaks.",
"In addition, there are more anomaly iterations in stage $s1$ then in the later stages.",
"This interesting pattern can be explained by the reduction of learning rate and the convergence of the model in the later stages.",
"At the same time, it also implies that the learning rate in stage $s1$ is slightly too high, leading to the instability in the model (the case in Sec.",
"REF indicates the same finding for the discovery of potential “dead” filters).",
"Details in classes.",
"To further examine what happens at the anomaly iterations for a class, the experts can further check the image-level information of the class (R5).",
"For example, the experts are curious about the abnormally large anomaly iteration in the class of “mushroom” (Fig.",
"REF d) that are captured by both the left-rule and the right-rule.",
"Then, they click and expand color stripe to see the pixel chart of images.",
"First, they confirm that this iteration is indeed special for this class, because nearly all images flip during particular that [rgb]0,0,0dumped interval (Fig.",
"REF e).",
"Thus, they may further investigate to find the layers or filters that cause such flips based on the filter updates around that iteration.",
"[rgb]0,0,0In particular, the experts comment that, it seems that after the iteration, the CNN model has jumped to a better local optimal for the class, because the green color is more stable after the iteration.",
"This may result from the reduction of the learning rate (from $s1$ to $s2$ ).",
"This kind of patterns appear frequently in many classes during the whole training process, many of them occurring not in the learning rate transition point.",
"The experts wonder that the model should be trying to jump from one local optimal to another better local for these classes continuously, so as to reduce the overall error rate gradually.",
"This insight has never been obtained because the experts initially thought that the error rate for one class should decrease steadily.",
"Besides, the experts also find that, at the bottom of the pixel chart, several images are mislabeled in the entire training process, although the class is easy to train overall(Fig.",
"REF d).",
"To understand why, the experts click on these images to examine them, and find that the contents in the mislabeled images have a clear color pattern different from that of the rest of the mushroom images.",
"The correctly labeled mushrooms are all red, while the mislabeled ones are white or orange.",
"This finding indicates color is a critical feature that the CNN has learned to classify this class of images.",
"Exploring Weight-Relevant Information Figure: (a, b, c) The sd values of each layer decrease slowly and have different scales in different CONV modules.",
"(d) The weight changes in filters are large at the beginning.",
"(e) One outlier filter is detected, whose weights never change during the entire training process.This scenario shows how to discover patterns in neuron weights via the Layer View (R1, R2).",
"First, the experts choose to show the sd (standard deviation) of the weights at the layer-level using horizontal graphs (Fig.",
"REF ).",
"As the experts expected, all the trends show a similar pattern of slow decrease, indicating the weights in the entire model is converging over iterations.",
"Besides, the experts also find that deeper layers (closer to the loss layer) tend to have smaller sd values.",
"[rgb]0,0,0In particular, by tuning the band number (finally to 3) of the horizon graphs, they found the sd values of a CONV module are usually twice as large as those of the one below it (a, b, and c in Fig.",
"REF ).",
"Given that we apply Xavier initializationREF and for ResNet-50, the input sizes of layers in a CONV module are twice as large as the ones in the layers of its previous CONV module, the observed result is not beyond the experts' expectation.",
"This suggests that there is no problem exist on the initialization approach.",
"Analogously, the experts find that the weight means of each layer become negative quickly (from green to blue instantly, Fig.",
"REF a) except for the FC layer (Fig.",
"REF b).",
"At first, the pattern looks strange to the experts.",
"Then, the experts realize that it is reasonable to have more negative weights than positive ones, since negative values are often used to filter out trivial features owing to the use of ReLU activations.",
"The increase of negative weights suggests that the network is trained to extract useful information from the original image layer by layer, and then finally remain the most relevant features.",
"[rgb]0,0,0As for the FC layer, it plays a function of shaping the extracted useful features into feature vectors of given dimension for further classification.",
"One strange phenomenon intrigues the experts, that is, the FC layer weight means are always positive in many-times training ResNet-50 (with different batch sizes and learning rates) on ImageNet Dataset, whereas becoming negative when training ResNet-164 on Cifar Dataset [22].",
"This finding is worth a further investigation.",
"Apart from layer-level values, the experts also explore the filter-level information (R1).",
"In our system, two different ways (i.e., filter-based or iteration-based) are used to normalize weight changes at the filter-level.",
"For filter-based normalization, changes are grouped and normalized by filters, which aims to help experts see the change distribution over iterations for individual filters.",
"Similarly, iteration-based normalization allows experts to examine the distribution over filters for individual iterations.",
"For example, Fig.",
"REF d visualizes the filter changes in one of the CONV layer belonging to the second CONV module using filter-based normalization.",
"The experts find that the changes are drastic in stage $s1$ and become relatively small in the later stages because of the decrease in learning rate and the convergence of the model.",
"However, the experts also identify two strange filters among 64 filters in the first CONV layer that have a constant deep blue color (Fig.",
"REF e).",
"By further checking, the experts find that the weights of these two filters never change during the entire training process.",
"Figure: (a, b) The means of weights in each CONV layer become negative quickly (from green to blue) except for the FC layer.",
"(c) Three filters are always more actively changed than the other filters in the later part of training progress.",
"[rgb]0,0,0This is a total surprise to the experts.",
"Excluding programming bugs, the most likely reason should be due to the dying-ReLU problem, namely, these two filters are inactive for essentially all inputs and no gradients flow backward through the neurons of the two filters.",
"The experts suspect the dying-ReLU problem results from the high learning rate in the beginning of train.",
"In fact, the experts usually follow a rule of thumb to set the hyper-parameter learning rate, that is, multiply the learning rate by k if the batch size is multiplied by k. This rule is currently formally introduced in a recent work [15].",
"In this experiment, we use 32 times larger batch size GPUs (32 GPUs) than the mini-batch size 32 for one GPU to train the model with the corresponding size of learning rate, dying-ReLU problem still occurs.",
"This reminds the experts that the rule may not so accurate for extremely large batch size sometimes, but the problem can be solved by carrying out a warmup strategy (i.e., using lower learning rate at the start of training [17]), which the experts haven't done in previous trainings.",
"One further interesting finding is that by inactivating these two \"dead\" filters (i.e., set their weights as 0 so that they are inactive for any inputs), the experts find the overall performance not affected at all, whereas if we inactivate other random-picked filters in the first CONV layer of the model, the number of mislabeled images in $D_v$ would increase few thousands.",
"Thus, the experts finally modified the network configure and eliminated these two filters, so that the model can run faster while costing less memory.",
"Fig.",
"REF c visualizes the weight changes in one middle layer using iteration-based normalization.",
"The experts find that a small number of filters are always more actively changed than the other filters (long deep blue lines in Fig.",
"REF c) in the later part of iterations.",
"This pattern implies that the updates inside a layer may be highly divergent.",
"[rgb]0,0,0In the later part of the training, where the learning rate is getting smaller and the model is converging, only a couple of filters are still continually actively updated for every iteration.",
"We have tried to inactivate these identified filters, the result showing that the overall performance is not affected.",
"It is not beyond the experts' expectation due to the ResNet's ensemble-like behavior [40] (even several entire layers can be removed without impacting performance).",
"In the end, the experts still cannot fully explain the behavior of these continually updating filters.",
"One possible reason could be that these special filters are not trained well (not converge) to extract some specific features, thus reacting violently for every iteration even in the later stages of the training.",
"Exploring Filter-Image Correlations In this scenario, we demonstrate how the experts use the Correlation View to explore correlations between images and filters.",
"Figure: A cube-style visualization that fuses three coordinated views together to reveal the rich dynamics in a CNN training process:(top) the Validation View shows the error rate changes of validation classes; (front) the Layer View shows the weight changes in CNNfilters; (right) the Correlation View shows the potential relationships between filters and validation classes.Shallow-layer filters vs. deep-layer filters.",
"[rgb]0,0,0At first, the experts choose to only show the top $k$ (100 in this case) changing filters in the layer view.",
"By checking the network structure visualization, the experts find that the activated shallow layers (the layers close to data input layer) are more than the activated deep layers, and most activated layers are the last basic CONV layers of bottlenecks for deep CONV modules.",
"Besides, Fig.",
"REF a shows that deep CONV modules tend to contain more anomaly filters (especially for the CONV modules 4).",
"The experts think that this kind of knowledge is of great importance for network compression [16].",
"Then, the experts go to the complicated version to examine the more detailed correlation information.",
"They filter the mini-sets with very few appearing times, finding that anomaly filters in shallow layers are generally shared by more anomaly classes (columns) and iterations (vertical lines in one column) than those in deep layers (Fig.",
"REF a).",
"The experts think that this pattern may relate to the fact [40] that shallow-layer filters are more likely to capture basic visual features than deep ones, thereby the huge change of these filters affecting more classes of images (e.g., the long and opaque lines marked by b in Fig.",
"REF ).",
"By contrast, a deep filter tends to learn higher-level features, thus only relating to specific classes of images.",
"To further explore the correlations, the experts select two mini-sets (b1 and b2 in Fig.",
"REF ), for comparison.",
"Both the horizontal lines of b1 and b2 are opaque and thick.",
"By tracking them in the Layer View and the Validation View, the experts can see that b1 is in the first CONV layer, and related to many classes.",
"The experts open these classes and discover that many images in them have a common feature, i.e., a large background of blue sky or ocean (b1 in Fig.",
"REF ).",
"This discovery suggests that these filters in E1 may target the basic pattern to help identify images that contain large blue areas.",
"By contrast, b2 is located at the fifth [rgb]0,0,0CONV module and related to only three classes.",
"Interestingly, the images in the three classes also share a more concrete feature, i.e., objects in a bush (b2 in Fig.",
"REF ).",
"[rgb]0,0,0In short, this case confirms that we are on the right track to reveal how the model weight changes relate to the classification result changes.",
"Important filters for a class.",
"To find stronger correlations between filters and classes, the expert focus on anomaly filters that appear more than once in a cell for a specific class.",
"For example, the experts find two appearances of the same mini-set (containing two anomaly filters) for the class of “gong” (c1 in Fig.",
"REF ).",
"Tracking horizontally (along with the pink-highlighted area), the experts find that the mini-set does not appear in other anomaly iterations, which also implies a strong correlation between filters in the mini-set and the class.",
"Then, the experts click on these two rectangular glyphs to highlight the corresponding iterations on the timeline (c2 in Fig.",
"REF ) and the filter locations in the Layer View (c3 in Fig.",
"REF ).",
"It is clear that the gong class is not a well trained class as it has a very large yellow area (indicate a relatively high error rate) in the Validation View.",
"However, the experts also find a period in the middle when this class has a relatively good performance (c4 in Fig.",
"REF ), which happens to contain the highlighted anomaly iterations.",
"Meanwhile, the Layer View shows that the highlighted filters are also updated dramatically during the period of good performance (c3 in Fig.",
"REF ).",
"Considering these patterns, the experts speculate that the filters in the mini-set have a strong impact on the classification of gong images.",
"As expected, we conduct experiments to inactivate these two filters, finding the overall performance and the performance on “gong” class are not impacted (see the reason in the last paragraph in Sec.",
"REF ).",
"Nevertheless, it provides the experts with a new manner to investigate the functions of filters co-working together for classifying one class of images.",
"That is, increase the threshold to find anomaly filters as many as possible, find the mini-sets containing many filters to some classes from multiple layers, and then inactivate them all to validate corresponding impacts.",
"Abnormal anomaly filters.",
"The experts are also attracted by two mini-sets (d1 and e1 in Fig.",
"REF ), because of their abnormal color patterns.",
"The filters in these two mini-sets exhibit large changes all the time in the latter part of the training, which are very different from the other anomaly filters.",
"Thus, the experts are interested in these filters and further check their correlated classes in the Correlation View (right).",
"Interestingly, each abnormal mini-set only appears with two classes (d2 and e2 in Fig.",
"REF ), and each pair of classes have very similar performances displayed in the Validation View (d3 and e3 in Fig.",
"REF ).",
"By checking the detailed images of these classes, the experts discover some common patterns.",
"For example, for mini-set e1, the corresponding classes are about mushrooms growing on grass and dogs playing on grass (e3 in Fig.",
"REF ).",
"For mini-set d1, the corresponding classes are related to curved shapes, such as parachutes and round textures (d3 in Fig.",
"REF ).",
"Although the experts are still unclear about why these two mini-sets have such a special behavior, they believe that these filters are likely to play important roles in identifying middle-level features such as grass and curved shapes.",
"We also conduct further experiments to validate the impact of inactivating these filters and the results are similar to the previous case (i.e., important filters for a class).",
"Expert Feedback Usability [rgb]0,0,0DeepTracker is built upon a close collaboration with three domain experts, who constantly underscore their requirements and provide suggestions during the implementation process.",
"After several iterations of refinement, the experts were happy with the current version.",
"They all praised our way of effectively exploring such extreme large-scale training log via a hierarchical manner.",
"$\\mathrm {E}_a$ and $\\mathrm {E}_b$ mentioned that the well-designed validation and layer views were very intuitive and helped them greatly.",
"For example, the layer view allowing the experts to effectively observe and compare layer-related information (e.g., weight/gradient distribution) can help them diagnose network structures.",
"The detecting of dying-ReLU problem in the early stage of a training is useful for tuning the hyper-parameters (e.g., learning rate).",
"This kind of knowledge can also be leveraged to conduct model compression [16], so as to improve the model in respect to computing speed and memory cost.",
"Although the experts still cannot figure out the exact reason that some filters are always more actively updated in the later training stages, they believe the insight that would be obtained from the future investigation will be helpful in diagnosing and improving network structures.",
"Besides, the divergent evolving patterns of classes and the numerous anomaly iterations found in validation view provide the experts with a new promising direction to train a better model.",
"Both $\\mathrm {E}_a$ and $\\mathrm {E}_c$ were particularly fond of the cube-style visualization and deemed it as a new perspective to observe the training of CNNs for them.",
"They both have found many interesting patterns with the cube visualization, some of which were reasonable or could be explained after thinking for a while.",
"However, the experts also failed to figure out some other patterns, notwithstanding they conducted several testing experiments.",
"Nevertheless, they were still excited that our system can help them identify potential subjects for further study.",
"Generality During the implementation, we were concerned about the generality of DeepTracker, that is, whether the design was biased to the specific requirements from these three experts.",
"Therefore, to check how our system is accepted by broader expert communities, we presented our system in a workshop, which involved about 20 experts in the machine learning and visualization fields.",
"In addition, we also interviewed another group of twelve experts, who worked on a large project about using CNNs to improve image search quality.",
"We presented the latest version of DeepTracker to the experts, encouraged them to experiment with the system, and collected their feedback in the process.",
"Exceeding our expectation, DeepTracker was well accepted by these experts.",
"Although they proposed several new requirements, the experts shared many major interests with our three collaborators, such as tracking class level performance, filter-level information, and the relationships between them.",
"After introducing our system, they immediately understood the purposes of each view and all appreciated this novel, intuitive, and expressive way to watch training processes.",
"Although the demo was performed on our experiment datasets, the experts saw its potential in their project and immediately asked for collaboration with us, so that they could plug in and analyze their own datasets.",
"Improvement Apart from this positive feedback, the experts also made several interesting suggestions to further improve DeepTracker.",
"For example, two experts suggested that our current system only differentiates correct or incorrect classifications for validation images (i.e., 1 and 0).",
"However, the exact incorrect labels should also be presented because such information can help identify confusing or similar classes.",
"One expert mentioned that he showed strong interest on what happens before the anomaly iteration and suggested dump data of every iteration at that abnormal interval for fine-grained analysis.",
"Another expert suggested that our system shoud be integrated with online dashboards, as visualizing the training log on the fly can allow them to terminate the training early and save time if the visualization shows anything undesired.",
"Discussion DeepTracker is our first step to open the “black box” of CNN training.",
"Although the experts have high expectations of this tool, we all agree to start with two fundamental pieces of information: neuron weights and validation results.",
"Considering our target users and the large scale of datasets, we try to avoid using sophisticated visual encodings to ensure a fluent exploration experience.",
"Unsurprisingly, our bare-to-metal visualizations are preferred by the experts, and they use it to find many patterns easily, either expected or unexpected.",
"However, we still have several limitations.",
"First and foremost, although our system can effectively help experts identify strange or interesting patterns, there is still a gap between finding patterns and accelerating CNN training.",
"The experts still have to reason about and understand what these patterns mean or how to use them to accelerate model training in future.",
"We think it is not a problem faced just by our system, but by all CNN visualizations in general.",
"Like previous work, DeepTracker may only peel a hole in the box and reveal limited information.",
"But we hope that, by peeling enough holes, all these strange patterns will start to connect and make sense by themselves, thus providing a clear picture of CNNs.",
"[rgb]0,0,0Second, we have adopted many space-efficient visualizations and interaction techniques (e.g., hierarchy, filtering, and aggregation) to address the scalability issue.",
"Our current design can well support showing dozens of layers and classes in the same time.",
"The correlation view shares all the filter strategies with the other two views, and vice versa.",
"Thus, our system can perform well in most cases.",
"Nevertheless, the worst scenario still requires to display hundreds or thousands of small multiples at the same time.",
"A possible solution is to employ task-specific aggregation or filtering methods to show the data of interests.",
"Third, we propose a rule-based anomaly detection method that requires experts to manually pick a reasonable window size $k$ and set the threshold for filtering.",
"The number and patterns of anomalies are sensitive to these settings.",
"One potential solution to this problem is to develop an automatic method to enumerate all potential parameter settings and identify those can detect a reasonable amount of significant anomalies and provide these settings to the experts as guidance.",
"Finally, we only conduct experiments on ResNet-50 [17], but our method can also be applied to other state-of-the-art deep CNN models, which often have similar hierarchical structures (e.g., “inception block” in google-inception-v4 [37]).",
"[rgb]0,0,0 Besides, the cube visualization is a general technique, which can be used to explore multiple heterogeneous time series data and their complex correlations.",
"However, to further generalize it, a strict user study has to be conducted to find the best manner to use it, such as the axis skew degree, and the minimum height/width for each row/column in the three faces.",
"Conclusion We propose a novel visual analytics solution to disclose the rich dynamics of CNN training processes.",
"Knowing such information can help machine learning experts better understand, debug, and optimize CNNs.",
"We develop a comprehensive system that mainly comprises the validation, layer, and correlation views to facilitate an interactive exploration of the evolution of image classification results, network parameters, and the correlation between them.",
"We conduct experiments by training a very deep CNN (ResNet-50) on ImageNet, one of the largest labeled image datasets that is commonly used in practice, to demonstrate the applicability of our system.",
"The positive feedback from our collaborating experts and the experts from an internal workshop validates the usefulness and effectiveness of our system.",
"Future studies may integrate some feature-oriented visualization techniques, which typically require recording the activation information for input instances.",
"Feature visualizations can provide insights on what features a filter in a given snapshot of CNN has learned.",
"Our system can track critical iterations to take snapshots for a CNN over training, and then use feature visualization techniques to analyze the learned features evolving patterns for the detected important filters.",
"The other urgent need is to deploy the system in a real-time environment.",
"To this end, we have to consider some new design and interaction requirements to fill the gap between finding patterns and accelerating CNN training.",
"The authors would like to thank Kai Yan for providing support in editing the relevant media materials.",
"The work is supported by the National Basic Research Program of China (973 program) under Grant No.",
"2014CB340304 and ITC Grant with No.",
"UIT/138."
],
[
"Use Examples",
"We derived these examples through the assistance of our collaborating experts, who were familiar with our designs and data.",
"As a remark, the following results are from the experiment with 8 times larger batch size and learning rate setting than the basic setting introduced in Sec.",
"."
],
[
"Exploring Validation Results",
"The first scenario demonstrates how the experts use DeepTracker to explore the image classification results (R3, R4, and R5).",
"Performance evolution patterns.",
"Fig.",
"REF a shows a typical visualization of training/validation errors that may appear on any popular training platform.",
"The timeline at the top shows a total of 1.2 million iterations.",
"Beneath the timeline, four line segments represent four stages ($s1$ , $s2$ , $s3$ , and $s4$ in Fig.",
"REF ) in the training process, [rgb]0,0,0where the later stage has one-tenth of the learning rate of the previous stage.",
"We can observe that two sudden drops in the curves match well with the boundaries of the training stages (Fig.",
"REF b).",
"However, this is a well-known pattern to the experts.",
"On the other hand, although the overall error rate continues to decrease, the class-level error rates show a more complicated story, that is new to the experts.",
"By quickly scanning the small multiples in cluster-level, the experts identify there are generally four types of class evolving patterns (Fig.",
"REF ).",
"From top to bottom, the four types are more and more difficult to train.",
"[rgb]0,0,0For example, for the type at the top, these classes are recognized correctly after a few iterations.",
"By contrast, the classes at the bottom always have high error rates in the entire training process, which means that the resulting network fails to recognize the related images.",
"From this, the experts learn that the model has spent most of time to improve its performance on the classes of middle-level classification, since the model has already performed well on the easy-trained classes at a very early stage and is always performing miserably on the hard-trained classes over the entire training.",
"From these patterns, the experts consider it promising to accelerate the training process and improve the overall performance by treating classes differently during the training process.",
"That is, stop feeding the easy-trained classes in an appropriate early stage, put more efforts on training the classes of middle difficulties for classification, and figure out why some classes always have extremely high error rates.",
"One similar attempt has been made in a recent work [25].",
"Anomaly iterations.",
"The experts are curious about the three sudden peaks in stage $s1$ , and then mark these three iterations with dotted lines (Fig.",
"REF c), which look like anomaly iterations (R4).",
"However, the colors in the small multiples do not have clear patterns related to these iterations.",
"Then, the experts turn on the anomaly detection and immediately find that many triangles are aligned well with the dotted lines (Fig.",
"REF f), thereby confirming our suspicion.",
"Then, the experts can click on the corresponding image icons to see the detailed images that contribute to the three peaks.",
"In addition, there are more anomaly iterations in stage $s1$ then in the later stages.",
"This interesting pattern can be explained by the reduction of learning rate and the convergence of the model in the later stages.",
"At the same time, it also implies that the learning rate in stage $s1$ is slightly too high, leading to the instability in the model (the case in Sec.",
"REF indicates the same finding for the discovery of potential “dead” filters).",
"Details in classes.",
"To further examine what happens at the anomaly iterations for a class, the experts can further check the image-level information of the class (R5).",
"For example, the experts are curious about the abnormally large anomaly iteration in the class of “mushroom” (Fig.",
"REF d) that are captured by both the left-rule and the right-rule.",
"Then, they click and expand color stripe to see the pixel chart of images.",
"First, they confirm that this iteration is indeed special for this class, because nearly all images flip during particular that [rgb]0,0,0dumped interval (Fig.",
"REF e).",
"Thus, they may further investigate to find the layers or filters that cause such flips based on the filter updates around that iteration.",
"[rgb]0,0,0In particular, the experts comment that, it seems that after the iteration, the CNN model has jumped to a better local optimal for the class, because the green color is more stable after the iteration.",
"This may result from the reduction of the learning rate (from $s1$ to $s2$ ).",
"This kind of patterns appear frequently in many classes during the whole training process, many of them occurring not in the learning rate transition point.",
"The experts wonder that the model should be trying to jump from one local optimal to another better local for these classes continuously, so as to reduce the overall error rate gradually.",
"This insight has never been obtained because the experts initially thought that the error rate for one class should decrease steadily.",
"Besides, the experts also find that, at the bottom of the pixel chart, several images are mislabeled in the entire training process, although the class is easy to train overall(Fig.",
"REF d).",
"To understand why, the experts click on these images to examine them, and find that the contents in the mislabeled images have a clear color pattern different from that of the rest of the mushroom images.",
"The correctly labeled mushrooms are all red, while the mislabeled ones are white or orange.",
"This finding indicates color is a critical feature that the CNN has learned to classify this class of images."
],
[
"Exploring Weight-Relevant Information",
"This scenario shows how to discover patterns in neuron weights via the Layer View (R1, R2).",
"First, the experts choose to show the sd (standard deviation) of the weights at the layer-level using horizontal graphs (Fig.",
"REF ).",
"As the experts expected, all the trends show a similar pattern of slow decrease, indicating the weights in the entire model is converging over iterations.",
"Besides, the experts also find that deeper layers (closer to the loss layer) tend to have smaller sd values.",
"[rgb]0,0,0In particular, by tuning the band number (finally to 3) of the horizon graphs, they found the sd values of a CONV module are usually twice as large as those of the one below it (a, b, and c in Fig.",
"REF ).",
"Given that we apply Xavier initializationREF and for ResNet-50, the input sizes of layers in a CONV module are twice as large as the ones in the layers of its previous CONV module, the observed result is not beyond the experts' expectation.",
"This suggests that there is no problem exist on the initialization approach.",
"Analogously, the experts find that the weight means of each layer become negative quickly (from green to blue instantly, Fig.",
"REF a) except for the FC layer (Fig.",
"REF b).",
"At first, the pattern looks strange to the experts.",
"Then, the experts realize that it is reasonable to have more negative weights than positive ones, since negative values are often used to filter out trivial features owing to the use of ReLU activations.",
"The increase of negative weights suggests that the network is trained to extract useful information from the original image layer by layer, and then finally remain the most relevant features.",
"[rgb]0,0,0As for the FC layer, it plays a function of shaping the extracted useful features into feature vectors of given dimension for further classification.",
"One strange phenomenon intrigues the experts, that is, the FC layer weight means are always positive in many-times training ResNet-50 (with different batch sizes and learning rates) on ImageNet Dataset, whereas becoming negative when training ResNet-164 on Cifar Dataset [22].",
"This finding is worth a further investigation.",
"Apart from layer-level values, the experts also explore the filter-level information (R1).",
"In our system, two different ways (i.e., filter-based or iteration-based) are used to normalize weight changes at the filter-level.",
"For filter-based normalization, changes are grouped and normalized by filters, which aims to help experts see the change distribution over iterations for individual filters.",
"Similarly, iteration-based normalization allows experts to examine the distribution over filters for individual iterations.",
"For example, Fig.",
"REF d visualizes the filter changes in one of the CONV layer belonging to the second CONV module using filter-based normalization.",
"The experts find that the changes are drastic in stage $s1$ and become relatively small in the later stages because of the decrease in learning rate and the convergence of the model.",
"However, the experts also identify two strange filters among 64 filters in the first CONV layer that have a constant deep blue color (Fig.",
"REF e).",
"By further checking, the experts find that the weights of these two filters never change during the entire training process.",
"Figure: (a, b) The means of weights in each CONV layer become negative quickly (from green to blue) except for the FC layer.",
"(c) Three filters are always more actively changed than the other filters in the later part of training progress.",
"[rgb]0,0,0This is a total surprise to the experts.",
"Excluding programming bugs, the most likely reason should be due to the dying-ReLU problem, namely, these two filters are inactive for essentially all inputs and no gradients flow backward through the neurons of the two filters.",
"The experts suspect the dying-ReLU problem results from the high learning rate in the beginning of train.",
"In fact, the experts usually follow a rule of thumb to set the hyper-parameter learning rate, that is, multiply the learning rate by k if the batch size is multiplied by k. This rule is currently formally introduced in a recent work [15].",
"In this experiment, we use 32 times larger batch size GPUs (32 GPUs) than the mini-batch size 32 for one GPU to train the model with the corresponding size of learning rate, dying-ReLU problem still occurs.",
"This reminds the experts that the rule may not so accurate for extremely large batch size sometimes, but the problem can be solved by carrying out a warmup strategy (i.e., using lower learning rate at the start of training [17]), which the experts haven't done in previous trainings.",
"One further interesting finding is that by inactivating these two \"dead\" filters (i.e., set their weights as 0 so that they are inactive for any inputs), the experts find the overall performance not affected at all, whereas if we inactivate other random-picked filters in the first CONV layer of the model, the number of mislabeled images in $D_v$ would increase few thousands.",
"Thus, the experts finally modified the network configure and eliminated these two filters, so that the model can run faster while costing less memory.",
"Fig.",
"REF c visualizes the weight changes in one middle layer using iteration-based normalization.",
"The experts find that a small number of filters are always more actively changed than the other filters (long deep blue lines in Fig.",
"REF c) in the later part of iterations.",
"This pattern implies that the updates inside a layer may be highly divergent.",
"[rgb]0,0,0In the later part of the training, where the learning rate is getting smaller and the model is converging, only a couple of filters are still continually actively updated for every iteration.",
"We have tried to inactivate these identified filters, the result showing that the overall performance is not affected.",
"It is not beyond the experts' expectation due to the ResNet's ensemble-like behavior [40] (even several entire layers can be removed without impacting performance).",
"In the end, the experts still cannot fully explain the behavior of these continually updating filters.",
"One possible reason could be that these special filters are not trained well (not converge) to extract some specific features, thus reacting violently for every iteration even in the later stages of the training."
],
[
"Exploring Filter-Image Correlations",
"In this scenario, we demonstrate how the experts use the Correlation View to explore correlations between images and filters.",
"Figure: A cube-style visualization that fuses three coordinated views together to reveal the rich dynamics in a CNN training process:(top) the Validation View shows the error rate changes of validation classes; (front) the Layer View shows the weight changes in CNNfilters; (right) the Correlation View shows the potential relationships between filters and validation classes.Shallow-layer filters vs. deep-layer filters.",
"[rgb]0,0,0At first, the experts choose to only show the top $k$ (100 in this case) changing filters in the layer view.",
"By checking the network structure visualization, the experts find that the activated shallow layers (the layers close to data input layer) are more than the activated deep layers, and most activated layers are the last basic CONV layers of bottlenecks for deep CONV modules.",
"Besides, Fig.",
"REF a shows that deep CONV modules tend to contain more anomaly filters (especially for the CONV modules 4).",
"The experts think that this kind of knowledge is of great importance for network compression [16].",
"Then, the experts go to the complicated version to examine the more detailed correlation information.",
"They filter the mini-sets with very few appearing times, finding that anomaly filters in shallow layers are generally shared by more anomaly classes (columns) and iterations (vertical lines in one column) than those in deep layers (Fig.",
"REF a).",
"The experts think that this pattern may relate to the fact [40] that shallow-layer filters are more likely to capture basic visual features than deep ones, thereby the huge change of these filters affecting more classes of images (e.g., the long and opaque lines marked by b in Fig.",
"REF ).",
"By contrast, a deep filter tends to learn higher-level features, thus only relating to specific classes of images.",
"To further explore the correlations, the experts select two mini-sets (b1 and b2 in Fig.",
"REF ), for comparison.",
"Both the horizontal lines of b1 and b2 are opaque and thick.",
"By tracking them in the Layer View and the Validation View, the experts can see that b1 is in the first CONV layer, and related to many classes.",
"The experts open these classes and discover that many images in them have a common feature, i.e., a large background of blue sky or ocean (b1 in Fig.",
"REF ).",
"This discovery suggests that these filters in E1 may target the basic pattern to help identify images that contain large blue areas.",
"By contrast, b2 is located at the fifth [rgb]0,0,0CONV module and related to only three classes.",
"Interestingly, the images in the three classes also share a more concrete feature, i.e., objects in a bush (b2 in Fig.",
"REF ).",
"[rgb]0,0,0In short, this case confirms that we are on the right track to reveal how the model weight changes relate to the classification result changes.",
"Important filters for a class.",
"To find stronger correlations between filters and classes, the expert focus on anomaly filters that appear more than once in a cell for a specific class.",
"For example, the experts find two appearances of the same mini-set (containing two anomaly filters) for the class of “gong” (c1 in Fig.",
"REF ).",
"Tracking horizontally (along with the pink-highlighted area), the experts find that the mini-set does not appear in other anomaly iterations, which also implies a strong correlation between filters in the mini-set and the class.",
"Then, the experts click on these two rectangular glyphs to highlight the corresponding iterations on the timeline (c2 in Fig.",
"REF ) and the filter locations in the Layer View (c3 in Fig.",
"REF ).",
"It is clear that the gong class is not a well trained class as it has a very large yellow area (indicate a relatively high error rate) in the Validation View.",
"However, the experts also find a period in the middle when this class has a relatively good performance (c4 in Fig.",
"REF ), which happens to contain the highlighted anomaly iterations.",
"Meanwhile, the Layer View shows that the highlighted filters are also updated dramatically during the period of good performance (c3 in Fig.",
"REF ).",
"Considering these patterns, the experts speculate that the filters in the mini-set have a strong impact on the classification of gong images.",
"As expected, we conduct experiments to inactivate these two filters, finding the overall performance and the performance on “gong” class are not impacted (see the reason in the last paragraph in Sec.",
"REF ).",
"Nevertheless, it provides the experts with a new manner to investigate the functions of filters co-working together for classifying one class of images.",
"That is, increase the threshold to find anomaly filters as many as possible, find the mini-sets containing many filters to some classes from multiple layers, and then inactivate them all to validate corresponding impacts.",
"Abnormal anomaly filters.",
"The experts are also attracted by two mini-sets (d1 and e1 in Fig.",
"REF ), because of their abnormal color patterns.",
"The filters in these two mini-sets exhibit large changes all the time in the latter part of the training, which are very different from the other anomaly filters.",
"Thus, the experts are interested in these filters and further check their correlated classes in the Correlation View (right).",
"Interestingly, each abnormal mini-set only appears with two classes (d2 and e2 in Fig.",
"REF ), and each pair of classes have very similar performances displayed in the Validation View (d3 and e3 in Fig.",
"REF ).",
"By checking the detailed images of these classes, the experts discover some common patterns.",
"For example, for mini-set e1, the corresponding classes are about mushrooms growing on grass and dogs playing on grass (e3 in Fig.",
"REF ).",
"For mini-set d1, the corresponding classes are related to curved shapes, such as parachutes and round textures (d3 in Fig.",
"REF ).",
"Although the experts are still unclear about why these two mini-sets have such a special behavior, they believe that these filters are likely to play important roles in identifying middle-level features such as grass and curved shapes.",
"We also conduct further experiments to validate the impact of inactivating these filters and the results are similar to the previous case (i.e., important filters for a class)."
],
[
"Expert Feedback",
"Usability [rgb]0,0,0DeepTracker is built upon a close collaboration with three domain experts, who constantly underscore their requirements and provide suggestions during the implementation process.",
"After several iterations of refinement, the experts were happy with the current version.",
"They all praised our way of effectively exploring such extreme large-scale training log via a hierarchical manner.",
"$\\mathrm {E}_a$ and $\\mathrm {E}_b$ mentioned that the well-designed validation and layer views were very intuitive and helped them greatly.",
"For example, the layer view allowing the experts to effectively observe and compare layer-related information (e.g., weight/gradient distribution) can help them diagnose network structures.",
"The detecting of dying-ReLU problem in the early stage of a training is useful for tuning the hyper-parameters (e.g., learning rate).",
"This kind of knowledge can also be leveraged to conduct model compression [16], so as to improve the model in respect to computing speed and memory cost.",
"Although the experts still cannot figure out the exact reason that some filters are always more actively updated in the later training stages, they believe the insight that would be obtained from the future investigation will be helpful in diagnosing and improving network structures.",
"Besides, the divergent evolving patterns of classes and the numerous anomaly iterations found in validation view provide the experts with a new promising direction to train a better model.",
"Both $\\mathrm {E}_a$ and $\\mathrm {E}_c$ were particularly fond of the cube-style visualization and deemed it as a new perspective to observe the training of CNNs for them.",
"They both have found many interesting patterns with the cube visualization, some of which were reasonable or could be explained after thinking for a while.",
"However, the experts also failed to figure out some other patterns, notwithstanding they conducted several testing experiments.",
"Nevertheless, they were still excited that our system can help them identify potential subjects for further study.",
"Generality During the implementation, we were concerned about the generality of DeepTracker, that is, whether the design was biased to the specific requirements from these three experts.",
"Therefore, to check how our system is accepted by broader expert communities, we presented our system in a workshop, which involved about 20 experts in the machine learning and visualization fields.",
"In addition, we also interviewed another group of twelve experts, who worked on a large project about using CNNs to improve image search quality.",
"We presented the latest version of DeepTracker to the experts, encouraged them to experiment with the system, and collected their feedback in the process.",
"Exceeding our expectation, DeepTracker was well accepted by these experts.",
"Although they proposed several new requirements, the experts shared many major interests with our three collaborators, such as tracking class level performance, filter-level information, and the relationships between them.",
"After introducing our system, they immediately understood the purposes of each view and all appreciated this novel, intuitive, and expressive way to watch training processes.",
"Although the demo was performed on our experiment datasets, the experts saw its potential in their project and immediately asked for collaboration with us, so that they could plug in and analyze their own datasets.",
"Improvement Apart from this positive feedback, the experts also made several interesting suggestions to further improve DeepTracker.",
"For example, two experts suggested that our current system only differentiates correct or incorrect classifications for validation images (i.e., 1 and 0).",
"However, the exact incorrect labels should also be presented because such information can help identify confusing or similar classes.",
"One expert mentioned that he showed strong interest on what happens before the anomaly iteration and suggested dump data of every iteration at that abnormal interval for fine-grained analysis.",
"Another expert suggested that our system shoud be integrated with online dashboards, as visualizing the training log on the fly can allow them to terminate the training early and save time if the visualization shows anything undesired."
],
[
"Discussion",
"DeepTracker is our first step to open the “black box” of CNN training.",
"Although the experts have high expectations of this tool, we all agree to start with two fundamental pieces of information: neuron weights and validation results.",
"Considering our target users and the large scale of datasets, we try to avoid using sophisticated visual encodings to ensure a fluent exploration experience.",
"Unsurprisingly, our bare-to-metal visualizations are preferred by the experts, and they use it to find many patterns easily, either expected or unexpected.",
"However, we still have several limitations.",
"First and foremost, although our system can effectively help experts identify strange or interesting patterns, there is still a gap between finding patterns and accelerating CNN training.",
"The experts still have to reason about and understand what these patterns mean or how to use them to accelerate model training in future.",
"We think it is not a problem faced just by our system, but by all CNN visualizations in general.",
"Like previous work, DeepTracker may only peel a hole in the box and reveal limited information.",
"But we hope that, by peeling enough holes, all these strange patterns will start to connect and make sense by themselves, thus providing a clear picture of CNNs.",
"[rgb]0,0,0Second, we have adopted many space-efficient visualizations and interaction techniques (e.g., hierarchy, filtering, and aggregation) to address the scalability issue.",
"Our current design can well support showing dozens of layers and classes in the same time.",
"The correlation view shares all the filter strategies with the other two views, and vice versa.",
"Thus, our system can perform well in most cases.",
"Nevertheless, the worst scenario still requires to display hundreds or thousands of small multiples at the same time.",
"A possible solution is to employ task-specific aggregation or filtering methods to show the data of interests.",
"Third, we propose a rule-based anomaly detection method that requires experts to manually pick a reasonable window size $k$ and set the threshold for filtering.",
"The number and patterns of anomalies are sensitive to these settings.",
"One potential solution to this problem is to develop an automatic method to enumerate all potential parameter settings and identify those can detect a reasonable amount of significant anomalies and provide these settings to the experts as guidance.",
"Finally, we only conduct experiments on ResNet-50 [17], but our method can also be applied to other state-of-the-art deep CNN models, which often have similar hierarchical structures (e.g., “inception block” in google-inception-v4 [37]).",
"[rgb]0,0,0 Besides, the cube visualization is a general technique, which can be used to explore multiple heterogeneous time series data and their complex correlations.",
"However, to further generalize it, a strict user study has to be conducted to find the best manner to use it, such as the axis skew degree, and the minimum height/width for each row/column in the three faces."
],
[
"Conclusion",
"We propose a novel visual analytics solution to disclose the rich dynamics of CNN training processes.",
"Knowing such information can help machine learning experts better understand, debug, and optimize CNNs.",
"We develop a comprehensive system that mainly comprises the validation, layer, and correlation views to facilitate an interactive exploration of the evolution of image classification results, network parameters, and the correlation between them.",
"We conduct experiments by training a very deep CNN (ResNet-50) on ImageNet, one of the largest labeled image datasets that is commonly used in practice, to demonstrate the applicability of our system.",
"The positive feedback from our collaborating experts and the experts from an internal workshop validates the usefulness and effectiveness of our system.",
"Future studies may integrate some feature-oriented visualization techniques, which typically require recording the activation information for input instances.",
"Feature visualizations can provide insights on what features a filter in a given snapshot of CNN has learned.",
"Our system can track critical iterations to take snapshots for a CNN over training, and then use feature visualization techniques to analyze the learned features evolving patterns for the detected important filters.",
"The other urgent need is to deploy the system in a real-time environment.",
"To this end, we have to consider some new design and interaction requirements to fill the gap between finding patterns and accelerating CNN training.",
"The authors would like to thank Kai Yan for providing support in editing the relevant media materials.",
"The work is supported by the National Basic Research Program of China (973 program) under Grant No.",
"2014CB340304 and ITC Grant with No.",
"UIT/138."
]
] | 1808.08531 |
[
[
"A versatile lattice Boltzmann model for immiscible ternary fluid flows"
],
[
"Abstract We propose a lattice Boltzmann color-gradient model for immiscible ternary fluid flows, which is applicable to the fluids with a full range of interfacial tensions, especially in near-critical and critical states.",
"An interfacial force for N-phase systems is derived based on the previously developed perturbation operator and is then introduced into the model using a body force scheme, which helps reduce spurious velocities.",
"A generalized recoloring algorithm is applied to produce phase segregation and ensure immiscibility of three different fluids, where a novel form of segregation parameters is proposed by considering the existence of Neumann's triangle and the effect of equilibrium contact angle in three-phase junction.",
"The proposed model is first validated with three typical examples, namely the interface capturing for two separate static droplets, the Young-Laplace test for a compound droplet, and the spreading of a droplet between two stratified fluids.",
"This model is then used to study the structure and stability of double droplets in a static matrix.",
"Consistent with the theoretical stability diagram, seven possible equilibrium morphologies are successfully reproduced by adjusting two ratios of the interfacial tensions.",
"By simulating Janus droplets in various geometric configurations, the model is shown to be accurate when three interfacial tensions satisfy a Neumann's triangle.",
"In addition, we also simulate the near-critical and critical states of double droplets where the outcomes are very sensitive to the model accuracy.",
"Our results show that the present model is advantageous to three-phase flow simulations, and allows for accurate simulation of near-critical and critical states."
],
[
"Introduction",
"An emulsion is a mixture of a dispersed phase as droplets in another immiscible fluid that forms a continuous phase.",
"Two basic types of emulsions are the oil-in-water (O/W) and water-in-oil (W/O) emulsions[1].",
"Recently, more complex systems referred to as double emulsions and Janus emulsions have received a rapidly growing interest due to their unique properties[2], [3], [4] and potential applications[5], [6], [7], [8], [9], [10].",
"Double emulsions, also known as `emulsion of emulsion' or `emulsion within emulsion', are emulsions with smaller droplets encapsulated in larger droplets.",
"The shell fluid can serve as a barrier between the core droplets and the outer environment, which makes double emulsions highly desirable for applications in controlled release, separation, and encapsulation[1], [2], [3], [4].",
"Janus emulsions, which are named after the two-faced Roman god Janus, are highly structured fluids consisting of emulsion droplets that have two distinct physical properties[11].",
"Because of their natural asymmetric ability in the compositions and the shapes, Janus emulsions are often used in the fields that need asymmetry in the shape and the materials.",
"In the applications of emulsions, morphology is one of the most important properties and closely related to other emulsion properties such as rheology, droplet size, relative stability, electrical conductivity and zeta potential[2], [3], [4], [12].",
"A number of theoretical and experimental studies have been devoted to identifying different equilibrium morphologies and their transformation.",
"For example, Torza and Mason[13] studied the droplet morphology in terms of spreading coefficients and obtained the theoretical relationship between the droplet morphology and spreading coefficients.",
"They experimentally observed three equilibrium morphologies of double droplets, i.e.",
"complete engulfing, partial engulfing and non-engulfing, which correspond to three sets of spreading coefficients.",
"Beyond these three equilibrium states, Pannacci et al.",
"[14] identified several new morphologies of double droplets, and found the non-equilibrium morphologies can have long lifetimes controlled by hydrodynamics, which facilitates the use of double droplets to produce encapsulated particles at early times and Janus particles at longer times.",
"Guzowski et al.",
"[15] presented a detailed theoretical analysis on the possible equilibrium morphologies of double droplets and designed the structure of double emulsions by tuning the volumes of the constituent segments experimentally.",
"As a supplement to theoretical and experimental studies, numerical modelling and simulations are becoming increasingly popular in investigation of the behavior of Janus/double emulsions, which are typical of three-phase flow problems.",
"Traditionally, three-phase flows are simulated by solving the macroscopic Navier-Stokes equations together with various approaches to capturing or tracking the interfaces between fluids.",
"Among these approaches, the front-tracking[16], volume-of-fluid (VOF)[17], level-set[18], [19], [20] and phase-field[21], [22], [23], [24], [25], [26] methods are commonly used.",
"However, the front-tracking method is not suitable for simulating interface breakup and coalescence; the VOF and level-set methods require either sophisticated interface reconstruction algorithms or unphysical re-initialization processes to represent the interfaces; and the phase-field method yields an interface thickness far greater than its actual value, which may lead to unphysical dissolution of small droplets and mobility-dependent numerical results[27].",
"It still remains an open question for the phase-field method to choose an optimal mobility, even for a two-phase flow problem[28].",
"In the past decades, the lattice Boltzmann (LB) method has developed into a promising alternative to the traditional Navier-Stokes-based solvers, for simulating complex flow problems.",
"It is a pseudo-molecular method tracking evolution of the distribution function of an assembly of molecules, built upon microscopic models and mesoscopic kinetic equations[29].",
"The LB method has several advantages over the traditional Navier-Stokes-based solvers, e.g.",
"the algorithm simplicity and parallelizability, and the ease of handling complex boundaries[30].",
"In addition, its kinetic nature allows a simple incorporation of microscopic physics without suffering from the limitations in terms of length and time scales typical of molecular dynamics simulations[31].",
"Thus, the LB method is particularly useful in the simulation of multiphase flows.",
"The existing LB models for multiphase flows can be generally classified into four categories: color-gradient model[32], [33], [28], interparticle-potential model[34], [35], [36], [37], [38], phase-field-based model[39], [40], [41], and mean-field theory model[42].",
"These models have shown great success as in dealing with two-phase flow problems, and all of them except the mean-field theory model have been extended to the modeling of immiscible ternary fluids, see, e.g.",
"Refs[43], [44], [45], [46], [47], [48], [49], [50], [51].",
"The ternary color-gradient models[49], [50] inherit a series of advantages of its two-phase counterpart, such as strict mass conservation for each fluid, flexibly tunable interfacial tensions, and the stability for a broad range of viscosity ratios, and they are well suited to exploring the dynamic processes occurring in ternary fluid systems as previously demonstrated by Fu et al.",
"[51] and Jiang et al.[52].",
"The existing color-gradient models, however, commonly suffer from a problem, i.e.",
"three interfacial tensions should satisfy a Neumann's triangle.",
"In industrial processes, surfactants are often added to emulsions to stabilize them against droplet coalescence.",
"The presence of surfactants could significantly modify the interfacial tensions so that the interfacial tensions do not always yield a Neumann's triangle.",
"To correctly predict the dynamical behavior of emulsions, thereby allowing precise control over the droplet geometry and composition, it is necessary for a numerical model to be capable of simulating ternary fluids with a full range of interfacial tensions.",
"On the other hand, it is challenging to simulate the near-critical and critical states of a ternary fluid system where the largest interfacial tension is close to the sum of the other two, as the outcomes are very sensitive to the model accuracy.",
"In this paper, we develop a LB color-gradient model for simulating immiscible ternary fluids with a full range of interfacial tensions.",
"Based on the perturbation operator developed by Leclaire et al.",
"[50], an interfacial force formulation is derived to describe the interactions among different fluids and is then introduced into the model using a body force scheme, which is found to effectively reduce spurious velocities.",
"In addition, the recoloring algorithm proposed by Spencer et al.",
"[49] is applied to maintain the interfaces and ensure immiscibility of three different fluids, where a new form of segregation parameters is proposed by considering both the existence of Neumann's triangle and the effect of equilibrium contact angle in three-phase junction.",
"The capability and accuracy of this model are first assessed by simulating the interface capturing for two separate static droplets, the Young-Laplace test for a compound droplet, and the spreading of a droplet between two stratified fluids.",
"It is then used to study the structure and stability of double droplets in a static matrix fluid, where we emphasize the model's capability for simulating ternary fluid flows in near-critical and critical states."
],
[
"Numerical method",
"The two-phase color-gradient LB model of Liu et al.",
"[28], [53] is extended to the simulation of immiscible ternary fluids.",
"The ternary color-gradient model consists of three steps, i.e.",
"the collision step, the recoloring step and the streaming step.",
"In the collision step, an interfacial force that describes the interactions among different fluids is derived from the perturbation operator presented in Leclaire et al.",
"[50], and is then introduced by the body force scheme of Guo et al.",
"[54] In the recoloring step, a novel form of segregation parameters is proposed to ensure accurate phase segregation in three-phase junction and allow for the states where three interfacial tensions between the fluids cannot form a triangle, known as the Neumann's triangle.",
"The distribution functions $f_{i,r}$ , $f_{i,g}$ and $f_{i,b}$ are introduced to represent three immiscible fluids, i.e.",
"red fluid, green fluid and blue fluid, where the subscript $i$ is the lattice velocity direction and ranges from 0 to ($n$ -1) for a given $m$ -dimensional D$m$ Q$n$ lattice model.",
"The total distribution function is defined as $f_{i}=\\sum _{k}f_{i,k}$ ($k=r$ , $g$ or $b$ ), which undergoes a collision step as $ f_{i}^{\\dag }\\left(\\mathbf {x},t\\right)=f_{i}\\left(\\mathbf {x},t\\right)+\\Omega _{i}\\left(\\mathbf {x},t\\right)+\\Phi _i\\left(\\mathbf {x},t\\right),$ where $f_i \\left( {{\\mathbf {x}},t} \\right)$ is the total distribution function in the $i$ -th velocity direction at the position $\\mathbf {x}$ and the time $t$ , $f_i^\\dagger $ is the post-collision distribution function, $\\Omega _i$ is the Bhatnagar-Gross-Krook (BGK) collision operator, and $\\Phi _i$ is the forcing term (also known as perturbation operator), which contributes to the mixed interfacial regions and creates the interfacial tensions between different fluids.",
"In the BGK collision operator, the total distribution functions are relaxed toward a local equilibrium with a single relaxation time: $\\Omega _i({\\mathbf {x}},t)=-\\frac{1}{\\tau _{f}}\\left[f_{i}(\\mathbf {x},t)-f_{i}^{eq}(\\mathbf {x},t)\\right],$ where $\\tau _f$ is the dimensionless relaxation time, and $f_i^{eq}$ is the equilibrium distribution function of $f_i$ .",
"The equilibrium distribution function is obtained by a second order Taylor expansion of Maxwell-Boltzmann distribution with respect to the local fluid velocity ${\\mathbf {u}}$ : $f_{i}^{eq}=w_{i}\\rho \\left[1+\\frac{\\mathbf {e}_{i}\\cdot \\mathbf {u}}{c_{s}^{2}}+\\frac{\\left(\\mathbf {e}_{i}\\cdot \\mathbf {u}\\right)^{2}}{2c_{s}^{4}}-\\frac{\\mathbf {u}\\cdot \\mathbf {u}}{2c_{s}^{2}}\\right],$ where $\\rho =\\sum _k \\rho _k$ is the total density and $\\rho _k$ is the density of the fluid $k$ ; $c_s$ is the speed of sound; ${\\mathbf {e}}_i$ is the lattice velocity in the $i$ -th direction; and $w_i$ is the weighting factor.",
"For the two-dimensional nine-velocity (D2Q9) model, $\\mathbf {e}_{i}$ is defined as $\\mathbf {e}_0=(0,0)$ , $\\mathbf {e}_{1,3}=(\\pm c,0)$ , $\\mathbf {e}_{2,4}=(0,\\pm c)$ , $\\mathbf {e}_{5,7}=(\\pm c,\\pm c)$ , and $\\mathbf {e}_{6,8}=(\\mp c,\\pm c)$ , where $c=\\delta _x/\\delta _t=\\sqrt{3}c_s$ with $\\delta _x$ and $\\delta _t$ being the lattice spacing and time step, respectively (for the sake of simplicity, $\\delta _x=\\delta _t=1$ is used hereafter); $w_i$ is given by $w_0 = 4/9$ , $w_{1-4} = 1/9$ and $w_{5-8} = 1/36$ .",
"Using the concept of a continuum surface force to model the interfacial tension along with the constraints of mass conservation and momentum conservation, Liu et al.",
"[28] derived a generalized expression for the perturbation operator in two-phase simulations.",
"This perturbation operator was later improved by Leclaire et al.",
"[50] to model the interfacial tensions between different fluids in three-phase simulations.",
"Following Leclaire et al.",
"[50], the perturbation operator is given by $\\Phi _i &=& \\sum _k \\Phi _{i,k}, \\\\\\Phi _{i,k} &=& \\sum _{l,l\\ne {k}}\\frac{A_{kl}C_{kl}}{2}\\left|\\mathbf {G}_{kl}\\right|\\left[w_{i}\\frac{\\left(\\mathbf {e}_{i}\\cdot \\mathbf {G}_{kl}\\right)^{2}}{\\left|\\mathbf {G}_{kl}\\right|^{2}}-B_{i}\\right],$ where $\\mathbf {G}_{kl}=\\frac{\\rho _{l}}{\\rho }\\mathbf {\\nabla }\\frac{\\rho _{k}}{\\rho }-\\frac{\\rho _{k}}{\\rho }\\mathbf {\\nabla }\\frac{\\rho _{l}}{\\rho }$ is the color gradient [50] and is introduced to identify the location of the $k$ -$l$ interface, i.e.",
"the interface between the fluid $k$ and the fluid $l$ .",
"$C_{kl}$ is a concentration factor that controls the activation of the interfacial tension at the $k$ -$l$ interface, and is given by [50] $C_{kl}=\\min \\left(10^{6}\\frac{\\rho _{k}\\rho _{l}}{\\rho _{k}^{0}\\rho _{l}^{0}},1\\right),$ where $\\rho _{k}^{0}$ is the density of the pure fluid $k$ , and $A_{kl}$ is a parameter related to the interfacial tension between the fluids $k$ and $l$ , i.e.",
"$\\sigma _{kl}=\\frac{1}{9}\\left(A_{kl}+A_{lk}\\right)\\tau _f$ .",
"The generalized expression for $B_i$ was given by Liu et al.",
"[28] and it was in particular taken as $B_0=-4/27$ , $B_{1-4}=2/27$ and $B_{5-8}=5/108$ in the work of Leclaire et al.[50].",
"It is worth noting that Eq.",
"() is not limited to the case with ternary fluids, and can be also applicable to $N$ -phase ($N>3$ ) systems.",
"Using the Chapman-Enskog multiscale analysis, it is shown that the perturbation operator, given by Eqs.",
"(REF ) and (), can lead to the following interfacial force: $\\mathbf {F}_s=-\\nabla \\cdot \\left(\\tau _{f}\\delta _{t}\\sum _{i}{\\Phi _{i}\\mathbf {e}_{i}\\mathbf {e}_{i}}\\right)=\\sum _k\\sum _{l,l\\ne k}\\nabla \\cdot \\left[\\frac{\\sigma _{kl}C_{kl}}{2}\\left|\\mathbf {G}_{kl}\\right|\\left(\\mathbf {I}-\\mathbf {n}_{kl}\\mathbf {n}_{kl}\\right)\\right],$ where $\\mathbf {n}_{kl}$ is the unit normal vector of the $k$ -$l$ interface and is defined by $\\mathbf {n}_{kl}=\\mathbf {G}_{kl}/\\left|\\mathbf {G}_{kl}\\right|$ .",
"Instead of using Eqs.",
"(REF ) and (), the effect of interfacial tension is realized through the body force scheme of Guo et al.",
"[54], which is able to reduce effectively spurious velocities while keeping high numerical accuracy [53], [55].",
"According to Guo et al.",
"[54], the forcing term $\\Phi _{i}$ in Eq.",
"(REF ) is written as $\\Phi _{i}\\left(\\mathbf {x},t\\right)=w_{i}\\left(1-\\frac{1}{2\\tau _{f}}\\right)\\left(\\frac{\\mathbf {e}_{i}-\\mathbf {u}}{c_{s}^{2}}+\\frac{\\mathbf {e}_{i}\\cdot \\mathbf {u}}{c_{s}^{4}}\\mathbf {e}_{i}\\right)\\cdot \\mathbf {F}_s\\left(\\mathbf {x},t\\right)\\delta _{t},$ where the local fluid velocity is defined by the averaged momentum before and after the collision, i.e., $\\rho {\\mathbf {u}}({\\mathbf {x}},t)=\\sum _i f_i({\\mathbf {x}},t){\\mathbf {e}}_i+\\frac{1}{2}{\\mathbf {F}}_s({\\mathbf {x}},t)\\delta _t.$ In this work, we assume equal densities for the red, green and blue fluids.",
"To allow for unequal viscosities of the three fluids, we determine the local kinematic viscosity $\\nu $ by a harmonic mean $\\frac{\\rho }{\\nu } = \\sum _k \\frac{\\rho _k}{\\nu _k},$ where $\\nu _k$ ($k=R$ , $G$ or $B$ ) is the kinematic viscosity of the fluid $k$ .",
"The local relaxation time $\\tau _f$ can be calculated from the local viscosity using the following equation: $\\nu = \\left(\\tau _f-\\frac{1}{2}\\right)c_s^2\\delta _t.$ The partial derivatives in the interfacial force ${\\mathbf {F}}_s$ should be evaluated through suitable difference schemes.",
"To minimize the discretization errors, the fourth-order isotropic finite difference scheme $\\partial _{\\alpha } \\varphi \\left( {\\bf {x}}, t\\right) =\\frac{1}{c_s^2}\\sum _i w_i \\varphi \\left( {\\bf {x}}+{\\bf {e}}_i\\delta _t, t\\right)e_{i\\alpha },$ is used to evaluate the derivatives of a variable $\\varphi $ .",
"Although the forcing term generates the interfacial tensions, it does not guarantee the immiscibility of different fluids.",
"In order to minimize the mixing of the fluids, a recoloring step is applied.",
"Based on the pioneering work of D'Ortona et al.",
"[56], Latva-Kokko and Rothman[57] developed a recoloring algorithm to demix two immiscible fluids, which can overcome the lattice pinning problem and creates a symmetric distribution of particles around the interface so that unphysical spurious velocities can be effectively reduced.",
"This recoloring algorithm was later generalized by Spencer et al.",
"[49] to three-phase fluid flows.",
"Following Spencer et al.",
"[49], the recolored distribution functions of the fluid $k$ ($k=r$ , $g$ or $b$ ) are $f_{i,k}^{\\ddagger }\\left(\\mathbf {x},t\\right)=\\frac{\\rho _{k}}{\\rho }f_{i}^{\\dagger }\\left(\\mathbf {x},t\\right)+\\sum \\limits _{l,l\\ne {k}}\\beta _{kl}w_{i}\\frac{\\rho _{k}\\rho _{l}}{\\rho }\\mathbf {n}_{kl}\\cdot \\mathbf {e}_{i},$ where $f_{i,k}^{\\ddagger }$ is the recolored distribution functions of the fluid $k$ , and $\\beta _{kl}$ is a segregation parameter related to the thickness of the $k$ -$l$ interface.",
"It should be noted that $\\beta _{kl}=\\beta _{lk}$ in order to conserve mass and momentum during the recoloring process.",
"Figure: Neumann's triangleFor the ternary fluids and when three interfacial tensions satisfy a Neumann's triangle (see Fig.",
"REF ), the equilibrium contact angle $\\varphi _{kl}$ will be formed between the fluids in three-phase junction, and it is related to the interfacial tensions by $\\cos (\\varphi _{kl})=\\frac{\\sigma _{mk}^{2}+\\sigma _{ml}^{2}-\\sigma _{kl}^{2}}{2\\sigma _{mk}\\sigma _{ml}}.$ Spencer et al.",
"[49] theoretically showed that in three-phase junction, there should be a relationship between $\\varphi _{kl}$ and the (relative) interface thickness, which is controlled by the segregation parameter $\\beta _{kl}$ .",
"Hence, it is of great importance to select a proper $\\beta _{kl}$ in three-phase simulations.",
"Several different forms of $\\beta _{kl}$ have been provided in literature.",
"Spencer et al.",
"[49] proposed the first expression for the segregation parameters, which is given by $\\left\\lbrace \\begin{aligned}\\beta _{rg}=&\\beta ^{0} \\\\\\beta _{rb}=&\\beta ^{0}\\left[1+\\frac{27\\rho _{r}\\rho _{g}\\rho _{b}}{\\rho ^{3}}\\left(\\sin {\\varphi _{gb}}-1\\right)\\right] \\\\\\beta _{gb}=&\\beta ^{0}\\left[1+\\frac{27\\rho _{r}\\rho _{g}\\rho _{b}}{\\rho ^{3}}\\left(\\sin {\\varphi _{rb}}-1\\right)\\right]\\end{aligned}\\right.,$ where $\\beta ^{0}$ is the reference segregation parameter.",
"Clearly, the segregation parameters in Eq.",
"(REF ) will degenerate into $\\beta _{kl}=\\beta ^{0}$ at an interface where only two fluids are present.",
"So it is suggested to take $\\beta ^{0}=0.7$ to be consistent with the segregation parameter in the two-phase color-gradient model[28].",
"Leclaire et al.",
"[50] improved the segregation parameters of Spencer et al.",
"[49] by setting $\\beta _{kl}=\\beta ^0$ for the largest $\\varphi _{kl}$ in the Neumann's triangle, $\\beta _{kl}=\\left\\lbrace \\begin{aligned}\\beta ^{0}& &\\text{$kl$ with $\\varphi _{max}$} \\\\\\beta ^{0}&+\\beta ^{0}C_{t}\\left[\\sin {\\left(\\pi -\\varphi _{max}-\\varphi _{kl}\\right)}-1\\right] &\\text{otherwise}\\end{aligned}\\right.,$ where $\\varphi _{max}=\\max (\\varphi _{kl})$ and $C_{t}=\\min \\left(\\frac{35\\rho _{r}\\rho _{g}\\rho _{b}}{\\rho ^{3}},1\\right)$ .",
"Leclaire et al.",
"[50] also mentioned to use $\\beta _{kl}=\\beta ^{0}$ when the Neumann's triangle does not exist.",
"Clearly, Eq.",
"(REF ) will degradate to Eq.",
"(REF ) when $\\varphi _{max}=\\varphi _{rg}$ .",
"Althogh Eqs.",
"(REF ) and (REF ) work to some extent especially when the Neumann's triangle exists, they cannot accurately simulate the critical state where the largest interfacial tension equals the sum of the other two, which will be shown later.",
"Recently, Fu et al.",
"[51] seemed to have also noticed that Eqs.",
"(REF ) and (REF ) do not always produce convincing results in three-phase simulations, so they simply selected a constant $\\beta _{kl}$ , i.e.",
"$\\beta _{kl}=\\beta ^{0}.$ It is evident that the dependence of $\\beta _{kl}$ on $\\varphi _{kl}$ is not considered in Eq.",
"(REF ), and thus incorrect results may be obtained, e.g.",
"in the critical state.",
"To overcome the aforementioned drawbacks associated with the existing $\\beta _{kl}$ , a novel form of segregation parameters is proposed.",
"First, we determine whether the Neumann's triangle exists by calculating $X_{kl}=\\frac{\\sigma _{mk}^{2}+\\sigma _{ml}^{2}-\\sigma _{kl}^{2}}{2\\sigma _{mk}\\sigma _{ml}}.$ It is easily seen from Eq.",
"(REF ) that the Neumann's triangle will exist if $\\left|X_{kl}\\right|<1$ for all $kl$ .",
"Then, the segregation parameter $\\beta _{kl}$ is defined as a continuous function of $X_{kl}$ : $\\beta _{kl}=\\beta ^{0}+\\beta ^{0}\\min \\left(\\frac{35\\rho _{r}\\rho _{g}\\rho _{b}}{\\rho ^{3}},1\\right)g\\left(X_{kl}\\right),$ where $g\\left(X_{kl}\\right)=\\left\\lbrace \\begin{aligned}&1 & X_{kl}<-1\\\\&1-\\sin {\\left(\\arccos \\left(X_{kl}\\right)\\right)} & -1\\le X_{kl}< 0 \\\\&\\sin {\\left(\\arccos \\left(X_{kl}\\right)\\right)}-1 & 0\\le X_{kl}\\le 1 \\\\&-1 & 1<X_{kl}\\end{aligned}\\right..$ It should be noted in three-phase junction that Eqs.",
"(REF ) and (REF ) are derived based on the following relationship: $\\frac{\\beta _{rg}}{\\sin (\\varphi _{rg})}=\\frac{\\beta _{rb}}{\\sin (\\varphi _{rb})}=\\frac{\\beta _{gb}}{\\sin (\\varphi _{gb})},$ which is consistent with the nature of diffuse interfaces, thus leading to more accurate results than using other forms of $\\beta _{kl}$ .",
"Moreover, the proposed $\\beta _{kl}$ works well no matter if the Neumann's triangle exists or not.",
"After the recoloring step, the red, green and blue distribution functions propagate to the neighboring lattice nodes, known as the propagation or streaming step: $f_{i,k}\\left(\\mathbf {x}+\\mathbf {e}_{i}\\delta _{t},t+\\delta _{t}\\right)=f_{i,k}^{\\ddagger }\\left(\\mathbf {x},t\\right), \\quad k=\\lbrace r,g,b\\rbrace $ with the post-propagation distribution functions used to compute the densities of colored fluids by $\\rho _k=\\sum _i f_{i,k}$ ."
],
[
"Interface capturing",
"We first consider two separate static droplets immersed in another fluid (say blue fluid) to validate the present model for interface capturing.",
"Initially, a red droplet and a green droplet, both having equal radius $R=20$ , are placed in a $200\\times 100$ lattice domain, and their centers are located at $(x_r,y_r)=(50,50)$ and $(x_g,y_g)=(150,50)$ , respectively.",
"Considering the distance between two droplets, each droplet interface is essentially a two-phase region, so the equilibrium density distributions at $y=50$ can be analytically expressed as[58] $\\frac{\\rho _{r}}{\\rho }\\left(x\\right)=0.5+0.5\\tanh \\left[\\frac{R-\\sqrt{(x-x_r)^{2}}}{\\xi }\\right],$ $\\frac{\\rho _{g}}{\\rho }\\left(x\\right)=0.5+0.5\\tanh \\left[\\frac{R-\\sqrt{(x-x_g)^{2}}}{\\xi }\\right],$ $\\frac{\\rho _{b}}{\\rho }\\left(x\\right)=1-\\frac{\\rho _{r}}{\\rho }\\left(x\\right)-\\frac{\\rho _{g}}{\\rho }\\left(x\\right),$ for the red, green and blue fluids, respectively.",
"Here, the parameter $\\xi $ is a measure of the interface thickness related to $\\beta ^{0}$ by $\\xi =1/(6k\\beta ^{0})$ [59], and $k$ is a geometric constant that is determined by [58] $k=\\frac{1}{2}\\sum _{i}{\\frac{w_{i}\\mathbf {e}_{i}\\mathbf {e}_{i}}{\\left|\\mathbf {e}_{i}\\right|}}.$ For the D2Q9 model, one can obtain $k\\approx 0.1504$ from Eq.",
"(REF ), and thus $\\xi \\approx 1.5831$ for $\\beta ^{0}=0.7$ .",
"The simulation is run with the interfacial tensions $\\sigma _{rg}=\\sigma _{rb}=\\sigma _{gb}=0.01$ and the viscosities $\\nu _r=\\nu _g=\\nu _b=0.1$ .",
"Periodic boundary conditions are applied in both the $x$ and $y$ directions.",
"Fig.",
"REF shows the simulated density distributions of the red, green and blue fluids along $y=50$ in the steady state, and the corresponding analytical solutions, given by Eq.",
"(REF ), are also shown for comparison.",
"Clearly, the simulated density distributions are all in good agreement with the analytical solutions, indicating that the present color-gradient LBM can correctly model and capture phase interfaces.",
"Figure: The equilibrium density distributions of three different fluids for two separate static droplets immersed in a third fluid."
],
[
"Young-Laplace test",
"A compound droplet, which consists of an inner droplet encapsulated by another immiscible fluid, suspended in a third fluid, is simulated to assess whether the interfacial tensions are correctly modelled.",
"The computational domain is taken as $160\\times 160$ , and it is filled with three different fluids, which are initialized as $\\left\\lbrace \\begin{aligned}\\rho _r=1, \\quad \\rho _g=\\rho _b=0 & & (x-80)^2+(y-80)^2\\le R_r^2 \\\\\\rho _g=1, \\quad \\rho _r=\\rho _b=0 && \\quad R_r^2<(x-80)^2+(y-80)^2\\le R_g^2 \\\\\\rho _b=1, \\quad \\rho _r=\\rho _g=0 && \\text{otherwise}\\end{aligned}\\right.$ with $R_g=2R_r$ .",
"This gives the initial condition that a compound droplet is located in the center of the computational domain.",
"The interfacial tensions and the fluid viscosities are all kept the same as those used in Section REF , and the periodic boundary conditions are used in both $x$ and $y$ directions.",
"According to the Young-Laplace's law, when the system reaches the equilibrium state, the pressure difference $\\Delta p$ across an interface is related to the interfacial tension $\\sigma $ by $\\Delta p=\\frac{\\sigma }{R},$ where $R$ is the radius of the interface curvature.",
"Eq.",
"(REF ) allows us to quantify the modeling accuracy of interfacial tensions through the relative error $\\epsilon =\\frac{\\left|\\Delta {p_{gb}}R_{g}+\\Delta {p_{rg}}R_{r}-\\left(\\sigma _{gb}+\\sigma _{rg}\\right)\\right|}{\\sigma _{gb}+\\sigma _{rg}}\\times 100\\%.$ Table: The relative errors of interfacial tensions for various values of R r R_r.Table: The maximum spurious velocities (𝐮 max \\left|\\mathbf {u}\\right|_{max}) obtained with two different forcing methods for various R r R_r.Table REF shows the relative errors of interfacial tensions for different values of $R_r$ .",
"All the relative errors $\\epsilon $ are below $1.5\\%$ , suggesting that our LBM results are in excellent agreement with the Young-Laplace's law.",
"In addition to the present forcing method, i.e.",
"Eqs.",
"(REF ) and (REF ), the interfacial tension effects can also be realized by the forcing method of Leclaire et al.",
"[50], i.e.",
"Eqs.",
"(REF ) and ().",
"It is of interest to compare the effect of these two different forcing methods on spurious velocities.",
"Table REF shows the maximum spurious velocities ($\\left|\\mathbf {u}\\right|_{max}$ ) for various $R_r$ , where the values of $\\left|\\mathbf {u}\\right|_{max}$ are magnified by $10^5$ times.",
"It is seen that the maximum spurious velocities are almost independent of $R_r$ for either forcing method, and that the present spurious velocities are always smaller than those obtained with the forcing method of Leclaire et al.",
"[50]."
],
[
"Spreading of a droplet between two stratified fluids",
"To assess the overall performance of the proposed model, we simulate the spreading of a droplet between two other immiscible fluids.",
"The computational domain is set to be $160\\times 160$ lattices.",
"Initially, a red circular droplet with the radius $R=20$ is placed in the center of the computational domain, and the green and blue fluids are allocated to the lower and upper halves of the computational domain outside the droplet.",
"Periodic boundary conditions are used in both the $x$ and $y$ directions.",
"Depending on the values of the interfacial tensions, two different spreading phenomena can be observed, i.e.",
"partial spreading and complete spreading.",
"Figure: The shape of a liquid lens at equilibrium.We first consider the partial spreading, where three interfacial tensions yield a Neumann's triangle.",
"In a partial spreading, the droplet can eventually reach a steady lens shape, which is often characterized by the lens length $D$ and the heights $h_1$ and $h_2$ (see Fig.",
"REF ).",
"The lens length and heights can be analytically given as [60], [52] $D=2\\sqrt{\\frac{A}{\\sum \\limits _{i=1}^{2}{\\frac{1}{\\sin \\theta _{i}}\\left(\\frac{\\theta _{i}}{\\sin \\theta _{i}}-\\cos \\theta _{i}\\right)}}},$ $h_{i}=\\frac{D}{2}\\left(\\frac{1-\\cos \\theta _{i}}{\\sin \\theta _{i}}\\right)\\quad \\text{with}\\quad i=1,2,$ where $A$ is the area of the red droplet; $\\theta _{1}=\\varphi _{rg}$ and $\\theta _{2}=\\varphi _{rb}$ are the equilibrium contact angles that can be calculated from Eq.",
"(REF ).",
"Four groups of interfacial tensions are simulated with a constant $\\sigma _{gb}$ of $0.01$ but varying $\\sigma _{rb}$ and $\\sigma _{rg}$ , i.e., (a) $\\sigma _{rb}=0.01$ and $\\sigma _{rg}=0.01$ , (b) $\\sigma _{rb}=0.0087$ and $\\sigma _{rg}=0.005$ , (c) $\\sigma _{rb}=0.0173$ and $\\sigma _{rg}=0.02$ , (d) $\\sigma _{rb}=0.0058$ and $\\sigma _{rg}=0.0115$ .",
"The fluid viscosities are all kept at $0.1$ , and the final fluid distributions are shown in Fig.",
"REF .",
"As expected, the droplet exhibits a lens shape in each of the cases considered, and the geometrical sizes ($D$ , $h_1$ and $h_2$ ) of the lens are case dependent.",
"Based on the fluid distributions, we also quantify the geometrical sizes of the lens, and compare the simulated results with the analytical predictions from Eq.",
"(REF ).",
"It is seen in Table REF that the simulated results (denoted by $D^{s}$ , $h_1^{s}$ and $h_2^{s}$ ) agree well with the analytical predictions (denoted by $D^{a}$ , $h_1^{a}$ and $h_2^{a}$ ) with the relative errors (defined by $E(\\chi )=\\frac{|\\chi ^{a}-\\chi ^{s}|}{\\chi ^{a}}\\times 100\\%$ , where $\\chi =D$ , $h_1$ or $h_2$ ) all around $1\\%$ except in the cases of small contact angles.",
"The increased errors at small contact angles are attributed to the low resolution in sharp corners, which were also found by Jiang and Tsuji[52].",
"Figure: Final fluid distributions in the cases of partial spreading for (a) σ rb =0.01\\sigma _{rb}=0.01, σ rg =0.01\\sigma _{rg}=0.01; (b) σ rb =0.0087\\sigma _{rb}=0.0087, σ rg =0.005\\sigma _{rg}=0.005; (c) σ rb =0.02\\sigma _{rb}=0.02, σ rg =0.0173\\sigma _{rg}=0.0173; (d) σ rb =0.0115\\sigma _{rb}=0.0115, σ rg =0.0058\\sigma _{rg}=0.0058.",
"The third interfacial tension is fixed at σ gb =0.01\\sigma _{gb}=0.01.Table: Comparison between the analytical predictions and simulated results for the geometrical sizes of the deformed droplet.We then consider the complete spreading, where three interfacial tensions cannot yield a Neumann's triangle.",
"Two different cases of complete spreading are simulated for $\\sigma _{rg}=0.01$ and $\\sigma _{rg}=0.015$ at $\\sigma _{gb}=\\sigma _{rb}=0.005$ .",
"Clearly, $\\sigma _{rg}=\\sigma _{gb}+\\sigma _{rb}$ in the first case, which corresponds to the critical state; whereas $\\sigma _{rg}>\\sigma _{gb}+\\sigma _{rb}$ in the second case, which corresponds to the supercritical state.",
"Fig.",
"REF shows the time evolution of the interface in both cases for a constant fluid viscosity of $0.05$ .",
"We can see that in the critical state, the red droplet sits exactly on the $gb$ interface in the end; whereas in the supercritical state, it bounces off the $gb$ interface and rises up to the blue fluid.",
"Figure: Time evolution of the interface in the cases of complete spreading for (a) σ rg =0.01\\sigma _{rg}=0.01 and (b) σ rg =0.015\\sigma _{rg}=0.015.",
"The other two interfacial tensions are fixed at σ gb =σ rb =0.005\\sigma _{gb}=\\sigma _{rb}=0.005.",
"Note that the system has reached the steady state at t=50000t=50000 in each case."
],
[
"Structure and stability of double droplets",
"Double emulsions have received considerable attention because of their potential applications in food science, cosmetics, pharmaceuticals and medical diagnostics.",
"Since emulsion properties and functions are related to the droplet geometry and composition, it is of great importance, from a numerical point of view, to accurately predict the topological structure of double droplets when dispersed in another immiscible fluid."
],
[
"Stability diagram for double droplets",
"Consider a pair of equal-sized droplets, consisting of red and green fluids and initially sitting next to each other, immersed in the third fluid (blue fluid).",
"Based on the theoretical analysis, Guzowski et al.",
"[15] presented a stability diagram that describes the possible topologies of double droplets and their transitions in terms of two ratios of the interfacial tensions (see the left panel of Fig.",
"REF ).",
"In the stability diagram, seven typical cases (represented by the solid points) are simulated to examine if the present model is able to reproduce the correct morphologies of double droplets.",
"These typical cases are (i) $\\frac{\\sigma _{gb}}{\\sigma _{rg}}=1.7$ and $\\frac{\\sigma _{rb}}{\\sigma _{rg}}=0.5$ , (ii) $\\frac{\\sigma _{gb}}{\\sigma _{rg}}=2$ and $\\frac{\\sigma _{rb}}{\\sigma _{rg}}=1$ , (iii) $\\frac{\\sigma _{gb}}{\\sigma _{rg}}=0.4$ and $\\frac{\\sigma _{rb}}{\\sigma _{rg}}=0.4$ , (iv) $\\frac{\\sigma _{gb}}{\\sigma _{rg}}=0.5$ and $\\frac{\\sigma _{rb}}{\\sigma _{rg}}=0.5$ , (v) $\\frac{\\sigma _{gb}}{\\sigma _{rg}}=1$ and $\\frac{\\sigma _{rb}}{\\sigma _{rg}}=1$ , (vi) $\\frac{\\sigma _{gb}}{\\sigma _{rg}}=1$ and $\\frac{\\sigma _{rb}}{\\sigma _{rg}}=2$ , and (vii) $\\frac{\\sigma _{gb}}{\\sigma _{rg}}=0.5$ and $\\frac{\\sigma _{rb}}{\\sigma _{rg}}=1.7$ , which cover all the possible morphologies identified by Guzowski et al.[15].",
"Figure: Stability diagram representing possible morphologies of double droplets (left panel) and equilibrium shapes of the droplets for the typical cases marked in the stability diagram (right panel).",
"The red lines represent the critical morphologies or the transitions between the regions of complete engulfing, partial engulfing and non-engulfing.The computational domain is taken to be $[1,120]\\times [1,120]$ , and the initial fluid distributions are $&\\rho _{r}(x,y)=0.5+0.5\\tanh \\left[\\frac{R-\\sqrt{(x-60.5)^{2}+(y-60.5-R)^{2}}}{\\xi }\\right], \\\\&\\rho _{g}(x,y)=0.5+0.5\\tanh \\left[\\frac{R-\\sqrt{(x-60.5)^{2}+(y-60.5+R)^{2}}}{\\xi }\\right], \\\\&\\rho _{b}(x,y)=1-\\rho _{r}(x,y)-\\rho _{g}(x,y),$ where the droplet radius $R=20$ lattices.",
"The periodic boundary conditions are used in both the $x$ and $y$ directions.",
"All the fluids are assumed to have equal viscosity of $0.1$ , and the interfacial tension $\\sigma _{rg}$ is fixed at $0.01$ .",
"The simulations are run until an equilibrium state is reached, and the equilibrium morphologies of double droplets for the seven cases are depicted in the right panel of Fig.",
"REF .",
"It is seen that seven different equilibrium morphologies are exhibited and they can be described as complete engulfing of green fluid by red fluid (i), critical engulfing of green fluid by red fluid (ii), separate dispersion or non-engulfing (iii), kissing (iv), partial engulfing (v), critical engulfing of red fluid by green fluid (vi), and complete engulfing of red fluid by green fluid (vii).",
"These simulation results are consistent with the theoretical predictions by Guzowski et al.",
"[15]."
],
[
"Janus droplet",
"Among the seven morphologies shown in Fig.",
"REF , the double droplets with partial engulfing morphology are often referred to as the Janus droplet.",
"When the interfacial tension between the constituent fluids is negligibly small, the Janus droplet forms a perfect circle, which is known as perfect Janus droplet (PJD) [15].",
"Differentiating from the PJD, the Janus droplet that does not exhibit a perfect circle is called as the general Janus droplet (GJD).",
"Figure: Equilibrium geometry and force balance at a three-phase junction for (a) a general Janus droplet (GJD) and (b) a perfect Janus droplet (PJD).A Janus droplet, consisting of red and green fluids, is immersed in a static blue fluid.",
"Fig.",
"REF shows the equilibrium geometries of a GJD and a PJD, as well as the corresponding force balances at three-phase junctions.",
"In this figure, $R_{r}$ , $R_{g}$ and $R_{b}$ are the curvature radii of the $rb$ , $gb$ and $rg$ interfaces respectively ($kl$ interface refers to the interface between fluid $k$ and fluid $l$ ); $\\theta _{r}$ , $\\theta _{g}$ and $\\theta _{b}$ are the half of the central angles subtended by the chord $AB$ ; and $d_{rg}$ ($d_{gb}$ ) is the distance between the centers $O_{g}$ and $O_{r}$ ($O_{b}$ ).",
"For a GJD, provided that four independent geometric parameters, e.g.",
"$R_{r}$ , $R_{g}$ , $R_{b}$ and $d_{rg}$ , are given, one can analytically obtain all the other geometric parameters, including $d_{gb}$ , $\\theta _{r}$ , $\\theta _{g}$ and $\\theta _{b}$ , and the relative magnitudes of $\\sigma _{rb}$ , $\\sigma _{gb}$ and $\\sigma _{rg}$ .",
"Specifically, the half of the central angle $\\theta _{g}$ can be first calculated by $\\theta _{g}=\\arccos {\\frac{R_{g}^{2}+d_{rg}^{2}-R_{r}^{2}}{2R_{g}d_{rg}}},$ which is used to calculate the other two angles, $\\theta _{r}$ and $\\theta _{b}$ , and $R_{b}$ according to $R_{g}\\sin {\\theta _{g}}=R_{b}\\sin {\\theta _{b}}=R_{r}\\sin {\\theta _{r}},$ and the distance $d_{gb}$ is then obtained as $d_{gb}=R_{g}\\cos {\\theta _{g}}+R_{b}\\cos {\\theta _{b}}.$ Next, we determine the angles $\\varphi _{rg}$ , $\\varphi _{rb}$ and $\\varphi _{gb}$ through the geometric relationship and the Neumann's triangle shown in Fig.",
"REF (a).",
"For example, when $R_{g}\\cos {\\theta _{g}}<{d_{gb}}$ and $R_{g}\\cos {\\theta _{g}}\\ge {d_{rg}}$ , these angles can be calculated by $\\left\\lbrace \\begin{aligned}\\varphi _{rb}=\\frac{\\pi }{2}-\\theta _{r}+\\theta _{g} \\\\\\varphi _{gb}=\\pi -\\theta _{r}-\\theta _{b} \\\\\\varphi _{rg}=\\pi -\\varphi _{rb}-\\varphi _{gb}\\end{aligned}\\right.",
";$ and on the other hand, when $R_{g}\\cos {\\theta _{g}}<{d_{gb}}$ and $R_{g}\\cos {\\theta _{g}}<{d_{rg}}$ , we have $\\left\\lbrace \\begin{aligned}\\varphi _{rb}=\\theta _{b}+\\theta _{g} \\\\\\varphi _{gb}=\\theta _{r}-\\theta _{b} \\\\\\varphi _{rg}=\\pi -\\varphi _{rb}-\\varphi _{gb}\\end{aligned}\\right..$ Finally, one can obtain the relative magnitudes of the interfacial tensions by the law of Sines: $\\frac{\\sigma _{rg}}{\\sin {\\varphi _{rg}}}=\\frac{\\sigma _{rb}}{\\sin {\\varphi _{rb}}}=\\frac{\\sigma _{gb}}{\\sin {\\varphi _{gb}}}.$ In other words, all the interfacial tensions can be determined from Eq.",
"(REF ) if one of them is also given, as we shall do below.",
"By contrast, the geometry of a PJD is only determined by two areas of the dispersed fluids, i.e.",
"$A_r$ and $A_g$ , and its analytical solution is given by $\\left\\lbrace \\begin{aligned}R_{g}=R_{r}=\\sqrt{\\frac{A_{g}+A_{r}}{\\pi }} \\\\R_{b}=R_{g}\\tan {\\theta _{g}} \\\\\\frac{A_{r}}{A_{r}+A_{g}}=\\frac{\\theta _{g}-\\sin {\\theta _{g}}\\cos {\\theta _{g}}+\\tan ^{2}{\\theta _{g}}\\left(\\frac{\\pi }{2}-\\theta _{g}-\\sin {\\theta _{g}}\\cos {\\theta _{g}}\\right)}{\\pi } \\\\d_{gb}=R_{g}\\cos {\\theta _{g}}+R_{b}\\cos {\\theta _{b}}\\end{aligned}\\right..$ The above equation suggests that one can obtain all the other geometric parameters, such as $R_{b}$ , $\\theta _{g}$ and $d_{gb}$ , if the area ratio $\\frac{A_{r}}{A_{r}+A_{g}}$ and $R_{g}$ are given.",
"To test the accuracy of the present model for Janus droplets, we conduct two groups of simulations with one for GJD and the other for PJD.",
"The size of the computational domain is set as $[1,300]\\times [1,300]$ , and the periodic boundary conditions are used at all the boundaries.",
"The kinematic viscosities for all the fluids are fixed at $\\nu _{k}=0.1$ .",
"In the GJD simulations, we select $\\sigma _{rg}=0.01$ , $R_{r}=60$ , $R_{g}=80$ and $R_{b}=160$ , and vary the distance $d_{rg}$ from 40 to 120 with an increment of 20.",
"Using these parameters, we can compute the geometric parameters $d_{rg}$ and $d_{gb}$ as well as the interfacial tensions $\\sigma _{gb}$ and $\\sigma _{rg}$ through Eqs.",
"(REF ) to (REF ), which are presented in Table REF .",
"We initialize the fluid distribution such that it follows the given and analytically computed geometric parameters described above, and assume that the circles for $gb$ , $rb$ , and $rg$ interfaces are initially centered at $(150.5,R_{g}+10)$ , $(150.5,R_{g}+10+d_{rg})$ and $(150.5,R_{g}+10+d_{gb})$ , respectively.",
"Table: The interfacial tensions σ gb \\sigma _{gb} and σ rb \\sigma _{rb} and the distance d gb d_{gb} calculated from Eqs.",
"()-() for GJDs with σ rg =0.01\\sigma _{rg}=0.01, R r =60R_{r}=60, R g =80R_{g}=80 and R b =160R_{b}=160 at different values of d rg d_{rg}.In the PJD simulations, we select $\\sigma _{rb}=\\sigma _{gb}=0.01$ , $\\sigma _{rg}=1\\times {10^{-8}}$ and $R_{r}=R_{g}=80$ , and vary the area fraction $\\frac{A_{r}}{A_{r}+A_{g}}$ from 0.1 to 0.5 with an increment of 0.1.",
"These parameters allow us to analytically compute all the other geometric parameters of a PJD, e.g.",
"$R_b$ and $d_{gb}$ , which are listed in Table REF .",
"We follow the analytical geometric parameters to initialize the fluid distribution, and assume that the circles for $gb$ , $rb$ and $rg$ interfaces are initially located at $(150.5,150.5)$ , $(150.5,150.5)$ and $(150.5,150.5+d_{gb})$ , respectively.",
"In particular, we note that $\\frac{A_{r}}{A_{r}+A_{g}}=0.5$ leads to $R_{b}\\rightarrow \\infty $ and $d_{gb}\\rightarrow \\infty $ , suggesting that the interface $rg$ is theoretically a straight line located at $y=150.5$ .",
"Table: The geometric parameters R b R_b and d gb d_{gb} calculated from Eq.",
"() for PJDs with R r =R g =80R_{r}=R_{g}=80 at different area fractions.All of the simulations are run until a steady state is reached.",
"Figs.",
"REF and REF show the comparison between the analytical and simulated results for the GJDs and PJDs.",
"In each of the figures, the analytical interface profiles are represented by the white lines of different patterns, while the red, green and blue fluids are indicated in red, green and blue, respectively.",
"It is seen that our simulation results agree well with the analytical ones for various geometry configurations of GJD and PJD.",
"Figure: Comparison between the analytical and simulated results for GJDs with σ rg =0.01\\sigma _{rg}=0.01, R r =60R_{r}=60, R g =80R_{g}=80 and R b =160R_{b}=160 at different values of d rg d_{rg}.",
"The analytical interface profiles are represented by the white lines of different patterns, while the simulated red, green and blue fluids are indicated in red, green and blue, respectively.Figure: Comparison between the analytical and simulated results for PJDs with R r =R g =80R_{r}=R_{g}=80 at different area fractions.",
"The analytical interface profiles are represented by the white lines of different patterns, while the simulated red, green and blue fluids are indicated in red, green and blue, respectively."
],
[
"Near-critical and critical states",
"For double droplets immersed in a static matrix, the critical state occurs when the largest interfacial tension equals the sum of the other two.",
"As previously shown in Fig.",
"REF , the critical state of double droplets can be subdivided into the kissing state ((iv) in Fig.",
"REF ) and the critical engulfing state ((ii) or (vi) in Fig.",
"REF ).",
"For the convenience of description, we define the near-critical state as the state where the largest interfacial tension is close to the sum of the other two.",
"It is challenging to accurately simulate the critical and near-critical states, where a slight inaccuracy in modeling could lead to significant simulation errors.",
"To highlight the strength of the present model for critical scenarios, we consider the kissing/near-kissing states and the critical/near-critical engulfing states in a rectangular domain of $[1,300]\\times [1,300]$ .",
"The boundary conditions and fluid viscosities are set the same as those in Section REF .",
"In the kissing/near-kissing states, a pair of equal-sized droplets with the radii of $R_{r}=R_{g}=60$ are initially placed with a distance of $d_{rg}$ , and they are symmetric with respect to the centerline $y=150.5$ .",
"The simulations are performed for a constant $\\sigma _{rg}$ of $0.01$ but varying $\\sigma _{gb}$ ($=\\sigma _{rb}$ ), which is varied around the critical value of $0.005$ with an increment of $2\\times 10^{-4}$ .",
"Note that the initial distance $d_{rg}$ depends on the value of $\\sigma _{gb}$ , and is given by its analytical value in equilibrium as $d_{rg}=\\left\\lbrace \\begin{aligned}&R_{r}\\sigma _{rg}/\\sigma _{gb} & \\mathrm {if}~ \\sigma _{gb}>0.005;\\\\&R_{r}+R_{g}=120 & \\mathrm {otherwise}.\\end{aligned}\\right.$ In the critical/near-critical engulfing states, we consider a green droplet with $R_{g}=80$ entirely or partially engulfing a red droplet with $R_{r}=60$ for $\\sigma _{gb}=\\sigma _{rg}=0.01$ .",
"$\\sigma _{rb}$ is varied around the critical value of 0.02 with an increment of $2\\times 10^{-4}$ .",
"With these parameters, we are able to analytically compute other geometric parameters, which are given by $d_{rg}=\\sqrt{R_{r}^{2}+R_{g}^{2}-2R_{r}R_{g}\\cos {\\alpha }}$ , $R_{b}=\\frac{R_{g}\\sin {\\theta _{g}}}{\\sin {\\theta _{b}}}$ and $d_{gb}=R_{g}\\cos {\\theta _{g}}-R_{b}\\cos {\\theta _{b}}$ for $\\sigma _{rb}<0.02$ , and by $d_{rg}=R_{g}-R_{r}=20$ for $\\sigma _{rb}\\ge 0.02$ .",
"Herein, $\\cos {\\alpha }=\\frac{\\sigma _{rb}}{2\\sigma _{gb}}$ , $\\theta _{g}=\\arccos {\\frac{R_{g}^{2}+d_{gr}^{2}-R_{r}^{2}}{2R_{g}d_{gr}}}$ and $\\theta _{b}=\\theta _{g}+2\\alpha $ .",
"Again, we initialize the fluid distribution such that it follows the analytical geometric parameters.",
"Figure: (a) The interface length L rg L_{rg} as a function of σ gb \\sigma _{gb} in the kissing/near-kissing states; (b) the interface length L rb L_{rb} as a function of σ rb \\sigma _{rb} in the critical/near-critical engulfing states.In addition to the present model, we also use the model of Fu et al.",
"[51] and the model of Leclaire et al.",
"[50] for the simulations.",
"When the simulations reach the steady state, we quantify the interface lengths $L_{rg}$ in the kissing/near-kissing states and $L_{rb}$ in the critical/near-critical engulfing states.",
"Fig.",
"REF compares the simulated results from the present model with those from the model of Fu et al.",
"[51] and the model of Leclaire et al.",
"[50], and the analytical solutions.",
"It is seen that for either kissing/near-kissing states or critical/near-critical engulfing states, the simulated results from the present model are in good agreement with the analytical solutions, while the simulated results from the other two models significantly deviate from the analytical solutions.",
"Fig.",
"REF shows the final fluid distributions obtained by the present model and the model of Fu et al.",
"[51], in both kissing and critical engulfing states.",
"Note that the fluid distribution from the model of Leclaire et al.",
"[50] is not shown in the figure, since it produces almost the same results as the model of Fu et al.[51].",
"Clearly, both critical states are correctly reproduced by the present model but not by the model of Fu et al.[51].",
"These results indicate that the present model is advantageous to simulate critical state in ternary fluids.",
"Figure: The final fluid distributions obtained by (a) the present model and (b) the model of Fu et al.",
"for the kissing state, and by (c) the present model and (d) the model of Fu et al.",
"for the critical engulfing state."
],
[
"conclusions",
"A LB color-gradient model is proposed to simulate immiscible ternary fluids with a full range of interfacial tensions.",
"An interfacial force formulation for $N$ -phase ($N\\ge 3$ ) systems is derived and then introduced into the model using a body force scheme, which is found to effectively reduce spurious velocities.",
"A recoloring algorithm proposed by Spencer et al.",
"[49] is applied to produce the phase segregation and ensure the immiscibility of three different fluids, where a novel form of segregation parameters is proposed by considering the existence of Neumann's triangle and the effect of equilibrium contact angle in three-phase junction.",
"The model's capability in capturing interfaces and modeling interfacial tensions is first validated by the simulation of the two separate static droplets and the Young-Laplace test for a compound droplet.",
"The overall performance of the model is then assessed by simulating the spreading of a droplet between two stratified fluids, and both the partial and complete spreadings are predicted with satisfactory accuracy.",
"Finally, the present model is used to study the stability and structure of double droplets in a static matrix over a wide range of interfacial tensions.",
"By changing two ratios of the interfacial tensions, seven possible equilibrium morphologies are successfully reproduced, which are consistent with the theoretical stability diagram by Guzowski et al.[15].",
"For various geometry configurations of general and perfect Janus droplets, good agreemento between simulated results and analytical solutions shows the present model is accurate when three interfacial tensions yield a Neumann's triangle.",
"In addition, we also simulate the near-critical and critical states of double droplets, which is challenging since the outcomes are very sensitive to the model accuracy.",
"It is found that the simulated results from the present model agree well with the analytical solutions, while the simulated results from the existing color-gradient models significantly deviate from the analytical solutions, especially in critical states.",
"In summary, the present work provides the first LB multiphase model that allows for accurate simulation of ternary fluid flows with a full range of interfacial tensions."
],
[
"Acknowledgements",
"This work was supported by the National Natural Science Foundation of China (Nos.",
"51506168, 51711530130), the National Key Research and Development Project of China (No.",
"2016YFB0200902), the China Postdoctoral Science Foundation (No.",
"2016M590943), Guangdong Provincial Key Laboratory of Fire Science and Technology (No.",
"2010A060801010) and Guangdong Provincial Scientific and Technological Project (No.",
"2011B090400518).",
"Y. Yu was supported by the China Scholarship Council for one year study at the University of Strathclyde, UK.",
"H. Liu gratefully acknowledges the financial supports from Thousand Youth Talents Program for Distinguished Young Scholars, the Young Talent Support Plan of Xi'an Jiaotong University."
]
] | 1808.08555 |
[
[
"Supercondutivity in SnSb with natural superlattice structure"
],
[
"Abstract We report the results of electrical resistivity, magnetic and thermodynamic measurements on polycrystalline SnSb, whose structure consists of stacks of Sb bilayers and Sn4Sb3 septuple layers along the c-axis.",
"The material is found to be a weakly coupled, fully gapped, type-II superconductor with a bulk Tc of 1.50 K, while showing a zero resistivity transition at a significantly higher temperature of 2.48 K. The Sommerfeld coefficient and upper critical field, obtained from specific heat measurements, are 2.29 mJ/mol K and 520 Oe, respectively.",
"Compositional inhomogeneity and strain effect at the grain boundaries are proposed as possible origins for the difference in resistive and bulk superconducting transitions.In addition, a comparison with the rock-salt structure SnAs superconductor is presented.",
"Our results provide the first clear evidence of bulk superconductivity in a natural superlattice derived from a topological semimetal."
],
[
"Introduction",
"Recently, superconductors derived from topological materials have attracted a lot of attention because of their potential as topological superconductors that are useful in fault tolerant quantum computing [1], [2], [3].",
"A common strategy to look for such kind of superconductors is to dope topological (crystalline) insulators [3], which increases the carrier concentration and induces superconductivity (SC).",
"Prominent examples include $A_{x}$ Bi$_{2}$ Se$_{3}$ ($A$ = Cu [4], [5], Sr [6], Nb [7]), and Sn$_{1-x}$ In$_{x}$ Te [8], [9].",
"On the other hand, it has been shown that natural superlattice structure can be used to tailor the properties of topological materials by changing the constituent building blocks [10].",
"For instance, Sb$_{2}$ Te, which is composed of [Sb$_{2}$ ] bilayers and [Sb$_{2}$ Te$_{3}$ ] quintuple layers, exhibits significantly different surface and bulk band structures from both the topological insulator Sb$_{2}$ Te$_{3}$ and the topological semimetal Sb [11].",
"In this context, it is of interest to see whether similar superlattices without intentional doping can support SC.",
"SnSb is one of the stable binary phases at room temperature in the Sn-Sb system [12], and has been studied intensively as an anode material for (Li/Na)-ion batteries over the past two decades [13], [14], [15], [16].",
"Initially, the material was reported to form a rhombohedrally distorted rock-salt-like structure [12].",
"Until recently, an incommensurate modulation is found along the $c$ -axis [17], and the structure of SnSb can be viewed as intercalation of Sn layers between adjacent Sb layers, leading to stacking of [Sb$_{2}$ ] bilayers and [Sn-Sb-Sn-Sb-Sn-Sb-Sn] septuple-layers shown in Fig.",
"1 [18].",
"Notably, it was mentioned in a paper by Geller and Hull in 1960s that SnSb exhibits SC as in rock-salt structure SnAs, albeit with two superconducting transitions [19].",
"While SnAs has been confirmed to be a type-I superconductor experimentally [20], the nature of SC in SnSb remains unclear to date.",
"In this paper, we present a comprehensive study of the physical properties of SnSb.",
"It is found that the material is an $n$ -type metal in the normal state, and undergoes a transition to zero resistivity at 2.48 K. However, magnetic susceptibility and specific heat measurements indicate that bulk type-II SC is established at a considerably lower temperature of 1.50 K. Furthermore, the electronic specific heat jump of SnSb follows a weak coupling BCS-like behavior, pointing to a fully gapped superconducting state.",
"The normal-state and superconducting parameters are extracted and compared with those of SnAs.",
"The origin of two superconducting transitions is also discussed."
],
[
"Experimental",
"Polycrystalline SnSb samples were synthesized by using a two-step method.",
"High-purity shots of Sn (99.99%) and Sb (99.999%) with the stoichiometric ratio of 1:1 were melted in sealed evacuated quartz tube at 900 $^{\\rm o}$ C for 48 h with intermittent shaking to ensure homogeneity, followed by quenching into cold water.",
"The resulting ingot was then annealed at 320 $^{\\rm o}$ C for another 48 h, and finally quenched into cold water.",
"The purity of the sample was checked by powder X-ray diffraction (XRD) using a PANalytical x-ray diffractometer with a monochromatic Cu-K$_{\\alpha 1}$ radiation at room temperature.",
"For consistency reason, all the physical property measurements were performed on samples obtained from the same ingot.",
"A part of the ingot was cut into regular-shaped samples for electrical resistivity, Hall coefficient and specific heat measurements, and the remaining part was crushed into powders.",
"The typical dimensions are 4 mm $\\times $ 0.8 mm $\\times $ 0.35 mm and 2 mm $\\times $ 2 mm $\\times $ 0.2 mm for transport (resistivity/Hall) and specific heat measurements, respectively.",
"The resistivity was measured by using a standard four-probe method.",
"Resistivity, Hall coefficient and specific heat measurements down to 1.8 K were carried out on a Quantum Design PPMS-9 Dynacool.",
"Specific heat measurements down to 0.5 K were preformed on a Quantum Design PPMS-9 Evercool II.",
"The two sets of specific heat data agree well within 5% in the overlapped temperature range.",
"The dc magnetization down to 0.4 K was done on crushed powders with a commercial SQUID magnetometer (Quantum Design MPMS3)."
],
[
"Crystal Structure",
"Figure 2(a) shows the XRD pattern of the SnSb sample at room temperature, together with structure refinement profile using the JANA2006 program [21].",
"All the diffraction peaks can be well fitted with the (3 + 1)-dimensional superspace group $R$$\\overline{3}$$m$ (00$\\gamma $ ) (No.166.1), where $\\gamma $ = 1.315 is the modulation $q$ -vector component.",
"As exemplified in Fig.",
"2(b), first-order satellite reflections due to the incommensurate modulation are clearly visible between the strong peaks of the $R$$\\overline{3}$$m$ average structure.",
"The refined lattice parameters are $a$ = 4.331(2) Å and $c$ = 5.352(2) Å in the hexagonal setting, or $a$ = 3.072(1) Å and $\\alpha $ = 89.65$^{\\circ }$ in the rhombohedral setting.",
"Note that the $\\alpha $ angel is very close to 90$^{\\circ }$ , hence the structure of SnSb deviates only slightly from the cubic symmetry.",
"These results are in good agreement with previous reports [17], [18], [12], and demonstrate high sample quality.",
"It is pointed out that the structure of Sb can also be described by the $R$$\\overline{3}$$m$ (00$\\gamma $ ) space group with $\\gamma $ = 1.5 [18].",
"Moreover, Sn is just before Sb in the periodic table, hence a strong spin-orbit coupling is expected in SnSb.",
"Taken together, it is tempting to speculate that SnSb and Sb have a similar topological character."
],
[
"Resistivity and Hall coefficient",
"The main panel of Fig.",
"3(a) shows the temperature dependence of resistivity ($\\rho $ ) for the SnSb sample under zero field.",
"The $\\rho $ value at room temperature is $\\sim $ 28 $\\mu $$\\Omega $ cm, which lies in between that of pure Sn and pure Sb [22].",
"With decreasing temperature, $\\rho $ decreases linearly down to $\\sim $ 50 K and then varies as $T^{2.4}$ at lower temperature, which is typical of a normal metal.",
"Nevertheless, the low residual resistivity ratio ($\\rho _{\\rm 300 K}$ /$\\rho _{\\rm 3 K}$ ) of $\\sim $ 2.6 suggests the presence of significant disorder scattering, likely due to antisite occupation between Sn and Sb.",
"On further cooling below 2.8 K, $\\rho $ drops rapidly to zero, evidencing a transition to the superconducting state [inset of Fig.",
"3(a)].",
"Here we define $T_{\\rm c}^{\\rho }$ = 2.48 K as the temperature corresponding to the midpoint of the resistive drop.",
"On the other hand, as shown in Fig.",
"3(b), the Hall resistivity depends linearly on the magnetic field, and the corresponding Hall coefficient is negative and nearly temperature independent.",
"These results are consistent with the metallic character of SnSb, and identify electrons as the dominant carriers.",
"Assuming a one-band model, we obtain an electron density of 1.9 $\\times $ 10$^{22}$ cm$^{-3}$ at 1.8 K."
],
[
"Magnetic susceptibility and magnetization",
"Figure 4(a) shows the temperature dependence of the magnetic susceptibility $\\chi $ measured on powder samples in both zero-field cooling (ZFC) and field cooling (FC) modes with an applied field of 4.8 Oe.",
"Here the demagnetization effect is taken into consideration assuming sphere-shaped grains with the demagnetization factor $N_{\\rm d}$ = 1/3.",
"A diamagnetic signal is observed in both $\\chi _{\\rm ZFC}$ and $\\chi _{\\rm FC}$ , and its onset temperature $\\sim $ 2.9 K is very close to that of the resistive transition.",
"Below 2.5 K, a divergence is present between the ZFC and FC curve, indicating the presence of trapped flux.",
"On closer examination, it is noted that the diamagnetic transition is rather broad at the beginning, leading to a low shielding fraction of 9.3 % at 1.7 K [inset of Fig.",
"4(a)].",
"With further decreasing temperature, however, both ZFC and FC data show a steep drop and become nearly flat below 1.3 K, which correspond to shielding and Meissner fractions of 106% and 48%, respectively.",
"Note that the linear interpolation of this strong diamagnetic signal intersects with that of the initial one at 1.66 K, which is one-third lower than $T_{\\rm c}^{\\rho }$ .",
"This strongly suggest that the zero resistivity transition is not of bulk nature, as will be shown below.",
"Fig.",
"4(b) shows the field dependence of magnetization ($M$ ) of SnSb at 0.4 K. The data exhibits a small hysteresis loop, which corroborates that SnSb is a type-II superconductor with weak pinning.",
"As can be seen in the inset, the initial magnetization curve is linear, as expected for a Meissner state.",
"The lower critical field $B_{\\rm c1}$ can be estimated from the deviation of the linearity, which gives the effective $B_{\\rm c1}^{\\ast }$ (0) $\\approx $ 8 Oe.",
"After corrected by the demagnetization factor, $B_{\\rm c1}$ (0) = $B_{\\rm c1}^{\\ast }$ (0)/(1 $-$ $N_{\\rm d}$ ) $\\approx $ 12 Oe is obtained."
],
[
"Specific heat",
"Figure 5 summarizes the results from the specific heat ($c_{\\rm p}$ ) measurements for SnSb.",
"As can be seen in Fig.",
"5(a), $c_{\\rm p}$ varies smoothly across $T_{\\rm c}^{\\rho }$ while exhibits a sharp jump at $\\sim $ 1.6 K. With increasing field, the specific heat jump shifts towards lower temperature and becomes broadened, consistent with a superconducting transition.",
"At 500 Oe, the anomaly is almost suppressed, and the data can be analyzed by the Debye model $c_{\\rm p}$ /$T$ = $\\gamma _{\\rm n}$ + $\\beta _{3}$$T^{2}$ + $\\beta _{5}$$T^{4}$ , where $\\gamma _{\\rm n}$ and $\\beta _{i}$ ($i$ = 3, 5) are the electronic and phonon specific-heat coefficients, respectively.",
"The best fit to the data below 3 K yields $\\gamma _{\\rm n}$ = 2.29 mJ mol$^{-1}$ K$^{-2}$ , $\\beta _{3}$ = 0.43 mJ mol$^{-1}$ K$^{-4}$ , and $\\beta _{5}$ = 0.009 mJ mol$^{-1}$ K$^{-6}$ .",
"Once $\\beta _{3}$ is known, the Debye temperature $\\Theta _{\\rm D}$ can be calculated using the equation $\\Theta _{\\rm D}$ = (12$\\pi ^{4}$$NR$ /5$\\beta _{3}$ )$^{\\frac{1}{3}}$ , where $N$ = 2 and $R$ is the molar gas constant 8.314 J/mol K. This gives $\\Theta _{\\rm D}$ = 208 K for SnSb.",
"On the other hand, as shown in the inset of Fig.",
"5(a), the $c_{\\rm p}$ value exceeds the Dulong-Petit limit of 3$N$$R$ = 49.88 J/mol K at temperatures above 200 K, suggesting that all the phonons are excited in this temperature range.",
"Hence one may estimate $\\Theta _{\\rm D}$ $\\sim $ 200 K, in good agreement with the above calculation.",
"Figure 5(b) shows the normalized electronic specific heat $C_{\\rm el}$ /$\\gamma _{\\rm n}$$T$ at zero field after subtraction of the phonon contribution.",
"The result further confirms the absence of a specific heat anomaly corresponding to the resistive transition.",
"More importantly, it turns out that the $C_{\\rm el}$ /$\\gamma _{\\rm n}$$T$ jump can be reproduced by the BCS theory [23], especially at low temperature region, and the entropy balance determines the bulk superconducting transition temperature $T_{\\rm c}^{\\rm bulk}$ = 1.50 K. This suggests that SnSb is fully gapped with a conventional $s$ -wave pairing symmetry and a zero temperature gap value $\\Delta $ (0) = 0.22 meV.",
"Nevertheless, since the data is limited to $T$ /$T_{\\rm c}^{\\rm bulk}$ $\\ge $ 0.34, the possibility of multiband SC with a very small gap in one of the bands cannot be ruled out.",
"Therefore, $c_{\\rm p}$ measurements at lower temperature would be necessary to draw a more concrete conclusion.",
"Assuming a phonon mediated pairing mechanism, the electron-phonon coupling constant, $\\lambda _{\\rm ph}$ , can be calculated by the inverted McMillan formula [24], $\\lambda _{\\rm ph} = \\frac{1.04 + \\mu ^{\\ast } \\rm ln(\\Theta _{\\rm D}/1.45\\emph {T}_{\\rm c})}{(1 - 0.62\\mu ^{\\ast })\\rm ln(\\Theta _{\\rm D}/1.45\\emph {T}_{\\rm c}) - 1.04},$ where $\\mu ^{\\ast }$ is the Coulomb repulsion pseudopotential.",
"With an empirical value of $\\mu ^{\\ast }$ = 0.13, $\\lambda _{\\rm ph}$ =0.52 and 0.58 are obtained for $T_{\\rm c}^{\\rho }$ and $T_{\\rm c}^{\\rm bulk}$ , respectively, which implies that SnSb is a weakly coupled superconductor.",
"In addition, since $\\gamma _{\\rm n}$ is related to the bare density of states $N$ (0) at the Fermi level through the relation $\\gamma _{\\rm n}$ = $\\frac{1}{3}$$\\pi $$^{2}$$k_{\\rm B}$$^{2}$$N$ (0)(1 + $\\lambda _{\\rm ph}$ ), one can obtain $N$ (0) = 0.64 states/eV per formula unit."
],
[
"Superconducting parameters",
"Figure 6 shows the magnetic field-temperature phase diagrams determined from resistivity and specific heat measurements.",
"Here the resistive-transition criterion for $T_{\\rm c}$ under different fields is the same as that at zero field (see the inset).",
"Using the Werthamer-Helfand-Hobenberg (WHH) thoery [25], the upper critical field $B_{\\rm c2}$ can be extrapolated to zero temperature, which gives $B_{\\rm c2}^{\\rho }$ (0) = 1660 Oe and $B_{\\rm c2}$ (0) = 520 Oe for the two measurements, respectively.",
"The Ginzburg-Landau (GL) coherence length $\\xi _{\\rm GL}$ (0) is then calculated as $\\xi _{\\rm GL}$ (0) = $\\sqrt{\\Phi _{0}/2\\pi B_{\\rm c2}(0)}$ , where $\\Phi _{0}$ = 2.07 $\\times $ 10$^{-15}$ Wb is the flux quantum.",
"This yields $\\xi _{\\rm GL}^{\\rho }$ (0) = 44.5 nm and $\\xi _{\\rm GL}^{\\rho }$ (0) = 79.5 nm.",
"Also the equations $B_{\\rm c1}$ (0)/$B_{\\rm c2}$ (0) = (ln$\\kappa _{\\rm GL}$ + 0.5)/(2$\\kappa _{\\rm GL}^{2}$ ) [26] and $B_{\\rm c1}$ (0) = ($\\Phi _{0}$ /4$\\pi $$\\lambda _{\\rm eff}^{2}$ )(ln$\\kappa _{\\rm GL}$ + 0.5) allow one to deduce the GL parameter $\\kappa _{\\rm GL}$ = 7.4 and the effective penetration depth $\\lambda _{\\rm eff}$ = 585 nm.",
"Note that this $\\lambda _{\\rm eff}$ is more than two orders of magnitude smaller than the typical grain size of about 150$-$ 200 $\\mu $ m, reassuring that the effect of penetration depth is negligible [27]."
],
[
"Origin of two superconducting transitions",
"From the above results, it is clear that there exists two superconducting transitions at $T_{\\rm c}^{\\rho }$ = 2.48 K and $T_{\\rm c}^{\\rm bulk}$ = 1.50 K in SnSb, respectively.",
"This is in agreement with the previous report [19], although the two $T_{c}$ values are slightly higher in the present case, which is likely due to the different sample preparation conditions.",
"We emphasize that since no additional peaks are observed in the XRD data, the SC at $T_{\\rm c}^{\\rho }$ is unlikely due to a secondary phase with different structure.",
"Actually, this marked difference between $T_{\\rm c}^{\\rho }$ and $T_{\\rm c}^{\\rm bulk}$ is reminiscent of what has been found in CeIrIn$_{5}$ [28], PrOs$_{4}$ Sb$_{12}$ [29], CePt$_{3}$ Si [30], La$_{3}$ Rh$_{4}$ Sn$_{13}$ [31], and SrTi$_{1-x}$ Nb$_{x}$ O$_{3}$ [32].",
"Note that the small $\\gamma _{\\rm n}$ value of SnSb is indicative of a weak electron correlation, which precludes the formation of a textured superconducting phase due to a competing order [33].",
"Instead, two possibilities are considered here.",
"One possibility is that the existence of two superconducting transitions is due to the micro-phase segregation with a spread in the Sn/Sb ratio, which has been observed in long-term annealed SnSb samples [17].",
"These off-stoichiometric regions usually have a characteristic length of a few hundred $\\mu $ m or smaller and a very similar structure to SnSb [12], and thus are difficult to distinguish by the XRD measurements.",
"Although a much shorter annealing time is adopted in the present study, it is possible that these inhomogeneous regions are already present, which is responsible for the resistive transition at $T_{\\rm c}^{\\rho }$ , as in La$_{3}$ Rh$_{4}$ Sn$_{3}$ [31].",
"Moreover, the initial diamagnetic signal in the susceptibility data could also be due to these regions.",
"Hence we take the shielding fraction of $\\sim $ 13% at 1.66 K, the temperature at which the linear interpolation of the strong diamagnetic signal intersects with the base line [see Fig.",
"4(a)], as an estimation of the upper limit of their volume faction.",
"This indicates that such inhomogeneous regions, if they exist, occupy only a small part of the sample, in consistent with the absence of an anomaly in the specific heat data near $T_{\\rm c}^{\\rho }$ .",
"Hence their presence has no practical effect in determining the intrinsic properties of SnSb.",
"Another possibility is that the zero resistivity transition results from the strain effect at the grain boundaries.",
"It is prudent to note that SnSb is not a line compound [12], and hence rapid quench is necessary to retain the phase to room temperature.",
"During this process, it is expected that the grain surface cools faster than the interior, which leads to the development of strain at the grain boundaries.",
"This may result in the increase of density of states and/or phonon softening, and consequently an enhanced $T_{\\rm c}$ .",
"Note that the grain boundaries constitute a small fraction of the sample, yet they are connected to form a continuous superconducting path, as observed experimentally.",
"In this respect, high pressure study and epitaxial film growth of SnSb, which may favor a higher $T_{\\rm c}$ [34], [35], will be of interest in future."
],
[
"Comparison with SnAs",
"Finally, we present a comparison of the normal-state and superconducting properties between SnSb and SnAs [20], which are summarized in Table 1.",
"Despite their different crystal structures, the $\\gamma _{\\rm n}$ and $\\Theta _{\\rm D}$ values are very similar for these compounds.",
"Nevertheless, compared with SnAs, SnSb has a much lower $T_{\\rm c}$ and a smaller $\\lambda _{\\rm ph}$ .",
"Actually, this is also the case when one compares the experimental results of SnSb with the theoretically predicted ones assuming the same crystal structure as SnAs [35].",
"This implies that, within the BCS framework, the lower $T_{\\rm c}$ value of SnSb in the rombohedral structure compared with that in the rock-salt structure is due to a decrease in the electron phonon coupling strength.",
"However, it is noted that the electronic band structure of SnAs measured by ARPES is not in full agreement with the calculated one, but bears signatures of the spin-orbit coupling (SOC) [36].",
"As for SnSb, the SOC effect is expected to be even stronger, and hence the possibility of odd-parity pairing [5], [8] cannot be excluded.",
"In this regard, to better understand the SC in SnSb, a combined theoretical and spectroscopic study of its band structure and Fermi surface is strongly called for."
],
[
"Conclusion",
"In summary, we have studied the electrical transport, magnetic and thermodynamic properties of SnSb, which confirms the occurrence of SC in this natural superlattice phase.",
"It is further shown that the material a bulk type-II superconductor below 1.50 K with a BCS-like gap.",
"Nevertheless, the zero resistivity transition takes place at a considerably higher temperature of 2.48 K, for which two possibilities are discussed.",
"One possibility is that this is due to the presence of inhomogeneous regions with off stoichiometric Sn/Sb compositions due to the micro-phase segregation.",
"Another possible scenario in that the enhanced $T_{\\rm c}$ in resistivity measurement results from the strain effect that develops at the grain boundaries during the sample quenching.",
"We also compare the properties of SnSb with those of the rock-salt structure SnAs superconductor, which suggests that the lower $T_{\\rm c}$ in SnSb is likely due to a weakening of the electron phonon coupling strength.",
"Since SnSb can be regarded as a close relative to the topological semimetal Sb, further band structure calculations and surface-sensitive spectroscopic studies are called for to assess the potential of this material as a topological superconductor."
],
[
"Acknowledgments",
"We thank Xiao Lin for useful discussion.",
"This work is supported by the National Key Research and Development Program of China (No.2017YFA0303002) and the Fundamental Research Funds for the Central Universities of China."
]
] | 1808.08500 |
[
[
"Localized solar power prediction based on weather data from local\n history and global forecasts"
],
[
"Abstract With the recent interest in net-zero sustainability for commercial buildings, integration of photovoltaic (PV) assets becomes even more important.",
"This integration remains a challenge due to high solar variability and uncertainty in the prediction of PV output.",
"Most existing methods predict PV output using either local power/weather history or global weather forecasts, thereby ignoring either the impending global phenomena or the relevant local characteristics, respectively.",
"This work proposes to leverage weather data from both local weather history and global forecasts based on time series modeling with exogenous inputs.",
"The proposed model results in eighteen hour ahead forecasts with a mean accuracy of $\\approx$ 80\\% and uses data from the National Ocean and Atmospheric Administration's (NOAA) High-Resolution Rapid Refresh (HRRR) model."
],
[
"Introduction",
"The power output from a PV array is known to depend on environmental variables such as irradiance, temperature (ambient and cell), wind velocity, relative humidity, air pressure, and sky conditions .",
"Previous works have shown that the lack of accurate information about these variables can affect the prediction error significantly .",
"Therefore, it is important to be able to forecast these fluctuations accurately.",
"Using such forecast plus a model that maps the environmental states to the renewable output, we can forecast the renewable output.",
"While there exist several forecast products from the National Oceanic and Atmospheric Administration (NOAA), the spatial resolution of each of these products are typically on the order of kilometers .",
"Resolution at such scales is inadequate for purposes of localized predictions over smaller spatial scales.",
"Therefore, a blend of local weather history along with global weather forecasts is critical to improve local weather forecast accuracy.",
"In this work, we are concerned with the problem of weather forecast-based solar power prediction.",
"Methods proposed in the literature to forecast solar power are either based on time-series power data, global weather forecast data, or local weather measurement data.",
"In models such as those in Pedro et al.",
", time-series power data with no exogenous inputs are used for short-term forecasting with a horizon of up to two hours into the future.",
"However, using only past power data for forecasting does not directly integrate globally induced weather-related changes into the forecasting model.",
"In , both time-series power data and global weather forecasts based on mesoscale models were used for forecasting.",
"Bacher et al.",
"conclude that for horizons up to less than six-hours, solar power is an important variable for prediction whereas over longer forecast horizons greater than nineteen hours ahead, only weather input was found adequate for prediction.",
"However, using only global weather forecast data does not incorporate local characteristics such as shadows due to trees, buildings, or birds into the weather forecast model.",
"On the contrary, there are also approaches that use local power and weather history for prediction as described by Chen et al.",
".",
"However, using only local data does not allow the model to account for impending global phenomena.",
"As a result it is imperative to develop models that incorporate local and global characteristics to improve prediction accuracy.",
"Consequently, this work proposes a two-step approach to predict local power data based on past local weather data and global weather forecast data.",
"In the first step, a local weather prediction problem is solved by employing a time-series model with past local weather data and global weather forecast data as inputs.",
"The local weather prediction results from the first step are used as inputs in the second step to predict the solar power output based on an existing weather-to-power map .",
"In this work, the solar irradiance, outside air temperature, and wind speed are the weather variables considered in the weather prediction problem.",
"Out of these variables, the solar irradiance and outside air temperature are considered in the power prediction problem.",
"The remainder of this paper is structured as follows.",
"Notation is presented in Section and the weather data is described in .",
"The two-step model is described in Section and is followed by results and discussion in Section .",
"Concluding remarks are presented in Section ."
],
[
"Notation",
"Let the elements of the sequence $\\lbrace t_i\\rbrace _{i=-\\infty }^\\infty $ represent time instances in the past ($i<0$ ), the present ($i=0$ ), and the future ($i>0$ ).",
"The weather variables such as solar irradiance, outside air temperature, and wind speed may either be measured or HRRR-forecasted or predicted.",
"Let these measured variables, j-hour ahead forecast variables, and variables predicted at the time instant $t_i$ be represented by the ordered triples $(I^M_i,T^M_i,W^M_i)$ , $(I^F_{i,j},T^F_{i,j},W^F_{i,j})$ , and $(\\hat{I}_i,\\hat{T}_i,\\hat{W}_i)$ , respectively.",
"Further, at the time instant $t_i$ , let the measured or analytically determined power be represented by $P_i$ and let the predicted power be represented by $\\hat{P}_i$ ."
],
[
"NOAA weather data",
"The NOAA's National Center for Environmental Prediction (NCEP) provides several forecast products differing in forecast horizon, spatio-temporal resolution, update frequency, forecast variables, and forecast method .",
"In this work, we consider the High Resolution Rapid Refresh (HRRR) product which offers weather data at a spatial resolution of three kilometers .",
"The temporal resolution of the HRRR data is either one hour or fifteen minute depending on whether an hourly or a subhourly product is used .",
"An archive of the HRRR hourly data is available from the University of Utah MesoWest HRRR archive .",
"This data is temporally organized into twenty four model cycles each reflecting an hourly update during the day.",
"Within each model cycle, forecast files are provided for up to eighteen hours ahead at a temporal resolution of one hour.",
"Each model cycle also contains a zero hour ahead forecast, which is an assimilation of observations from several primary sources.",
"For this work, we consider this assimilated value as the reference or measured value of the corresponding weather variable at the location of interest.",
"In line with the available HRRR archive data, we let the time instances $t_i \\forall i \\in \\mathbb {Z}$ to be an hour apart from each other so that $\\Delta t = (t_{i+1}-t_i)$ = 3600 seconds.",
"Accordingly, during the present hour $t_i (i=0)$ , the weather measurements $(I^M_0,T^M_0,W^M_0)$ and forecasts up to eighteen hours ahead $(I^F_{i,j},T^F_{i,j},W^F_{i,j}) \\forall i \\in \\lbrace 1,\\cdots ,18\\rbrace , 1\\le j\\le 18$ are well-defined.",
"In this work, the forecast horizon is set to eighteen hours ahead while noting similar analyses may be performed for shorter forecast horizons with the HRRR forecast data.",
"Henceforth, the eighteen hour ahead forecasts $(I^F_{i,18},T^F_{i,18},W^F_{i,18})$ for the time instant $t_i$ will be represented succinctly as $(I^F_i,T^F_i,W^F_i)$ ."
],
[
"Model",
"We employ the following two-step model to predict the eighteen hour ahead solar power output based on the measured weather data and the eighteen hour ahead forecast weather data."
],
[
"Step 1: Weather Prediction",
"In the weather prediction problem we seek to predict the eighteen hour ahead weather data $(\\hat{I}_i,\\hat{T}_i,\\hat{W}_i)$ at the instant $t_i$ based on the 24-hour behind measured data $(I^M_{i-24},T^M_{i-24},W^M_{i-24})$ and 18-hour ahead forecast data $(I^F_i,T^F_i,W^F_i)$ .",
"We resort to time-series modeling and develop an autoregressive model ARX(1,1) with past weather measurements and exogenous weather forecasts as inputs.",
"A comparison to the reference model (AR) without the exogenous forecast input will then illustrate the utility of weather forecasts in the otherwise local history-based predictions.",
"This comparison is discussed Section .",
"Accordingly, let the ARX(1,1) model be represented by: $\\hat{X}_i = \\alpha X^M_{i-24} + \\beta X^F_i + \\gamma + \\epsilon _i$ where, $X$ represents each of the weather variables, $(\\alpha ,\\beta )$ represent the model coefficients, and $(\\gamma ,\\epsilon _i)$ represent the bias and the error term.",
"Correspondingly, the reference model can be degenerated from the above ARX(1,1) model as shown in equation REF .",
"$\\hat{\\underline{X}}_i = \\underline{\\alpha } X^M_{i-18} + \\underline{\\gamma } + \\underline{\\epsilon _i}$ where, $\\hat{\\underline{X}}_i$ represents the weather prediction from the reference model and $(\\underline{\\alpha },\\underline{\\gamma },\\underline{\\epsilon _i})$ represent the model parameters along with the error term.",
"Both these models are trained over the HRRR dataset to learn the corresponding model parameters.",
"The trained model is used to predict local weather variables in the future."
],
[
"Step 2: Solar Power Prediction",
"Once the weather prediction model is developed, a weather-to-power mapping is used to translate the weather predictions $(\\hat{I}_i,\\hat{T}_i,\\hat{W}_i)$ into solar power predictions $\\hat{P}_i$ .",
"While accurate physics-based or sophisticated data-based mappings can be employed in principle , we resort to a linear model to demonstrate the concept using the relation from as shown in the equation below: $\\hat{P}_i = \\eta S \\hat{I}_i [1-0.05(\\hat{T}_i-298.15)]$ where, $\\eta $ represents the panel efficiency, $S$ represents the panel area, and $298.15$ is the temperature under standard conditions.",
"In this work, the usable area for a medium office building $S\\approx 1660 sqm.$ and a solar panel efficiency $\\eta =16\\%$ are considered as stated in Davidson et al.",
"."
],
[
"Results and Discussion",
"The dataset used in this work consisted of weather variables such as solar irradiance, temperature, and windspeed spanning over six months from Dec 2017 - May 2018 at Moffett Field, California.",
"The error characteristics of the forecast against the measurements are shown in the Figure REF and in the histogram REF .",
"The mean error metrics comparing the forecast dataset against the zero-hour ahead measurements are summarized in the Table .",
"Figure: Forecast Error CharacteristicsFigure: Forecast Error HistogramtableError metrics from weather datasets Table: NO_CAPTIONFor the weather modeling, the data points for each weather variable were classified into the hour of the day and the corresponding hourly model parameters were learnt based on the functional forms specified in equations REF and REF .",
"Accordingly, twenty four hourly models were learnt each with and without exogenous inputs for all the weather variables of interest.",
"The hourly datasets were split in the ratio 3:1 for training and testing purposes.",
"The parameters of the models were estimated using the MATLAB System Identification Toolbox and are presented for ARX hourly model in the table .",
"Table: Estimated Parameters (Mean) for the ARX(1,1) model (α h ,β h ,σ(ϵ h ))(\\alpha _h,\\beta _h,\\sigma (\\epsilon _h))"
]
] | 1808.08657 |
[
[
"Uniform hypergraphs with the first two smallest spectral radii"
],
[
"Abstract The spectral radius of a uniform hypergraph $G$ is the the maximum modulus of the eigenvalues of the adjacency tensor of $G$.",
"For $k\\ge 2$, among connected $k$-uniform hypergraphs with $m\\ge 1$ edges, we show that the $k$-uniform loose path with $m$ edges is the unique one with minimum spectral radius, and we also determine the unique ones with second minimum spectral radius when $m\\ge 2$."
],
[
"Introduction",
"Let $G$ be a hypergraph with vertex set $V(G)$ and edge set $E(G)$ , where $E(G)$ is a set whose elements are subsets of $V(G)$ .",
"For an integer $k\\ge 2$ , if each edge of $G$ contains exactly $k$ distinct vertices, then $G$ is a $k$ -uniform hypergraph.",
"Two vertices $u$ and $v$ are adjacent if $u$ and $v$ are contained in some edge.",
"An $e$ is incident with vertex $v$ if $v\\in e$ .",
"An alternating sequence of vertices and edges is called a path if all vertices and edges are distinct, and a cycle if the first and last vertices are the same, the other vertices and all edges are distinct.",
"If there exists a path between any two vertices of $G$ , then $G$ is connected.",
"A hypertree is a connected acyclic hypergraph.",
"A vertex of degree one is called a pendant vertex.",
"For a $k$ -uniform hypergraph $G$ with vertex set $V(G)=\\lbrace 1, \\dots , n\\rbrace $ , its adjacency tensor is the tensor $\\mathcal {A}(G)=(a_{i_1\\ldots i_k})$ of order $k$ and dimension $n$ with $a_{i_1\\ldots i_k}=\\frac{1}{(k-1)!",
"}$ if $\\lbrace i_1\\ldots i_k\\rbrace \\in E(G)$ , 0 otherwise, where $i_j\\in \\lbrace 1, \\dots , n\\rbrace $ and $j\\in \\lbrace 1, \\dots , k\\rbrace $ .",
"For some complex $\\lambda $ , if there exists a nonzero vector $x=(x_1,\\ldots ,x_n)^T$ such that $\\mathcal {A}(G)x=\\lambda x^{k-1}$ , then $\\lambda $ is called an eigenvalue of $G$ and $x$ is called the eigenvector of $G$ corresponding to $\\lambda $ , where $\\mathcal {A}(G)x$ is a $n$ -dimensional vector whose $i$ -th component is $(\\mathcal {A}(G)x)_i=\\sum \\limits _{i_2,\\ldots , i_k=1}^{n}a_{ii_2\\ldots i_k}x_{i_2}\\cdots x_{i_k}$ and $x^{k-1}=(x_1^{k-1},\\ldots ,x_n^{k-1})^T$ , Moreover, if $\\lambda $ and $x$ are both real, then we call $\\lambda $ an $H$ -eigenvalue of $G$ .",
"Note that $x^T(\\mathcal {A}(G)x)=k\\sum _{e\\in E(G)}\\prod _{v\\in e}x_v$ .",
"The spectral radius of a $k$ -uniform hypergraph $G$ , denoted by $\\rho (G)$ , is defined as the maximum modulus of eigenvalues of $\\mathcal {A}(G)$ .",
"Since $\\mathcal {A}(G)$ is symmetric and thus $\\rho (G)$ is the largest $H$ -eigenvalue of $\\mathcal {A}(G)$ , see [13].",
"It is proved in [4], [12] that for a connected $k$ -uniform hypergraph $G$ , $\\mathcal {A}(G)$ has a unique positive eigenvector $x$ with $\\sum _{v\\in V(G)}x_v^k=1$ corresponding to $\\rho (G)$ , which is called the principal eigenvector of $G$ .",
"The problem to determine the hypergraphs in some given classes of hypergraphs with maximum spectral radius received much attention.",
"Li et al.",
"[5] determined the uniform hypertree with maximum spectral radius.",
"Yuan et al.",
"[17] extended to this to determine the first eight uniform hypertrees with maximum spectral radius.",
"Fan et al.",
"[3] determined the hypergraphs with maximum spectral radius among uniform hypergraphs with few edges.",
"Xiao et al.",
"[15] determined the hypertree with maximum spectral radius among uniform hypertrees with a given degree sequence.",
"See for more work on hypergraphs with maximum spectral radius in someclasses of uniform hypergraphs.",
"However, there is much less work on the problem to determine the hypergraphs in some given classes of hypergraphs with minimum spectral radius.",
"Li et al.",
"[5] determined the unique one with minimum spectral radius among $k$ -uniform power hypertrees with fixed number of edges.",
"Here a $k$ -uniform power hypertree is a $k$ -uniform hypertree in which every edge contains at least $k-2$ pendant vertices.",
"Lu and Man [7] classified all connected $k$ -uniform hypergraphs with spectral radius at most $\\@root k \\of {4}$ .",
"There seems no more result in this line.",
"In this note, for $k\\ge 2$ , we show that the $k$ -uniform loose path with $m$ edges is the unique one with minimum spectral radius among connected $k$ -uniform hypergraphs with $m\\ge 1$ edges, and we also determine the unique ones with second minimum spectral radius among connected $k$ -uniform hypergraphs with $m\\ge 2$ edges.",
"To obtain our main result, we use the nice result (Lemma REF ) from [7] and propose a hypergraph transformation that decreases the spectral radius (in Lemma REF )."
],
[
"Preminaries",
"Let $r\\ge 1$ , $G$ be a hypergraph with $u\\in V(G)$ and $e_1,\\ldots ,e_r\\in E(G)$ .",
"Suppose that $v_i\\in e_i$ and $u\\notin e_i$ for $i=1,2,\\ldots ,r$ .",
"Let $e_i^{\\prime }=(e_i\\backslash \\lbrace v_i\\rbrace )\\cup \\lbrace u\\rbrace $ for $i=1,\\ldots ,r$ .",
"Suppose that $e_i^{\\prime }\\notin E(G)$ for $i=1,\\ldots ,r$ .",
"Let $G^{\\prime }$ be the hypergraph obtained from $G$ by deleting $e_1,e_2,\\ldots ,e_r$ and adding $e_1^{\\prime },e_2^{\\prime },\\ldots ,e_r^{\\prime }$ .",
"Then we say that $G^{\\prime }$ is obtained from $G$ by moving $(e_1,e_2,\\ldots ,e_r)$ from $(v_1,v_2,\\ldots ,v_r)$ to $u$ .",
"Lemma 1 [5] Let $r\\ge 1$ , $G$ be a hypergraph with $u\\in V(G)$ and $e_1,\\ldots ,e_r\\in E(G)$ .",
"If $G^{\\prime }$ is obtained from $G$ by moving $(e_1,e_2,\\ldots ,e_r)$ from $(v_1,v_2,\\ldots ,v_r)$ to $u$ , and $x_u\\ge \\max \\limits _{1\\le i\\le r}x_{v_i}$ , then $\\rho (G^{\\prime })>\\rho (G)$ .",
"Lemma 2 [15] Let $G$ be a connected $k$ -uniform hypergraph, and $e=U_1\\cup U_2$ , $f=V_1\\cup V_2$ be two edges of $G$ , where $e\\cap f=\\emptyset $ , and $1\\le |U_1|=|V_1|<k$ .",
"Let $e^{\\prime }=U_1\\cup V_2$ and $f^{\\prime }=V_1\\cup U_2$ .",
"Suppose that $e^{\\prime },f^{\\prime }\\notin E(G)$ .",
"Let $G^{\\prime }$ be the hypergraph obtained from $G$ by deleting edges $e$ and $f$ and adding edges $e^{\\prime }$ and $f^{\\prime }$ .",
"Let $x$ be the principle eigenvector of $G$ .",
"If $x_{U_1}\\ge x_{V_1}$ , $x_{U_2}\\le x_{V_2}$ , and one is strict, then $\\rho (G)<\\rho (G^{\\prime })$ .",
"A path $(u_0,e_1,u_1,\\ldots ,e_p,u_p)$ in a $k$ -unoform hypergraph $G$ is called a pendant path at $u_0$ if $d(u_0)\\ge 2, d(u_i)=2$ for $i=1,2,\\ldots ,p-1$ , $d(u_p)=1$ and $d(u)=1$ for any $u\\in e_i\\setminus \\lbrace u_{i-1},u_i\\rbrace $ with $i=1,2,\\ldots ,p$ .",
"If $p=1$ , then it is a pendant edge at $u_0$ .",
"For a $k$ -uniform hypergraph $G$ with a pendant path $P$ at $u$ , we say that $G$ is obtained from $H$ by attaching a pendant path $P$ at $u$ , where $H=G[V(G)\\setminus (V(P)\\setminus \\lbrace u\\rbrace )]$ .",
"We write $G=H(u,p)$ if the length of $P$ is $p$ .",
"Let $H(u,0)=H$ .",
"For a $k$ -unifoorm hypergraph $G$ with $u\\in V(G)$ , and $p\\ge q\\ge 0$ , let $G_u(p,q)=(G_u(p))_u(q)$ .",
"Lemma 3 [16] Let $u$ be a vertex of a connected $k$ -uniform hypergraph $G$ with $|E(G)|\\ge 1$ .",
"If $p\\ge q\\ge 1$ , then $\\rho (G_u(p,q))>\\rho (G_u( p+1,q-1))$ .",
"Let $G$ be a connected $k$ -uniform hypergraph with $u,v\\in V(G)$ , and $p\\ge q\\ge 0$ , let $G_{u,v}(p,q)=(G_u(p))_v(q)$ .",
"Lemma 4 Let $G$ be a $k$ -uniform hypergraph with $k\\ge 3$ .",
"Let $e$ be a pendant edge of $G$ , and $u$ and $v$ be two pendant vertices in $e$ .",
"If $p\\ge q\\ge 1$ , then $\\rho (G_{u,v}(p,q))>\\rho (G_{u,v}(p+1,q-1))$ .",
"Proof.",
"Suppose that $P=(u_1,e_1,u_2,e_2,\\ldots $ , $u_{p+1}, e_{p+1},u_{p+2})$ and $Q=(v_1,f_1,$ $v_2,f_2,\\ldots , v_{q-1}, f_{q-1}, v_{q})$ are two pendant paths of $G_{u,v}(p+1,q-1)$ at $u$ of length $p+1$ and at $v$ of length $q-1$ , respectively, where $u=u_1$ and $v=v_1$ .",
"If $q=1$ , then let $Q=(v_1)$ .",
"Suppose that $\\rho (G_{u,v}(p,q))\\le \\rho (G_{u,v}(p+1,q-1))$ .",
"Let $x$ be the principal eigenvector of $G_{u,v}(p+1,q-1)$ .",
"Suppose that $x_{u_{p+1}}\\le x_{v_{q}}$ .",
"Let $G^{\\prime }$ be the hypergraph obtained from $G_{u,v}(p+1,q-1)$ by moving edge $e_{p+1}$ from $u_{p+1}$ to $v_{q}$ .",
"It is obvious that $G^{\\prime }\\cong G_{u,v}(p,q)$ .",
"By Lemma REF , we have $\\rho (G_{u,v}(p,q))=\\rho (G^{\\prime })> \\rho (G_{u,v}(p+1,q-1))$ , a contradiction.",
"Hence $x_{u_{p+1}}> x_{v_{q}}$ .",
"Suppose that $i\\ge 1$ and $x_{u_{p+1-i}}>x_{v_{q-i}}$ for $i\\le q-2$ .",
"We want to show that $x_{u_{p-i}}>x_{v_{q-i-1}}$ .",
"Suppose that $x_{u_{p-i}}\\le x_{v_{q-i-1}}$ .",
"If $x_{e_{p-i}\\setminus \\lbrace u_{p-i},u_{p-i+1}\\rbrace }>x_{f_{q-i-1}\\setminus \\lbrace v_{q-i-1},v_{q-i}\\rbrace }$ .",
"Let $U_1=e_{p-i}\\setminus \\lbrace u_{p-i}\\rbrace $ and $V_1=f_{q-i-1}\\setminus \\lbrace v_{q-i-1}\\rbrace $ .",
"Let $G^{\\prime }$ be the hypergraph $G^{\\prime }$ obtained from $G_{u,v}(p+1,q-1)$ by deleting edges $e_{p-i}$ and $f_{q-i-1}$ and adding edges $e^{\\prime }_{p-i}$ and $f^{\\prime }_{q-i-1}$ , where $e^{\\prime }_{p-i}=U_1\\cup (f_{q-i-1}\\setminus V_1)$ and $f^{\\prime }_{q-i-1}=V_1\\cup (e_{p-i}\\setminus U_1)$ .",
"Note that $G^{\\prime }\\cong G_{u,v}(p,q)$ .",
"We have by Lemma REF that $\\rho (G_{u,v}(p,q))=\\rho (G^{\\prime })> \\rho (G_{u,v}(p+1,q-1))$ , a contradiction.",
"Thus $x_{e_{p-i}\\setminus \\lbrace u_{p-i},u_{p-i+1}\\rbrace }\\le x_{f_{q-i-1}\\setminus \\lbrace v_{q-i-1},v_{q-i}\\rbrace }$ .",
"Now let $U_1=\\lbrace u_{p-i+1}\\rbrace $ and $V_1=\\lbrace v_{p-i}\\rbrace $ .",
"Let $G^{\\prime }$ be the hypergraph obtained from $G_{u,v}( p+1,q-1)$ by deleting edges $e_{p-i}$ and $f_{q-i-1}$ and adding edges $e^{\\prime }_{p-i}$ and $f^{\\prime }_{q-i-1}$ , where $e^{\\prime }_{p-i}=V_1\\cup (e_{p-i}\\setminus U_1)$ and $f^{\\prime }_{q-i-1}=U_1\\cup (f_{q-i-1}\\setminus V_1)$ .",
"Note that $G^{\\prime }\\cong G_{u,v}(p,q)$ .",
"By Lemma REF , we have $\\rho (G^{\\prime })=\\rho (G_{u,v}(p,q))> \\rho (G_{u,v}(p+1,q-1))$ , also a contradiction.",
"It follows that $x_{u_{p-i}}>x_{v_{q-i-1}}$ , as desired.",
"Let $e=\\lbrace u_1,v_1,w_1,w_2,\\dots , w_{k-2}\\rbrace $ with $d_G(w_1)\\ge 2$ and let $e_{p-q+1}=\\lbrace u_{p-q+1},u_{p-q+2}, w^{\\prime }_1,$ $w^{\\prime }_2,\\dots , w^{\\prime }_{k-2}\\rbrace $ .",
"Clearly, $x_{w_2}=\\cdots =x_{w_{k-2}} $ and $x_{w^{\\prime }_1}=\\cdots =x_{w^{\\prime }_{k-3}}$ .",
"If $x_{w_1}\\le x_{w_1^{\\prime }}$ , then we can obtain a $G^{\\prime }$ from $G_{u,v}(p+1,q-1)$ by moving all the edges except $e$ incident with $w_1$ from $w_1$ to $w_1^{\\prime }$ .",
"By Lemma REF and the fact that $G^{\\prime }\\cong G_{u,v}(p,q)$ , we have $\\rho (G_{u,v}(p,q))> \\rho (G_{u,v}(p+1,q-1))$ , a contradiction.",
"Hence $x_{w_1}>x_{w_1^{\\prime }}$ .",
"Suppose that $x_{e\\setminus \\lbrace v_1,w_1\\rbrace }\\ge x_{e_{p-q+1}\\setminus \\lbrace u_{p-q+2},w_1^{\\prime }\\rbrace }$ .",
"Let $U_1=e\\setminus \\lbrace v_1\\rbrace $ and $V_1=e_{p-q+1}\\setminus \\lbrace u_{p-q+2}\\rbrace $ .",
"Thus we can form a hypergraph $G^{\\prime }$ from $G_{u,v}(p+1,q-1)$ by deleting edges $e$ and $e_{p-q+1}$ and adding edges $e^{\\prime }$ and $e^{\\prime }_{p-q+1}$ , where $e^{\\prime }=U_1\\cup (e_{p-q+1}\\setminus V_1)$ and $e^{\\prime }_{p-q+1}=V_1\\cup (e\\setminus U_1)$ .",
"It is obvious that $G^{\\prime }\\cong G_{u,v}(p,q)$ .",
"By Lemma REF , we have $\\rho (G_{u,v}(p,q))> \\rho (G_{u,v}(p+1,q-1))$ , a contradiction.",
"Thus $x_{e\\setminus \\lbrace v_1,w_1\\rbrace }< x_{e_{p-q+1}\\setminus \\lbrace u_{p-q+2},w_1^{\\prime }\\rbrace }$ .",
"Now let $U_1=\\lbrace w_1\\rbrace $ and $V_1=\\lbrace w_1^{\\prime }\\rbrace $ .",
"Let $G^{\\prime }$ be the hypergraph obtained from $G_{u,v}(p+1,q-1)$ by deleting edges $e$ and $e_{p-q+1}$ and adding edges $e^{\\prime }$ and $e^{\\prime }_{p-q+1}$ , where $e^{\\prime }=V_1\\cup (e\\setminus U_1)$ and $e^{\\prime }_{p-q+1}=U_1\\cup (e_{p-q+1}\\setminus V_1)$ .",
"Note that $G^{\\prime }\\cong G_{u,v}(p,q)$ .",
"By Lemma REF , we have $\\rho (G^{\\prime })=\\rho (G_{u,v}(p,q))> \\rho (G_{u,v}(p+1,q-1))$ , also a contradiction.",
"We complete the proof.",
"$\\Box $ A hypergraph $G$ is said to be reducible if every edge contains at least one pendant vertex.",
"For a reducible $k$ -uniform hypergraph $G$ with $e\\in E(G)$ , let $v_e$ be a pendant vertex in $e$ , and let $G^{\\prime }$ be the hypergraph with $V(G^{\\prime })=V(G)\\setminus \\lbrace v_e: e\\in E(G)\\rbrace $ and $E(G^{\\prime })=\\lbrace e\\setminus \\lbrace v_e\\rbrace : e\\in E(G)\\rbrace $ .",
"We say that $G^{\\prime }$ is reduced from $G$ .",
"Lemma 5 [7] Let $G$ be a reducible $k$ -uniform hypergraph.",
"If $G^{\\prime }$ is reduced from $G$ , then $\\rho ^k(G)=\\rho ^{k-1}(G^{\\prime })$ .",
"For a $k$ -uniform hypertree $G$ with $E(G)=\\lbrace e_1, \\dots , e_m\\rbrace $ , if $V(G)=\\lbrace v_1, \\dots , v_n\\rbrace $ with $n=(k-1)m+1$ , and $e_i=\\lbrace v_{(i-1)(k-1)+1}, \\dots , v_{(i-1)(k-1)+k}\\rbrace $ for $i=1, \\dots , m$ , then we call $G$ a $k$ -uniform loose path, denoted by $P_m^{(k)}$ .",
"For $k\\ge 2$ and $m\\ge 3$ , let $D_m^{(k)}$ be the $k$ -uniform hypertree obtained from a $k$ -uniform loose path $P_{m-1}^{(k)}=\\left(u_0,e_1,u_1,\\dots ,e_{m-1},u_{m-1}\\right)$ by attaching a pendant edge at $u_1$ .",
"For $k\\ge 3$ and $m\\ge 3$ , let $D_m^{\\prime (k)}$ be the $k$ -uniform hypertree obtained from a $k$ -uniform loose path $P_{m-1}^{(k)}=\\left(u_0,e_1,u_1,\\dots ,e_{m-1},u_{m-1}\\right)$ by attaching a pendant edge at a vertex of degree 1 in $e_2$ .",
"Lemma 6 [7] Let $G$ be a $k$ -uniform hypergraph with $k\\ge 3$ .",
"$(i)$ If $k=3$ and $\\rho (G)<\\@root k \\of {4}$ , then $G$ is isomorphic to one of the following hypergraphs: $P_m^{(3)}$ for $m\\ge 1$ , $D_m^{(3)}$ for $m\\ge 3$ , $D_m^{\\prime (3)}$ for $m\\ge 4$ , $B_m^{(3)}$ for $m\\ge 5$ , $B_m^{\\prime (3)}$ for $m\\ge 6$ , $\\bar{B}_m^{(3)}$ for $m\\ge 7$ , $BD_m^{(3)}$ for $m\\ge 5$ , and thirty-one additional hypergraphs: $E_{1,2,2}^{(3)}$ , $E_{1,2,3}^{(3)}$ , $E_{1,2,4}^{(3)}$ , $F_{2,3,3}^{(3)}$ , $F_{2,2,l}^{(3)}$ $($ for $2\\le l\\le 6$$)$ , $F_{1,3,l}^{(3)}$ $($ for $3\\le l\\le 13$$)$ , $F_{1,4,l}^{(3)}$ $($ for $4\\le l\\le 7$$)$ , $F_{1,5,5}^{(3)}$ , and $G_{1,1:l:1,3}^{(3)}$ $($ for $0\\le l\\le 5$$)$ $($ see Figure REF $)$ .",
"$(ii)$ If $k=4$ , $G$ is not reducible, and $\\rho (G)\\le \\@root k \\of {4}$ , then $G\\cong H_{1,1,1, t}$ with $t=1,2,3,4$ $($ see Figure REF $)$ .",
"$(iii) $ If $k\\ge 5$ and $\\rho (G)\\le \\@root k \\of {4}$ , then $G$ is reducible.",
"Figure: Hypergraphs in Lemma (i)Figure: HypergraphsH 1,1,1,i H_{1,1,1,i} for i=1,2,3,4i=1,2,3,4 in Lemma (ii).We remark that in Figure REF , $E^{(3)}_{i,j,l}$ consists of three pendant paths of length $i$ , $j$ and $l$ at a common vertex, $F^{(3)}_{i,j,l}$ consists of three pendant paths of length $i$ , $j$ and $l$ at three different vertices of a single edge, and $G_{i,j:l:p,q}^{(3)}$ consists of a 3-uniform loose path of length $l+2$ with two pendant paths of length $i$ and $j$ at two pendant vertices in the first edge and two pendant paths of length $p$ and $q$ at two pendant vertices in the last edge."
],
[
"Result",
"Now we are ready to show our main result.",
"Theorem 1 Let $G$ be a connected $k$ -uniform hypergraph with $m\\ge 4$ edges, where $k\\ge 3$ .",
"Suppose that $G\\lnot \\cong P_m^{(k)}$ .",
"Then $\\rho (G)\\ge \\rho (D^{\\prime (k)}_m)$ with equality if and only if $G\\cong D^{\\prime (k)}_m$ .",
"Proof.",
"By Lemma REF (i) , $\\rho ( D_m^{\\prime (3)})<\\@root 3 \\of {4}$ .",
"Then by Lemma REF , we have $\\rho ^k(D_m^{\\prime (k)})=\\rho ^3( D_m^{\\prime (3)})<4$ , and thus $\\rho (D_m^{\\prime (k)})<\\@root k \\of {4}$ .",
"Let $G$ be a connected $k$ -uniform hypergraph with $m\\ge 4$ edges and $G\\lnot \\cong P_m^{(k)}$ having minimum spectral radius.",
"We need only to show that $G\\cong D^{\\prime (k)}_m$ .",
"Since $\\rho (D_m^{\\prime (k)})<\\@root k \\of {4}$ , we have $\\rho (G)<\\@root k \\of {4}$ .",
"Case 1.",
"$k=3$ .",
"By Lemma REF (i) , $G$ is isomorphic to one of the following hypergraphs: $P_m^{(3)}$ for $m\\ge 1$ , $D_m^{(3)}$ for $m\\ge 3$ , $D_m^{\\prime (3)}$ for $m\\ge 4$ , $B_m^{(3)}$ for $m\\ge 5$ , $B_m^{\\prime (3)}$ for $m\\ge 6$ , $\\bar{B}_m^{(3)}$ for $m\\ge 7$ , $BD_m^{(3)}$ for $m\\ge 5$ , and thirty-one additional hypergraphs: $E_{1,2,2}^{(3)}$ , $E_{1,2,3}^{(3)}$ , $E_{1,2,4}^{(3)}$ , $F_{2,3,3}^{(3)}$ , $F_{2,2,l}^{(3)}$ $($ for $2\\le l\\le 6$$)$ , $F_{1,3,l}^{(3)}$ $($ for $3\\le l\\le 13$$)$ , $F_{1,4,l}^{(3)}$ $($ for $4\\le l\\le 7$$)$ , $F_{1,5,5}^{(3)}$ , and $G_{1,1:l:1,3}^{(3)}$ $($ for $0\\le l\\le 5$$)$ .",
"By Lemma REF , we have $\\rho (D_m^{(3)})>\\rho (D^{\\prime (3)}_m)$ .",
"By Lemma REF , we have $\\rho (BD_m^{(3)})>\\rho (D^{\\prime (3)}_m)$ , and for $E^{(3)}_{1,2,l}$ with $l=2,3,4$ , we have $m=l+3$ , and thus $\\rho (E^{(3)}_{1,2,l})>\\rho (D_{l+3}^{(3)})>\\rho (D^{\\prime (3)}_{m})$ .",
"By Lemma REF , we have $\\rho (B_m^{(3)})>\\rho (D^{\\prime (3)}_m)$ , $\\rho (B^{\\prime (3)}_m)>\\rho (D^{\\prime (3)}_m)$ , and $\\rho (\\bar{B}_m^{(3)})>\\rho (D^{\\prime (3)}_m)$ .",
"For $F_{i,j,l}^{(3)}$ , we have $m=i+j+l+1$ .",
"By Lemma REF , we have $\\rho (F_{i,j,l}^{(3)})>\\rho (F_{i,1,m-i-2}^{(3)})>\\rho (F_{1,1,m-3}^{(3)})=\\rho (D^{\\prime (3)}_{m})$ .",
"Thus $\\rho (F_{2,3,3}^{(3)})>\\rho (D^{\\prime (3)}_{9})$ , $\\rho (F_{2,2,m-5}^{(3)})>\\rho (D^{\\prime (3)}_{m})$ for $(7\\le m\\le 11)$ , $\\rho (F_{1,3,m-5}^{(3)})>\\rho (D^{\\prime (3)}_{m})$ (for $8\\le m\\le 18$ ), $\\rho (F_{1,4,m-6}^{(3)})>\\rho (D^{\\prime (3)}_{m})$ (for $10\\le m\\le 13$ ), and $\\rho (F_{1,5,5}^{(3)})>\\rho (D^{\\prime (3)}_{12})$ .",
"By Lemma REF , we have $\\rho (G_{1,1:m-8:1,3}^{(3)})>\\rho (F_{1,1,m-3}^{(3)})=\\rho (D^{\\prime (3)}_{m})$ for $8\\le m\\le 13$ .",
"Therefore, if $G\\lnot \\cong D^{\\prime (k)}_m$ , then $\\rho (G)>\\rho (D^{\\prime (3)}_m)$ .",
"It follows that $G\\cong D^{\\prime (3)}_m$ .",
"Case 2.",
"$k=4$ .",
"If $G$ is reducible, then by Lemma REF , for the hypergraph $G_1$ reduced from $G$ , we have $\\rho (G_1)<\\@root 3 \\of {4}$ , and by the proof in Case 1, we have $G_1\\cong D^{\\prime (3)}_m$ , implying that $G\\cong D^{\\prime (4)}_m$ .",
"Next suppose that $G$ is not reducible.",
"Then by Lemma REF (ii), $G\\cong H_{1,1,1, i}$ with $i=1,2,3,4$ .",
"We will show that these are impossible.",
"Since $\\rho (D_m^{(3)})>\\rho (D^{\\prime (3)}_m)$ , by Lemma REF we have $\\rho (D_m^{(4)})>\\rho (D^{\\prime (4)}_m)$ .",
"Thus it suffices to show that $\\rho (G)>\\rho (D^{(4)}_{m})$ .",
"Suppose that $G\\cong H_{1,1,1,4}$ .",
"Then $m=8$ .",
"From the table of [2] we have $\\rho (D_{8}^{(2)})=1.962$ .",
"By Lemma REF we have $\\rho (D_{8}^{(4)})=(\\rho (D_{8}^{(2)})^{\\frac{2}{4}}$ , implying that $(\\rho (D_{8}^{(4)}))^4$ $=3.8494$ .",
"Let $x$ be the principal eigenvector of $G$ , and let $u_i,w_i$ for $i=1,2,3,4,5$ and $v_1,v_2$ be the vertices of $H_{1,1,1,4}$ as labeled in Figure REF ).",
"Let $\\rho (G)=\\rho $ and $x_{u_i}=x_i$ for $i=1,2,3,4,5$ .",
"Then $x_{v_1}=x_{v_2}=x_{u_1}=x_1$ .",
"We have $\\rho x_{w_1}^4=x_{w_1}^3x_1$ , and then $x_{w_1}=\\frac{x_1}{\\rho }$ .",
"Similarly, we have that $x_{w_5}=\\frac{x_5}{\\rho }$ , $x_{w_i}=\\sqrt{\\frac{x_ix_{i+1}}{\\rho }}$ for $i=2,3,4$ .",
"Thus $\\begin{aligned}\\rho x_1^4=\\frac{x_1^3}{\\rho ^3}x_1+x_1^3x_2,\\\\\\rho x_2^4=x_1^3x_2+\\frac{x_2^2x_3^2}{\\rho },\\\\\\rho x_3^4=\\frac{x_2^2x_3^2}{\\rho }+\\frac{x_3^2x_4^2}{\\rho },\\\\\\rho x_4^4=\\frac{x_3^2x_4^2}{\\rho }+\\frac{x_4^2x_5^2}{\\rho },\\\\\\rho x_5^4=\\frac{x_4^2x_5^2}{\\rho }+\\frac{x_5^3}{\\rho ^3}x_5.\\\\\\end{aligned}$ From the first two equations, we have $(\\rho ^2-\\frac{\\rho ^{10}}{(\\rho ^4-1)^3})x_2^2=x_3^2$ , and from the the other three equations, we have $x_2^2=\\rho ^2x_3^2-x_4^2=(\\rho ^6-3\\rho ^2+\\frac{1}{\\rho ^2})x_5^2$ and $x_3^2=\\rho ^2x_4^2-x_5^2=(\\rho ^4-2)x_5^2$ .",
"Thus $(\\rho ^4)^5-8(\\rho ^4)^4+21(\\rho ^4)^3-23(\\rho ^4)^2+13(\\rho ^4)-3=0.$ Since $P_{6}^{(4)}$ is a subhypergraph of $G$ , we have $\\rho ^4>\\rho ^4(P_6^{(4)})=(\\sqrt{2\\cos \\frac{\\pi }{8}})^4=2+\\sqrt{2}$ .",
"Let $f(t)=t^5-8t^4+21t^3-23t^2+13t-3$ .",
"Note that $f(\\frac{1}{2})=-\\frac{3}{2^5}<0, f(2-\\sqrt{2})=-7+5\\sqrt{2}>0, f(1)=1>0,f(2+\\sqrt{2})=-7-5\\sqrt{2}<0,f(3.9)=-4.94181<0$ , and $f(4)=1>0$ .",
"Thus $f(t)=0$ has three real roots $t_1, t_2$ and $t_3$ satisfying $\\frac{1}{2}<t_3<2-\\sqrt{2}$ , $1<t_2<2+\\sqrt{2}$ , and $3.9<t_1<4$ .",
"Let $t_4$ and $t_5$ be the remaining two roots of $f(t)=0$ .",
"Then $t_4t_5>0$ , $t_4+t_5>8-(2-\\sqrt{2})-(2+\\sqrt{2})-4=0$ , and $t_4+t_5<8-\\frac{1}{2}-1-3.9=2.6<2+\\sqrt{2}$ .",
"Note that $\\rho ^4>2+\\sqrt{2}$ .",
"So whether $t_4$ and $t_5$ are real or not, they can not be equal to $\\rho $ .",
"Thus $\\rho ^4=t_1>3.9>3.8494= (\\rho (D^{(4)}_8)^4$ , i.e., $\\rho (G)>\\rho (D^{(4)}_8)$ , as desired.",
"In the following, we consider the cases when $G\\cong H_{1,1,1,i}$ for $i=1,2,3$ .",
"From the table of [2] we have $\\rho (D_{5}^{(2)})=1.902$ , $\\rho (D_{6}^{(2)})=1.932$ , and $\\rho (D_{7}^{(2)})=1.950$ .",
"Then by Lemma REF we have $\\rho (D_{5}^{(4)})=1.3791$ , $(\\rho (D_{6}^{(4)}))^4=3.733$ , $(\\rho (D_{7}^{(4)}))^4=3.8025$ .",
"By similar but simpler argument as above, we have $\\rho (H_{1,1,1,1})$ , $\\rho (H_{1,1,1,2}^{(4)})$ , and $\\rho (H_{1,1,1,3})$ are roots of $\\rho ^4-\\rho ^3-1=0$ , $(\\rho ^4)^4-6(\\rho ^4)^3+10(\\rho ^4)^2-7(\\rho ^4)+2=0$ , and $(\\rho ^4)^5-7(\\rho ^4)^4+15(\\rho ^4)^3-13(\\rho ^4)^2+6(\\rho ^4)-1=0$ , respectively.",
"And we may check that $\\begin{aligned}&\\rho (G)> 1.38>1.3791= \\rho (D^{(4)}_5) & \\mbox{ if }i=1, \\\\&\\rho ^4(G)> 3.8>3.733=\\rho ^4(D^{(4)}_6) & \\mbox{ if }i=2,\\\\&\\rho ^4(G)> 3.9>3.8025=\\rho ^4(D^{(4)}_7) & \\mbox{ if }i=3,\\end{aligned}$ i.e., $\\rho (G)>\\rho (D^{(4)}_m)$ , as desired.",
"Case 3.",
"$k\\ge 5$ .",
"By Lemma REF (iii) , $G$ is reducible.",
"By Lemma REF , for the hypergraph $G_1$ reduced from $G$ , we have $\\rho (G_1))<\\@root k-1 \\of {4}$ .",
"Repeating this process by using Lemmas REF (iii) and REF , we have a hypergraph sequence $G_0, G_1, \\dots , G_{k-4}$ with $G_0=G$ , where $G_{i+1}$ is reduced from $G_i$ for $i=0, \\dots , k-5$ .",
"It is easily seen that $G_{k-4}$ is 4-uniform and $\\rho ^k(G)=\\rho ^4(G_{k-4})<4$ .",
"By the proof of Case 2, $G_{k-4}\\cong D^{(4)}_m$ , and thus $G\\cong D^{(k)}_m$ .",
"$\\Box $ Among connected 2-uniform hypergraphs with $m$ edges, the ones with spectral radius less than 2 are determined in [8] to be the trees $P_m^{(2)}$ , $D_m^{(2)}$ , and three additional trees with $m=5,6,7$ , obtained from $D_{m-1}^{(2)}$ by attaching a pendant edge at a pendant vertex that is adjacent to a vertex of degree 3, and by Lemma REF , it is easy to see that $P_m^{(2)}$ for $m\\ge 1$ is the unique one with minimum spectral radius, while $D_m^{(2)}$ for $m\\ge 3$ is the unique one with second minimum spectral radius.",
"Let $G$ be a connected $k$ -uniform hypergraph with 2 edges, where $k\\ge 3$ .",
"Let $a$ be the number of common vertices of the two edges.",
"Obviously, $1\\le a \\le k-1$ .",
"By direct calculation, $\\rho (G)=2^{\\frac{a}{k}}$ .",
"Therefore $P_2^{(k)}$ and the hypergraph in which two edges share two vertices in common are the unique hypergraphs with minimum and second minimum spectral radii among connected $k$ -uniform hypergraphs with exactly 2 edges.",
"Let $G$ be a connected $k$ -uniform hypergraphs with 3 edges, where $k\\ge 3$ .",
"If there is a subhypergraph consisting two edges containing at least two vertices in common, then $\\rho (G)\\ge \\@root k \\of {4}$ .",
"If any two edges of $G$ contain at most one common vertex, then $G$ is a cycle of length 3, $D_3^{(k)}$ or $P_3^{(k)}$ .",
"If $G$ is a cycle of length 3, then $\\rho (G)=\\@root k \\of {4}$ .",
"By Lemma REF , $\\rho (D_3^{(k)})>\\rho (P_3^{(k)})$ .",
"By Lemmas REF and REF (i) , $\\rho ^k(D_3^{(k)})=\\rho ^3(D_3^{(3)})<4$ .",
"Therefore $P_3^{(k)}$ and $D_3^{(k)}$ are the unique hypergraphs with minimum and second minimum spectral radii among connected $k$ -uniform hypergraphs with exactly 3 edges.",
"For $m\\ge 4$ and $k\\ge 3$ , by Lemma REF , we have $\\rho (D^{\\prime (k)}_m)>\\rho (P_m^{(k)})$ .",
"Combining the above facts and Theorem REF , we have Theorem 2 Among connected $k$ -uniform hypergraphs with $m$ edges, $P_m^{(k)}$ for $m\\ge 1$ is the unique one with minimum spectral radius, and the hypergraph in which two edges share two vertices in common for $m=2$ and $k\\ge 3$ , $D_3^{(k)}$ for $k=2$ or $m=3$ , and $D^{\\prime (k)}_m$ for $m\\ge 4$ and $k\\ge 3$ are the unique ones with second minimum spectral radius.",
"Acknowledgements This work was supported by National Natural Science Foundation of China (No.",
"11071089 and No.",
"11701102), Natural Science Foundation of Guangdong Province (No.",
"2017A030313032 and No.",
"2017A030310441)."
]
] | 1808.08590 |
[
[
"Understanding the effect of the base oil on the physical adsorption\n process of organic additives using molecular dynamics"
],
[
"Abstract Organic friction modifiers (OFMs) are widely added to oil to reduce the boundary friction in many kinds of lubricants such as vehicle engine oils.",
"At the contact area in machine elements, the OFMs form a self-assembled organic monolayer.",
"Although the friction properties of the monolayer are widely studied on a molecular level, the formation process is not well-known.",
"In this study, the initial adsorbing process of additive molecules in explicit base oil molecules are calculated using molecular dynamics.",
"The adsorption time depends on the structure of the base oils.",
"Another effect of the base oil other than \"chain matching\" is found."
],
[
"Introduction",
"In machine elements, there are many interfaces that perform relative motions, and it is necessary to control the friction and wear generated between the two surfaces.",
"The control of friction and wear leads to the suppression of energy loss, vibration, improvement of function, performance, and reliability at the entire mechanical system.",
"In order to inhibit the contact between the two surfaces, hydrodynamic lubrication by oil may be the best solution.",
"There are, however, many systems that cannot maintain the hydrodynamic lubrication film due to the sliding condition.",
"A boundary lubrication film is the candidate to decrease the friction in such solid contacts.",
"Organic friction modifiers (OFMs) or oiliness agents are some of the most popular additives which produce a boundary lubrication film [1].",
"OFM additive molecules are composed of alkyl chains and polar groups on the end.",
"Fatty acids are the first used OFMs [2].",
"The polar group adsorbs on the metal surface and the molecules form a self-assembled monolayer (SAM).",
"The mechanical and physiochemical stability of the monolayer is important to protect the metal surface and to reduce friction.",
"In the long history of the friction modifiers, OFMs are the most common, which have been used since the 1920s, and widely studied [2].",
"The most important idea is the formation of an adsorbed self-organized layer by Bowden and Tabor [1].",
"After this model, the concept of \"chain matching” by Askwith et al.",
"[3] is the important finding in order to consider the effect of the base oil (solvent).",
"If the chain length of the fatty acid additive is same as the chain length of the base oil, the system shows a lower friction than other base oils.",
"Although this idea is supported by experiments, such as the experiments focused on nucleation and crystal growth [4], the precise mechanism of low friction is still an open question [5].",
"Molecular dynamics (MD) simulations are used to understand the friction phenomena of SAMs, since the MD is the most useful tool to understand the dynamics of SAMs and comparable with the experimental data obtained by a surface force apparatus (SFA) or atom force microscope (AFM) [2].",
"The MD simulation demonstrated that the friction behavior of the SAMs obtained by AFM experiments is well described by the simulation between the hard AFM tip and the soft SAMs [6], [7], [8], [9].",
"This was one of the most studied friction systems using MD during the early stage of the nano tribology studies.",
"Recent nonequilibrium molecular dynamics (NEMD) studies show moer detailed structure and friction behavior of OFMs [10], [11], [12], and NEMD of OFM reverse micelles adsorption behaviour under shear [13], [13].",
"Tight-binding quantum chemistry calculations [15], [16] and density functional theory claculations [17].",
"have also been used to study OFM adsorption [15], [16], [17].",
"These methods are required to accurately reproduce strong chemisorption observed in XPS experiments [16], but are computationally expensive and limited in terms of accessible system size and timescales.",
"The formation process of the monolayers, on the other hand, is not yet well known.",
"In experiments, adsportion process is observed by polarized neutron reflectometry [18] or AFM [19], however, the observation of the initial process is difficult.",
"The major part of the tribological use of the OFMs used in the solution of the base oil is more than 90 %, and a small amount of additives.",
"In the engine oil of the automobile, the percentage of the OFMs are only a few percent.",
"It is known that only 0.5 % OFMs makes the friction lowering effect significantly [2].",
"When simulating the solution including 4 % additives by MD, about 96 % of the calculation cost is devoted to the calculation of the fluctuation of the base oil molecules.",
"The formation process of the SAMs using a coarse-grained MD simulation is proposed [21].",
"Both the adsorption process of the additives and the formation of SAM by aggregation of the adsorbed molecules are simulated.",
"In the simulation, however, the solvent molecules are described as a single sphere molecule using the Lennard-Jones attractive or soft-core repulsive interactions.",
"The effect of the structure of solvent molecules can not be discussed by the sphere model in the coarse-grained simulation.",
"In order to directly know the solvent effect, such as chain matching, the all-atom MD simulation is more appropriate.",
"All-atom force-fields have also important to accurately model OFM film strucure and friction [20].",
"In this study, the all-atom MD simulation is used to solve the solvent (base oil) effect on the formation process of the OFM film.",
"Since the calculation cost of the base oil is high, only the beginning process of the adsorprtion is discussed here.",
"However, an interesting effect, i.e., the adsorption time varies with the structure of the layer of the base oil molecules is found."
],
[
"Simulation Methods",
"All-atom MD simulations of the adsorption process of additives (OFMs) are done in the following manner.",
"Schematic picture of the simulation model is shown in Fig.",
"REF , and step by step procedure for preparing the model solution is shown in Fig.",
"REF .",
"The solution with the hydrocarbon base oil and additives is confined by two solid layers.",
"The top solid layer is a neutral Fe wall and the bottom solid layer is a charged Fe wall (Fig.",
"REF ).",
"The Fe walls are modeled as a universal model of solid materials.",
"Then the charge on the bottom solid layer is put, then the adsorption process was simulated.",
"The precise conditions are described as follows.",
"First, three types of solutions of the base oil with additives under standard conditions is prepared.",
"The base oil (solvent) of n-hexadecane, 2, 4-dimethyltetradecane, and 3, 5-dimethydodecane with the additive (solute) of palmitic acid was chosen as the linear and branched base oils with the solute of the same number of carbons.",
"The latter branched base oils are taken from the structure of the Olefin Copolymer, or Poly $\\alpha $ olefin.",
"In the MD simulation, the organic molecules are dynamically treated using the Dreiding force field [22].",
"This all-atom force field include bonds, angles, dihiderals (torsions) and improper torsions for the intra-molecular forces, and Lennard-Jones and Coulomb for the inter-molecular forces.",
"The partial atomic charges on the organic molecules are determined using the MOPAC6 [23] semi-empirical molecular orbital calculation with the Hamiltonian:AM1.",
"A set of organic molecules is first arranged in a lattice configuration.",
"96 base oil molecules and 4 solute molecules are arranged in order to set the concentration of the additive at 4 % (Fig.",
"REF (a)), which is the commonly used condition such as in engine oil for automobiles.",
"The size of the simulation box was 27.31 Å along the $x$ axis, 32.77 Å along the $y$ axis and an arbitrary length for the $z$ axis, since the system is then pressed.",
"The molecules are then moved by the MD simulation under the periodic boundary condition in the $x,y$ directions, and reflective wall in the $z$ direction, for 500 ps at the constant temperature of 1,000 K in order to anneal the system (Fig.",
"REF (b)).",
"The MD simulation is then done for 10 ns in order to cool the system to 300 K. The temperature is controlled using the Nose-Hoover thermostat [26], [27], [28].",
"Then the system is pressed to the $z$ direction so that the density of the base oil is the same as the experimental value [29].",
"The solution in a thermal equilibrium is obtained (Fig.",
"REF (c)).",
"Next, 3 sets of thermal equilibrium fluids are arranged in the $z$ direction between the two solid plates, (Fig.",
"REF (d)), in order to form a semi-bulk region in the center of the oil film.",
"The final thickness of the solution in the $z$ direction is 194.19 Å.",
"On the $x$ and $y$ axes, periodic boundary conditions are adopted.",
"On the $z$ axe, non-periodic boundary conditions are adopted.",
"All the organic molecules are connected to a Nose-Hoover thermostat.",
"The long-range Coulombic interactions are calculated by the Multi-Summation Method [30].",
"This method enables to simulate a non-periodic system.",
"The equations of motion were integrated using a velocity-Verlet method [31] with a time step of 2.0 fs.",
"This time step is long for all-atom simulation.",
"During the simulation, a drift of conserved quantity such as temperature are not found.",
"Since the phenomena is governed by long-range coulomb interaction between the surface and the functional group, and the dynamics of not the hydrogen-carbon but carbon-carbon bonds are critical, we think this phenomena will reproduce at least qualitatively in shorter time step.",
"Each system was calculated five times, while changing the initial speed.",
"The adsorption dynamics changes drastically by initial speed.",
"The universal solid plate (wall) to analyze the adsorption process is made by the following procedure.",
"We propose this in order to treat both charged oxidized metal surfaces and the neutral metal surface with the least difference, i.e., the existance of a charge on the surface.",
"If the chemical reality of the surface, including the chemical reactions is in interest, a more realistic model such as copper and copper oxide surface treated by reactive force field should be used [32].",
"We are now interested in the physical nature of the interface and this model succeeded in reproducing the formation of the elastohydrodynamic lubrication oil film [33], [34], [35].",
"The parameter for the solid is taken from an alpha-ferrous crystal, a solid atom layer with a lattice of 11 $\\times $ 7 $\\times $ 5 atoms in the $x,y$ and $z$ directions, respectively, and the lattice parameter is set to 2.87 Å.",
"The fluid-solid interface is the (110) surface, and the vibrations of each solid atom are suppressed.",
"These atoms are not connected to the Nose-Hoover thermostat.",
"The parameter for the solid atoms is then set to 0.2853 Kcal/mol [33].",
"This parameter is taken from previous study [36] and it is known that the macroscopic slip between solid atoms and fluid molecules under sliding motion are supressed [33].",
"We then arranged only the bottom solid wall which is charged.",
"The charge distribution on the surface of the solid wall is shown in Fig.",
"REF .",
"The outermost layer is charged at +1e, and the second layer is charged at -1e in order to make system electrical neutral."
],
[
"Results and Discussion",
"Figure REF shows snapshots of the adsorption process of the palmitic acid molecule in the n-hexadecane solution on the charged surface.",
"It is obvious that the base oil molecules construct the absorbed layer in the vicinity of the solid wall.",
"The phenomena are suggested by many studies [33], [34], [37], [38], [39].",
"The fluid molecules show an oscillating density profile at the interface of the solid atoms [38] due to the cohesive force arising from the attraction potential of the solid atoms to the fluids.",
"In this solution system, since the number of base oil atoms is much higher than the solute palmitic acid, the adsorbed layer of the base oils behave as the wall to inhibit the adsorption of the solute molecules.",
"Subsequently, the palmitic acid molecule breaks through the absorbed layer of the base oils at t = 5.56 ns (this time depends on the initial condition) shown in Fig.",
"REF (d).",
"This is due to the long-range Coulomb force between the solid atoms and the carboxyl group of the palmitic acid.",
"The strength of the layer formation of the base oil molecules is also shown by the time development of the fluctuation in the $z$ direction of the palmitic acid molecules.",
"Figure REF shows the time history of two palmitic acid molecules selected from the same ensemble of the solution.",
"In the time development of the finaly adsorbed molecule shown in Fig.",
"REF (b), the molecule first stays at $z = 20.0$ Å, then stays around this height.",
"After 1.5 ns, it start to descend from $z = 20.0$ Å to $z = 10.0$ Å at t = 4.3 ns and suddenly touches the solid surface.",
"In the time development of the finally not adsorbed molecule shown in Fig.",
"REF (c), the molecule fluctuates around $z = 20.0$ Å for about 2 ns, then moves in the opposite direction from the surface.",
"This results shows that about a 20 Å thick layer inhibits the palmitic acid molecules to form touching the surface of the solid layer by forming the adsorbed layer.",
"In the initial condition, in all case, the nearest molecule to the solid surface position are in $z = 20.0$ Å.",
"However, this does not mean the nearest molecule first reach to the solid surface.",
"Even if the initial distribution of the molecules are same, the movement of molecules are very different due to the difference of the initial velocity.",
"Figure REF also shows that the time for the first molecule to reach the surface takes about 4.5 ns.",
"The question then arises whether this time will change due to the structure of the base oil molecule.",
"Figure REF shows the adsorption time, which is the time the palmitic acid molecules first reaches the surface of the solid layer in the 3 base oils, i.e., n-hexadecane and 2, 4-dimethyltetradecane, and 3, 5-dimethydodecane.",
"In each solution, 5 simulations are done by changing the seed of the random number which decides the initial velocity of each atom in the solution.",
"The graph even shows that the difference in the adsorption time in the same base oil molecules is high and the adsorption time between the base oil molecules remarkably differs.",
"The branhced molecules show half the adsorption time vs. the linear molecules, and the time also differs between the two branched molecules.",
"If the length of the branched chain is long (3, 5-dimethydodecane), the adsorption time is long.",
"Therefore, we confirmed that not only the structure [40], but the adsoption time due to the structure of the base oil molecules can be detected using the molecular dynamics simulation.",
"The difference in the adsorption time may due to structuring of the base oils.",
"Structuring of the base oil in the systems of a branched alkane is weaker than that in the system of a linear alkane.",
"It is thought that the branched alkane side-chains reduce the pair-potential vibration [37].",
"In order to understand the mechanism, density profiles of each of the base oil molecules are plotted in Fig.",
"REF .",
"The oscillation of the density is clearly shown in the vicinity of the solid atom, and the peak differs between the base oil molecules.",
"The graph also shows that even the system in the $z$ direction is asymmetric due to the difference between the charged and not-charged walls, while the distribution the of base oil is not almost different.",
"This is because the base oil molecules, which are made of alkyl chains, are not polarized, so the charge on the solid layer does not affect the structuring, and the van der Waals interaction becomes dominant.",
"In the center of the oil film, a plateau region of the density profiles is found.",
"This means that the film thickness of our simulation is large enough so that the adsorption process from the bulk region to the structured region can be discussed [10], [12].",
"In order to see the precise distribution of the base oil molecules, the enlarged view of Fig.",
"REF (a) is plotted in Fig.",
"REF (b).",
"It is clearly shown that the order of the structuring of the base oil is n-hexadecane, 2, 4-dimethyltetradecane, and 3, 5-dimethydodecane, which is consistent with the length of the adsorption time.",
"The Fourier power spectrum of the density profile can be used to analyze the strength of the order and periodicity of the molecular distribution [33].",
"Figure REF shows the Fourier power spectrum obtained from Fig.",
"REF .",
"The spectrum is calculated using the Octave FFT analyzer [41], and the number of data points is 512 using the rectanglular window as the window function.",
"The peak of the wavelength is about 5 to 6 Å, which is equal to the width of the alkyl chain.",
"The order of the structuring of the base oil is n-hexadecane and 2, 4-dimethyltetradecane, then 3, 5-dimethydodecane.",
"The peak of 3, 5-dimethydodecane is very broad compared to the peak of n-hexadecane.",
"Therefore, we can confirm that the structuring of the base oil is very different between the structure of the base oil molecules, and the more structured system takes more time for the additive molecules to adsorb onto the surface.",
"The structure of the base oil at the interface is also understood by the order parameter.",
"The order parameter is defined as the sum of the angle $\\theta $ between the direction of the $xy$ plane and the direction of the end-to-end vector of the base oil molecules.",
"The end-to-end vector is defined as the vector between the most topologically separated carbon atoms of each molecules.",
"In 2, 4-dimethyltetradecane, one carbon atom at the end of alkyl chain in the branched side (see Fig.REF ) is randomly chosen as the one side.",
"The order parameter $P (z)$ is then defined as follows: $P(z) = ( 3 \\langle \\cos ^2 \\theta \\rangle - 1) / 2$ where $<>$ denotes the ensemble average and the $z$ is taken from the center of the mass of each molecule.",
"$P(z)=1.0$ when the end-to-end vector is in the same direction of the $xy$ plane.",
"When the end-to-end vector is random, $P(z)$ is zero since $\\cos ^2 \\theta = 1/3$ in the random distribution of $\\theta $ .",
"Figure REF shows the order parameter $P (z)$ in the vicinity of the solid surface.",
"In all types of base oil molecules, $P (z)$ is almost 1, since the center of mass is the lowest $z$ value, and almost all the carbon atoms are in the lowest $z$ coordinate.",
"The $P (z)$ suddenly decreases with the increase in $z$ and shows a minimum when $z$ is about 5.0 Å.",
"Comparing this graph with Fig.",
"REF (b), this region corresponds to the intermediate part of the molecular layer.",
"When the center of mass is in this region, molecules are bridging between the two neighbor molecular layers.",
"The $\\theta $ then shows a higher value, which means the molecules are more against the $xy$ plane.",
"The $P (z)$ then increases again, since this region is the second molecular layer counted from the surface.",
"The $P (z)$ then periodically oscillates as the $z$ increases and reaches to the plateau value.",
"Comparing the base oil molecules in this region, n-hexadecane shows the most ordered structure and 2, 4-dimethyltetradecane and 3, 5-dimethydodecane are less structured.",
"Therefore, we can understand from this viewpoint that the linear alkane molecules are much more structured than the branched molecules.",
"The steric hindrance of the linear alkane molecules are much larger than the branched molecules.",
"If the order parameter is small which mean the molecules are randomly distributed, there may be a gap which the additive molecule can pass through, but when the base oil make a a more structured layer as in the linear alkane, such as n-hexadecane, the additive molecule is hard to go through a small gap.",
"The adsorption process is a dynamic process in which the additive molecules diffuse through the base oil layers and reach the solid surface.",
"In order to understand the dynamics, analyzing the diffusion of the additive molecules is the best way to show the adsorption mechanism.",
"The number of additive molecules, however, is too low in the MD simulations, thus it is very hard to directly obtain the diffusion coefficient of the additive molecules.",
"Even if the avaraged diffusion constant of additive molecules are obtained, the critical diffusion process is the diffusion in the vicinity of the surface.",
"Therefore, the distribution of diffusion coefficient is needed to discuss.",
"In the dilute solution under considering, the diffusion coefficient of additive molecules shows large fluctuation.",
"Therefore, we calculated the diffusion coefficients of the solvent base oil molecule, in order to clarify the mechanism.",
"The diffusion coefficient of the additive moleule and linear alkane may be similar, since the molecular weight and structure is similar, and interactino between polar groups would not affect in dilute solution.",
"The diffusion coefficient of the additive moleule and branched alkane may be different.",
"Therefore, this analysis of diffusion may be completed in the later studies using more powerful computers.",
"Figure REF shows the self-diffusion coefficients $D_{{s}} (z)$ as the function of the $z$ coordinate.",
"$D_{{s}} (z)$ is calculated by the following equation.",
"$D_{{s}} (z) = \\lim _{t \\rightarrow \\infty } \\frac{1}{2 t} \\langle | z(t) - z(0) |^2 \\rangle ,$ where $t$ is the time and $z(t)$ is the $z$ coordinate at time $t$ .",
"$D_{{s}} (z)$ is calculated for every 20 ps of motion of the base oil molecules so that the long-range dynamics are taken.",
"In Fig.",
"REF , $D_{{s}} (z)$ decreases as the $z$ coordinate decreases from the center of the oil film.",
"This is because the structured oil film in the vicinity of the interface affects not only the static structure but the dynamics is different in the interface.",
"The peak shown at the very low $z$ is due to the rebound motion of the reflective solid layer.",
"Each base oil molecule almost stops the motion at $z$ = 2.5 Åwhich is the peak of the first molecular layer shown in Fig.",
"REF .",
"$D_{{s}} (z)$ then shows the minimum at the intermediate region of the layers and shows maximum at the peak of the layers.",
"Comparing the three base oils, the $D_{{s}} (z)$ value of n-hexadecane is the highest except in the first layer.",
"This means that the motion of the molecules are not suppressed in the whole solution, but the structured molecular layer, especially at the first layer, acts as the hardest barrier to the adsorption process.",
"The diffusion coefficients in bulk calculated by the MD simulation are not well predicted in some studies [42], [43], [44].",
"However, our calculated diffusion coefficient is on the same order (1.0$\\times 10^{-5} {cm}^2 {s}^{-1}$ ).",
"Usually, the adsorption dynamics of molecules are separated into two process, i.e., the adsorption limited process and diffusion limited process [21].",
"The former is the difficulty of the physical adsorption using the surface interaction.",
"The latter is the diffusion of the additive molecule in the base oil.",
"In a previous study using coarse grained molecular dynamics [21], it showed that the diffusion limited adsorption is the major process.",
"In our study, however, the motion of the additive molecule to break through the first structured layer of the base oil is the limiting process, which mean although the ratio of the importance of both process can not be determined, since the simulation time and space is limited, adsorption limited process is also important.",
"The diffusion coefficient is rather higher in the linear alkane but the adsorption time is longer in it.",
"The difference in the phenomena is understood by the difference in the base oil model we used.",
"In real phenomena, this means that not only the structure of additive molecules [15], but the structure of the base oil is important to evaluate the efficiency of the surface protection.",
"The formation time of the adsorbed additive layer, as well as the coverage ratio [11], is important in the case such as the sliding speed of the machine element is high and the pressure is high enough to break the adsorbed layer.",
"If the sliding condition is mild, the stability of the adsorbed layer is more important, which mean the chain matching phenomena may be dominant.",
"There are few studies which take into consideration the formation process of the adsorbed additive layer in tribology.",
"We would like to note that structuring of the base oil on the surface is very important for the friction control using OFMs.",
"A further study may include the succeeding process of adsorbed additive layers, concentration dependence of additive molecules, and the chemical effect of surface and additives."
],
[
"Conclusions",
"Using an all-atom molecular dynamics with an explicit base oil, the phenomena that the molecules of the additives break through the adsorbed layer of the base oil is observed.",
"The relative adsorption time depends on the structure of the base oil molecules.",
"The adsorption process is hence not a diffusion limited but adsorption limited process.",
"The result reveals that the structuring of the base oil molecules near the solid wall causes a limitation for the adsorbing process.",
"This phenomena will result in the boundary lubrication effect of the machine elements."
],
[
"Acknowledgements",
"This study was supported by JSPS (Japan Society for the Promotion of Science) KAKENHI (Grants-in-Aid for Scientific Research) Challenging Research (Exploratory) Grant Number 18K18813.",
"We also thank Dr. Hiroaki Koshima and Dr. Kazuhiro Yagishita for their usefull discussions.",
"Figure: Snapshots of the adsorption process ofpalmitic acid molecule in n-hexadecane solutionto the charged surface.The Fe solid atoms are charged in t = 0 ns.The colors of the atoms are the same as inFig..Figure: Time evolution of the position of the additive moleculein direction zz perpendicular to the solid surface.Figure: Adsorption time in each base oil.",
"Each bar shows thetime which the first additive molecule approached thesurface of the solid using the same initial configuration anddifferent set of initial velocities.",
"The dashed line showsthe average adsoption time in the 5 trajectories.In n-hexadecane, since 2 system does not show an adsorption,the average time is calculated from the former 3 trajectories.Figure: (a) Profiles of carbon atom density of the threebase oil moleculesin direction zz perpendicular to the solid surface.",
"(b) Enlarged view of (a).Figure: Fourier transform spectrum of molecular density profilesof the three base oil molecules,taken from Fig.",
".Figure: Distribution of the order parameter of the three base oil moleculesin the vicinity of the surface.Figure: Diffusion coefficients of the three base oil moleculesin the vicinity of the surface."
]
] | 1808.08666 |
[
[
"Bayesian Hypothesis Testing: Redux"
],
[
"Abstract Bayesian hypothesis testing is re-examined from the perspective of an a priori assessment of the test statistic distribution under the alternative.",
"By assessing the distribution of an observable test statistic, rather than prior parameter values, we provide a practical default Bayes factor which is straightforward to interpret.",
"To illustrate our methodology, we provide examples where evidence for a Bayesian strikingly supports the null, but leads to rejection under a classical test.",
"Finally, we conclude with directions for future research."
],
[
"Introduction",
"Bayesians and Classicists are sharply divided on the question of hypothesis testing.",
"Hypothesis testing is a cousin to model selection and in a world of high dimensional selection problems, hypothesis testing is as relevant today as it ever has been.",
"We contrast these two approaches, by re-examining the construction of a hypothesis test, motivated by the seminal paper of Edwards, Lindman and Savage (1996) (hereafter ELS) who provide the following contrast: We now show informally, as much as possible from a classical point of view, how evidence that leads to classical rejection of a null hypothesis at the 0.05 level can favor that null hypothesis.",
"The loose and intuitive argument can easily be made precise.",
"Consider a two-tailed t test with many degrees of freedom.",
"If a true null hypothesis is being tested, $t$ will exceed $1.96$ with probability 2.5% and will exceed 2.58 with probability 0.5%.",
"(Of course, 1.96 and 2.58 are the 5% and 1% two-tailed significance levels; the other 2.5% and 0.5% refer to the possibility that t may be smaller than -1.96 or -2.58.)",
"So on 2% of all occasions when true null hypotheses are being tested, $t$ will lie between 1.96 and 2.58.",
"How often will $t$ lie in that interval when the null hypothesis is false?",
"That depends on what alternatives to the null hypothesis are to be considered.",
"Frequently, given that the null hypothesis is false, all values of $t$ between, say, $-20$ and $+20$ are about equally likely for you.",
"Thus, when the null hypothesis is false, t may well fall in the range from 1.96 to 2.58 with at most the probability (2.58 - 1.96)/ [+20 - (-20)] = 1.55%.",
"In such a case, since 1.55 is less than 2 the occurrence of t in that interval speaks mildly for, not vigorously against, the truth of the null hypothesis.",
"This argument, like almost all the following discussion of null hypothesis testing, hinges on assumptions about the prior distribution under the alternative hypothesis.",
"The classical statistician usually neglects that distribution in fact, denies its existence.",
"He considers how unlikely a t as far from 0 as 1.96 is if the null hypothesis is true, but he does not consider that a $t$ as close to 0 as 1.96 may be even less likely if the null hypothesis is false.",
"In terms of a decision ruleHere $\\Omega (A)$ is the prior odds of the null.",
"$\\Omega (A|D)$ is the posterior odds given datum $D$ , and $L(A;D)$ is the likelihood ratio (a.k.a.",
"Bayes factor, BF)., ELS go on to say: If you need not make your guess until after you have examined a datum $D$ , you will prefer to guess $A$ if and only if $ \\Omega (A|D) $ exceeds $J/I$ , that is $ L( A;D) > J / I \\Omega (A) = \\Lambda $ where your critical likelihood ratio $\\Lambda $ is denied by the context.",
"Classical Statisticians were the first to conclude that there must be some $\\Lambda $ such that you will guess $A$ if $ L( A;D)> \\Lambda $ and guess $\\bar{A}$ if $ L( A;D)< \\Lambda $ .",
"By and large, classical statisticians say the choice of $\\Lambda $ is an entirely subjective one which no one but you can make (e.g.",
"Lehman, 1959, p.62).",
"Bayesians agree; $\\Lambda $ is inversely proportional to your current odds for $A$ , an aspect of your personal opinion.",
"The classical statisticians, however, have overlooked a great simplification, namely that your critical $\\Lambda $ will not depend on the size or structure of the experiment and will be proportional to $J/I$ .",
"As Savage (1962) puts it: the subjectivist's position is more objective than the objectivist's, for the subjectivist finds the range of coherent or reasonable preference patterns much narrower than the objectivist thought it to be.",
"How confusing and dangerous big words are (p.67)!",
"Given this discussion, we build on the idea that a hypothesis test can be constructed by focusing on the distribution of the test statistic, denoted by $t$ , under the alternative hypothesis.",
"Bayes factors can then be calculated once the researcher is willing to assess a prior predictive interval for the $t$ statistic under the alternative.",
"In most experimental situations, this appears to be the most realistic way of assessing a priori information.",
"For related discussion, see Berger and Sellke (1987) and Berger (2003) who pose the question of whether Fisher, Jeffreys and Neyman could have agreed on testing and provide illuminating examples illustrating the differences (see Etz and Wagenmakers, 2017).",
"The rest of our paper is outlined as follows.",
"Section 2 provides a framework for the differences between Classical and Bayesian hypothesis testing.",
"Section 3 uses a probabilistic interval assessment for the test statistic distribution under the alternative to assess a Bayes factor.",
"Jeffrey's (1957, 1961) Cauchy prior and the Bartlett-Lindley paradox ( Lindley, 1957, and Bartlett, 1957) are discussed in this context.",
"Extensions to regression and $R^2$ , $\\chi ^2$ and $F$ tests (see Connely, 1991, and Johnson, 2005, 2008) are also provided.",
"Section 4 concludes with further discussion and with directions for future research."
],
[
"Bayesian vs Classical Hypothesis Testing",
"Suppose that you wish to test a sharp null hypothesis $ H_0 : \\theta =0$ against a non-sharp composite alternative $ H_1 : \\theta \\ne 0 $ .",
"We leave open the possibility that $H_0$ and $H_1$ could represent models and the researcher wishes to perform model selection.",
"A classical test procedure uses the sampling distribution, denoted by $p(\\hat{\\theta }|\\theta )$ , of a test statistic ${\\hat{\\theta }}$ , given the parameter $\\theta $ .",
"A critical value, $c$ , is used to provide a test procedure of the form ${\\rm Reject} \\; H_0 \\; {\\rm if} \\; | \\hat{\\theta } | > c.$ There are two types of errors that can arise.",
"Either the hypothesis maybe rejected even though it is true (a Type I error) or it maybe accepted even though it is false (Type II).",
"Typically, the critical value $c$ is chosen so as to make the probability of a type I error, $\\alpha $ , to be of fixed size.",
"We write $ \\alpha (c) = 1 - \\int _{-c}^c p( \\hat{\\theta } | \\theta ) d \\hat{\\theta } $ .",
"Bayes factor, denoted by BF, which is simply a density ratio (as opposed to a tail probability) is defined by a likelihood ratio $BF = \\frac{p(\\hat{\\theta } |H_0 )}{p(\\hat{\\theta }|H_1 )}.$ Here $p({\\hat{\\theta }}|H_0)=\\int p({\\hat{\\theta }}|\\theta ,H_0)p(\\theta |H_0)d\\theta $ is a marginal distribution of the test statistic and $p(\\theta |H_0)$ an a priori distribution on the parameter.",
"For a simple hypothesis, $(\\theta |H_0) \\sim \\delta _{\\theta _0}$ is a Dirac measure at the null value.",
"The difficulty comes in specifying $p(\\theta |H_1)$ , the prior under the alternative.",
"A Bayesian Hypothesis Test can then be constructed in conjunction with the a priori odds ratio $p(H_0)/p(H_1)$ , to calculate a posterior odds ratio, via Bayes rule, $\\frac{p(H_0|\\hat{\\theta } )}{p(H_1|\\hat{\\theta } )} = \\frac{p(\\hat{\\theta } |H_0 )}{p(\\hat{\\theta }|H_1 )}\\frac{p(H_0 )}{p(H_1 )}.$ As $ H_0 : \\theta =\\theta _0$ and $ H_1 : \\theta \\ne \\theta _0 $ , the Bayes factor calculates $p({\\hat{\\theta }}|\\theta =\\theta _0)/p({\\hat{\\theta }}|\\theta \\ne \\theta _0)$ .",
"We will focus on the test statistic distribution under the alternative hypothesis, namely $p({\\hat{\\theta }}|\\theta \\ne \\theta _0)$ .",
"See also Held and Ott (2018) for additional discussion on $p$ -values and Bayes factors."
],
[
"A Default Bayes Factor",
"Our approach is best illustrated with the usual $t$ -ratio test statistic in a normal means problem.",
"As ELS illustrate, the central question that a Bayesian must a priori address is the question: How often will $t$ lie in that interval when the null hypothesis is false?",
"To do this we need an assessment to the distribution of the $t$ -ratio test statistic under the alternative.",
"As ELS further observe: This argument, like almost all the following discussion of null hypothesis testing, hinges on assumptions about the prior distribution under the alternative hypothesis.",
"The classical statistician usually neglects that distribution in fact, denies its existence.",
"He considers how unlikely a $t$ as far from 0 as $1.96$ is if the null hypothesis is true, but he does not consider that a t as close to 0 as $1.96$ may be even less likely if the null hypothesis is false.",
"First, we calculate prior predictive distribution of the test statistic under the alternative and then show how such assessment can lead to a default Bayes factor."
],
[
"Predictive distribution, $Pr(T=t|H_1)$",
"A simple default approach to quantifying a priori opinion is to assess a hyperparameter, denoted by $A$ , such that the following probability statements hold true: $Pr \\left( - 1.96 \\sqrt{A} < T < 1.96 \\sqrt{A} | H_1 \\right) & = 0.95\\\\Pr \\left( - 1.96 < T < 1.96 | H_0 \\right) & = 0.95.$ Under the null, $H_0$ , both the Bayesian and Classicist agree that $A=1$ .",
"All that is needed to complete the specification is the assessment of $A$ .",
"In the normal mean testing problem we have an iid sample $( y_i |\\theta ) \\sim N(\\theta , \\sigma ^2)$ , for $i=1,\\ldots ,n$ , with $\\sigma ^2$ known and $n{\\bar{y}}=\\sum _{i=1}^n y_i$ .",
"Under the null, $H_0: \\theta =0$ , the distribution of $T = \\sqrt{n} \\bar{y} / \\sigma $ , is the standard normal distribution, namely $T \\sim N(0,1)$ .",
"The distribution of $T$ under the alternative, $H_1: \\theta \\ne 0$ , is a mixture distribution $p(T=t|H_1 ) = \\int _\\Theta p( T=t|\\theta )p(\\theta |H_1)d \\theta ,$ where $p(\\theta |H_1)$ denotes the prior distribution of the parameter under the alternative.",
"Under a normal sampling scheme, this is a location mixture of normals $p( T=t|H_1) = \\int _{-\\infty }^\\infty P(T=t|H_1,\\theta )p(\\theta |H_1)d \\theta $ where $T|H_1,\\theta $ is normal with mean $\\sqrt{n}\\theta /\\sigma $ and variance one; or $T=\\sqrt{n}\\theta /\\sigma + \\varepsilon $ , where $\\varepsilon \\sim N(0,1)$ .",
"Under a normal prior, $\\theta \\sim N( 0 , \\tau ^2)$ , the distribution $p(T=t|H_1)$ can be calculated in closed form as $T \\sim N(0,A) $ where $A=1+n\\tau ^2/\\sigma ^2$ .",
"Hence an assessment of $A$ will depend on the design (through $n$ ) and the relative ratio of measurement errors (through $\\tau ^2/\\sigma ^2$ ).",
"The gain in simplicity of the Bayes test is off-set by the difficulty in assessing $A$ .",
"The Bayes factor is then simply the ratio of two normal ordinates $B = \\frac{ \\phi (t)}{ \\phi ( t/\\sqrt{A}) } = \\sqrt{A} \\exp \\left\\lbrace -\\frac{1}{2} t^2(1-A^{-1})\\right\\rbrace \\;.$ The factor $A$ is often interpreted as the Occam factor (Berger and Jefferys, 1992, Jefferys and Berger, 1992, Good, 1992).",
"See Hartigan (2003) for a discussion of default Akaike-Jeffreys priors and model selection.",
"Our approach requires the researcher to “calibrate” $A$ ahead of time.",
"One simple approach is to perform a what if analysis and assess what posterior odds we would believe if we saw $t=0$ .",
"This assessment directly gives the quantity $\\sqrt{A}$ ."
],
[
"Dickey-Savage.",
"The Bayes factor $BF$ for testing $ H_0 $ versus $ H_1 $ can be calculated using the Dickey-Savage density ratio.",
"This relates the posterior model probability $p( \\theta = \\theta _0 | y)$ to the marginal likelihood ratio via Bayes rule $\\frac{Pr( \\theta = \\theta _0 | y) }{Pr( \\theta = \\theta _0 ) } = \\frac{ p( y | \\theta = \\theta _0 ) }{ p( y ) }.$"
],
[
"Bayes Factor Bounds.",
"Let ${\\hat{\\theta }}_{MLE}$ denote the maximum likelihood estimate, then $p( y | H_1) = \\int p( y | \\theta ) p( \\theta | H_1 ) d \\theta \\le p( y | \\hat{\\theta }_{MLE} ).$ This implies that, for $H_0: \\theta =0$ , $BF \\ge \\frac{p(T=t|\\theta =0)}{p(T=t|\\hat{\\theta })}.$ In a normal means testing context, this leads to a bound, $\\frac{ p( y | H_0 ) }{ p( y | H_1 )} \\ge \\exp \\lbrace -0.5(1.96^2-0^2)\\rbrace = 0.146\\;.$ Under an a priori $1/2$ -$1/2$ weight on either side of zero, the bound increases to $0.292$ .",
"Hence, at least 30% of the hypotheses that the classical approach rejects are true in the Bayesian world.",
"Amongst the experiments with $p$ -values of $0.05$ at least 30% will actually turn out to be true!",
"Put another way, the probability of rejecting the null conditional on the observed $p$ -value of $0.05$ is at least 30%.",
"You are throwing away good null hypothesis and claiming you have found effects!",
"In terms of posterior probabilities, with $p(H_0)=p(H_1)$ , we have a bound $Pr( H_0 | y ) = \\left[1 + \\frac{ p( y | H_1 ) }{ p( y| H_0 )} \\frac{ Pr( H_1 ) }{ Pr( H_0 )} \\right]^{-1} \\ge 0.128\\;.$ Hence, there is at least $12.8$ percent chance that the null is still true even in the one-sided version of the problem!",
"Clearly at odds with a $p$ -value of 5 percent.",
"One of the key issues, as discussed by ELS, is that the classicist approach is based on an observed $p$ -value is not a probability in any real sense.",
"The observed $t$ -value is a realization of a statistic that happens to be $N(0,1)$ under the null hypothesis.",
"Suppose that we observe $t=1.96$ .",
"Then the maximal evidence against the null hypothesis which corresponds to $t=0$ will be achieved by evaluating the likelihood ratio at the observed $t$ ratio, which is distributed $N(0,1)$ ."
],
[
"Normal means Bayes factors",
"We have the following set-up for the normal means case (see Berger and Delampaday, 1989, for the full details): Let ${\\bar{y}}|\\theta \\sim N(\\theta ,\\sigma ^2/n)$ , where $\\sigma ^2$ is known and let $t =\\sqrt{n}({\\bar{y}}-\\theta _0)/\\sigma $ the t-ratio test statistic when testing the null hypothesis $H_0: \\theta = \\theta _0$ against the alternative hypothesis $H_0: \\theta \\ne \\theta _0$ .",
"Also, let $ \\rho = \\sigma / ( \\sqrt{n} \\tau ) $ and $ \\eta = ( \\theta _0 - \\mu )/\\tau $ , derived from a normal prior in the alternative $ \\theta \\sim N (\\mu , \\tau ^2 ) $ .",
"Usually, we take a symmetric prior and set $\\mu = \\theta _0 $ , such that $ \\eta = 0$ and the Bayes factor simplifies to $BF = \\sqrt{ 1 + \\rho ^{-2} } \\exp \\left( - \\frac{1}{2 (1 + \\rho ^2 )} t^2 \\right).$ We can use the Dickey-Savage density ratio as follows to derive the above Bayes factor: $p(\\theta _0 | {\\bar{y}}) & = \\frac{1}{\\sqrt{2 \\pi } \\tau \\sqrt{ 1 + \\rho ^{-2} }}\\exp \\left( - \\frac{1}{2 (1 + \\rho ^2 )} t^2 \\right) \\\\p(\\theta _0) & = \\frac{1}{\\sqrt{2 \\pi } \\tau }$ The posterior distribution under the alternative is $( \\theta | y ) \\sim \\mathcal {N} \\left( \\left( \\frac{n}{\\sigma ^2} + \\frac{1}{\\tau ^2} \\right)^{-1}\\left( \\frac{n \\bar{y}}{\\sigma ^2} + \\frac{\\theta _0}{\\tau ^2} \\right), \\left( \\frac{n}{\\sigma ^2} + \\frac{1}{\\tau ^2} \\right)^{-1} \\right)$ with quantities $t^2 = \\frac{n ( \\bar{y} - \\theta _0 )^2 }{\\sigma ^2} \\;{\\rm and} \\; \\left( \\frac{n}{\\sigma ^2} + \\frac{1}{\\tau ^2} \\right)^{-1} = \\tau ^2 ( 1 + \\rho ^{-2} )^{-1}.$ The posterior mean $E( \\theta | y)$ can be written as $\\theta _0 + \\left( \\frac{T}{\\sigma ^2} + \\frac{1}{\\tau ^2} \\right)^{-1} \\frac{T ( \\bar{y} - \\theta _0 ) }{\\sigma ^2}.$ Substituting into the ratio of ordinates $ p( \\theta = \\theta _0 | y) /p( \\theta = \\theta _0) $ gives the result.",
"In the case where $ \\tau $ is moderate to large, this is approximately $BF = \\frac{ \\sqrt{n} \\tau }{\\sigma } \\exp \\left( - \\frac{1}{2} t^2 \\right).$ Clearly, the prior variance $\\tau $ has a dramatic effect on the answer.",
"First, we can see that the “noninformative” prior $ \\tau ^2 \\rightarrow \\infty $ makes little sense (Lindley, 1957, Bartlett, 1957).",
"For instance, when $\\sigma =\\tau $ and $t=2.567$ (a $p$ -value of 0.01), then the Bayes factor equals $0.16$ , $1.15$ , $3.62$ and $36.23$ for $n$ equal to 10, 100, 1000 and 1000000, respectively (see Section 3.3 for more details about the Bartlett-Lindley Paradox).",
"Secondly, the large effect is primarily due to the thinness of the normal prior in the tails.",
"Jeffreys (1961) then proposed the use of a Cauchy $(0,\\sigma ^2)$ prior (see Section 3.4 for further details)."
],
[
"Bartlett-Lindley Paradox",
"See Lindley (1957) and Bartlett (1957) for the full details.",
"The Barlett-Lindley paradox occurs when you let $ \\tau ^2 \\rightarrow \\infty $ .",
"This has the “appropriate” behaviour at the origin of flattening out the marginal distribution of $T$ .",
"So when comparing equal length intervals $Pr( a < T < b )$ and $Pr( c < T < d ) $ , where $ a-b=c-d$ , one would get approximately a Bayes factor of one.",
"The so-called paradox arises when the Bayes factor places all its weight on the alternative hypothesis $ H_1$ .",
"Thought of via the marginal predictive of $T$ this is not surprising.",
"As $ \\tau ^2 \\rightarrow \\infty $ implies $ A \\rightarrow \\infty $ , and your belief a priori that you expect an incredibly large value of $T$ values under the alternative.",
"Now, when you actually observe $ 1.96<T<2.56$ this is unlikely under the null approximately 2%, but nowhere near as likely under the alternative.",
"The Bayes factor correctly identifies the null as having the most posterior mass."
],
[
"Cauchy Prior",
"Jeffreys (1961) proposed a Cauchy (centered at $\\theta _0$ and scale 1) to allow for fat-tails whilst simultaneously avoiding having to specify a scale to the normal prior.",
"Using the asymptotic, large $n$ , form of the posterior $(\\sqrt{n}/\\sqrt{2 \\pi }\\sigma )\\exp \\lbrace -0.5 t^2\\rbrace $ for the usual $t$ -ratio test statistic and the fact that the prior density ordinate from the Cauchy prior is $p(\\theta _0 ) = 1/(\\pi \\sigma )$ , the Bayes Factor is $BF = \\frac{(\\sqrt{n}/\\sqrt{2 \\pi }\\sigma )\\exp \\lbrace -0.5 t^2\\rbrace }{1/(\\pi \\sigma )} = \\sqrt{0.5\\pi n}\\exp \\lbrace -0.5 t^2\\rbrace .$ We have the interval probability $Pr( - 1.96 \\sqrt{A}< T < - 1.96 \\sqrt{A} | H_1) \\approx 0.95,$ for $A \\approx 40$ , when $n=1$ and $\\sigma ^2=1$ .",
"Exact answer given by cdf of hypergeometric Beta You can also see this in the Bayes factor approximations.",
"Therefore, very different from letting $A \\rightarrow \\infty $ , in a normal prior."
],
[
"Coin tossing: $p$ -values and Bayes",
"Suppose that you routinely reject two-sided hypotheses at a fixed level of significance, say $\\alpha =0.05$ .",
"Furthermore, suppose that half the experiments under the null are actually true, i.e.",
"$Pr(H_0)=Pr(H_1)=0.5$ .",
"The experiment will provide data, $y$ , here we standardize the mean effect and obtain a $t$ -ratio."
],
[
"Example: Coin Tossing (ELS).",
"Let us start with a coin tossing experiment where you want to determine whether the coin is “fair\", $H_0: Pr(Head) = Pr(Tail)$ , or the coin is not fair, $H_1: Pr(Head) \\ne Pr(Tail)$ .",
"ELS discuss at length the following four experiments where, in each case, the test statistics is $t=1.96$ .",
"We reproduce below of their Table 1.",
"Table: The quantities nn and rr are, respectively, number of tosses of the coin and the number of heads thatbarely leads to rejection of the null hypothesis, H 0 :Pr(Head)=Pr(Tail)H_0: Pr(Head) = Pr(Tail), by a classical two-tailed test at the 5 percent level.For $n$ coin tosses and $r$ heads, the Bayes factor, $BF = \\left(\\frac{1}{2}\\right)^n/\\int _0^1 \\theta ^r(1-\\theta )^{n-r}p(\\theta |H_1)d\\theta ,$ which grows to infinity and so there is overwhelming evidence in favor of $H_0: Pr(Head) = Pr(Tail)$ .",
"This is a clear illustration of Lindley's paradox.",
"There are a number of ways of assessing the odds.",
"One is to use a uniform prior.",
"Another useful approach which gives a lower bound is to use the maximally informative prior which puts all its mass on the parameter value at the mle, $\\hat{\\theta }=r/n$ .",
"For example, in the $r=60$ versus $n=100$ example, we have $ \\hat{\\theta } = 0.6 $ .",
"Then we have $ p( y | H_1 ) \\le p( y | \\hat{\\theta } ) $ and for the odds ratio $\\frac{ p( y | H_0 ) }{ p( y | H_1 )} \\ge \\frac{ p( y | \\theta = \\theta _0 ) }{ p( y| \\hat{\\theta } ) }.$ For example, with $ n=100$ and $r=60$ , we have $\\frac{ p( y | H_0 ) }{ p( y | H_1 )} \\ge \\frac{0.5^{100} }{ 0.6^{60} 0.4^{40} } = 0.134.$ In terms of probabilities, if we start with a $50/50$ prior on the null, then the posterior probability of the null is at least $0.118$ : $Pr(H_0|y) = \\left(1 + \\frac{ p( y| H_1 ) }{ p( y | H_0 )} \\frac{Pr(H_1)}{Pr(H_0)} \\right)^{-1} \\ge 0.118.$"
],
[
"Regression",
"A number of authors have provided extensions to traditional classical tests, for example Johnson (2008) shows that $R^2$ , deviance, $t$ and $F$ can all be interpreted as Bayes factors.",
"See also Gelman et al (2008) for weakly informative default priors for logistic regression models.",
"In the case of nested models, Connelly (1991) proposes the use of $BF = n^{ - \\frac{d}{2} } \\left( 1 + \\frac{d}{n-k} F \\right)^{ \\frac{n}{2} }$ where $ F$ is the usual $F$ -statistic, $k$ is the number of parameters in the larger model and $d$ is the difference in dimensionality between the two models.",
"In the non-nested case, first it helps to nest them if you can, otherwise MCMC comparisons.",
"Zellner and Siow (1980) extend this to the Cauchy prior case, see Connelly (1991).",
"Essentially, introduces a constant out-front that depends on the prior ordinate $p(\\theta =\\theta _0)$ .",
"See Efron and Gous (2001) for additional discussion of model selection in the Fisher and Jeffreys approaches.",
"Additionally, Polson and Roberts (1994) and Lopes and West (2004) study model selection in diffusion processes and factor analysis, respectively.",
"Scott and Berger (2010) compare Bayes and empirical-Bayes in the variable selection context."
],
[
"Discussion",
"The goal of our paper was to revisit ELS.",
"There are a number of important take-aways from comparing the Bayesian paradigm to frequentist ones.",
"Jeffreys (1937) provided the foundation for Bayes factors (see Kass and Raftery, 1995, for a review).",
"Berkson (1938) was one of the first authors to point out problems with p-values.",
"The Bayesian viewpoint is clear: you have to condition on what you see.",
"You also have to make probability assessments about competing hypotheses.",
"The observed $y$ can be highly unlikely under both scenarios!",
"It is the relative oods that is important.",
"The $p$ -value under both hypotheses are then very small, but the Bayes posterior probability is based on the relative odds of observing the data plus the prior, that is $p(y| H_0) $ and $p(y|H_1)$ can both be small, but its $ p( y| H_0)/p(y|H_1) $ that counts together with the prior $p(H_0)/p(H_1)$ .",
"Lindley's paradox shows that a Bayes test has an extra factor of $\\sqrt{n} $ which will asymptotically favor the null and thus lead to asymptotic differences between the two approaches.",
"There is only a practical problem when $ 2 < t < 4 $ – but this is typically the most interesting case!",
"Jeffreys (1961), page 385, said that “what the use of P implies ...is that a hypothesis that may be true may be rejected because it has not predicted observable results that have not occurred.",
"This seems a remarkable procedure.” We conclude with two quotes on what is wrong with classical p-values with some modern day observations from two Bayesian statisticians."
],
[
"Jim Berger:",
"$p$ -values are typically much smaller than actual error probabilities $p$ -values do not properly seem to reflect the evidience in the data.",
"For instance, suppose one pre-selected $\\alpha =0.001$ .",
"This then is the error one must report whether $p=0.001$ or $p=0.0001$ , in spite of the fact that the latter would seem to provide much stronger evidence against the null hypothesis."
],
[
"Bill Jefferys:",
"The Lindley paradox goes further.",
"It says, assign priors however you wish.",
"You don't get to change them.",
"Then take data and take data and take data ...",
"There will be times when the classical test will reject with probability $(1-\\alpha )$ where you choose $\\alpha $ very small in advance, and at the same time the classical test will reject at a significance level $\\alpha $ .",
"This will not happen, regardless of priors, for the Bayesian test.",
"The essence of the Lindley paradox is that “sampling to a foregone conclusion” happens in the frequentist world, but not in the Bayesian world.",
"As we pointed out at the outset, hypothesis testing is still a central issue in modern-day statistics and machine learning, in particular, its relationship with high dimensional model selection.",
"Finding default regularization procedures in high dimensional settings is still an attractive area of research."
],
[
"References",
" Bartlett, M. S. (1957) Comment on “A Statistical Paradox” by D.V.",
"Lindley.",
"Biometrika, 44, 533-534.",
"Berger, J. O.",
"(2003) Could Fisher, Jeffreys and Neyman have agreed on testing?",
"Statistical Science, 18, 1-32.",
"Berger, J. O. and Delampady, M. (1987) Testing Precise Hypothesis (with Discussion).",
"Statistical Science, 2, 317-335.",
"Berger, J. O. and Jefferys, W. H. (1992) The application of robust Bayesian analysis to hypothesis testing and Occam's Razor.",
"Journal of the Italian Statistical Society, 1, 17-32.",
"Berger, J. O. and Sellke, T. (1987) Testing of a point null hypothesis: the irreconcilability of significance levels and evidence (with Discussion).",
"Journal of the American Statistical Association, 82, 112-139.",
"Berkson, J.",
"(1938) Some difficulties of interpretation encountered in the application of the chi-square test.",
"Journal of the American Statistical Association, 33, 526-542.",
"Connelly, R. A.",
"(1991) A Posterior Odds Analysis of the Weekend Effect.",
"Journal of Econometrics, 49, 51-104.",
"Edwards, W., Lindman, H. and Savage, L. J.",
"(1963) Bayesian Statistical Inference for Psychological Research.",
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"Efron, B. and Gous, A.",
"(2001) Scales of Evidence for Model Selection: Fisher versus Jeffreys.",
"Model selection, 208-246, Institute of Mathematical Statistics, Beachwood.",
"Etz, A. and Wagenmakers, E.-J.",
"(2017) J.",
"B. S. Haldane's Contribution to the Bayes Factor Hypothesis Test.",
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"Gelman, A., Jakulin, A., Pitau, M. G. and Su, Y.-S. (2008) A Weakly Informative Default Prior Distribution for Logistic and Other Regression Models.",
"The Annals of Applied Statistics, 2(4), 1360-1383.",
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"(1992) The Bayes/Non-Bayes Compromise: A Brief Review.",
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"Jefferys, W. H. and Berger, J. O.",
"(1992) Ockham's Razor and Bayesian Analysis.",
"American Scientist, 80, 64-72.",
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"Cambridge: Cambridge University Press.",
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"Johnson, V. (2005) Bayes factors based on test statistics.",
"Journal of the Royal Statistical Society, Series B, 67, 689-701.",
"Johnson, V. (2008) Properties of Bayes Factors Based on Test Statistics.",
"Scandinavian Journal of Statistics, 35, 354-368.",
"Kass, R. E. and Raftery, A. E. (1995) Bayes factors.",
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"Lehmann, E. L. (1959) Testing statistical hypotheses, New York: Wiley.",
"Lindley, D. V. (1957).",
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"Lopes, H. F. and West, M. (2004) Bayesian model assessment in factor analysis.",
"Statistica Sinica, 14, 41-67.",
"Polson, N. G. and Roberts, G. O.",
"(1994) Bayes Factors for discrete observations from diffusion processes.",
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"Savage, L. J.",
"(1962) Subjective probability and statistical practice.",
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"(2010) Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem.",
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"(1979) Posterior Odds Ratio for Selected Regression Hypotheses.",
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"585-603.",
"Valencia: University Press."
]
] | 1808.08491 |
[
[
"Finding a Zipf distribution and cascading propagation metric in utility\n line outage data"
],
[
"Abstract Observed transmission line outage data is grouped into successive generations of events.",
"The empirical distribution of the number of generations in the cascades follows a Zipf distribution that implies the increasing propagation as cascades progress.",
"The slope of the Zipf distribution gives a System Event Propagation Slope Index (SEPSI).",
"This new metric quantifies the cascade propagation, varies as expected, and determines the probabilities of small, medium, and large cascades."
],
[
"Introduction",
"Long sequences of cascading outages occasionally cause large blackouts of power transmission systems.",
"The cascade sequence starts with initial outages and is followed by propagating outages [1].",
"Cascading risk mitigation should address both the initiation and propagation of outages, but there has been no single scalar metric that quantifies the propagation.",
"We find in historical utility data that the distribution of the number of cascading generations characterizes the propagation, follows a Zipf distribution, and gives a new scalar metric of cascading propagation."
],
[
"Processing utility data into generations",
"The transmission line outage data consists of 10942 automatic line outages recorded over 14 years by a North American utility [2], [3].",
"The data includes the line outage start time to the nearest minute.",
"This data is standard and routinely collected by utilities worldwide, such as in the Transmission Availability Data System (TADS) in North America [4], [5].",
"The historical outage data is grouped into cascades and generations based on the outage start time using the simple method described in [6].",
"An outage occurring more than one hour after the preceding outage is assumed to start a new cascade, and within each cascade a series of outages less than one minute apart are grouped into the same generation.",
"Thus each cascade consists of a series of generations of outage events, with each generation containing one or more line outages that occur closely spaced in time.",
"For example, outages caused by protection within one minute are grouped together in the same generational event.",
"This processing produces 6687 cascades.",
"The power system is usually resilient, so that most of these cascades are a single generation of outages that does not propagate further.",
"Figure: Generation kk propagation ρ k \\rho _k as a function of kk.",
"Dots are utility data.",
"Solid curve is calculated from the Zipf distribution solid line in Fig.",
".Figure: Probability distribution of the number of generations on log-log plot.",
"Dots are utility data.",
"Solid line indicates Zipf distribution best fitting data."
],
[
"Propagation of generations and the Zipf distribution of the number of generations",
"Instead of focussing on line outages as in previous work [6], here we analyze the propagation of generations of line outages.",
"Suppose a cascade has at least $k$ generations of outages and let $\\rho _k$ be the probability that generation $k$ propagates to produce a further generation $k+1$ .",
"The dots in Fig.",
"REF show how the propagation $\\rho _k$ increases with $k$ in the data.",
"It is hard to characterize the increasing propagation in Fig.",
"REF with a single number.",
"Let us look at the propagation of generations in a different way.",
"In Fig.",
"REF the dots show the empirical distribution of the number of generations $G$ .",
"The distribution of $G$ is linear on this log-log plot and is a Zipf distribution (or zeta distribution) of the form $P[G=k]=\\frac{1}{\\zeta (s)}\\frac{1}{k^{s}}, \\quad k=1,2,3,...$ where the slope of the line is $-s$ and $\\zeta $ is the Riemann zeta function.",
"Indeed, the maximum likelihood method of [7] fits the data with the line of slope $-3.02$ shown in Fig.",
"REF and a Pearson $\\chi ^2$ test of goodness of fit shows consistency with the Zipf distribution with p-value 0.88.",
"Figs.",
"REF and REF are different descriptions of the same propagation information, because the hazard function of $G$ (the probability of the cascade stopping at generation $k$ , given that generation $k$ has been reached) is $1-\\rho _k$ .",
"Indeed, the propagation $\\rho _k$ implied by the fitted Zipf distribution of $G$ is indicated by the solid curve in Fig.",
"REF .",
"Note how this solid curve interpolates the more erratic estimates from the sparse data for the higher generations."
],
[
"System Event Propagation Slope Index",
"We propose using the negative of the slope of the line fitted to the distribution of the number of generations on the log-log plot as a cascading metric called the System Event Propagation Slope Index (SEPSI).",
"For example, the slope of the fitted line in Fig.",
"REF is $-3.02$ , so that SEPSI $=$ $3.02$ .",
"A lower value of SEPSI indicates a shallower slope and an increased probability of large cascades.",
"The data for all automatic outages can be divided into two parts according to whether the cascade occurs during a NOAA storm condition in the same weather zone, or during the summer months June to September, or during peak load hours 3 to 8 pm [8].",
"Table REF shows SEPSI calculated for each condition.",
"As expected, SEPSI is smaller and cascading is more severe for the stressed cases, with the lowest value of SEPSI$=$$2.2$ achieved when there are storms.",
"Moreover, as the condition varies, the distribution of the number of generations remains linear on the log-log plot, as shown by the Pearson $\\chi ^2$ p-values in Table REF for the fit of the Zipf distribution.",
"We can use the Zipf distribution to deduce the probabilities of small, medium, and large cascades from SEPSI.",
"These probabilities are conditional on a cascade starting.",
"Define a small cascade as 3 or less generations, a medium cascade as 4 to 9 generations, and a large cascade as 10 or more generations (different cut-offs can be chosen).",
"Then, substituting SEPSI for $s$ in (REF ), we compute the probabilities $p_{\\rm small}=\\sum _{k=1}^3 P[G$$=$$k]$ , $p_{\\rm medium}=\\sum _{k=4}^9 P[G$$=$$k]$ , and $p_{\\rm large}=1-p_{\\rm medium}-p_{\\rm large}$ as shown in Table REF .",
"As expected, $p_{\\rm large}$ varies with SEPSI by the largest factor.",
"To determine how many cascades need to be observed for a given accuracy of SEPSI, we bootstrapped samples from the Zipf distribution to find the approximate variation of the estimates of SEPSI.",
"Suppose that there are $n$ cascades and SEPSI$=$ 3.0 so that $p_{\\rm large}$$=$$0.005$ .",
"Then we find that the 95% confidence interval of SEPSI is approximately $\\pm 5/\\sqrt{n}$ .",
"For example, with 95% confidence, 1000 cascades determine SEPSI within $\\pm 0.16$ and $p_{\\rm large}$ within a factor of 1.5.",
"1000 cascades are needed for this accuracy because the larger cascades are rare.",
"There are 782 cascades per year in our data, so that accumulating 1000 cascades takes 1.3 years."
],
[
"Conclusions",
"The number of generations of events in a cascade is an indication of cascade size and severity.",
"We discover in utility line outage historical data that the distribution of the number of generations of events closely follows a Zipf distribution.",
"This intriguing pattern in the data suggests using the negative of the slope of the Zipf distribution as a metric of cascading propagation, the System Event Propagation Slope Index or SEPSI.",
"For our data the overall SEPSI is 3.0 and this reduces to 2.2 in the presence of storms.",
"The probabilities of small, medium and large cascades can be computed from SEPSI.",
"Observing roughly 1000 cascades seems to determine SEPSI and the probability of large cascades to a useful accuracy.",
"The results in this paper rely on line outages recorded by one utility.",
"But there is a clear and testable possibility of generalization to cascading events of a variety of components in other power systems and other infrastructures.",
"More general cascading events should be grouped into generations of events, and the empirical distribution of the number of the generations of cascading events should be obtained.",
"If the distribution of the number of generations can be satisfactorily approximated by a straight line on a log-log plot, then the cascading propagation and the probabilities of small, medium, and large cascades can be quantified with SEPSI.",
"Table: SEPSI varies with conditions and gives cascade probabilitiesWe gratefully acknowledge funding from NSF grants 1609080, 1735354 and thank BPA for making outage data publicly available.",
"The analysis and conclusions are strictly the author's and not BPA's."
]
] | 1808.08434 |
[
[
"Odderon effects in the differential cross-sections at Tevatron and LHC\n energies"
],
[
"Abstract In the present paper, we extend the Froissaron-Maximal Odderon (FMO) approach at $t$ different from 0.",
"Our extended FMO approach gives an excellent description of the 3266 experimental points considered in a wide range of energies and momentum transferred.",
"We show that the very interesting TOTEM results for proton-proton differential cross-section in the range 2.76-13 TeV, together with the Tevatron data for antiproton-proton at 1.8 and 1.96 TeV give further experimental evidence for the existence of the Odderon.",
"One spectacular theoretical result is the fact that the difference in the dip-bump region between $\\bar pp$ and $pp$ differential cross-sections is diminishing with increasing energies and for very high energies (say 100 TeV), the difference between $\\bar pp$ and $pp$ in the dip-bump region is changing its sign: $pp$ becomes bigger than $\\bar pp$ at $|t|$ about 1 GeV$^2$.",
"This is a typical Odderon effect.",
"Another important - phenomenological - result of our approach is that the slope in $pp$ scattering has different behavior in $t$ than the slope in $\\bar pp$ scattering.",
"This is also a clear Odderon effect."
],
[
"Introduction",
"The Odderon is certainly one the most important problems in strong interaction physics.",
"It was introduced [1] in 1973 on the basis of asymptotic theorems [2], [3] and was rediscovered later in QCD [4], [5], [6], [7], [8].",
"In spite of the fact that its theoretical status is very solid, its experimental evidence from half a century is still scarce.",
"This situation is not astonishing, The clear evidence for Odderon has to come by comparing the data at the same energy in hadron-hadron and antihadron-hadron scatterings.",
"But we have not such accelerators!",
"We therefore have to limit our search for evidence for the Odderon only in an indirect way.",
"The search for the Odderon is crucial in order to confirm the validity of QCD.",
"It is very fortunate that the TOTEM datum $\\rho ^{pp} = 0.1\\pm 0.01$ at 13 TeV [9] is the first experimental discovery of the Odderon at $t=0$ , namely in its maximal form [10].",
"Moreover, we checked recently that just the Maximal Odderon in FMO approach is preferred by the experimental data.",
"We generalized the FMO approach by relaxing the $\\ln ^2s$ constraints both in the even- and odd-under-crossing amplitude and we show that, in spite of a considerable freedom of a large class of amplitudes, the best fits bring us back to the maximality of strong interaction [11].",
"In the present paper, we extend the FMO approach at $t$ different from 0.",
"We show that the very interesting TOTEM results for proton-proton differential cross-section in the range 2.76-13 TeV, together with the D0 data for antiproton-proton at 1.96 TeV give further experimental evidence for the existence of the Odderon."
],
[
"Extension of the FMO approach at $t$ different from zero - General definitions",
"In general amplitude of $pp$ forward scattering is $F_{pp}(s,t) = F_+(s,t) + F_-(s,t)$ and the amplitude of antiproton-proton scattering is $F_{\\bar{p}p}(s,t) = F_+(s,t) - F_-(s,t).$ In this model we used the following normalization of the physical amplitudes.",
"$\\begin{array}{ll}\\sigma _t(s)&=\\dfrac{1}{\\sqrt{s (s-4m^2 )}}\\text{Im} F(s,0), \\\\\\dfrac{d\\sigma _{el}}{dt}&=\\dfrac{1}{64\\pi ks(s-4m^2)}|F(s,t)|^2\\end{array}$ where $k=0.3893797\\,\\, \\text{mb}\\cdot \\text{GeV}^2$ .",
"With this normalization the amplitudes have dimension $\\text{mb}\\cdot \\text{GeV}^2$ .",
"Strictly speaking crossing-even (CE), $F_+(s,t)$ , and crossing-odd (CO), $F_-(s,t)$ , parts of amplitudes are defined as functions of $z_t=(t+2s-4m^2)/(4m^2-t)$ , where $m$ is proton mass, with the property $F_{\\pm } (-z_t,t)=\\pm F_{\\pm } (z_t,t).$ In the FMO model CE and CO terms of amplitudes are defined as sums of the asymptotic contributions $F^H(s,t)$ , $F^{MO}(s,t)$ and Regge pole contributions which are important at the intermediate and relatively low energies $\\begin{array}{l}F_+(z_t,t)=F^H(z_t,t)+F^{R_+}(z_t,t), \\\\F_-(z_t,t)=F^{MO}(z_t,t)+F^{R_-}(z_t,t)\\end{array}$ where $F^H(z_t,t)$ denotes the Froissaron contribution and $F^{MO}(z_t,t)$ denotes the Maximal Odderon contribution.",
"Their specified form will be defined below."
],
[
"Regge poles and their double rescatterings",
"In the FMO model in the terms $F^{R_{\\pm }}(s,t)$ we consider not only single Regge pole contributions but also their double rescatterings or double cuts.",
"Their contributions, $F^R_{pp}(z_t,t), F^R_{\\bar{p}p}(z_t,t)$ , are the following $\\begin{array}{ll}F^R_{pp}(z_t,t)=F^{_+}(z_t,t)+F^{_-}(z_t,t),\\\\F^R_{\\bar{p}p}(z_t,t)=F^{_+}(z_t,t)-F^{_-}(z_t,t)\\\\\\end{array}$ where $z_t=-1+2s/(4m^2-t)\\approx 2s/(4m^2-t)$ .",
"For convenience in further work with parameterizations in FMO model at $t=0$ and $t\\ne 0$ contrary to standard definition of $z_t$ we put opposite sign for it.",
"$\\begin{array}{ll}F^{_+}(z_t,t)&=F^P(z_t,t)+F^{R_+}(z_t,t)+F^{PP}(z_t,t)\\\\&+F^{OO}(z_t,t),\\\\F^{_-}(z_t,t)&=F^O(z_t,t)+F^{R_-}(z_t,t)+F^{PO}(z_t,t).\\end{array}$ Here $F^P(z_t,t), F^O(z_t,t)$ are simple $j$ -pole Pomeron and Odderon contributions and $F^{R_+}(z_t,t), F^{R_-}(z_t,t)$ are effective $f$ and $\\omega $ simple $j$ -pole contributions, where $j$ is an angular momenta of these reggeons.",
"$F^{PP}(z_t,t)$ , $F^{OO}(z_t,t),$ $F^{PO}(z_t,t),$ are double $PP, OO, PO$ cuts, correspondingly.",
"We consider the model at $t\\ne 0$ and at energy $\\sqrt{s}> 19$ GeV, so we neglect the rescatterings of secondary reggeons with $P$ and $O$ .",
"In the considered kinematical region they are small.",
"Besides, because $f$ and $\\omega $ are effective, they can take into account small effects from the cuts.",
"The standard Regge pole contributions have the form $F^{R_\\pm }(z_t,t)=-\\binom{1}{i}2m^2C^{R_\\pm }(t)(-iz_t)^{\\alpha _\\pm (t)}$ where $R_\\pm = P,O,R_+,R_-$ and $\\alpha _P(0)=\\alpha _O(0)=1$ .",
"The factor $2m^2$ is inserted in amplitudes $F^{R_\\pm }(z_t,t)$ in order to have the normalization for amplitudes and dimension of coupling constants (in mb) coinciding with those in [10].",
"The same is made for all other amplitudes, including Froissaron and Maximal Odderon (see below).",
"For the coupling function $C^{R_\\pm }(t)$ we have considered two possibilities.",
"The first one is a simple exponential form.",
"It is used for the secondary reggeons, because we did not consider low energies where terms $R_{\\pm }(s,t)$ are more important.",
"$C^{R_\\pm }(t)=C^{R_\\pm }e^{b^{R_{\\pm }}t}, \\qquad C^{R_\\pm }(0)=C^{R_\\pm }.$ The second case is a linear combination of exponents for Standard Pomeron and Odderon terms which allow to take into account some possible effects of non-exponential behavior of coupling function.",
"$\\begin{array}{ll}C^{P,O}(t)&=C^{P,O}\\left[\\Psi ^{P,O}(t) \\right]^2, \\\\\\Psi ^{P,O}(t)&=d_{p,o}e^{b_1^{P,O}t}+(1-d_{p,o})e^{b_2^{P,O}t}.\\end{array}$ We have added as well the double pomeron and odderon cuts, $PP, OO, PO$ in their exact form without any new parameters.",
"Namely, $\\begin{array}{ll}F^{PP}(z_t,t)&=-i\\dfrac{(z_tC^P)^2}{16\\pi s\\sqrt{1-4m^2/s}} \\left\\lbrace \\dfrac{d_p^2}{2B_1^p}\\exp (tB_1^p/2)\\right.\\\\&+\\dfrac{2d_p(1-d_p)}{B_1^p+B_2^p}\\exp \\left(t\\dfrac{B_1^pB_2^p}{B_1^p+B_2^p} \\right)\\\\&\\left.+\\dfrac{(1-d_p)^2}{2B_2^p}\\exp (tB_2^p/2) \\right\\rbrace \\end{array}$ $\\begin{array}{ll}F^{OO}(z_t,t)&=-i\\dfrac{(z_tC^O)^2}{16\\pi s\\sqrt{1-4m^2/s}}\\left\\lbrace \\dfrac{d_o^2}{2B_1^o}\\exp (tB_1^o/2)\\right.",
"\\\\&+\\dfrac{2d_o(1-d_o)}{B_1^o+B_2^o}\\exp \\left(t\\dfrac{B_1^oB_2^o}{B_1^o+B_2^o} \\right)+\\\\&\\left.\\dfrac{(1-d_o)^2}{2B_2^o}\\exp (tB_2^o/2) \\right\\rbrace \\end{array}$ where $B_k^{p,o}=b_k^{P.O}+\\alpha ^{\\prime }_{P,0}\\ln (-iz_t),\\quad k=1,2 , \\quad b_k^{P,O}$ are the constants from single pomeron and odderon contributions.",
"$\\begin{array}{ll}F^{PO}(z_t,t)&=\\dfrac{z_t^2C^PC^O}{16\\pi s\\sqrt{1-4m^2/s}} \\\\&\\times \\left\\lbrace \\dfrac{d_pd_o}{B_1^p+B_1^o}\\exp \\left(t\\dfrac{B_1^pB_1^o}{B_1^p+B_1^o}\\right)\\right.\\\\&+ \\dfrac{d_p(1-d_o)}{B_1^p+B_2^o}\\exp \\left(t\\dfrac{B_1^pB_2^o}{B_1^p+B_2^o}\\right)\\\\& +\\dfrac{(1-d_p)d_o}{B_2^p+B_1^o}\\exp \\left(t\\dfrac{B_2^pB_1^o}{B_2^p+B_1^o}\\right)+\\\\&\\left.",
"\\dfrac{(1-d_p)(1-d_o)}{B_2^p+B_2^o}\\exp \\left(t\\dfrac{B_2^pB_2^o}{B_2^p+B_2^o}\\right) \\right\\rbrace \\end{array}$ We have found that for a better description of the data it is reasonable to add to the amplitudes the contributions which mimic some properties of ”hard“ pomeron ($P^H$ ) and odderon ($O^H$ ).",
"We take them in the simplest form $P^H(t)=i\\dfrac{C^{PH}z_t}{(1-t/t_P)^{\\mu _P} }, \\quad \\mu _P\\le 4.$ $P^O(t)=\\dfrac{C^{OH}z_t}{(1-t/t_O)^{\\mu _OP} }, \\quad \\mu _O\\le 4.$"
],
[
"Partial amplitudes for Froissaron and Odderon",
"Let us start from the Froissaron amplitude in ($s,t$ )-representation at high $s$ .",
"The amplitude can be expanded in the series of partial amplitudes $\\phi (\\omega ,t)$ .",
"In accordance with the standard definition of partial amplitude $F(z_t,t)=16\\pi \\sum \\limits _{j=0}^{\\infty }(2j+1)P_j(-z_t)\\phi (j,t).$ With such definition partial amplitude satisfies the unitarity equation in the form $\\begin{array}{ll}\\text{Im}\\phi (j,t)& = \\rho (t)|\\phi (j,t)|^2+\\text{inelastic contribution},\\\\\\\\\\rho (t)&=\\sqrt{1-4m^2/t}\\end{array}$ We use of the Sommerfeld-Watson transform amplitude (here and in what follows $\\omega =j-1$ and $j$ is complex angular momentum) which can be written as follows $\\begin{array}{ll}F^\\zeta (z_t,t)&=16\\pi \\sum \\limits _{\\xi =-1,1} \\int \\limits _C\\dfrac{d\\omega }{2\\pi i} (2\\omega +3)\\dfrac{1-\\xi e^{-i\\pi \\omega }}{-\\sin (\\pi \\omega ) }\\\\&\\times \\phi ^\\xi (\\omega ,t) P_{1+\\omega }(z_t) \\\\&=16\\pi \\sum \\limits _{\\xi =-1,1} \\int \\limits _C\\dfrac{d\\omega }{2\\pi i} (2\\omega +3)\\\\&\\times e^{-i\\pi \\omega /2}\\dfrac{e^{i\\pi \\omega /2}-\\xi e^{-i\\pi \\omega /2}}{-\\sin (\\pi \\omega )}\\phi ^\\xi (\\omega ,t)P_{1+\\omega }(z_t)\\\\&=z_t\\sum \\limits _{\\xi =-1,1} \\int \\limits _C\\dfrac{d\\omega }{2\\pi i}e^{\\omega \\zeta }\\varphi ^\\xi (\\omega ,t).\\end{array}$ where $\\omega =j-1$ , $\\xi $ is the signature of the term, contour $C$ is a straight line parallel to imaginary axis and at the right of all singularities of $\\phi ^\\xi (\\omega ,t)$ , $\\zeta =\\ln (z_t)-i\\pi /2 \\equiv \\ln (-iz_t)$ and $\\begin{array}{ll}\\varphi ^\\xi (\\omega ,t)&=16\\pi (2\\omega +3)\\dfrac{e^{i\\pi \\omega /2}-\\xi e^{-i\\pi \\omega /2}}{-\\sin (\\pi \\omega )}\\pi ^{-1/2}2^{ \\omega +1}\\\\&\\times \\dfrac{\\Gamma (\\omega +3/2)}{\\Gamma (\\omega +2)}\\phi ^\\xi (\\omega ,t)\\end{array}$ Thus for crossing even amplitude ($\\xi $ =+1) we have $\\varphi ^+(\\omega ,t)=i32\\sqrt{\\pi }(2\\omega +3)\\dfrac{\\Gamma (\\omega +3/2)}{\\Gamma (\\omega +2)}2^\\omega \\dfrac{\\phi ^+(\\omega ,t)}{\\cos (\\pi \\omega /2)}$ and for crossing odd amplitude ($\\xi $ =-1) $\\varphi ^-(\\omega ,t)=-32\\sqrt{\\pi }(2\\omega +3)\\dfrac{\\Gamma (\\omega +3/2)}{\\Gamma (\\omega +2)}2^\\omega \\dfrac{\\phi ^-(\\omega ,t)}{\\sin (\\pi \\omega /2)}.$ Inverse transformation is $\\varphi ^\\pm (\\omega ,t)=\\int \\limits _0^\\infty d\\zeta e^{-\\omega \\zeta } F^\\pm (z_t,t), \\qquad z_t=e^\\zeta .$ One can show that in order to have maximal growth of total cross section $\\sigma _{tot}(s)\\propto \\xi ^2$ at $s\\rightarrow \\infty $ , to have a growing elastic cross section bounded by $ \\sigma _{el}(s)/\\sigma _{tot}(s)\\rightarrow const \\quad \\text{at}\\quad s\\rightarrow \\infty $ and to provide the correct analytical properties of amplitude at $t\\approx 0$ necessary to write the partial amplitude $\\phi (\\omega ,t)$ in the following form (more details are given in the Appendics, Section ) $\\varphi ^\\pm (\\omega ,t )=\\binom{i}{-1}\\dfrac{\\beta ^\\pm (\\omega , t)}{[\\omega ^2+r_\\pm ^2q_{\\perp }^2]^{3/2}}.$ where $r_\\pm $ are some constants, $q_{\\perp }^2=-t$ and $\\beta (\\omega ,t)$ has not singularity at $\\omega ^2+R^2q_{\\perp }^2=0$ .",
"In fact a choice of the sign in $\\phi ^-(\\omega ,t)$ does nor matter because the crossing odd terms contribute to $pp$ and $\\bar{p}p$ amplitude with the opposite signs.",
"In order to have agreement with parametrization and parameters which we used in the papers devoted to analysis of the data at $t=0$ , we should replace -1 for for +1 in front of $\\phi ^-(\\omega ,t)$ .",
"At $\\omega =0$ , function $\\varphi ^-(\\omega ,t)$ has singularity in $t$ if $\\beta ^-(0,t) \\ne 0 $ , namely, $\\phi ^-(0,t)\\propto (-t)^{3/2}$ .",
"One of arguments against the Maximal Odderon is that this singularity in partial amplitude means the existing of massless particle in the model.",
"However as we seen above $\\varphi ^-(\\omega ,t) $ is not the real physical partial amplitude which is $\\begin{array}{ll}\\phi ^-(\\omega ,t)&=\\left[32\\sqrt{\\pi }(2\\omega +3)\\dfrac{\\Gamma (\\omega +3/2)}{\\Gamma (\\omega +2)}2^\\omega \\right]^{-1}\\\\&\\times \\sin (\\pi \\omega /2)\\varphi ^-(\\omega ,t)\\end{array}$ and it equals to 0 at $\\omega =0$ because of $\\sin (\\pi \\omega /2)$ coming from signature factor.",
"Now let us suppose that in accordance with the structure of the singularity of $\\varphi _\\pm (\\omega ,t)$ at $\\omega ^2+\\omega _{0\\pm }^2=0$ ($\\omega _{0\\pm }^2= R_\\pm ^2q_{\\perp }^2)$ the functions $\\beta _\\pm (\\omega ,t)$ , depending on $\\omega $ through the variable $\\kappa _\\pm =(\\omega ^2+\\omega _{0\\pm }^2) ^{1/2}$ , can be expanded in powers of $\\kappa _\\pm $ $\\varphi ^\\pm (\\omega ,t)=\\binom{i}{1}\\dfrac{\\beta _1^{\\pm }(t)+\\kappa _\\pm \\beta _2^{\\pm }(t)+ \\kappa _\\pm ^2 \\beta _3^{\\pm }(t)}{\\kappa _\\pm ^{3}}$ Then making use of the table integrals (see the Section ) we obtain the expressions for $F^\\pm (z_t,t)$ which are written in the next Section."
],
[
"Froissaron and Maximal Odderon in ($s, t$ )-representation",
"At $t=0$ Froissaron and Maximal Odderon have the universal form independently of any extension to $t\\ne 0$ : $F^H(z_t,t=0)=iz[H_1\\ln ^2(-iz_t)+H_2\\ln (-iz_t)+H_3],$ $F^{MO}(z_t,t=0)=z[O_1\\ln ^2(-iz_t)+O_2\\ln (-iz_t)+O_3]$ where $z=2m^2z_t$ .",
"At $t=0$ we have $z_t=(s-2m^2)/(2m^2)$ .",
"The Froissaron and the Maximal Odderon defined at $t=0$ by above Eqs.",
"(REF , REF ) allow various extensions to analytical $t$ -dependences.",
"Probably it is impossible a priory to choose the best of them.",
"In the present work we consider an extension of Eqs.",
"(REF , REF ) in accordance with Eq.",
"(REF ).",
"$\\begin{array} {ll}\\dfrac{-1}{iz}F^H(z_t,t)=H_1\\zeta ^2\\dfrac{2J_{1}(r_{+}\\tau \\zeta )}{r_{+}\\tau \\zeta }\\Phi ^{2}_{H,1}(t) \\\\+H_2\\zeta \\dfrac{\\sin (r_{+}\\tau \\zeta )}{r_+\\tau \\zeta }\\Phi ^{2}_{H,2}(t)+H_3J_0(r_{+}\\tau \\zeta )\\Phi ^{2}_{H,3}(t) ,\\\\\\Phi _{H,i}(t)=\\exp (b^H_iq_+),\\quad i=1.2,3 \\\\q_+=2m_{\\pi }-\\sqrt{4m_{\\pi }^2-t} .\\end{array}$ $\\begin{array}{ll}\\dfrac{1}{z}F^{MO}(z_t,t)=O_1\\zeta ^2\\dfrac{2J_{1}(r_{-}\\tau \\zeta )}{r_{-}\\tau \\zeta }\\Phi ^{2}_{O,1}(t) \\\\+ O_2\\zeta \\dfrac{\\sin (r_{-}\\tau \\zeta )}{r_{-}\\tau \\zeta }\\Phi ^{2}_{O,2}(t)+O_3J_0(r_{-}\\tau \\zeta )\\Phi ^{2}_{O,3}(t),\\\\\\Phi _{O,i}(t)=\\exp (b^O_iq_-), \\quad i=1,2,3,\\\\q_-=3m_{\\pi }-\\sqrt{9m_{\\pi }^2-t}.\\end{array}$ where $z=2m^2z_t, \\quad \\zeta =\\ln (-iz_t), \\quad \\tau =\\sqrt{-t/t_0}, \\quad t_0=1\\text{GeV}^2$ .",
"Due to the factor $z$ (instead of $z_t$ ) the amplitudes $F^H(z_t,t)$ and $F^{MO}(z_t,t)$ have the required normalization with additional factor $2m^2$ ."
],
[
"Comparison of the FMO model with the data",
"We give here the results of the fit to the data in the following region of $s$ and $|t|$ .",
"$\\begin{array}[]{llllll}\\text{for} \\quad \\sigma _{tot}(s), \\rho (s) \\quad & \\text{at} \\quad 5 & \\text{GeV} & \\le \\sqrt{s} & \\le 13 & \\text{TeV}, \\\\\\end{array}\\text{for} \\quad d\\sigma (s,t)/dt \\quad & \\text{at} \\quad 9 & \\text{GeV} & \\le \\sqrt{s} & \\le 13 & \\text{TeV}\\\\\\nonumber \\text{and} & \\text{at} \\quad 10^{-4} & \\text{GeV}^2 & \\le |t| & \\le 5 & \\text{GeV}^2.$ We add the recent data at $t=0$ of TOTEM Collaboration [9], [12], [13], [14] to data set published by Particle Data Group [15].",
"We have performed two alternative fits of the FMO model and experimental data from the above mentioned kinematic region.",
"In the Fit I we take into account all the data at t=0, i.e.",
"$\\sigma _{tot}$ and $\\rho $ are calculated from the FMO model, free parameters are determined from the fit to all $t$ , chosen in such a way that we can ignore in given region the contribution of the Coulomb part of amplitudes which is less than 1% of the nuclear amplitude.",
"Thus, $t$ -region $0.0<|t|<0.05$ GeV$^2$ is excluded in the Fit I, and the Coulomb part of amplitudes put to zero.",
"In the Fit II all experimental data on $\\sigma _{tot}$ and $\\rho $ are excluded and fit is made at energies $\\sqrt{s}> 19$ GeV and $0<|t|< 5 $ GeV$^2$ .",
"Taking into account that in this kinematic region parameters of CE and CO secondary reggeons are badly determined, we put all the parameters of these contributions as fixed from the results of Fit I.",
"For 13 TeV TOTEM data we used the data at $t=0$ for $\\sigma _{tot}$ [12] and $\\rho $ [9], as well as the data on differential cross sections [16], [17], [18], [19].",
"We add also recently published data on $d\\sigma /dt$ at $\\sqrt{s}=2.76$ TeV obtained by TOTEM [20]."
],
[
"Coulomb amplitude, one of the simplest parameterizations",
"Coulomb terms in the $pp$ and $\\bar{p}p$ amplitudes are written in the well known form ${\\cal F}_{CN}(s,t)=\\pm 8\\pi s\\dfrac{\\alpha }{t}F_1^2(t)\\exp (i\\alpha \\phi (s,t))$ where $\\alpha =7.297352 \\times 10^{-3} = 1/137.035$ is the fine-structure constant and $\\begin{array}{rl}F_1(t)&=\\dfrac{4m_p^2-\\mu _pt}{4m_p^2-t}\\dfrac{1}{(1-t/0.71)^2},\\\\&\\\\\\mu _p& =2.7928473446\\end{array}$ where $\\mu _p$ is the magnetic momemtum of proton.",
"For the phase $\\phi (s,t)$ we nave $\\phi (s,t)=\\pm \\left[ \\ln \\left( \\dfrac{B(s)}{2}|t|\\right)+\\gamma \\right]$ where $\\gamma =0.5772156649$ is the Euler constant.",
"The slope $B(s)$ is calculated through a fit making use of the equation $\\begin{array}{ll}<B(s)>&=(1/\\Delta _t)\\int \\limits _{t_{max}}^{t_{min}}dt \\,\\dfrac{d}{dt}(\\ln (d\\sigma (t)/dt))\\\\&= (1/\\Delta _t )\\left[\\ln \\left(\\dfrac{d\\sigma (t_{min})/dt}{d\\sigma (t_{max})/dt}\\right)\\right]\\end{array}$ where $\\Delta _t=t_{max}-t_{min}$ .",
"We put (in accordance with the TOTEM estimations [17]), $t_{max}=-0.07$ GeV$^2$ , $t_{min}=-0.005$ GeV$^2$ .",
"Figure: Total cross sections and ratios ρ\\rho in FMO model with the PP,PO,OOPP,PO, OO terms added"
],
[
"$pp$ and {{formula:ba71d818-c650-4e91-bf2a-0c34721207f8}} differential cross sections {{formula:f67ea034-89d6-4141-8279-c1c6fe96ff54}}",
"Here we present results for both methods of the data description.",
"Fit I: FMO model without Coulumb term fitted to the whole set of data excluding lowest $|t|<0.05$ GeV$^2$ .",
"Fit II: FMO model with Coulomb term fitted to the whole set of the data at $t\\ne 0$ .",
"In the legends of Fig.",
"REF -REF these fits are labeled as \"FMO\" and \"FMO+C\", correspondingly.",
"The curves shown at the Figs.",
"REF - REF were calculated in the FMO model without Coulomb terms (Fit I).",
"Number of experimental points in $pp$ and $\\bar{p}p$ total cross sections $\\sigma _t^{pp}, \\sigma _t^{\\bar{p}p}$ , ratios $\\rho ^{pp}, \\rho ^{\\bar{p}p}$ and differential cross sections used in the Fit I and quality of fit are shown in the Table 1.",
"Numbers of the data points and obtained values of $\\chi {^2}$ in the Fit II are given in the Table 2.",
"Table: Number of experimental points and the quality of their description when the usual minimization in FMO model is appliedTable: Number of experimental points and the quality of their description when the fit with FMO+Coulomb terms is made.",
"The data on σ tot (s)\\sigma _{tot}(s) and ρ(s)\\rho (s) has been excluded from this fitThe values of parameters and their errors obtained in these two fits within the FMO model are given in the Table 3 (parameters of the Froissaron and Maximal Odderon terms, of the standard Pomeron and Odderon, of the ”hard“Pomeron and Odderon, and of the secondary reggeons).",
"To avoid a possible negative cross sections in the large partial waves, $j$ , (at the edge of the disk) we put in the fit the restriction $r_-\\le r_+$ .",
"However, we observed that in the various considered modifications of the FMO model these parameters are almost equal each to other.",
"Based on this fact we put $r_-= r_+$ in the model presented here.",
"Also we have fixed the parameters $b^{\\pm }$ at 0 because in all considered fits $b^+$ has the error comparable with the value of parameter and $b^-$ has value close to 0.",
"Fig.",
"REF demonstrates a behavior of the $pp$ and $\\bar{p}p$ total cross sections and ratios real part to imaginary part of the amplitudes at $t=0$ obtained in the both Fit I and Fit II.",
"We would like to notice the interesting odderon effect: the change of sign in the differences between total cross section and $\\rho $ ’s between $\\sqrt{s}\\approx 50$ and $\\sqrt{s}\\approx 500$ GeV.",
"Such a spectacular effect is allowed by asymptotic theorems.",
"A detailed dynamic model for this effect was not yet invented.",
"In Figs.",
"REF and REF we show the differential cross-sections at energies bigger than 19 GeV.",
"In Fig.",
"REF we show the differential cross-sections at the LHC energies 7, 8 and 13 TeV and in Figs.",
"REF , REF , REF we show differential $pp$ and $\\bar{p}p$ at lowest $|t|$ .",
"In Fig.",
"REF , we show in a magnified way the differential cross-sections at 53 GeV.",
"As one can see from these figures our description of the data in a wide range of energies is very good.",
"In Fig.",
"REF we show the evolution of the dip-bump structure in $pp$ and $\\bar{p}p$ differential cross sections with increasing energy.",
"In Fig.",
"REF we show in a magnified way the dip-bump region at different energies and in Fig.",
"REF we show the evolution of the ratio $R_\\sigma = (d\\sigma (\\bar{p}p) /dt) / (d\\sigma (pp)/dt)$ with increasing energy.",
"A remarkable prediction can be seen from these last three figures: the difference in the dip-bump region between $\\bar{p}p$ and $pp$ differential cross sections is diminishing with increasing energies and, for very high energies (say 100 TeV, see Fig.",
"REF ), the ratio in the dip-bump region goes to 1.",
"At ISR energies until $\\sim 60$ GeV the ratio $R_\\sigma > 1$ and then it becomes less than 1 but increases to maximum at some $t_m$ .",
"After maximum the value of $R_\\sigma $ is decreasing and equals to 1 at some $t_1$ which is going to lower $t$ with increasing energy.",
"At higher $t$ however $R_\\sigma $ is oscillating around of 1 when $t$ increases.",
"This is a spectacular Odderon effect.",
"One can see also the clear Odderon effects and their evolution with energy in Fig.",
"REF .",
"Figure: pppp differential cross sections at s>19\\sqrt{s}>19 GeVTable: Parameters of standard Pomeron and Odderon, of their double rescatterings, of secondary Reggeons and their errors in FMO model determined from the fits to the data on dσ/dtd\\sigma /dt.",
"Total cross sections σ tot \\sigma _{tot} and ratios ρ\\rho were included in the fit without the Coulomb termFigure: p ¯p\\bar{p}p differential cross sections at s\\sqrt{s} from 19 GeV up to 1.96 TeV"
],
[
"Slope $B(s, t)$",
"The slope $B(s, t)$ is a very interesting quantity in the search for Odderon effects.",
"It is defined by $B(s,t)=\\dfrac{d}{dt}\\ln (d\\sigma /dt).$ If we consider the dependence of slope on energy and compare this dependence with available experimental data we have to take into account that slopes in any realistic model depend on $t$ .",
"Dependence of slope on $t$ at various energies in the FMO model is illustrated in Fig.",
"REF (left panel).",
"Therefore we must to calculate the slope $<B(s)>$ averaged in some interval of $t$ .",
"We did that in the interval $|t|\\in (0.05, 0.2) \\text{GeV}^2$ for GeV energies which approximately is in agreement to the intervals from which the experimental data on $B$ are determined.",
"$\\begin{array}{ll}<B(s)>&=(1/\\Delta _t)\\int \\limits _{t_{max}}^{t_{min}}dt \\,\\dfrac{d}{dt}(\\ln (d\\sigma (t)/dt))\\\\&= (1/\\Delta _t )\\left[\\ln \\left(\\dfrac{d\\sigma (t_{min})/dt}{d\\sigma (t_{max})/dt}\\right)\\right]\\end{array}$ where $\\Delta _t=t_{max}-t_{min}$ .",
"Figure: pppp differential cross sections at s=7,8,13\\sqrt{s}=7, 8, 13 TeVFigure: Differential pppp cross sections at the lowest |t||t| and at ISR energiesFigure: Differential pppp cross sections at the lowest |t||t| and at LHC energiesFigure: Differential p ¯p\\bar{p}p cross sections at the lowest |t||t|Figure: pppp and p ¯p\\bar{p}p differential cross sections at s=53\\sqrt{s}=53 GeVFigure: Evolution of pppp and p ¯p\\bar{p}p differential cross sections with increasing energyFigure: pppp and p ¯p\\bar{p}p differential cross sections in and around the dip regionFigure: Evolution of the ratio of differential cross sections R σ =(dσ(p ¯p)/dt)/(dσ(pp)/dt)R_{\\sigma }=(d\\sigma (\\bar{p}p)/dt)/(d\\sigma (pp)/dt) with energyFigure: Partial contributions of the real and imaginary parts of even and odd terms to pppp and p ¯p\\bar{p}p scattering amplitudes at various energiesWe show in Table 4 our predictions for the averaged slopes in the TeV region of energy as compared with experiments at Tevatron and LHC.",
"Table: Experimetal values of slopes of pppp and p ¯p\\bar{p}p differential cross sections at TeV energies and the averaged slopes calculated in FMO modelIn Fig.",
"REF (right panel) we show the increasing of the averaged slopes at t=0 with increasing energy.",
"One can see that the slopes are approaching the $\\ln ^2s$ increase at high energies.",
"Figure: Slopes B pp (t)B^{pp}(t) and B p ¯p (t)B^{\\bar{p}p}(t) at increasing energy(left panel) and the ss-dependence of the averaged slopes <B pp (s)><B^{pp}(s)>, <B p ¯p (s)><B^{\\bar{p}p}(s)> together with experimental data (right panel)Figure: Slope B(s,t)B(s,t) for pppp (left panel) and p ¯p)\\bar{p}p) (right panel) at selected energiesFigure: Dependence on tt of the slopes B(s,t)B(s,t) for pppp scattering at 7 and 13 TeV andfor p ¯p\\bar{p}p scattering at 1.96 TeVFigure: p ¯p\\bar{p}p differential cross section at 1.18-1.96 TeV and pppp differential cross section at 2.76 TeVIn Fig.",
"REF we plot the slopes as function of $t$ in $pp$ and $\\bar{p}p$ scatterings.",
"We discover from the $t$ -dependence of the slopes an extremely interesting phenomenon.",
"The slope in $pp$ scattering has a different behaviour in $t$ than the slope in $\\bar{p}p$ scattering.",
"In the left panel of Fig.",
"REF we see that in $pp$ scattering the slopes are first nearly constant and after that they fall sharply, they cut a first time the $B(t)=0$ line, reach a deep minimum negative value, after that they increase and cut a second time the $B(t)=0$ line and finally they reach an approximately constant value for higher $t$ .",
"The two crossing points of the $B(t)=0$ line move towards smaller $t$ when energy increases.",
"In the right panel of Fig.",
"REF we see a very different behaviour in $\\bar{p}p$ scattering.",
"In this case, at energies higher than ISR ones, B(t) marginally crosses zero, but no so deeply and sharply as in pp scattering.",
"For completeness, we show in Fig.",
"REF the slope parameter for $pp$ scattering at 7 and 13 TeV as compared with the slope parameter in $\\bar{p}p$ scattering at 1.96 TeV, where we can see the same phenomenon.",
"This phenomenon is a clear Odderon effect.",
"The odd-under crossing amplitude makes the difference between $pp$ and $\\bar{p}p$ scatterings and this amplitude is dominated at high energy by the Maximal Odderon."
],
[
"Comparison with other approaches",
"To our knowledge, the present model is the only model which fits forward and non forward data in a wide range of energies (including TeV region), without theoretical defects (like the violation of the unitarity).",
"However, it is important to note that our results concerning the slopes are in complete agreement with those obtained recently by Csörgö et al.",
"[21], who performed a very useful mirroring between the discontinuous experimental data (points) and continuous analytic functions (scattering amplitudes) by using an expansion in terms of Lévy polynomials.",
"In such a way they get a very clear Odderon effect concerning the slopes.",
"Their analysis have no dynamical content: it is a parametrization of experimental data in terms of big number of parameters.",
"This agreement is very important from two points of view.",
"On one side, the Odderon existence is reinforced by two quite different analysis, one model-independent and the other one having a dynamical content.",
"On another side, the fact that the Maximal Odderon is in agreement with a model-independent analysis reinforce the status of the Maximal Odderon."
],
[
"Conclusion",
"In our paper we present an extension of the Froissaron-Maximal Odderon (FMO) approach for $t$ different from zero, which satisfies rigorous theoretical constraints.",
"Our extended FMO approach gives an excellent description of the 3266 Experimental data at $t=0$ were taken from [15], with the recent TOTEM and ATLAS points being added.",
"Set of data at $t\\ne 0$ will be send after personal request to E. Martynov.",
"experimental points considered in a wide range of energies and momentum transferred.",
"One spectacular theoretical result is the fact that the difference in the dip-bump region between $\\bar{p}p$ and $pp$ differential cross sections is diminishing with increasing energies and for very high energies (say 100 TeV), the difference in the dip-bump region between $\\bar{p}p$ and $pp$ is changing its sign: $pp$ becomes bigger than $\\bar{p}p$ at $|t|$ about 1 GeV$^2$ .",
"This is a typical Odderon effect.",
"Another important - phenomenological - result of our approach is that the slope in $pp$ scattering has a different behaviour in $t$ than the slope in $\\bar{p}p$ scattering.",
"This is a clear Odderon effect.",
"Let us emphasize that the FMO model is in a good agreement with the data in a wide interval of energy.",
"However, there is a some discrepancy of the data and model in a region around $\\sqrt{s}$ =2 TeV (it is illustrated in the Fig.",
"REF ).",
"At the same time agreement with the data at lower and at higher energies is really very good.",
"This problem requires a special investigation which we will perform after the publication of the common TOTEM/D0 paper [23].",
"New ways of detecting Odderon effects, e. g. in an Electron-Ion Collider, were recently explored on the basis of a general QCD light front formalism [24].",
"Acknowledgment.",
"The authors thank Prof. Simone Giani for a careful reading of the manuscript.",
"One of us (E.M.) thanks the Department of Nuclear Physics and Power Engineering of the National Academy of Sciences of Ukraine for support (continuation of the project No 0118U005343)."
],
[
"General constraints",
"Let us reiterate here that the model with $\\sigma _{t}(s)\\propto \\ln ^{2}s$ is not compatible with a linear pomeron trajectory having the intercept 1.",
"Indeed, let us assume that $\\alpha _{P}(t)=1+\\alpha _{P}^{\\prime }t$ and the partial wave amplitude has the form $\\varphi (j,t)=\\eta (j)\\frac{\\beta (j,t)}{\\left[j-1-\\alpha _{P}^{\\prime }t\\right]^{n}}\\approx \\frac{i\\beta (1,t)}{\\left[j-1-\\alpha _{P}^{\\prime }t\\right]^{n}},$ $\\eta (j)=\\frac{1+\\xi e^{-i\\pi j}}{-\\sin \\pi j}.$ For Pomeron (simple or double pole) and Froissaron signature is positive, $\\xi =+1$ .",
"In ($s,t$ )-representation amplitude $\\varphi (j,t)$ is transformed to $a(s,t)=\\frac{1}{2\\pi i}\\int dj \\varphi (j,t)e^{\\xi j}, \\quad \\xi =\\ln (s/s_{0}).$ Then, we have pomeron contribution at large $s$ as $a(s,t)\\approx -\\tilde{\\beta }(t)[\\ln (-is/s_{0})]^{n-1}(-is/s_{0})^{1+\\alpha ^{\\prime }_{P}t}$ where $\\tilde{\\beta }(t)=\\beta (t)/\\sin (\\pi \\alpha _{P}(t)/2).$ If as usually $\\tilde{\\beta }(t)=\\tilde{\\beta }\\exp (bt)$ then we obtain $\\sigma _{t}(s)&&\\propto \\ln ^{n-1}s,\\nonumber \\\\\\sigma _{el}(s)&&\\propto \\frac{1}{s^{2}}\\int \\limits _{-\\infty }^{0}dt|a(s,t)|^{2}\\propto \\ln ^{2n-3}s.$ According to the obvious inequality, $\\sigma _{el}(s)\\le \\sigma _{t}(s)$ we have $2n-3\\le n-1 \\qquad \\Rightarrow \\qquad n\\le 2.$ Thus we come to the conclusion that the a model with $\\sigma _{t}(s)\\propto \\ln ^{2}s$ (n=3) is incompatible with a linear pomeron trajectory.",
"In other words the partial amplitude Eq.",
"(REF ) with $n=3$ is incorrect.",
"If $n=1$ we have a simple $j$ -pole leading to constant total cross section and vanishing at $s\\rightarrow \\infty $ elastic cross section.",
"However such a behaviour of the cross sections is not supported by experimental data.",
"If $n=2$ we have the model of dipole pomeron ($\\sigma _{t}(s)\\propto \\ln (s)$ ) and would like to emphasize that double $j$ -pole is the maximal singularity of partial amplitude settled by unitarity bound (REF ) if its trajectory is linear at $t\\approx 0$ .",
"We would like to notice here that TOTEM data for the $pp$ total cross section exclude the dipole pomeron model which is unable to describe with a reasonable $\\chi ^2$ the high values of $\\sigma ^{pp}_{tot}(s)$ at LHC energies.",
"Thus, constructing the model leading to cross section which increases faster than $\\ln (s)$ , we need to consider a more complicated case (we consider at the moment a region of small $t$ and $j\\approx 1$ ): $\\begin{array}{ll}\\varphi _+(j,t)&=\\dfrac{\\beta (j,t)}{\\left[j-1+r(-t)^{1/\\mu }\\right]^{n}}\\\\&\\approx \\dfrac{i\\beta (1,t)}{\\left[j-1+r(-t)^{1/\\mu }\\right]^{n}}.\\end{array}$ Making use of the same arguments as above, we obtain $\\sigma _{t}(s)\\propto \\ln ^{n-1}s,$ $\\sigma _{el}(s)\\propto \\ln ^{2n-2-\\mu }s \\qquad \\mbox{and} \\qquad \\mu \\ge n-1.$ However in this case amplitude $a(s,t)$ has a branch point at $t=0$ which is forbidden by analyticity of amplitude $a(s,t)$ .",
"A proper form of amplitude leading to $t_{eff}$ $t_{eff}$ can be defined by behaviour of elastic scattering amplitude at $s\\rightarrow \\infty $ .",
"If $a(s,t)\\approx sf(s)F(t/t_{eff}(s))$ then $\\sigma _{el}(s)\\propto |f(s)|^{2}\\int _{-\\infty }^{0}dt|F(t/t_{eff})|^{2}=t_{eff}|f(s)F(1)|^{2}$ .",
"decreasing faster than $\\ln ^{-1}s$ (it is necessary for $\\sigma _{t}$ rising faster than $\\ln s$ ) is the following $\\varphi _+ (j,t)=\\dfrac{\\beta (j,t)}{\\left[(j-1)^{m}-rt\\right]^{n}}.$ Now we have $m$ branch points colliding at $t=0$ in $j$ -plane and creating the pole of order $mn$ at $j=1$ (but there is no branch point in $t$ at $t=0$ ).",
"At the same time $t_{eff}\\propto 1/\\ln ^{m}s$ and from $\\sigma _{el}\\propto \\ln ^{2mn-2-m}s\\le \\sigma _{t}\\propto \\ln ^{mn-1}s\\le \\ln ^{2}s$ one obtains $\\left\\lbrace \\begin{array}{ll}mn &\\le m+1, \\\\mn &\\le 3.\\end{array}\\right.$ If $\\sigma _{el}\\propto \\sigma _{t}$ then $n=1+1/m$ .",
"Furthermore, if $\\sigma _{t}\\propto \\ln s$ then $m=1$ and $n=2$ which corresponds just to the dipole pomeron model.",
"In the Froissaron (or tripole pomeron) model $m=2$ and $n=3/2$ .",
"It means that $\\sigma _{t}\\propto \\ln ^{2}s$ ."
],
[
"Partial amplitudes",
"As it follows from Eq.",
"(REF ) for the dominating at $s\\rightarrow \\infty $ contribution in a Froissaron model with $\\sigma _{t}(s)\\propto \\ln ^{2}(s)$ , i.e.",
"$n=2$ , $m=3/2$ , we have to take (here and in what follows we used a more convenient notations $\\omega =j-1$ and $\\omega _{0\\pm }=r_\\pm \\tau =r_\\pm \\sqrt{-t/t_0},\\quad t_0=1 \\text{GeV}^2$ ).",
"Then $\\begin{array}{ll}\\varphi _{\\pm }(\\omega ,t)&= \\eta _{\\pm } (\\omega )\\dfrac{\\beta _{\\pm }(\\omega ,t)}{(\\omega ^{2}+\\omega _{0\\pm }^2)^{3/2}}\\\\&=\\binom{i}{1}e^{-i\\pi \\omega /2} \\dfrac{\\tilde{\\beta }_\\pm (\\omega ,t)}{(\\omega ^{2}+\\omega _{0\\pm }^2)^{3/2}}\\end{array}$ where $\\eta _\\pm (\\omega )=\\frac{1\\mp e^{-i\\pi \\omega } }{\\sin \\pi \\omega }.$ For even signature $\\tilde{\\beta }_+(\\omega ,t)=\\beta _+(\\omega ,t)/\\cos (\\omega \\pi /2)$ and for odd signature $\\tilde{\\beta }_-(\\omega ,t)=\\beta _-(\\omega ,t)/\\sin (\\omega \\pi /2).$ Now let us suppose that in agreement with the structure of the singularity of $\\phi _\\pm (\\omega ,t)$ at $\\omega ^2+\\omega _{0\\pm }^2=0$ the functions $\\tilde{\\beta }_\\pm (\\omega ,t)$ depend on $\\omega $ through the variable $\\kappa _\\pm =(\\omega ^2+\\omega _{0\\pm }^2) ^{1/2}$ and it can be expanded in powers of $\\kappa _\\pm $ $\\phi _\\pm (\\omega ,t)=\\binom{i}{1}e^{-i\\pi \\omega /2}\\dfrac{\\tilde{\\beta }_{1\\pm }(t)+\\kappa _\\pm \\tilde{\\beta }_{2\\pm }(t)+ \\kappa _\\pm ^2 \\tilde{\\beta }_{3\\pm }(t)}{\\kappa _\\pm ^{3}}.$ There are a different ways to add to partial amplitude $\\varphi (j,t)$ terms which at $s\\rightarrow \\infty $ are small corrections (they can be named as subasymptotic terms).",
"Thus we can expand the “residue” $\\beta (\\omega ,t) $ in powers of $\\omega $ (if $\\beta (\\omega ,t) $ has not branch point in $\\omega $ at $\\omega =0$ ) or in powers of $(\\omega ^2+\\omega _0^2)^{1/2}$ .",
"Then, for the first case $\\tilde{\\beta }(\\omega ,t)=\\tilde{\\beta }_1(t)+\\omega \\tilde{\\beta }_2(t)+\\omega ^2\\tilde{\\beta }_3(t),$ and in the second case we have (just this case is explored in the Section REF ) $\\tilde{\\beta }(\\omega ,t)=\\tilde{\\beta }_1(t)+(\\omega ^2+\\omega _0^2)^{1/2} \\tilde{\\beta }_2(t)+(\\omega ^2+\\omega _0^2) \\tilde{\\beta }_3(t).$ Let us notice that the main terms in $\\varphi (j,t)\\equiv \\varphi (\\omega ,t)$ for both cases are coinciding having a pair of branch points colliding at $\\omega _0=0\\quad (t=0)$ and generating a triple pole in partial amplitude.",
"Taking into account the table integrals $\\int \\limits _{0}^{\\infty } dx x^{\\alpha -1}e^{-\\omega x}\\textit {J}_{\\nu }(\\omega _{0}x)=I_{\\nu }^{\\alpha }(\\omega , \\omega _0)$ where $\\begin{array}{ll}I_{\\nu }^{\\nu +1}&=\\dfrac{(2\\omega _{0})^{\\nu }}{\\sqrt{\\pi }\\dfrac{\\Gamma (\\nu +1/2)}{(\\omega ^{2}+\\omega _{0}^{2})^{\\nu +1/2}}},\\\\ \\\\I_{\\nu }^{\\nu +2}&=2\\omega \\dfrac{(2\\omega _{0})^{\\nu }}{\\sqrt{\\pi }\\dfrac{\\Gamma (\\nu +3/2)}{(\\omega ^{2}+\\omega _{0}^{2})^{\\nu +3/2}}},\\end{array}$ one can find $\\begin{array}{rl}\\dfrac{1}{(\\omega ^{2}+\\omega _{0}^{2})^{3/2}}&=\\dfrac{1}{\\omega _{0}}\\int \\limits _{0}^{\\infty }dx xe^{-x\\omega }J_{1}(\\omega _{0}x), \\\\\\\\\\int \\limits _C \\dfrac{d\\omega }{2\\pi i} \\dfrac{e^{\\xi \\omega }}{(\\omega ^2+\\omega _0^2)^{3/2}}& = \\dfrac{J_1(\\omega _0\\xi )}{\\omega _0\\xi }.\\end{array}$ $\\begin{array}{rl}\\dfrac{1}{\\omega ^{2}+\\omega _{0}^{2}}&=\\dfrac{1}{\\omega _{0}}\\int \\limits _{0}^{\\infty }dx e^{-x\\omega }\\sin (x\\omega _{0}), \\\\\\\\\\int \\limits _C \\dfrac{d\\omega }{2\\pi i} \\dfrac{e^{\\xi \\omega }}{\\omega ^2+\\omega _0^2} &= \\dfrac{\\sin (\\omega _0\\xi )}{\\omega _0\\xi }.\\end{array}$ $\\begin{array}{rl}\\dfrac{1}{(\\omega ^{2}+\\omega _{0}^{2})^{1/2}}&=\\int \\limits _{0}^{\\infty }dx e^{-x\\omega }J_{0}(\\omega _{0}x), \\\\ \\\\\\int \\limits _C \\dfrac{d\\omega }{2\\pi i} \\dfrac{e^{\\xi \\omega }}{(\\omega ^2+\\omega _0^2)^{1/2}}& = J_0(\\omega _0\\xi ).\\end{array}$"
]
] | 1808.08580 |
[
[
"Topological generation and matrix models for quantum reflection groups"
],
[
"Abstract We establish several new topological generation results for the quantum permutation groups $S^+_N$ and the quantum reflection groups $H^{s+}_N$.",
"We use these results to show that these quantum groups admit sufficiently many \"matrix models\".",
"In particular, all of these quantum groups have residually finite discrete duals (and are, in particular, hyperlinear), and certain \"flat\" matrix models for $S_N^+$ are inner faithful."
],
[
"Introduction",
"The central objects of study in this paper are quantum permutations and their generalizations, quantum reflections.",
"Given $N \\in \\mathbb {N}$ and an Hilbert space $H$ , an $N\\times N$ matrix $P = [P_{ij}]_{1 \\le i,j \\le N} \\in M_N(B(H))$ ($B(H)$ being the C$^\\ast $ -algebra of bounded linear operators on $H$ ) is called a quantum permutation matrix (or magic unitary) if its entries $P_{ij}$ are self-adjoint projections satisfying the relations $\\sum _i P_{ij} = 1_{B(H)} = \\sum _{j} P_{ij}$ for each $1 \\le i,j \\le N$ .",
"The simplest examples of quantum permutation matrices are of course the classical permutation matrices (which correspond to those quantum permutations $P$ associated to a one-dimensional Hilbert space $H$ ).",
"In fact, more generally any quantum permutation $P$ with commuting entries $\\lbrace P_{ij}\\rbrace _{i,j}$ corresponds to a subset $X \\subseteq S_N$ of permutation matrices.",
"Indeed, in this case the commutative C$^\\ast $ -algebra $C^*(\\lbrace P_{ij}\\rbrace _{i,j})$ generated by the $P_{ij}$ 's is by Gelfand duality isomorphic to $C(X)$ , the C$^\\ast $ -algebra of complex functions on some subset $X \\subseteq S_N$ .",
"In particular $P$ is identified this way with the identity function on $X \\subset S_N \\subset M_N(\\mathbb {C})$ .",
"On the other hand, if one now considers quantum permutations $P$ whose entries do not commute, the structure of these objects becomes much less well-understood.",
"From an operator algebraic point of view, this should come as no surprise, as the C$^\\ast $ -algebras $C^*(P_{ij} \\ | \\ 1 \\le i,j \\le N) \\subset B(H)$ generated by the entries of a quantum permutation matrix $P$ with non-commuting entries can be highly non-trivial (e.g., can contain the free group C$^\\ast $ -algebras as quotients [46]).",
"Nonetheless, such “genuinely quantum” quantum permutations arise naturally in a variety of contexts.",
"For example, in quantum information theory, quantum permutation matrices arise naturally in the framework of non-local games and go under the name “projective permutation matrices” [4], [36], [34], [35].",
"From the perspective of non-commutative geometry and quantum group theory, $N \\times N$ quantum permutation matrices were discovered by Wang [46] to be precisely the structure that encode the quantum symmetries of a finite set of $N$ points.",
"More precisely, Wang considered the universal unital C$^\\ast $ -algebra $A = C^*\\Big (u_{ij}, 1 \\le i,j \\le N \\ \\big | \\ u_{ij} = u_{ij}^* = u_{ij}^2 \\ \\& \\ \\sum _{i} u_{ij} = \\sum _j u_{ij} = 1 \\ \\forall i,j\\Big ),$ generated by the coefficients of a “universal” $N \\times N$ quantum permutation matrix.",
"Wang then showed that there exists a compact quantum group, $S_N^+$ , acting universally and faithfully on the set of $N$ points in such a way that $A$ gets identified with the C$^\\ast $ -algebra $C(S_N^+)$ of “continuous functions” on the “quantum space” $S_N^+$ .",
"The quantum group $S_N^+$ is called the quantum permutation group or quantum symmetry group of $N$ points.",
"In contrast to its classical counterpart, $S_N^+$ (or more precisely $A = C(S_N^+)$ ) is a highly non-commutative and infinite-dimensional object.",
"It is one of our main goals in this paper is to investigate to what extent the quantum permutation groups $S_N^+$ (and the quantum reflection groups) can be approximated by elementary finite-dimensional structures.",
"We elaborate briefly on this now.",
"C$^\\ast $ -algebraic compact quantum groups as introduced in [48] form the basis of what by now is a rich theory, developing rapidly in a number of different directions.",
"As indicated above, the perspective we adopt in this paper is that the Hopf (C$^*$ -)algebras studied in [48], [49], [50] play the role of function algebras on a “compact quantum space” $\\mathbb {G}$ which is equipped with a group structure, and can equivalently be viewed as the complex group algebras of the Pontryagin dual “discrete quantum group” $\\Gamma $ .",
"To keep matters simple, throughout this introduction we write $\\mathbb {C}\\Gamma $ for the group algebra of a discrete quantum group $\\Gamma $ (see se.prel below for details).",
"One aspect of the theory of discrete quantum groups that presents itself naturally from this point of view is that of approximation properties.",
"This typically refers to the “accessibility” of the quantum group (or its associated algebras) via finite structures of some type.",
"The phrase `finite structure' is purposely vague, and lends itself to a variety of interpretations, for example: The amenability of a discrete quantum group implies the nuclearity of $C^*(\\Gamma )$ , the universal C$^*$ -completion of $\\mathbb {C}\\Gamma $ .",
"(I.e., the identity map on $C^*(\\Gamma )$ can be point-norm approximated by finite-rank completely positive contractions).",
"The converse is also true, at least for Kac type discree quantum groups [14].",
"The Haagerup approximation property of $\\Gamma $ corresponds to the point-norm approximation of the identity map on the reduced C$^\\ast $ -algebra $C^*_{r}(\\Gamma )$ by certain well-behaved $L^2$ -compact, contractive completely positive maps [26].",
"The weak amenbility of $\\Gamma $ corresponds to the point-norm approximation of the identity map on the reduced C$^\\ast $ -algebra $C^*_{r}(\\Gamma )$ by certain well-behaved finite rank, uniformly bounded completely bounded maps.",
"The above are only a few scattered examples, as we cannot possibly do justice to the vast literature here.",
"We refer to the survey [21] and its sources for a more expansive discussion on approximation properties for discrete (in fact locally compact) quantum groups.",
"The above list (and all those considered in [21]) can be thought of as instances internal approximation of a quantum group by finite structures, since all the above approximating maps are from a given object to itself.",
"In (quantum) group theory there are of course approximation properties which have an external flavor in the sense that one approximates a given large object by mapping it into smaller auxilliary objects.",
"For example, A Kac type discrete quantum group $\\Gamma $ is hyperlinear if its quantum group von Neumann algebra $\\mathcal {L}(\\Gamma )$ admits-finite dimensional matrix models relative to its Haar trace [22].",
"A Kac type discrete quantum group $\\Gamma $ has the Kirchberg factorization property if there is a net $\\varphi _k:C^*( \\Gamma ) \\rightarrow M_{n(k)}(\\mathbb {C})$ of contractive completely positive maps which are asymptotically trace-norm multiplicative and satisfy $h = \\lim _k \\text{tr}_{n(k)} \\circ \\varphi _k$ pointwise, where $h$ is the Haar trace and $\\text{tr}_{n(k)}$ is the normalized matrix trace.",
"A (finitely generated) discrete quantum group $\\Gamma $ is residually finite if the points of $\\mathbb {C}\\Gamma $ are separated by its finite-dimensional $\\ast $ -representations [24], [16].",
"A stronger form of residual finiteness of interest for us is the existence of a faithful or inner faithful matrix model $\\pi :\\mathbb {C}\\Gamma \\rightarrow M_N(C(X))$ for some compact Hausdorff $X$ [9], [11], [12], [7]; we will have more to say about this concept below.",
"It is the above list of external approximation properties that we are interested in establishing for (the duals of) Wang's quantum permutation groups $S^+_N$ and their generalizations $H^{s+}_N$ (the so-called quantum reflection groups [13]).",
"We recall some of the details in the preparatory se.prel below, and for now content ourselves to only remind the reader that $H^{s+}_N$ is a quantum version of the classical subgroup $H^s_N\\subset GL_N$ consisting of $N\\times N$ monomial matrices whose non-zero entries are $s^{th}$ roots of unity (so in particular $H^1_N\\cong S_N$ , the symmetric group on $N$ symbols).",
"The non-commutative topology of $H^{s+}_N$ as a compact quantum group plays a central role in our study of finiteness and approximation properties for their discrete duals $\\widehat{H^{s+}_N}$ , and hence the types of results proven in the paper.",
"We elaborate briefly: Let $\\mathbb {G}$ be a compact quantum group and let $\\mathbb {G}_i<\\mathbb {G}$ be a family of closed quantum subgroups of $\\mathbb {G}$ .",
"The condition that $\\mathbb {G}_i$ topologically generate $\\mathbb {G}$ was introduced in [22] for a pair of subgroups, but generalizes readily to arbitrary families.",
"In that paper, topological generation is used in the same fashion we do here: as a tool for lifting finiteness properties from the duals $\\Gamma _i = \\widehat{\\mathbb {G}_i}$ to $ \\Gamma = \\widehat{\\mathbb {G}}$ .",
"For that reason, we prove a number of topological generation results (th.top-gen,th.refl-tg) that can be summarized as: Theorem 1 For all $1\\le s\\le \\infty $ and $N\\ge 6$ , the quantum reflection group $H^{s+}_N$ is topologically generated by its quantum subgroups $S_N$ and $H^{s+}_{N-1}$ .",
"For $s=1$ the result also holds for $N=5$ .",
"$\\blacksquare $ This fits into a recurring pattern of topological generation results for infinite families of compact quantum groups.",
"For instance, [24] says in different terms that for $N\\ge 5$ the quantum unitary group $U^+_N$ is topologically generated by its quantum subgroup $\\mathbb {S}^1\\times U^+_{N-1}$ (product in the category of compact quantum groups, dual to the coproduct $C(\\mathbb {S}^1)*C(U^+_{N-1})$ of C$^*$ -algebras) and the classical unitary subgroup $U_N<U^+_N$ .",
"As for residual finiteness results, we use these topological generation results to prove (see th.rf,th.hns): Theorem 2 For $N\\ge 4$ and $1\\le s\\le \\infty $ the discrete duals $\\widehat{H^{s+}_N}$ of the quantum reflection groups are residually finite.",
"$\\blacksquare $ Here too there are precedents for other families: [24] treats the discrete duals of free unitary and orthogonal quantum groups.",
"We regard the above theorem as one of the main results of the paper, but it has a number of powerful consequences, including the hyperlinearity and Kirchberg factorization property for the selfsame discrete quantum groups [16], as well as improved estimates for the free entropy dimension of the generators of the associated von Neumann algebras $L^\\infty (H^{s+}_N)$ - see Section REF .",
"At this point it is worth highlighting that although our strategy for proving residual finiteness results for the quantum groups $H_N^{s+}$ (by means of inductive topological generation methods) is the same as that used in prior works, there is one critical difference here.",
"Unlike in the case of the free unitary/free orthogonal quantum groups which rely on the existence of “large” smoooth Lie subgroups (namely $U_N$ and $O_N$ , respectively), the quantum reflection groups only admit finite classical subgroups.",
"This difference turns out to be a fundamental obstruction to a straightforward extension of the inductive arguments of [24], [22].",
"To bypass this issue, we make essential use of a recent remarkable result of Banica [5] which establishes that there is no intermediate quantum subgroup for the inclusion $S_5 < S_5^+$ .",
"The maximality of the inclusion $S_N < S_N^+$ is widely conjectured to be true for all $N$ , and the case $N=5$ solved by Banica in [5] represents a major advancement on this conjecture.",
"It is also interesting to note that Banica's proof of the maximality of the inclusion $S_5 < S_5^+$ is based on a reduction of this problem to the seemingly different problem of classifying II$_1$ -subfactors at index 5.",
"This latter problem, however, has recently been solved [28], [27].",
"The authors find this connection to the classification of subfactors highly intriguing.",
"The other major set of results in this paper pertains to the quantum permutation groups $S_N^+$ .",
"In this case it turns out that we can say quite a lot more at the level of finite-dimensional representations.",
"While our residual finiteness results ensure the existence of enough finite-dimensional $\\ast $ -representations to separate points in the group algebras $\\mathcal {A}(S_N^+) = \\mathbb {C}\\widehat{S^+_N}$ , it is often desirable to have a single representation $\\pi :\\mathcal {A}(S_N^+) \\rightarrow B$ , where $B$ is some “nice” C$^\\ast $ -algebra (e.g.",
"finite-dimensional, or of the form $M_N(C(X))$ ) which encodes enough information about $S_N^+$ so as to generate an asymptotically faithful sequence of finite-dimensional representations $(\\pi _k)_{k \\in \\mathbb {N}}$ of $\\mathcal {A}(S_N^+)$ .",
"The relevant concept we are after here is that of an inner faithful represention $\\pi :\\mathcal {A}(S_N^+) \\rightarrow B$ .",
"We defer the precise definition of inner faithfulness to Section but note here that a representation $\\pi :\\mathcal {A}(S_N^+) \\rightarrow B$ is inner faithful if and only if the sequence of representations $\\pi _k:\\mathcal {A}(S_N^+) \\rightarrow B^{\\otimes k}; \\qquad \\pi _k = \\pi ^{\\otimes k} \\circ \\Delta ^{(k)};$ is asymptotically faithful in the sense that $\\bigcap _{k} \\ker \\pi _k = \\lbrace 0\\rbrace $ [39].",
"In the above, $\\Delta : \\mathcal {A}(S_N^+) \\rightarrow \\mathcal {A}(S_N^+) \\otimes \\mathcal {A}(S_N^+)$ denotes the coproduct and $\\Delta ^{(k)} = (\\operatorname{id}\\otimes \\Delta ^{(k-1)}) \\circ \\Delta $ for all $k \\ge 2$ .",
"In particular, this means that $\\mathcal {A}(S_N^+)$ faithfully embeds into a C$^\\ast $ -ultraproduct of the sequence of algebras $(B^{\\otimes k})_{k \\in \\mathbb {N}}$ .",
"One particular “minimal ” representation of $\\mathcal {A}(S_N^+)$ that has been conjectured to be inner faithful is Banica's universal flat representation [12], [7].",
"This particular representation takes the form $\\pi :\\mathcal {A}(S_N^+) \\rightarrow M_N(C(X_N))$ , where $X_N \\subset M_N(M_N(\\mathbb {C}))$ is the compact space of all $N \\times N$ bistochastic matrices $P= (P_{ij})_{i,j}$ whose entries are rank-one projections in $P_{ij} \\in M_N(\\mathbb {C})$ .",
"In this paper, we use modifications of our topological generation results to verify the conjectured inner faithfulness of the representation $\\pi $ for almost all values of $N$ (cf.",
"Corollary REF ).",
"Theorem 3 For all $N \\le 5$ and $N \\ge 10$ , the universal flat matric model $\\pi :\\mathcal {A}(S_N^+) \\rightarrow M_N(C(X_N))$ is inner faithful.",
"Piggybacking on the proof of this result, we are able to moreover show that one can reduce the base space $X_N$ to only contain at most 3 points and still achieve an inner faithful finite-dimensional representation.",
"In other words, we have (cf.",
"Theorem REF ) Theorem 4 For all $N \\le 5$ and $N \\ge 10$ , the quantum permutation group algebras $\\mathcal {A}(S_N^+)$ are inner unitary: they admit inner faithful $\\ast $ -homomorphisms $\\pi :\\mathcal {A}(S_N^+) \\rightarrow B$ , with $B$ a finite dimensional C$^\\ast $ -algebra.",
"It is an artifact of our proof that we are unable to settle the cases $N \\in [6,9]$ in the above theorems.",
"Our arguments rely heavily on our topological generation results for $S_N^+$ together with the crucial result of Banica [5] stating that the inclusion $S_5 < S_5^+$ is maximal.",
"The remainder of the paper is organized as follows.",
"se.prel gathers a number of prerequisites to be used later.",
"In se.rf we prove some of the main results, th.rf,th.hns, to the effect that the discrete Pontryagin duals of the quantum reflection groups are residually finite.",
"This then also implies that they are hyperlinear and have the Kirchberg factorization property.",
"The proofs rely in large part on an inductive argument, turning on the fact that quantum permutation groups $S^+_N$ are topologically generated by their quantum subgroups $S_N$ and $S^+_{N-1}$ (th.top-gen).",
"We also prove a similar result for quantum reflection groups in th.refl-tg; though strictly speaking not needed for residual finiteness of the quantum reflection groups, it might nevertheless be of some independent interest.",
"In the final section , we study inner faithful representations of Hopf $\\ast $ -algebras and prove that the universal flat representations are inner faithful for $N \\le 5$ and $N\\ge 10$ .",
"In subsection subse.inner-unitary we prove th.inner-unitary, confirming that for sufficiently large $N$ the duals $\\mathcal {A}(S_N^+)$ admit finite-dimensional inner faithful representations."
],
[
"Acknowledgements",
"M. Brannan and A. Chirvasitu are partially supported by the US National Science Foundation with grants DMS-1700267 and DMS-1801011 respectively."
],
[
"Generalities",
"Let us start by recalling some facts about compact quantum groups from [50]: Definition 2.1 A compact quantum group is a unital C$^*$ -algebra $C(\\mathbb {G})$ equipped with a unital $\\ast $ -morphism $\\Delta : C(\\mathbb {G})\\rightarrow C(\\mathbb {G}) \\otimes C(\\mathbb {G})$ (minimal tensor product) such that $\\mathrm {span}\\lbrace (a\\otimes 1)\\Delta (b)\\ |\\ a,b\\in C(\\mathbb {G})\\rbrace \\text{ and }\\mathrm {span}\\lbrace (1\\otimes a)\\Delta (b)\\ |\\ a,b\\in C(\\mathbb {G})\\rbrace $ are dense in $C(\\mathbb {G}) \\otimes C(\\mathbb {G})$ .",
"As is customary, we regard $C(\\mathbb {G})$ as the algebra of continuous functions on the fictitious “compact quantum space” $\\mathbb {G}$ .",
"For this reason, the category of compact quantum groups is dual to that of C$^*$ -algebras as in def.cqg.",
"For a compact quantum group $\\mathbb {G}$ we denote the unique dense Hopf $*$ -subalgebra of $C(\\mathbb {G})$ by $\\mathcal {A}(\\mathbb {G})$ ; it can be regarded alternatively as the complex group algebra $\\mathbb {C}\\widehat{\\mathbb {G}}$ of the discrete quantum group $\\widehat{\\mathbb {G}}$ whose Pontryagin dual is $\\mathbb {G}$ , for which reason we might occasionally revert to that alternative notation for it.",
"It is often also rendered as $\\mathrm {Pol}(\\mathbb {G})$ or $\\mathcal {O}(\\mathbb {G})$ in the literature.",
"The C*-algebra $C(G)$ is equipped with a unique state $h:C(\\mathbb {G})\\rightarrow \\mathbb {C}$ (its Haar state) that is left and right-invariant in the sense that the diagram $\\begin{tikzpicture}[baseline=(current bounding box.center),anchor=base,cross line/.style={preaction={draw=white,-,line width=6pt}}](0,0) node (1) {C(\\mathbb {G})} +(3,.5) node (2) {C(\\mathbb {G}) \\otimes C(\\mathbb {G})} +(6,0) node (3) {C(\\mathbb {G})} +(3,-.5) node (4) {\\mathbb {C}};[->] (1) to[bend left=6] node[pos=.5,auto]{\\scriptstyle \\Delta } (2) ;[->] (2) to[bend left=6] node[pos=.5,auto] {\\scriptstyle h\\otimes \\mathrm {id}} (3);[->] (1) to[bend right=6] node[pos=.5,auto,swap] {\\scriptstyle h} (4);[->] (4) to[bend right=6] node[pos=.5,auto,swap] {} (3);\\end{tikzpicture}$ and the analogous mirror diagram (obtained by substituting $\\mathrm {id}\\otimes h$ for the upper right hand arrow) both commute.",
"We will also occasionally refer to the quantum group von Neumann algebra $L^{\\infty }(\\mathbb {G})$ , which is by definition the von Neumann algebra generated by the GNS representation of $C(\\mathbb {G})$ associated to $h$ .",
"Often in the literature the von Neumann algebra $L^\\infty (\\mathbb {G})$ is written as $\\mathcal {L}(\\widehat{\\mathbb {G}})$ in view of it being a generalization of the von Neumann algebra generated by the left-regular representation of a discrete group.",
"Definition 2.2 A representation of the compact quantum group $\\mathbb {G}$ is a finite-dimensional comodule over the Hopf $*$ -algebra $\\mathcal {A}(\\mathbb {G})$ .",
"We write $\\mathrm {Rep}(\\mathbb {G})$ for the category of representations of $\\mathbb {G}$ .",
"We refer to [38] for the relatively small amount of background needed here on comodules over coalgebras or Hopf algebras.",
"Definition 2.3 Given two compact quantum groups $\\mathbb {H}$ and $\\mathbb {G}$ , we say that $\\mathbb {H}$ is a (closed) quantum subgroup of $\\mathbb {G}$ if there exists a surjective Hopf $\\ast $ -algebra morphism $\\pi : \\mathcal {A}(\\mathbb {G}) \\rightarrow \\mathcal {A}(\\mathbb {H})$ .",
"In this case, we write $\\mathbb {H}< \\mathbb {G}$ .",
"Note that if $\\mathbb {H}< \\mathbb {G}$ as above and $V$ is a $\\mathbb {G}$ -comodule, then $V$ automatically becomes a $\\mathbb {H}$ -comodule in a natural way.",
"Indeed if $\\alpha :V \\rightarrow V \\otimes \\mathcal {A}(\\mathbb {G})$ is the associated corepresentation defining the comodule $V$ and $\\pi : \\mathcal {A}(\\mathbb {G}) \\rightarrow \\mathcal {A}(\\mathbb {H})$ is the surjective Hopf $\\ast $ -morphism from above, then $V$ becomes an $\\mathbb {H}$ -comodule via the corepresentation $(\\operatorname{id}\\otimes \\pi )\\alpha : V \\rightarrow V \\otimes \\mathcal {A}(\\mathbb {H})$ .",
"This “restriction” of representations of $\\mathbb {G}$ to representations of $\\mathbb {H}$ , induces, at the level of Hom-spaces, natural inclusions $\\mathrm {hom}_{\\mathbb {G}}(V,W)\\hookrightarrow \\mathrm {hom}_{\\mathbb {H}}(V,W),$ for any pair of $\\mathbb {G}$ comodules $V,W$ ."
],
[
"Quantum reflection groups",
"Quantum reflection groups were introduced in [13] based on earlier work done in [6].",
"Following the former reference we denote them by $H^{s+}_N$ .",
"Definition 2.4 Let $N\\ge 2$ and $1\\le s\\le \\infty $ .",
"The underlying Hopf $*$ -algebra $\\mathcal {A}=\\mathcal {A}(H^{s+}_N)$ of the quantum reflection group $H^{s+}_N$ is generated as a $*$ -algebra by $N^2$ normal elements $u_{ij}$ , $1\\le i,j\\le N$ such that the matrices $u=(u_{ij})_{i,j}$ and $u^t = (u_{ji})_{i,j} $ are both unitary in $M_N(\\mathcal {A})$ ; $p_{ij}=u_{ij}u^*_{ij}$ is a self-adjoint idempotent for each $1 \\le i,j \\le N$ ; $u^s_{ij}=p_{ij}$ for each $1 \\le i,j \\le N$ , with the last relation absent when $s=\\infty $ .",
"The coproduct $\\Delta : \\mathcal {A}\\rightarrow \\mathcal {A}\\otimes \\mathcal {A}$ is determined by $\\Delta (u_{ij}) = \\sum _{k=1}^N u_{ik} \\otimes u_{kj} \\qquad (1 \\le i,j \\le N).$ The quantum groups $H^{s+}_N$ are meant to be quantum analogues of their classical versions $H^s_N$ consisting of monomial $N\\times N$ matrices whose non-zero entries are $s^{th}$ roots of unity.",
"In particular, $s=1$ recovers the symmetric group $S_N$ ; this is also the case in the quantum setting, recovering the quantum groups introduced in [46]: Definition 2.5 Let $N\\ge 2$ .",
"The quantum symmetric group $S^+_N$ is the quantum reflection group $H^{1+}_N$ from def.refl.",
"Its Hopf $\\ast $ -algebra $\\mathcal {A}$ is freely generated as a $*$ -algebra by an $N \\times N$ magic unitary matrix $u=(u_{ij})_{ij}\\in M_N(\\mathcal {A})$ , in the sense that all of the entries of $u$ are projections and the entries from each row and column add up to $1\\in \\mathcal {A}$ .",
"Now let $N\\ge 2$ and $1\\le s,t<\\infty $ be positive integers, as in def.refl.",
"The generators $u_{ij}$ of $\\mathcal {A}(H^{s+}_N)$ clearly satisfy the defining relations of $\\mathcal {A}(H^{st+}_N)$ as well, so we get a surjective Hopf $*$ -algebra morphism $\\mathcal {A}(H^{st+}_N)\\ni u_{ij}\\mapsto u_{ij}\\in \\mathcal {A}(H^{s+}_N).$ In other words, we have natural embeddings $H^{s+}_{N} < H^{st+}_N$ .",
"These embeddings will feature again below.",
"Also for future use, we briefly recall some background on the representation theory of the quantum groups $H^{s+}_N$ from [13].",
"Let $F_s$ be the free monoid on the symbols in $\\mathbb {Z}_s$ (understood as $\\mathbb {Z}$ when $s=\\infty $ ), equipped with the following operations: The involution $x\\mapsto \\overline{x}$ defined by $a_1\\cdots a_k \\mapsto (-a_k)\\cdots (-a_1),\\ a_i\\in \\mathbb {Z}_s;$ The fusion binary operation `$\\cdot $ ' defined by $(a_1\\cdots a_k)\\cdot (b_1\\cdots b_\\ell ) = a_1\\cdots a_{k-1}(a_k+b_1)b_2\\cdots b_\\ell .$ Then, according to [13], the Grothendieck ring $R_s$ of the category of finite-dimensional representations of $H^{s+}_N$ has a basis $\\lbrace x_f\\rbrace $ over $\\mathbb {Z}$ indexed by $f\\in F_s$ , and the multiplication resulting from the tensor product $x_f x_g = \\sum _{f=vz,g=\\overline{z}w}\\left(x_{vw}+x_{v\\cdot w}\\right).$ We remark that [13] is not clearly stated in this manner for $s=\\infty $ , but the proof goes through essentially unchanged for both finite and infinite $s$ .",
"A more general argument can be found in [32].",
"Restriction via the inclusion $H^{s+}_N< H^{st+}_N$ induces a ring morphism $R_{st}\\rightarrow R_s$ sending the generator $x_1$ , $1\\in \\mathbb {Z}_{st}$ to $x_1$ , $1\\in \\mathbb {Z}_s$ .",
"In particular, we have Lemma 2.6 The natural map $R_{\\infty }\\rightarrow \\varprojlim _s R_s$ is an embedding, where the directed inverse limit is taken over the positive integers ordered by division: $s\\le st$ .",
"$\\blacksquare $"
],
[
"Finiteness properties and topological generation",
"The following is a combination of [22] and [16].",
"Definition 2.7 A discrete quantum group $\\widehat{\\mathbb {G}}$ is finitely generated if $\\mathbb {C}\\widehat{\\mathbb {G}}$ is finitely generated as an algebra.",
"If $\\widehat{\\mathbb {G}}$ is finitely generated then we say it is residually finite if $\\mathbb {C}\\widehat{\\mathbb {G}}$ embeds as a $*$ -algebra into a (possibly infinite) direct product of matrix algebras.",
"Moreover, $\\widehat{\\mathbb {G}}$ is said to have the Connes embedding property (or is Connes-embeddable or hyperlinear) if it is of Kac type and the von Neumann algebra $(L^{\\infty }(\\mathbb {G})$ admits a Haar state-preserving embedding into the ultrapower $R^{\\omega }$ of the hyperfinite II$_1$ -factor.",
"Remark 2.8 Note that residual finiteness implies the Kirchberg factorization property of [17] by [16].",
"In turn, Kirchberg factorization implies hyperlinearity (e.g.",
"[16]); in conclusion, residual finiteness is stronger than Connes embeddability for a discrete quantum group.",
"We also recall from [22] the notion of topological generation for compact quantum groups: Definition 2.9 A family of compact quantum subgroups $(\\mathbb {G}_i < \\mathbb {G})_{i \\in I}$ topologically generate $\\mathbb {G}$ if, for every pair of representations $V$ , $W$ of $\\mathbb {G}$ , the natural inclusion map $\\mathrm {hom}_{\\mathbb {G}}(V,W)\\hookrightarrow \\bigcap _{i \\in I} \\mathrm {hom}_{\\mathbb {G}_i}(V,W)$ is an isomorphism.",
"In this case, we write $\\mathbb {G}= \\langle \\mathbb {G}_i\\rangle _{i \\in I}$ We will need the following alternative description of topological generation, which is almost immediate given def.top-gen; see also [22].",
"Lemma 2.10 A family of quantum subgroups $(\\mathbb {G}_i < \\mathbb {G})_{i \\in I}$ topologically generates $\\mathbb {G}$ if and only if for every $\\mathbb {G}$ -representation $V$ a map $f:V\\rightarrow \\mathbb {C}$ that is a morphism over every $\\mathbb {G}_i$ is a morphism over $\\mathbb {G}$ (i.e., $f \\in \\cap _{i \\in I} \\mathrm {hom}_{\\mathbb {G}_i}(V,\\mathbb {C}) \\Rightarrow f \\in \\mathrm {hom}_{\\mathbb {G}}(V,\\mathbb {C}) $ ).",
"Equivalently, it suffices to check this for all irreducible representations $V$ .",
"$\\blacksquare $ In particular, we have the following sufficient criterion for topological generation: Corollary 2.11 Let $(\\mathbb {G}_i< \\mathbb {G})_{i \\in I}$ be a family of quantum subgroups of a compact quantum group and assume that for every irreducible non-trivial $\\mathbb {G}$ -representation $V$ there is some $i$ such that the restriction of $V$ to $\\mathbb {G}_i$ contains no trivial summands.",
"Then, $\\mathbb {G}$ is topologically generated by the $\\mathbb {G}_i$ .",
"Proof 1 This follows from le.alt-top-gen: a morphism $f:V\\rightarrow \\mathbb {C}$ of representations witnesses an embedding of the trivial representation into $V$ , and the hypothesis ensures that a non-trivial irreducible $V\\in \\mathrm {Rep}(\\mathbb {G})$ retains the property of having no trivial summands over $\\mathbb {G}_i$ for some $i$ , meaning that $\\bigcap _{i \\in I} \\mathrm {hom}_{\\mathbb {G}_i}(V,\\mathbb {C}) = \\lbrace 0\\rbrace = \\mathrm {hom}_{\\mathbb {G}}(V,\\mathbb {C}).$ Topological generation appears under a different name in [24].",
"Rephrasing Corollary 2.16 therein more appropriately for our setting yields Lemma 2.12 A finitely generated discrete quantum group $\\widehat{\\mathbb {G}}$ is residually finite if and only if its dual $\\mathbb {G}$ is topologically generated by a family of subgroups $\\mathbb {G}_i< \\mathbb {G}$ with residually finite $\\widehat{\\mathbb {G}_i}$ .",
"$\\blacksquare $ Recall the embeddings $H^{s+}_N< H^{st+}_N$ from subse.refl.",
"The remark we will need in the sequel is Lemma 2.13 For $N\\ge 2$ the quantum group $H^{\\infty +}_N$ is topologically generated by its quantum subgroups $H^{s+}_N$ for finite $s$ .",
"Proof 2 According to le.limr the criterion of cor.tg-suff is satisfied: indeed, the former result ensures that every non-trivial irreducible $H^{\\infty +}_N$ -representation remains irreducible and non-trivial over some $H^{s+}_N$ ."
],
[
"Residual finite-dimensionality",
"We focus here on $*$ -algebras satisfying the condition required of $\\mathbb {C}\\widehat{\\mathbb {G}}$ in def.fin: Definition 2.14 A $*$ -algebra $\\mathcal {A}$ is residually finite-dimensional or RFD if it embeds as a $*$ -algebra in a product of matrix algebras.",
"Remark 2.15 The notion is used frequently in the context of C$^*$ -algebras, but here we are interested in the purely $*$ -algebraic version.",
"We gather a number of general observations on the RFD property for later use.",
"First, since residual finite-dimensionality obviously passes to $*$ -subalgebras, we will need to know that certain natural morphisms between free products with amalgamation (or pushouts, as we will also refer to them) are embeddings.",
"The following result is likely well known, but we include it here for completeness.",
"Lemma 2.16 Suppose we have the following commutative diagram of complex algebras, all of whose arrows are embeddings.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center),anchor=base,cross line/.style={preaction={draw=white,-,line width=6pt}}](0,0) node (1l) {\\mathcal {D}} +(2,1) node (2l) {\\mathcal {A}} +(2,-1) node (3l) {\\mathcal {B}}+(4,0) node (1r) {\\mathcal {D}} +(6,1) node (2r) {\\mathcal {A}^{\\prime }} +(6,-1) node (3r) {\\mathcal {B}^{\\prime }};[->] (1l) to[bend left=6] (2l);[->] (1l) to[bend right=6] (3l);[->] (1r) to[bend left=6] (2r);[->] (1r) to[bend right=6] (3r);[->] (1l) to node[pos=.5,auto] {\\scriptstyle \\cong } (1r);[->] (2l) to (2r);[->] (3l) to (3r);\\end{tikzpicture}$ If $\\mathcal {D}$ is finite-dimensional and semisimple then the canonical map $\\mathcal {A}*_{\\mathcal {D}}\\mathcal {B}\\rightarrow \\mathcal {A}^{\\prime }*_{\\mathcal {D}}\\mathcal {B}^{\\prime }$ is one-to-one.",
"Proof 3 Denote by $\\mathcal {D}^{\\circ }$ the opposite algebra of $\\mathcal {D}$ .",
"Under the hypothesis on $\\mathcal {D}$ the enveloping algebra $\\mathcal {D}\\otimes \\mathcal {D}^{\\circ }$ is semisimple, and hence the category of $\\mathcal {D}$ -bimodules (which are simply $\\mathcal {D}\\otimes \\mathcal {D}^{\\circ }$ -modules) is semisimple.",
"It follows that the inclusions $\\mathcal {D}\\rightarrow \\mathcal {A}\\rightarrow \\mathcal {A}^{\\prime }$ are split as $\\mathcal {D}$ -bimodule maps, and hence we have direct sum decompositions $\\mathcal {A}= \\mathcal {D}\\oplus \\mathcal {A}_1,\\ \\mathcal {A}^{\\prime }=\\mathcal {D}\\oplus \\mathcal {A}^{\\prime }_1 = \\mathcal {D}\\oplus \\mathcal {A}_1\\oplus \\mathcal {A}_2$ and similarly for the $\\mathcal {B}$ side of the diagram.",
"According to [15] the pushout $\\mathcal {A}*_{\\mathcal {D}}\\mathcal {B}$ decomposes as a direct sum of tensor products of the form $T_1\\otimes T_2\\cdots \\otimes T_k,\\quad T_i\\text{ chosen alternately from } \\lbrace \\mathcal {A}_1,\\mathcal {B}_1\\rbrace $ in the category of $\\mathcal {D}$ -bimodules.",
"The analogous decomposition holds for $\\mathcal {A}^{\\prime }*_{\\mathcal {D}}\\mathcal {B}^{\\prime }$ , and due to eq:dab the tensor product eq:ts is a summand in its counterpart $T^{\\prime }_1\\otimes \\cdots \\otimes T^{\\prime }_k,\\quad T^{\\prime }_i\\in \\lbrace \\mathcal {A}^{\\prime }_1,\\mathcal {B}^{\\prime }_1\\rbrace \\text{ alternately},\\quad T^{\\prime }_i=\\mathcal {A}^{\\prime }_1 \\iff T_i=\\mathcal {A}_1.$ The conclusion that $\\mathcal {A}*_{\\mathcal {D}}\\mathcal {B}\\rightarrow \\mathcal {A}^{\\prime }*_{\\mathcal {D}}\\mathcal {B}^{\\prime }$ is an embedding follows.",
"There are analogues of this in the C$^*$ -algebra literature: see e.g.",
"[3] and [37].",
"We will only use le.emb-push in the context of $*$ -algebras, with all embeddings being $*$ -algebra morphisms.",
"An immediate consequence of le.emb-push is Corollary 2.17 Under the hypotheses of le.emb-push, suppose furthermore that all maps are $*$ -algebra morphisms.",
"If $\\mathcal {A}^{\\prime }*_{\\mathcal {D}}\\mathcal {B}^{\\prime }$ is RFD, then so is $\\mathcal {A}*_{\\mathcal {D}}\\mathcal {B}$ .",
"$\\blacksquare $ We now turn to the technical result of this section, which will be needed later on.",
"Proposition 2.18 If $\\mathcal {A}$ is an RFD $*$ -algebra and $\\mathcal {D}\\subset \\mathcal {A}$ is a finite-dimensional commutative C$^*$ -subalgebra then $\\mathcal {A}*_{\\mathcal {D}}\\mathcal {A}$ is RFD.",
"Before going into the proof we treat a particular case.",
"Lemma 2.19 pr.aa holds when $\\mathcal {A}$ is a finite-dimensional C$^*$ -algebra.",
"This in turn requires some preparation.",
"More precisely, we first build a faithful Hilbert space representation of the pushout $\\mathcal {A}*_{\\mathcal {D}}\\mathcal {A}$ .",
"Throughout the present discussion $\\mathcal {D}\\subset \\mathcal {A}$ are as in the statement of le.aa-fin.",
"Consider the canonical conditional expectation $E:\\mathcal {A}\\rightarrow \\mathcal {D}$ that preserves an arbitrary but fixed faithful tracial state $\\tau $ on $\\mathcal {A}$ .",
"This means that $\\begin{tikzpicture}[baseline=(current bounding box.center),anchor=base,cross line/.style={preaction={draw=white,-,line width=6pt}}](0,0) node (1) {\\mathcal {A}} +(2,.5) node (2) {\\mathcal {D}} +(4,0) node (3) {\\mathbb {C}};[->] (1) to[bend left=6] node[pos=.5,auto] {\\scriptstyle E} (2);[->] (2) to[bend left=6] node[pos=.5,auto] {\\scriptstyle \\tau |_{\\mathcal {D}}} (3);[->] (1) to[bend right=6] node[pos=.5,auto,swap] {\\scriptstyle \\tau } (3);\\end{tikzpicture}$ Moreover, the map $E$ is contractive, completely positive, and splits the inclusion $\\mathcal {D}\\rightarrow \\mathcal {A}$ in the category of $\\mathcal {D}$ -bimodules (see e.g.",
"[40] for background on expectations on operator subalgebras).",
"We denote by $\\mathcal {A}_{\\ell }$ and $\\mathcal {A}_r$ the copies of $\\ker (E)$ in the left and respectively right hand side free factor of $\\mathcal {A}*_{\\mathcal {D}}\\mathcal {A}$ .",
"Then, as a consequence of [15], we have $\\mathcal {A}*_{\\mathcal {D}}\\mathcal {A}= \\bigoplus _{{\\bf i}} T_{{\\bf i}}$ where ${\\bf i}$ ranges over the words on ${\\ell ,r}$ with no consecutive repeating letters and e.g.",
"$T_{\\ell r\\ell \\cdots } = \\mathcal {A}_{\\ell }\\otimes \\mathcal {A}_r\\otimes \\mathcal {A}_{\\ell }\\otimes \\cdots $ with tensor products over $\\mathcal {D}$ .",
"The empty word is allowed, with $T_{\\emptyset }=\\mathcal {D}$ .",
"Each $T_{\\bf i}$ is then naturally a Hilbert $\\mathcal {D}$ -module (e.g.",
"[47]).",
"Composing the $\\mathcal {D}$ -valued inner product on $T_{\\bf i}$ further with the inner product $\\mathinner {\\langle {x|y}\\rangle } = \\tau (x^*y),\\ x,y\\in \\mathcal {D}$ makes each $T_{\\bf i}$ into a Hilbert space.",
"Left multiplication makes the algebraic direct sum $\\bigoplus _{\\bf i}T_{\\bf i}$ a faithful module over $\\mathcal {A}*_{\\mathcal {D}}\\mathcal {A}$ (it is simply the left regular representation of the algebra in question).",
"The Hilbert space completion of $\\bigoplus _{\\bf i} T_{\\bf i}$ is thus a faithful Hilbert space representation of $\\mathcal {A}*_{\\mathcal {D}} \\mathcal {A}$ .",
"Proof of le.aa-fin 1 The fact that the full C$^*$ -pushout $\\overline{\\mathcal {A}*_{\\mathcal {D}}\\mathcal {A}}$ is residually finite-dimensional follows from [3].",
"It thus remains to show that the algebraic pushout $\\mathcal {A}*_{\\mathcal {D}}\\mathcal {A}$ embeds in its C$^*$ -envelope.",
"In other words, we want to argue that $\\mathcal {A}*_{\\mathcal {D}}\\mathcal {A}$ admits a faithful representation on some Hilbert space.",
"This, however, is precisely what we constructed above in the discussion preceding the proof: the Hilbert space direct sum of $T_{\\bf i}$ for ${\\bf i}$ ranging over words on $\\lbrace \\ell ,r\\rbrace $ with no repeating consecutive letters is such a representation.",
"Remark 2.20 The construction of the Hilbert modules $T_{\\bf i}$ sketched above features prominently in free probability; see e.g.",
"[43] and [42].",
"Proof of pr.aa 1 The RFD condition implies that there is an embedding $\\mathcal {A}\\rightarrow \\prod _{i\\in I} M_{n_i}$ for an index set $I$ (or arbitrary cardinality).",
"By cor.rfd-psh, it thus suffices to assume that $\\mathcal {A}$ is such a product of matrix algebras to begin with.",
"We therefore make this assumption throughout the rest of the proof.",
"We first appeal once more to [15] to conclude that since the embedding $\\mathcal {D}\\rightarrow \\mathcal {A}$ splits as $\\mathcal {A}= \\mathcal {D}\\oplus \\mathcal {A}_1$ in the category of $\\mathcal {D}$ -bimodules (because $\\mathcal {D}\\otimes \\mathcal {D}^{\\circ }$ is semisimple), we have a decomposition $\\mathcal {A}*_{\\mathcal {D}}\\mathcal {A}\\cong \\bigoplus T_1\\otimes \\cdots \\otimes T_k$ where $T_i$ are chosen from among the two copies of $\\mathcal {A}_1$ .",
"An arbitrary element in $\\mathcal {A}*_{\\mathcal {D}}\\mathcal {A}$ can thus be expressed as a sum of elements of finitely many tensor products as in eq:t1k.",
"It follows that if $x$ is non-zero then it maps to a non-zero element of a pushout $\\mathcal {B}*_{\\mathcal {D}}\\mathcal {B}$ through some projection $\\mathcal {A}=\\prod _{i\\in I}M_{n_i}\\rightarrow \\prod _{i\\in F}M_{n_i}=\\mathcal {B}$ for a finite subset $F\\subseteq I$ .",
"In conclusion, it will be enough to further assume that $\\mathcal {A}$ is a finite product of matrix algebras, i.e.",
"a finite-dimensional C$^*$ -algebra.",
"This is le.aa-fin, hence the conclusion.",
"Remark 2.21 Once more, there are versions of pr.aa applicable to C$^*$ -algebras; [33] is one example.",
"We conclude this section with one more proposition which will have direct application in the next section to the residual finiteness of the duals of quantum reflection groups.",
"Proposition 2.22 Let $\\mathcal {B}$ be an RFD $*$ -algebra, $\\mathcal {D}\\subset \\mathcal {B}$ a finite-dimensional commutative C$^*$ -subalgebra, and $\\mathcal {C}$ a finite-dimensional C$^*$ -algebra.",
"Then, $\\mathcal {C}*\\mathcal {B}/[\\mathcal {C},\\mathcal {D}]$ is RFD.",
"Proof 4 The algebra in question is isomorphic to the pushout $(\\mathcal {C}\\otimes \\mathcal {D})*_{\\mathcal {D}} \\mathcal {B}.$ Since we have rightward embeddings $\\begin{tikzpicture}[baseline=(current bounding box.center),anchor=base,cross line/.style={preaction={draw=white,-,line width=6pt}}](0,0) node (1l) {\\mathcal {D}} +(2,1) node (2l) {\\mathcal {C}\\otimes \\mathcal {D}} +(2,-1) node (3l) {\\mathcal {B}}+(4,0) node (1r) {\\mathcal {D}} +(6,1) node (2r) {\\mathcal {C}\\otimes \\mathcal {B}} +(6,-1) node (3r) {\\mathcal {C}\\otimes \\mathcal {B}};[->] (1l) to[bend left=6] (2l);[->] (1l) to[bend right=6] (3l);[->] (1r) to[bend left=6] (2r);[->] (1r) to[bend right=6] (3r);[->] (1l) to node[pos=.5,auto] {\\scriptstyle \\cong } (1r);[->] (2l) to (2r);[->] (3l) to (3r);\\end{tikzpicture}$ cor.rfd-psh shows that it will be enough to prove residual finite-dimensionality for $(\\mathcal {C}\\otimes \\mathcal {B})*_{\\mathcal {D}}(\\mathcal {C}\\otimes \\mathcal {B})$ In turn, this follows from pr.aa and the simple observation that tensor products of RFD $*$ -algebras are RFD."
],
[
"Topological generation and residual finiteness",
"Our first goal in this section is to treat the case of quantum permutation groups.",
"Before getting to the main results, we first recall some basic facts about the description of the invariant theory for $S_N^+$ in terms of non-crossing partitions [10]."
],
[
"Non-crossing partition maps",
"Definition 3.1 Fix $k \\in \\mathbb {N}$ and consider the ordered set $[k] = \\lbrace 1, \\ldots , k\\rbrace $ .",
"A partition of $[k]$ is a decomposition $p$ of $[k]$ into a disjoint union of non-empty subsets, called the blocks of $p$ .",
"A partition $p$ of $[k]$ has a crossing if there exist $a < b < c <d \\in [k]$ such that $\\lbrace a,c\\rbrace $ and $\\lbrace b,d\\rbrace $ belong to different blocks.",
"A partition $p$ is called non-crossing if it has no crossings.",
"The collection of all non-crossing partions of $[k]$ is denoted by $NC(k)$ .",
"Given a function $i:[k] \\rightarrow [N]$ (i.e.",
"a multi-index $i = (i(1), \\ldots , i(k)) \\in [N]^k$ ), we let $\\ker i$ be the partition of $[k]$ given by declaring that $r,s$ belong to the same block of $\\ker i$ if and only if $i(r) = i(s)$ .",
"Given $i$ as above and $p \\in NC(k)$ , we define $\\delta _p(i) =\\Bigg \\lbrace \\begin{matrix} 1& \\ker i \\ge p \\\\0& \\text{otherwise}\\end{matrix},$ where $\\ge $ denotes the refinement partial order on the lattice partitions of $[k]$ .",
"Now let $V$ be an $N$ -dimensional Hilbert space with distinguished orthonormal basis $e_j$ , $1\\le j\\le N$ .",
"Given $k,l \\in \\mathbb {N}_0$ and $p \\in NC(k + l)$ , we form the linear map $T_p^{k,l,N}: V^{\\otimes k} \\rightarrow V^{\\otimes l}; \\qquad T_p^{k,l,N}(e_{i(1)} \\otimes \\ldots \\otimes e_{i(k)}) = \\sum _{j:[l] \\rightarrow [N]}\\delta _p(ij) e_{j(1)}\\otimes \\ldots \\otimes e_{j(l)},$ where $ij:[k+l] \\rightarrow [N]$ is the concatenation of $i$ and $j$ .",
"We call such linear maps non-crossing partition maps.",
"The fundamental result that will be of use to us here is the following description of $S_N^+$ invariants in terms of these non-crossing partition maps.",
"Theorem 3.2 ([10]) For each $k,l \\in \\mathbb {N}_0$ , $N \\in \\mathbb {N}$ we have linear isomorphisms $\\textrm {hom}_{S_N^+}(V^{\\otimes k}, V^{\\otimes l}) = \\text{span}\\lbrace T_p^{k,l,N}: p \\in NC(k,l)\\rbrace .$ Moreover, if $N \\ge 4$ , then $\\lbrace T_p^{k,l,N}: p \\in NC(k,l)\\rbrace $ forms a linear basis for $\\text{hom}_{S_N^+}(V^{\\otimes k}, V^{\\otimes l}) $ ."
],
[
"Quantum permutation groups",
"Let $V = \\text{span}(e_i)_{i=1}^N$ be as above, regarded as the fundamental representation of $S_N^+$ .",
"We regard $S_{N-1}^+$ as also acting on $V$ via its fundamental representation on the $(N-1)$ -dimensional subspace $V_N$ spanned by $e_i$ , $1\\le i\\le N-1$ , and via the trivial representation on the orthogonal complement $\\mathbb {C}e_N$ .",
"In this way, we regard $S_{N-1}^+ < S_N^+$ .",
"For future reference, we denote by $V_i$ the subspace of $V$ spanned by all $e_j$ , $j\\ne i$ , and by $W_i$ the span of $e_i$ alone.",
"Our first main result in this section reads as follows.",
"Theorem 3.3 For $N\\ge 5$ the quantum permutation group $S^+_N$ is topologically generated by its subgroups $S_N$ and $S_{N-1}^+$ .",
"Before embarking on the proof we record the following immediate strengthening of the statement: Corollary 3.4 Let $N\\ge 5$ , $4\\le M\\le N$ and $S^+_M<S^+_N$ the embedding of the quantum subgroup fixing $N-M$ of the standard basis vectors of the defining representation of $S^+_N$ on $\\mathbb {C}^N$ .",
"Then, $S^+_N$ is topologically generated by $S_N$ and $S^+_M$ .",
"Proof 5 This is a repeated application of th.top-gen: $S_M < S_N$ and $S^+_M$ topologically generate $S^+_{M+1}$ and the result follows by induction.",
"The cases $N=5$ and $N\\ge 6$ will be treated differently.",
"The latter requires pr.coinv below, where $V$ is the $N$ -dimensional Hilbert space with distinguished orthonormal basis $e_j$ , $1\\le j\\le N$ carrying the defining representation of $S^+_N$ .",
"Proposition 3.5 Suppose $N\\ge 6$ .",
"For any $k\\ge 0$ , any linear map $f:V^{\\otimes k}\\rightarrow \\mathbb {C}$ that is invariant under both $S_{N-1}^+$ and the classical permutation group $S_N$ is invariant under all of $S_N^+$ .",
"Proof 6 In other words, we have to show a functional $f:V^{\\otimes k}\\rightarrow \\mathbb {C}$ that is both an $S_{N-1}^+$ -coinvariant and an $S_N$ -coinvariant must also be an $S_N^+$ -coinvariant.",
"Note that the $S_N$ -invariance ensures that $f$ respects the action of each one among the $N-1$ choices of quantum subgroup $S_{N-1}^+< S_N^+$ obtained by acting on the subspaces $V_i \\subset V$ .",
"Moreover, the invariance under the original copy of $S_{N-1}^+< S_N^+$ ensures that when restricted to $V_N^{\\otimes k}$ , $f$ acts as a linear combination of the non-crossing partition maps $\\lbrace T^{k,0,N-1}_p\\rbrace _{p \\in NC(k)}$ .",
"Replacing the maps $\\lbrace T^{k,0,N-1}_p\\rbrace _{p \\in NC(k)}$ in this linear combination with $\\lbrace T^{k,0,N}_p\\rbrace _{p \\in NC(k)}$ (which belong to $\\textrm {hom}_{S_N^+}(V^{\\otimes k}, \\mathbb {C})$ ) and subtracting this new linear combination from $f$ , we may as well assume that $f|_{V_N^{\\otimes k}}$ vanishes and then try to prove that $f$ itself is zero.",
"Once more, the $S_N$ -invariance ensures that the restriction of $f$ to $V_i^{\\otimes k}\\subset V^{\\otimes k}$ vanishes for every $1\\le i\\le N$ .",
"What we have to show, however, is that it also vanishes on summands of $V^{\\otimes k}$ obtained by tensoring some copies of $V_N$ with some copies of $W_N$ .",
"To simplify notation and fix ideas, we will show that the restriction of $f$ to, say, $U=V_N^{\\otimes (k-l)}\\otimes W_N^{\\otimes l} \\subset V$ vanishes.",
"The general case is perfectly analogous, with only notational difficulties making the presentation more cumbersome.",
"As the action of the original copy of $S_{N-1}^+$ that we considered is trivial on $W_N$ , $U$ can be identified with the $(k-l)^{\\text{th}}$ tensor power of the fundamental representation of $S_{N-1}$ , and hence any $S_{N-1}^+$ -coinvariant $U\\rightarrow \\mathbb {C}$ will be some linear combination of non-crossing partitions maps associated to $NC(k-l)$ .",
"Now let $V_{1,N}\\subset V_N$ be the span of $e_j$ , $j\\ne 1,N$ .",
"Because $N\\ge 6$ , $V_{1,N}$ is at least 4-dimensional and hence the linear forms associated to non-crossing partitions are linearly independent on it.",
"This means that if a linear combination of non-crossing partition functionals on $V_{1,N}^{\\otimes (k-l)}\\rightarrow \\mathbb {C}$ vanishes, then the linear combination itself must be trivial.",
"But note now that $f$ restricted to $V_{1,N}^{\\otimes (k-l)}\\otimes W_N^{\\otimes l}$ vanishes, because the space in question is a subspace of $V_1^{\\otimes k}$ .",
"By the paragraph above, $f$ must therefore vanish on all of $U$ .",
"Proof of th.top-gen 1 As mentioned above, we treat the cases $N=5$ and $N>5$ separately.",
"(Case 1: $N\\ge 6$ ) Recall e.g.",
"from [8] that every finite-dimensional $S^+_N$ -representation appears as a summand of $V^{\\otimes k}$ where $V$ is the $N$ -dimensional defining representation and $k$ is some positive integer.",
"By le.alt-top-gen the conclusion is now a paraphrase of pr.coinv.",
"(Case 2: $N=5$ ) According to [5] the inclusion $S_5< S^+_5$ admits no intermediate quantum groups.",
"Since $S^+_4< S^+_5$ is not a quantum subgroup of $S_5$ , we indeed have $S^+_5=\\langle S_5,S^+_4\\rangle $ .",
"As a consequence of the above we have Theorem 3.6 The discrete duals $\\widehat{S^+_N}$ of the free quantum permutation groups are residually finite.",
"Proof 7 By le.gen-rf,th.top-gen we can proceed inductively once we know that $\\widehat{S_N}$ is residually finite (the group algebra is finite-dimensional); $\\widehat{S^+_4}$ is residually finite.",
"For the latter, recall from [11] that $\\mathcal {A}(S^+_4)$ embeds into the C$^*$ -algebra $C(SU_2, M_4)$ , and hence has enough 4-dimensional representations.",
"Since, as observed in Remark REF , hyperlinearity and the Kirchberg factorization property are weaker than residual finiteness, we also have Corollary 3.7 The discrete duals $\\widehat{S^+_N}$ have the Kirchberg factorization property and are hyperlinear.",
"$\\blacksquare $ It will be of some interest to have alternative topological generation results which we now state and prove.",
"Let $4\\le M\\le N$ be a pair of positive integers, and write $N=M+T$ .",
"We then have Hopf $\\ast $ -algebra surjections $\\mathcal {A}(S^+_N)\\rightarrow \\mathcal {A}(S^+_M)*\\mathcal {A}(S^+_T)$ that annihilate $u_{ij}$ for $i,j$ in distinct parts of any partition of $[N]$ into two parts of sizes $M$ and $T$ .",
"We will be somewhat vague on which partitions to use; sometimes we need to refer to arbitrary ones, but when we do not, the reader can simply assume the partition is $[N] = [M] \\sqcup \\lbrace M+1,\\cdots ,M+T\\rbrace ,$ corresponding to the upper left-hand corner embedding $S_M^+ < S_N^+$ corresponding to action of $S_M^+$ on $V = \\text{span}\\lbrace e_1, \\ldots , e_N\\rbrace $ which fixes $e_{M+1}, \\ldots , e_{M+T}$ .",
"We now come to a critical notion.",
"Definition 3.8 An compact quantum group embedding $\\mathbb {G}< S^+_N$ is $(M,N)$ -large if it factors the upper path in the diagram $\\begin{tikzpicture}[baseline=(current bounding box.center),anchor=base,cross line/.style={preaction={draw=white,-,line width=6pt}}](0,0) node (1) {\\mathcal {A}(S^+_N)} +(4,-.5) node (2) {\\mathcal {A}(S^+_M)*\\mathcal {A}(S^+_T)} +(7,-.5) node (3) {\\mathcal {A}(\\mathbb {G})} +(0,-1) node (1bis) {\\mathcal {A}(S^+_M)};[->] (1) to [bend right=6] (2);[->] (2) -- (3);[->] (1bis) to[bend left=6] (2);\\end{tikzpicture}$ so as to make its lower path one-to-one.",
"Example 3.9 The obvious examples of $(M,N)$ -large embeddings are those corresponding to the standard surjections $\\mathcal {A}(S^+_N)\\rightarrow \\mathcal {A}(S^+_M)$ .",
"Slightly less obvious examples can be obtained as follows: Suppose $N=KM$ for some positive integer $K$ .",
"The diagonal embedding $S^+_M\\le S^+_N$ is obtained at the level of Hopf algebras as the surjection $\\mathcal {A}(S^+_N)\\rightarrow \\mathcal {A}(S^+_M)^{*K}\\rightarrow \\mathcal {A}(S^+_M)$ where the left hand arrow annihilates the generators $u_{ij}$ that are off the diagonal consisting of $M\\times M$ blocks and the right hand arrow is the identity on each free factor.",
"Diagonal embeddings are $(M,N)$ -large in the sense of def.large.",
"The embedding of Hopf $\\ast $ -algebras $\\mathcal {A}(S^+_M)\\rightarrow \\mathcal {A}(\\mathbb {G})$ forming the lower half of eq:vee corresponds to a morphism of quantum groups $\\mathbb {G}\\rightarrow S^+_M$ , which in turn gives rise to restriction and induction functors $\\begin{tikzpicture}[baseline=(current bounding box.center),anchor=base,cross line/.style={preaction={draw=white,-,line width=6pt}}](0,0) node (1) {\\mathrm {Rep}(S^+_M)} +(4,0) node (2) {\\mathrm {Rep}(\\mathbb {G})};[->] (1) to [bend left=6] node[pos=.5,auto] {\\scriptstyle \\mathrm {res}} (2);[->] (2) to [bend left=6] node[pos=.5,auto] {\\scriptstyle \\mathrm {ind}} (1);\\end{tikzpicture}$ with restriction being the left adjoint to induction.",
"These remarks will recur below.",
"Proposition 3.10 Let $5\\le M\\le N$ and $\\mathbb {G}< S^+_N$ an $(M,N)$ -large embedding.",
"Then, $S^+_N$ is topologically generated by $\\mathbb {G}$ and $S_N$ .",
"Proof 8 Throughout the proof we will fix a basis $\\lbrace e_i\\rbrace $ for the $N$ -dimensional carrier space $V$ of the defining representation of $S^+_N$ , and assume that the upper left hand arrow in eq:vee is the standard one corresponding to the partition eq:part.",
"The proof will be very similar to the that of pr.coinv: we fix a map $f:V^{\\otimes k}\\rightarrow \\mathbb {C}$ that is a morphism both over $S_N<S^+_N$ and $\\mathbb {G}$ and seek to show that $f$ is also an $S^+_N$ -morphism.",
"Decompose $V=V_1\\oplus V_2$ where $V_1=\\mathrm {span}\\lbrace e_1,\\cdots ,e_M\\rbrace ,\\quad V_2=\\mathrm {span}\\lbrace e_{M+1},\\cdots ,e_{M+T}\\rbrace .$ The tensor power $V^{\\otimes k}$ then decomposes as $\\bigoplus _{{\\bf i}} V_{i_1}\\otimes \\cdots \\otimes V_{i_k}$ with summands ranging over all tuples ${\\bf i}=(i_j)$ , $i_j\\in \\lbrace 1,2\\rbrace $ .",
"The summand $V_1^{\\otimes k}$ is a comodule over $\\mathcal {A}(S^+_M)\\subset \\mathcal {A}(\\mathbb {G})$ (the embedding being the lower path in eq:vee), and hence $f|_{V_1^{\\otimes k}}$ is a linear combination of non-crossing partitions.",
"Subtracting the same linear combination of non-crossing partitions on $V^{\\otimes k}$ , we can assume $f|_{V_1^{\\otimes k}}=0$ .",
"The goal now is to show that in fact $f=0$ globally (i.e.",
"on the entirety of $V^{\\otimes k}$ ).",
"We do this iteratively, proving by the following $t$ -dependent claim by induction on $t$ : Claim($t$ ): The restriction of $f$ to any summand of the type $V_{i_1}\\otimes \\cdots \\otimes V_{i_k},\\ t\\text{ of the }i_j\\text{ are }2.$ is zero.",
"The base case $t=0$ of the induction is in place (claiming simply that $f|_{V_1^{\\otimes k}}=0$ , which we know).",
"We now turn to the induction step, assuming Claim($s$ ) for all $s\\le t-1$ and seeking to prove Claim($t$ ).",
"In order to lessen the notational load of the argument we will focus on $V_1^{\\otimes (k-t)}\\otimes V_2^{\\otimes t}$ (i.e.",
"we assume the last $t$ indices in eq:vis are 2).",
"Equivalently, we have to show that $f$ is zero on $V_1^{\\otimes (k-t)}\\otimes \\mathbb {C}e_j\\otimes V_2^{\\otimes (t-1)}$ for any $j\\in [M+1,M+T]$ .",
"To do this, first note that because $V_2$ is self-dual over $S^+_M* S^+_T$ , hence also over its quantum subgroup $\\mathbb {G}$ , the $\\mathbb {G}$ -morphisms $V_1^{\\otimes (k-t)}\\otimes V_2^{\\otimes t}\\rightarrow \\mathbb {C}$ are in bijection with the $\\mathbb {G}$ -morphisms $V_1^{\\otimes (k-t)}\\rightarrow V_2^{\\otimes t}.$ Since moreover $V_1$ is the induction of the corresponding $S^+_M$ -representation, the adjunction eq:frob reads $\\mathrm {hom}_{\\mathbb {G}}\\left(V_1^{\\otimes (k-t)},V_2^{\\otimes t}\\right)\\cong \\mathrm {hom}_{S^+_M}\\left(V_1^{\\otimes (k-t)},\\mathrm {ind} V_2^{\\otimes t}\\right)$ Now, the finite-dimensional $S^+_M$ -representation $\\mathrm {ind} V_2^{\\otimes t}$ embeds into some tensor power $V_1^{\\otimes s}$ , and hence elements in the right hand side of eq:frob2 are spanned by non-crossing partitions of $[s+k-t]$ .",
"Since non-crossing partitions are linearly independent on spaces of dimension $\\ge 4$ , morphisms in eq:frob2 vanish if they do so when restricted to $(V_1^{\\prime })^{\\otimes (k-t)},\\ V_1^{\\prime } = \\mathrm {span}\\lbrace e_1,\\cdots ,e_{M-1}\\rbrace .$ In our setting, what this means is that it is enough to prove that $f$ vanishes on $(V_1^{\\prime })^{\\otimes (k-t)}\\otimes \\mathbb {C}e_j\\otimes V_2^{\\otimes (t-1)}.$ Set $V_2^{\\prime } = \\mathrm {span}\\lbrace e_\\ell \\ |\\ \\ell \\in [M+1,M+T]-\\lbrace j\\rbrace \\rbrace .$ Then, $(V_1^{\\prime })^{\\otimes (k-t)}\\otimes \\mathbb {C}e_j\\otimes (V_2^{\\prime })^{\\otimes (t-1)}$ is contained in the image of $V_1^{\\otimes (k-t+1)}\\otimes V_2^{\\otimes (t-1)}$ through a the permutation of the $e_i$ interchanging $e_M$ and $e_j$ .",
"Since $f$ is an $S_N$ -morphism, Claim($t-1$ ) now implies that $f$ vanishes on $(V_1^{\\prime })^{\\otimes (k-t)}\\otimes \\mathbb {C}e_j\\otimes (V_2^{\\prime })^{\\otimes (t-1)}.$ Similarly, $f$ vanishes on the other summands of eq:otimes-target resulting from the decomposition $V_2=V^{\\prime }_2\\oplus \\mathbb {C}e_j$ by the other instances Claim($s$ ), $s\\le t-2$ and this concludes the proof."
],
[
"Quantum reflection groups",
"We now turn to the quantum reflection groups $H_N^{s+}$ for $1\\le s\\le \\infty $ .",
"The main result of the subsection is Theorem 3.11 For $N\\ge 4$ and $1\\le s\\le \\infty $ the dual $\\widehat{H_N^{s+}}$ is residually finite and hence also hyperlinear.",
"Proof 9 We fix $N$ throughout, and denote by $\\mathcal {A}_s$ and $\\mathcal {A}$ the Hopf $*$ -algebras associated to $H_N^{s+}$ and $S^+_N$ respectively.",
"(Case 1: $s<\\infty $ ) Recall from [13] that we have a free wreath product decomposition $\\mathcal {A}_s\\cong C(\\mathbb {Z}_s)*_w \\mathcal {A},$ where the right hand side is by definition the free $*$ -algebra generated by $\\mathcal {A}$ and $n$ copies of $C(\\mathbb {Z}_s)$ with the constraint that the $i^{th}$ copy of $C(\\mathbb {Z}_s)$ commutes with the $i^{th}$ row of generators $u_{ij}$ , $1\\le j\\le N$ of $\\mathcal {A}$ .",
"It follows that $\\mathcal {A}_s$ can be realized as a succession of extensions of the form $\\mathcal {B}\\mapsto C(\\mathbb {Z}_s)*\\mathcal {B}/[C(\\mathbb {Z}_s),\\mathcal {D}]$ for a commutative finite-dimensional $*$ -subalgebra $\\mathcal {D}\\subset \\mathcal {B}$ .",
"More concretely, the various algebras $\\mathcal {D}$ are the $N$ -dimensional subalgebras of $\\mathcal {A}$ generated by a row $u_{ij}$ , $1\\le j\\le N$ of generators.",
"The residual finite-dimensionality of $\\mathcal {A}_s$ then follows inductively from pr.push-rf.",
"(Case 2: $s=\\infty $ ) Given that $H^{\\infty +}_N$ is topologically generated by all of the finite $H^{s+}_N$ embedded therein (by le.hn-top-gen), the conclusion follows from le.gen-rf.",
"While the proof given above for th.hns does not proceed inductively on $N$ or require topological generation, there is nevertheless an analogue of th.top-gen for quantum reflection groups that we record here.",
"Just as we did for quantum permutation groups, we regard $H^{s+}_{N-1}$ as a quantum subgroup of $H^{s+}_{N}$ via the map $\\mathcal {A}(H^{s+}_N)\\rightarrow \\mathcal {A}(H^{s+}_{N-1})$ that sends the generators $u_{ij}$ to $\\delta _{ij}$ if $N\\in \\lbrace i,j\\rbrace $ .",
"Theorem 3.12 For $N\\ge 6$ and $1\\le s\\le \\infty $ the quantum reflection group $H_N^{s+}$ is topologically generated by its quantum subgroups $H^{s+}_{N-1}$ and $S_N$ .",
"Proof 10 We once more separate the finite and infinite-$s$ cases.",
"(1: finite $s$ ) This follows from a slight adaptation of pr.coinv, modified as follows based on the representation theory of $H_N^{s+}$ as developed in [13].",
"Instead of a single $n$ -dimensional fundamental representation $V$ we have $s$ of them, labeled $V^{(i)}$ for $i\\in \\mathbb {Z}_s$ .",
"As for morphisms, $f:V^{(i_1)}\\otimes \\cdots \\otimes V^{(i_k)}\\rightarrow \\mathbb {C}$ is $H^{s+}_{N}$ -invariant precisely when it is a linear combination of non-crossing partitions whose blocks are of the form $\\lbrace i_{a_1},\\ \\cdots ,\\ i_{a_t}\\rbrace ,\\ \\sum _j i_{a_j} = 0\\in \\mathbb {Z}_s.$ The argument in the proof of pr.coinv then goes through virtually unchanged.",
"(2: $s=\\infty $ ) This follows as in the proof of th.hns, from the claim for $s<\\infty $ and the fact that $H^{s+}_N$ topologically generate $H^{\\infty +}_{N}$ (by le.hn-top-gen).",
"Remark 3.13 Note the bound on $N$ : unlike th.top-gen, th.refl-tg does not apply to $N=5$ hence does not yield an inductive proof of residual finite-dimensionality; we do not know whether the result is still valid in that case, but believe it to be."
],
[
"Remarks on free entropy dimension",
"In this section we make some brief remarks on what is currently known about the free entropy dimension of the canonical generators of the finite von Neumann algebras $L^\\infty (H_N^{s+})$ .",
"We refer the reader to the survey [44] and the references therein for details on the various versions of free entropy dimension that exist.",
"Fix $N \\in \\mathbb {N}$ and $s \\in \\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ and consider the self-adjoint family $X(N,s) = \\lbrace u_{ij}, u_{ij}^*\\rbrace _{1 \\le i,j \\le N}$ of generators of $\\mathcal {A}(H_N^{s+}) \\subset L^\\infty (H_N^{s+})$ .",
"Associated to the sets $X(N,s)$ we have the (modified) microstates free entropy dimension $\\delta _0(X(N,s)) \\in [0, n]$ and the non-microstates free entropy dimension $\\delta ^*(X(N,s))\\in [0, n]$ .",
"From [18] it is known that the general inquality $\\delta _0(\\cdot ) \\le \\delta ^*(\\cdot )$ always holds, and from [25] an upper bound for $\\delta ^*$ exists in terms of the $L^2$ -Betti numbers of the discrete dual quantum groups $\\widehat{H_N^{s+}}$ : $\\delta ^*(X(N,s)) \\le \\beta _1^{(2)}(\\widehat{H_N^{s+}}) - \\beta _0^{(2)}(\\widehat{H_N^{s+}}) +1.$ Here $\\beta _k^{(2)}(\\cdot )$ is the $k$ th $L^2$ -Betti number of a discerete quantum group.",
"See for example [41], [31], [19].",
"Now, in [31], we have the following computations $\\beta _1^{(2)}(\\widehat{H_N^{s+}}) = 1-\\frac{1}{s} \\quad \\&\\quad \\beta _0^{(2)}(\\widehat{H_N^{s+}}) = 0 \\qquad (N \\ge 4).$ Finally, since $L^\\infty (H_N^{s+})$ is Connes embeddable by Theorem REF , it follows from [29] that $\\delta _0(X(N,s)) \\ge 1$ whenever $L^\\infty (H_N^{s+})$ is diffuse.",
"The question of when exactly $L^\\infty (H_N^{s+})$ is diffuse still seems to be open in complete generality.",
"However, it is known that $L^\\infty (H_N^{s+})$ is a II$_1$ -factor (and in particular diffuse) $N \\ge 8$ [20], [32], [45].",
"Combing all the above inequalities together, we finally obtain Corollary 3.14 For $N \\ge 8$ and $s \\in \\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ , we have $1 \\le \\delta _0(X(N,s)) \\le \\delta ^*(X(N,s)) \\le 2-\\frac{1}{s}.$ In particular, the generators $X(N) = \\lbrace u_{ij}\\rbrace _{1 \\le i,j \\le N}$ of $L^\\infty (S_N^+)$ satisfy $\\delta _0(X(N)) = \\delta ^*(X(N)) = 1$ for $N \\ge 8$ .",
"Remark 3.15 The situation for $S_N^+$ in Corollary REF is similar to what happens for the free orthogonal quantum groups $O_N^+$ [22].",
"However, for $O_N^+$ , even more is known: In [23] it was shown that in fact $L^\\infty (O_N^+)$ is a strongly 1-bounded von Neumann algebra for all $N \\ge 3$ .",
"The notion of strong 1-boundedness was introduced by Jung in [30] and entails that $\\delta _0(X) \\le 1$ for any self-adjoint generating set $X \\subset L^\\infty (O_N^+)$ .",
"In particular, it follows that $L^\\infty (O_N^+)$ is never isomorphic to an interpolated free group factor.",
"In this context, it is natural to ask whether similar results hold for quantum permuation groups: Is $L^\\infty (S_N^+)$ a strongly 1-bounded von Neumann algebra for all $N \\ge 5$ ?"
],
[
"Inner faithful matrix models for quantum permutation groups",
"th.rf shows that the quantum group algebras $\\mathcal {A}(S_N^+) = \\mathbb {C}\\widehat{S^+_N}$ have enough finite-dimensional $*$ -representations, i.e.",
"map faithfully into a product of matrix algebras.",
"In the present section we prove that a specific, canonical collection of “elementary” representations is faithful in a certain sense.",
"First, let us clarify the appropriate notion of faithfulness here.",
"See e.g.",
"[9] for more details.",
"Definition 4.1 A $\\ast $ -homomorphism $\\pi :\\mathcal {A}\\rightarrow B$ from a Hopf $\\ast $ -algebra $\\mathcal {A}$ into a $\\ast $ -algebra $B$ is inner faithful if $\\ker \\pi $ contains no non-trivial Hopf $\\ast $ -ideals.",
"Equivalently, for any factorization $\\pi =\\tilde{\\pi }\\circ \\rho $ with $\\rho : \\mathcal {A}\\rightarrow \\mathcal {A}^{\\prime }$ , a surjective morphism of Hopf $\\ast $ -algebras, we have in fact that $\\rho $ is an isomorphism.",
"More generally, the Hopf image of $\\pi :\\mathcal {A}\\rightarrow B$ is the “smallest” quotient Hopf $\\ast $ -algeba $\\mathcal {A}^{\\prime }$ such that $\\pi $ factors through the quotient map $\\rho : \\mathcal {A}\\rightarrow \\mathcal {A}^{\\prime }$ .",
"Note that the Hopf image always exists and is unique (up to isomorphism) [9].",
"In this paper, we will only be concerned with the cases where our Hopf $\\ast $ -algebras are of the form $\\mathcal {A}(\\mathbb {G})$ for a compact quantum group $\\mathbb {G}$ .",
"In this case, the Hopf image of $\\pi :\\mathcal {A}(\\mathbb {G}) \\rightarrow B$ is $\\mathcal {A}(\\mathbb {H})$ , where $\\mathbb {H}< \\mathbb {G}$ is the “smallest” quantum subgroup such that $\\pi $ factors through $\\rho : \\mathcal {A}(\\mathbb {G}) \\rightarrow \\mathcal {A}(\\mathbb {H})$ .",
"Moreover, $\\pi $ is inner faithful if and only if $\\mathbb {H}= \\mathbb {G}$ (up to isomorphism)."
],
[
"Flat matrix models",
"Following [12], we denote by $X_N$ the space of $N\\times N$ matrices $P = (P_{ij})_{ij} \\in M_N(M_N(\\mathbb {C}))$ , whose entries $P_{ij} \\in M_N(\\mathbb {C})$ , are rank-one projections with the property that for all $1\\le i,j\\le N$ , $\\sum _j P_{ij} = 1 = \\sum _i P_{ij}$ In other words, $X_N \\subset M_N(M_N(\\mathbb {C}))$ is the compact set of bistochastic $N\\times N$ matrices of rank one projections in $M_N$ .",
"It is clear that each $P \\in X_N$ gives rise to a $\\ast $ -homomorphism $\\pi _P:\\mathcal {A}(S_N^+) \\rightarrow M_N(\\mathbb {C}); \\qquad \\pi _P(u_{ij}) = P_{ij} \\qquad (1 \\le i,j \\le N),$ and we call $\\pi _P$ a flat matrix model for the quantum group $S_N^+$ .",
"If we package all these flat matrix models $\\pi _P$ into one single representation by allowing $P \\in X_N$ to vary, we arrive at a construction that features prominently in [12], [7].",
"Definition 4.2 The universal flat matrix model of $S^+_N$ is the morphism $\\pi : \\mathcal {A}(S^+_N)\\rightarrow M_N(C(X_N))\\cong C(X_N,M_N(\\mathbb {C})); \\qquad \\pi (u_{ij}) = \\lbrace P \\mapsto \\pi _P(u_{ij}) = P_{ij}\\rbrace .$ One of the main conjectures in [12] is the inner faithfulness of the universal flat matrix models.",
"Conjecture 4.3 [12] The universal flat matrix model $\\mathcal {A}(S^+_N)\\rightarrow M_N(C(X_N))$ is faithful for $N=4$ and inner faithful for $N\\ge 5$ .",
"Remark 4.4 Note that for $N=4$ the conjecture is true by [11] (or [12]).",
"Indeed, the latter result produces some faithful representation of the form $\\mathcal {A}(S^+_4)\\rightarrow M_4(C(X))$ for compact $X$ (in fact $X=SU_2$ ), which must factor as $\\mathcal {A}(S^+_4)\\rightarrow M_4(C(X_4))\\rightarrow M_4(C(X))$ by universality.",
"The main result of this section is that this conjecture is true for at least almost all $N$ .",
"To begin, we first need a few remarks and observations.",
"Consider the closed subspace $X_{N}^{class}\\subset X_{N}$ of matrices $P \\in X_N$ for which the entries $P_{ij}$ pairwise commute.",
"Then, the classical permutation group $S_{N}$ also has a universal flat matrix model $\\pi ^{class} : \\mathcal {A}(S_{N})\\rightarrow M_{N}(C(X_{N}^{class}))$ .",
"Moreover, if $q : \\mathcal {A}(S_{N}^{+})\\rightarrow \\mathcal {A}(S_{N})$ is the canonical quotient map and $r : C(X_{N})\\rightarrow C(X_{N}^{class})$ is the restriction map, then by construction $\\pi ^{class}\\circ q= r\\circ \\pi $ .",
"Lemma 4.5 If $N\\ge 4$ then the inclusion $X_N^{class}\\subset X_N$ is proper.",
"Proof 11 To produce examples of $N\\times N$ bistochastic matrices whose entries do not all commute we proceed as follows.",
"First, fix a basis $e_i$ , $1\\le i\\le N$ for $\\mathbb {C}^N$ and let $L \\in M_N(\\mathbb {N})$ be a Latin square of size $N\\times N$ (meaning that each row and column is a permutation of $\\lbrace 1,\\cdots ,N\\rbrace $ ).",
"Assume furthermore that the upper left hand $2\\times 2$ corner of $L$ is $\\begin{pmatrix}1&2\\\\2&1\\end{pmatrix}$ We can then form the bistochastic and commutative matrix whose $(i,j)$ entry is the projection on the one-dimensional span of $e_{L_{ij}}$ , and then modify it slightly by changing its upper left hand $2\\times 2$ corner to $\\begin{pmatrix}P_u & P_v\\\\P_v & P_u\\end{pmatrix}$ where $u=e_1+e_2$ and $v=e_1-e_2$ .",
"The resulting bistochastic matrix contains, say, the projections $P_u$ and $P_{e_1}$ , which do not commute.",
"It remains to argue that a Latin square $L$ as above exists if $N\\ge 4$ (we have not used this hypothesis thus far).",
"To see this observe that for $N\\ge 4$ we can complete the square eq:22 to a $2\\times n$ Latin rectangle, in the sense that the two rows are permutations of $\\lbrace 1,\\cdots ,N\\rbrace $ and no two entries in the same column coincide.",
"We can then use the result that any Latin rectangle can be completed to a Latin square (e.g.",
"[1]).",
"This already yields one instance of cj.if.",
"Proposition 4.6 $S^+_5$ satisfies cj.if.",
"Proof 12 Let $\\pi , \\pi ^{class}, r, q$ be as above and assume $N=5$ .",
"Let $I$ be a Hopf $*$ -ideal contained in $\\ker (\\pi )$ and observe that $\\pi ^{class}(q(I)) = r\\circ \\pi (I) = 0$ .",
"Since $\\pi ^{class}$ is inner faithful (see e.g., [7]), this forces $q(I) = (0)$ , i.e.",
"$I\\subset \\ker (q)$ .",
"But because there is no intermediate quantum group between $S_5$ and $S_5^+$ by [5], it follows that either $I = (0)$ or $I = \\ker (q)$ .",
"In the second case we get $\\pi = \\pi ^{class}\\circ q = r\\circ \\pi .$ In particular, this implies that any family $(P_{ij})_{1\\leqslant i, j\\leqslant N}$ of rank-one projections which are pairwise orthogonal on rows and columns commutes, i.e.",
"$X_{N} = X_{N}^{class}$ .",
"This equality, however, is invalid for $N\\geqslant 4$ by le.not-cls.",
"Remark 4.7 In fact, the proof of le.N=5 shows that cj.if is satisfied whenever the inclusion $S_N < S^+_N$ is maximal.",
"We are now ready for the main technical result of this section.",
"Proposition 4.8 Let $M \\ge 5$ and $N\\ge 2M$ .",
"If $S^+_M$ satisfies cj.if then so does $S^+_N$ .",
"Proof 13 Let $\\mathcal {A}=\\mathcal {A}(S^+_N)$ .",
"We have to argue that under the hypothesis, the Hopf image $\\mathcal {A}\\rightarrow \\mathcal {A}_{\\pi }$ of eq:pi is all of $\\mathcal {A}$ .",
"Since the quantum group attached to $\\mathcal {A}_{\\pi }$ clearly contains $S_N$ , pr.gen-diag reduces the problem to showing that it also contains a quantum subgroup $\\mathbb {G}<S^+_N$ that is $(M,N)$ -large in the sense of def.large.",
"For this, fix a collection of rank-one projections $P_i$ , $0\\le i\\le N-1$ in $M_N$ summing up to 1.",
"We form an $N\\times N$ Latin square $(\\mathcal {L}_{ij})_{0\\le i,j\\le N-1}$ all of whose entries are the projections $P_i$ as follows: if $i,j\\le M-1$ we set $\\mathcal {L}_{ij}=P_{(i-j)\\;\\mathrm {mod}\\;M}$ ; we fill the rest of the first $M$ rows with $P_i$ 's arbitrarily so as to retain the Latin rectangle property (this is possible because $2M\\le N$ ); complete the above Latin rectangle to a Latin square, once more using [1].",
"Having constructed $\\mathcal {L}$ as above, consider the subspace $Y\\subset X_N$ consisting of those $N\\times N$ bistochastic matrices $\\mathcal {M}$ of rank-one projections that are identical to $\\mathcal {L}$ outside of the upper left hand $M\\times M$ corner.",
"Setting $P=\\sum _{i=0}^{M-1} P_i,$ all operators appearing as entries of matrices $\\mathcal {M}\\in Y$ commute with $P$ .",
"Restricting these operators to the range of $P$ (which is in turn isomorphic to $\\mathbb {C}^M$ ), we obtain the upper right hand arrow in the composition $\\begin{tikzpicture}[baseline=(current bounding box.center),anchor=base,cross line/.style={preaction={draw=white,-,line width=6pt}}](0,0) node (1) {\\mathcal {A}(S^+_N)} +(3,.5) node (2) {M_N(C(X_N))} +(6,.5) node (3) {M_N(C(Y))} +(9,0) node (4) {M_M(C(Y))} +(4.5,-.5) node (d) {\\mathcal {A}(S^+_M)*\\mathcal {A}(S^+_{N-M})};[->] (1) to[bend left=6] node[pos=.5,auto]{\\scriptstyle \\pi } (2);[->] (2) to[bend left=6] (3);[->] (3) to[bend left=6] (4);[->] (1) to[bend right=6] (d);[->] (d) to[bend right=6] node[pos=.5,auto,swap] {\\scriptstyle \\eta } (4);\\end{tikzpicture}$ where the lower factorization occurs because by construction the off-block-diagonal entries $\\mathcal {L}_{ij}$ with precisely one of $i,j$ in $\\lbrace 0,\\cdots ,M-1\\rbrace $ are projections orthogonal to $P$ and hence vanish on $\\mathrm {Im}\\;P$ .",
"Our goal is now to show that the Hopf image of $\\eta $ in the above diagram contains an $(M,N)$ -large quantum subgroup $\\mathbb {G}< S^+_M * S^+_T < S^+_N,\\ T:=N-M.$ Equivalently, this means proving that the composition $\\begin{tikzpicture}[baseline=(current bounding box.center),anchor=base,cross line/.style={preaction={draw=white,-,line width=6pt}}](0,0) node (1) {\\mathcal {A}(S^+_M)} +(3,0) node (2) {\\mathcal {A}(S^+_M)*\\mathcal {A}(S^+_T)} +(6,0) node (3) {M_M(C(Y))};[->] (1) -- (2);[->] (2) --(3)node[pos=.5,auto]{\\scriptstyle \\eta };\\end{tikzpicture}$ is inner faithful.",
"To verify this, recall that by construction the upper left hand $M\\times M$ corners of matrices in $Y$ are arbitrary bistochastic matrices in $M_M\\cong \\mathrm {End}(\\mathrm {Im}\\;P).$ Since the other entries of matrices in $Y$ are identical to those of the fixed Latin square $\\mathcal {L}$ , we have an isomorphism $M_M(C(Y))\\cong M_M(C(X_M))$ .",
"The isomorphism described here renders eq:eta-comp identical to the canonical universal flat representation of $\\mathcal {A}(S^+_M)$ , which is inner faithful by hypothesis.",
"As a consequence, we can prove cj.if for almost all $N$ .",
"Corollary 4.9 All $S^+_N$ with $N \\le 5$ and $N\\ge 10$ satisfy cj.if.",
"Proof 14 As explained above, we already know the conjecture to hold in the cases $N \\le 4$ .",
"For the remaining cases , it is enough to prove that $S_{5}^{+}$ satisfies cj.if in view of pr.cj-ind, and this is taken care of by le.N=5."
],
[
"Inner unitary Hopf $\\ast $ -algebras",
"cor.div5 shows that the universal flat matrix model is inner faithful for most quantum permutation groups.",
"In this final section we show that we can do even better: it turns out that for the same values of the parameter $N$ a single finite-dimensional representation suffices to achieve inner faithfulness.",
"We first recall the relevant concept from [2].",
"Definition 4.10 A Hopf $*$ -algebra $\\mathcal {A}$ is inner unitary if it has an inner faithful $*$ -homomorphism into a finite-dimensional C$^*$ -algebra.",
"The main result of this subsection is the following improvement on cor.div5.",
"Theorem 4.11 The Hopf $\\ast $ -algebra $\\mathcal {A}=\\mathcal {A}(S^+_N)$ is inner unitary for all $N$ outside the range $[6,9]$ .",
"Proof 15 We first tackle the smaller-$N$ cases.",
"(Case 1: $N\\le 3$ ) $S^+_N$ is classical, and hence the conclusion follows from [9].",
"(Case 2: $N=4,5$ ) Let $x\\in X_N$ be any of the bistochastic matrices whose entries generate a non-commutative subalgebra of $M_N(\\mathbb {C})$ and let $y\\in X_{N}^{class}\\subset X_N$ be such that the corresponding flat representation $\\pi ^{class}_y$ is inner faithful on $\\mathcal {A}(S_N)$ (such $x$ and $y$ exist by le.not-cls) and [7], respectively).",
"The Hopf image of the representation $\\pi _x\\oplus \\pi _y:\\mathcal {A}(S^+_N)\\rightarrow M_N(\\mathbb {C})\\oplus M_N(\\mathbb {C})$ is then a non-commutative quotient Hopf $*$ -algebra of $\\mathcal {A}$ containing $S_N$ .",
"Since for $N=4,5$ we know that there are no intermediate quantum groups $S_N<\\mathbb {G}<S^+_N,$ the Hopf image of $\\pi _x\\oplus \\pi _y$ is all of $\\mathcal {A}$ , as desired.",
"(Case 3: $N\\ge 10$ ) The proof of pr.cj-ind in fact shows that if $5\\le M\\le \\frac{N}{2}$ and $\\mathcal {A}(S^+_M)$ admits a finite inner faithful family $\\lbrace \\pi _{z_1}, \\ldots , \\pi _{z_n}\\rbrace $ ($z_i \\in X_M$ ) of flat $M$ -dimensional representations, then $\\mathcal {A}$ admits a finite family $\\lbrace \\pi _{z_1^{\\prime }}, \\ldots , \\pi _{z_n^{\\prime }}\\rbrace $ ($z_i^{\\prime } \\in X_N$ ) of flat representations whose joint Hopf image surjects onto $\\mathcal {A}(S^+_M)*\\mathcal {A}(S^+_{N-T})$ .",
"Since for $M=5$ we do have such a family $\\lbrace \\pi _x,\\pi _y\\rbrace $ by the previous step of the current proof, we have such an $x^{\\prime }, y^{\\prime }\\in X_N$ .",
"Further choosing any inner faithful flat representation $\\pi ^{class}_w:\\mathcal {A}(S_N)\\rightarrow M_N(\\mathbb {C})$ ($w \\in X_N^{class}$ ), the resulting direct sum representation $\\pi _{x^{\\prime }} \\oplus \\pi _{y^{\\prime }} \\oplus \\pi _{w}: \\mathcal {A}(S^+_N)\\rightarrow M_N(\\mathbb {C})^{\\oplus 3}$ is inner faithful.",
"Indeed, the quantum subgroup of $S^+_N$ dual to its Hopf image contains both $S_N$ and $S^+_M * S^+_{N-M}$ and hence coincides with $S^+_N$ by cor.top-gen-bis.",
"tocsectionReferences Michael Brannan, Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA E-mail address: mbrannan@math.tamu.edu Alexandru Chirvasitu, Department of Mathematics, University at Buffalo, Buffalo, NY 14260-2900, USA E-mail address: achirvas@buffalo.edu Amaury Freslon, Laboratoire de Mathématiques d’Orsay, Univ.",
"Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France E-mail address: amaury.freslon@math.u-psud.fr"
]
] | 1808.08611 |
[
[
"MADARi: A Web Interface for Joint Arabic Morphological Annotation and\n Spelling Correction"
],
[
"Abstract In this paper, we introduce MADARi, a joint morphological annotation and spelling correction system for texts in Standard and Dialectal Arabic.",
"The MADARi framework provides intuitive interfaces for annotating text and managing the annotation process of a large number of sizable documents.",
"Morphological annotation includes indicating, for a word, in context, its baseword, clitics, part-of-speech, lemma, gloss, and dialect identification.",
"MADARi has a suite of utilities to help with annotator productivity.",
"For example, annotators are provided with pre-computed analyses to assist them in their task and reduce the amount of work needed to complete it.",
"MADARi also allows annotators to query a morphological analyzer for a list of possible analyses in multiple dialects or look up previously submitted analyses.",
"The MADARi management interface enables a lead annotator to easily manage and organize the whole annotation process remotely and concurrently.",
"We describe the motivation, design and implementation of this interface; and we present details from a user study working with this system."
],
[
"Introduction",
"Annotated corpora have been vital for research in the area of natural language processing (NLP).These resources provide the necessary training and evaluation data to build automatic annotation systems and benchmark them.",
"The task of human manual annotation, however, is rather difficult and tedious; and as such a number of annotation interface tools have been created to assist in such effort.",
"These tools tend to be specialized to optimize for specific tasks such as spelling correction, part-of-speech (POS) tagging, named-entity tagging, syntactic annotation, etc.",
"Certain languages bring additional challenges to the annotation task.",
"Compared with English, Arabic annotation introduces a need for diacritization of the diacritic-optional orthography, frequent clitic segmentation, and a richer POS tagset.",
"Although the goal of language-independence is something most researchers and interface developers keep in mind, it is rather hard to achieve without a tradeoff with utility and efficiency.",
"In this paper, we focus on a tool targeting Arabic dialect morphological annotation.",
"Arabic dialects introduce yet more complexity than standard Arabic in that the input text has noisy orthography.",
"For example, the last word in the sentence used as example in Figure REF .",
"(a), <wyaabuwhA-Alhliyj> wyAbwhAAlxlyjAll transliteration is in the Buckwalter scheme [9].",
"involves two spelling errors (a word merge and character replacement) which can be corrected as <wjaabuwhA Alhliyj> wjAbwhA Alxlyj `and they brought it to the Gulf'.",
"Furthermore, the first of the two corrected words includes two clitics that when segmented produce the form: <hA>+ <jaabuwA> +<w> w+ jAbwA +hA `and+ they-brought +it'.",
"Previous work on Arabic morphology annotation interfaces focused either on the problem of manual annotations for POS tagging, or diacritization, or spelling conventionalization.",
"In this paper we present a tool that allows one to do all of these tasks together, eliminating the possibility of error propagation from one annotation level to another.",
"Our tool is named MADARi<madaary> madAriy means `my orbit' in Arabic.",
"after the project under which it was created: Multi-Arabic Dialect Annotations and Resources (MADAR).",
"Next, we present related work to this effort.",
"In Section 3, we discuss the MADARi task description and design concerns.",
"In section 4 and 5, we discuss the annotation and management interfaces, respectively.",
"Section 6 presents some details on a user study of working with MADARi."
],
[
"Related Work",
"Several annotation tools and interfaces were proposed for many languages and to achieve various annotation tasks such as the general purpose annotation tools BRAT [15], WebAnno [17].",
"For task specific annotation tools, we can cite the post-editing and error correction tools such as the work of aziz+2012:pet, Stymne:2011:BTE:2002440.2002450, conflrec, and dickinson.",
"For Arabic, there are several existing annotation tools, however, they are designed to handle a specific NLP task and it is not easy to adapt them to our project.",
"We can cite tools for semantic annotation such as the work of saleh2009aratation and el2014proposed and the work on dialect annotation by benajiba2010web and Diab10colaba:arabic.",
"AttiaRA09 built a morphological annotation tool and more recently MADAD [2], a general-purpose online collaborative annotation tool for Arabic text was designed during a readability assessments project.",
"In the COLABA initiative [6], the authors built tools and resources to process Arabic social media data such as blogs, discussion forums, and chats.",
"Above all, most, if not all of these tools are not designed to handle the peculiarities of the dialectal Arabic, which is a very specific task.",
"Moreover, the existing tools, do not provide facilities for managing thousands of documents and they often do not permit the distribution of tasks to tens of annotators while evaluating the inter-annotator agreement (IAA).",
"Our interface borrows ideas from three other annotation tools: DIWAN, QAWI, and MANDIAC.",
"Here we describe each of these tools and how they have influenced the design of our system."
],
[
"DIWAN",
"DIWAN is an annotation tool for Arabic dialectal texts [1].",
"It provides annotators with a set of tools for reducing duplicate effort including the use of morphological analyzers to precompute analyses, and the ability to apply analyses to multiple occurrences simultaneously.",
"However it requires installation on a Windows machine and the user interface is not very friendly to newcomers."
],
[
"QAWI",
"The QALB Annotation Web Interface (QAWI) first introduced the concept of token-based text edits for annotating parallel corpora used in text correction tasks [11], [18].",
"It allowed for the exact recording of all modifications performed by the annotator which previous tools did not.",
"As we show later on, we utilize this token-based editing system for minor text corrections that transform text of a given dialect into the appropriate CODA format."
],
[
"MANDIAC",
"MANDIAC [12] utilized the token-based editor used in QAWI to perform text diacritization tasks.",
"More importantly, it introduced a flexible hybrid data storage system that allows for adding new features to the annotation front-end with little to no modifications to the back-end.",
"Our annotation system utilizes this design to provide the same utility."
],
[
"Task Description",
"The MADARi interface will be used by human annotators to create a morphologically annotated corpus of Arabic text.",
"The text we work with comes from social media and is highly dialectal and as such, it has a lot of spelling errors.",
"The annotators will carefully correct the spelling of the words in the text and also annotate the words' morphology.",
"The in-context morphology annotation includes tokenization, POS tagging, lemmatization and English glossing."
],
[
"Desiderata",
"In order to manage and process the annotation of the large scale dialectal Arabic corpus, we needed to create a tool to streamline the annotation process.",
"The desiderata for developing the MADARi annotation tool include the following: No installation time and very minimal requirements on the annotators.",
"The tool must allow off-site data management of documents to allow annotation leaders to assign and grade documents from anywhere in the world and to allow hiring annotators anywhere in the world.",
"The tool must allow easily customizable POS tag sets by annotation leads.",
"The tool must allow easy access to other user annotations of similar texts.",
"The tool must allow for easy navigation between spelling changes and morphological disambiguation."
],
[
"Design and Architecture",
"The design of our interface borrows heavily from the design of MANDIAC [12].",
"In particular, we utilized the client-server architecture, as well as the flexible hybrid SQL/JSON storage system used by MANDIAC.",
"This allows us to easily extend our annotation interface with minor changes, if any, to the back-end.",
"Like, DIWAN and MANDIAC, we also utilize MADAMIRA , a state-of-the-art morphological analyzer for Arabic to precompute analyses.",
"Figure: The MADARi Annotation Interface"
],
[
"Annotation Interface",
"The Annotation Interface (Figure REF ) is where annotators perform the annotation tasks assigned to them.",
"Here we describe the different components and utilities this interface provides."
],
[
"Text Editing",
"Annotators are able to edit a sentence anytime during the annotation process.",
"This is primarily used to make sure all text is in the CODA format of the dialect of choice.",
"We adopted the same token-based editing system used by QAWI.",
"Our token-based editor (Figure REF ) only allows for modifying, splitting, and merging tokens where QAWI also allows for adding and deleting tokens as well as moving tokens around.",
"The operations we allow are sufficient for CODA formatting without allowing the text to be changed substantially."
],
[
"POS Tagging",
"The essential component of our interface is the POS tagging system.",
"Here, all words are annotated in their tokenized form which divides a word into its base word, enclitics, and proclitics.",
"Each of these are assigned a POS tag as well as a morphological feature where applicable.",
"Annotators also assign the gloss and lemma for each word.",
"For the convenience of annotators, we provide precomputed values for each field using MADAMIRA's morphological analyzers."
],
[
"Utilities",
"We have added utility features to make the annotation process easier and more efficient for annotators.",
"Basic utilities include undo and redo buttons, access to the original text for reference, and color-coding edited tokens for quick navigation as seen in Figure REF .",
"We also allow annotators to update multiple tokens with the same orthography instantaneously.",
"Additionally, we provide annotators with a search utility to look up previously submitted annotations of the same word as well as query MADAMIRA for out-of-context analyses in different dialects in real-time (Figure REF )."
],
[
"Management Interface",
"The Annotation Management Interface enables the lead annotator to easily manage and organize the whole annotation process remotely and concurrently.",
"The management interface contains: (a) a user management tool for creating new annotator accounts and viewing annotator progress; (b) a document management tool for uploading new documents, assigning them for annotation, and viewing submitted annotations; and (c) a monitoring tool for viewing overall annotation progress; (d) an inter-annotator agreement (IAA) evaluation tool to compare the annotations produced by each annotator to a gold reference in order to monitor the quality of the annotations; and (e) a data repository and annotation export feature."
],
[
"User Study",
"Our tool is being used as part of an ongoing annotation project on Gulf Arabic (forthcoming).",
"In this paper, we describe the experience of one annotator who has done annotations in different settings previously.",
"The annotator morphologically disambiguated 80 sentences totaling in 1,355 raw tokens of Gulf Arabic text.",
"We noted that the annotator preferred, based on her experience, to convert the orthography of the text to CODA first, which made the disambiguation task more efficient.",
"It took about 52 minutes to complete this task (corresponding to a rate of 1,563 words/hour).",
"The annotator made a few minor fixes later on, which is an advantage of our tool to minimize error propagation.",
"The total number of words that were changed from the raw tokens to CODA was 288 (21%).",
"Changes were mostly spelling adjustments and the rest is word splitting (44 cases or 15% of all changes) and no merges.",
"The final word count is 1,398 words.",
"Following the CODA conversion, the annotator worked on tokenization, POS tagging, lemmatization and English glossing.",
"This more complex task took around 6 hours (at a rate of 277 words/hour).",
"This makes the cumulative time spent to finish the spelling adjustment and the full disambiguation tasks for this set of data about 7 hours (at a rate of 200 words/hour).",
"Since the tool provides initial guesses for all the annotation components, the annotator was able to use many of the valid decisions as is, and modify them in other cases.",
"In the event of a word split, the tool currently removes the raw word predictions, but the analysis search utility allows fast access to alternatives to select from.",
"We compared the final tokenization, POS tag and lemma choices to the ones suggested by the tool on the CODA version of the text.",
"We found that the tool gave correct suggestions 74% of the time on tokenization, 69% of the time on baseword POS tags and 70% of the time on lemmas.",
"The annotator indicated that their favorite utilities were the ability to annotate multiple tokens of the same type in different contexts simultaneously, and the ability to use the `Analysis Search' box to annotate multiple fields simultaneously."
],
[
"Conclusion and Outlook",
"We presented an overview of our web-based annotation framework for joint morphological annotation and spelling correction of Arabic.",
"We plan to release the tool and make it freely available to the research community so it can be used in other related annotation tasks.",
"In the future, we will continue extending the tool to work on different dialects and genres of Arabic."
],
[
"Acknowledgments",
"This publication was made possible by grant NPRP7-290-1-047 from the Qatar National Research Fund (a member of the Qatar Foundation).",
"The statements made herein are solely the responsibility of the authors."
]
] | 1808.08392 |
[
[
"Predicting Semantic Relations using Global Graph Properties"
],
[
"Abstract Semantic graphs, such as WordNet, are resources which curate natural language on two distinguishable layers.",
"On the local level, individual relations between synsets (semantic building blocks) such as hypernymy and meronymy enhance our understanding of the words used to express their meanings.",
"Globally, analysis of graph-theoretic properties of the entire net sheds light on the structure of human language as a whole.",
"In this paper, we combine global and local properties of semantic graphs through the framework of Max-Margin Markov Graph Models (M3GM), a novel extension of Exponential Random Graph Model (ERGM) that scales to large multi-relational graphs.",
"We demonstrate how such global modeling improves performance on the local task of predicting semantic relations between synsets, yielding new state-of-the-art results on the WN18RR dataset, a challenging version of WordNet link prediction in which \"easy\" reciprocal cases are removed.",
"In addition, the M3GM model identifies multirelational motifs that are characteristic of well-formed lexical semantic ontologies."
],
[
"Introduction",
"Semantic graphs, such as WordNet [7], encode the structural qualities of language as a representation of human knowledge.",
"On the local level, they describe connections between specific semantic concepts, or synsets, through individual edges representing relations such as hypernymy (`is-a') or meronymy (`is-part-of'); on the global level, they encode emergent regular properties in the induced relation graphs.",
"Local properties have been subject to extensive study in recent years via the task of relation prediction, where individual edges are found based mostly on distributional methods that embed synsets and relations into a vector space , , [34], [17].",
"In contrast, while the structural regularity and significance of global aspects of semantic graphs is well-attested [28], global properties have rarely been used in prediction settings.",
"In this paper, we show how global semantic graph features can facilitate in local tasks such as relation prediction.",
"Figure: Probable (a) and improbable (b-c) structures in a hypothetical hypernym graph.To motivate this approach, consider the hypothetical hypernym graph fragments in fig:motifs: in (a), the semantic concept (synset) `catamaran' has a single hypernym, `boat'.",
"This is a typical property across a standard hypernym graph.",
"In (b), the synset `cat' has two hypernyms, an unlikely event.",
"While a local relation prediction model might mistake the relation between `cat' and `boat' to be plausible, for whatever reason, a high-order graph-structure-aware model should be able to discard it based on the knowledge that a synset should not have more than one hypernym.",
"In (c), an impossible situation arises: a cycle in the hypernym graph leads each of the participating synsets to be predicted by transitivity as its own hypernym, contrary to the relation's definition.",
"However, a purely local model has no explicit mechanism for rejecting such an outcome.",
"In this paper, we examine the effect of global graph properties on the link structure via the WordNet relation prediction task.",
"Our hypothesis is that features extracted from the entire graph can help constrain local predictions to structurally sound ones [9].",
"Such features are often manifested as aggregate counts of small subgraph structures, known as motifs, such as the number of nodes with two or more outgoing edges, or the number of cycles of length 3.",
"Returning to the example in fig:motifs, each of these features will be affected when graphs (b) and (c) are evaluated, respectively.",
"To estimate weights on local and global graph features, we build on the Exponential Random Graph Model (ERGM), a log-linear model over networks utilizing global graph features [11].",
"In ERGMs, the likelihood of a graph is computed by exponentiating a weighted sum of the features, and then normalizing over all possible graphs.",
"This normalization term grows exponentially in the number of nodes, and in general cannot be decomposed into smaller parts.",
"Approximations are therefore necessary to fit ERGMs on graphs with even a few dozen nodes, and the largest known ERGMs scale only to thousands of nodes [26].",
"This is insufficient for WordNet, which has an order of $10^5$ nodes.",
"We extend the ERGM framework in several ways.",
"First, we replace the maximum likelihood objective with a margin-based objective, which compares the observed network against alternative networks; we call the resulting model the Max-Margin Markov Graph Model (m3gm), drawing on ideas from structured prediction [33].",
"The gradient of this loss is approximated by importance sampling over candidate negative edges, using a local relational model as a proposal distribution.",
"The complexity of each epoch of estimation is thus linear in the number of edges, making it possible to scale up to the $10^5$ nodes in WordNet.Although in principle the number of edges could grow quadratically with the number of nodes, steyvers2005large show that semantic graphs like WordNet tend to be very sparse, so that the number of observed edges grows roughly linearly with the number of nodes.",
"Second, we address the multi-relational nature of semantic graphs, by incorporating a combinatorial set of labeled motifs.",
"Finally, we link graph-level relational features with distributional information, by combining the m3gm with a dyad-level model over word sense embeddings.",
"We train m3gm as a re-ranker, which we apply to a a strong local-feature baseline on the WN18RR dataset [6].",
"This yields absolute improvements of 3-4 points on all commonly-used metrics.",
"Model inspection reveals that m3gm assigns importance to features from all relations, and captures some interesting inter-relational properties that lend insight into the overall structure of WordNet.Our code is available at http://www.github.com/yuvalpinter/m3gm."
],
[
"Relational prediction in semantic graphs.",
"Recent approaches to relation prediction in semantic graphs generally start by embedding the semantic concepts into a shared space and modeling relations by some operator that induces a score for an embedding pair input.",
"We use several of these techniques as base models [20], [4], [36]; detailed description of these methods is postponed to ssec:local.",
"SocherChenManningNg2013 generalize over the approach of nickel2011three by using a bilinear tensor which assigns multiple parameters for each relation; shi2017proje project the node embeddings in a translational model similar to that of bordes2013translating; dettmers2018conve apply a convolutional neural network by reshaping synset embeddings to 2-dimensional matrices.",
"None of these embedding-based approaches incorporate structural information; in general, improvements in embedding-based methods are expected to be complementary to our approach.",
"Some recent works compose single edges into more intricate motifs, such as guu-miller-liang:2015:EMNLP, who define a task of path prediction and compose various functions to solve it.",
"They find that compositionalized bilinear models perform best on WordNet.",
"minervini2017adversarial train link-prediction models against an adversary that produces examples which violate structural constraints such as symmetry and transitivity.",
"Another line of work builds on local neighborhoods of relation interactions and automatic detection of relations from syntactically parsed text [23], [35].",
"gcn use Graph Convolutional Networks to predict relations while considering high-order neighborhood properties of the nodes in question.",
"In general, these methods aggregate information over local neighborhoods, but do not explicitly model structural motifs.",
"Our model introduces interaction features between relations (e.g., hypernyms and meronyms) for the goal of relation prediction.",
"To our knowledge, this is the first time that relation interaction is explicitly modeled into a relation prediction task.",
"Within the ERGM framework, lu2010supervised train a limited set of combinatory path features for social network link prediction."
],
[
"Scaling exponential random graph models.",
"The problem of approximating the denominator of the ERGM probability has been an active research topic for several decades.",
"Two common approximation methods exist in the literature.",
"In Maximum Pseudolikelihood Estimation ;strauss1990pseudolikelihood, a graph's probability is decomposed into a product of the probability for each edge, which in turn is computed based on the ERGM feature difference between the graph excluding the edge and the full graph.",
"Monte Carlo Maximum Likelihood Estimation ;snijders2002markov follows a sampling logic, where a large number of graphs is randomly generated from the overall space under the intuition that the sum of their scores would give a good approximation for the total score mass.",
"The probability for the observed graph is then estimated following normalization conditioned on the sampling distribution, and its precision increases as more samples are gathered.",
"Recent work found that applying a parametric bootstrap can increase the reliability of MPLE, while retaining its superiority in training speed [26].",
"Despite this result, we opted for an MCMLE-based approach for m3gm, mainly due to the ability to keep the number of edges constant in each sampled graph.",
"This property is important in our setup, since local edge scores added or removed to the overall graph score can occasionally dominate the objective function, giving unintended importance to the overall edge count."
],
[
"Max-Margin Markov Graph Models",
"Consider a graph $G = (V,E)$ , where $V$ is a set of vertices and $E = \\lbrace (s_i,t_i)\\rbrace _{i=1}^{|E|}$ is a set of directed edges.",
"The ERGM scoring function defines a probability over $\\mathcal {G}_{|V|}$ , the set of all graphs with $|V|$ nodes.",
"This probability is defined as a log-linear function, $P_{\\text{ERGM}}(G) \\propto \\psi _{\\footnotesize {\\textsc {ergm}}}(G) = \\exp {}\\left(\\theta ^T \\mathbf {f}(G)\\right),$ where $\\mathbf {f}$ is a feature function, from graphs to a vector of feature counts.",
"Features are typically counts of motifs — small subgraph structures — as described in the introduction.",
"The vector $\\theta $ is the parameter to estimate.",
"In this section we discuss our adaptation of this model to the domain of semantic graphs, leveraging their idiosyncratic properties.",
"Semantic graphs are composed of multiple relation types, which the feature space needs to accommodate; their nodes are linguistic constructs (semantic concepts) associated with complex interpretations, which can benefit the graph representation through incorporating their embeddings in ${R}^d$ into a new scoring model.",
"We then present our m3gm framework to perform reliable and efficient parameter estimation on the new model."
],
[
"Graph Motifs as Features",
"Based on common practice in ERGM feature extraction [16], we select the following graph features as a basis: Total edge count; Number of cycles of length $k$ , for $k \\in \\lbrace 2,3\\rbrace $ ; Number of nodes with exactly $k$ outgoing (incoming) edges, for $k \\in \\lbrace 1,2,3\\rbrace $ ; Number of nodes with at least $k$ outgoing (incoming) edges, for $k \\in \\lbrace 1,2,3\\rbrace $ ; Number of paths of length 2; Transitivity: the proportion of length-2 paths $u\\rightarrow v \\rightarrow w$ where an edge $u \\rightarrow w$ also exists.",
"Semantic graphs are multigraphs, where multiple relationships (hypernymy, meronymy, derivation, etc.)",
"are overlaid atop a common set of nodes.",
"For each relation $r$ in the relation inventory $\\mathcal {R}$ , we denote its edge set as $E_r$ , and redefine $E = \\bigcup _{r\\in \\mathcal {R}} E_r$ , the union of all labeled edges.",
"Some relations do not produce a connected graph, while others may coincide with each other frequently, possibly in regular but intricate patterns: for example, derivation relations tend to occur between synsets in the higher, more abstract levels of the hypernym graph.",
"We represent this complexity by expanding the feature space to include relation-sensitive combinatory motifs.",
"For each feature template from the basis list above, we extract features for all possible combinations of relation types existing in the graph.",
"Depending on the feature type, these could be relation singletons, pairs, or triples; they may be order-sensitive or order-insensitive.",
"For example: A combinatory `transitivity' feature will be extracted for the proportion of paths $u \\xrightarrow{} v \\xrightarrow{} w$ where an edge $u \\xrightarrow{} w$ also exists.",
"A combinatory `2-outgoing' feature will be extracted for the number of nodes with exactly one derivation and one has_part.",
"The number of features thus scales in $O(|\\mathcal {R}|^K)$ for a feature basis which involves up to $K$ edges in any feature, and so our 17 basis features (with $K=3$ ) generate a combinatory feature set with roughly 3,000 features for the 11-relation version of WordNet used in our experiments (see ssec:wnrr)."
],
[
"Local Score Component",
"In classical ERGM application domains such as social media or biological networks, nodes tend to have little intrinsic distinction, or at least little meaningful intrinsic information that may be extracted prior to applying the model.",
"In semantic graphs, however, the nodes represent synsets, which are associated with information that is both valuable to predicting the graph structure and approximable using unsupervised techniques such as embedding into a common $d$ -dimensional vector space based on copious amounts of available data.",
"We thus modify the traditional scoring function from eq:score to include node-specific information, by introducing a relation-specific association operator $\\mathcal {A}^{(r)} : V \\times V \\rightarrow {R}$ : $\\begin{split}& \\psi _{\\footnotesize {\\textsc {ergm}+}}(G) = \\\\& = \\exp {}\\left(\\theta ^T \\mathbf {f}(G) + \\sum _{r\\in \\mathcal {R}}\\sum _{(s,t)\\in E_r} \\mathcal {A}^{(r)}(s,t)\\right).\\end{split}$ The association operator generalizes various models from the relation prediction literature: TransE [4] embeds each relation $r$ into a vector in the shared space, representing a `difference' between sources and targets, to compute the association score under a translational objective, $\\mathcal {A}_{\\textsc {TransE}}^{(r)}(s,t) = - \\Vert \\mathbf {e}_s + \\mathbf {e}_r - \\mathbf {e}_t \\Vert .",
"$ BiLin [20] embeds relations into full-rank matrices, computing the score by a bilinear multiplication, $ \\mathcal {A}_{\\textsc {BiLin}}^{(r)}(s,t) = \\mathbf {e}_s^T \\mathbf {W}_r \\mathbf {e}_t .",
"$ DistMult [36] is a special case of BiLin where the relation matrices are diagonal, reducing the computation to a ternary dot product, $ ~~~\\mathcal {A}_{\\textsc {DistMult}}^{(r)}(s,t) = \\langle \\mathbf {e}_s, \\mathbf {e}_r, \\mathbf {e}_t \\rangle = \\sum _{i=1}^{d} e_{s_i}~e_{r_i}~e_{t_i}.",
"$"
],
[
"Parameter Estimation",
"The probabilistic formulation of ERGM requires the computation of a normalization term that sums over all possible graphs with a given number of nodes, $\\mathcal {G}_N$ .",
"The set of such graphs grows at a rate that is super-exponential in the number of nodes, making exact computation intractable even for networks that are orders of magnitude smaller than semantic graphs like WordNet.",
"One solution is to approximate probability using a variant of the Monte Carlo Maximum Likelihood Estimation (MCMLE) produce, $\\log P(G) \\approx \\log \\psi (G) - \\log \\frac{|\\mathcal {G}_{|V|}|}{M} \\sum _{\\tilde{G}\\sim \\mathcal {G}_{|V|}}^{M} \\psi (\\tilde{G}),$ where $M$ is the number of networks $\\tilde{G}$ sampled from $\\mathcal {G}_{|V|}$ , the space of all (multirelational) edge sets on nodes $V$ .",
"Each $\\tilde{G}$ is referred to as a negative sample, and the goal of estimation is to assign low scores to these samples, in comparison with the score assigned to the observed network $G$ .",
"Network samples can be obtained using edge-wise negative sampling.",
"For each edge $s\\xrightarrow{}t$ in the training network $G$ , we remove it temporarily and consider $T$ alternative edges, keeping the source $s$ and relation $r$ constant, and sampling a target $\\tilde{t}$ from a proposal distribution $Q$ .",
"Every such substitution produces a new graph $\\tilde{G}$ , $\\tilde{G}= &{} G \\cup \\lbrace s\\xrightarrow{}\\tilde{t}\\rbrace \\setminus \\lbrace s\\xrightarrow{}t\\rbrace .$"
],
[
"Large-margin objective.",
"Rather than approximating the log probability, as in MCMLE estimation, we propose a margin loss objective: the log score for each negative sample $\\tilde{G}$ should be below the log score for $G$ by a margin of at least 1.",
"This motivates the hinge loss, $\\mathcal {L}(\\Theta , \\tilde{G}; G) = \\Big ( 1 & - \\log \\psi _{\\footnotesize {\\textsc {ergm}+}}(G)\\\\&+ \\log \\psi _{\\footnotesize {\\textsc {ergm}+}}(\\tilde{G})\\Big )_{+},$ where $(x)_+ = \\max (0, x)$ .",
"Recall that the scoring function $\\psi _{\\footnotesize {\\textsc {ergm}+}}$ includes both the local association score for the alternative edge and the global graph features for the resulting graph.",
"However, it is not necessary to recompute all association scores; we need only subtract the association score for the deleted edge $s \\xrightarrow{} t$ , and add the association score for the sampled edge $s \\xrightarrow{}\\tilde{t}.$ The overall loss function is the sum over $N=|E|\\times T$ negative samples, $\\lbrace \\tilde{G}^{(i)}\\rbrace _{i=1}^N$ , plus an $L_2$ regularizer on the model parameters, $\\mathcal {L}(\\Theta ; G) =\\lambda || \\Theta ||_2^2 + \\sum _{i=1}^N \\mathcal {L}(\\Theta , \\tilde{G}^{(i)}).$"
],
[
"Proposal distribution.",
"The proposal distribution $Q$ used to sample negative edges is defined to be proportional to the local association scores of edges not present in the training graph: $\\begin{split}Q(\\tilde{t}\\mid s,r,G) \\propto {\\left\\lbrace \\begin{array}{ll}0 & s\\xrightarrow{}\\tilde{t}~\\in G \\\\\\mathcal {A}^{(r)}(s,\\tilde{t}) & s\\xrightarrow{}\\tilde{t}~\\notin G~.\\end{array}\\right.}",
"\\\\\\end{split}$ By preferring edges that have high association scores, the negative sampler helps push the m3gm parameters away from likely false positives."
],
[
"Relation Prediction",
"We evaluate m3gm on the relation graph edge prediction task.Sometimes referred to as Knowledge Base Completion, e.g.",
"in SocherChenManningNg2013.",
"Data for this task consists of a set of labeled edges, i.e.",
"tuples of the form $( s, r, t )$ , where $s$ and $t$ denote source and target entities, respectively.",
"Given an edge from an evaluation set, two prediction instances are created by hiding the source and target side, in turn.",
"The predictor is then evaluated on its ability to predict the hidden entity, given the other entity and the relation type.We follow prior work in excluding the following from the ranked lists: the known entity (no self loops); entities from the training set which fit the instance; other entities in the evaluation set."
],
[
"WN18RR Dataset",
"A popular relation prediction dataset for WordNet is the subset curated as WN18 [4], [3], containing 18 relations for about 41,000 synsets extracted from WordNet 3.0.",
"It has been noted that this dataset suffers from considerable leakage: edges from reciprocal relations such as hypernym / hyponym appear in one direction in the training set and in the opposite direction in dev / test [30], [6].",
"This allows trivial rule-based baselines to achieve high performance.",
"To alleviate this concern, dettmers2018conve released the WN18RR set, removing seven relations altogether.",
"However, even this dataset retains four symmetric relation types: also see, derivationally related form, similar to, and verb group.",
"These symmetric relations can be exploited by defaulting to a simple rule-based predictor."
],
[
"Metrics",
"We report the following metrics, common in ranking tasks and in relation prediction in particular: MR, the Mean Rank of the desired entity; MRR, Mean Reciprocal Rank, the main evaluation metric; and H@$k$, the proportion of Hits (true entities) found in the top $k$ of the lists, for $k\\in \\lbrace 1,10\\rbrace $ .",
"Unlike some prior work, we do not type-restrict the possible relation predictions (so, e.g., a verb group link may select a noun, and that would count against the model)."
],
[
"Systems",
"We evaluate a single-rule baseline, three association models, and two variants of the m3gm re-ranker trained on top of the best-performing association baseline."
],
[
"We include a single-rule baseline that predicts a relation between $s$ and $t$ in the evaluation set if the same relation was encountered between $t$ and $s$ in the training set.",
"All other models revert to this baseline for the four symmetric relations."
],
[
"Association Models",
"The next group of systems compute local scores for entity-relation triplets.",
"They all encode entities into embeddings $\\mathbf {e}$ .",
"Each of these systems, in addition to being evaluated as a baseline, is also used for computing association scores in m3gm, both in the proposal distribution (see ssec:learn) and for creating lists to be re-ranked (see below): TransE, BiLin, DistMult.",
"For detailed descriptions, see ssec:local."
],
[
"Max-Margin Markov Graph Model",
"The m3gm is applied as a re-ranker.",
"For each relation and source (target), the top $K$ candidate targets (sources) are retrieved based on the local association scores.",
"Each candidate edge is introduced into the graph, and the score $\\psi _{\\footnotesize {\\textsc {ergm}+}}(G)$ is used to re-rank the top-$K$ list.",
"We add a variant to this protocol where the graph score and association score are weighted by $\\alpha $ and $1-\\alpha $ , repsectively, before being summed.",
"We tune a separate $\\alpha _r$ for each relation type, using the development set's mean reciprocal rank (MRR).",
"These hyperparameter values offer further insight into where the m3gm signal benefits relation prediction most (see sec:analysis).",
"Since we do not apply the model to the symmetric relations (scored by the Rule baseline), they are excluded from the sampling protocol described in eq:loss, although their edges do contribute to the combinatory graph feature vector $\\mathbf {f}$ .",
"Our default setting backpropagates loss into only the graph weight vector $\\theta $ .",
"We experiment with a model variant which backpropagates into the association model and synset embeddings as well."
],
[
"Synset Embeddings",
"For the association component of our model, we require embedding representations for WordNet synsets.",
"While unsupervised word embedding techniques go a long way in representing wordforms [5], [14], [21], they are not immediately applicable to the semantically-precise domain of synsets.",
"We explore two methods of transforming pre-trained word embeddings into synset embeddings."
],
[
"Averaging.",
"A straightforward way of using word embeddings to create synset embeddings is to collect the words representing the synset as surface form within the WordNet dataset and average their embeddings [30].",
"We apply this method to pre-trained GloVe embeddings [21] and pre-trained FastText embeddings [1], averaging over the set of all wordforms in all lemmas for each synset, and performing a case-insensitive query on the embedding dictionary.",
"For example, the synset `determine.v.01' lists the following lemmas: `determine', `find', `find_out', `ascertain'.",
"Its vector is initialized as $\\frac{1}{5}(\\mathbf {e}_{determine} + 2\\cdot \\mathbf {e}_{find} + \\mathbf {e}_{out} + \\mathbf {e}_{ascertain}).$"
],
[
"AutoExtend retrofitting + Mimick.",
"AutoExtend is a method developed specifically for embedding WordNet synsets [24], in which pre-trained word embeddings are retrofitted to the tripartite relation graph connecting wordforms, lemmas, and synsets.",
"The resulting synset embeddings occupy the same space as the word embeddings.",
"However, some WordNet senses are not represented in the underlying set of pre-trained word embeddings.We use the out-of-the-box vectors supplied in http://www.cis.lmu.de/~sascha/AutoExtend.",
"To handle these cases, we trained a character-based model called Mimick, which learns to predict embeddings for out-of-vocabulary items based on their spellings [22].",
"We do not modify the spelling conventions of WordNet synsets before passing them to Mimick, so e.g.",
"`mask.n.02' (the second synset corresponding to `mask' as a noun) acts as the input character sequence as is."
],
[
"Random initialization.",
"In preliminary experiments, we attempted training the association models using randomly-initialized embeddings.",
"These proved to be substantially weaker than distributionally-informed embeddings and we do not report their performance in the results section.",
"We view this finding as strong evidence to support the necessity of a distributional signal in a type-level semantic setup."
],
[
"Setup",
"Following tuning experiments, we train the association models on synset embeddings with $d=300$ , using a negative log-likelihood loss function over 10 negative samples and iterating over symmetric relations once every five epochs.",
"We optimize the loss using AdaGrad with $\\eta =0.01$ , and perform early stopping based on the development set mean reciprocal rank.",
"m3gm is trained in four epochs using AdaGrad with $\\eta =0.1$ .",
"We set m3gm's re-rank list size $K=100$ and, following tuning, the regularization parameter $\\lambda =0.01$ and negative sample count per edge $T=10$ .",
"Our models are all implemented in DyNet [18]."
],
[
"Results",
"tab:dev presents the results on the development set.",
"Lines 1-3 depict the results for local models using averaged FastText embedding initialization, showing that the best performance in terms of MRR and top-rank hits is achieved by TransE.",
"Mean Rank does not align with the other metrics; this is an interpretable tradeoff, as both BiLin and DistMult have an inherent preference for correlated synset embeddings, giving a stronger fallback for cases where the relation embedding is completely off, but allowing less freedom for separating strong cases from correlated false positives, compared to a translational objective."
],
[
"Effect of global score.",
"There is a clear advantage to re-ranking the top local candidates using the score signal from the m3gm model (line 4).",
"These results are further improved when the graph score is weighted against the association component per relation (line 5).",
"We obtain similar improvements when re-ranking the predictions from DistMult and BiLin.",
"The m3gm training procedure is not useful in fine-tuning the association model via backpropagation: this degrades the association scores for true edges in the evaluation set, dragging the re-ranked results along with them to about a 2-point drop relative to the untuned variant.",
"tab:test-res shows that our main results transfer onto the test set, with even a slightly larger margin.",
"This could be the result of the greater edge density of the combined training and dev graphs, which enhance the global coherence of the graph structure captured by m3gm features.",
"To support this theory, we tested the m3gm model trained on only the training set, and its test set performance was roughly one point worse on all metrics, as compared with the model trained on the training+dev data.",
"Table: Main results on test set.",
"† ^\\dagger These models were not re-implemented, and are reported as in kbc-cnn and indettmers2018conve."
],
[
"Synset embedding initialization.",
"We trained association models initialized on AutoExtend+Mimick vectors (see ssec:embs).",
"Their performance, inferior to averaged FastText vectors by about 1-2 MRR points on the dev set, is somewhat at odds with findings from previous experiments on WordNet [10].",
"We believe the decisive factor in our result is the size of the training corpus used to create FastText embeddings, along with the increase in resulting vocabulary coverage.",
"Out of 124,819 lemma tokens participating in 41,105 synsets, 118,051 had embeddings available (94.6%; type-level coverage 88.1%).",
"Only 530 synsets (1.3%) finished this initialization process with no embedding and were assigned random vectors.",
"AutoExtend, fit for embeddings from mikolov2013efficient which were trained on a smaller corpus, offers a weaker signal: 13,377 synsets (32%) had no vector and needed Mimick initialization."
],
[
"Graph Analysis",
"As a consequence of the empirical experiment, we aim to find out what m3gm has learned about WordNet.",
"tab:top-feats presents a sample of top-weighted motifs.",
"Lines 1 and 2 demonstrate that the model prefers a broad scattering of targets for the member_meronym and has_part relationsExample edges: `America' $\\rightarrow $ `American', `face' $\\rightarrow $ `mouth', respectively., which are flat and top-downwards hierarchical, respectively, while line 4 shows that a multitude of unique hypernyms is undesired, as expected from a bottom-upwards hierarchical relation.",
"Line 5 enforces the asymmetry of the hypernym relation.",
"Lines 3, 6, and 7 hint at deeper interactions between the different relation types.",
"Line 3 shows that the model assigns positive weights to hypernyms which have derivationally-related forms, suggesting that the derivational equivalence classes in the graph tend to exist in the higher, more abstract levels of the hypernym hierarchy, as noted in sec:feat-eng.",
"Line 6 captures a semantic conflict: synsets located in the lower, specific levels of the graph can be specified either as instances of abstract conceptsExample instance_hypernym edge: `Rome' $\\rightarrow $ `national capital'., or as members of less specific concrete classes, but not as both.",
"Line 7 may have captured a nodal property – since part_of is a relation which holds between nouns, and verb_group holds between verbs, this negative weight assignment may be the manifestation of a part-of-speech uniqueness constraint.",
"In addition, in features 3 and 7 we see the importance of symmetric relations (here derivationally_related_form and verb_group, respectively), which manage to be represented in the graph model despite not being directly trained on.",
"Table: Successful m3gm re-ranking examples.Table: Graph score weights found for relations on the dev set.",
"Zero means graph score is not considered at all for this relation, one means only it is considered.tab:rerank presents examples of relation targets successfully re-ranked thanks to these features.",
"The first false connection created a new unique hypernym, `garden lettuce', downgraded by the graph score through incrementing the count of negatively-weighted feature 4.",
"In the second case, `vienna' was brought from rank 10 to rank 1 since it incremented the count for the positively-weighted feature 2, whereas all targets ranked above it by the local model were already has_part-s, mostly of `europe'.",
"The $\\alpha _r$ values weighing the importance of m3gm scores in the overall function, found per relation through grid search over the development set, are presented in tab:alphas.",
"It appears that for all but two relations, the best-performing model preferred the signal from the graph features to that from the association model ($\\alpha _r > 0.5$ ).",
"Based on the surface properties of the different relation graphs, the decisive factor seems to be that synset_domain_topic_of and has_part pertain mostly to very common concepts, offering good local signal from the synset embeddings, whereas the rest include many long-tail, low-frequency synsets that require help from global features to detect regularity."
],
[
"Conclusion",
"This paper presents a novel method for reasoning about semantic graphs like WordNet, combining the distributional coherence between individual entity pairs with the structural coherence of network motifs.",
"Applied as a re-ranker, this method substantially improves performance on link prediction.",
"Our analysis of results from tab:top-feats, lines 6 and 7, suggests that adding graph motifs which qualify their adjacent nodes in terms of syntactic function or semantic category may prove useful.",
"From a broader perspective, m3gm can do more as a probabilistic model than predict individual edges.",
"For example, consider the problem of linking a new entity into a semantic graph, given only the vector embedding.",
"This task involves adding multiple edges simultaneously, while maintaining structural coherence.",
"Our model is capable of scoring bundles of new edges, and in future work, we plan to explore the possibility of combining m3gm with a search algorithm, to automatically extend existing knowledge graphs by linking in one or more new entities.",
"We also plan to explore multilingual applications.",
"To some extent, the structural parameters estimated by m3gm are not specific to English: for example, hypernymy cannot be symmetric in any language.",
"If the structural parameters estimated from English WordNet are transferable to other languages, then the combination of m3gm and multilingual word embeddings could facilitate the creation and extension of large-scale semantic resources across many languages [8], [2], [12]."
],
[
"Acknowledgments",
"We would like to thank the anonymous reviewers for their helpful comments.",
"We discussed fast motif-counting algorithms with Polo Chau and Oded Green, and received early feedback from Jordan Boyd-Graber, Erica Briscoe, Martin Hyatt, Bryan Leslie Lee, Martha Palmer, and Oren Tsur.",
"This research was funded by the Defense Threat Research Agency under award HDTRA1-15-1-0019."
]
] | 1808.08644 |
[
[
"Modified Erd\\\"os--Ginzburg--Ziv Constants for $\\mathbb Z/n\\mathbb Z$ and\n $(\\mathbb Z/n\\mathbb Z)^2$"
],
[
"Abstract For an abelian group $G$ and an integer $t > 0$, the \\emph{modified Erd\\\"os--Ginzburg--Ziv constant} $s_t'(G)$ is the smallest integer $\\ell$ such that any zero-sum sequence of length at least $\\ell$ with elements in $G$ contains a zero-sum subsequence (not necessarily consecutive) of length $t$.",
"We compute $s_t'(G)$ for $G = \\mathbb Z/n\\mathbb Z$ and for $t = n$, $G = (\\mathbb Z/n\\mathbb Z)^2$."
],
[
".0.5em name="
],
[
"[runin].0.5em[.]",
"name="
],
[
"[runin].0.5em[.]",
"Modified Erdös–Ginzburg–Ziv constants for $\\mathbb {Z}/n\\mathbb {Z}$ and $(\\mathbb {Z}/n\\mathbb {Z})^2$ Aaron BergerDanielle Wang Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307bergera@mit.edudiwang@mit.edu For an abelian group $G$ and an integer $t > 0$ , the modified Erdös–Ginzburg–Ziv constant $s_t^{\\prime }(G)$ is the smallest integer $\\ell $ such that any zero-sum sequence of length at least $\\ell $ with elements in $G$ contains a zero-sum subsequence (not necessarily consecutive) of length $t$ .",
"We compute $s_t^{\\prime }(G)$ for $G = \\mathbb {Z}/n\\mathbb {Z}$ and for $t = n$ , $G = (\\mathbb {Z}/n\\mathbb {Z})^2$ .",
"Keywords: Zero-sum sequence, Zero-sum subsequence, Erdös–Ginzburg–Ziv Constant."
],
[
"Introduction",
"In 1961, Erdös, Ginzburg, and Ziv proved the following classical theorem.",
"Theorem 2.1 (Erdös–Ginzburg–Ziv [6]) Any sequence of length $2n - 1$ in $\\mathbb {Z}/n\\mathbb {Z}$ contains a zero-sum subsequence of length $n$ .",
"Here, a subsequence need not be consecutive, and a sequence is zero-sum if its elements sum to 0.",
"This theorem has lead to many problems involving zero-sum sequences over groups.",
"In general, let $G$ be an abelian group, and let $G_0 \\subseteq G$ be a susbset.",
"Let $\\mathcal {L}\\subseteq \\mathbb {N}$ .",
"Then $s_{\\mathcal {L}}(G_0)$ is defined to be the minimal $\\ell $ such that any sequence of length $\\ell $ with elements in $G_0$ contains a zero-sum subsequence whose length is in $\\mathcal {L}$ .",
"When $G_0 = G$ and $\\mathcal {L} = \\lbrace \\exp (G)\\rbrace $ , this constant is called the Erdös–Ginzburg–Ziv constant.",
"When $G = \\mathbb {Z}$ , this problem turns out to be not very interesting — if $G_0$ contains a nonzero element, then $s_{\\mathcal {L}}(G_0) = \\infty $ .",
"This has lead to [2] the study of the modified Erdös–Ginzburg–Ziv constant $s_{\\mathcal {L}}^{\\prime }(G_0)$ , defined as the smallest $\\ell $ such that any zero-sum sequence of length at least $\\ell $ with elements in $G_0$ contains a zero-sum subsequence whose length is in $\\mathcal {L}$ .",
"When $\\mathcal {L} = \\lbrace t\\rbrace $ is a single element, we omit the set brackets for convenience.",
"In [3], the first author determined modified EGZ constants in the infinite cyclic case.",
"Here we treat the finite cyclic case and extensions.",
"Problem 2.2 ([3]) Compute $s_t^{\\prime }(G)$ for $G = \\mathbb {Z}/n\\mathbb {Z}$ and $(\\mathbb {Z}/n\\mathbb {Z})^2$ .",
"In this paper, we answer Problem REF for $G = \\mathbb {Z}/n\\mathbb {Z}$ and for $t = n$ , $G = (\\mathbb {Z}/n\\mathbb {Z})^2$ .",
"Note that in both cases, when $n$ does not divide $t$ , the quantity $s_t(G)$ is infinite.",
"thmmcyclic The modified EGZ constant of $\\mathbb {Z}/n\\mathbb {Z}$ is given by $s_{nt}^{\\prime }(\\mathbb {Z}/n\\mathbb {Z})= (t+1)n - \\ell + 1$ , where $\\ell $ is the smallest integer such that $\\ell \\nmid n$ .",
"thmmsquare We have $s_n^{\\prime }((\\mathbb {Z}/n\\mathbb {Z})^2) = 4n - \\ell + 1$ where $\\ell $ is the smallest integer such that $d \\ge 4$ and $\\ell \\nmid n$ ."
],
[
"The cyclic case",
"In this section we give the proof of Theorem .",
"As in [10], if $J$ is a sequence of elements of $\\mathbb {Z}/n\\mathbb {Z}$ or $(\\mathbb {Z}/n\\mathbb {Z})^2$ , we use $(k \\mid J)$ to denote the number of zero-sum subsequences of $J$ of size $k$ .",
"Proposition 3.1 If $d \\mid n$ and $J$ is a zero-sum sequence in $\\mathbb {Z}/n\\mathbb {Z}$ of length $2n - d$ , then $(n \\mid J) > 0$ .",
"By Theorem REF , we can break off subsequences of $J$ of size $d$ with sum $0 \\pmod {d}$ until we have fewer than $2d-1$ remaining.",
"In fact, since $d \\mid n$ , we will have exactly $d$ remaining.",
"But since the sum was zero-sum to begin with, the last $d$ must also sum to zero, so we have $2(n/d) - 1$ blocks of size $d$ with sums $dx_1, \\dots , dx_{2(n/d) - 1}$ for some $x_i$ .",
"By Theorem REF , some $n/d$ of these must sum to 0 in $\\mathbb {Z}/(n/d)\\mathbb {Z}$ , so the union of these blocks gives a subsequence of length $n$ whose sum is zero in $\\mathbb {Z}/n\\mathbb {Z}$ .",
"Corollary 3.2 Let $\\ell $ be the smallest positive integer such that $\\ell \\nmid n$ , and let $t \\ge 1$ .",
"If $J$ is a zero-sum sequence in $\\mathbb {Z}/n\\mathbb {Z}$ of length at least $(t + 1)n - \\ell + 1$ , then $(nt \\mid J) > 0$ .",
"We induct on $t$ .",
"The case $t = 1$ follows from Proposition REF since $\\ell - 1, \\dots , 1$ all divide $n$ .",
"Suppose the result is true for positive integers less than $t > 1$ .",
"Then $J$ contains a zero-sum subsequence of length $(t-1)n$ .",
"Remove these elements from $J$ .",
"We are left with a zero-sum sequence of length $2n - \\ell + 1$ .",
"This is the $t = 1$ case, so we can find another zero-sum subsequence of length $n$ .",
"Combine this with the $(t-1)n$ to get the desired subsequence of length $nt$ .",
"Proposition 3.3 Suppose $\\ell \\nmid n$ and $t \\ge 1$ .",
"Then there exists a zero-sum subsequence in $\\mathbb {Z}/n\\mathbb {Z}$ of length $(1 + t)n - \\ell $ which contains no zero-sum subsequence of length $nt$ .",
"Consider a sequence of 0's and 1's with multiplicities $a \\le tn - 1$ , $b \\le n - 1$ respectively where $a + b = (t+1)n - \\ell $ .",
"Such a sequence will have no zero-sum subsequence of length $nt$ .",
"It suffices to find $a$ , $b$ such that $g = \\gcd (n, \\ell ) \\mid b$ , because then we can add some constant to every term of the sequence to make it zero-sum.",
"Note that adding a constant to every term does not introduce any new zero-sum subsequences.",
"It suffices to take $b = tn - g$ and $a = n - \\ell + g\\le n - \\ell /2 \\le n - 1$ .",
"Corollary REF and Proposition REF together imply Theorem ."
],
[
"The case $(\\mathbb {Z}/n\\mathbb {Z})^2$",
"In this section we prove Theorem .",
"We first prove some preliminary lemmas.",
"The following results from [10] are key.",
"Lemma 4.1 ([10]) Let $p$ be a prime, and let $J$ be a sequence of elements in $(\\mathbb {Z}/p\\mathbb {Z})^2$ .",
"If $|J| = 3p-2$ or $|J| = 3p-1$ , then $(p \\mid J) = 0$ implies $(2p \\mid J) \\equiv - 1 \\pmod {p}$ .",
"Lemma 4.2 ([10]) Let $p$ and $J$ be as in Lemma REF .",
"If $|J|$ is a zero-sum sequence with exactly $3p$ elements, then $(p \\mid J) > 0$ .",
"Theorem 4.3 ([10]) If $J$ is a sequence of length $4n -3$ in $(\\mathbb {Z}/n\\mathbb {Z})^2$ then $(n \\mid J) > 0$ .",
"We generalize Lemma REF to non-prime $n$ .",
"Lemma 4.4 If $J$ is a zero-sum sequence of length $3n$ in $(\\mathbb {Z}/n\\mathbb {Z})^2$ , then $(n \\mid J) > 0$ .",
"We induct on $n$ .",
"The base case $n = 1$ is clear.",
"Assume the the lemma is true for all positive integers less than $n$ .",
"Let $n = pm$ with $p$ prime and $m < n$ .",
"Since $3n > 4m - 3$ , we can find some $m$ elements of $J$ whose sum is $0 \\pmod {m}$ .",
"Say their sum is $mx_1$ and remove these $m$ elements.",
"We can continue doing this until there remain only $3m$ elements.",
"But since $J$ was a zero-sum sequence, the remaining $3m$ elements must sum to $0 \\pmod {m}$ , so by the induction hypothesis, we can remove another $m$ with sum a multiple of $m$ .",
"This gives us $3p-2$ blocks of size $m$ whose sums are $mx_1, \\dots , mx_{3p-2}$ for some $x_i$ .",
"If some $p$ of the $x_i$ sum to $0 \\pmod {p}$ , then combining the blocks would give us $n$ elements whose sum is $0 \\pmod {n}$ , as desired.",
"If not, by Lemma REF , we must have some $2p$ of the $x_i$ summing to $0 \\pmod {p}$ , so we have $2n$ elements whose sum is $0 \\pmod {n}$ .",
"But since $J$ itself is zero-sum and has size $3n$ , the complement is zero-sum as well and has size $n$ .",
"Proposition 4.5 If $d\\mid n$ , and $J$ is a zero-sum sequence in $(\\mathbb {Z}/n\\mathbb {Z})^2$ of length $4n - d$ , then $(n \\mid J ) > 0$ .",
"Note that $4n - d \\ge 3m$ .",
"By Theorem REF , we can break off subsequences of size $d$ with sum $0 \\pmod {d}$ until we have only $3d$ elements remaining.",
"Then by Lemma REF we can break off another $d$ elements, to obtain $4(n/d) - 3$ blocks of size $d$ , with sums $dx_1, \\dots , dx_{4(n/d) - 3}$ for some $x_i$ .",
"By Theorem REF , some $n/d$ of the $x_i$ must sum to 0 in $(\\mathbb {Z}/(n/d)\\mathbb {Z})^2$ .",
"Combining the corresponding blocks gives a subsequence of length $n$ whose sum is zero in $(\\mathbb {Z}/n\\mathbb {Z})^2$ .",
"The following corollary is clear from Proposition REF and Theorem REF .",
"Corollary 4.6 Let $\\ell $ be the smallest integer greater than or equal to 4 such that $\\ell \\nmid n$ .",
"If $J$ is a zero-sum sequence in $(\\mathbb {Z}/n\\mathbb {Z})^2$ of length at least $4n - \\ell + 1$ , then $(n \\mid J) > 0$ .",
"Proposition 4.7 Suppose $4 \\le \\ell \\nmid n$ .",
"There exists a zero-sum sequence in $(\\mathbb {Z}/n\\mathbb {Z})^2$ of length $4n - \\ell $ which contains no zero-sum subsequences of length $n$ .",
"First, consider a sequence of the form $(0,0)\\quad & a \\le n - 1 \\\\(0,1)\\quad & b \\le n - 1\\\\(1,0)\\quad & c \\le n - 1\\\\(1,1)\\quad & d \\le n - 1,$ where $a$ denotes the number of $(0,0)$ 's, etc., and $a + b + c + d = 4n - \\ell $ .",
"It is easy to check that this sequence contains no zero-sum subsequence of length $n$ .",
"Now, we claim that there exists $(r,s) \\in (\\mathbb {Z}/n\\mathbb {Z})^2$ such that adding $(r,s)$ to each term of the above sequence will result in a zero-sum sequence.",
"Note that adding $(r,s)$ to each term does not change the fact that there is no zero-sum subsequence of length $n$ .",
"In fact, all we need is $g \\gcd (n, \\ell ) \\mid c + d,b + d.$ We claim that the following $a, b, c, d$ work.",
"$a &= n - \\ell + g + 1 \\text{ (or } n - \\ell + 2g + 1\\text{ if } g = 1) \\\\b &= n - 1\\\\c &= n - 1 \\\\d &= n - g + 1 \\text{ (or } n - 2g + 1 \\text{ if } g = 1).$ Note that $g \\le \\ell /2$ because $\\ell \\nmid n$ , so $a \\le n - \\ell /2 + 1 \\le n - 1$ if $g \\ne 1$ , and $a = n - \\ell + 3 \\le n - 1$ if $g = 1$ .",
"It is easy to show that we always have $a, d \\ge 0$ and $d \\le n - 1$ , and that these $a, b, c, d$ satisfy the divisibility relation.",
"Now, Corollary REF and Proposition REF imply Theorem ."
],
[
"Open problems",
"Harborth [8] first considered the problem of computing $s_n((\\mathbb {Z}/n\\mathbb {Z})^d)$ for higher dimensions.",
"He proved the following bounds.",
"Theorem 5.1 (Harborth [8]) We have $(n-1)2^d + 1 \\le s_n((\\mathbb {Z}/n\\mathbb {Z})^d) \\le (n-1)n^d + 1.$ For $d > 2$ the precise value of $s_n((\\mathbb {Z}/n\\mathbb {Z})^d)$ is not known.",
"See [4], [5] for some better lower bounds and [1], [9] for some better upper bounds.",
"In general the lower bound in Theorem REF is not tight, but Harborth showed that it is an equality for $n = 2^k$ a power of 2.",
"Conjecture 5.2 If $n = 2^k$ and $d \\ge 1$ , we have $s_n^{\\prime }((\\mathbb {Z}/n\\mathbb {Z})^d) = 2^d n - \\ell + 1,$ where $\\ell $ is the smallest integer such that $\\ell \\ge 2^d$ and $\\ell \\nmid n$ .",
"By an argument similar to the $(\\mathbb {Z}/n\\mathbb {Z})^2$ case, we can reduce this conjecture to the case $n = 2^d$ , in which case $\\ell = 2^d + 1$ .",
"We also have not determined the modified EGZ constants for $(\\mathbb {Z}/n\\mathbb {Z})^2$ for subseqences of length greater than $n$ .",
"Problem 5.3 Compute $s_{nt}^{\\prime }((\\mathbb {Z}/n\\mathbb {Z})^2)$ for $t > 1$ .",
"The constant $s_n(\\mathbb {Z}/m\\mathbb {Z}\\times \\mathbb {Z}/n\\mathbb {Z})$ is known to be $2m + 2n - 3$ for $m \\mid n$ [7].",
"Problem 5.4 Compute $s_{nt}^{\\prime }(\\mathbb {Z}/m\\mathbb {Z}\\times \\mathbb {Z}/n\\mathbb {Z})$ for $t \\ge 1$ and $m \\mid n$ ."
],
[
"Acknowledgements",
"This research was conducted at the University of Minnesota Duluth REU and was supported by NSF / DMS grant 1650947 and NSA grant H98230-18-1-0010.",
"We would like to thank Joe Gallian for running the program."
]
] | 1808.08486 |
[
[
"Bernstein Functions and Radial Limits of Prescribed Mean Curvature\n Surfaces"
],
[
"Abstract The radial limits at a point ${\\bf y}$ of the boundary of the domain $\\Omega\\subset {\\bf R}^{2}$ of a bounded variational solution $f$ of Dirichlet or contact angle boundary value problems for a prescribed mean curvature equation are studied with an emphasis on the effects of assumptions about the curvatures of the boundary $\\partial\\Omega$ on each side of the point ${\\bf y}.$ For example, at a nonconvex corner ${\\bf y},$ we previously proved that all nontangential radial limits of $f$ at ${\\bf y}$ exist, here we provide sufficient conditions for the tangential radial limits to exist, even when the Dirichlet data $\\phi\\in L^{\\infty}(\\partial\\Omega)$ has no one-sided limits at ${\\bf y}$ or the contact angle $\\gamma\\in L^{\\infty}(\\partial\\Omega:[0,\\pi])$ is not bounded away from $0$ or $\\pi.$ We also provide a complement to a 1976 Theorem by Leon Simon on least area surfaces."
],
[
"Introduction",
"Let $\\Omega $ be a locally Lipschitz domain in ${\\rm I\\hspace{-1.99997pt}R}^{2}$ and define $Nf = \\nabla \\cdot Tf = {\\rm div}\\left(Tf\\right),$ where $f\\in C^{2}(\\Omega )$ and $Tf= \\frac{\\nabla f}{\\sqrt{1+\\left|\\nabla f\\right|^{2}}}.$ Consider the Dirichlet problem $Nf & = & H(\\cdot ,f(\\cdot )) \\mbox{ \\ in \\ } \\Omega \\\\f & = & \\phi \\mbox{ \\ on \\ } \\partial \\Omega $ and the contact angle problem $Nf & = & H(\\cdot ,f(\\cdot )) \\mbox{ \\ in \\ } \\Omega \\\\Tf \\cdot {\\bf \\nu } & = & \\cos \\gamma \\mbox{ \\ on \\ } \\partial \\Omega ,$ where $\\phi :\\partial \\Omega \\rightarrow {\\rm I\\hspace{-1.99997pt}R},$ $\\gamma :\\partial \\Omega \\rightarrow [0,\\pi ],$ and $H:\\Omega \\times {\\rm I\\hspace{-1.99997pt}R}\\rightarrow {\\rm I\\hspace{-1.99997pt}R}$ are prescribed functions, $H({\\bf x},t)$ is nondecreasing in $t$ for each ${\\bf x}\\in \\Omega $ (cf.",
"[6]) and $\\nu $ is the exterior unit normal to $\\partial \\Omega .$ For a smooth domain, some type of boundary curvature condition (which depends on $H$ ) must be satisfied in order to guarantee that a classical solution of (REF )-() exists for each $\\phi \\in C^{0}\\left( \\partial \\Omega \\right);$ when $H\\equiv 0,$ this curvature condition is that $\\partial \\Omega $ must have nonnegative curvature (with respect to the interior normal direction of $\\Omega $ ) at each point (e.g.",
"[17]).",
"However, Leon Simon ([30]) has shown that if $\\Gamma _{0}\\subset \\partial \\Omega $ is smooth (i.e.",
"$C^{4}$ ), $H\\equiv 0,$ $\\phi \\in C^{0,1}(\\partial \\Omega ),$ the curvature $\\Lambda $ of $\\partial \\Omega $ is negative on $\\Gamma _{0}$ and $\\Gamma $ is a compact subset of $\\Gamma _{0},$ then the variational solution $z=f({\\bf x}),$ ${\\bf x}\\in \\Omega ,$ extends to $\\Omega \\cup \\Gamma $ as a Hölder continuous function with Lipschitz continuous trace, even though $f$ may not equal $\\phi $ on $\\Gamma ;$ Simon's result holds for least area hypersurfaces in ${\\rm I\\hspace{-1.99997pt}R}^{n},$ $n\\ge 2$ when the mean curvature of $\\partial \\Omega $ has a negative upper bound on $\\Gamma \\subset \\partial \\Omega $ (see also [1], [27]).",
"One can look at this in a different way.",
"In the case $H\\equiv 0,$ the requirement that $\\Lambda ({\\bf p})<0$ at a point ${\\bf p}\\in \\partial \\Omega $ implies that $Nf=0$ has a (continuous) Bernstein function $\\psi $ at ${\\bf p}$ for $\\Omega $ (see Definition (REF ) and Definition (REF )).",
"In [8], Bernstein functions for the minimal surface equation in ${\\rm I\\hspace{-1.99997pt}R}^{2}$ are constructed for $C^{2,\\alpha }$ domains $\\Omega \\subset {\\rm I\\hspace{-1.99997pt}R}^{2}$ whose curvature $\\Lambda $ (with respect to $-\\nu $ ) vanishes at a finite number of points and satisfies $\\Lambda \\le 0$ on a segment of $\\partial \\Omega .$ Using these Bernstein functions, we will prove the following generalization of [30] when $n=2.$ Corollary 1 Let $\\Omega $ be a domain in ${\\rm I\\hspace{-1.99997pt}R}^{2},$ $\\Gamma $ is a $C^{2,\\lambda }$ open subset of $\\partial \\Omega $ and the curvature $\\Lambda $ (with respect to $-\\nu $ ) of $\\Gamma $ is nonpositive and vanishes at only a finite number of points of $\\Gamma ,$ for some $\\lambda \\in (0,1).$ Suppose $\\phi \\in L^{\\infty }(\\partial \\Omega ),$ ${\\bf y}\\in \\Gamma ,$ either $f$ is symmetric with respect to a line through ${\\bf y}$ or $\\phi $ is continuous at ${\\bf y},$ and $f\\in BV(\\Omega )$ minimizes $J(u)=\\int _{\\Omega } \\sqrt{1+|Du|^{2}} d{\\bf x} + \\int _{\\partial \\Omega } |u-\\phi | ds$ for $u\\in BV(\\Omega ).$ Then $f\\in C^{0}(\\Omega \\cup \\lbrace {\\bf y}\\rbrace ).$ If $\\phi \\in C^{0}(\\Gamma ),$ then $f\\in C^{0}(\\Omega \\cup \\Gamma ).$ Example 1 Let $\\Omega =\\lbrace (x,y)\\in {\\rm I\\hspace{-1.99997pt}R}^{2} : 1< (x+1)^{2}+y^{2}<\\cosh ^{2}(1)\\rbrace $ and $\\phi (x,y)=\\sin \\left(\\frac{\\pi }{x^2+y^2}\\right)$ for $(x,y)\\ne (0,0)$ (see Figure REF for a rough illustration of the graph of $\\phi $ ).",
"Set ${\\cal O}=(0,0).$ Let $f\\in C^{2}(\\Omega )$ minimize (REF ) over $BV(\\Omega )$ (i.e.",
"$f$ is the variational solution of (REF )-() with $H \\equiv 0$ ).",
"Then Corollary REF (with ${\\bf y}={\\cal O}$ ) implies $f\\in C^{0}\\left(\\overline{\\Omega }\\right),$ even though $\\phi $ has no limit at ${\\cal O}.$ Figure: Ω\\Omega and part of the graph of φ\\phi Variational solutions of (REF )-() will exist in some sense (e.g.",
"§7.3 of [12]) but they need not be finitely valued (e.g.",
"the discussion of extremal curves in Chapter 6 of [12]), bounded (e.g.",
"[12], Corollary 5.5) or continuous at each point of the boundary (e.g.",
"[18]).",
"Variational solutions of (REF )-() will be bounded if $\\phi \\in L^{\\infty }(\\Omega )$ but need not be continuous at each point of the boundary.",
"Many authors (e.g.",
"[7], [9], [13], [22], [28], [30], [31]) have investigated the boundary behavior at corners of variational solutions of (REF )-() and a number of authors (e.g.",
"[4], [10], [12], [11], [15], [18], [23], [24], [25], [29]) have done so for variational solutions of (REF )-().",
"We shall investigate the existence and behavior of the radial limits of nonparametric prescribed mean curvature surfaces at corners of the domain, including “smooth corners” (e.g.",
"Corollary REF ).",
"In particular, we shall use Bernstein functions to investigate the behavior of variational solutions of (REF )-() or (REF )-() at points of $\\partial \\Omega .$"
],
[
"Radial Limit Theorems",
"Let $Q$ be the operator on $C^{2}(\\Omega )$ given by $Qf({\\bf x}) \\mathrel {\\overset{\\makebox{[}0pt]{\\mbox{\\normalfont \\tiny \\sffamily def}}}{=}}Nf({\\bf x}) - 2H({\\bf x},f({\\bf x})), \\ \\ \\ \\ {\\bf x}\\in \\Omega ,$ where $H:\\Omega \\times {\\rm I\\hspace{-1.99997pt}R}\\rightarrow {\\rm I\\hspace{-1.99997pt}R}$ is prescribed and $H({\\bf x},t)$ is weakly increasing in $t$ for each ${\\bf x}\\in \\Omega .$ Let $\\nu $ be the exterior unit normal to $\\partial \\Omega ,$ defined almost everywhere on $\\partial \\Omega .$ We assume that for almost every ${\\bf y}\\in \\partial \\Omega ,$ there is a continuous extension $\\hat{\\nu }$ of $\\nu $ to a neighborhood of ${\\bf y}.$ For each point ${\\bf y}\\in \\partial \\Omega ,$ polar coordinates relative to ${\\bf y}$ are denoted by $r_{\\bf y}$ and $\\theta _{\\bf y}.$ We shall assume that for each ${\\bf y}\\in \\partial \\Omega ,$ there exists a $\\delta >0$ such that $\\partial \\Omega \\cap B_{\\delta }({\\bf y})\\setminus \\lbrace {\\bf y}\\rbrace $ consists of two (open) arcs ${\\partial }_{\\bf y}^{1}\\Omega $ and $\\partial _{\\bf y}^{2}\\Omega ,$ whose tangent rays approach the rays $L^{1}_{\\bf y}: \\: \\theta _{\\bf y} = \\alpha ({\\bf y})$ and $L^{2}_{\\bf y}: \\: \\theta _{\\bf y} = \\beta ({\\bf y})$ respectively, as the point ${\\bf y}$ is approached, with $\\alpha ({\\bf y})<\\beta ({\\bf y})<\\alpha ({\\bf y})+2\\pi ,$ in the sense that the tangent cone to $\\overline{\\Omega }$ at ${\\bf y}$ is $\\lbrace \\alpha ({\\bf y})\\le \\theta _{\\bf y} \\le \\beta ({\\bf y}), 0\\le r_{\\bf y}<\\infty \\rbrace .$ (In particular, $\\lbrace \\alpha ({\\bf y})<\\theta _{\\bf y} <\\beta ({\\bf y}), 0<r_{\\bf y}<\\epsilon (\\theta _{\\bf y})\\rbrace $ is a subset of $\\Omega $ for some $\\epsilon \\in C^{0}((\\alpha ({\\bf y}),\\beta ({\\bf y}))),$ $\\epsilon (\\cdot )>0,$ and $\\lbrace \\beta ({\\bf y})<\\theta _{\\bf y} <\\alpha ({\\bf y})+2\\pi , 0<r_{\\bf y}<\\epsilon (\\theta _{\\bf y})\\rbrace \\cap \\Omega =\\emptyset $ for some $\\epsilon \\in C^{0}((\\beta ({\\bf y}),\\alpha ({\\bf y})+2\\pi )),$ $\\epsilon (\\cdot )>0.$ ) When $\\beta ({\\bf y})-\\alpha ({\\bf y}) < \\pi ,$ $\\partial \\Omega $ is said to have a convex corner at ${\\bf y}$ and when $\\beta ({\\bf y})-\\alpha ({\\bf y}) > \\pi ,$ $\\partial \\Omega $ is said to have a nonconvex corner at ${\\bf y}.$ The radial limit of $f$ at ${\\bf y}=(y_{1},y_{2})\\in \\partial \\Omega $ in the direction $\\omega (\\theta )=(\\cos \\theta ,\\sin \\theta ),$ $\\theta \\in \\left(\\alpha ({\\bf y}),\\beta ({\\bf y})\\right),$ is $Rf(\\theta ,{\\bf y}) \\mathrel {\\overset{\\makebox{[}0pt]{\\mbox{\\normalfont \\tiny \\sffamily def}}}{=}}\\lim _{r\\downarrow 0} f(y_{1}+r\\cos (\\theta ),y_{2}+r\\sin (\\theta )).$ $Rf(\\alpha ({\\bf y}),{\\bf y})$ will be defined as the limit at ${\\bf y}$ of the trace of $f$ restricted to ${\\partial }_{\\bf y}^{1}\\Omega $ and $Rf(\\beta ({\\bf y}),{\\bf y})$ as the limit at ${\\bf y}$ of the trace of $f$ restricted to ${\\partial }_{\\bf y}^{2}\\Omega .$ Notice that if $f$ is a generalized (e.g.",
"variational or Perron) solution of (REF )-(), $f$ need not equal $\\phi $ on portions of $\\partial \\Omega $ and the tangential radial limits $Rf(\\alpha ({\\bf y}),{\\bf y})$ and $Rf(\\beta ({\\bf y}),{\\bf y})$ may, for example, differ from $\\phi ({\\bf y})$ when $\\phi $ is continuous at ${\\bf y}.$ Definition 1 Given a domain $\\Omega $ as above, a upper Bernstein pair $\\left(U^{+},\\psi ^{+}\\right)$ for a curve $\\Gamma \\subset \\partial \\Omega $ and a function $H$ is a $C^{1}$ domain $U^{+}$ and a function $\\psi ^{+}\\in C^{2}(U^{+})\\cap C^{0}\\left(\\overline{U^{+}}\\right)$ such that $\\Gamma \\subset \\partial U^{+},$ $\\nu $ is the exterior unit normal to $\\partial U^{+}$ at each point of $\\Gamma $ (i.e.",
"$U^{+}$ and $\\Omega $ lie on the same side of $\\Gamma ;$ see Figure REF ), $Q\\psi ^{+}\\le 0$ in $U^{+},$ and $T\\psi ^{+}\\cdot \\nu =1$ almost everywhere on an open subset of $\\partial U^{+}$ containing $\\overline{\\Gamma }$ in the same sense as in [3]; that is, for almost every ${\\bf y}\\in \\Gamma ,$ $\\lim _{U^{+}\\ni {\\bf x}\\rightarrow {\\bf y}} \\frac{\\nabla \\psi ^{+}({\\bf x})\\cdot \\hat{\\nu }({\\bf x})}{\\sqrt{1+|\\nabla \\psi ^{+}({\\bf x})|^{2}}} = 1.$ Definition 2 Given a domain $\\Omega $ as above, a lower Bernstein pair $\\left(U^{-},\\psi ^{-}\\right)$ for a curve $\\Gamma \\subset \\partial \\Omega $ and a function $H$ is a $C^{1}$ domain $U^{-}$ and a function $\\psi ^{-}\\in C^{2}(U^{-})\\cap C^{0}\\left(\\overline{U^{-}}\\right)$ such that $\\Gamma \\subset \\partial U^{-},$ $\\nu $ is the exterior unit normal to $\\partial U^{-}$ at each point of $\\Gamma $ (i.e.",
"$U^{-}$ and $\\Omega $ lie on the same side of $\\Gamma $ ), $Q\\psi ^{-}\\ge 0$ in $U^{-},$ and $T\\psi ^{-}\\cdot \\nu =-1$ almost everywhere on an open subset of $\\partial U^{-}$ containing $\\overline{\\Gamma }$ in the same sense as in [3].",
"In the following theorem, we consider a domain with a nonconvex corner ${\\bf y}$ and prove that the radial limits of $f$ at ${\\bf y}$ exist and behave as in [7], [20], [21], [25].",
"In [20], $\\Omega $ was required to be locally convex at points of ${\\partial }_{\\bf y}^{1}\\Omega $ and $\\partial _{\\bf y}^{2}\\Omega $ and, in [7], [21], the curvatures of ${\\partial }_{\\bf y}^{1}\\Omega $ and $\\partial _{\\bf y}^{2}\\Omega $ were required to have an appropriate positive lower bound when these curves were smooth.",
"In [9], no such curvature requirement was imposed but only nontangential radial limits were shown to exist.",
"This theorem strengthens Theorem 1 of [9] when the curvatures of ${\\partial }_{\\bf y}^{1}\\Omega $ and $\\partial _{\\bf y}^{2}\\Omega $ imply Bernstein functions exist (see §).",
"Theorem 1 Let $f\\in C^{2}(\\Omega )\\cap L^{\\infty }(\\Omega )$ satisfy $Qf=0$ in $\\Omega $ and let $H^{*}\\in L^{\\infty }({\\rm I\\hspace{-1.99997pt}R}^{2})$ satisfy $H^{*}({\\bf x})=H({\\bf x},f({\\bf x}))$ for ${\\bf x}\\in \\Omega .$ Suppose that ${\\bf y}\\in \\partial \\Omega ,$ $\\beta ({\\bf y})-\\alpha ({\\bf y}) > \\pi ,$ and there exist $\\delta >0$ and upper and lower Bernstein pairs $\\left(U^{\\pm }_{1},\\psi ^{\\pm }_{1}\\right)$ and $\\left(U^{\\pm }_{2},\\psi ^{\\pm }_{2}\\right)$ for $(\\Gamma _{1},H^{*})$ and $(\\Gamma _{2},H^{*})$ respectively, where $\\Gamma _{1}=B_{\\delta }({\\bf y})\\cap {\\partial }_{\\bf y}^{1}\\Omega $ and $\\Gamma _{2}=B_{\\delta }({\\bf y})\\cap {\\partial }_{\\bf y}^{2}\\Omega .$ Then the limits $\\lim _{\\Gamma _{1}\\ni {\\bf x}\\rightarrow {\\bf y}} f({\\bf x})=z_{1} \\ {\\rm and} \\ \\lim _{\\Gamma _{2}\\ni {\\bf x}\\rightarrow {\\bf y}} f({\\bf x})=z_{2}$ exist, the radial limit $Rf(\\theta ,{\\bf y})$ exists for each $\\theta \\in [\\alpha ({\\bf y}),\\beta ({\\bf y})],$ $Rf(\\alpha ({\\bf y}),{\\bf y})=z_{1},$ $Rf(\\beta ({\\bf y}),{\\bf y})=z_{2},$ and $Rf(\\cdot ,{\\bf y})$ is a continuous function on $[\\alpha ({\\bf y}),\\beta ({\\bf y})]$ which behaves in one of the following ways: (i) $Rf(\\cdot ,{\\bf y})=z_{1}$ is a constant function and $f$ is continuous at ${\\bf y}.$ (ii) There exist $\\alpha _{1}$ and $\\alpha _{2}$ so that $\\alpha ({\\bf y}) \\le \\alpha _{1}< \\alpha _{2} \\le \\beta ({\\bf y}),$ $Rf=z_{1}$ on $[\\alpha ({\\bf y}), \\alpha _{1}],$ $Rf=z_{2}$ on $[\\alpha _{2}, \\beta ({\\bf y})]$ and $Rf$ is strictly increasing (if $z_{1}<z_{2}$ ) or strictly decreasing (if $z_{1}>z_{2}$ ) on $[\\alpha _{1}, \\alpha _{2}].$ (iii) There exist $\\alpha _{1}, \\alpha _{L}, \\alpha _{R}, \\alpha _{2}$ so that $\\alpha ({\\bf y}) \\le \\alpha _{1} < \\alpha _{L} < \\alpha _{R} < \\alpha _{2} \\le \\beta ({\\bf y}),$ $\\alpha _{R}= \\alpha _{L} + \\pi $ , and $Rf$ is constant on $[\\alpha ({\\bf y}), \\alpha _{1}],[ \\alpha _{L}, \\alpha _{R}]$ , and $[ \\alpha _{2}, \\beta ({\\bf y})]$ and either strictly increasing on $[\\alpha _{1}, \\alpha _{L}]$ and strictly decreasing on $[ \\alpha _{R}, \\alpha _{2}]$ or strictly decreasing on $[\\alpha _{1}, \\alpha _{L}]$ and strictly increasing on $[\\alpha _{R},\\alpha _{2}]$ .",
"Figure: Ω\\Omega (left) U 2 ± U^{\\pm }_{2} (middle) U 1 ± U^{\\pm }_{1} (right)In the second theorem, we consider a domain with a smooth corner ${\\bf y}$ (i.e.",
"$\\beta ({\\bf y})-\\alpha ({\\bf y}) = \\pi $ ) and show that the radial limits of $f$ at ${\\bf y}$ exist and behave as expected.",
"Corollary REF follows from this theorem and an additional argument.",
"Theorem 2 Let $f\\in C^{2}(\\Omega )\\cap L^{\\infty }(\\Omega )$ satisfy $Qf=0$ in $\\Omega $ and let $H^{*}\\in L^{\\infty }({\\rm I\\hspace{-1.99997pt}R}^{2})$ satisfy $H^{*}({\\bf x})=H({\\bf x},f({\\bf x}))$ for ${\\bf x}\\in \\Omega .$ Suppose that ${\\bf y}\\in \\partial \\Omega ,$ $\\beta ({\\bf y})-\\alpha ({\\bf y}) = \\pi ,$ and there exist $\\delta >0$ and upper and lower Bernstein pairs $\\left(U^{\\pm },\\psi ^{\\pm }\\right)$ for $(\\Gamma ,H^{*}),$ where $\\Gamma =B_{\\delta }({\\bf y})\\cap {\\partial }\\Omega .$ Then the limits $\\lim _{\\Gamma _{1}\\ni {\\bf x}\\rightarrow {\\bf y}} f({\\bf x})=z_{1} \\ {\\rm and} \\ \\lim _{\\Gamma _{2}\\ni {\\bf x}\\rightarrow {\\bf y}} f({\\bf x})=z_{2}$ exist, $Rf(\\theta ,{\\bf y})$ exists for each $\\theta \\in [\\alpha ({\\bf y}),\\beta ({\\bf y})],$ $Rf(\\cdot ,{\\bf y})\\in C^{0}([\\alpha ({\\bf y}),\\beta ({\\bf y})])$ , $Rf(\\alpha ({\\bf y}),{\\bf y})=z_{1},$ $Rf(\\beta ({\\bf y}),{\\bf y})=z_{2},$ and $Rf(\\cdot ,{\\bf y})$ behaves as in (i) or (ii) of Theorem REF .",
"In the third theorem, we consider a domain with a convex corner ${\\bf y}$ and prove that the radial limits of $f$ at ${\\bf y}$ exist and behave as expected.",
"This theorem strengthens Theorem 2 of [9].",
"Theorem 3 Let $f\\in C^{2}(\\Omega )\\cap L^{\\infty }(\\Omega )$ satisfy $Qf=0$ in $\\Omega $ and let $H^{*}\\in L^{\\infty }({\\rm I\\hspace{-1.99997pt}R}^{2})$ satisfy $H^{*}({\\bf x})=H({\\bf x},f({\\bf x}))$ for ${\\bf x}\\in \\Omega .$ Suppose that ${\\bf y}\\in \\partial \\Omega $ and there exist $\\delta >0$ and upper and lower Bernstein pairs $\\left(U^{\\pm }_{2},\\psi ^{\\pm }_{2}\\right)$ for $(\\Gamma _{2},H^{*}),$ where $\\Gamma _{2}=B_{\\delta }({\\bf y})\\cap {\\partial }_{\\bf y}^{2}\\Omega .$ Suppose further that $z_{1}=\\lim _{\\Gamma _{1}\\ni {\\bf x}\\rightarrow {\\bf y} } f\\left({\\bf x}\\right)$ exists, where $\\Gamma _{1}=B_{\\delta }({\\bf y})\\cap {\\partial }_{\\bf y}^{1}\\Omega .$ Then $\\lim _{\\Gamma _{2}\\ni {\\bf x}\\rightarrow {\\bf y}} f({\\bf x})=z_{2}$ exists, $Rf(\\theta ,{\\bf y})$ exists for each $\\theta \\in [\\alpha ({\\bf y}),\\beta ({\\bf y})],$ $Rf(\\cdot ,{\\bf y})\\in C^{0}([\\alpha ({\\bf y}),\\beta ({\\bf y})])$ , $Rf(\\alpha ({\\bf y}),{\\bf y})=z_{1},$ $Rf(\\beta ({\\bf y}),{\\bf y})=z_{2},$ and $Rf(\\cdot ,{\\bf y})$ behaves as in (i), (ii) or (iii) of Theorem REF .",
"In the fourth theorem, we generalize Theorem 2 of [10].",
"Theorem 4 Let $f\\in C^{2}(\\Omega )$ satisfy $Qf=0$ in $\\Omega $ and let $H^{*}\\in L^{\\infty }({\\rm I\\hspace{-1.99997pt}R}^{2})$ satisfy $H^{*}({\\bf x})=H({\\bf x},f({\\bf x}))$ for ${\\bf x}\\in \\Omega .$ Suppose that ${\\bf y}\\in \\partial \\Omega ,$ $\\beta ({\\bf y})-\\alpha ({\\bf y}) < \\pi ,$ and there exist $\\delta >0$ and upper and lower Bernstein pairs $\\left(U^{\\pm }_{2},\\psi ^{\\pm }_{2}\\right)$ for $(\\Gamma _{2},H^{*}),$ where $\\Gamma _{2}=B_{\\delta }({\\bf y})\\cap {\\partial }_{\\bf y}^{2}\\Omega .$ Suppose further that $f\\in C^{1}\\left(\\Omega \\cup \\partial ^{1}_{{\\bf y}}\\Omega \\cup \\partial ^{2}_{{\\bf y}}\\Omega \\right),$ $Tf({\\bf x})\\cdot \\nu ({\\bf x})= \\cos (\\gamma ({\\bf x})) \\mbox{\\ for \\ } {\\bf x}\\in \\partial ^{1}_{{\\bf y}}\\Omega ,$ and $\\lim _{\\partial ^{1}_{{\\bf y}}\\Omega \\ni {\\bf x}\\rightarrow {\\cal O} } \\gamma \\left({\\bf x}\\right)=\\gamma _{2}.$ Suppose also that there exist $\\lambda _{1},\\lambda _{2}\\in [0,\\pi ]$ with $0<\\lambda _{2}-\\lambda _{1}<2\\left(\\beta ({\\bf y})-\\alpha ({\\bf y})\\right)$ such that $\\lambda _{1}\\le \\gamma ({\\bf x})\\le \\lambda _{2}$ for ${\\bf x}\\in \\partial ^{2}_{{\\bf y}}\\Omega $ and $\\pi -2\\alpha -\\lambda _{1}<\\gamma _{2}<\\pi +2\\alpha -\\lambda _{2}.$ Then the conclusions of Theorem REF hold.",
"In the fifth theorem, we generalize Theorem 1 of [25] at the cost of extra boundary assumptions; Theorem 1 of [5] also generalizes the Lancaster-Siegel theorem but only obtains nontangential radial limits while here the existence of all radial limits is established while not requiring the contact angle to be bounded away from zero or $\\pi .$ Theorem 5 Let $f\\in C^{2}(\\Omega )\\cap L^{\\infty }(\\Omega )$ satisfy (REF ) and () almost everywhere on $\\partial \\Omega .$ Let $H^{*}\\in L^{\\infty }({\\rm I\\hspace{-1.99997pt}R}^{2})$ satisfy $H^{*}({\\bf x})=H({\\bf x},f({\\bf x}))$ for ${\\bf x}\\in \\Omega .$ Let ${\\bf y}\\in \\partial \\Omega $ and suppose there exist $\\delta >0$ and upper and lower Bernstein pairs $\\left(U_{1}^{\\pm },\\psi ^{\\pm }_{1}\\right)$ and $\\left(U^{\\pm }_{2},\\psi ^{\\pm }_{2}\\right)$ for $(\\Gamma _{1},H^{*})$ and $(\\Gamma _{2},H^{*})$ respectively, where $\\Gamma _{1}=B_{\\delta }({\\bf y})\\cap {\\partial }_{\\bf y}^{1}\\Omega $ and $\\Gamma _{2}=B_{\\delta }({\\bf y})\\cap {\\partial }_{\\bf y}^{2}\\Omega .$ If $\\beta ({\\bf y})-\\alpha ({\\bf y}) \\le \\pi ,$ assume there exist constants $\\underline{\\gamma }^{\\, \\pm },\\overline{\\gamma }^{\\, \\pm }, 0 \\le \\underline{\\gamma }^{\\, \\pm } \\le \\overline{\\gamma }^{\\, \\pm } \\le \\pi ,$ satisfying $\\pi - (\\beta ({\\bf y})-\\alpha ({\\bf y})) < \\underline{\\gamma }^{+} + \\underline{\\gamma }^{-}$ $\\le \\overline{\\gamma }^{\\, +} + \\overline{\\gamma }^{\\, -} < \\; \\pi + \\beta ({\\bf y})-\\alpha ({\\bf y})$ such that $\\underline{\\gamma }^{\\pm }\\le \\gamma ^{\\pm }(s) \\le \\overline{\\gamma }^{\\, \\pm }$ for all $s\\in (0,s_{0}),$ for some $s_{0}>0.$ Then the conclusions of Theorem REF hold.",
"Example 2 Let $\\Omega = \\lbrace (r\\cos \\theta ,r\\sin \\theta ) : 0<r<1, -\\alpha <\\theta <\\alpha \\rbrace $ with $\\alpha >\\frac{\\pi }{2}.$ (see Figure REF (a)).",
"Let $\\phi (x,y)=\\sin \\left(\\frac{\\pi }{x^2+y^2}\\right)$ for $(x,y)\\ne (0,0)$ (see Figure REF (b) for a rough illustration of the graph of $\\phi .$ ).",
"Let $f$ satisfy (REF ) in $\\Omega $ with $H\\equiv 0$ and $f=\\phi $ on $\\partial \\Omega \\setminus \\lbrace {\\cal O} \\rbrace .$ Then [9] shows that $Rf(\\theta )$ exists when $|\\theta |<\\alpha .$ Since $\\Omega $ is locally convex at each point of $\\partial \\Omega \\setminus \\lbrace {\\cal O}\\rbrace ,$ we see that $f\\in C^{0}(\\overline{\\Omega }\\setminus \\lbrace {\\cal O}\\rbrace )$ and $f=\\phi $ on $\\partial \\Omega \\setminus \\lbrace {\\cal O}\\rbrace .$ Since $\\phi $ has no limit at ${\\cal O},$ $Rf(\\pm \\alpha )$ do not exist; however $\\lim _{\\theta \\downarrow -\\alpha } Rf(\\theta )$ and $\\lim _{\\theta \\uparrow \\alpha } Rf(\\theta )$ both exist (e.g.",
"from the behavior of $Rf(\\theta )$ established in [9], [21], [25]) and, by symmetry, are equal.",
"Suppose we replace $\\Omega $ with a slightly larger (and still symmetric) domain $\\Omega _{1},$ $\\Omega \\subset \\Omega _{1}\\subset B_{1}({\\cal O}),$ such that $\\partial \\Omega _{1}\\cap B_{1}({\\cal O})$ has negative curvature (with respect to the exterior normal to $\\Omega _{1}$ ) and $\\partial \\Omega $ and $\\partial \\Omega _{1}$ are tangent at ${\\cal O}$ (see Figure REF (c) for an illustration of $\\Omega _{1}$ ).",
"Let $f_{1}\\in C^{2}(\\Omega )$ minimize (REF ) over $BV(\\Omega _{1}),$ so that $f_{1}$ is the variational solution of (REF )-() in $\\Omega _{1}$ with $H\\equiv 0.$ Then Theorem REF implies $Rf_{1}(\\theta )$ exists when $|\\theta |\\le \\alpha $ and symmetry implies $Rf_{1}(-\\alpha )=Rf_{1}(\\alpha ).$ One wonders, for example, about the relationship between $Rf_{1}(\\alpha )$ and $\\lim _{\\theta \\uparrow \\alpha } Rf(\\theta ).$ Figure: (a) Ω\\Omega (b) The graph of φ\\phi over ∂Ω\\partial \\Omega (c) Ω 1 \\Omega _{1}"
],
[
"Proofs",
"Remark 1 The proofs of these Theorems are similar to those in [9] (and [5]).",
"One difference is that the results in [5], [9] were only concerned with nontangential radial limits at one point, ${\\cal O},$ and so restricting the solution (“$f$ ”) to a subdomain which is tangent to the domain $\\Omega $ at ${\\cal O}$ and therefore assuming $f\\in C^{0}(\\overline{\\Omega }\\setminus \\lbrace {\\cal O} \\rbrace )$ caused no difficulties.",
"Since we wish to show that tangential radial limits also exist and describe the behavior of $f$ on $\\partial \\Omega ,$ we cannot make such simplifying assumptions and so we have to modify the proofs in [5], [9].",
"Proof of Theorem REF : We may assume $\\Omega $ is a bounded domain.",
"Set $S_{0} = \\lbrace ({\\bf x},f({\\bf x})) : {\\bf x} \\in \\Omega \\rbrace .$ From the calculation on page 170 of [25], we see that the area of $S_{0}$ is finite; let $M_{0}$ denote this area.",
"For $\\delta \\in (0,1),$ set $p(\\delta ) = \\sqrt{\\frac{8\\pi M_{0}}{\\ln \\left(\\frac{1}{\\delta }\\right)}}.$ Let $E= \\lbrace (u,v) : u^{2}+v^{2}<1 \\rbrace .$ As in [7], [25], there is a parametric description of the surface $S_{0},$ $Y(u,v) = (a(u,v),b(u,v),c(u,v)) \\in C^{2}(E:{{\\rm I\\hspace{-1.99997pt}R}}^{3}),$ which has the following properties: $\\left(a_{1}\\right)$ $Y$ is a diffeomorphism of $E$ onto $S_{0}$ .",
"$\\left(a_{2}\\right)$ Set $G(u,v)=(a(u,v),b(u,v)),$ $(u,v)\\in E.$ Then $G \\in C^{0}(\\overline{E} : {{\\rm I\\hspace{-1.99997pt}R}}^{2}).$ $\\left(a_{3}\\right)$ Set $\\sigma ({\\bf y})=G^{-1}\\left(\\partial \\Omega \\setminus \\lbrace {\\bf y} \\rbrace \\right);$ then $\\sigma ({\\bf y})$ is a connected arc of $\\partial E$ and $Y$ maps $\\sigma ({\\bf y})$ onto $\\partial \\Omega \\setminus \\lbrace {\\bf y} \\rbrace .$ We may assume the endpoints of $\\sigma ({\\bf y})$ are ${\\bf o}_{1}({\\bf y})$ and ${\\bf o}_{2}({\\bf y}).$ (Note that ${\\bf o}_{1}({\\bf y})$ and ${\\bf o}_{2}({\\bf y})$ are not assumed to be distinct.)",
"$\\left(a_{4}\\right)$ $Y$ is conformal on $E$ : $Y_{u} \\cdot Y_{v} = 0, Y_{u}\\cdot Y_{u} = Y_{v}\\cdot Y_{v}$ on $E$ .",
"$\\left(a_{5}\\right)$ $\\triangle Y := Y_{uu} + Y_{vv} = H^{*}\\left(Y\\right)Y_{u} \\times Y_{v}$ on $E$ .",
"Notice that for each $C\\in {\\rm I\\hspace{-1.99997pt}R},$ $Q(\\psi ^{+}_{j}+C)=Q(\\psi ^{+}_{j})\\le 0$ on $\\Omega \\cap U^{+}_{j}$ and $Q(\\psi ^{-}_{j}+C)=Q(\\psi ^{-}_{j})\\ge 0$ on $\\Omega \\cap U^{-}_{j},$ $j=1,2,$ and so $N(\\psi ^{+}_{j}+C)({\\bf x}) \\le 2H({\\bf x},f({\\bf x}))=Nf({\\bf x}) \\ \\ {\\rm for} \\ \\ {\\bf x}\\in \\Omega \\cap U^{+}_{j}, \\ \\ j=1,2$ and $N(\\psi ^{-}_{j}+C)({\\bf x}) \\ge 2H({\\bf x},f({\\bf x}))=Nf({\\bf x}) \\ \\ {\\rm for} \\ \\ {\\bf x}\\in \\Omega \\cap U^{-}_{j}, \\ \\ j=1,2.$ Let $q$ denote a modulus of continuity for $\\psi ^{\\pm }_{1}$ and $\\psi ^{\\pm }_{2}.$ Let $\\zeta ({\\bf y})=\\partial E\\setminus \\sigma ({\\bf y});$ then $G(\\zeta ({\\bf y}))=\\lbrace {\\bf y}\\rbrace $ and ${\\bf o}_{1}({\\bf y})$ and ${\\bf o}_{2}({\\bf y})$ are the endpoints of $\\zeta ({\\bf y}).$ There exists a $\\delta _{1}>0$ such that if ${\\bf w}\\in E$ and ${\\rm dist}\\left({\\bf w}, \\zeta ({\\bf y})\\right)\\le 2\\delta _{1},$ then $G({\\bf w})\\in \\left(U^{+}_{1}\\cup U^{+}_{2}\\right)\\cap \\left(U^{-}_{1}\\cup U^{-}_{2}\\right).$ Now $T\\psi ^{\\pm }_{j}\\cdot \\nu =\\pm 1$ (in the sense of [3]) almost everywhere on an open subset $\\Upsilon ^{\\pm }_{j}$ of $\\partial U^{\\pm }_{j}$ which contains $\\overline{\\Gamma _{j}};$ there exists a $\\delta _{2}>0$ such that $\\left(\\partial U^{\\pm }_{j} \\setminus \\Upsilon ^{\\pm }_{j}\\right) \\cap \\lbrace {\\bf x}\\in {\\rm I\\hspace{-1.99997pt}R}^{2} : |{\\bf x}-{\\bf y}|\\le 2p(\\delta _{2})\\rbrace =\\emptyset .$ Set $\\delta ^{*}=\\min \\lbrace \\delta _{1},\\delta _{2}\\rbrace $ and $V^{*}= \\lbrace {\\bf w}\\in E : {\\rm dist}({\\bf w},\\zeta ({\\bf y}))<\\delta ^{*} \\rbrace .$ Notice if ${\\bf w}\\in V^{*},$ then $G({\\bf w})\\in U^{+}_{1}\\cup U^{+}_{2}$ and $G({\\bf w})\\in U^{-}_{1}\\cup U^{-}_{2}.$ Claim: $Y$ is uniformly continuous on $V^{*}$ and so extends to a continuous function on $\\overline{V^{*}}.$ Pf: Let $\\epsilon >0.$ Choose $\\delta \\in \\left(0,\\left(\\delta ^{*}\\right)^{2}\\right)$ such that $p(\\delta )+2q(p(\\delta ))<\\epsilon .$ Let ${\\bf w}_{1},{\\bf w}_{2}\\in V^{*}$ with $|{\\bf w}_{1}-{\\bf w}_{2}|<\\delta ;$ then $G({\\bf w}_{1}), G({\\bf w}_{2})\\in \\left(U^{+}_{1}\\cup U^{+}_{2}\\right) \\cap \\left(U^{-}_{1}\\cup U^{-}_{2}\\right) .$ Set $C_{r}({\\bf w}) = \\lbrace {\\bf u} \\in E : |{\\bf u} - {\\bf w}| = r \\rbrace $ and $B_{r}({\\bf w}) = \\lbrace {\\bf u} \\in E : |{\\bf u} - {\\bf w}| < r \\rbrace .$ From the Courant-Lebesgue Lemma (e.g.",
"Lemma $3.1$ in [2]), we see that there exists $\\rho =\\rho (\\delta )\\in \\left(\\delta ,\\sqrt{\\delta }\\right)$ such that the arclength $l_{\\rho }({\\bf w}_{1})$ of $Y(C_{\\rho }({\\bf w}_{1}))$ is less than $p(\\delta ).$ Notice that ${\\bf w}_{2}\\in B_{\\rho (\\delta )}({\\bf w}_{1}).$ Let $k(\\delta )({\\bf w}_{1})= \\inf _{{\\bf u}\\in C_{\\rho (\\delta )}({\\bf w}_{1})}c({\\bf u}) = \\inf _{ {\\bf x}\\in G(C_{\\rho (\\delta )}({\\bf w}_{1})) } f({\\bf x})$ and $m(\\delta )({\\bf w}_{1})= \\sup _{{\\bf u}\\in C_{\\rho (\\delta )}({\\bf w}_{1})}c({\\bf u}) = \\sup _{ {\\bf x}\\in G(C_{\\rho (\\delta )}({\\bf w}_{1})) } f({\\bf x});$ then $m(\\delta )({\\bf w}_{1})-k(\\delta )({\\bf w}_{1})\\le l_{\\rho } < p(\\delta ).$ Fix ${\\bf x}_{0}\\in C^{\\prime }_{\\rho (\\delta )}({\\bf w}_{1}).$ For $j=1,2,$ set $C^{+}_{j}=\\inf _{{\\bf x}\\in U^{+}_{j}\\cap C^{\\prime }_{\\rho (\\delta )}({\\bf w}_{1}) } \\psi ^{+}_{j}({\\bf x}) \\ \\ {\\rm and} \\ \\ C^{-}_{j}=\\sup _{{\\bf x}\\in U^{-}_{j}\\cap C^{\\prime }_{\\rho (\\delta )}({\\bf w}_{1}) } \\psi ^{-}_{j}({\\bf x}).$ Then $\\psi ^{+}_{j}-C^{+}_{j}\\ge 0$ on $U^{+}_{j}\\cap C^{\\prime }_{\\rho (\\delta )}({\\bf w}_{1})$ and $\\psi ^{-}_{j}-C^{-}_{j}\\le 0$ on $U^{-}_{j}\\cap C^{\\prime }_{\\rho (\\delta )}({\\bf w}_{1}).$ Therefore, for $j,l \\in \\lbrace 1,2\\rbrace $ and ${\\bf x}\\in U^{+}_{j}\\cap U^{-}_{l}\\cap C^{\\prime }_{\\rho (\\delta )}({\\bf w}_{1}),$ we have $k(\\delta )({\\bf w}_{1})+\\left(\\psi ^{-}_{l}({\\bf x})-C^{-}_{l}\\right) \\le f({\\bf x}) \\le m(\\delta )({\\bf w}_{1})+\\left(\\psi ^{+}_{j}({\\bf x})-C^{+}_{j}\\right).$ For $j=1,2,$ set $b_{j}^{+}({\\bf x})= m(\\delta )({\\bf w}_{1})+\\left(\\psi ^{+}_{j}({\\bf x})-C^{+}_{j}\\right) \\ \\ \\ \\ {\\rm for} \\ \\ {\\bf x}\\in U^{+}_{j}\\cap \\overline{G\\left(B_{\\rho (\\delta )}({\\bf w}_{1})\\right)}$ and $b_{j}^{-}({\\bf x})= k(\\delta )({\\bf w}_{1})+\\left(\\psi ^{-}_{j}({\\bf x})-C^{-}_{j}\\right) \\ \\ \\ \\ {\\rm for} \\ \\ {\\bf x}\\in U^{-}_{j} \\cap \\overline{G\\left(B_{\\rho (\\delta )}({\\bf w}_{1})\\right)}.$ Now $\\rho (\\delta )<\\sqrt{\\delta }<\\delta ^{*}\\le \\delta _{2};$ notice that if ${\\bf w}\\in \\overline{B_{\\rho (\\delta )}({\\bf w}_{1})},$ then $|{\\bf w}-{\\bf w}_{1}|<\\delta _{2}$ and $|G({\\bf w})-{\\bf y}|<2p(\\delta _{2})$ and thus if ${\\bf x}\\in \\overline{G\\left(B_{\\rho (\\delta )}({\\bf w}_{1})\\right)} \\cap \\partial U_{j}^{\\pm },$ then ${\\bf x}\\in \\Upsilon _{j}^{\\pm }.$ From (REF ), (REF ), the facts that $b_{l}^{-}\\le f$ on $U^{-}_{l} \\cap C^{\\prime }_{\\rho (\\delta )}({\\bf w}_{1})$ and $f\\le b_{j}^{+}$ on $U^{+}_{j} \\cap C^{\\prime }_{\\rho (\\delta )}({\\bf w}_{1})$ for $j,l=1,2,$ and the general comparison principle (Theorem 5.1, [12]), we have (see Figure REF ) $b_{l}^{-}\\le f\\ {\\rm on} \\ U^{-}_{l} \\cap \\overline{G\\left(B_{\\rho (\\delta )}({\\bf w}_{1})\\right)} \\ \\ {\\rm for} \\ \\ l=1,2$ and $f\\le b_{j}^{+}\\ {\\rm on} \\ U^{+}_{j}\\cap \\overline{G\\left(B_{\\rho (\\delta )}({\\bf w}_{1})\\right)} \\ \\ {\\rm for} \\ \\ j=1,2.$ Figure: General comparison principle applied on U 2 ± U^{\\pm }_{2} (left) and U 1 ± U^{\\pm }_{1} (right)Since the diameter of $G\\left(B_{\\rho (\\delta )}({\\bf w}_{1})\\right)\\le p(\\delta ),$ we have $\\left|\\psi ^{\\pm }_{j}({\\bf x})-C^{\\pm }_{j}\\right|\\le q(p(\\delta ))$ for ${\\bf x}\\in U^{\\pm }_{j}\\cap G\\left(B_{\\rho (\\delta )}({\\bf w}_{1})\\right).$ Thus, whenever ${\\bf x}_{1},{\\bf x}_{2}\\in \\overline{G\\left(B_{\\rho (\\delta )}({\\bf w}_{1})\\right)},$ at least one of the cases (a) ${\\bf x}_{1},{\\bf x}_{2} \\in U^{+}_{1}\\cap U^{-}_{1},$ (b) ${\\bf x}_{1},{\\bf x}_{2} \\in U^{+}_{2}\\cap U^{-}_{2},$ (c) ${\\bf x}_{1} \\in U^{+}_{1}$ and ${\\bf x}_{2}\\in U^{-}_{2}$ or (d) ${\\bf x}_{1} \\in U^{+}_{2}$ and ${\\bf x}_{2}\\in U^{-}_{1}$ holds.",
"Since $c({\\bf w})=f\\left(G({\\bf w})\\right),$ $G({\\bf w}_{1})\\in U^{+}_{i}\\cap U^{-}_{j}$ for some $i=1,2$ and $j=1,2,$ and $G({\\bf w}_{2})\\in U^{+}_{l}\\cap U^{-}_{n}$ for some $l=1,2$ and $n=1,2,$ we have $b_{j}^{-}\\left(G({\\bf w}_{1})\\right)-b_{l}^{+}\\left(G({\\bf w}_{2})\\right) \\le c({\\bf w}_{1})-c({\\bf w}_{2}) \\le b_{i}^{+}\\left(G({\\bf w}_{1})\\right)-b_{n}^{-}\\left(G({\\bf w}_{2})\\right)$ or $-\\left[m(\\delta )({\\bf w}_{1})-k(\\delta )({\\bf w}_{1}) +\\left(\\psi ^{+}_{l}(G({\\bf w}_{2}))-C^{+}_{l}\\right) - \\left(\\psi ^{-}_{j}(G({\\bf w}_{1}))+C^{-}_{j}\\right)\\right]$ $\\le c({\\bf w}_{1})-c({\\bf w}_{2}) \\le $ $\\left[m(\\delta )({\\bf w}_{1})-k(\\delta )({\\bf w}_{1}) +\\left(\\psi ^{+}_{i}(G({\\bf w}_{1}))-C^{+}_{i}\\right) - \\left(\\psi ^{-}_{n}(G({\\bf w}_{2}))+C^{-}_{n}\\right)\\right].$ Since $|\\psi ^{\\pm }_{j}(G({\\bf w}))-C^{\\pm }_{j}| \\le q(p(\\delta ))$ for ${\\bf w}\\in B_{\\rho (\\delta )}({\\bf w}_{1})\\cap U^{\\pm }_{j},$ we have $|c({\\bf w}_{1})-c({\\bf w}_{2})| \\le p(\\delta )+2q(p(\\delta ))<\\epsilon .$ Thus $c$ is uniformly continuous on $V^{*}$ and, since $G \\in C^{0}(\\overline{E} : {{\\rm I\\hspace{-1.99997pt}R}}^{2}),$ we see that $Y$ is uniformly continuous on $V^{*}.$ Therefore $Y$ extends to a continuous function, still denote $Y,$ on $\\overline{V^{*}}.$ $\\Box $ Notice that $\\lim _{\\Gamma _{1}\\ni {\\bf x}\\rightarrow {\\bf y}} f({\\bf x})=\\lim _{\\partial E\\ni {\\bf w}\\rightarrow {\\bf o}_{1}({\\bf y})} c({\\bf w})=c({\\bf o}_{1}({\\bf y}))$ and $\\lim _{\\Gamma _{2}\\ni {\\bf x}\\rightarrow {\\bf y}} f({\\bf x})=\\lim _{\\partial E\\ni {\\bf w}\\rightarrow {\\bf o}_{2}({\\bf y})} c({\\bf w})=c({\\bf o}_{2}({\\bf y}))$ and so, with $z_{1}=c({\\bf o}_{1}({\\bf y}))$ and $z_{2}=c({\\bf o}_{2}({\\bf y})),$ we see that (REF ) holds.",
"Now we need to consider two cases: $\\left(A\\right)$ ${\\bf o}_{1}({\\bf y})= {\\bf o}_{2}({\\bf y}).$ $\\left(B\\right)$ ${\\bf o}_{1}({\\bf y})\\ne {\\bf o}_{2}({\\bf y}).$ These correspond to Cases 5 and 3 respectively in Step 1 of the proof of Theorem 1 of [25].",
"Case (A): Suppose ${\\bf o}_{1}({\\bf y})= {\\bf o}_{2}({\\bf y});$ set ${\\bf o}={\\bf o}_{1}({\\bf y})= {\\bf o}_{2}({\\bf y}).$ Then $f$ extends to a function in $C^{0}\\left(\\Omega \\cup \\lbrace {\\bf y}\\rbrace \\right)$ and case (i) of Theorem REF holds.",
"Pf: Notice that $G$ is a bijection of $E\\cup \\lbrace {\\bf o}\\rbrace $ and $\\Omega \\cup \\lbrace {\\bf y}\\rbrace .$ Thus we may define $f=c\\circ G^{-1},$ so $f\\left(G({\\bf w})\\right)=c({\\bf w})$ for ${\\bf w}\\in E\\cup \\lbrace {\\bf o}\\rbrace ;$ this extends $f$ to a function defined on $\\Omega \\cup \\lbrace {\\bf y}\\rbrace .$ Let $\\lbrace \\delta _{i}\\rbrace $ be a decreasing sequence of positive numbers converging to zero and consider the sequence of open sets $\\lbrace G(B_{\\rho (i)}({\\bf o}))\\rbrace $ in $\\Omega ,$ where $\\rho (i)=\\rho (\\delta _{i}({\\bf o})).$ Now ${\\bf y}\\notin G(C_{\\rho (i)}({\\bf o}))$ and so there exist $\\sigma _{i}>0$ such that $P(i) =\\lbrace {\\bf x}\\in \\Omega : |{\\bf x}-{\\bf y}|<\\sigma _{i}\\rbrace \\subset G(B_{\\rho (i)}({\\bf o}))$ for each $i\\in {\\rm I\\hspace{-1.99997pt}N}.$ Thus if ${\\bf x}\\in P(i),$ we have $|f({\\bf x}) - f({\\bf y})|<p(\\delta _{i})+2q(p(\\delta _{i})).$ The continuity of $f$ at ${\\bf y}$ follows from this.",
"$\\Box $ Case (B): Suppose ${\\bf o}_{1}({\\bf y})\\ne {\\bf o}_{2}({\\bf y}).$ Then one of case (ii) or (iii) of Theorem REF holds.",
"Pf: As at the end of Step 1 of the proof of Theorem 1 of [25], we define $X:B\\rightarrow {\\rm I\\hspace{-1.99997pt}R}^{3}$ by $X=Y\\circ g$ and $K:B\\rightarrow {\\rm I\\hspace{-1.99997pt}R}^{2}$ by $K=G\\circ g,$ where $B=\\lbrace (u,v)\\in {\\rm I\\hspace{-1.99997pt}R}^{2} : u^{2}+v^{2}<1, \\ v>0\\rbrace $ and $g:\\overline{B}\\rightarrow \\overline{E}$ is either a conformal or an indirectly conformal (or anticonformal) map from $\\overline{B}$ onto $\\overline{E}$ such that $g(1,0)= {\\bf o}_{1}({\\bf y}),$ $g(-1,0)= {\\bf o}_{2}({\\bf y})$ and $g(u,0)\\in {\\bf o}_{1}({\\bf y}){\\bf o}_{2}({\\bf y})$ for each $u\\in [-1,1],$ where ${\\bf ab}$ denotes the (appropriate) choice of arc in $\\partial E$ with ${\\bf a}$ and ${\\bf b}$ as endpoints.",
"Notice that $K(u,0)={\\bf y}$ for $u\\in [-1,1].$ Set $x=a\\circ g,$ $y=b\\circ g$ and $z=c\\circ g,$ so that $X(u,v)=(x(u,v),y(u,v),z(u,v))$ for $(u,v)\\in B.$ Now, from Step 2 of the proof of Theorem 1 of [25], $X\\in C^{0}\\left(\\overline{B}:{\\rm I\\hspace{-1.99997pt}R}^{3}\\right)\\cap C^{1,\\iota }\\left(B\\cup \\lbrace (u,0):-1<u<1\\rbrace :{\\rm I\\hspace{-1.99997pt}R}^{3}\\right)$ for some $\\iota \\in (0,1)$ and $X(u,0)=({\\bf y},z(u,0))$ cannot be constant on any nondegenerate interval in $[-1,1].$ Define $\\Theta (u)= {\\rm arg}\\left( x_{v}(u,0)+iy_{v}(u,0) \\right).$ From equation (12) of [25], we see that $\\alpha _{1}=\\lim _{u\\downarrow -1} \\Theta (u) \\ \\ \\ \\ {\\rm and} \\ \\ \\ \\ \\alpha _{2}=\\lim _{u\\uparrow 1} \\Theta (u);$ here $\\alpha _{1}<\\alpha _{2}.$ As in Steps 2-5 of the proof of Theorem 1 of [25], we see that $Rf(\\theta )$ exists when $\\theta \\in \\left(\\alpha _{1},\\alpha _{2}\\right),$ $\\overline{G^{-1}\\left( L(\\alpha _{2}) \\right)} \\cap \\partial E = \\lbrace {\\bf o}_{1}({\\bf y}) \\rbrace \\ (\\& \\ \\overline{K^{-1}\\left( L(\\alpha _{2}) \\right)} \\cap \\partial B = \\lbrace (1,0) \\rbrace ) \\ {\\rm when} \\ \\alpha _{2}<\\beta ({\\bf y})$ $\\overline{G^{-1}\\left( L(\\alpha _{1}) \\right)} \\cap \\partial E = \\lbrace {\\bf o}_{2}({\\bf y}) \\rbrace \\ (\\& \\ \\overline{K^{-1}\\left( L(\\alpha _{1}) \\right)} \\cap \\partial B = \\lbrace (-1,0) \\rbrace ) \\ {\\rm when} \\ \\alpha _{1}>\\alpha ({\\bf y})$ where $L(\\theta )= \\lbrace {\\bf y}+(r\\cos (\\theta ),r\\sin (\\theta ))\\in \\Omega : 0<r<\\delta ^{*} \\rbrace ,$ and one of the following cases holds: (a) $Rf$ is strictly increasing or strictly decreasing on $(\\alpha _{1}, \\alpha _{2})$ .",
"(b) There exist $\\alpha _{L}, \\alpha _{R}$ so that $\\alpha _{1} < \\alpha _{L} < \\alpha _{R} < \\alpha _{2},$ $\\alpha _{R}= \\alpha _{L} + \\pi $ , and $Rf$ is constant on $[ \\alpha _{L}, \\alpha _{R}]$ and either increasing on $(\\alpha _{1}, \\alpha _{L}]$ and decreasing on $[\\alpha _{R}, \\alpha _{2})$ or decreasing on $(\\alpha _{1}, \\alpha _{L}]$ and increasing on $[\\alpha _{R}, \\alpha _{2})$ .",
"We may argue as in Case A to see that $f$ is uniformly continuous on $\\Omega ^{+} =\\lbrace {\\bf y}+(r\\cos (\\theta ),r\\sin (\\theta ))\\in \\Omega : 0<r<\\delta , \\alpha _{2}\\le \\theta <\\beta ({\\bf y})+\\epsilon \\rbrace $ and $f$ is uniformly continuous on $\\Omega ^{-} =\\lbrace {\\bf y}+(r\\cos (\\theta ),r\\sin (\\theta ))\\in \\Omega : 0<r<\\delta , \\alpha ({\\bf y})-\\epsilon < \\theta \\le \\alpha _{1}\\rbrace $ for some small $\\epsilon >0$ and $\\delta >0,$ since $G$ is a bijection of $E\\cup \\lbrace {\\bf o}_{1}({\\bf y})\\rbrace $ and $\\Omega \\cup \\lbrace {\\bf y}\\rbrace $ and a bijection of $E\\cup \\lbrace {\\bf o}_{2}({\\bf y})\\rbrace $ and $\\Omega \\cup \\lbrace {\\bf y}\\rbrace .$ (Also see [5], [10].)",
"Theorem REF then follows, as in [9], from Steps 2-5 of the proof of Theorem 1 of [25] (replacing Step 3 with [6]).",
"$\\Box $ Proof of Theorem REF : The proof of this theorem is essentially the same as that of Theorem REF .",
"Proof of Corollary REF : From pp.1064-5 in [8], we see that there exist upper and lower Bernstein pairs $\\left(U^{\\pm },\\psi ^{\\pm }\\right)$ for $(\\Gamma ,H^{*}).$ From Theorem REF , we see that the radial limits $Rf(\\theta ,{\\bf y})$ exist for each $\\theta \\in [\\alpha ({\\bf y}),\\beta ({\\bf y})].$ (Since $\\beta ({\\bf y})-\\alpha ({\\bf y})=\\pi ,$ case (iii) of Theorem REF cannot occur.)",
"Set $z_{1}=Rf(\\alpha ({\\bf y}),{\\bf y}),$ $z_{2}=Rf(\\beta ({\\bf y}),{\\bf y})$ and $z_{3}=\\phi ({\\bf y}).$ If $z_{1}=z_{2},$ then case (i) of Theorem REF holds.",
"(If $f$ is symmetric with respect to a line through ${\\bf y},$ then $z_{1}=z_{2}$ and we are done.)",
"Suppose otherwise that $z_{1}\\ne z_{2};$ we may assume that $z_{1}<z_{3}$ and $z_{1}<z_{2}.$ Then there exist $\\alpha _{1}, \\alpha _{2}\\in [\\alpha ({\\bf y}),\\beta ({\\bf y})]$ with $\\alpha _{1}<\\alpha _{2}$ such that $Rf(\\theta ,{\\bf y}) \\ \\ {\\rm is}\\left\\lbrace \\begin{array}{ccc}{\\rm constant}(=z_{1}) & {\\rm for} & \\alpha ({\\bf y})\\le \\theta \\le \\alpha _{1}\\\\{\\rm strictly \\ increasing} & {\\rm for} & \\alpha _{1}\\le \\theta \\le \\alpha _{2}\\\\{\\rm constant}(=z_{2}) & {\\rm for} & \\alpha _{2}\\le \\theta \\le \\beta ({\\bf y}).",
"\\\\\\end{array}\\right.$ From Theorem REF , we see that $Rf(\\theta ,{\\bf y})$ exists for each $y\\in \\Gamma $ and $\\theta \\in [\\alpha ({\\bf y}),\\beta ({\\bf y})]$ and $f$ is continuous on $\\Omega \\cup \\Gamma \\setminus \\Upsilon $ for some countable subset $\\Upsilon $ of $\\Gamma .$ Let $z_{0}\\in \\left(z_{1},\\min \\lbrace z_{2},z_{3}\\rbrace \\right)$ and $\\theta _{0}\\in (\\alpha _{1},\\alpha _{2})$ satisfy $Rf(\\theta _{0},{\\bf y})=z_{0}.$ Let $C_{0}\\subset \\Omega $ be the $z_{0}-$ level curve of $f$ which has ${\\bf y}$ and a point ${\\bf y}_{0}\\in \\partial \\Omega \\setminus \\lbrace {\\bf y}\\rbrace $ as endpoints.",
"Let ${\\bf y}_{1}\\in \\partial _{{\\bf y}}^{1}\\Omega \\cap \\Gamma \\setminus \\Upsilon $ and ${\\bf y}_{2}\\in C_{0}$ such that the (open) line segment $L$ joining ${\\bf y}_{1}$ and ${\\bf y}_{2}$ is entirely contained in $\\Omega .$ Let $M=\\inf _{L} f,$ $\\Pi $ be the plane containing $({\\bf y},z_{0})$ and $L\\times \\lbrace M\\rbrace ,$ and $h$ be the affine function on ${\\rm I\\hspace{-1.99997pt}R}^{2}$ whose graph is $\\Pi .$ Let $\\Omega _{0}$ be the component of $\\Omega \\setminus \\left(C_{0}\\cup L\\right)$ whose closure contains $B_{\\delta }({\\bf y})\\cap \\partial _{{\\bf y}}^{1}\\Omega $ for some $\\delta >0.$ Then there is a curve $C\\subset \\Omega _{0}$ on which $f=h$ whose endpoints are ${\\bf y}_{3}$ and ${\\bf y},$ for some ${\\bf y}_{3}\\in \\partial _{{\\bf y}}^{1}\\Omega $ between ${\\bf y}_{1}$ and ${\\bf y},$ such that $h>f$ in $\\Omega _{1},$ where $\\Omega _{1}\\subset \\Omega _{0}$ is the open set bounded by $C$ and the portion of $\\partial _{{\\bf y}}^{1}\\Omega $ between ${\\bf y}$ and ${\\bf y}_{3}.$ Notice that $h<f$ in $L\\cup C_{0}.$ (In Figure REF , on the left, $\\lbrace \\left({\\bf x},h\\left({\\bf x}\\right)\\right): {\\bf x}\\in C\\rbrace $ is in red, $L$ is in dark blue, $C_{0}$ is in yellow, and the light blue region is a portion of $\\partial _{{\\bf y}}^{1}\\Omega \\times {\\rm I\\hspace{-1.99997pt}R},$ and, on the right, $\\Omega _{0}$ is in light green and $\\partial _{{\\bf y}}^{2}\\Omega $ is in magenta.)",
"Now let $g\\in C^{2}(\\Omega )$ be defined by $g=f$ on $\\Omega \\setminus \\overline{\\Omega _{1}}$ and $g=h$ on $\\Omega _{1}$ and observe that $J(g)<J(f),$ which contradicts the fact that $f$ minimizes $J.$ Thus it must be the case that $z_{1}=z_{2},$ case (i) of Theorem REF holds and $f$ is continuous at ${\\bf y}.$ $\\Box $ Figure: Side View of Π∩Ω×IR\\Pi \\cap \\left(\\Omega \\times {\\rm I\\hspace{-1.99997pt}R}\\right) (left) and Ω 1 \\Omega _{1} (right)Remark 2 Corollary REF can be generalized to minimizers of $J(u)=\\int _{\\Omega } \\sqrt{1+|Du|^{2}} d{\\bf x} + \\int _{\\Omega } \\left( \\int _{c}^{u({\\bf x})} H({\\bf x},t) \\ dt \\right) d{\\bf x}+ \\int _{\\partial \\Omega } |u-\\phi | ds$ for $u\\in BV(\\Omega )$ and the conclusion remains the same; here $c$ is a reference height (e.g.",
"$c=0$ ).",
"In the proof of Corollary REF , the only change is a replacement of the plane $\\Pi $ with an appropriate surface (e.g.",
"a portion of a sphere) over a subdomain like $\\Omega _{1}$ such that the test function $g$ satisfies $J(g)<J(f).$ Proof of Example REF : By Corollary REF , $f$ is continuous on $\\Omega \\cup \\lbrace (0,0)\\rbrace .$ Clearly $f$ is continuous at $(x,y)$ when $(x+1)^2+y^2=\\cosh ^{2}(1).$ By [30], $f$ is continuous at $(x,y)$ when $(x+1)^2+y^2=1$ and $(x,y)\\ne (0,0).$ The parametrization (REF ) of the graph of $f$ (restricted to $\\Omega \\setminus \\lbrace (x,0):x<0\\rbrace $ ) satisfies $Y\\in C^{0}(\\overline{E}).$ Notice that $\\zeta ((0,0))=\\lbrace {\\bf o} \\rbrace $ (since $\\beta ((0,0))-\\alpha ((0,0))=\\pi $ and $z_{1}=z_{2}$ ) for some ${\\bf o}\\in \\partial E.$ Suppose $G$ in $(a_{2})$ is not one-to-one.",
"Then there exists a nondegenerate arc $\\zeta \\subset \\partial E$ such that $G(\\zeta )=\\lbrace {\\bf y}_{1}\\rbrace $ for some ${\\bf y}_{1}\\in \\partial \\Omega $ and therefore $f$ is not continuous at ${\\bf y}_{1},$ which is a contradiction.",
"Thus $f=g\\circ G^{-1}$ and so $f\\in C^{0}\\left(\\overline{\\Omega }\\right).$ (The continuity of $G^{-1}$ follows, for example, from Lemma $3.1$ in [2].)",
"$\\Box $ Proof of Theorem REF : The proof of Theorem 2 of [9] uses unduloids as Bernstein functions (i.e.",
"comparison surfaces) on subdomains of $\\Omega $ (see Figure 7 of [9]).",
"The proof of Theorem REF is essentially the same, using the Bernstein pairs $\\left(U^{\\pm },\\psi ^{\\pm }\\right)$ rather than unduloids, staying on $\\partial _{\\bf y}^{2}\\Omega $ rather than on an arc of a circle inside $\\Omega ,$ and arguing as in the proof of Theorem REF .",
"$\\Box $ Proof of Theorem REF : The proof of Theorem 2 of [10] uses portions of tori as Bernstein functions (i.e.",
"comparison surfaces) on subdomains of $\\Omega $ (see Figure 7 of [10]).",
"The proof of Theorem REF is essentially the same, using the Bernstein pairs $\\left(U^{\\pm },\\psi ^{\\pm }\\right)$ rather than tori, staying on $\\partial _{\\bf y}^{2}\\Omega $ rather than on an arc of a circle inside $\\Omega ,$ and arguing as in the proof of Theorem REF .",
"$\\Box $ Proof of Theorem REF : The proof of Theorem 1 of [5] uses Theorem 2 of [10]; the proof of Theorem REF is essentially the same, using Theorem REF in place of Theorem 2 of [10] and arguing as in the proof of Theorem REF .",
"$\\Box $"
],
[
"Bernstein Functions",
"The value of Theorems REF - REF is dependent on the existence of Bernstein functions.",
"The results of [8] provide Bernstein pairs for minimal surfaces.",
"Proposition 1 Let $a<b,$ $\\lambda \\in (0,1),$ $\\psi \\in C^{2,\\lambda }([a,b])$ and $\\Gamma =\\lbrace (x,\\psi (x))\\in {\\rm I\\hspace{-1.99997pt}R}^{2} : x\\in [a,b]\\rbrace $ such that $\\psi ^{\\prime }(x)< 0$ for $x\\in [a,b],$ $\\psi ^{\\prime \\prime }(x)<0$ for $x\\in [a,b]\\setminus J,$ there exist $C_{1}>0$ and $\\epsilon _{1}>0$ such that if $\\bar{x}\\in J$ and $|x-\\bar{x}|<\\epsilon _{1},$ then $\\psi ^{\\prime \\prime }(x)\\le -C_{1}|x-\\bar{x}|^{\\lambda },$ and $t\\psi (x_{1})+(1-t)\\psi (x_{2})<\\psi \\left(tx_{1}+(1-t)x_{2}\\right)$ for each $t\\in (0,1)$ and $x_{1},x_{2}\\in [a,b]$ with $x_{1}\\ne x_{2},$ where $J$ is a finite subset of $(a,b).$ Then there exists an open set $U\\subset {\\rm I\\hspace{-1.99997pt}R}^{2}$ with $\\Gamma \\subset \\partial U$ and a function $h\\in C^{2}\\left(U\\right)\\cap C^{0}\\left(\\overline{U}\\right)$ such that $\\partial U$ is a closed, $C^{2,\\lambda }$ curve, $\\Gamma $ lies below $U$ in ${\\rm I\\hspace{-1.99997pt}R}^{2}$ (i.e.",
"the exterior unit normal $\\nu =(\\nu _{1}(x),\\nu _{2}(x))$ to $\\partial U$ satisfies $\\nu _{2}(x)<0$ for $a\\le x\\le b$ ), $Nh=0$ in $U$ and (REF ) holds for each ${\\bf y}\\in \\Gamma ,$ where $\\hat{\\nu }$ is a continuous extension of $\\nu $ to a neighborhood of $\\Gamma .$ Proof: We may assume that $a,b>0.$ There exists $c>b$ and $k\\in C^{2,\\lambda }([-c,c])$ with $k(-x)=k(x)$ for $x\\in [0,c]$ such that $k(x)=-\\psi (x)$ for $x\\in [a,b],$ $k^{\\prime \\prime }(x)>0$ for $x\\in [-c,c]\\setminus J,$ where $J$ is a finite set, $k^{\\prime \\prime }(0)>0,$ and the set $K=\\lbrace (x,k(x))\\in {\\rm I\\hspace{-1.99997pt}R}^{2} : x\\in [-c,c]\\rbrace $ is strictly concave (i.e.",
"$tk(x_{1})+(1-t)k(x_{2})>k\\left(tx_{1}+(1-t)x_{2}\\right)$ for each $t\\in (0,1)$ and $x_{1},x_{2}\\in [-c,c]$ with $x_{1}\\ne x_{2}$ ).",
"From [8] (pp.1063-5), we can construct a domain $\\Omega (K,l)$ such that $K\\subset \\partial \\Omega (K,l)$ and $\\Omega (K,l)$ lies below $K$ (i.e.",
"the outward unit normal to $\\Omega (K,l)$ at $(x,k(x))$ is $\\nu (x)=\\frac{\\left(-k^{\\prime }(x),1\\right)}{\\sqrt{1+\\left(k^{\\prime }(x)\\right)^{2}}};$ see Figure 4 of [8]) and a function $F^{+}\\in C^{2}\\left(\\Omega (K,l)\\right)\\cap C^{0}\\left(\\overline{\\Omega (K,l)}\\right)$ such that $\\mu ({\\bf x})\\mathrel {\\overset{\\makebox{[}0pt]{\\mbox{\\normalfont \\tiny \\sffamily def}}}{=}}\\frac{\\left(\\nabla F^{+}({\\bf x}),-1\\right)}{\\sqrt{1+|\\nabla F^{+}({\\bf x})|^{2}}}, \\ \\ \\ {\\bf x}\\in \\Omega (K,l),$ extends continuously to a function on $\\Omega (K,l)\\cup K$ and $\\mu (x,k(x))\\cdot \\nu (x)=1$ for $x\\in [-c,c].$ Now let $V$ be an open subset of $\\Omega $ with $C^{2,\\lambda }$ boundary such that $\\lbrace (x,-\\psi (x)) : x\\in [a,b]\\rbrace \\subset \\partial V$ and $\\partial V\\cap \\left(\\partial \\Omega (K,l)\\setminus K\\right)=\\emptyset $ and then let $U=\\lbrace (x,-y) : (x,y)\\in V\\rbrace $ and $h(x,y)=F^{+}(x,-y)$ for $(x,y)\\in \\overline{U}.$ $\\Box $ Remark 3 Let $\\Omega \\subset {\\rm I\\hspace{-1.99997pt}R}^{2}$ be an open set, $\\Gamma \\subset \\partial \\Omega $ be a $C^{2,\\lambda }$ curve and ${\\bf y}\\in \\Gamma $ be a point at which we wish to have upper and lower Bernstein pairs for $H\\equiv 0.$ Let $\\Sigma \\subset \\Gamma $ be the intersection of $\\partial \\Omega $ with a neighborood of ${\\bf y}$ and suppose there is a rigid motion $\\zeta :{\\rm I\\hspace{-1.99997pt}R}^{2}\\rightarrow {\\rm I\\hspace{-1.99997pt}R}^{2}$ such that $\\zeta \\left(\\Sigma \\right)$ and $\\zeta \\left(\\Omega \\right)$ satisfy the hypotheses of Proposition REF .",
"Then $\\left(\\zeta ^{-1}(U),h\\circ \\zeta \\right)$ will be an upper Bernstein pair for $\\Sigma $ and $H\\equiv 0$ and $\\left(\\zeta ^{-1}(U),-h\\circ \\zeta \\right)$ will be a lower Bernstein pair for $\\Sigma $ and $H\\equiv 0.$ When $H({\\bf x},z)$ is independent of $z,$ the existence of (bounded) Bernstein functions is tied to boundary curvature conditions; in Theorem 3.1 of [15] (and Theorem 6.6 of [12]), we see that Proposition 2 Suppose $\\Omega $ is a $C^{2}$ domain in ${\\rm I\\hspace{-1.99997pt}R}^{2}$ such that $|\\int \\int _{A} H({\\bf x}) d{\\bf x}| < \\int |D\\chi _{A}| \\ \\ \\ {\\rm for \\ all\\ } A\\subset \\Omega , \\ A\\ne \\emptyset ,\\Omega $ and $\\int \\int _{\\Omega } H({\\bf x}) d{\\bf x} = \\int |D\\chi _{\\Omega }|;$ that is, $\\Omega $ is an extremal domain.",
"Let ${\\bf y}\\in \\partial \\Omega $ and suppose $\\Lambda ({\\bf y})< 2H({\\bf y}),$ where $\\Lambda ({\\bf y})$ is the (signed) curvature of $\\partial \\Omega $ at ${\\bf y}$ with respect to the interior normal direction.",
"Then the (unique up to vertical translations) solution $g$ of $Ng({\\bf x})=H({\\bf x})$ for ${\\bf x}\\in \\Omega $ is bounded and continuous in $W=\\overline{\\Omega }\\cap B_{\\epsilon }({\\bf y}),$ $Tg$ extends continuously to a function on $W$ and $Tg({\\bf x})=\\nu ({\\bf x})$ for each ${\\bf x}\\in B_{\\epsilon }({\\bf y}) \\cap \\partial \\Omega $ for some $\\epsilon >0,$ where $\\nu $ is the exterior unit normal to $\\Omega .$ Using Proposition REF and a similar procedure to that in the proof of Proposition REF , we can obtain Bernstein pairs near ${\\bf y}$ when $\\partial \\Omega \\cap B_{\\epsilon }({\\bf y})$ is a subset of the boundary of an extremal domain $W$ for some $\\epsilon >0$ such that $\\Omega $ and $W$ are on the same side of $\\partial \\Omega \\cap B_{\\epsilon }({\\bf y})$ and the boundary curvature condition $\\Lambda _{W}({\\bf y})< 2|H({\\bf y})|$ is satisfied.",
"In the same manner, we can obtain Bernstein pairs near ${\\bf y},$ illustrated in Figure REF by the sets $U^{\\pm }_{1}$ and $U^{\\pm }_{2},$ when ${\\partial }_{\\bf y}^{1}\\Omega \\cap B_{\\epsilon }({\\bf y})$ and ${\\partial }_{\\bf y}^{2}\\Omega \\cap B_{\\epsilon }({\\bf y})$ are subsets of the boundaries of extremal domains $W_{1}$ and $W_{2}$ for some $\\epsilon >0,$ $\\Omega $ and $W_{j}$ are on the same side of ${\\partial }_{\\bf y}^{j}\\Omega \\cap B_{\\epsilon }({\\bf y})$ for $j=1,2,$ $\\Lambda _{W_{1}}({\\bf y})< 2|H({\\bf y})|$ and $\\Lambda _{W_{2}}({\\bf y})< 2|H({\\bf y})|,$ where $\\Lambda _{W_{j}}({\\bf y})$ denotes (signed) curvature of $\\partial W_{j}.$ Remark 4 In Proposition REF , the sets $A$ are Caccioppoli sets; that is, Borel sets such that the distributional (first) derivatives of the characteristic function $\\chi _{A}$ of $A$ are Radon measures.",
"The notation $A\\ne \\emptyset ,\\Omega $ means that neither $A$ nor $\\Omega \\setminus A$ has (two-dimensional) measure zero and the notation $\\int |D\\chi _{\\Omega }|$ means the total variation of $\\chi _{A}\\in BV(\\Omega )$ (e.g.",
"§6.3 of [12]).",
"Determining when hypothesis (REF ) is satisfied can be difficult; Giusti includes an Appendix in [15] which discusses the case of constant $H.$ We may use §14.4 of [14] (also see Corollary 14.13) to obtain Bernstein functions in a neighborhood $U$ of a point ${\\bf y}\\in \\Gamma $ when $\\Gamma \\subset \\partial \\Omega $ is a $C^{2}$ curve satisfying $\\Lambda ({\\bf x})<2|H({\\bf x})|$ for ${\\bf x}\\in \\Gamma \\cap U$ and $H\\in C^{0}\\left(\\overline{U\\cap \\Omega }\\right)$ is either non-positive or non-negative in $U\\cap \\Omega .$ Lemma 1 Suppose $\\Omega $ is a $C^{2,\\lambda }$ domain in ${\\rm I\\hspace{-1.99997pt}R}^{2}$ for some $\\lambda \\in (0,1).$ Let ${\\bf y}\\in \\partial \\Omega $ and $\\Lambda ({\\bf y})$ denote the (signed) curvature of $\\partial \\Omega $ at ${\\bf y}$ with respect to the interior normal direction (i.e.",
"$-\\nu $ ).",
"Suppose $\\Lambda ({\\bf y})< 2|H({\\bf y})|$ and $H\\in C^{0}\\left(\\overline{U\\cap \\Omega }\\right)$ is either non-positive or non-negative in $U\\cap \\Omega ,$ where $U$ is some neighborhood of ${\\bf y}.$ Then there exist $\\delta >0$ and upper and lower Bernstein pairs $\\left(U^{\\pm },\\psi ^{\\pm }\\right)$ for $(\\Gamma ,H),$ where $\\Gamma =B_{\\delta }({\\bf y})\\cap {\\partial }\\Omega .$ Proof: There exists $\\delta _{1}>0$ such that $B_{\\delta _{1}}({\\bf y})\\subset U$ and $\\Lambda ({\\bf x})< 2|H({\\bf x})|$ for each ${\\bf x}\\in \\partial \\Omega \\cap B_{\\delta _{1}}({\\bf y}).$ There exists a $\\delta _{2}\\in (0,\\delta _{1}/2)$ such that $\\Lambda _{0}\\mathrel {\\overset{\\makebox{[}0pt]{\\mbox{\\normalfont \\tiny \\sffamily def}}}{=}}\\sup \\lbrace \\Lambda ({\\bf x}) : {\\bf x}\\in \\partial \\Omega \\cap B_{\\delta _{2}}({\\bf y})\\rbrace <\\inf \\lbrace 2|H({\\bf x})| : {\\bf x}\\in \\partial \\Omega \\cap B_{\\delta _{2}}({\\bf y})\\rbrace \\mathrel {\\overset{\\makebox{[}0pt]{\\mbox{\\normalfont \\tiny \\sffamily def}}}{=}}2H_{0}.$ If $\\Lambda _{0}>0,$ set $R=\\frac{1}{\\Lambda _{0}};$ otherwise let $R$ be a small positive number.",
"Now let $W$ be a $C^{2,\\lambda }$ domain in ${\\rm I\\hspace{-1.99997pt}R}^{2}$ such that $\\partial \\Omega \\cap B_{\\delta _{1}}({\\bf y})\\subset \\partial W,$ $\\Omega $ and $W$ lie on the same side of $\\partial \\Omega \\cap B_{\\delta _{1}}({\\bf y})$ and $W$ satisfies an interior sphere condition of radius $R$ at each point of $\\partial \\Omega \\cap B_{\\delta _{2}}({\\bf y}).$ Continuously extend $H$ outside $U$ to $\\overline{W}$ in such a manner that $H$ is either non-positive or non-negative in $W.$ From inequality (14.73) of [14], there exists $L>0$ such that $u({\\bf x})-u_{0}({\\bf x})\\le L \\ \\ \\ {\\rm for} \\ {\\bf x}\\in \\partial W \\cap B_{\\delta _{2}}({\\bf y}),$ where $u$ is any solution of (REF ) in $W$ and $u_{0}({\\bf x})=\\sup \\lbrace u({\\bf t}) : {\\bf t}\\in \\partial W\\setminus B_{R}({\\bf x}) \\rbrace .$ We may assume $2\\delta _{2}<R$ and set $u^{*}=\\sup \\lbrace u({\\bf t}) : {\\bf t}\\in \\partial W\\setminus B_{R-\\delta _{2}}({\\bf y}) \\rbrace .$ Then $u_{0}({\\bf x})\\le u^{*}$ for each ${\\bf x}\\in \\partial W \\cap B_{\\delta _{2}}({\\bf y})$ and $u\\le L+u^{*}$ on $\\partial \\Omega \\cap B_{\\delta _{2}}({\\bf y}).$ Now let $\\phi \\in C^{\\infty }(\\partial W)$ such that $\\phi =0$ on $\\partial W\\setminus B_{R-\\delta _{2}}({\\bf y})$ and $\\phi >L$ on $\\partial W \\cap B_{\\delta _{2}}({\\bf y})$ and let $h\\in C^{2}(W)$ be the solution of (REF )-() in $W$ with Dirichlet data $\\phi .$ (Just as [14] ignores in Theorem 14.11 the question of whether $u=\\phi $ on $\\partial \\Omega \\setminus B_{R}({\\bf y}),$ we may assume that $W$ satisfies curvature conditions (i.e.",
"$\\Lambda _{W}\\ge 2|H|$ ) on $\\partial W\\setminus B_{R-\\delta _{2}}({\\bf y})$ so that $h=\\phi $ on $\\partial \\Omega \\setminus B_{R}({\\bf y})$ and so $h^{*}=0.$ ) It then follows (e.g.",
"[1]) that $h\\in C^{0}(\\overline{W})$ and $\\frac{\\partial h}{\\partial \\nu }=+\\infty \\ \\ \\ \\ {\\rm on} \\ \\ B_{\\delta _{2}}({\\bf y})\\cap {\\partial }\\Omega .$ Thus $h$ is an upper Bernstein function.",
"The existence of a lower Bernstein function is similar.",
"$\\Box $ Remark 5 In a similar manner, given ${\\bf y}\\in \\partial \\Omega $ we can establish the existence of upper and lower Bernstein pairs for the intersections of ${\\partial }_{\\bf y}^{1}\\Omega $ and $\\partial _{\\bf y}^{2}\\Omega $ with a neighborhood of ${\\bf y}$ when these sets are each subsets of the boundaries of smooth (i.e.",
"$C^{2,\\lambda }$ ) domains $W_{1}$ and $W_{2}$ which satisfy appropriate boundary curvature conditions at ${\\bf y}.$ (For capillary surfaces in positive gravity (and prescribed mean curvature surfaces with $\\frac{\\partial H}{\\partial z}({\\bf x},z)\\ge \\kappa >0),$ one can examine Theorem 2 of [19].)"
],
[
"Curvature Conditions on ${\\partial }_{\\bf y}^{1}\\Omega $ and {{formula:74f74ba1-9689-4453-8216-1c712c01709d}}",
"In [9], the existence of nontangential radial limits of bounded, nonparametric prescribed mean curvature surfaces at nonconvex corners was proven; in Theorem REF , we showed that all radial limits of such surfaces at nonconvex corners exist when Bernstein functions exist.",
"On the other hand, [22] and Theorem 3 of [25] provide examples in which no radial limit exists at a point ${\\bf y}$ of $\\partial \\Omega $ at which the boundary of $\\Omega $ is smooth.",
"In this section, we shall focus on the points ${\\bf y}\\in \\partial \\Omega $ at which $\\beta ({\\bf y})-\\alpha ({\\bf y}) \\le \\pi $ and ask which type of behavior (i.e.",
"(a) no radial limits exist, (b) nontangential radial limits exist or (c) all radial limits exist) occurs, depending essentially on the curvatures of ${\\partial }_{\\bf y}^{1}\\Omega $ and $\\partial _{\\bf y}^{2}\\Omega .$ The following lemma shows that (a), (b) and (c) are the only possible behaviors of radial limits when $H({\\bf x},t)$ is weakly increasing in $t$ for each ${\\bf x}\\in \\Omega ,$ provided that we include in (b) all of the cases in which $Rf(\\theta ,{\\bf y})$ exists for $\\theta $ in one of the three intervals $(\\alpha ({\\bf y}),\\beta ({\\bf y})),$ $[\\alpha ({\\bf y}),\\beta ({\\bf y})),$ and $(\\alpha ({\\bf y}),\\beta ({\\bf y})].$ Lemma 2 Let $f\\in C^{2}(\\Omega )\\cap L^{\\infty }(\\Omega )$ satisfy $Qf=0$ in $\\Omega $ and let $H^{*}\\in L^{\\infty }({\\rm I\\hspace{-1.99997pt}R}^{2})$ satisfy $H^{*}({\\bf x})=H({\\bf x},f({\\bf x}))$ for ${\\bf x}\\in \\Omega .$ Let ${\\bf y}\\in \\partial \\Omega $ and suppose there exists a $\\theta _{1}\\in [\\alpha ({\\bf y}),\\beta ({\\bf y})]$ such that $Rf(\\theta _{1},{\\bf y})$ exists.",
"Then $Rf(\\theta ,{\\bf y})$ exists for each $\\theta \\in (\\alpha ({\\bf y}),\\beta ({\\bf y})),$ $Rf(\\cdot ,{\\bf y})\\in C^{0}((\\alpha ({\\bf y}),\\beta ({\\bf y})))$ and $Rf(\\cdot ,{\\bf y})$ behaves as in Theorem 1 of [9].",
"Suppose, in addition, that there exist $\\delta >0$ and upper and lower Bernstein pairs $\\left(U^{\\pm }_{1},\\psi ^{\\pm }_{1}\\right)$ and $\\left(U^{\\pm }_{2},\\psi ^{\\pm }_{2}\\right)$ for $(\\Gamma _{1},H^{*})$ and $(\\Gamma _{2},H^{*})$ respectively, where $\\Gamma _{1}=B_{\\delta }({\\bf y})\\cap {\\partial }_{\\bf y}^{1}\\Omega $ and $\\Gamma _{2}=B_{\\delta }({\\bf y})\\cap {\\partial }_{\\bf y}^{2}\\Omega .$ Then the conclusions of Theorem REF hold.",
"Proof: The first part follows from Theorem 2 of [9].",
"The second part follows from Theorem REF .",
"$\\Box $ Now suppose $f\\in C^{2}(\\Omega )$ satisfies $Nf({\\bf x})=H({\\bf x})$ for ${\\bf x}\\in \\Omega $ and ${\\bf y}\\in \\partial \\Omega $ satisfies $\\beta ({\\bf y})-\\alpha ({\\bf y}) \\le \\pi .$ Under what conditions do types of behavior (a), (b) or (c) occur?",
"Lemma 3 Suppose $\\Omega ,$ ${\\bf y}$ and $H$ are as above and $\\Lambda ({\\bf x})\\ge 2|H({\\bf x})|$ for almost all ${\\bf x}\\in B_{\\epsilon }({\\bf y})\\cap {\\partial }_{\\bf y}^{1}\\Omega \\cup \\partial _{\\bf y}^{2}\\Omega ,$ for some $\\epsilon >0.$ Then there exists $\\phi \\in L^{\\infty }(\\partial \\Omega )$ such that the solution $f\\in C^{2}(\\Omega )\\cap L^{\\infty }(\\Omega )$ of $Nf=H$ in $\\Omega $ and $f=\\phi $ almost everywhere on $\\partial \\Omega $ has no radial limits at ${\\bf y}.$ Proof: This follows from Theorem 16.9 of [14] and the “gliding hump” argument in [22].",
"$\\Box $ Lemma 4 Suppose $\\Omega ,$ ${\\bf y}$ and $H$ are as above and $\\Lambda ({\\bf x})< 2\\left|H({\\bf x})\\right|$ for almost all ${\\bf x}\\in B_{\\epsilon }({\\bf y})\\cap {\\partial }_{\\bf y}^{1}\\Omega \\cup \\partial _{\\bf y}^{2}\\Omega ,$ for some $\\epsilon >0.$ Then there exist $\\delta >0$ and upper and lower Bernstein pairs $\\left(U^{\\pm }_{j},\\psi ^{\\pm }_{j}\\right)$ for $(\\Gamma _{j},H),$ where $\\Gamma _{j}=B_{\\delta }({\\bf y})\\cap {\\partial }_{\\bf y}^{j}\\Omega ,$ for $j=1,2,$ and the conclusions of Theorems REF –REF hold when their other hypotheses are satisfied.",
"Proof: This follows from Remark REF .",
"$\\Box $ Theorem 6 Suppose $\\Omega $ is a $C^{2,\\lambda }$ domain in ${\\rm I\\hspace{-1.99997pt}R}^{2}$ and $f\\in C^{2}(\\Omega )\\cap L^{\\infty }(\\Omega )$ is a variational (i.e.",
"BV) solution of (REF )-() for some $\\phi \\in L^{\\infty }(\\Omega )$ and $\\lambda \\in (0,1).$ Let ${\\bf y}\\in \\partial \\Omega $ and let $\\Lambda ({\\bf y})$ denote the (signed) curvature of $\\partial \\Omega $ at ${\\bf y}$ with respect to the interior normal direction (i.e.",
"$-\\nu $ ).",
"(i) Suppose $\\Lambda ({\\bf y})< 2|H({\\bf y})|.$ Then the conclusions of Theorem REF hold.",
"(ii) Suppose $\\Lambda ({\\bf y})> 2|H({\\bf y})|.$ Then the conclusions of Theorem REF hold if $\\phi $ restricted to ${\\partial }_{\\bf y}^{j}\\Omega $ has a limit $z_{j}$ at ${\\bf y}$ for $j=1,2,$ while for certain $\\phi \\in L^{\\infty }(\\Omega ),$ $Rf(\\cdot ,{\\bf y})$ does not exist for any $\\theta \\in [\\alpha ({\\bf y}),\\beta ({\\bf y})].$ Proof: The first part follows from Lemma REF .",
"The second part follows from Theorem 16.9 of [14], [21] (see also [7], [25]) and Lemma REF .",
"$\\Box $ Remark 6 One can state a theorem similar to Theorem REF when $f\\in C^{2}(\\Omega )$ is a variational solution of (REF )-() for some $\\phi \\in L^{\\infty }(\\Omega )$ and ${\\bf y}\\in \\partial \\Omega $ satisfies $\\beta ({\\bf y})-\\alpha ({\\bf y})<\\pi ,$ ${\\partial }_{\\bf y}^{1}\\Omega $ and ${\\partial }_{\\bf y}^{2}\\Omega $ are smooth, and boundary curvature conditions apply on ${\\partial }_{\\bf y}^{j}\\Omega ,$ $j=1,2.$ If, for example, $\\Lambda ({\\bf x})\\ge 2|H({\\bf x})|$ for ${\\bf x}\\in {\\partial }_{\\bf y}^{1}\\Omega $ near ${\\bf y},$ $\\Lambda ({\\bf x})< 2|H({\\bf x})|$ for ${\\bf x}\\in {\\partial }_{\\bf y}^{2}\\Omega $ near ${\\bf y},$ and $\\phi $ restricted to ${\\partial }_{\\bf y}^{1}\\Omega $ has a limit $z_{1}$ at ${\\bf y},$ then the conclusions of Theorem REF hold (e.g.",
"Theorem 2 of [9])."
]
] | 1808.08599 |
[
[
"Core-Halo Collective Instabilities"
],
[
"Abstract At strong space charge, transverse modes of the bunch core may effectively couple with those of the halo, leading to instabilities well below the core-only transverse mode-coupling threshold."
],
[
"Core-Halo Collective Instabilities Alexey Burov burov@fnal.gov Fermilab, PO Box 500, Batavia, IL 60510-5011 At strong space charge, transverse modes of the bunch core may effectively couple with those of the halo, leading to instabilities well below the core-only transverse mode-coupling threshold.",
"00.00.Aa , 00.00.Aa , 00.00.Aa , 00.00.Aa Suggested keywords Introduction.",
"Collective instabilities limit intensity of charged particle beams in many accelerators.",
"Such instabilities, caused by mutual interaction of the beam particles, lead either to density degradation or to beam loss, full or partial.",
"Both, Coulomb fields of the bunch space charge (SC) and wake fields left behind the particles, are generally important in this respect.",
"While the former change the spectra of the individual and collective modes, making them more or less prone to instabilities, the latter, being non-Hamiltonian, are able to drive the instabilities.",
"Hereafter, a beam in a circular machine is modeled as a single bunch, with short-range wakes only (no multi-turn wakes), and with fully linear focusing.",
"When wake and SC fields can be neglected, frequencies of the bunch transverse collective modes represent a series of equidistant sidebands around the main transverse frequency, the betatron frequency $\\omega _\\beta $ .",
"Distances between the neighbor sidebands are equal to the longitudinal frequency of the bunch particles inside the potential well, the synchrotron frequency $\\omega _s$ .",
"In reference to the betatron frequency, frequencies of the transverse collective modes $\\Omega _k$ represent a series, $\\Omega _k=k \\omega _s$ , where integers $k=0,\\; \\pm 1,\\; \\pm 2,\\,...$ serve as the mode numbers.",
"For circular accelerators, it is conventional to normalize all the beam frequencies to the revolution frequency; the resulting values are called tunes.",
"However, for analysis of the collective motion, the revolution frequency does not mean much; it makes more sense to normalize all the sidebands $\\Omega _k$ to the synchrotron frequency instead, dealing with properly normalized collective tunes $\\nu _k=\\Omega _k/\\omega _s$ .",
"With negligible interaction between the particles, the collective tunes $\\nu _k=k$ ; however, SC and wake both shift the tunes from these unperturbed values.",
"The wake force $F$ is conventionally represented by the wake function $W(s)$ between two unit charges separated by a distance $s$ : $\\; F(s) = W(s)\\,y/C_0$ , where $y$ is an offset of the leading particle and $C_0$ is the ring circumference, see e.g. [1].",
"With these forces and without SC, the bunch particles represent a set of identical harmonic oscillators, acting on each other by linear forces $F(s)$ .",
"The wakes are normally causal, i.e.",
"the trailing (or tail) particle does not act back on the leading (or head) one, $F(s)=0$ at $s<0$ , so Newton's third law does not apply; wakes make the system non-Hamiltonian.",
"Nevertheless, this sort of bunch is stable below a certain threshold intensity, since the equations of its collective dynamics reduce to an eigensystem problem of a real non-degenerate matrix; at zero wake it is diagonal and filled with consecutive integers $k=0,\\; \\pm 1,\\; \\pm 2,\\,...$ .",
"The collective tunes are given by the eigenvalues of that matrix.",
"With a non-zero wake, some of the eigenvalues may become complex, coming in pairs: if $\\nu _k$ is the eigenvalue, then its complex conjugate $\\nu _k^*$ is the eigenvalue too.",
"Due to this fact, the complex tunes with $\\Im \\nu \\ne 0$ may appear only after coupling of two modes, whose real tunes become identical at the threshold and get opposite imaginary parts only above it.",
"That is why the instability is called TMCI, the transverse mode coupling instability.",
"With all the collective frequencies measured in units of the synchrotron frequency, the same rule applied to the SC frequency shift $\\Delta \\omega _\\mathrm {sc}$ requires the introduction of the space charge parameter $q = \\Delta \\omega _\\mathrm {sc}/\\omega _s$ as the measure of its strength.",
"The wake parameter $w$ can be introduced in a similar way, as $\\sim N_p\\,W_0 \\max _k d\\nu _k/d(N_pW_0)$ , where $W_0$ is an amplitude of the wake function, $N_p$ is the number of particles per bunch, and the derivative is taken at zero intensity; in other words, the wake parameter represents the maximal collective tune shift in the units of the synchrotron frequency.",
"Influence of SC on the TMCI was considered in a series of publications, starting from a pioneer work of M. Blaskiewicz [2], followed by more detailed analysis of Refs.",
"[3], [4], [5], [6].",
"It has been shown that the wake threshold $w_\\mathrm {th}$ almost always grows linearly with SC, $w_\\mathrm {th} \\propto q \\,,$ as soon as SC is strong, $q \\gg 1$ .",
"For years, nobody published anything about a certain strangeness of this stability condition: its total insensitivity to the number of particles.",
"Since both terms of Eq.",
"(REF ) are proportional to the bunch intensity $N_p$ , the latter simply cancels out.",
"Thus, were it correct, such a beam with sufficiently strong SC, being stable at some intensity, would remain stable at higher intensity, regardless of how much!",
"This fantastic conclusion followed from strict and thorough independent analyses by the authors mentioned above in this respect.",
"Resolution of the conundrum was recently suggested in Ref. [7].",
"According to that, a mistake, or rather unreflected prejudice, of the author himself and the related community consisted in a tacit equating of instabilities in general with absolute instabilities, i.e.",
"instabilities in general were equated with the existence of a collective mode with the positive imaginary part of the tune, $\\Im \\nu > 0$ .",
"Well, this seemingly obvious equality is deceptive.",
"Collective instabilities do not reduce to absolute ones; this family of beasts includes another genus as well, namely, convective instabilities.",
"According to Landau and Lifshitz [8], a distinction between the two genera of instabilities can be described as follows.",
"For an absolute instability, initial perturbations cause an unrestricted exponential growth everywhere in the medium; to suppress it, a damping rate must exceed the nonzero growth rate.",
"For the convective instability, instead, there is only a spacial amplification, and the perturbation eventually decays everywhere when a dissipation is added, no matter how tiny.",
"Generally speaking, convective instabilities are not less dangerous than absolute ones.",
"Even if all the modes are stable in the absolute sense, convective amplifications along the bunch can be so large, that for all practical purposes the beam would and should be considered unstable.",
"Due to the absolute stability, these convectively unstable modes have their amplitudes still bounded; that is why such modes may be characterized as saturating convective instabilities, or SCI, in distinction with unbounded convective instabilities, UCI; the latter are known in linacs as beam breakups.",
"Resolution of the conundrum consists in a demonstration that at strong SC, there is a significant interval of the wake parameters corresponding to an absolutely stable and convectively unstable bunch, with large head-to-tail amplifications.",
"The lower limit of this interval is about the same as the TMCI threshold at zero SC, $w^0_\\mathrm {th}$ , while the upper limit is the TMCI threshold $w_\\mathrm {th}(q)$ at the given SC parameter $q \\gg 1$ ; typically $w_\\mathrm {th}(q)/w^0_\\mathrm {th} \\simeq q$ .",
"Since the amplification coefficient of the SCI depends exponentially on the wake parameter, the amplification quickly reaches a practically intolerable level of hundreds and thousands.",
"A convective instability with large amplification can be considered as a special metastable state: even a tiny, otherwise totally negligible, tail-to-head feedback, provided by a multi-revolution or a coupled-bunch wake could be sufficient to drive an absolute instability.",
"Thus, the latter can be called absolute-convective instability, ACI.",
"With sufficiently large amplification, any damper, including the conventional resistive one, turns, in principle, into an ACI generator.",
"Keeping all this in mind, we still may imagine a bunch without all these feedbacks, injection errors and aperture limitations, and ask the question: could there be anything at all in the bunch itself that can still limit the amplifications of the convective instabilities at wake parameters considerably below the TMCI threshold at strong SC?",
"Core-Halo Interaction.",
"Before trying to answer that question, let us show what sort of reason is behind the strange stability condition (REF ); what features of SC provide such a dramatic elevation of the TMCI wake threshold, as it suggests.",
"Well, this SC ability is caused by a fact that both SC and wake tune shifts are typically of the same sign; they are both defocusing.",
"At strong SC, all the modes with intra-slice motion are strongly detuned from the rigid-slice modes, mostly coupled with wake; modes of the opposite groups cannot cross, and modes with non-rigid slices are insensitive to wake.",
"Another important point is that typically wake mostly shifts (down) the mode 0, responsible for bunch motion as a whole, while other modes are shifted to a lesser extent, thus wake works on the mode divergence.",
"In short, that is why TMCI vanishes at strong SC: the negative modes with considerable intra-slice motion are SC-separated from $0^\\mathrm {th}$ and positive modes, while wake separates positive modes even more.",
"If the modes cannot couple and there are no feedbacks, absolute instabilities are impossible.",
"At this point, however, we may recall that even at strong SC the described reasoning breaks for some particles with large individual amplitudes, the halo particles, which feel SC to lesser extent.",
"Although the relative number of such particles may be small, this small percentage could be compensated by a large amplification coefficient of the core convective instability.",
"An important feature of the halo is that it is sensitive to wake but less sensitive to SC, and its reduced sensitivity to SC is of a different sort than that of the core.",
"Specifically, the rarified halo practically does not feel its own SC, only that of the core, so all of its modes can be equally easily or non-easily excited by the motion of the core; essentially they are similar to no-SC modes, just shifted down a bit according to the halo's reduced SC tune shift, $q_h \\ll q \\equiv q_c$ .",
"Nothing essentially prevents coupling between the core and the halo modes, which effect can be dramatically enhanced by SCI of the core.",
"Thus, we are coming to the idea of the core-halo mode-coupling instability at strong SC as an absolute instability, which wake threshold may be well below the conventional core-only TMCI threshold.",
"To check this idea, we need a reasonable quantitative model of the bunch collective dynamics, where both core and halo modes can be taken into account.",
"ABS Model.",
"For that matter, our single bunch in a circular accelerator can be modeled similarly to Ref.",
"[7], assuming the same airbag square well (ABS) model of M. Blaskiewicz [2].",
"The longitudinal potential well, which keeps the beam bunched, is assumed to be square, while all the bunch particles are supposed to have the same synchrotron (longitudinal) frequency.",
"The only novelty we have to introduce here with this model is to represent the bunch as consisting of two fractions, core and halo, with different SC tune shifts and different populations.",
"Before doing this, though, let us write down the conventional core-only ABS equations of motion in the form of Ref.",
"[7]: $\\begin{split}& \\frac{\\partial \\,x}{\\partial \\theta } + \\frac{\\partial \\,x}{\\partial \\psi } = i q (x - \\bar{x}) + i\\,F\\, ; \\\\& F(\\psi ) = w \\int _0^{|\\psi |} \\frac{ \\mathrm {d}\\psi ^{\\prime }}{\\pi } W(\\psi ^{\\prime }/\\pi ) \\bar{x}(\\psi -\\psi ^{\\prime }) \\, ;\\\\& \\bar{x}(\\psi ) \\equiv x(\\psi )/2 + x(-\\psi )/2 \\,.\\\\\\end{split}$ Here $x=x(\\theta ,\\psi )$ is a slow amplitude of the transverse oscillations at time $\\theta $ , measured in the synchrotron radians, and of the synchrotron phase $\\psi $ ; the searched-for function $x$ is periodical on the latter variable, supposed to change from $-\\pi $ to 0 for the tail-to-head moving particles, or for the $+$ flux, and from 0 to $\\pi $ for the $-$ flux, moving back to the tail; the bunch length is 1 in these units.",
"The term $\\propto q$ is the SC force, and $F$ is the wake force with $W(s)$ as a dimensionless wake function; $\\bar{x}(\\psi )$ is the local centroid.",
"The dimensionless wake parameter $w$ keeps in itself all dimensional values of the problem; it is defined as $w = \\frac{N_{\\text{p}}W_0 r_0 R_0}{4\\,\\pi \\,\\gamma \\,\\beta ^2\\, Q_{\\beta } Q_s}\\,,$ with $N_{\\text{p}}$ as the number of particles, $W_0$ as the amplitude of the wake function in conventional units of Ref [1], $r_0$ as the particle classical radius, $R_0$ as the average radius of the machine, $\\gamma $ and $\\beta $ as the relativistic factors, $ Q_{\\beta }$ and $ Q_s$ as the conventional betatron and synchrotron tunes.",
"Due to its phase space periodicity, the amplitude $x$ can be Fourier-expanded over the phase $\\psi $ , leading to a set of ordinary homogeneous linear equations on the harmonics $A_n \\equiv (2\\pi )^{-1}\\int _{-\\pi }^{\\pi } x \\exp (-in\\psi )\\,\\mathrm {d}\\psi $ : $\\begin{split}& i\\dot{A}_n = n A_n - q(A_n - \\bar{A}_n) - w \\sum _{m=-\\infty }^{\\infty } U_{nm} \\bar{A}_m \\,; \\\\& U_{nm} \\equiv {\\int _0^1 \\mathrm {d}s \\int _0^s \\mathrm {d}s^{\\prime } W(s-s^{\\prime }) \\cos (\\pi ns) \\cos (\\pi m s^{\\prime })}\\,,\\end{split}$ with the centroid's harmonic $\\bar{A}_n=(A_n+A_{-n})/2$ .",
"This form of the dynamic equations makes its generalization for any number of bunch fractions fairly obvious: the Fourier amplitude of every fraction satisfies the same Eq.",
"(REF ) with its own relative shift of the synchrotron frequency $\\Delta \\omega _s/\\omega _s$ and its own SC tune shift, i.e.",
"with $n \\rightarrow n (1+\\Delta \\omega _s/\\omega _s)$ and its own parameter $q$ .",
"Each fraction, weighed with its relative intensity, contributes to the centroid $\\bar{A}$ .",
"As a result, the problem can be solved with a reasonable accuracy in a reasonable CPU time, leading to a straightforward analysis of continuous and discrete van Kampen spectra [9], similar to Refs.",
"[10], [11].",
"Here, however, a different approach is suggested and realized.",
"Border of the Halo.",
"Instead of presenting the bunch by a sufficiently large number of fractions, let us consider it as consisting of just two: the core and the halo, marked with the corresponding indices $c$ and $h$ : $\\begin{split}& i\\dot{A}_n^c = n A_n^c +G^c_n - w \\sum _{m=-\\infty }^{\\infty } U_{nm} \\bar{A}_m \\,; \\\\& G^c_n \\equiv -q_c(A_n^c - \\bar{A}_n^c)-q_h p_h(A_n^c - \\bar{A}_n^h)\\,; \\\\& i\\dot{A}_n^h = n A_n^h + G^h_n - w \\sum _{m=-\\infty }^{\\infty } U_{nm} \\bar{A}_m \\,;\\\\& G^h_n \\equiv - q_h(A_n^h - \\bar{A}_n^c)\\,; \\\\& \\bar{A}_n \\equiv \\bar{A}_n^c + p_h \\bar{A}_n^h \\,,\\end{split}$ where $p_h \\ll 1$ is a relative population of the halo.",
"For zero wake, Eqs.",
"(REF ) describe a Hamiltonian system, stable for any choice of the remaining parameters.",
"The suggested two-fractional bunch model meets an obvious question: how to define the halo tune shift $q_h$ ?",
"Where should the borderline between the core and the halo be drawn?",
"Seemingly, there is no border in reality; thus, wouldn't whatever border of the model be as arbitrary and artificial as any other?",
"This paper suggests a natural solution to this problem: let the system itself make the decision!",
"For the given wake and (core) SC parameters, $w$ and $q \\equiv q_c$ , let the halo relative SC tune shift $\\tilde{q} \\equiv q_h/q_c$ be a free variable, and let us then find such a value for it that corresponds to the highest instability growth rate.",
"That special value of the halo SC tune shift would point to the most effective, most powerful, and hence most important collective interaction, motivating to take such halo border as the natural border.",
"With this dynamic definition, the border would be a function of the wake and SC parameters.",
"The above implies that the halo relative population $p_h$ is a known function of its parameter $\\tilde{q}$ ; indeed, as soon as the bunch distribution function is given, the population function can be computed.",
"For a transversely Gaussian bunch inside the longitudinal square well, the partial SC tune shift $\\tilde{q}$ versus two transverse actions $J_{x,y}$ was found by Lopez [12]: $\\tilde{q}_L=\\int _0^1 \\mathrm {d}z \\frac{\\mathrm {I}_0(J_x z/2)-\\mathrm {I}_1(J_x z/2)}{\\exp (J_x z/2+J_y z/2)}\\mathrm {I}_0(J_y z/2).$ Here, the $x$ direction is the one of the tune shift $q$ ; the actions $J_{x,y}$ are measured in units of their beam-average values, or emittances, assumed to be equal here; $\\mathrm {I}_{0,1}$ are the modified Bessel functions.",
"Now, the portion of particles $p(\\tilde{q})$ whose tune shifts do not exceed $\\tilde{q}$ is obtained right away: $p(\\tilde{q})=\\int _0^{\\infty }\\int _0^{\\infty } \\mathrm {d}J_x \\mathrm {d}J_y \\mathrm {\\Theta }(\\tilde{q}_L(J_x,\\,J_y)-\\tilde{q}) e^{-J_x - J_y} \\,,$ where $\\mathrm {\\Theta }$ stands for the Heaviside theta-function.",
"For these general core-halo considerations, an approximation of large actions suffices, leading to $\\tilde{q}_L \\simeq 3/(2J_x+J_y)$ , and, further, to the asymptotic expression of the halo population function, $p(\\tilde{q}) \\simeq 2 \\exp (-1.5/ \\tilde{q}) \\,.$ At a sufficiently small $\\tilde{q}$ , the distribution $p$ is sharp; its relative width $\\delta \\tilde{q}/\\tilde{q} \\simeq 0.7\\tilde{q}$ ; this fact supports the idea to consider all the halo particles as having the same SC tune shift, as soon as the halo border is sufficiently far.",
"Figure: Instability growth rate versus the halo parameter q ˜=q h /q c \\tilde{q}=q_h/q_c for the SC and wake q c =10q_c=10 and w=4.w=4.Figure: Centroid stroboscopic images of the core and halo components of the most unstable core-halo mode for the same qq and ww as in Fig.",
", at the most unstable q ˜=0.29\\tilde{q}=0.29\\,.",
"Waists instead of nodes in the halo image tell about an absolute instability.Figure: Amplitudes and phases for the two fluxes of the core and the halo for the same modes, +2+2 and +3+3 correspondingly, as in Fig. .",
"The core mode is convectively unstable, with its ++ and -- fluxes in phase, while the halo mode is similar to a typical no-SC modes having the ++ and -- phases steadily running with opposite signs.Figure: Growth rates of the most unstable modes versus wake parameter for three different SC parameters.",
"Note the conventional TMCI threshold for q c =5q_c=5 at w≈15w \\approx 15.Results.",
"To demonstrate the general properties of the instability, the simplest model wake function is chosen, the Heaviside theta, $W(s)=\\Theta (s)$ .",
"After that, the eigensystem problems of Eqs.",
"(REF ) are solved for the given wake and SC parameters, with the halo parameter $\\tilde{q}$ initially free, but later assigned its natural value.",
"Figure REF shows an example of how the calculated growth rate of the most unstable mode may depend on the halo parameter; the exemplifying wake is $\\sim 4$ times above its no-SC TMCI threshold, and almost an order of magnitude below its TMCI threshold at this SC.",
"The sequence of partially overlapped resonances of the core and halo modes is dominated by the resonance with the center at $\\tilde{q} \\approx 0.29$ , the latter corresponding to the halo relative population $p_h \\approx 1\\%$ .",
"For the lower values of $\\tilde{q}$ , the halo would be too tiny, while for its higher values, SC depression of the mode coupling would be stronger; that is why the maximally unstable halo parameter $\\tilde{q}$ is somewhere in between but not too close to 0 and 1.",
"Figure REF demonstrates stroboscopic images of the core and halo centroids of the most unstable eigenvector of Eqs.",
"(REF ) at these parameters, $\\Re (\\bar{x}_c \\exp (-i l \\phi ))$ , $l=0,1,2,...$ , taken with an arbitrary phase advance $\\phi $ , as if the oscillating centroids were observed a certain number of times, revolution after revolution, and the images superimposed.",
"Figure REF suggests another representation of the same modes, showing the absolute values and complex arguments of the positive and negative fluxes for their core and halo components.",
"The plots show that the core mode $+2$ is coupled with the halo mode $+3$ .",
"At this core and halo SC parameters, the tune of the halo mode $\\nu ^h_3 \\approx 3-q_h=0.1$ , which indeed is very close to the tune of the core mode $+2$ , computed with the core-only model.",
"The core mode is convectively amplified, showing the typical cobra shape and ACI phases in Fig.",
"REF .",
"This figure shows the halo mode with almost constant amplitude and steadily running phase, which is typical to no-SC modes; its waists instead of nodes at the right part of Fig.REF indicate an instability.",
"Generally speaking, the halo's ability to play a feedback role is reduced by its low population; however, amplification of the core mode, considerable wake parameter and some halo SC tune shift enhance the core's sensitivity to the halo's oscillations, and thus may restore this ability to a level sufficient for driving the ACI.",
"Growth rate of the most unstable mode versus wake, with the natural halo parameters $\\tilde{q}$ , is presented in Fig.",
"REF for three values of SC.",
"Growth rates that are too small at $w \\lessapprox 1$ likely exceed the model accuracy, and should be rather considered as indistinguishable from zero; apparently, the instability threshold cannot be correctly predicted by this core-halo model, which accuracy is limited by the simplification of the two-fractional approach.",
"What is clear though, is that the instability is already significant for the wake parameters well below the TMCI threshold.",
"Note that the latter is clearly seen for $q=5$ at $w \\approx 15$ , in agreement with the no-halo calculations [2].",
"The core-halo mode coupling suggests a new type of collective instabilities of bunched beams at strong SC, combining in themselves features of the TMCI and ACI.",
"Contrary to the pure convective modes of the core itself, this instability is absolute; it may lead to the halo loss and the core emittance growth.",
"On the other hand, the core-halo instability may be less limiting than the pure convective instabilities of the core, which typically have larger amplifications.",
"Acknowledgements.",
"I am thankful to Yuri Alexahin for letting me know that in his computations with rather strong SC, as in Ref.",
"[11], the instability thresholds were found to be significantly lower when the rigidity of the bunch slices was not forced.",
"Fermilab is operated by Fermi Research Alliance, LLC under Contract No.",
"DE-AC02-07CH11359 with the United States Department of Energy."
]
] | 1808.08498 |
[
[
"A Stern-Gerlach separator of chiral enantiomers based on the\n Casimir-Polder potential"
],
[
"Abstract We propose a method to separate enantiomers using parity violation in the Casimir-Polder potential between chiral mirrors and chiral molecules.",
"The proposed setup involves a molecular beam composed of chiral molecules passing through a planar cavity consisting of two chiral mirrors.",
"Enantiomers of opposite handedness are deflected differently due to a chiral dependence of the Casimir-Polder potential resulting in the separation of the enantiomers.",
"Our setup provides an alternative experimental tool for enantiomer separation, as well as shedding light on the fundamental properties of the Casimir-Polder potential."
],
[
"Introduction",
"Many molecules are chiral which can exist in left- and right-handed forms (i.e., non-superimposable mirror images).",
"These two forms of a chiral molecule are known as enantiomers.",
"Distinguishing two types of enantiomers is of great practical importance.",
"For example, in designing pharmaceuticals, it is necessary to choose the right enantiomer to obtain the desired effects since the other enantiomer is less active, inactive, or can even have adverse side effects, including high toxicity.",
"Therefore, developing technology which separates enantiomers can improve pharmacy by introducing medicines composed of only one enantiomer which enhances the desired effects, while eliminating the side effects [1].",
"Common methods for chiral separation include chromatography [2] and crystalization [3].",
"Chiral separations using chromatography now represent a popular, robust routine technique utilized in laboratories.",
"However, the selection of columns still remains a matter of trial and error, and it is difficult to find materials that show both high efficiency and high enantio-selectivity.",
"Stern–Gerlach type separators for enantiomers have been proposed that are based on inhomogeneous laser fields [4], [5].",
"The trajectories of the emerging molecular beams depend on both spin and handedness.",
"In such schemes, the orientation and rotation of the molecules needs to be carefully addressed [6].",
"Here we propose a method for the separation of chiral enantiomers that is based on the Casimir-Polder (CP) potential.",
"CP forces are effective quantum electrodynamical forces between neutral, unpolarised molecules and macroscopic bodies which arise from the interaction of the objects' charge and current densities with the vacuum electromagnetic field.",
"Originally predicted by Casimir and Polder, they are typically attractive and purely monotonic for ground-state molecules and bodies with a purely electric response [7].",
"It was later seen that both the direction and distance dependence of the potential changes when considering excited molecules.",
"Here, the possibility that the molecule emits a real photon which gets reflected off the body and subsequently reabsorbed leads to a potential whose sign depends on the relative phase of emitted and reabsorbed photons [8], [9].",
"Depending on the detuning between the radiation emitted by the molecule and the plasmonic resonances of the body, the potential can be attractive or repulsive in the near zone [10] while oscillating with distance in the far zone [11]; and the potential may strongly exceed its ground-state counterpart.",
"Figure: (Color online)A beam of chiral molecules driven by a laser field passes through a planar cavity consisting of two chiral mirrors.",
"Enantiomers of opposite handedness are deflected differently due to the chiral component of the CP potential.In addition, the sign of the CP force may depend on the electromagnetic response of the interacting objects.",
"Motivated by similar findings for the Casimir force by Boyer [12], it was found that CP forces due to interactions of electric atoms with magnetic surfaces—or vice versa—are repulsive [13], [14], [15] in contrast to the attractive electric–electric force.",
"Similar repulsive forces have been predicted for circularly polarised atoms interacting with axionic topological insulators [16] and—most crucially for this work—for chiral molecules interacting with chiral surfaces.",
"Here the forces are attractive for objects of the same handedness and repulsive for opposite handedness [17].",
"The chiral component of the CP potential is hence sensitive to the handedness of molecules and can be attractive or repulsive depending on their chirality.",
"This is because the CP potential between a chiral molecule and a chiral mirror depends on the optical rotatory strength of the molecule which is a pseudoscalar that changes sign under a parity inversion [18].",
"The potential could hence be used to differentiate enantiomers of opposite handedness.",
"Using this property of the CP potential, it would be possible to identify the handedness of the chiral molecule by setting up a Stern-Gerlach type discriminator for enantiomers.",
"Although the electric component of the CP force has been already observed [19], here we aim to measure its chiral component which has not yet been observed.",
"The closely related chiral van der Waal force between two molecules has been discussed earlier, where again, enantiomer-discriminatory interactions are predicted when both particles are chiral [20], [21], [22], [23].",
"Since these forces are very weak, a stronger force between a chiral molecule and an achiral molecule has been proposed which is mediated by a nearby chiral surface [24].",
"Our proposed setup uses the molecular beam deflection technique.",
"We let a molecular beam composed of chiral molecules pass through a planar cavity consisting of two chiral mirrors (FIG.",
"REF ).",
"Although the electric component of the CP force between the cavity and the molecules acts on both enantiomers in the same way, the chiral component can be attractive or repulsive depending on the handedness of the molecules and can deflect enantiomers of opposite handedness differently and separate them.",
"To exploit the enhanced excited-state CP force, the molecules are subject to a weakly detuned driving laser while passing through the cavity.",
"The method proposed here is more universal than the above-mentioned conventional schemes, since we only utilize the interaction between a chiral mirror and chiral molecules induced by photo excitations and emissions.",
"All molecules have characteristic excitation spectra, and therefore a suitable, resonant light source (either UV or IR) is the only prerequisite for the present method.",
"The same chiral mirror can be used for enantiomer separation of a wide range of molecules.",
"The article is organized as follows: In Sec.",
", we study the behavior of the CP potential between the cavity and the chiral molecule driven by a laser field within the framework of macroscopic quantum electrodynamics.",
"We compare the CP potential experienced by 3-methyl-cyclopentanone (3-MCP) molecules with electric circular dichroism (ECD) and propylene-oxide molecules with vibrational circular dichroism (VCD).",
"We find that the CP potential for propylene-oxide molecules with VCD can depend on temperature.",
"We also observe the enhancement of the CP potential due to the cavity structure when the cavity is small enough that it contains only a few molecular transition wavelengths.",
"However, experimentally we use a realistic setup where the cavity width is $1\\,$ mm and such enhancement can not be observed.",
"In Sec.",
", we simulate separation of enantiomers using our setup by studying trajectories of the molecules in the cavity.",
"Since the strength of the CP potential depends on the frequency of photons and the electric and chiral components depend on the electric dipole strength and the optical rotatory strength of the molecules respectively, we use chiral molecules with ECD whose optical rotatory strength is large compared to its electric dipole moment in the simulation.",
"In this article, we use 3-methyl-cyclopentanone as an example for a quantitative analysis.",
"Finally we conclude that our proposed setup can be used for separating enantiomers of opposite handedness and detection of the chiral dependence of the CP potential within reach of current technology (Sec.",
").",
"Figure: (Color online) The electric and chiral component (experienced by the 3-MCP molecule with the positive R 01 R_{01}) of the CP potential with different reflection coefficients: r e =0.01r_{e}=0.01, 0.050.05 and 0.10.1, r c =0.5r_{c}=0.5, 0.70.7 and 0.90.9.",
"The solid lines: the excited-state potential, the dotted lines: the ground-state potential."
],
[
"Behavior of the Casimir-Polder Potential",
"Within the framework of macroscopic quantum electrodynamics [25], [26], the CP potential $U_{CP}$ between the cavity and the chiral molecule (approximated as two-level system) driven by a laser field is given by [27], [28] $U_{CP}=p_0 (t) U_0 +p_1 (t) U_1$ where the populations of the ground state $p_0 (t)$ and that of the excited state $p_1 (t)$ are written as $p_0 (t)&=& \\frac{\\Omega ^2}{\\Delta ^2 +\\Omega ^2}\\cos ^2 \\left(\\frac{1}{2} \\sqrt{\\Delta ^2 +\\Omega ^2} \\mbox{ }t\\right)+\\frac{\\Delta ^2}{\\Delta ^2+\\Omega ^2},\\nonumber \\\\p_1 (t)&=& \\frac{\\Omega ^2}{\\Delta ^2+\\Omega ^2}\\sin ^2 \\left(\\frac{1}{2} \\sqrt{\\Delta ^2 +\\Omega ^2} \\mbox{ }t\\right).$ Note that the coherence of the two states gives rise to an additional term which oscillates with optical frequencies and is hence unobservable.",
"Here $\\Omega $ is the Rabi frequency and the detuning $\\Delta =\\omega _{L}-\\omega _{10}$ .",
"$\\omega _{10}$ is the molecular transition frequency and $\\omega _{L}$ is the driven frequency of the laser.",
"We choose $\\omega _{L} \\approx \\omega _{10}$ to ensure a large excited-state population.",
"The potential for the molecule in the ground state is $U_{0}=U_{0e}+U_{0c}$ , and that for the molecule in the excited state is $U_{1}=U_{1e}+U_{1c}$ , where the electric components of the potential are given by $U_{0e}&=&\\frac{\\hbar \\mu _0}{2\\pi }\\int ^{\\infty }_0 d\\xi \\xi ^2 \\alpha (i\\xi ) \\mbox{tr}\\mathbf {G} (\\mathbf {r},\\mathbf {r}, i\\xi ),\\nonumber \\\\U_{1e}&=&-\\frac{\\hbar \\mu _0}{2\\pi }\\int ^{\\infty }_0 d\\xi \\xi ^2 \\alpha (i\\xi ) \\mbox{tr}\\mathbf {G} (\\mathbf {r},\\mathbf {r}, i\\xi )\\nonumber \\\\&&-\\frac{\\mu _0}{3}\\omega ^2_{10}|\\mathbf {d}_{01}|^2\\mbox{tr}[\\mbox{Re}\\mathbf {G}(\\mathbf {r},\\mathbf {r},\\omega _{10})],$ while its chiral components are given by [17] $U_{0c}&=&-\\frac{\\hbar \\mu _0}{\\pi }\\int _0^{\\infty }d\\xi \\xi \\Gamma (i\\xi ) \\mbox{tr}[\\nabla \\times \\mathbf {G} (\\mathbf {r},\\mathbf {r}, i\\xi )],\\nonumber \\\\U_{1c}&=&\\frac{\\hbar \\mu _0}{\\pi }\\int _0^{\\infty }d\\xi \\xi \\Gamma (i\\xi ) \\mbox{tr}[\\nabla \\times \\mathbf {G} (\\mathbf {r},\\mathbf {r}, i\\xi )]\\nonumber \\\\&&+\\frac{2\\mu _0\\omega _{10}R_{01}}{3}\\mbox{tr}[\\nabla \\times \\mbox{Re}[\\mathbf {G} (\\mathbf {r},\\mathbf {r},\\omega _{10})]],$ and $\\alpha (i\\xi ) =\\frac{2}{3\\hbar }\\frac{\\omega _{10}|\\mathbf {d}_{01}|^2}{\\omega ^2_{10}+\\xi ^2},\\quad \\Gamma (i\\xi ) =-\\frac{2}{3\\hbar }\\frac{\\xi R_{01}}{\\omega _{10}^2+\\xi ^2}.$ Here $\\hbar $ is Planck's constant, $\\mu _0$ is vacuum permeability, $\\mathbf {d}_{01}$ is the electric dipole transition matrix element, and the optical rotatory strength $R_{01}=\\mbox{Im} (\\mathbf {d}_{01}\\cdot \\mathbf {m}_{10})$ where $\\mathbf {m}_{10}$ is the corresponding magnetic dipole moment matrix element.",
"We have ignored the magnetic component of the potential which is a factor of $(|\\mathbf {m}_{10}|/|\\mathbf {d}_{01}|)^2\\sim \\alpha ^2_{f}$ smaller than the electric component, and is a factor of $|\\mathbf {m}_{10}|/|\\mathbf {d}_{01}|\\sim \\alpha _{f}$ smaller than the chiral component, with $\\alpha _{f}$ being the fine-structure constant.",
"Since the optical rotatory strengths of enantiomers are generally identical in magnitude but opposite in sign, the chiral components of the CP potential (REF ) take opposite signs for enantiomers.",
"Note that, when the average thermal photon number $n(\\omega _{10})=\\frac{1}{e^{\\hbar \\omega _{10}/(k_{B}T)}-1}$ is not negligible, (REF ) and (REF ) can be replaced by the thermal CP potentials which are written as [26], [29], [30], [31] $U_{0e}^{\\rm therm}&=&\\mu _0 k_{B} T\\displaystyle \\sum _{j=0}^{\\infty }\\left(1-\\frac{1}{2}\\delta _{j0}\\right)\\xi _{j}^{\\prime 2} \\alpha (i\\xi ^{\\prime }_{j}) \\mbox{tr}\\mathbf {G} (\\mathbf {r},\\mathbf {r},i\\xi ^{\\prime }_{j})\\nonumber \\\\&&+\\frac{\\mu _0}{3} n (\\omega _{10})\\omega ^2_{10}|\\mathbf {d}_{01}|^2\\mbox{tr}[\\mbox{Re}\\mathbf {G}(\\mathbf {r},\\mathbf {r},\\omega _{10})],\\nonumber \\\\U_{1e}^{\\rm therm}&=&-U_{0e}^{\\rm therm}\\nonumber \\\\&&-\\frac{\\mu _0}{3}[n(\\omega _{10})+1]\\omega _{10}^2|\\mathbf {d}_{01}|^2 \\mbox{tr}[\\mbox{Re} \\mathbf {G} (\\mathbf {r},\\mathbf {r},\\omega _{10})],\\nonumber \\\\U_{0c}^{\\rm therm}&=&-2\\mu _0 k_{B}T\\displaystyle \\sum _{j=0}^{\\infty }\\left(1-\\frac{1}{2}\\delta _{j0}\\right)\\xi _{j}^{\\prime } \\Gamma (i\\xi ^{\\prime }_{j}) \\nonumber \\\\&&\\times \\mbox{tr}[\\nabla \\times \\mathbf {G} (\\mathbf {r},\\mathbf {r},i\\xi )]\\nonumber \\\\&&-\\frac{2\\mu _0\\omega _{10}R_{01}}{3}n(\\omega _{10})\\mbox{tr}[\\nabla \\times \\mbox{Re}[\\mathbf {G} (\\mathbf {r},\\mathbf {r},\\omega _{10})],\\nonumber \\\\U_{1c}^{\\rm therm}&=&-U_{0c}^{\\rm therm}\\nonumber \\\\&&+\\frac{2\\mu _0\\omega _{10}R_{01}}{3}[n(\\omega _{10})+1]\\mbox{tr}[\\nabla \\times \\mbox{Re}[\\mathbf {G} (\\mathbf {r},\\mathbf {r},\\omega _{10})]],\\nonumber \\\\$ where $\\xi ^{\\prime }_{j}=2\\pi k_{B} Tj/\\hbar $ $(j=0,1,2,\\ldots )$ are Matsubara frequencies [32].",
"The thermal CP potentials have contributions proportional to $n(\\omega _{10})$ and $n(\\omega _{10})+1$ due to the absorption and emission of photons by the molecule respectively.",
"Different from (REF ) and (REF ), we see that a ground-state molecule also exhibits a resonant contribution associated with absorption of thermal photons.",
"The scattering Green's tensor $\\mathbf {G}(\\mathbf {r},\\mathbf {r},\\omega )$ for our cavity setup can be written as [33] $&&\\mathbf {G}(\\mathbf {r},\\mathbf {r},\\omega )=\\frac{i}{8\\pi ^2}\\int \\frac{d^2 k^{||}}{k^{\\perp }}\\nonumber \\\\&&\\times \\Biggr [\\displaystyle \\sum _{\\sigma _1 \\sigma _2=s,p}e^{2ik^{\\perp }z}(\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(\\sigma _1,\\sigma _2)}\\mathbf {e}_{\\sigma _1}(k^{\\perp })\\mathbf {e}_{\\sigma _2}(-k^{\\perp })\\nonumber \\\\&&+\\displaystyle \\sum _{\\sigma _1\\sigma _2=s,p}e^{2ik^{\\perp }(a-z)}(\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(\\sigma _1,\\sigma _2)}\\mathbf {e}_{\\sigma _1}(-k^{\\perp })\\mathbf {e}_{\\sigma _2}(k^{\\perp })\\Biggr ]\\nonumber \\\\$ where the polarization sum runs over $s-$ and $p-$ polarizations, $\\mathbf {D}=\\mathbf {I}-e^{2iak^{\\perp }}\\mathbf {R}\\cdot \\mathbf {R}^{\\prime }$ , $k^{||}$ and $k^{\\perp }$ are the components of the wave vector parallel and perpendicular to the surface of mirrors respectively and $k^{\\perp 2}=\\omega ^2/c^2-k^{||2}$ , $a$ is the distance between two mirrors (i.e., the width of the cavity).",
"We discarded terms that are independent of the position of the molecule $z$ ($0<z<a$ ).",
"Mirror A is located at $z=0$ while mirror B is located at $z=a$ .",
"$\\mathbf {R}=\\begin{pmatrix}r_{ss} & r_{sp} \\\\r_{ps} & r_{pp}\\end{pmatrix}=\\begin{pmatrix}-r_{e} & r_{c} \\\\-r_{c} & r_{e}\\end{pmatrix}$ and $\\mathbf {R}^{\\prime }=\\begin{pmatrix}r^{\\prime }_{ss} & r^{\\prime }_{sp} \\\\r^{\\prime }_{ps} & r^{\\prime }_{pp}\\end{pmatrix}=\\begin{pmatrix}-r_{e} & -r_{c} \\\\r_{c} & r_{e}\\end{pmatrix}$ are the reflection matrices of mirrors A and B, respectively, where the superscript $^{(\\sigma _1\\sigma _2)}$ represents the element $(\\sigma _1\\sigma _2)$ of the matrix.",
"The mirrors are assumed to have opposite chirality.",
"We adopt the convention that a left-handed mirror or a mirror of positive chirality rotates the polarisation of an electromagnetic wave in a counter-clockwise direction upon reflection when travelling alongside the wave, while a right-handed mirror of negative chirality rotates it in a clockwise direction.",
"With this convention, $r_c>0$ implies that mirror A is right-handed or of negative chirality while mirror B is left-handed or of positive chirality.",
"Note that we are neglecting the dependence of the reflection coefficients on frequency and on the in-plane component of the wave vector $\\textbf {k}^\\parallel $ .",
"This is justified as we are mainly interested in the dominant resonant part of the Casimir-Polder potential which is governed by a single transition frequency and travelling-wave contributions at normal incidence.",
"In practice, we will obtain the entries of the reflection matrix from laser reflection experiments on chiral materials currently under development.",
"Figure: (Color online) The electric and chiral component (experienced by the (R)(R)-propylene-oxide molecule with the positive R 01 R_{01}) of the CP potential with different temperatures.",
"r e =0.05,r_{e}=0.05, r c =0.8r_{c}=0.8.",
"The solid lines: the excited-state potential, the dotted lines: the ground-state potential.Figure: (Color online) The CP potential experienced by the driven 3-MCP molecule in a superposition of half-excited and half-ground state.",
"Solid red line: 3-MCP molecule with the positive R 01 R_{01}, dotted blue line: 3-MCP molecule with the negative R 01 R_{01}.In the case of linearly polarized light, the $p$ - and $s$ - polarized waves $\\mathbf {E}_{s}$ , $\\mathbf {E}_{p}$ are reflected as $\\mathbf {E}_{s}\\rightarrow r_{sp} \\mathbf {E}_{p}+r_{ss} \\mathbf {E}_{s}$ and $\\mathbf {E}_{p}\\rightarrow r_{pp} \\mathbf {E}_{p} + r_{ps} \\mathbf {E}_{s}$ by the chiral mirror respectively.",
"Normal mirrors have small or zero values for $r_{sp}$ and $r_{ps}$ .",
"Perfect chiral mirrors have $|r_{sp}|=|r_{ps}|=1$ and $|r_{ss}|=|r_{pp}|=0$ , which exhibits the strongest chiral dependence on the CP potential.",
"For circularly polarized light, normal mirrors in general reverse its polarization upon reflection, but the chiral mirrors conserve the polarization with positive or negative phase shifts."
],
[
"Electric Circular Dichroism",
"The 3-MCP molecules exhibit electric circular dichroism (ECD) by absorbing left- and right-circularly polarized light differently by electric transitions in the ultraviolet region.",
"We have four isomers of 3-MCP molecules with an equatorial (eq) or axial (ax) methyl group in the $(R)-$ or $(S)-$ configurations.",
"For the $S_0\\rightarrow S_1$ excitation of the 3-MCP molecule with an equatorial methyl group, we have the electric transition dipole $|\\mathbf {d}_{01}|=2.44\\times 10^{-31}$ Cm, the optical rotatory strength $|R_{01}/c|=8.07\\times 10^{-63}\\mbox{C}^2\\mbox{m}^2$ , and the vertical $S_0-S_1$ excitation energy $\\hbar \\omega _{10} = E_1-E_0=4.24\\,$ eV ($\\omega _{10}=6.44\\times 10^{15}\\mbox{s}^{-1}$ ) [34], [35].",
"Since the optical rotatory strength of the molecule with an equatorial methyl group is slightly larger than that of the molecule with an axial methyl group, we use the molecule with an equatorial methyl group for a quantitative analysis in this article.",
"Since the enantiomers have the optical rotatory strength of opposite sign and equal in magnitude for every transitions, the chiral component of the CP potential (REF ) can be used to separate them.",
"Here an $(R)$ -3-(eq)-MCP molecule and an $(S)$ -3-(eq)-MCP molecule have a positive and negative optical rotatory strength $R_{01}$ respectively.",
"Since the ratio $|\\mathbf {d}_{01}|^2/|R_{01}/c|=7.38$ is relatively small, the chiral component of the potential is comparable to its electric component for the molecules.",
"The average thermal photon number is generally very small for the case of ECD (i.e., $n(\\omega _{10})\\approx 10^{-72}$ for the $S_0\\rightarrow S_1$ excitation of the 3-MCP molecule at room temperature $T=298$ K), and the temperature dependence of the CP potential is usually negligible for the case of ECD.",
"The molecule in the excited state has the resonant contributions to the CP potential caused by real photons.",
"When the molecule is far away from the mirrors (i.e., $z, a-z>\\lambda _{10}=\\frac{2\\pi c}{\\omega _{10}}$ ), they are generally dominant over the non-resonant contributions from virtual photons.",
"For this reason, we introduced a laser field to drive the molecules to excite them.",
"In FIG.",
"REF , the electric component and the chiral component of the CP potential experienced by 3-MCP molecule near mirror A with the parameters above are plotted respectively with the cavity width $a=1\\,$ mm (Appendix A).",
"The chiral component in the figure is the one acting on the molecule with the positive $R_{01}$ (i.e., $(R)$ -3-(eq)-MCP molecule), and it changes sign for the molecule with the negative $R_{01}$ (i.e., $(S)$ -3-(eq)-MCP molecule).",
"Since mirror B has opposite chirality from mirror A, the molecule sees the chiral component with opposite sign near mirror B.",
"The magnitude of the electric component and that of the chiral component depend linearly on the reflection coefficient $r_{e}$ and $r_{c}$ respectively.",
"For our purposes, it is desirable to introduce nearly perfect chiral mirrors with $r_{e}\\approx 0$ and $r_{c}\\approx 1$ .",
"For the 3-MCP molecules, the magnitude of the chiral component is similar to that of the electric component when $r_{c}$ is around 10 times as large as $r_{e}$ .",
"In [36], an enhancement of the the electric component of the CP potential was observed when the cavity width $a$ is equal to a half integer multiple of the molecular transition wavelength, i.e., $a=\\nu \\lambda _{10}/2$ , $\\nu \\in \\mathbb {N}$ .",
"For the chiral component, it is found that such enhancement occurs when the cavity width $a=\\nu \\lambda _{10}/4$ where $\\nu $ is even/odd natural number when the two mirrors of the cavity have the same/opposite chirality respectively.",
"In FIG.",
"REF , the electric and chiral component of the CP potential with $r_{e}=0.05$ , $r_{c}=0.8$ and the cavity width $a=\\nu \\lambda _{10}/4$ is plotted.",
"It can be seen that the chiral component is enhanced when $\\nu $ is odd, i.e., $\\nu =7$ , while the electric component increases as $a$ becomes close to a half integer multiple of the molecular transition wavelength.",
"However this enhancement can only be seen if we have a very small cavity whose width contains only a few molecular transition wavelengths, and a strong enhancement only occurs when the reflection coefficients are very close to 1.",
"In this article, we choose a realistic setup with $a=1\\,$ mm where such enhancement can not be observed and rather the CP potential behaves as if it is the sum of two independent potentials created by two mirrors.",
"Then the potential far away from the mirrors is negligible, and it only becomes significant when the molecule is very close to the mirrors as shown in FIG.",
"REF .",
"Therefore our results presented in this article can be used even for the setup with a single chiral mirror, and the cavity setup here is only used for the purpose of constraining the molecular beam inside the cavity."
],
[
"Vibrational Circular Dichrosim",
"In contrast to cases of ECD, we often observe the temperature dependence of the CP potential caused by thermal photons in the infrared region when we use molecules with vibrational circular dichroism (VCD).",
"In FIG.",
"REF , the temperature dependence of the CP potential experienced by the $(R)$ -propylene-oxide molecule with $|\\mathbf {d}_{01}|=8.82\\times 10^{-32}$ Cm, $R_{01}/c=3.89\\times 10^{-67}\\mbox{C}^2\\mbox{m}^2$ (positive optical rotatory strength), and $\\omega _{10}=3.8\\times 10^{13}\\mbox{s}^{-1}$ [37] is shown as an example.",
"The reflection coefficients are set to be $r_{e}=0.05$ , $r_{c}=0.8$ .",
"Here we have the average thermal photon number $n(\\omega _{10})=0.6$ at room temperature and the CP potential increases with temperature.",
"As it can be seen in the figure, the use of molecules with vibrational circular dichroism has the advantage that the peaks of the CP potential appear far from the mirror due to the long wavelength of photons involved.",
"However it has challenges due to the weakness of the CP potential (REF ) which depends on the frequency of photons.",
"Therefore, in the next section, we use the 3-MCP molecules with the stronger CP potential (FIG.",
"REF ) in order to analyze our setup to separate enantiomers."
],
[
"Separation of enantiomers",
"Next, we study trajectories of the 3-MCP molecules in the cavity.",
"The CP force can be obtained by $F_{CP}=-dU_{CP}/dz$ where $U_{CP}$ is given in (REF ).",
"We set $r_{c}=0.8$ , $r_{e}=0.05$ and the width of the cavity $a=1\\,$ mm.",
"The mass of one 3-MCP molecule is approximately equal to $1.63\\times 10^{-25}$ kg.",
"We choose a laser intensity $I=5$ W/$\\mbox{cm}^2$ , a Rabi frequency $\\Omega =2|\\mathbf {d}_{01}|(2\\pi I/c)^{1/2}/\\hbar =1.42\\times 10^{7}\\mbox{s}^{-1}$ [38], and the detuning $\\Delta =\\omega _{L}-\\omega _{10}=2\\pi \\times 0.1$ MHz.",
"When $\\Delta \\le 0.1\\Omega $ , approximately half of the molecules are in the excited state, and the other half are in the ground state.",
"FIG.",
"REF shows the CP potential experienced by the molecule in such a superposition state.",
"Near mirror A, the molecules with the positive $R_{01}$ and those with the negative $R_{01}$ are subject to a high potential barrier $V_{+}$ ($1.26\\times 10^{-31}$ J) and a low potential barrier $V_{-}$ ($3.76\\times 10^{-32}$ J) respectively.",
"The differences in the potential barriers seen by these enantiomers come from the chiral component of the CP potential, and the behavior of the potential becomes opposite near mirror B which has the chirality opposite to that of mirror A.",
"$1.26\\times 10^{-31}$ J and $3.76\\times 10^{-32}$ J correspond to kinetic energy of the molecule with speed $1.2\\,$ mm/s and $0.7\\,$ mm/s respectively.",
"Therefore, near mirror A, if the initial speed of the molecule $|v_{z}|$ in the $\\hat{z}$ -direction (perpendicular to the plane of mirrors) is in between $0.7\\,$ mm/s and $1.2\\,$ mm/s, the molecule with the negative $R_{01}$ climbs up the low potential barrier and gets attracted to mirror A, while the one with the positive $R_{01}$ can not overcome the high potential barrier and gets repelled back to the opposite side of the cavity (FIG.",
"REF (a, below)).",
"The opposite behavior of the enantiomers can be observed near mirror B (FIG.",
"REF (a, above)).",
"When $|v_{0z}|$ is smaller than $0.7\\,$ mm/s, both enantiomers can not overcome the potential barrier and get reflected back to the center of the cavity (FIG.",
"REF (b)).",
"In this way, we aim to collect the molecules with the positive $R_{01}$ and those with the negative $R_{01}$ near mirror B and mirror A respectively.",
"Figure: (Color online) Trajectories of (a) the molecules with the positive R 01 R_{01} and (b) those with the negative R 01 R_{01} whose initial velocities are Gaussian distributed with mean 0.80.8\\,mm/s and standard deviation 0.10.1\\,mm/s.",
"(100 trajectories)We now consider the molecular beam initially injected to the center of the cavity (i.e., initial position of molecule $z_0=0.5\\,$ mm) with distribution for initial velocities in $\\hat{z}$ -direction being a Gaussian with mean $0\\,$ mm/s and standard deviation $0.4\\,$ mm/s.",
"The time evolution of positions of molecules can be obtained numerically (FIG.",
"REF (c)).",
"As can be seen, $\\sim 10\\%$ of the molecules with the positive $R_{01}$ and those with the negative $R_{01}$ get collected in the vicinity of mirror B and of mirror A respectively after $\\sim 1$ second.",
"A more efficient separation can be realized when the molecules have initial velocities toward one of the mirrors.",
"For example, if the distribution for initial velocities is a Gaussian with mean $0.8\\,$ mm/s and a standard deviation $0.1\\,$ mm/s so that the molecules move toward mirror B initially, around $90\\%$ of the molecules with the positive $R_{01}$ and those with the negative $R_{01}$ get collected near mirror B and mirror A respectively after $\\sim 1.5$ seconds, with the remaining $10\\%$ staying around the center of the cavity (FIG.",
"REF )."
],
[
"Conclusion",
"To summarise, we have presented a realistic proposal for a Stern–Gerlach type separator for chiral enantiomers based on discriminatory CP forces.",
"We have shown that the enantiomer-selective excited-state CP force due to electric molecular dichroism is most suitable for this purpose.",
"It exhibits pronounced repulsive potential barriers when using chiral mirrors which selectively repel only one of the two enantiomers while attracting the other.",
"By sending a molecular beam through a cavity formed by two mirrors of opposite chirality, each enantiomer gets attracted to one mirror and repelled from the other, leading to an efficient separation.",
"Our analysis shows that our setup can be used for separation of enantiomers, as well as for the detection of the chiral dependence of the CP potential within the reach of current technology.",
"Compared to conventional methods for chiral separations such as chromatography, our method proposed here is more universal since the same chiral mirror can be used for chiral separation of a wide range of molecules.",
"Two alternatives might be considered in order to potentially improve our scheme.",
"First, while we use an exciting laser parallel to the plates, one could also consider an evanescent laser field emerging from the chiral mirrors themselves.",
"The resulting light force could possibly be designed to enhance chiral effects.",
"Secondly, the use of curved mirrors would lead to a stronger mode confinement and associated enhancement of the CP potential.",
"This could be particularly useful for molecules with vibrational dichroism, where the potential is weak to start with, but cavity enhancement can be made particularly strong as the respective transitions are in the microwave regime.",
"We would like to thank P. Barcellona, R. Bennett, S. Fuchs and A. Salam for discussions.",
"This work was supported by the German Research Council (grants BU1803/3-1 and GRK 2079/1), a National Science and Engineering Research Discovery Grant in Canada, funds from Canada Foundation for Innovation for the Centre for Research on Ultra-Cold Systems (CRUCS) and Chirality Research on Origins and Separation (CHIROS) at UBC.",
"F. S. thanks DAAD Research Grants for support.",
"S.Y.B is grateful for support by the Freiburg Institute of Advanced Studies."
],
[
"Derivation of the Casimir-Polder Potential",
"Here we derive the electric and the chiral components of the Casimir-Polder (CP) potential experienced by a molecule in a planar cavity.",
"Taking the trace of the Green's function is equivalent to taking the dot product between the dyads where $\\mathbf {e}_{s}\\cdot \\mathbf {e}_{p} (-k^{\\perp })=\\mathbf {e}_{p}(k^{\\perp })\\cdot \\mathbf {e}_{s}=0$ and $\\mathbf {e}_{s}\\cdot \\mathbf {e}_{s}=1, \\quad \\mathbf {e}_{p}(k^{\\perp })\\cdot \\mathbf {e}_{p}(-k^{\\perp })=-1+\\frac{2k^{||2}c^2}{\\omega ^2}.$ Since $\\int d^2 k^{||}=2\\pi \\int _0^{\\infty } k^{||}dk^{||}$ , we have $&&\\mbox{tr}\\mathbf {G}(\\mathbf {r},\\mathbf {r},\\omega )=\\frac{i}{4\\pi }\\int \\frac{d k^{||}k^{||}}{k^{\\perp }}\\nonumber \\\\&&\\times \\Biggr [e^{2ik^{\\perp }z}\\left((\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(s,s)}-\\left(1-\\frac{2k^{||2}c^2}{\\omega ^2}\\right)(\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(p,p)}\\right)\\nonumber \\\\&&+e^{2ik^{\\perp }(a-z)}\\nonumber \\\\&&\\times \\left((\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(s,s)}-\\left(1-\\frac{2k^{||2}c^2}{\\omega ^2}\\right)(\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(p,p)}\\right)\\Biggr ]$ where we discarded the terms that are independent of the position of the molecule $z$ since we are eventually interested in the CP force given by $F_{CP}=-d U_{CP}/d z$ .",
"When we introduce $\\omega =i\\xi $ and $k^{\\perp }=i\\kappa ^{\\perp }$ ($\\kappa ^{\\perp }=\\sqrt{\\xi ^2/c^2+k^{||2}}$ ), we have $\\int _0^{\\infty } dk^{||} k^{||}/\\kappa ^{\\perp }=\\int _{\\xi /c}^{\\infty }d\\kappa ^{\\perp }$ and $&&\\mbox{tr}\\mathbf {G}(\\mathbf {r},\\mathbf {r},i\\xi )=\\frac{1}{4\\pi }\\int _{\\xi /c}^{\\infty } d\\kappa ^{\\perp }\\nonumber \\\\&&\\times \\Biggr [e^{-2\\kappa ^{\\perp }z}\\left((\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(s,s)}+\\left(1-\\frac{2\\kappa ^{\\perp 2}c^2}{\\xi ^2}\\right)(\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(p,p)}\\right)\\nonumber \\\\&&+e^{-2\\kappa ^{\\perp }(a-z)}\\nonumber \\\\&&\\times \\left((\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(s,s)}+\\left(1-\\frac{2\\kappa ^{\\perp 2}c^2}{\\xi ^2}\\right)(\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(p,p)}\\right)\\Biggr ].$ Therefore, $&&U_{0e}=\\frac{\\hbar \\mu _0}{8\\pi ^2}\\int _0^{\\infty }d\\xi \\xi ^2 \\alpha (i\\xi )\\int _{\\xi /c}^{\\infty } d\\kappa ^{\\perp }\\nonumber \\\\&&\\times \\Biggr [e^{-2\\kappa ^{\\perp }z}\\left((\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(s,s)}+\\left(1-\\frac{2\\kappa ^{\\perp 2}c^2}{\\xi ^2}\\right)(\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(p,p)}\\right)\\nonumber \\\\&&+e^{-2\\kappa ^{\\perp }(a-z)}\\nonumber \\\\&&\\times \\left((\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(s,s)}+\\left(1-\\frac{2\\kappa ^{\\perp 2}c^2}{\\xi ^2}\\right)(\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(p,p)}\\right)\\Biggr ],$ and $&&U_{1e}=-U_{0e}\\nonumber \\\\&&-\\frac{\\mu _0}{12\\pi }\\omega ^2_{10}|\\mathbf {d}_{01}|^2\\mbox{Re}\\Biggr \\lbrace i\\int _0^{\\infty }\\frac{d k^{||}k^{||}}{k^{\\perp }}\\nonumber \\\\&&\\times \\Biggr [e^{2ik^{\\perp }z}\\left((\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(s,s)}-\\left(1-\\frac{2k^{||2}c^2}{\\omega ^2_{10}}\\right)(\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(p,p)}\\right)\\nonumber \\\\&&+e^{2ik^{\\perp }(a-z)}\\nonumber \\\\&&\\times \\left((\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(s,s)}-\\left(1-\\frac{2k^{||2}c^2}{\\omega ^2_{10}}\\right)(\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(p,p)}\\right)\\Biggr ]\\Biggr \\rbrace .\\nonumber \\\\$ For the chiral components of the potential, we use $\\mathbf {a}\\times \\mathbf {bc}=(\\mathbf {a}\\times \\mathbf {b})\\mathbf {c}$ [39], then $\\nabla \\times \\mathbf {e}_{s}=-i\\frac{\\omega }{c} \\mathbf {e}_{p} (k^{\\perp }), \\quad \\nabla \\times \\mathbf {e}_{p}(k^{\\perp })=i\\frac{\\omega }{c}\\mathbf {e}_{s}.$ Therefore, $&&\\mbox{tr}[\\nabla \\times \\mathbf {G}(\\mathbf {r},\\mathbf {r},\\omega )]=\\frac{\\omega }{4\\pi c}\\int \\frac{d k^{||}k^{||}}{k^{\\perp }}\\nonumber \\\\&&\\times \\Biggr [e^{2ik^{\\perp }z}\\left((\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(s,p)}\\left(\\frac{2k^{||2}c^2}{\\omega ^2}-1\\right)-(\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(p,s)}\\right)\\nonumber \\\\&&+e^{2ik^{\\perp }(a-z)}\\Bigr ((\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(s,p)}\\left(\\frac{2k^{||2}c^2}{\\omega ^2}-1\\right)\\nonumber \\\\&&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad -(\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(p,s)}\\Bigr )\\Biggr ],$ and $&&\\mbox{tr}[\\nabla \\times \\mathbf {G}(\\mathbf {r},\\mathbf {r},i\\xi )]=-\\frac{\\xi }{4\\pi c}\\int ^{\\infty }_{\\xi /c}d\\kappa ^{\\perp }\\nonumber \\\\&&\\times \\Biggr [e^{-2\\kappa ^{\\perp }z}\\left((\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(s,p)}\\left(\\frac{2\\kappa ^{\\perp 2}c^2}{\\xi ^2}-1\\right)+(\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(p,s)}\\right)\\nonumber \\\\&&+e^{-2\\kappa ^{\\perp }(a-z)}\\Bigr ((\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(s,p)}\\left(\\frac{2\\kappa ^{\\perp 2}c^2-1}{\\xi ^2}\\right)\\nonumber \\\\&&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad +(\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(p,s)}\\Bigr )\\Biggr ].$ Finally we obtain $&&U_{0c}=\\frac{\\hbar \\mu _0}{4\\pi ^2c}\\int _0^{\\infty }d\\xi \\xi ^2 \\Gamma (i\\xi )\\int ^{\\infty }_{\\xi /c}d\\kappa ^{\\perp }\\nonumber \\\\&&\\times \\Biggr [e^{-2\\kappa ^{\\perp }z}\\left((\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(s,p)}\\left(\\frac{2\\kappa ^{\\perp 2}c^2}{\\xi ^2}-1\\right)+(\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(p,s)}\\right)\\nonumber \\\\&&+e^{-2\\kappa ^{\\perp }(a-z)}\\Bigr ((\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(s,p)}\\left(\\frac{2\\kappa ^{\\perp 2}c^2-1}{\\xi ^2}\\right)\\nonumber \\\\&&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad +(\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(p,s)}\\Bigr )\\Biggr ],$ and $&& U_{1c}=-U_{0c}\\nonumber \\\\&&+\\frac{\\mu _0\\omega _{10}^2R_{01}}{6\\pi c}\\mbox{Re}\\Biggr \\lbrace \\int \\frac{d k^{||}k^{||}}{k^{\\perp }}\\nonumber \\\\&&\\times \\Biggr [e^{2ik^{\\perp }z}\\left((\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(s,p)}\\left(\\frac{2k^{||2}c^2}{\\omega ^2_{10}}-1\\right)-(\\mathbf {D}^{-1}\\cdot \\mathbf {R})^{(p,s)}\\right)\\nonumber \\\\&&+e^{2ik^{\\perp }(a-z)}\\Bigr ((\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(s,p)}\\left(\\frac{2k^{||2}c^2}{\\omega ^2_{10}}-1\\right)\\nonumber \\\\&&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad -(\\mathbf {R}^{\\prime }\\cdot \\mathbf {D}^{-1})^{(p,s)}\\Bigr )\\Biggr ]\\Biggr \\rbrace .$"
]
] | 1808.08642 |
[
[
"Perfect Optical Nonreciprocity in a Double-Cavity Optomechanical System"
],
[
"Abstract Nonreciprocal devices are indispensable for building quantum networks and ubiquitous in modern communication technology.",
"Here, we use optomechanical interaction and linearly-coupled interaction to realize optical nonreciprocal transmission in a double-cavity optomechanical system.",
"The scheme relies on the interference between the two interactions.",
"We derive the essential conditions to realize perfect optical nonreciprocity in the system, and analyse the properties of optical nonreciprocal transmission and the output fields from mechanical mode.",
"These results can be used to control optical transmission in quantum information processing."
],
[
"Introduction",
"Nonreciprocal optical devices, such as isolators and circulators, allow transmission of signals to exhibit different characteristics if source and observer are interchanged.",
"They are essential to several applications in quantum signal processing and communication, as they can suppress spurious modes and unwanted signals [1].",
"For example, they can protect devices from noise emanating from readout electronics in quantum superconducting circuits.",
"To violate reciprocity and obtain asymmetric transmission, breaking time-reversal symmetry is required in any such device.",
"Traditionally, nonreciprocal transmission has relied on applied magnetic bias fields to break time-reversal symmetry and Lorentz reciprocity [2], [3].",
"These conventional devices are typically bulky and incompatible with ultra-low loss superconducting circuits.",
"Many alternative methods recently have been proposed to replace conventional nonreciprocal schemes, such as usage of coupled-mode systems [4], [5], Brillouin scattering [6], and spatiotemporal modulation of the refractive index [7].",
"These schemes are particularly promising because they can be integrated on-chip with existing superconducting technology.",
"In recent years, the rapidly growing field of cavity optomechanics [8], [9], [10], where optical fields and mechanical resonators are coupled through radiation pressure, has shown promising potential for applications in quantum information processing and communication.",
"So far, many interesting quantum phenomena have been studied in this field, such as mechanical ground-state cooling [11], [12], [13], optomechanically induced transparency [14], [15], [16], [17], [18], [19], [20], entanglement [21], [22], [23], [24], [25], [26], [27], [28], nonlinear effects [29], [30], and coherent perfect absorption [31], [32], [33].",
"Very recently, it has been realized that optomechanical coupling can lead to nonreciprocal transmission and optical isolation.",
"In Refs.",
"[34], [35], [36], [37], [38], nonreciprocal optical responses are theoretically predicted through optomechanical interactions, and nonreciprocal transmission spectra were recently observed in Refs.",
"[39], [40], [41], [42], [43], [44], [45].",
"In Refs.",
"[48], [46], [47], [49], it was recognized that the mechanically-mediated quantum-state transfer between two cavity modes can be made nonreciprocal with suitable optical driving.",
"In Ref.",
"[50], nonreciprocal quantum-limited amplification of microwave signals has been proposed in an optomechanical system.",
"Besides, in Refs.",
"[51], [52], phonon circulators and thermal diodes are theoretically predicted through optomechanical coupling.",
"In most of these references, perfect optical nonreciprocity can be achieved under the conditions of equal damping rate (mechanical damping rate $\\gamma $ is equal to cavity damping rate $\\kappa $ ) or nonreciprocal phase difference $\\theta =\\pm \\frac{\\pi }{2}$ .",
"Figure: Adouble-cavity optomechanical system with a mechanical resonator interactedwith two cavities.",
"Two strong coupling fields (probe fields) with amplitudes ε c \\varepsilon _{c} andε d \\varepsilon _{d} (ε L \\varepsilon _{L} and ε R \\varepsilon _{R}) are used to drivethe system from the left and right fixed mirror respectively.",
"Meanwhile, the two cavities are linearly coupled to each other with coupling strength JJ.Here, we show that perfect optical nonreciprocity can be achieved under more general conditions, using the example of a double-cavity system in Fig.",
"1.",
"This setup has been realized in several recent experiments [53], [54], [55], and quantum nonlinearity [56] has been theoretically studied in this setup.",
"With this simple model, we can easily capture the essential mechanisms about nonreciprocity, i.e., quantum interference of signal transmission between two possible paths corresponding to two interactions (optomechanical interaction and linearly-coupled interaction).",
"From the expressions of output fields, we derive essential conditions to achieve perfect optical nonreciprocity, and find some interesting results.",
"One of them is that mechanical decay rate does not influence the appearance of perfect optical nonreciprocity, which means perfect optical nonreciprocity can still occur in the realistic parameter regime ($\\gamma \\ll \\kappa $ ) in cavity optomechanics.",
"Another interesting result is that perfect optical nonreciprocity can be achieved with any phase difference $\\theta $ ($\\theta \\ne 0,\\pi $ ) as long as rotating wave approximation is valid.",
"We believe the results of this paper can be used to control optical transmission in modern communication technology."
],
[
"System model and equations",
"The system we considered here is depicted in Fig.",
"1, where a mechanical membrane is placed in the middle of an optical cavity.",
"The operators $c_{1}$ and $c_{2}$ denote geometrically distinct optical modes with same frequency $\\omega _{\\text{0}}$ and decay rates $\\kappa _{1}$ and $\\kappa _{2}$ respectively.",
"The coupling $J$ describes photon tunneling through the membrane, and the interaction between the two cavities is described by $\\hbar J(c^{\\dag }_{1}c_{2}+c^{\\dag }_{2}c_{1})$ .",
"The mechanical resonator with frequency $\\omega _{\\text{m}}$ and decay rate $\\gamma $ is described by operator $b$ .",
"Two strong coupling fields (probe fields) with same frequency $\\omega _{c}$ ($\\omega _{p}$ ) and amplitudes $\\varepsilon _{c}$ and $\\varepsilon _{d}$ ($\\varepsilon _{L}$ and $\\varepsilon _{R}$ ) are used to drive the double-cavity system from the left and right fixed mirror respectively.",
"Then the total Hamiltonian in the rotating-wave frame of coupling frequency $\\omega _{c}$ can be written as ($\\hbar =1$ ) $H & =\\Delta _{c}(c_{1}^{\\dag }c_{1}+c_{2}^{\\dag }c_{2})+\\omega _{\\text{m}}b^{\\dag }b+g_{0}(c_{2}^{\\dag }c_{2}-c_{1}^{\\dag }c_{1})(b^{\\dag }+b)\\nonumber \\\\& +J(c_{1}^{\\dag }c_{2}+c_{2}^{\\dag }c_{1})+i(\\varepsilon _{c}c_{1}^{\\dag }-\\varepsilon ^{\\ast }_{c}c_{1})+i(\\varepsilon _{d}c_{2}^{\\dag }-\\varepsilon ^{\\ast }_{d}c_{2})\\nonumber \\\\& +i\\varepsilon _{L}(c_{1}^{\\dag }e^{-i\\delta t}-c_{1}e^{i\\delta t})+i\\varepsilon _{R}(c_{2}^{\\dag }e^{-i\\delta t}-c_{2}e^{i\\delta t}) .$ Here, $\\Delta _{c}=\\omega _{0}-\\omega _{c}$ ($\\delta =\\omega _{p}-\\omega _{c}$ ) is the detuning between cavity modes (probe fields) and coupling fields, and $g_{0}$ is the single photon coupling constant between mechanical resonator and optical modes.",
"The dynamics of the system is described by the quantum Langevin equations for the relevant operators of the mechanical and optical modes $\\dot{c}_{1} & =-[i\\Delta _{c}+\\frac{\\kappa _{1}}{2}-ig_{0}(b^{\\dag }+b)]c_{1}+\\varepsilon _{c}+\\varepsilon _{L}e^{-i\\delta t}-iJc_{2},\\nonumber \\\\\\dot{c}_{2} & =-[i\\Delta _{c}+\\frac{\\kappa _{2}}{2}+ig_{0}(b^{\\dag }+b)]c_{2}+\\varepsilon _{d}+\\varepsilon _{R}e^{-i\\delta t}-iJc_{1},\\nonumber \\\\\\dot{b} & =-i\\omega _{m}b-\\frac{\\gamma }{2}b-ig_{0}(c_{2}^{\\dag }c_{2}-c_{1}^{\\dag }c_{1}).",
"$ In the absence of probe fields $\\varepsilon _{L}$ , $\\varepsilon _{R}$ and with the factorization assumption $\\langle bc_{i}\\rangle =\\langle b\\rangle \\langle c_{i}\\rangle $ , we can obtain the steady-state mean values $\\langle b\\rangle & =b_{s}=\\frac{-ig_{0}(\\left|c_{2s}\\right|^{2}-\\left|c_{1s}\\right|^{2})}{\\frac{\\gamma }{2}+i\\omega _{m}},\\nonumber \\\\\\left\\langle c_{1}\\right\\rangle & =c_{1s}=\\frac{(\\frac{\\kappa _{2}}{2}+i\\Delta _{2})\\varepsilon _{c}-iJ\\varepsilon _{d}}{J^{2}+(\\frac{\\kappa _{1}}{2}+i\\Delta _{1})(\\frac{\\kappa _{2}}{2}+i\\Delta _{2})},\\nonumber \\\\\\left\\langle c_{2}\\right\\rangle & =c_{2s}=\\frac{(\\frac{\\kappa _{1}}{2}+i\\Delta _{1})\\varepsilon _{d}-iJ\\varepsilon _{c}}{J^{2}+(\\frac{\\kappa _{1}}{2}+i\\Delta _{1})(\\frac{\\kappa _{2}}{2}+i\\Delta _{2})} $ with $\\Delta _{1,2}=\\Delta _{c}\\mp g_{0}(b_{s}+b_{s}^{\\ast })$ denoting the effective detunings between cavity modes and coupling fields.",
"In the presence of both probe fields, however, we can write each operator as the sum of its mean value and its small fluctuation, i.e., $b=b_{s}+\\delta b$ , $c_{1}=c_{1s}+\\delta c_{1}$ , $c_{2}=c_{2s}+\\delta c_{2}$ to solve Eq.",
"(2) when both coupling fields are sufficiently strong.",
"Then keeping only the linear terms of fluctuation operators and moving into an interaction picture by introducing $\\delta b\\rightarrow \\delta be^{-i\\omega _{m}t}$ , $\\delta c_{1}\\rightarrow \\delta c_{1}e^{-i\\Delta _{1}t}$ , $\\delta c_{2}\\rightarrow \\delta c_{2}e^{-i\\Delta _{2}t}$ , we obtain the linearized quantum Langevin equations $\\delta \\dot{c}_{1} & =-\\frac{\\kappa _{1}}{2}\\delta c_{1}+iG_{1}(\\delta b^{\\dag }e^{i(\\omega _{m}+\\Delta _{1})t}+\\delta be^{-i(\\omega _{m}-\\Delta _{1})t})\\nonumber \\\\& +\\varepsilon _{L}e^{-i(\\delta -\\Delta _{1})t}-iJ\\delta c_{2}e^{i(\\Delta _{1}-\\Delta _{2})t},\\nonumber \\\\\\delta \\dot{c}_{2} & =-\\frac{\\kappa _{2}}{2}\\delta c_{2}-iG_{2}e^{i\\theta }(\\delta b^{\\dag }e^{i(\\omega _{m}+\\Delta _{2})t}+\\delta be^{-i(\\omega _{m}-\\Delta _{2})t})\\nonumber \\\\& +\\varepsilon _{R}e^{-i(\\delta -\\Delta _{2})t}-iJ\\delta c_{1}e^{i(\\Delta _{2}-\\Delta _{1})t},\\nonumber \\\\\\delta \\dot{b} & =-\\frac{\\gamma }{2}\\delta b+iG_{1}(\\delta c_{1}e^{i(\\omega _{m}-\\Delta _{1})t}+\\delta c^{\\dag }_{1}e^{i(\\omega _{m}+\\Delta _{1})t})\\nonumber \\\\& -iG_{2}(e^{-i\\theta }\\delta c_{2}e^{i(\\omega _{m}-\\Delta _{2})t}+e^{i\\theta }\\delta c^{\\dag }_{2}e^{i(\\Delta _{2}+\\omega _{m})t}) $ with $G_{1}=g_{0}c_{1s}$ and $G_{2}=g_{0}c_{2s}e^{-i\\theta }$ .",
"The phase difference $\\theta $ between effective optomechanical coupling $g_{0}c_{1s}$ and $g_{0}c_{2s}$ can be controlled by adjusting the coupling fields amplitudes $\\varepsilon _{c}$ and $\\varepsilon _{d}$ according to Eq.",
"(3).",
"It will be seen that the phase difference $\\theta $ is a critical factor to attain optical nonreciprocity.",
"Without loss of generality, we take $G_{i}$ and $J$ as positive number (not negative to avoid introducing unimportant phase difference $\\pi $ ).",
"If each coupling field drives one cavity mode at the mechanical red sideband $(\\Delta _{1}\\approx \\Delta _{2}\\approx \\omega _{m})$ , and the mechanical frequency $\\omega _{m}$ is much larger than $g_{0}|c_{1s}|$ and $g_{0}|c_{2s}|$ , then Eq.",
"(4) will be simplified to $\\delta \\dot{c}_{1} & =-\\frac{\\kappa _{1}}{2}\\delta c_{1}+iG_{1}\\delta b-iJ\\delta c_{2}+\\varepsilon _{L}e^{-ixt},\\nonumber \\\\\\delta \\dot{c}_{2} & =-\\frac{\\kappa _{2}}{2}\\delta c_{2}-iG_{2}e^{i\\theta }\\delta b-iJ\\delta c_{1}+\\varepsilon _{R}e^{-ixt},\\nonumber \\\\\\delta \\dot{b} & =-\\frac{\\gamma }{2}\\delta b+iG_{1}\\delta c_{1}-iG_{2}e^{-i\\theta }\\delta c_{2} $ with $x=\\delta -\\omega _{m}$ .",
"For simplicity, we set equal damping rate $\\kappa _{1}=\\kappa _{2}=\\kappa $ and equal coupling $G_{1}=G_{2}=G$ in the following (actually, it can be proven that $G_{1}$ must equal $G_{2}$ if $\\kappa _{1}=\\kappa _{2}$ when the system exhibits perfect optical nonreciprocity).",
"By assuming $\\delta s=\\delta s_{+}e^{-ixt}+\\delta s_{-}e^{ixt}$ ($s=b,c_{1},c_{2}$ ), we can solve Eq.",
"(5) as follows $\\delta b_{+} & =\\frac{4G[(i\\kappa _{x}-2Je^{-i\\theta })\\varepsilon _{L}+(2J-i\\kappa _{x}e^{-i\\theta })\\varepsilon _{R}]}{8G^{2}\\kappa _{x}+(4J^{2}+\\kappa ^{2}_{x})\\gamma _{x}+16iG^{2}J\\cos \\theta },\\nonumber \\\\\\delta c_{1+} & =\\frac{2(4G^{2}+\\gamma _{x}\\kappa _{x})\\varepsilon _{L}+(8G^{2}e^{-i\\theta }-4iJ\\gamma _{x})\\varepsilon _{R}}{8G^{2}\\kappa _{x}+(4J^{2}+\\kappa ^{2}_{x})\\gamma _{x}+16iG^{2}J\\cos \\theta },\\nonumber \\\\\\delta c_{2+} & =\\frac{2(4G^{2}+\\gamma _{x}\\kappa _{x})\\varepsilon _{R}+(8G^{2}e^{i\\theta }-4iJ\\gamma _{x})\\varepsilon _{L}}{8G^{2}\\kappa _{x}+(4J^{2}+\\kappa ^{2}_{x})\\gamma _{x}+16iG^{2}J\\cos \\theta } $ where $\\gamma _{x}=\\gamma -2ix$ , $\\kappa _{x}=\\kappa -2ix$ , and $\\delta s_{-}=0$ .",
"To study optical nonreciprocity, we must study the output optical fields $\\varepsilon ^{out}_{L}$ and $\\varepsilon ^{out}_{R}$ which can be obtained according to the input-output relation [31], [32], [57] $\\varepsilon ^{out}_{L}+\\varepsilon ^{in}_{L}e^{-ixt} & =\\sqrt{\\kappa }\\delta c_{1}\\nonumber \\\\\\varepsilon ^{out}_{R}+\\varepsilon ^{in}_{R}e^{-ixt} & =\\sqrt{\\kappa }\\delta c_{2},$ here, $\\varepsilon ^{in}_{L,R}=\\varepsilon _{L,R}/\\sqrt{\\kappa }$ .",
"Still following the assumption $\\delta s=\\delta s_{+}e^{-ixt}+\\delta s_{-}e^{ixt}$ , the output fields can be obtained as $\\varepsilon ^{out}_{L+} & =\\sqrt{\\kappa }\\delta c_{1+}-\\varepsilon _{L}/\\sqrt{\\kappa }\\nonumber \\\\\\varepsilon ^{out}_{R+} & =\\sqrt{\\kappa }\\delta c_{2+}-\\varepsilon _{R}/\\sqrt{\\kappa }$ and $\\varepsilon ^{out}_{L-}=\\varepsilon ^{out}_{R-}=0$ ."
],
[
"Perfect optical nonreciprocity",
"Perfect optical nonreciprocity can be achieved if transmission amplitudes $T_{i\\rightarrow j}$ ($i,j=L,R$ ) satisfy $T_{L\\rightarrow R}=\\left|\\frac{\\varepsilon _{R}^{out}}{\\varepsilon ^{in}_{L}}\\right|_{\\varepsilon ^{in}_{R}=0}=1,T_{R\\rightarrow L}=\\left|\\frac{\\varepsilon _{L}^{out}}{\\varepsilon ^{in}_{R}}\\right|_{\\varepsilon ^{in}_{L}=0}=0,$ or $T_{L\\rightarrow R}=\\left|\\frac{\\varepsilon _{R}^{out}}{\\varepsilon ^{in}_{L}}\\right|_{\\varepsilon ^{in}_{R}=0}=0,T_{R\\rightarrow L}=\\left|\\frac{\\varepsilon _{L}^{out}}{\\varepsilon ^{in}_{R}}\\right|_{\\varepsilon ^{in}_{L}=0}=1.$ It means that the input signal from one side can be completely transmitted to the other side, but not vice versa.",
"What the Eq.",
"(9a) and (9b) represent is the two different directions of isolation.",
"Here, we just discuss the case of Eq.",
"(9a), as the case of Eq.",
"(9b) is similar.",
"The subscript $\\varepsilon ^{in}_{R/L}=0$ indicates there is not signal injected into the system from right/left side.",
"We omit the subscripts because, in general, nonreciprocity is only related to one-way input, and write transmission amplitudes $T_{i\\rightarrow j}$ as $T_{ij}$ for simplicity in the following.",
"According to Eq.",
"(6) and Eq.",
"(8), the two optical output fields can be obtained as $\\frac{\\varepsilon _{R+}^{out}}{\\varepsilon ^{in}_{L}}&=\\frac{4\\kappa (2G^{2}e^{i\\theta }-iJ\\gamma _{x})}{8G^{2}\\kappa _{x}+(4J^{2}+\\kappa ^{2}_{x})\\gamma _{x}+16iG^{2}J\\cos \\theta },\\nonumber \\\\\\frac{\\varepsilon _{L+}^{out}}{\\varepsilon ^{in}_{R}}&=\\frac{4\\kappa (2G^{2}e^{-i\\theta }-iJ\\gamma _{x})}{8G^{2}\\kappa _{x}+(4J^{2}+\\kappa ^{2}_{x})\\gamma _{x}+16iG^{2}J\\cos \\theta }.$ When $\\theta =n\\pi $ ($n$ is an integer), the two output fields are equal, which indicates the photon transmission is reciprocal.",
"But in the other cases, where $\\theta \\ne n\\pi $ , the system will exhibit a nonreciprocal response.",
"It can be clearly seen from the numerator of Eq.",
"(10) that the optical nonreciprocity comes from quantum interference between the optomechanical interaction $G$ and the linearly-coupled interaction $J$ .",
"With Eq.",
"(10), we find perfect optical nonreciprocity Eq.",
"(9a) can be achieved only when $J=-\\frac{e^{\\mp i\\theta }(\\gamma \\cot \\theta \\pm i\\kappa )}{2}$ which can take positive real number only if $\\theta =-\\frac{\\pi }{2},$ or $\\kappa =\\gamma .",
"$ In the following, we will discuss perfect optical nonreciprocity in two cases, Eq.",
"(12a) and (12b), respectively."
],
[
"Phase difference $\\theta =-\\frac{\\pi }{2}$",
"With nonreciprocal phase difference $\\theta =-\\frac{\\pi }{2}$ , the two optical output fields in Eq.",
"(10) now become $\\frac{\\varepsilon _{R+}^{out}}{\\varepsilon ^{in}_{L}}&=\\frac{-4i\\kappa (2G^{2}+J\\gamma _{x})}{8G^{2}\\kappa _{x}+(4J^{2}+\\kappa ^{2}_{x})\\gamma _{x}},\\nonumber \\\\\\frac{\\varepsilon _{L+}^{out}}{\\varepsilon ^{in}_{R}}&=\\frac{4i\\kappa (2G^{2}-J\\gamma _{x})}{8G^{2}\\kappa _{x}+(4J^{2}+\\kappa ^{2}_{x})\\gamma _{x}}.$ According to Eq.",
"(13), the perfect optical nonreciprocity Eq.",
"(9a) can be achieved only when $x&=0, \\nonumber \\\\J&=\\frac{\\kappa }{2},\\nonumber \\\\G&=\\frac{\\sqrt{\\kappa \\gamma }}{2}.$ It is surprising that there is not any restriction on mechanical decay rate $\\gamma $ in Eq.",
"(14).",
"In other words, mechanical decay rate $\\gamma $ does not influence perfect optical nonreciprocity, which means that perfect optical nonreciprocity can still occur even in the case of $\\gamma /\\kappa \\rightarrow 0$ as long as the conditions Eq.",
"(14) is satisfied.",
"This is important because, in general, mechanical decay rate $\\gamma $ is much less than cavity decay rate $\\kappa $ in cavity optomechanics.",
"In addition, even with very weak optomechanical coupling $(G\\ll \\kappa )$ , perfect optical nonreciprocity can still occur as $\\gamma \\ll \\kappa $ according to Eq.",
"(14).",
"In Fig.",
"2(a)–2(d), we plot transmission amplitudes $T_{LR}$ (red line) and $T_{RL}$ (black line) vs normalized detuning $x/\\kappa $ with $J=\\frac{\\kappa }{2}$ , $G=\\frac{\\sqrt{\\kappa \\gamma }}{2}$ for $\\gamma /\\kappa =2$ , 1, $1/5$ , $1/100$ respectively.",
"It can be clearly seen from Fig.",
"2 that mechanical decay rate $\\gamma $ indeed does not affect the appearance of perfect optical nonreciprocity, but can strongly affect the width of transmission spectrum, especially for the case of $\\gamma \\ll \\kappa $ .",
"The two curves of transmission amplitudes $T_{LR}$ and $T_{RL}$ tend to coincide outside the vicinity of resonance frequency ($x=0$ ) when $\\gamma /\\kappa \\rightarrow 0$ .",
"But the system always exhibits optical nonreciprocity near resonance frequency in the case, such as $\\gamma /\\kappa =1/100$ [see Fig.",
"2(d)].",
"By the way, the perfect optical nonreciprocity Eq.",
"(9b) will occur if $\\theta =\\frac{\\pi }{2}$ .",
"Figure: Normalized coupling strengths G/γG/\\gamma (J=GJ=G) (blue line) and detuning x/γx/\\gamma (yellow line) are plotted vs phase difference θ\\theta according to Eq.",
"(16)."
],
[
"Equal damping rate $\\kappa =\\gamma $",
"With equal damping rate $\\kappa =\\gamma $ , the two optical output fields Eq.",
"(10) now become $\\frac{\\varepsilon _{R+}^{out}}{\\varepsilon ^{in}_{L}}&=\\frac{4\\gamma (2e^{i\\theta }G^{2}-iJ\\gamma _{x})}{(8G^{2}+4J^{2}+\\gamma _{x}^{2})\\gamma _{x}+16iG^{2}J\\cos \\theta },\\nonumber \\\\\\frac{\\varepsilon _{L+}^{out}}{\\varepsilon ^{in}_{R}}&=\\frac{4\\gamma (2e^{-i\\theta }G^{2}-iJ\\gamma _{x})}{(8G^{2}+4J^{2}+\\gamma _{x}^{2})\\gamma _{x}+16iG^{2}J\\cos \\theta }.$ Figure: Transmission amplitudes T LR T_{LR} (red line) and T RL T_{RL} (black line) are plotted vs normalized detuning x/γx/\\gamma for different phase difference: (a) θ=-3π 4\\theta =-\\frac{3\\pi }{4}, (b) θ=-π 4\\theta =-\\frac{\\pi }{4}, (c) θ=π 4\\theta =\\frac{\\pi }{4}, and (d) θ=3π 4\\theta =\\frac{3\\pi }{4}.",
"The coupling strengths G=±γcscθ 2G=\\pm \\frac{\\gamma \\csc \\theta }{2} (J=GJ=G) according to Eq.",
"(16).From Eq.",
"(15), we can obtain the conditions for perfect optical nonreciprocity as follows $x&=\\pm \\frac{\\gamma \\cot \\theta }{2},\\nonumber \\\\ J&=\\pm \\frac{\\gamma \\csc \\theta }{2},\\nonumber \\\\G&=\\pm \\frac{\\gamma \\csc \\theta }{2}$ where the negative sign and $\\theta \\in (\\pi ,2\\pi )$ meet Eq.",
"(9a), and positive sign and $\\theta \\in (0,\\pi )$ meet Eq.",
"(9b).",
"It means that we can change the direction of isolation by adjusting the nonreciprocal phase difference $\\theta \\in (0,\\pi )$ or $(\\pi ,2\\pi )$ .",
"In Fig.",
"3, we plot the normalized coupling strengths $G/\\gamma $ , $J/\\gamma $ ($J=G$ ) (blue line) and detuning $x/\\gamma $ (yellow line) vs phase difference $\\theta $ according to Eq.",
"(16).",
"For the special case of $\\theta =\\pm \\frac{\\pi }{2}$ , the coupling strength $G$ ($J$ ) takes the minimum value $\\frac{\\gamma }{2}$ and detuning $x=0$ (see Fig.",
"3), and the transmission spectrums $T_{LR}$ and $T_{RL}$ take a symmetric form with respect to detuning $x$ [see Fig.",
"2(b)].",
"From Eq.",
"(16), we can see that perfect optical nonreciprocity can occur with any phase $\\theta $ ($\\theta \\ne n\\pi $ ) as long as $G\\ll \\omega _{m}$ ($|\\sin \\theta |\\gg \\frac{\\gamma }{2\\omega _{m}}$ ) where rotating wave approximation is valid.",
"It means the strongest quantum interference takes place at detuning $x=\\pm \\frac{\\gamma \\cot \\theta }{2}$ in the case of $\\kappa =\\gamma $ .",
"In Fig.",
"4(a)–4(d), we plot the transmission amplitudes $T_{LR}$ (red line) and $T_{RL}$ (black line) vs normalized detuning $x/\\gamma $ with $G=\\pm \\frac{\\gamma \\csc \\theta }{2}$ ($J=G$ ) for $\\theta =-\\frac{3\\pi }{4}$ , $-\\frac{\\pi }{4}$ , $\\frac{\\pi }{4}$ , $\\frac{3\\pi }{4}$ , respectively.",
"It can be seen from Fig.",
"4, the transmission spectrums $T_{LR}$ and $T_{RL}$ will not take the symmetric form anymore as $\\theta \\ne \\pm \\frac{\\pi }{2}$ , and $T_{LR}>T_{RL}$ for $\\theta \\in (-\\pi ,0)$ , $T_{LR}<T_{RL}$ for $\\theta \\in (0,\\pi )$ ."
],
[
"conclusion",
"In summary, we have theoretically studied how to achieve perfect optical nonreciprocity in a double-cavity optomechanical system.",
"In this paper, we focus on the conditions where the system can exhibit perfect optical nonreciprocity.",
"From the expressions of condition, we can draw three important conclusions: (1) when nonreciprocal phase difference $\\theta =\\pm \\frac{\\pi }{2}$ , the mechanical damping rate has no effect on the appearance of perfect optical nonreciprocity as long as Eq.",
"(14) is satisfied; (2) even with very weak optomechanical coupling $(G\\ll \\kappa )$ , perfect optical nonreciprocity can still occur according to Eq.",
"(14); (3) the system can exhibit perfect optical nonreciprocity with any nonreciprocal phase difference $\\theta $ $(\\theta \\ne 0,\\pi )$ if $\\kappa =\\gamma $ and Eq.",
"(16) is satisfied.",
"Our results can also be applied to other parametrically coupled three-mode bosonic systems, besides optomechanical systems.",
"L. Yang is supported by National Natural Science Foundation of China (Grant No.",
"11804066), the China Postdoctoral Science Foundation (Grant No.",
"2018M630337), and Fundamental Research Funds for the Central Universities (Grant No.",
"3072019CFM0405)."
]
] | 1808.08527 |
[
[
"Detection and Mitigation of Attacks on Transportation Networks as a\n Multi-Stage Security Game"
],
[
"Abstract In recent years, state-of-the-art traffic-control devices have evolved from standalone hardware to networked smart devices.",
"Smart traffic control enables operators to decrease traffic congestion and environmental impact by acquiring real-time traffic data and changing traffic signals from fixed to adaptive schedules.",
"However, these capabilities have inadvertently exposed traffic control to a wide range of cyber-attacks, which adversaries can easily mount through wireless networks or even through the Internet.",
"Indeed, recent studies have found that a large number of traffic signals that are deployed in practice suffer from exploitable vulnerabilities, which adversaries may use to take control of the devices.",
"Thanks to the hardware-based failsafes that most devices employ, adversaries cannot cause traffic accidents directly by setting compromised signals to dangerous configurations.",
"Nonetheless, an adversary could cause disastrous traffic congestion by changing the schedule of compromised traffic signals, thereby effectively crippling the transportation network.",
"To provide theoretical foundations for the protection of transportation networks from these attacks, we introduce a game-theoretic model of launching, detecting, and mitigating attacks that tamper with traffic-signal schedules.",
"We show that finding optimal strategies is a computationally challenging problem, and we propose efficient heuristic algorithms for finding near optimal strategies.",
"We also introduce a Gaussian-process based anomaly detector, which can alert operators to ongoing attacks.",
"Finally, we evaluate our algorithms and the proposed detector using numerical experiments based on the SUMO traffic simulator."
],
[
"Introduction",
"The evolution of traffic signals from standalone hardware devices to complex networked systems has provided society with many benefits, such as reducing wasted time and environmental impact.",
"However, it has also exposed traffic signals to a variety of cyber-attacks.",
"While traditional hardware systems were susceptible only to attacks based on direct physical access, modern systems are vulnerable to attacks through wireless interfaces or even to remote attacks through the Internet.",
"To assess the severity of these threats in practice, Ghena et al.",
"recently analyzed the security of real-world traffic infrastructure in cooperation with a road agency located in Michigan [1].",
"This agency operates around a hundred traffic signals, which are all part of the same wireless network, but the signals at every intersection operate independently of the other intersections.",
"The study found three major weaknesses in the traffic infrastructure: lack of encryption for the wireless network, lack of secure authentication due to the use of default usernames and passwords on the devices, and the presence of exploitable software vulnerabilities.",
"While all of these known weaknesses could be eliminated, it is extremely difficult to ensure that devices are free of any unknown weaknesses.",
"In general, it is virtually impossible—or prohibitively expensive—to ensure that a system is perfectly secure.",
"In addition to the general difficulty of attaining perfect security, traffic-control devices pose further challenges.",
"Similar to other distributed cyber-physical systems, traffic-control systems have large attack surfaces, and they often have long system lifetime and complicated software-upgrade procedures, which makes fixing vulnerabilities difficult.",
"Consequently, operators cannot hope to stop all cyber-attacks since a determined and sophisticated attacker might always find a way to compromise some of the devices.",
"Therefore, instead of focusing solely on the first line of defense, operators must also consider minimizing the impact of successful cyber-attacks.",
"Due to hardware-based failsafes, compromising a traffic signal does not allow an attacker to set the signal to an unsafe configuration that would lead to traffic accidents, such as giving green light to two intersecting directions.",
"However, compromising a signal does enable tampering with its schedule, which allows the attacker to cause disastrous traffic congestion.",
"Such malicious cyber-attacks may be launched by any adversary whose interest is to case disruption and damage, ranging from cyber-terrorists to disgruntled ex-employees.",
"For instance, during the Los Angeles traffic engineers' 2006 strike, two disgruntled employees allegedly penetrated the traffic-control system of the city, and reprogrammed the traffic lights of four intersections to cause congestion: “[t]he red signal would be on too long for the critical approach and the green signal would be on too long for the noncritical approach, thus resulting in long backups into the airport and other key intersections around the city” [2].",
"Furthermore, terrorists could also mount these attacks in conjunction with physical attacks, thereby increasing their impact (e.g., delaying ambulances and firefighters).",
"To minimize the impact of attacks tampering with traffic signals, operators must be able to detect and mitigate them promptly and effectively.",
"In practice, the detection of novel cyber-attacks poses multiple challenges.",
"Since signature-based detectors are ineffective against novel attack, operators must employ anomaly-based detectors.",
"However, these detectors are prone to raising false alarms, which must be investigated manually, resulting in a waste of manpower and resources.",
"Considering the relative scarcity of attacks, the cost of these investigations may exceed the benefit of early attack detection and mitigation.",
"Therefore, when configuring the sensitivity of detectors, operators must carefully balance the cost of false alarms and the risk from delayed detection.",
"Moreover, sophisticated attackers can act strategically by mounting stealthy attacks, which delay detection but still cause significant impact.",
"In light of this, operators must also plan their defense strategically, by anticipating the attackers' responses."
],
[
"Contributions",
"In [3], we introduced an approach for evaluating the vulnerability of transportation networks to cyber-attacks that tamper with traffic-control devices.",
"In this paper, we extend this approach by considering detectors and countermeasures that operators can implement to mitigate these attacks.",
"In particular, we introduce a game-theoretic model, in which an operator can setup anomaly-based detectors and mitigate ongoing attacks by reconfiguring traffic control.",
"Similar to [3], we build on the cell-transmission model introduced by Daganzo [4], [5].",
"To the best of our knowledge, our work is the first to consider the problem of designing and deploying systems based on traffic-sensors measurements to detect tampering attacks against traffic control.",
"Our main contributions in this paper are the following: We formulate a multi-stage security game that models the detection and mitigation of cyber-attacks against transportation networks.",
"We propose an efficient metaheuristic search algorithm for finding detector configurations that minimize losses in the face of strategic attacks.",
"We introduce an anomaly-based detector for attacks against traffic control, which is built on a Gaussian-process based model of normal traffic.",
"We evaluate our detector and algorithms based on detailed simulations of traffic using SUMO.",
"The remainder of this paper is organized as follows.",
"In Section , we introduce our game-theoretic model of detecting and mitigating attacks against transportation networks.",
"In Section , we present computational results on our model and propose efficient heuristic algorithms.",
"In Section , we introduce a Gaussian-process based detector for attacks against traffic control.",
"In Section , we use detailed simulations of transportation networks to evaluate our detector and the heuristic algorithms.",
"In Section , we discuss related work on the vulnerability of transportation networks, configuration of attack detectors, and game theory for security of cyber-physical systems.",
"Finally, in Section , we offer concluding remarks."
],
[
"Game-Theoretic Model of Attacks on Traffic Signals",
"In this section, we introduce our model of launching, detecting, and mitigating cyber-attacks against traffic control in transportation networks.",
"Our model includes two agents: an attacker who can launch cyber-attacks and a defender who attempts to detect and mitigate them.",
"Since these agents may anticipate and react to each other's actions, we formulate our model using game theory, which enables us to capture the strategic interactions between the two agents.",
"For a list of symbols used in our model, see Table REF .",
"Table: List of Symbols"
],
[
"Traffic Model",
"First, we introduce Daganzo's cell transmission model, the traffic model on which our game-theoretic model, our analysis, and our numerical evaluation are built.",
"Here, we provide only a very brief summary of this traffic model, focusing on the notation that will be used throughout the paper.",
"For a detailed description of the model, we refer the reader to [4], [5], [6].",
"For readers who are familiar with the cell-transmission model, we recommend to continue with Section REF .",
"The cell transmission model divides a road network into cells, which represent homogeneous road segments, and divides time into uniform intervals.",
"The length of a road segment corresponding to a cell is equal to the distance traveled in light traffic by a typical vehicle in one time interval.",
"Each cell $i$ has three sets of parameters: $N_i^t$ is the maximum number of vehicles that can be present in cell $i$ at time $t$ ; $Q_i^t$ is the maximum number of vehicles that can flow into or out of cell $i$ during time interval $t$ ; and $\\delta _i^t$ is the ratio between the free-flow speed and the backward propagation speed of cell $i$ at time $t$ (see [6] for a detailed explanation).This constant is used to quantify how the speed of traffic decreases as the cell becomes congested, and can model traffic phenomena such as shockwaves.",
"Cells that model road segments where vehicles can enter traffic are called source cells, and each source cell $i$ has a traffic demand parameter $d_i^t$ , which is the number of vehicles entering traffic at cell $i$ in time interval $t$ .",
"Cells where vehicles may exit traffic are called sink cells.",
"Every cell is connected to one or more other cells: cells that correspond to consecutive road segments or road segments that are joined by an intersection are connected.",
"The set of cells from which vehicles can move into cell $i$ is called the set of predecessor cells, denoted by $\\Gamma ^{-1}(i)$ .",
"Cell that have multiple predecessors are called merging cells, while cells that are the predecessors of multiple cells are called sink cells.",
"To model signal control at intersections, we follow Daganzo's proposition [5] and introduce the time-dependent parameters $p_{ki}^t$ controlling the inflow proportions of merging cell $i$ .",
"We let $\\mathcal {I}$ denote the set of merging cells that model signalized intersections.",
"To solve the traffic model (i.e., to determine the traffic flow for a given network and set of parameters), we use Ziliaskopoulos's linear programming approach [6].",
"The objective of this linear program is the sum of the number of vehicles traveling (i.e., number of vehicles on the road) over time, which is clearly equal to the total travel time of all the vehicles.",
"As a consequence, we can use the value of the linear program—which can be computed efficiently for a given instance—as a measure of network congestion.",
"Finally, we consider relatively short-term attack scenarios, in which the parameters of the cells and the default (i.e., unattacked) schedules of the traffic signals are constant.",
"Hence, in our game-theoretic model, we will omit the superscript $^t$ from $Q_i^t$ , $N_i^t$ , $\\delta _i^t$ , and $p_{ki}^t$ ."
],
[
"Multi-Stage Security Game",
"We model defensive countermeasures and attacks in a transportation network as a two-player multi-stage security game between a defender and an attacker [7].",
"The defender represents the operator of the transportation network, who can configure traffic-control devices and aims to minimize congestion in the network.",
"The attacker represents any strategic adversary that can compromise and tamper with traffic signals and aims to maximize congestion.",
"In a nutshell, our game consists of the following three stages.",
"Detector Configuration: In the first stage, the defender configures detectors, which are deployed in the transportation network, to detect cyber-attacks against traffic control.",
"The detectors may be traffic-anomaly based detectors or conventional cyber-security intrusion detection systems (IDS).",
"When configuring these detectors, the defender should anticipate the attacker's possible adversarial actions in the second stage.",
"Attack on Traffic Control: In the second stage, the attacker mounts a cyber-attack against the transportation network by compromising traffic signals and tampering with their schedule to cause congestion.",
"When choosing its attack, the attacker must take into account both the detector configuration chosen by the defender in the first stage as well as the defender's possible mitigation actions in the third stage.",
"Mitigation of Attack: In the third stage, the defender attempts to mitigate the attack by changing the configuration of uncompromised traffic-control devices to minimize congestion.",
"As this is the final stage, mitigation simply needs to respond to the particular attack that was launched in the second stage.",
"Note that once the defender detects an ongoing attack, it should also try to regain control of the compromised devices as soon as possible.",
"However, since the devices may be physically scattered throughout the transportation network, regaining control of them can take a long time.",
"For instance, an attacker could have changed remote login passwords, severed communication-network connections, etc., forcing the defender to physically reset or reinstall compromised devices.",
"Meanwhile, disastrous traffic congestions may form in the transportation network, which the defender must mitigate immediately."
],
[
"Stages and Strategic Choices",
"Next, we provide a detailed description of the three stages of the game and the players' action spaces.",
"To detect stealthy cyber-attacks, detectors are deployed on the traffic-control devices at a subset $\\mathcal {I}_D$ of the signalized intersections $\\mathcal {I}$ .",
"These detectors can be either traffic-anomaly based detectors, such as the one that we will introduce in Section , or conventional cyber-security intrusion detection systems.",
"We assume that detectors are imperfect, which means that they may raise false alarms (when there is no attack in progress) and they may detect actual attacks with some delay.",
"The rate of false-positive errors and detection delay both depend on how sensitive a detector is: a more sensitive detector is more likely to raise false alarms but detects actual attacks earlier, and vice versa.",
"We assume that the operator can configure the sensitivity of every one of the $|\\mathcal {I}_D|$ detectors individually.",
"Specifically, in the first stage of the game, the defender chooses a sensitivity configuration $D$ for the detectors.",
"For ease of presentation, we let the sensitivity of the detector at each intersection $i \\in \\mathcal {I}_D$ be represented by the false-positive rate of the detector.",
"Formally, a detector configuration $D$ is an $|\\mathcal {I}_D|$ -dimensional non-negative vector, where $D_i$ is the rate of false alarms raised by the detector at intersection $i \\in \\mathcal {I}_D$ ."
],
[
"Stage II: Cyber-Attack on Traffic Control",
"[color=orange!40, linecolor=black!30!orange!60!white]Aron: Describe attacker model in more detail and motivate it!",
"In the second stage of the game, the attacker compromises a subset of the traffic signals and changes their schedule.",
"We let $\\mathcal {I}_A \\subseteq \\mathcal {I}$ denote the set of traffic signals that the attacker chooses to compromise.",
"We assume that the attacker is resource bounded, which means that it can compromise signals in at most $B \\le |\\mathcal {I}|$ intersections at the same time.",
"Hence, the attacker's choice $\\mathcal {I}_A$ must satisfy $|\\mathcal {I}_A| \\le B$ .",
"Once the attacker has compromised a set of traffic signals $\\mathcal {I}_A$ , it can reconfigure every one of them.",
"However, most traffic control devices have hardware-level safety mechanisms in practice (see, e.g., [1]), which constrain the configurations that may be set by an adversary.",
"In particular, traffic signals typically use malfunction management units as a safety feature against controller faults (e.g., overriding a faulty controller that would give green lights to two intersecting directions).",
"These hardware-based failsafes also limit the impact of cyber attacks (e.g., preventing the attacker from causing traffic accidents) by overriding invalid configurations.",
"In our traffic model, the attacker's reconfiguration corresponds to setting new inflow proportions $\\hat{p}_{ki}$ for the cells in $\\mathcal {I}_A$ .",
"Therefore, we can model hardware-level failsafes by requiring the inflow proportions chosen by the attacker to constitute a valid configuration.",
"Specifically, we assume that the inflow proportions set by a feasible attack must sum up to 1 for each compromised intersection: $\\forall i \\in \\mathcal {I}_A: \\sum _{k \\in \\Gamma ^{-1}(i)} \\hat{p}_{ki} = 1 .$ In sum, we can represent a feasible attack action $A$ as a pair $A= (\\mathcal {I}_A, \\hat{p})$ that satisfies $|\\mathcal {I}_A| \\le B$ and $\\sum _{k \\in \\Gamma ^{-1}(i)} \\hat{p}_{ki} = 1$ for every $i \\in \\mathcal {I}_A$ , where $\\mathcal {I}_A$ is the set of compromised signals, and $\\hat{p}$ are the tampered signal schedules."
],
[
"Between Stages II and III: Detection",
"Once the attacker has perpetrated its attack in the second stage, the compromised signals begin to operate with tampered schedules, which results in increased congestion in the transportation network.",
"Eventually, the detectors deployed by the defender will detect the attack (based on either traffic or cyber anomalies).",
"We let $\\Delta _D$ denote the detection delay, that is, the amount of time between the launch and detection of the attack.",
"The detection delay depends on both the configuration $D$ chosen by the defender and the attack $A$ chosen by the attacker, which we express by representing delay as a function $\\Delta _D(D, A)$ of $D$ and $A$ .",
"Once the attack is detected, the game progresses to the third stage."
],
[
"Stage III: Mitigation of Attack",
"In the third stage, the defender mitigates the detected attack by reconfiguring traffic-control devices to alleviate congestion.",
"We assume that the defender can reconfigure any device that is still under its control, that is, any traffic signal that is not compromised by the attacker.",
"Since the attacker has compromised signals $\\mathcal {I}_A$ , the defender can set new inflow proportions $\\hat{p}_{ki}$ for the cells $i$ in $\\mathcal {I}\\setminus \\mathcal {I}_A$ .",
"We again require the new configuration to be valid, which means that the inflow proportions must sum up to 1 for each reconfigured intersection: $\\forall i \\in \\mathcal {I}\\setminus \\mathcal {I}_A: \\sum _{k \\in \\Gamma ^{-1}(i)} \\hat{p}_{ki} = 1 .$ Finally, we let $M= \\lbrace \\hat{p}_{ki} \\,|\\, i \\in \\mathcal {I}\\setminus \\mathcal {I}_A\\rbrace $ denote the defender's mitigation (i.e., reconfiguration) action."
],
[
"Player's Utilities",
"We now define the defender's and the attacker's utilities resulting from the various strategic choices that they can make in the game.",
"First, we let $T$ denote the level of congestion in the transportation network with the default configuration of traffic-control devices (i.e., with inflow proportions $p$ ).",
"In practice, we can quantify congestion $T$ as, e.g., the average or total travel time of the vehicles in the transportation network between their origin and destination.",
"Recall from Section REF that we can efficiently compute travel time with default proportions $p$ in our traffic model using a linear program.",
"Next, we let $T_A(A)$ denote the level of congestion after the attack but before the mitigation, which depends on the attacker's action $A$ chosen in the second stage.",
"Similar to $T$ , we can compute $T_A(A)$ using our traffic model with the default proportion $p_{ki}$ for every cell $i \\in \\mathcal {I}\\setminus \\mathcal {I}_A$ but with the adversarial proportion $\\hat{p}_{ki}$ for every cell $i \\in \\mathcal {I}_A$ .",
"Finally, we let $T_M(A, M)$ denote the level of congestion after the attack has been mitigated, which depends on both attacker's action $A$ and the defender's mitigation action $M$ .",
"We let the attacker's gain $\\mathcal {G}$ (i.e., positive utility) for actions $(D, A, M)$ be the total impact of the attack in terms of increased congestion level: $\\mathcal {G}(&D, A, M) \\nonumber \\\\ &= \\left(T_A(A) - T\\right) \\cdot \\Delta _D(D, A) + \\left(T_M(A, M) - T\\right) \\cdot \\Delta _M ,$ where $\\Delta _M$ is the amount of time between mitigation and the transportation network returning to normal operation (e.g., manually resetting compromised devices).",
"The first term quantifies the impact of the attack before mitigation, while the second term quantifies impact after mitigation but before returning to normal operation.",
"Next, we define the defender's loss (i.e., negative utility) resulting from actions $(D, A, M)$ .",
"Recall that the detectors deployed in the transportation network are imperfect, and each detector $i \\in \\mathcal {I}_D$ is continuously generating false alerts (i.e., false-positive errors) at rate $D_i$ .",
"Since the defender cannot tell which alerts are false, it has to investigate every single alert, which costs manpower and resources.",
"Hence, we define the defender's loss considering both the total impact of attacks and the cost of investigating false alerts: $\\mathcal {L}(D, A, M) = \\mathcal {G}(D, A, M) + \\sum _{i \\in \\mathcal {I}_D} D_i \\cdot C ,$ where $C$ is the cost of investigating an alert.",
"The first term quantifies the total impact of the attack, while the second term captures the cost of investigating false alerts."
],
[
"Solution Concept and Problem Formulation",
"We assume that both players have perfect information: in the second stage, the attacker knows the detector configuration $D$ chosen by the defender in the first stage; and in the third stage, the defender knows the attack action $A$ chosen by the attacker in the second stage.",
"We assume that the attacker has perfect information because we are considering a sophisticated, worst-case attacker, who has extensive knowledge of its target (i.e., Kerckhoffs's principle) and may know the algorithms or techniques employed by the defender for configuring the detectors.",
"On the other hand, we assume that the defender has perfect information because we are considering a smart transportation network with monitoring capabilities; however, this assumption could be relaxed.",
"[color=orange!40, linecolor=black!30!orange!60!white]Aron: Check and revise if necessary!",
"Under this assumption, we can model the players' optimal choices most naturally using the solution concept of subgame perfect equilibrium.",
"Our goal is to find an optimal strategy for the defender by solving the game.",
"We can do so by finding the players' best-response actions for every stage using backward induction, that is, by solving each stage starting with the third and finishing with the first, in each stage building on the solutions for the subgame formed by the subsequent stages.",
"Given detector configuration $D$ and attack $A$ , a best-response mitigation is $\\operatornamewithlimits{argmin}_{M} \\mathcal {L}(D, A, M) .$ Note that the best-response mitigation does not actually depend on the detector configuration $D$ .",
"To prove this, observe that the only term of $\\mathcal {L}(D, A, M)$ that depends on mitigation action $M$ is $\\left(T_M(A, M) - T\\right) \\cdot \\Delta _M$ .",
"Since this term does not depend on detector configuration $D$ , neither does the best-response mitigation.",
"Intuitively, the explanation for this is that once the defender has detected the attack, it does not matter how it was detected (and all costs associated with detection are sunk).",
"Given detector configuration $D$ , a best-response attack is $&\\operatornamewithlimits{argmax}_{A} \\mathcal {G}(D, A, M) \\,\\big |_{M\\in \\, \\operatornamewithlimits{argmin}_{M^{\\prime }} \\mathcal {L}(D, A, M^{\\prime }) } \\nonumber \\\\&=\\operatornamewithlimits{argmax}_{A} \\mathcal {G}(D, A, M) \\,\\big |_{M\\in \\, \\operatornamewithlimits{argmin}_{M^{\\prime }} \\mathcal {G}(D, A, M^{\\prime }) + \\sum _{i \\in \\mathcal {I}_D} D_i \\cdot C } \\\\&=\\operatornamewithlimits{argmax}_{A} \\mathcal {G}(D, A, M) \\,\\big |_{M\\in \\, \\operatornamewithlimits{argmin}_{M^{\\prime }} \\mathcal {G}(D, A, M^{\\prime }) } \\\\&=\\operatornamewithlimits{argmax}_{A} \\min _{M} \\mathcal {G}(D, A, M) .$ Note that the attacker must anticipate the defender's mitigation action in the next stage; however, since the game is strategically equivalent to a zero-sum game, the attacker's problem simplifies to a maximin optimization.",
"Our threat model assumes a worst-case attacker, whose goal is to minimize the defender's utility.",
"This is a safe assumption since the defender's utility can only be higher if the attacker behaves differently.",
"In contrast, if our threat model assumed a particular attacker behavior, then the defender's strategy would be vulnerable to deviations from that assumed behavior.",
"Finally, an optimal (i.e., equilibrium) detector configuration is $&\\operatornamewithlimits{argmin}_{D} \\min _{M} \\mathcal {L}(D, A, M) \\,\\big |_{A\\in \\operatornamewithlimits{argmax}_{A^{\\prime }} \\min _{M^{\\prime }} \\mathcal {G}(D, A^{\\prime }, M^{\\prime }) } \\nonumber \\\\&=\\operatornamewithlimits{argmin}_{D} \\max _{A} \\min _{M} \\mathcal {L}(D, A, M) .$ Note that the defender needs to anticipate the attacker's attack action in the next stage; however, since the game is strategically equivalent to a zero-sum game, the defender's problem simplifies to a minimax optimization (i.e., minimaximin if we consider the mitigation choice as well).",
"Even though we can express the players' optimal strategies as relatively simple maximin and minimax optimization problems, actually finding optimal strategies is computationally challenging due to the sizes of the players' strategy spaces.",
"Hence, we focus our analysis on the computational aspects of solving the transportation security game.",
"First, in Section REF , we show that finding an optimal action for the attacker is a computationally hard problem.",
"Although we study the complexity of only the attacker's problem in this paper, our computational-complexity argument could be easily extended to the defender's problem.",
"Due to the complexity of solving the game, we focus the remainder of this section on providing efficient heuristic algorithms: we introduce a greedy heuristic for the attacker in Section REF , and a metaheuristic search algorithm for the defender in Section REF ."
],
[
"Computational Complexity",
"We begin our analysis by showing that the attacker's problem (i.e., finding a worst-case attack) is computationally hard.",
"Given a detector configuration $D$ , the attacker's problem is to find an optimal attack $A^*$ that maximizes $\\min _{M} \\mathcal {G}(D, A^*, M)$ .",
"Following the backward-induction approach, we assume that we have an oracle that finds the optimal mitigation action $M$ for any attack $A$ , and we study the attacker's problem by building on this oracle.",
"In practice, the oracle can be replaced with, for example, a linear program for finding optimal traffic control.",
"Further, for ease of presentation, we overload the notation $\\mathcal {G}$ as follows $\\mathcal {G}(D, A) = \\min _{M} \\mathcal {G}(D, A, M) .$ First, we formulate a decision version of the attacker's problem as follows.",
"Attacker's Decision Problem: Given a transportation network, a budget $B$ , a detector configuration $D$ , and a threshold gain $\\mathcal {G}^*$ , determine if there exists an attack $A^*$ satisfying the budget constraint such that $\\mathcal {G}(D, A^*) > \\mathcal {G}^*$ .",
"We show that the above problem is computationally hard by reducing a well-known NP-hard problem, the Set Cover Problem, to the above problem.",
"Set Cover Problem: Given a base set $U$ , a collection $\\mathcal {C}$ of subsets of $U$ , and a number $k$ , determine if there exists a subcollection $\\mathcal {C}^{\\prime } \\subseteq \\mathcal {C}$ of at most $k$ subsets such that every element of $U$ is contained by at least one subset in $\\mathcal {C}^{\\prime }$ .",
"The following theorem establishes the computational complexity of the attacker's problem.",
"Theorem 1 The Attacker's Decision Problem is $N\\!P$ -hard.",
"Figure: Illustration for the proof of Theorem .Given an instance of the Set Cover Problem (i.e., a set $U$ , a collection $\\mathcal {C}$ of subsets, and a number $k$ ), we construct an instance of the Attacker's Decision Problem as follows: let the transportation network be the following (see Figure REF for an illustration): there is one source cell $r$ , with $Q_r = k + 1$ , $d_r^1 = k + 1$ , and $d_r^t = 0$ for $t > 1$ ; there is one sink cell $s$ ; for every element $u \\in U$ , there is a merging cell $u$ ; for every subset $C \\in \\mathcal {C}$ , there is a diverging cell $C$ ; each diverging cell $C$ is connected to every merging cell $u \\in C$ ; for every cell $i$ , $N_i = k + 1$ and $\\delta _i = 1$ ; for every merging cell $u$ , $Q_u = k + 1$ ; for every diverging cell $C$ , $Q_C = 1$ ; let the attacker's budget be $B = |U|$ ; let the detector configuration be such that $\\forall A: \\Delta _D(D, A) \\equiv 1$ let the default congestion be $T = 0$ , let the congestion after the attack $T_A(A)$ be equal to the total travel time of the vehicles, and let the mitigation time be $\\Delta _M = 0$ ; let the threshold gain be $\\mathcal {G}^* = 3 (k + 1)$ .",
"Clearly, the above reduction can be carried out in time that is polynomial in the size of the Set Cover Problem instance.",
"It remains to show that the above instance of the Attacker's Decision Problem has a solution $A^*$ if and only if the given instance of the Set Cover Problem has a solution $\\mathcal {C}^{\\prime }$ .",
"Before we proceed to prove this equivalence, notice that the values $Q_r$ , $N_i$ and $\\delta _i$ for every cell $i$ , and $Q_u$ for every merging cell $u$ will not play any role, since they are high enough to allow any traffic to pass through.",
"Furthermore, since $B = |U|$ and $\\Delta _D \\equiv 1$ , the attacker will be able to reconfigure every traffic signal without decreasing detection time.",
"Hence, the attacker's problem is simply to pick the values $\\hat{p}_{Cu}$ for every $u \\in C$ so that the total travel time is at least $\\mathcal {G}^* = 3 (k + 1)$ .",
"First, suppose that there exists a set cover $\\mathcal {C}^{\\prime }$ of size at most $k$ .",
"Then, we construct an attack as follows: for every merging cell $u$ , choose one diverging cell $C$ from $\\mathcal {C}^{\\prime }$ that is connected to $u$ (if there are multiple, then choose an arbitrary one), and let $\\hat{p}_{Cu} = 1$ .",
"We have to show that the total travel time in the transportation network is greater than $3 (k + 1)$ after the attack.",
"Since the distance between the source cell and the sink cell is 3 hops and there are $k + 1$ vehicles, all the vehicles must move one step closer to the sink in every time interval in order for the total travel time to be at most $3 (k + 1)$ .",
"However, from the source cell, the vehicles may only move to the cells in $\\mathcal {C}^{\\prime }$ ; otherwise, they would get “stuck” at one of the diverging cells that are not in $\\mathcal {C}^{\\prime }$ .",
"Consequently, in the second time interval, at most $k$ of the $k + 1$ vehicles may move on, which means that the total travel time has to be greater than $3 (k + 1)$ .",
"Second, suppose that there does not exist a set cover $\\mathcal {C}^{\\prime }$ of size at most $k$ .",
"Then, we have to prove that there cannot exist an attack which increases the total travel time to more than $3 (k + 1)$ .",
"Firstly, we show that there exists an optimal attack which assigns either 0 or 1 to every $\\hat{p}_{Cu}$ .",
"To prove this, consider an attack in which there is a merging cell $v$ with a $\\hat{p}_{Cv}$ value other than 0 or 1.",
"If none of its predecessor cells $C$ has a positive $\\hat{p}_{Cw}$ value for some other merging cell $w$ , then the assignment for $v$ can clearly be changed to 0 and 1 values without changing the total travel time.",
"Next, suppose that one (or more) of the predecessor cells $C$ of the merging cell has a positive $\\hat{p}_{Cw}$ value for some other merging cell $w$ .",
"Then, the total travel time maximizing assignment is clearly one which assigns $\\hat{p}_{Cv} = 1$ to a predecessor cell $C$ for which $\\sum _{u \\in C} \\hat{p}_{Cu}$ is maximal, since this “wastes” the most “merging capacity.” Thus, for the remainder of the proof, it suffices to consider only attacks where every $\\hat{p}_{Cu}$ value is either 0 or 1.",
"Now, consider an optimal attack $A^*$ against the transportation network, and let $\\mathcal {C}^*$ be the set of diverging cells $C$ for which there exists a merging cell $u$ such that $\\hat{p}_{Cu} = 1$ .",
"Clearly, $\\mathcal {C}^*$ forms a set cover of $U$ since for every element $u$ , there is a subset $C \\in \\mathcal {C}^*$ such that $u \\in C$ (i.e., $C$ is connected to $u$ ).",
"From our initial supposition, it follows readily that the cardinality of set $\\mathcal {C}^*$ must be at least $k + 1$ .",
"However, this also implies that the total travel time after the attack is equal to $3 (k + 1)$ : in the second time interval, all $k + 1$ vehicles may move forward to the diverging cells in set $\\mathcal {C}^*$ ; in the third time interval, all the vehicles may again move forward to the merging cells (since every cell in $\\mathcal {C}$ has at least one “enabled” connection); and all the vehicles may leave the network by the next interval through the sink cell.",
"Since the total travel time after an optimal attack $A^*$ is equal to $T^* = 3 (k + 1)$ , the attacker's problem does not have a solution.",
"Therefore, the constructed instance of the Attacker's Decision Problem has a solution if and only if the given instance of the Set Cover Problem has one, which concludes our proof."
],
[
"Algorithms",
"Mitigation—in our model—means adapting the schedule of uncompromised traffic signals given the schedule of compromised signals, which is equivalent to optimizing traffic control in a non-adversarial setting, with the compromised signals acting as fixed-schedule signals.",
"Since optimizing traffic control in non-adversarial settings has been studied in prior work, we focus on providing efficient algorithms for solving the first two stages of the game."
],
[
"Greedy Algorithm for Attacks",
"Since the attacker's problem is $N\\!P$ -hard, we cannot hope for a polynomial-time algorithm that always finds a worst-case attack (unless $P = N\\!P$ ).",
"Hence, to provide an alternative to computationally infeasible exhaustive search, we turn our attention to designing an efficient heuristic algorithm.",
"[h!]",
"Polynomial-Time Greedy Heuristic for Finding an Attack transportation network security game, detector configuration $D$ attack $A^*$ $\\mathcal {I}_A^* \\leftarrow \\emptyset $ , $\\hat{p}^* \\leftarrow p$ $b = 1, \\ldots , B$ $\\mathcal {I}_A^{\\prime } \\leftarrow \\mathcal {I}_A^*$ , $\\hat{p}^{\\prime } \\leftarrow \\hat{p}^*$ $i \\in \\mathcal {I}$ $\\mathcal {I}_A \\leftarrow \\mathcal {I}_A^* \\cup \\lbrace i\\rbrace $ $k \\in \\Gamma ^{-1}(i)$ $\\forall l, j: ~ \\hat{p}_{lj} \\leftarrow {\\left\\lbrace \\begin{array}{ll}1 & \\text{ if } j = i \\wedge l = k \\\\0 & \\text{ if } j = i \\wedge l \\in \\Gamma ^{-1}(i) \\setminus \\lbrace k\\rbrace \\\\\\hat{p}^*_{lj} & \\text{ otherwise.}\\end{array}\\right.",
"}$ $\\mathcal {G}(D, (\\mathcal {I}_A, \\hat{p})) \\ge \\mathcal {G}(D, (\\mathcal {I}_A^{\\prime }, \\hat{p}^{\\prime }))$ $\\mathcal {I}_A^{\\prime } \\leftarrow \\mathcal {I}_A$ , $\\hat{p}^{\\prime } \\leftarrow \\hat{p}$ $\\mathcal {I}_A^* \\leftarrow \\mathcal {I}_A^{\\prime }$ , $\\hat{p}^* \\leftarrow \\hat{p}^{\\prime }$ output $A^* = (\\mathcal {I}_A^*, \\hat{p}^*)$ The attacker's problem can be viewed as the composition of two problems: finding a subset $\\mathcal {I}_A$ of at most $B$ signalized intersections and finding new inflow proportions $\\hat{p}_{ki}$ for the cells $i \\in \\mathcal {I}_A$ .",
"For finding a subset $\\mathcal {I}_A$ , we propose to use a greedy heuristic, which starts with an empty set and adds new cells to it one-by-one, always picking the one that leads to the greatest increase in the attacker's gain.",
"Finding new inflow proportions $\\hat{p}_{ki}$ is particularly challenging, since the set of possible choices is continuous.",
"However, we observe that in most networks, the worst-case configuration is an “extreme” one, which assigns proportion $\\hat{p}_{ki} = 1$ to one predecessor cell $k$ and proportion $\\hat{p}_{ji} = 0$ to every other predecessor cell $j$ [3].",
"In fact, we have tested this property on hundreds of networks that we generated according to the Grid model with Random Edges (see Section REF ), resembling road networks from the U.S. and Europe, and we have not found a single counterexample.",
"Based on these numerical results, we conjecture that this property holds in general, that is, for any network.We leave the theoretical proof of this claim for future work.",
"Hence, for every new cell $i$ added to the set of attacked intersections, we propose to search over the possible extreme configurations by iterating over the predecessors of cell $i$ .",
"Based on the above ideas, we formulate Algorithm REF .",
"It is fairly easy to see that we can implement Algorithm REF as a polynomial-time algorithm.",
"Due to the three nested iterations, the running time of Algorithm REF is $O\\big (B \\cdot |\\mathcal {I}|$ $\\cdot \\left(\\max _{i \\in \\mathcal {I}} |\\Gamma ^{-1}(i)|\\right)\\big )$ times the running time of computing $\\mathcal {G}$ .",
"Since we can compute $\\mathcal {G}(D, A)$ for any attack $A$ using a linear program, it follows readily that the running time of the algorithm can be upper bounded by a polynomial function of the input size (i.e., size of the transportation network and budget $B$ ).",
"We can formally state this observation as the following proposition.",
"Proposition 1 With a polynomial-time oracle for computing $\\mathcal {G}$ , the running time of Algorithm REF is a polynomial function of the input size."
],
[
"Metaheuristic Search Algorithm for Detector Configuration",
"Next, we present an algorithm for finding a detector configuration (i.e., false-positive rates) based on a metaheuristic approach.",
"In particular, we use simulated annealing to find a near-optimal detector configuration $D$ .",
"The basic idea of this approach is to start with an arbitrary configuration $D$ , which we then improve iteratively.",
"In each iteration, we generate a new solution $D^{\\prime }$ in the neighborhood of $D$ .",
"If the new configuration $D^{\\prime }$ is better in terms of minimizing the defender's loss (against an attacker playing a best response), then the current configuration $D$ is replaced with the new one.",
"On the other hand, if the new configuration $D^{\\prime }$ increases the defender's loss, then new configuration replaces the current one with only a small probability.",
"This probability depends on the difference between the two solutions in terms of loss as well as a parameter commonly referred to as “temperature,” which is a decreasing function of the number of iterations.",
"These random replacements prevent the search from “getting stuck” in a local minimum.",
"The algorithm is presented below as Algorithm REF .",
"[h!]",
"Polynomial-Time Metaheuristic for Finding a Detector Configuration transportation network security game, iterations $k_{\\max }$ , initial temperature $T_0$ , cooling parameter $\\beta $ detector configuration $D^*$ $D\\leftarrow 1$ $L \\leftarrow \\max _{A} \\mathcal {G}(D, A) + \\sum _{i \\in \\mathcal {I}_D} D_i \\cdot C$ $k = 1, \\ldots , k_{\\max }$ $D^{\\prime } \\leftarrow \\mathtt {Perturb}(D)$ $L^{\\prime } \\leftarrow \\max _{A} \\mathcal {G}(D, A) + \\sum _{i \\in \\mathcal {I}_D} D^{\\prime }_i \\cdot C$ $T \\leftarrow T_0 \\cdot e^{-\\beta k}$ $pr \\leftarrow e^{(L^{\\prime } - L) / T}$ $(L^{\\prime } < L) \\; \\vee \\; (\\mathtt {rand}(0, 1) \\le pr)$ $D\\leftarrow D^{\\prime }$ , $L \\leftarrow L^{\\prime }$ output $D$ In Algorithm REF , $\\mathtt {Perturb}(D)$ picks a random configuration $D^{\\prime }$ from the neighborhood of $D$ .",
"In particular, we implement $\\mathtt {Perturb}(D)$ as choosing a value for each $D_i^{\\prime }$ uniformly at random from $[D_i \\cdot (1 - \\varepsilon ), D_i \\cdot (1 + \\varepsilon )]$ , where $\\varepsilon $ is a small constant (e.g., $0.1$ ).",
"For solving $\\max _{A} \\mathcal {G}(D, A)$ in practice, we can use the greedy heuristic (Algorithm REF ).",
"The temperature $T$ is decreasing exponentially with iteration number $k$ , and the rate of the decrease is controlled by the “cooling” parameter $\\beta $ .",
"Finally, we note that a simpler algorithm could also be obtained, in which $D$ is updated with $D^{\\prime }$ in each iteration if and only if $D^{\\prime }$ is strictly better than $D$ .",
"This heuristic search, commonly known as hill climbing, also works well for our problem; however, Algorithm REF gives better results."
],
[
"Anomaly-based Detector",
"Now, we introduce a traffic-anomaly based detector against stealthy attacks that tamper with traffic control.",
"The core idea of anomaly-based detection is to build a probabilistic model of normal traffic conditions, which can then be used to estimate the likelihood that observed traffic conditions are normal.",
"Note that we must employ a probabilistic model to account for the uncertainty in parameter values since many parameters (e.g., traffic demand) can only be estimated in practice.",
"We can estimate the likelihood that the observed traffic is normal as the probability that our model of normal traffic would generate the observed traffic.",
"We can then compare the likelihood value to a threshold, and if the likelihood is lower, we can raise an alarm.",
"In our detector, the model of normal traffic is based on Gaussian processes, which have been successfully used in prior work for traffic volume forecasting [8], [9].",
"Note that we cannot use macroscopic traffic models, such as the cell transmission model, to detect attacks because these models abstract away details for the sake of tractability (e.g., inflow proportions instead of actual traffic light schedules); however, such details can be crucial for the detection of stealthier attacks that alter traffic control only slightly."
],
[
"Gaussian Processes",
"We begin giving a very brief overview of Gaussian processes.",
"For a comprehensive discussion of Gaussian processes in machine learning, we refer the reader to [10].",
"In principle, Gaussian processes are an extension of multivariate Gaussian distributions to infinite collections of random variables.",
"Formally, a Gaussian process is a stochastic process such that any finite collection of variables $(X_1, \\ldots , X_n)$ follows a multivariate Gaussian distribution.",
"A Gaussian process is typically described using a mean function $\\mu (X) = \\mathbb {E}(X)$ and a covariance function $k(X_1, X_2) = \\mathbb {E}\\left[(X_1 - m(X_1)) (X_2 - m(X_2))\\right] .$ The covariance function is often chosen to be some well-known kernel function, such as squared exponential, whose parameters can be estimated from a training dataset $(x_1,$ $\\ldots , x_n)$ .",
"A common application of Gaussian processes is regression: given the values of a set of training variables $(x_1, \\ldots , x_n)$ , we can easily compute the expected value and variance of a target variable $Y$ using the mean and covariance functions.",
"Gaussian-process based regression models have been successfully applied to a wide range of problem, such as traffic volume forecasting [8], [9], spatial modeling of extreme snow depth [11], wind power forecasting [12], estimation of water chlorophyll concentration [13], and spectrum sensing [14]."
],
[
"Model",
"We assume that traffic sensors, such as induction loop sensors, have been deployed for monitoring the transportation network.",
"Since modeling an entire network would be computationally challenging—and would certainly not scale well—we divide sensors into subsets, and we build a separate model and detector for each one of these subsets.",
"For example, traffic sensors that are deployed next to the same intersection $i \\in \\mathcal {I}$ may be grouped together and provide traffic data for one detector (see, e.g., Figure REF in Section REF ).",
"The outputs of all the detectors deployed in a transportation network can then be combined together to form a single detector for the entire network.",
"We assume that sensors measure and report traffic values, such as traffic flow or occupancy, in fixed-length intervals (e.g., report one measurement value for every 15-minute intervals).Note that the length of these measurement intervals is independent of the time intervals of the cell-transmission model.",
"However, for ease of presentation, we will reuse notation $t$ to identify measurement intervals.",
"In our Gaussian-process model, we model each one of these measurements as a random variable.",
"Formally, for every sensor $s$ and time interval $t$ , there exists a random variable $X_s^t$ whose value is equal to the traffic value measured by sensor $s$ in interval $t$ .",
"Hence, our model has a discrete but—due to the time dimension—potentially infinite set of variables.",
"A key part of modeling is establishing mean and covariance functions for these variables.",
"For each sensor $s$ , we model the mean values $\\mu (X_s^t)$ of variables $X_s^t$ using a periodic function: $\\mu \\left(X_s^t\\right) \\equiv \\mu \\left(X_s^{t + P}\\right) ,$ where $P$ is the length of the period.",
"We call this time period, which is measured in number of discrete time intervals, the detector window $W$ .",
"Similarly, for sensors $s_1$ and $s_2$ , we model the covariance values $k\\left(X_{s_1}^{t_1}, X_{s_2}^{t_2}\\right)$ between variables $X_{s_1}^{t_1}$ and $X_{s_2}^{t_2}$ using a periodic function: $k\\left(X_{s_1}^{t_1}, X_{s_2}^{t_2}\\right) \\equiv k\\left(X_{s_1}^{t_1 + P}, X_{s_2}^{t_2 + P}\\right) .$ Further, we assume that $k\\left(X_{s_1}^{t_1}, X_{s_2}^{t_2}\\right) \\equiv 0$ if $|t_1 - t_2| > P$ .",
"To train the model, the actual values of these functions must be learned from traffic values observed during normal operation."
],
[
"Training",
"Before training, our model has $S \\cdot P + \\frac{S \\cdot (S - 1) \\cdot P \\cdot (2P + 1)}{2} + S \\cdot P \\cdot (2P + 1)$ unknown values, where $S$ is the number of sensors: $S \\cdot P$ mean values: for each sensor $s$ , the mean function $\\mu $ can be described by $P$ values since its period length is $P$ ; $\\frac{S \\cdot (S - 1) \\cdot P \\cdot (2P + 1)}{2}$ covariance values: for each distinct pair of sensors $s_1$ and $s_2$ , the covariance function $k$ can be described by $P \\cdot (2 \\cdot P + 1)$ values since its period length is $P$ , the maximum difference between $t_1$ and $t_2$ is $P$ , and covariance values are symmetric; and $S \\cdot P \\cdot (2P + 1)$ variance values: for each sensor $s$ , the covariance function $k$ can be described by $P \\cdot (2 \\cdot P + 1)$ .",
"Training the model means learning these mean and covariance values for normal traffic.",
"In practice, we can train the model by observing sensor measurements $(x_{s_1}^{t_1}, x_{s_1}^{t_2}, x_{s_2}^{t_1}, \\ldots )$ of traffic under normal conditions, and then simply estimating the most likely mean and covariance values from these observations (i.e., maximum likelihood estimation)."
],
[
"Detection",
"Once we have trained the model, we can use it to detect attacks against traffic control.",
"First, we take sensor measurements $(x_s^t, \\ldots )$ , which are observed in the network that might be under attack, and we use the Gaussian process to compute the likelihood of these measurement values being generated by our model of normal traffic.",
"Due computational limitations, we restrict detection to observations from a single detector window (i.e., measurement values from some range $(t, t + P - 1)$ ).",
"If more observations are available, we can evaluate the detector multiple times.",
"Since the measurement values are continuous, we can use the probability density of the Gaussian distribution $(X_s^t, \\ldots )$ at $(x_s^t, \\ldots )$ as the likelihood value.",
"We then interpret this likelihood as the likelihood of the network operating under normal control (i.e., not being under attack).",
"Finally, we compare the likelihood to a detector threshold $\\tau _i$ , and raise an alarm if the likelihood is lower than the threshold.",
"Thus, our detector has two parameters, the detector window $W$ and the detector threshold $\\tau _i$ , which together determine the rate of false alarms $D_i$ and the detection delay.",
"Note that the detector threshold $\\tau _i$ and the rate of false alarms $D_i$ are closely related to each other: lower thresholds result in fewer false alarms since more observations are accepted as likely; and vice versa.",
"Hence, we can express $\\tau _i(D_i)$ and $D_i(\\tau _i)$ as increasing functions, and we may specify the configuration of the detector either as the desired false-positive rate $D_i$ or as the threshold $\\tau _i$ .",
"We chose to use representation $D_i$ in our game-theoretic model and analysis for ease of presentation.",
"The exact relation $\\tau _i(D_i)$ can be determined experimentally by evaluating the detector with various configurations on normal traffic."
],
[
"Numerical Results",
"In this section, we present numerical results on our heuristic algorithms from Section and our anomaly-based detector from Section ."
],
[
"Anomaly-based Attack Detection",
"We begin by training and evaluating our detector based on simulated flows of traffic under normal conditions and under attacks.",
"We will then use the results of this evaluation to instantiate our game-theoretic model.",
"In particular, we will use the measured false-alarms rate and detection delay values as numerical parameters for our game-theoretic model.",
"For these experiments, we simulate the signalized four-way intersection shown by Figure REF using SUMO (Simulation of Urban MObility) http://sumo.dlr.de/wiki/Main_Page, a well-known and widely-used micro simulator [15], [16].",
"Signals are deployed on both the incoming and outgoing lanes (represented by yellow rectangles in the figure), and these signals measure traffic flow in 15-second intervals.",
"We generate one month of traffic data with the original signal schedule (45 seconds for both roads, including left-turning and yellow phases) for training the Gaussian-process based model.",
"In these simulations, vehicles enter the intersection from all four directions, and each car turns left, continues straight, or turns right with probability 5.3%, 73.7%, and 21.1%, respectively.",
"From each direction, 0.19 vehicles arrive each second on average.",
"We also generate one month of test traffic-data with the original schedule for measuring the false-positive rate of the detector.",
"Finally, for each of the attacks considered below, we generate one day of traffic data, which includes 1 hour with the original schedule and then 23 hours with the tampered schedule.",
"To confirm that our Gaussian-process based traffic model fits observations well, we perform posterior predictive checking [17].",
"The idea of posterior predictive checking is to draw simulated samples from the joint posterior predictive distribution and compare them to the observed sample.",
"If there were significant systematic differences between the simulated and observed samples, that would indicate that our model did not fit well.",
"Note that the applicability of our model is also demonstrated by the low false-positive rate exhibited by our detector.",
"Table: Posterior Predictive Checking pp-ValuesThe significance of difference can be quantified as the classical $p$ -value [17]: $p = \\Pr \\left[ T\\left(x^{\\textnormal {rep}}\\right) \\ge T\\left(x\\right) \\,|\\, \\mu , k\\right] ,$ where $x^{\\textnormal {rep}}$ is the replicated data generated according to our model, $x$ is the observed data, $\\mu $ and $k$ are the model parameters (see Section ), and $T$ is a test statistic.",
"In our checks, we compute $p$ values for various standard test statistics, including mean, variance (with Bessel's correction), median, and quantiles.",
"Note that a perfectly fitting model will yield $p$ -values around $0.5$ .",
"To reliably estimate the probability $p$ , we generate and evaluate the test statistics on 10,000 replicated samples (drawn independently according to our trained model).",
"Table REF shows the $p$ -values for various statistical tests ($p$ -values around $0.5$ indicate a perfect fit).",
"Since our samples are multidimensional (i.e., one value for each sensor around the intersection), we apply the statistical tests to the marginal distributions corresponding to the individual sensors.",
"We list sensors in clockwise order, starting with the sensors on the incoming and outgoing lanes of the road eastward of the intersection (see Figure REF ), denoted `east in' and `east out.'",
"For mean and variance statistics, we see that our model produces an almost perfect fit for all sensors, which is important since these play key role in determining likelihood, on which our detector is built.",
"Further, we see that our model produces a good fit for the median statistic as well, which indicates that there is no significant asymmetry that could not be captured by our model.",
"Finally, we test quantiles $Q(0.3)$ and $Q(0.7)$ , and see a reasonable fit for most sensors.",
"The worst fit is for sensor `north in' (i.e., inward lane of the road northwards) and $Q(0.3)$ ; however, we see an almost perfect fit for the same sensor for $Q(0.7)$ , indicating a skewed distribution.",
"Next, we study the detection performance of our model, showing that observations under normal conditions and under attacks exhibit significantly different likelihood values."
],
[
"Detector Configuration",
"We first consider the defender's problem of balancing the number of false-positive errors and the detection delay.",
"For this experiment, we assume an attack which changes 4.4% of the traffic-signal schedule.",
"Figure: Trade-off between false-positive rate and detection delay.Figure REF shows the trade-off between the false-positive rate and the detection delay.",
"Each point on the curve is a Pareto optimal point that is attainable with some detector window $W$ and threshold $\\tau $ .",
"The figure shows that with a negligible false-positive rate, even the stealthy attack considered in this example can be detected in approximately one hour.",
"The configuration of the detector when the false-positive rate reaches zero is detector window being equal to $W = 48$ minutes and log-likelihood threshold being equal to $\\ln \\tau = -112.58$ .",
"Figure: Likelihood values output by the Guassian-process based model.Figure REF shows the likelihood values output by the Gaussian-process model for traffic data resulting from the original and the attacked traffic-signal schedules.",
"For this figure, we set the detector window to be $W = 3$ minutes, which results in highly variable likelihood values.",
"The figure shows that after one hour (i.e., when the attack starts), the likelihood values for the tampered schedule become much lower than for the original one.",
"In other words, the detector correctly estimates that the traffic with tampered schedule is less likely to be normal."
],
[
"Stealthy Attacks",
"Next, we consider the attacker's problem of balancing stealthiness and impact.",
"If stealthy attacks could avoid detection for extended periods of time while having substantial impact on traffic, they could pose a significant threat to the transportation network.",
"To show that stealthy attacks are not effective against our detector, we compare a wide range of attacks, from stealthy ones that change control only slightly to non-stealthy ones that change control fundamentally.",
"To maximize the advantage of stealthiness, we consider a low detection threshold, which allows attacks to remain undetected for long periods of time.",
"In particular, for this experiment, we assume a detector window of $W = 48$ minutes and a log-likelihood threshold of $\\ln \\tau = -112.58$ , which result in zero false-positive rate for the one-month test interval (see the discussion of Figure REF ).",
"Figure: Impact and detection delay for various attacks.Figure REF shows detection delay and impact for attacks of various magnitudes.",
"The impact of an attack is measured as the fraction of traffic that is “blocked” by the attack, i.e., the decrease in the number of vehicles passing through the intersection compared to the normal traffic-signal schedule; the magnitude of an attack is the fraction of the traffic-signal schedule that is modified by the attacker.",
"The figure shows that attacks with higher magnitude may be less stealthy (i.e., detected earlier), but they cause much more significant impact.",
"In fact, the total impact of attacks, measured as the number of vehicles that could not pass through the intersection due to the attack until its detection, is a strictly increasing function of the attack magnitude.",
"This means that our detector can make stealthy attacks essentially pointless in this example."
],
[
"Multi-Stage Security Game Strategies",
"Next, we provide numerical results on the algorithms that we proposed in Section REF for finding strategies in practice.",
"We first compare the proposed greedy heuristic for finding attacks (Algorithm REF ) to an exhaustive search, and then study the metaheuristic search algorithm for finding detector configurations (Algorithm REF )."
],
[
"Setup",
"To provide meaningful numerical results, we have to evaluate our algorithms on a large number of transportation networks.",
"Every point plotted in the figures of this subsection represents a mean value computed over a large number of random networks with the same parameters.",
"To obtain these networks, we use the Grid model with Random Edges (GRE) to generate random network topologies [18], which closely resemble real-world transportation networks.",
"For a detailed description of this model, we refer the reader to [18], [19].",
"We set both the width and height of the generated grids to be 4, and let the bottom-left corner be a source and the upper-right corner be a sink.",
"For the parameters controlling the randomness of the generation, we use the values from [18], which were derived from measurements on actual road networks from the USA.",
"We let the inflow at the source cell be $d^0 = 8$ , $d^1 = 12$ , $d^2 = 8$ , and $d^{t} = 0$ for $t \\ge 3$ .",
"For every other cell $i$ , we let the parameters be $Q_i = 6$ , $\\delta _i = 1.0$ , and $N_i = 10$ .",
"Finally, we let every merging cell be a signalized intersection, and optimize the inflow proportions for every intersection using a linear program.",
"We assume that there is an anomaly detector deployed in each intersection (i.e., $\\mathcal {I}_D = \\mathcal {I}$ ).",
"We also assume that every one of these detectors exhibits the false-positive rate and detection delay characteristics observed in Section REF .",
"In other words, for each intersection, the defender chooses one of the Pareto optimal configurations that were identified in the experiments of Section REF by choosing a false-positive rate $D_i$ ; the delay of this detector is then determined by the magnitude of the attack against the corresponding intersection.",
"Finally, we assume that the attack is detected as soon as one detector raises an alarm, that attack mitigation takes $\\Delta _M = 20$ minutes, and that congestion levels $T$ , $T_A$ , and $T_M$ are measured in travel time."
],
[
"Attacks",
"We begin by comparing the greedy attack heuristic to an exhaustive search.",
"To perform an exhaustive search, we quantize the space of possible schedules for each intersection, so that we have a finite and discrete search space.",
"For this experiment, we assume that the defender uses a detector configuration that sets the false-positive rate of every intersection $i \\in \\mathcal {I}_D$ to $D_i = 1$ .",
"Figure: Impact resulting from attacks found by exhaustive and greedy searches.Figure REF shows the impact of attacks found by exhaustive and greedy (Algorithm REF ) searches for various budget values.",
"The vertical axis shows the total impact $\\mathcal {G}$ of the attacks, which includes impact that was caused both before and after detection.",
"The figure shows that the attacks found by the greedy search are very close to the ones found by the exhaustive search in terms of total impact, with the largest difference being 5%.",
"Figure: Running time of exhaustive and greedy searches.Figure REF compares the greedy heuristic (Algorithm REF ) to the exhaustive search in terms of running time.",
"Note that we used fairly small problem instances for our experiments in order to be able to apply the algorithms to a large number of networks.",
"We observe that the running time of the greedy heuristic is much lower than that of the exhaustive search, and it grows slower as the attacker's budget increases."
],
[
"Detector Configuration",
"Next, we evaluate the metaheuristic search algorithm for finding detector configurations.",
"We compare our strategic configurations to a non-strategic baseline represented by uniform configurations, which assign the same false-positive rate to all detectors.",
"We find quasi-optimal uniform configurations using the same search algorithm, but restricting the search space to a single scalar value, which is used for all detectors.",
"For these experiments, we let the attacker's budget be enough to compromise $B = 2$ intersections; we assume that the attacker always mounts a best-response attack; and we let the unit cost of false positives be equal to $C = 10$ .",
"Figure: Defender's loss resulting from uniform and strategic detector configurations output by the metaheuristic search algorithm.Figure REF shows the defender's total loss—which includes both the cost of investigating false alarms and the total impact of the attack—with strategic and uniform detector configurations.",
"The horizontal axis shows the number of iterations $k_{\\max }$ for which the search algorithms was run.",
"We can learn two important lessons from this figure.",
"First, strategic thresholds result in much lower losses than uniform ones, which suggests that game-theoretic optimization can have a significant practical impact.",
"Second, losses decrease rapidly in the first 100 or 500 hundred iterations, but they do not decrease further even after a significant number of additional iterations We actually run the search with $k_{\\max } = 100,000$ iterations, but plot only the first 2,000 for clarity, since losses do not decrease significantly after 2,000 iterations., which suggests that the search algorithm is a very efficient practical approach for finding near optimal detector configurations.",
"Figure: Defender's false-positive cost and attack impact resulting from uniform and strategic configurations output by the metaheuristic search algorithm.Figure REF shows the cost of false positives and the total impact of attacks with strategic and uniform detector configurations found by the search algorithm.",
"We observe that strategic detector configurations may result in slightly more false-positive errors, but they can significantly decrease the impact of attacks."
],
[
"Related Work",
"In this section, we briefly survey related work on the vulnerability of transportation networks, the optimal configuration of attack detectors, and game theory for security of cyber-physical systems."
],
[
"Vulnerability of Transportation Networks",
"We first give a brief overview of the related work on the vulnerability of transportation networks.",
"A number of research efforts have studied the vulnerability of transportation networks to natural disasters and attacks.",
"However, to the best of our knowledge, our work is the first one to consider traffic-signal tampering attacks against general transportation networks.",
"In a closely related work, Reilly et al.",
"consider the vulnerability of freeway control systems to attacks on the sensing and control infrastructure [20].",
"They present an in-depth analysis on the takeover of a series of onramp-metering traffic lights using a methodology based on finite-horizon optimal control techniques and multi-objective optimization.",
"Prior work has studied the impact of other disruptive events as well.",
"Sullivan et al.",
"study short-term disruptive events, such as partial flooding, and propose an approach that employs various link-based capacity-disruption values [21].",
"The proposed approach can be used to identify and rank the most critical links and to quantify transportation network robustness (i.e., inverse vulnerability).",
"Jenelius and Mattson introduce an approach for systematically analyzing the robustness of road networks to disruptions affecting extended areas, such as floods and heavy snowfall [22].",
"Their methodology is based on covering the area of interest with grids of uniformly shaped and sized cells, where each cell represents the extent of an event.",
"The authors apply their approach to the Swedish road network, and find that the impact of area-covering disruptions are largely determined by the internal, outbound, and inbound travel demands of the affected area itself.",
"In addition to assessing vulnerability, prior work has also considered the problems of identifying critical links and studying other aspects of vulnerabiltiy.",
"Scott et al.",
"propose a comprehensive, system-wide approach for identifying critical links and evaluating network performance [23].",
"Using three hypothetical networks, the authors demonstrate that their approach yields different highway planning solutions than traditional approaches, which rely on volume/capacity ratios to identify congested or critical links.",
"Jenelius proposes a methodology for vulnerability analysis of road networks and considers the impact of road-link closures [24].",
"The author considers different aspects of vulnerability, and explores the dichotomy between system-wide efficiency and user equity.",
"Prior work has also considered game-theoretic models of attacks against transportation.",
"Alpcan and Buchegger investigate the resilience aspects of vehicular networks using a game-theoretic model, in which defensive measures are optimized with respect to threats posed by intentional attacks [25].",
"The game is formulated in an abstract manner, based on centrality values computed by mapping the centrality values of the car communication network onto the road topology.",
"The authors consider multiple formulations based on varying assumptions on the players' information, and evaluate their models using numerical examples.",
"Bell introduces a two-player non-cooperative game between a network user, who seeks to minimize expected travel cost, and an adversary, who chooses link performance scenarios to maximize the travel cost [26], [27].",
"The Nash equilibrium of this game can be used to measure network performance when users are pessimistic and, hence, may be used for cautious network design.",
"Wu and Amin study normal-form and sequential attacker-defender games over transportation networks to understand how a defender should prioritize its investment in securing a set of facilities [28]."
],
[
"Configuration of Detectors",
"The problem of configuring the sensitivity of intrusion detection systems in the presence of strategic attackers has been studied in a variety of different ways in the academic literature [29].",
"For example, Alpcan and Basar study distributed intrusion detection in access control systems as a security game between an attacker and an IDS, using a model that captures the imperfect flow of information from the attacker to the IDS through a network [30], [31].",
"The authors investigate the existence of a unique Nash equilibrium and best-response strategies under specific cost functions, and analyze long-term interactions using repeated games and a dynamic model.",
"As another example, Dritsoula et al.",
"consider the problem of setting a threshold for classifying an attacker into one of two categories, spammer and spy, based on its intrusion attempts [32].",
"They give a characterization of the Nash equilibria in mixed strategies, and show that the equilibria can be computed in polynomial time.",
"More recently, Lisỳ et al.",
"study randomized detection thresholds using a general model of adversarial classification, which can be applied to e-mail filtering, intrusion detection, steganalysis, etc. [33].",
"The authors analyze both Nash and Stackelberg equilibria based on the true-positive to false-positive curve of the classifier, and find that randomizing the detection threshold may force a strategic attacker to design less efficient attacks.",
"Finally, Zhu and Basar study the problem of optimal signature-based IDS configuration under resource constraints [34].",
"The strategic configuration of the sensitivity of e-mail filtering against spear-phishing and other malicious e-mail is also closely related to the problem considered in this paper.",
"Laszka et al.",
"study a single defender who has to protect multiple users against targeted and non-targeted malicious e-mail [35].",
"The authors focus on characterizing and computing optimal filtering thresholds, and they use numerical results to demonstrate that optimal thresholds can lead to substantially lower losses than naïve ones.",
"Zhao et al.",
"study a variant of the previous model: they assume that the attacker can mount an arbitrary number of costly spear-phishing attacks in order to learn a secret, which is known only by a subset of the users [36], [37].",
"They also focus on the computational aspects of finding optimal filtering thresholds; however, their variant of the model does not capture non-targeted malicious e-mails, such as spam."
],
[
"Game Theory for Security of Cyber-Physical Systems",
"[color=orange!40, linecolor=black!30!orange!60!white]Aron: Check and revise if necessary!",
"Beyond the configuration of detectors, prior efforts have also used game-theory to study a variety of other security problems in cyber-physical systems.",
"For instance, Zhu and Basar introduce a game-theoretic framework for resilient control design and studying the trade-off between robustness, security, and resilience [38].",
"They employ a hybrid model, which integrates a discrete-time Markov model that captures the evolution of cyberstates with continuous-time dynamics that capture the underlying controlled physical process.",
"Backhaus et al.",
"consider the problem of designing attack-resilient power grids and control systems [39].",
"They use game theory to model the conflict between a cyber-physical intruder and a system operator, and use simulation results to assess design options.",
"Pawlick et al.",
"consider Advanced Persistent Threats (APTs) against cloud-controlled cyber-physical systems and design a framework that specifies when a devices should trust commands from the cloud [40].",
"They model this scenario as a three player game between the cloud administrator, the attacker, and the device by combining the FlipIt game [41], [42] with a signaling game.",
"Li et al.",
"consider jamming attacks against remote state estimation[43].",
"In particular, they assume that a sensor and an estimate have to communicate over a wireless, formulate a game-theoretic model, and provide Nash equilibrium strategies."
],
[
"Conclusion",
"As traffic-control devices in practice evolve into complex networks of smart devices, the risks posed to transportation networks by cyber-attacks increases.",
"Thus, it is imperative for traffic-network operators to be prepared to detect and mitigate attacks against traffic control.",
"To provide theoretical foundations for planning and implementing countermeasures, we introduced a game-theoretic model of cyber-attacks against traffic control.",
"Our security game model consists of three stages: defender configuring detectors, attacker mounting a tampering attack against traffic signals, and defender mitigating the attack.",
"Since mitigation—in our model—means adapting the schedules of some traffic signals given the schedules of other signals (set by the adversary in the previous stage), it is equivalent to optimizing traffic control in a non-adversarial setting, which has been studied in prior work.",
"In light of this, we focused on the computational problem of finding optimal actions in the first two stages.",
"We showed that this is a computationally hard problem, which prompted us to propose efficient heuristic algorithms.",
"Using numerical results, we demonstrated that the proposed algorithms are practical.",
"In particular, we first showed that the greedy algorithm for attackers is close to optimal and computationally very efficient.",
"Second, we showed that the metaheuristic search algorithm for detector configuration is effective, and it can significantly decrease losses compared to non-strategic detector configuration.",
"We also introduced and studied a Gaussian-process based traffic-anomaly detector, which we showed to be very effective at detecting tampering attacks against traffic signals."
],
[
"Acknowledgements",
"This work was supported in part by the National Science Foundation under Grant IIS-1905558."
]
] | 1808.08349 |
[
[
"Approach for Video Classification with Multi-label on YouTube-8M Dataset"
],
[
"Abstract Video traffic is increasing at a considerable rate due to the spread of personal media and advancements in media technology.",
"Accordingly, there is a growing need for techniques to automatically classify moving images.",
"This paper use NetVLAD and NetFV models and the Huber loss function for video classification problem and YouTube-8M dataset to verify the experiment.",
"We tried various attempts according to the dataset and optimize hyperparameters, ultimately obtain a GAP score of 0.8668."
],
[
"Introduction",
" Video traffic from video sites such as YouTube has increased in recent years.",
"The growth of personal media through technological development is particularly remarkable.",
"With the development of smartphones, media is now brought to the consumer’s hand, and individuals are no longer only consumers of multimedia but are now producers as well.",
"This trend may be confirmed by internet traffic statistics and other global data.",
"As a result, it is becoming increasingly difficult for consumers to identify desirable media.",
"Accordingly, there is a growing need for techniques to recommend videos or automatically classify subjects.",
"Much effort has been made to process video.",
"Many recent advancements in artificial neural networks have been applied to video processing in an attempt to understand each frame of the video using a convolutional neural network (CNN) [7].",
"Other methods used include VLAD [6] for processing time series data, recurrent neural network network (RNN) series and long short-term memory (LSTM) [4] or GRU [3] for processing time series data.",
"The skipLSTM and skipGRU [2], which add a skip connection to the RNN network, have been proven effective ways to process time series data.",
"The present paper uses NetVLAD and NetFV, which are known as effective methods for video processing, to find the optimal network by adjusting various hyperparameters used in the network.",
"A single model was sought to solve the problem rather than an ensemble technique.",
"In this process, YouTube-8M dataset was used, and a GAP rating of 0.8668 was obtained for the test set.",
"The total number of video in YouTube-8M dataset is 6.1 million.",
"The training set consists of 3.9 million videos.",
"The test and validation sets are each 1.1 million.",
"All video has an average of three labels, and each label is composed of 3,862 multi-labels.",
"Every video is between 120 and 500 seconds in length.",
"This paper use frame-level and audio features.",
"The frame-level features are 1,024-dimensional vectors in which selects one frame per second in video and extracts through Inception V3 model.",
"The audio features are extracted with 128-dimensional vectors drawn through a VGG-inspired acoustic model."
],
[
"Models",
" This paper used NetVLAD [1] and NetFV [10], which were the most successful models used by Willow, the first place–winning team of the YouTube-8M video understanding challenge [8].",
"A hyperparameter was identified to match the 2nd YouTube-8M video understanding challenge limit (model size < 1 GB).",
"NetVLAD and NetFV model uses integrated frame-level features and audio features."
],
[
"Loss Function",
" This paper used the Huber loss function [5], which was used by SNUVL X SKT when the team earned 8th place in the YouTube-8M video understanding challenge [9].",
"The Huber loss combines L2 loss and L1 loss as shown in Equation REF .",
"As the YouTube-8M dataset is substantially imbalanced by a label, the Huber loss function was used to somewhat reduce the noise.",
"$L_\\delta (a) = \\delta ^2\\left(\\sqrt{1+(a/\\delta )^2}-1\\right)$"
],
[
"Evaluation Metric",
" In this paper, the global average precision (GAP) is used as an evaluation method.",
"The GAP is calculated with the top N predictions sorted by confidence score as shown in Equation REF .",
"$GAP = \\sum \\limits _{i=1}^N p(i)\\Delta r(i)$ In Equation REF , $p(i)$ is the precision and $r(i)$ is the recall.",
"In the 2nd YouTube-8M video understanding challenge, N is set to 20 and this paper is calculated accordingly also."
],
[
"Experiments",
" The following experiments were performed: performance comparison of epoch, performance comparison by learning rate, performance comparison of modified dataset preprocessing."
],
[
"Epoch",
" Of particular interest is the evaluation performed of each validation dataset for each epoch.",
"Usually, dozens of epochs are trained in the image training process.",
"However, that was not necessary for the YouTube-8M dataset.",
"The validation results for each epoch are shown in Fig.",
"REF .",
"In the case of the YouTube-8M dataset, the training of one epoch was performed in approximately 25,000 steps when setting the batch size was 80 and used 2-GPU (total 160-batch).",
"Although the GAP difference was slight, it was possible to observe an optimal training performance between approximately 2.5 and 3 epochs.",
"In addition, the GAP for the training set increased in the additional training, but for the validation set decreased.",
"Similar trends were found for various parameters of various models.",
"In this way, it was discovered that not many epochs were needed in the training process of this dataset, thus training was completed after 2.5 epochs.",
"Figure: GAP per epoch curve.",
"The dotted line represents GAP per epoch in the training set; the solid line is GAP per epoch in the validation dataset.",
"As the epoch increases, the training GAP curve also increases.",
"However, the validation GAP curve shows a trend of declining after about 2.5 epochs."
],
[
"Learning Rate",
" The epoch experiment revealed that overfitting of the training set occurs when the model continues to train more than 3 epochs.",
"This paper resolves this problem by adjusting the learning rate to be more effective.",
"In the early part of training, it is good to provide a relatively high learning rate to ensure quick training.",
"Conversely, at the end of the training, the learning rate should be decreased.",
"Thus, the experiment began with a high learning rate, which was set to diminish over time.",
"The learning rate decay per epoch was set to 1/10 of the baseline.",
"It also increased the initial learning rate by 10 times that of the baseline.",
"These ways have helped reduce overfitting.",
"The learning rate decay was 0.8, for the purpose of keeping the learning rate of the two methods similar when the first epoch is complete.",
"So that the learning rate was greater than the baseline even if the training data is in the latter half.",
"This ensured training about these data.",
"The learning rate change at this method is shown in Fig.",
"REF .",
"The resulting performance improvement is shown in Table REF .",
"Figure: Comparison of baseline and experimental learning rates.",
"The initial learning rate of the present method was set at 10 times that of the baseline and the decay per epoch at 1/10 the baseline.",
"The learning rate decay was modified so that the learning rate when the first epoch passed was similar to that of the baseline.Table: Hyperparameters and its GAP score.",
"Hyperparameters were modified.",
"GAP increased by 0.002."
],
[
"Data Preprocessing",
" There was an attempt to improve performance through dataset modifications.",
"First, the imbalance of the dataset was identified and addressed.",
"Second, the false values of the results obtained were analyzed by validating the data trained with the default training set."
],
[
"Overfit to Non-dominant Pattern",
"Fig.",
"REF illustrates that the top 900 labels in the training set accounted for 89% of the multi-label video data, or 10,445,267 of the 11,711,620 total labels for video data.",
"The remaining 2,962 labels represent only 10% of 1,266,353 individuals.",
"To solve this data imbalance, a small training set was constructed with a label index > 977.",
"In this small training set, one epoch of training is performed at 7,300 steps with 2-GPU and each 80-batch (160-batch in total).",
"This small training set was used in two ways.",
"The first method was to train the small training set when the train GAP converged to 1.0 and retrain it as the existing default training set.",
"However, this performance was lower than that of the existing GAP of 0.86 (see Fig.",
"REF ).",
"In the second method, 2.5 epochs were trained with the default training set and retrained with a small training set (see Fig.",
"REF ).",
"But, performance dropped when trained with a small training set.",
"Figure: Visualization of training set imbalance.",
"The top 300 labels represent 76% of the total multi-labels.",
"Expanding the range to the top 900 labels takes up 89% of the total.Figure: GAP curve about validation set: Training with the default training set and retraining with the small training set.",
"0.0 Epoch is when 2.5 epochs were trained with the default training set."
],
[
"Analysis of Validation Results and Additional Experiment",
"The correct answer was compared to the top 20 prediction results of the model.",
"The model validation GAP is 0.86.",
"The analysis showed that 117,410 in a total of 1,112,357 validation set did not include some or all of the correct answers in the top 20 predictions.",
"Of these, 21,828 were single-label, and 55,470 had more than 4 labels.",
"Those with 1 label involved a unique feature of the video label, and those with 4 or more labels had overlapping label features and did not train well.",
"So, training data was selected with only 1 or 4-plus labels.",
"These data were added to the training data by tripling them from other data.",
"In all, about 8,500,000 large training set was created and trained.",
"Unfortunately, there was no significant performance improvement; with a difference of only about 0.0001 according to the GAP, no performance improvement was found through dataset preprocessing.",
"A clearer interpretation method is needed."
],
[
"Final Submission Model",
" An optimal hyperparameter for the NetVLAD and NetFV models was found through the above methods.",
"The results of the test set are shown in Table REF .",
"Table: The performance (GAP) of each model.",
"Optimal cluster size, hidden size and GAP are shown.",
"In the end, a GAP score of 0.8668 was obtained at the 2nd YouTube-8M video understanding challenge.",
"This paper used video classification of the YouTube-8M dataset, applying the NetVLAD and NetFV models with reference to previous research data and using the Huber loss function.",
"Experimental verification is effective for improving performance by adjusting the training epoch, learning rate, and training set.",
"Unlike in conventional training for classification problem, the performance of 2.5 epochs is found to be optimal, as the training set is sufficiently large.",
"The learning rate was also adjusted for optimal training.",
"Even though no performance improvement was found, an attempt was made to train with the set that emphasized frequently-wrong patterns."
],
[
"Acknowledgement",
" This work was partly supported by Institute for Information & communications Technology Promotion(IITP) grant funded by the Korea government(MSIT) (2017-0-01772, Development of QA system for video story understanding to pass Video Turing Test), Information & communications Technology Promotion(IITP) grant funded by the Korea government(MSIT) (2017-0-01781, Data Collection and Automatic Tuning System Development for the Video Understanding), and Institute for Information & communications Technology Promotion(IITP) grant funded by the Korea government(MSIT) (No.2017-0-00271, Development of Archive Solution and Content Management Platform)"
]
] | 1808.08671 |
[
[
"Twisted magnetization states and inhomogeneous resonance modes in a\n Fe/Gd ferrimagnetic multilayer"
],
[
"Abstract Static and dynamic magnetic properties of a ferrimagnetic [Fe(35A)/Gd(50A)]x12 superlattice were investigated in a wide 4-300 K temperature range using magneto-optical Kerr effect (MOKE) and ferromagnetic resonance (FMR) techniques.",
"The multilayer structure was sputtered on a transparent glass substrate which made it possible to perform MOKE measurements on both Fe and Gd terminated sides of the superlattice.",
"These experiments allowed us to detect a transition between field-aligned and canted magnetic states on both sides of the film and to distinguish between the bulk and surface twisted phases of the superlattice.",
"As a result, the experimental H-T magnetic phase diagram of the system was obtained.",
"FMR studies at frequencies 7-36 GHz demonstrated a complex evolution of absorption spectra as temperature decreased from room down to 4 K. Two spectral branches were detected in the sample.",
"Theoretical simulations show that the observed spectral branches correspond to different types of inhomogeneous resonance modes in the multilayer with non-uniform magnetization precession inside Gd layers."
],
[
"Introduction",
"Layered structures based on transition (TM) and rare-earth (RE) ferromagnetic (FM) metals, like Fe/Gd, are model ferrimagnetic systems demonstrating a rich magnetic phase diagram with complex types of magnetic ordering [1], [2].",
"Due to an antiferromagnetic (AFM) coupling at Fe-Gd interfaces and essentially different Curie temperatures of Fe and Gd (for bulk materials, $T_\\mathrm {C}^\\mathrm {Fe}=1043$ K and $T_\\mathrm {C}^\\mathrm {Gd}=293$ K) a so-called \"compensation point\" $T_\\mathrm {comp}$ can exist in the system.",
"At $T=T_\\mathrm {comp}$ magnetic moments of Fe and Gd layers are equal to each other and the total magnetization of the system vanishes.",
"Below $T_\\mathrm {comp}$ , the magnetic moment in Gd subsystem exceeds that in Fe subsystem, while above $T_\\mathrm {comp}$ , opposite situation takes place.",
"As a result, in weak fields applied in the film plane, a collinear magnetic phase is realized with Fe magnetization vector oriented parallel (at $T > T_\\mathrm {comp}$ ) or antiparallel to the field direction (at $T < T_\\mathrm {comp}$ ).",
"As the magnetic field increases and exceeds some critical value, such field-aligned phases become unstable and a transition to canted magnetic state occurs.",
"Moreover, due to a relatively weak exchange stiffness of Gd, the external magnetic field initiates essentially non-uniform distribution of magnetization inside Gd layers (twisted state).",
"The above-discussed complex behaviour of the Fe/Gd system was described theoretically by Camley et al., using the mean-field approach [3–5], and observed experimentally by different techniques in a number of works [6–9].",
"At the same time it was predicted theoretically that even more complicated situation takes place when a finite Fe/Gd superlattice is considered.",
"In this case, two types of twisted magnetic states are possible in the system: surface twist and bulk twist [10].",
"Starting from the field-aligned state in weak external field, an increase of the field leads first to distortion of the collinear state near the superlattice surface (at $H=H_\\mathrm {s}$ ).",
"At higher fields ($H>H_\\mathrm {b}$ ) the bulk twisted state is realized.",
"Fig.",
"REF represents schematically the corresponding magnetization distributions calculated for different field values at $T>T_\\mathrm {comp}$ [11].",
"It is important to note that the surface twist phase arises at the outermost layer of the superlattice when its magnetization is directed opposite to the applied field.",
"Thus, the surface twist phase arises on Gd-terminated side of the superlattice at $T>T_\\mathrm {comp}$ and on Fe-terminated side of the superlattice at $T<T_\\mathrm {comp}$ .",
"Figure: Different types of magnetization vector distribution in a Fe/Gd superlattice at T>T comp T>T_\\mathrm {comp} (calculated using the mean-field model ).Direct experimental observation of such surface twisted states comes to difficulties since it requires simultaneous probing bulk and surface magnetic states of the superlattice.",
"A few works were devoted to this problem.",
"Haskel et al.",
"[12] demonstrated surface twist effects in a Fe-terminated [Fe/Gd]$_{15}$ /Fe multilayer, using grazing-incidence x-ray magnetic circular dichroism.",
"Kravtsov et al.",
"[9] used simultaneous refinement of polarized neutron and resonant x-ray magnetic reflectometry data to directly obtain magnetization depth profiles in a [Fe/Gd]$_5$ multilayer.",
"In both cases the complexity of the used methods makes it difficult to perform detailed studies of stability regions of bulk and surface twisted phases as a function of temperature and magnetic field.",
"Magneto-optical Kerr effect (MOKE) is a relatively simple and sensitive method to obtain direct information about the surface magnetic state of the multilayer.",
"The penetration depth of visible light into metal is about $\\sim 100$ Å which is comparable with typical thickness of individual layers in the superlattice.",
"Thus, MOKE signal provides information about magnetization in several upper layers of the superlattice.",
"Hahn et al.",
"[13] used MOKE to study surface magnetic states in a [Fe/Gd]$_{15}$ structure.",
"Since the samples were sputtered on non-transparent Si substrates, authors compared MOKE signals from superlattices terminated by Fe and Gd layers.",
"The difference of the MOKE curves for two samples was explained by the surface magnetic twist arising in case of the surface layer magnetization oriented opposite to the field direction.",
"In our previous work [11] we studied static magnetization curves of a [Fe/Gd]$_{12}$ multilayer.",
"Comparing the experimental data with mean-field calculations, we found indications of field-dependent phase transitions between field-aligned, surface- and bulk twisted states.",
"However, the static magnetometry provides only the net magnetic moment of the entire multilayer and the surface effects are manifested too weakly.",
"In this work we use MOKE to obtain more precise knowledge on the surface magnetic states in the superlattice.",
"The investigated [Fe/Gd]$_{12}$ multilayer is grown on a transparent glass substrate which allows direct probing magnetic states on both sides of the structure.",
"As a result, we determine the stability regions of bulk and surface twisted states in the superlattice, depending on temperature and magnetic field.",
"The experimental phase diagram is compared with calculations based on the mean-field model [11].",
"Studies of magnetization dynamics in RE/TM systems attract attention due to a recent idea to use such materials for realization of ultrafast magnetic switching, promising for potential applications in magnetic storage devices [14–16].",
"A number of works were devoted to investigations of ferromagnetic resonance (FMR) in TM/Gd multilayers [17–26].",
"Room temperature studies [17–20] demonstrated the importance of spin pumping into RE metal to explain a large FMR line width in TM/RE systems.",
"Several groups reported about the effect of line broadening and shift of the absorption peak to lower fields at cooling the system below room temperature.",
"Such behaviour was observed for Co/Gd [21], [22], Py/Gd [23], Fe/Gd [11] and Fe/Cr/Gd [24], [25] multilayers.",
"In most of the cited works only one \"high-temperature\" resonance peak was detected.",
"This peak became much weaker or even disappeared as temperature decreased below $T_\\mathrm {C}^\\mathrm {Gd}$ which effect was explained by non-local damping mechanisms in the system [11], [23].",
"In a short letter [26], Svalov et al.",
"reported about experimental observation of a second absorption peak below $T_\\mathrm {C}^\\mathrm {Gd}$ in a Co/Gd multilayer.",
"Similar behaviour was observed in our previous works for the Fe/Gd system [11].",
"Theoretical simulations showed that the observed absorption peaks corresponded to different types of inhomogeneous resonance modes in the multilayer.",
"In this work we perform more detailed investigation of temperature evolution of the resonance spectra in the Fe/Gd superlattice.",
"In contrast to the work [11], here we pay special attention to the transformation of the spectra in the vicinity of $T_\\mathrm {C}^\\mathrm {Gd}$ .",
"In particular, we note that the behaviour of the high-temperature resonance peak is strongly dependent on the pumping frequency.",
"To explain this result and identify the observed resonance modes, the experimental data are compared with model calculations based on Landau-Lifshitz equations describing magnetization dynamics in the system."
],
[
"Sample and experimental details",
"The [Fe(35 Å)/Gd(50 Å)]$_{12}$ superlattice was prepared on a glass substrate using high vacuum magnetron sputtering technique.",
"Two chromium layers with thickness 50 Å and 30 Å served as buffer and cap layers respectively.",
"X-ray diffraction studies performed in [11] demonstrated well-defined layered structure of the sample with interfacial root mean square roughness of about 1–2 atomic monolayers.",
"Magnetic properties of the multilayer were studied using MOKE and FMR techniques in the $4-300$ K temperature range in magnetic fields up to 10 kOe applied in the film plane.",
"Longitudinal MOKE studies of the surface magnetization were performed on both sides of the film, using a 635 nm semiconductor laser.",
"In our experimental geometry the MOKE signal was proportional to the component of magnetization parallel to the applied field.",
"FMR measurements were carried out using a conventional field-sweep technique on a laboratory developed transmission type spectrometer at different frequencies in the range $7-36$ GHz.",
"Figure: MOKE curves measured at 155 K from two sides of the film: 1) from the glass substrate side (Fe-terminated side of the superlattice) and 2) from the film surface (Gd-terminated side of the superlattice).",
"Comparing the curves, different types of magnetic ordering can be identified.Figure: MOKE curves obtained at different temperatures on Gd-terminated (a) and Fe-terminated (b) sides of the superlattice.",
"Black arrows show transitions from field-aligned to canted state of the surface magnetization."
],
[
"Magneto-optical Kerr effect",
"Static magnetometry of the investigated sample performed in [11] showed that Gd layers had reduced Curie temperature, $T_\\mathrm {C}^\\mathrm {Gd}\\approx 200$ K, comparing with the bulk value 293 K. The system demonstrated the compensation point at $T_\\mathrm {comp}\\approx 90$ K. Testing MOKE experiments on Fe and Gd thin films showed that both Fe and Gd layers should contribute to the total Kerr effect for the combined Fe/Gd layered system.",
"Under our experimental conditions, the MOKE signal from Gd is comparable with that from Fe (about two times smaller at low temperature) but has opposite sign.",
"Thus, we expect different signs of MOKE for Gd- and Fe-aligned states in the investigated multilayer.",
"Fig.",
"REF shows the experimental MOKE hysteresis loops measured at $T=155$ K from two sides of the superlattice.",
"For both curves, a flat part in the region of weak fields means that the magnetic moment of the outermost layer remains collinear to the external field.",
"Positive sign of the MOKE signal at $H>0$ indicates the Fe-aligned state.",
"At some higher field the MOKE signal decreases, indicating that the magnetization of the outermost layer begins to rotate.",
"Note that on Gd-terminated side this rotation starts in weaker field ($H=H_\\mathrm {s}$ ) than on Fe-terminated side ($H=H_\\mathrm {b}$ ).",
"Thus, we can conclude that in magnetic fields $H_\\mathrm {s}<H<H_\\mathrm {b}$ the surface twist state is realized on Gd-terminated side of the superlattice.",
"In higher fields $H>H_\\mathrm {b}$ , a transformation to the bulk twisted phase occurs.",
"Similar analysis of the MOKE curves was performed for different temperatures in the range 4–300 K (see Fig.",
"REF ) and the resulting phase diagram of the system was obtained (Fig.",
"REF ).",
"At $T>T_\\mathrm {C}^\\mathrm {Gd}$ we observe simple rectangular hysteresis loops without any signs of possible phase transitions.",
"At lower temperatures the shape of the MOKE curves changes.",
"The compensation point $T_\\mathrm {comp}\\approx 90$ K can be clearly detected as temperature where an inversion of the hysteresis loop occurs (Fig.",
"REF ), i.e.",
"different orientation of Fe magnetization is realized in weak fields above and below $T_\\mathrm {comp}$ .",
"It is also clearly seen that at $T>T_\\mathrm {comp}$ the rotation of magnetization starts in weaker fields on Gd-terminated side of the superlattice.",
"On the contrary, at $T<T_\\mathrm {comp}$ this rotation begins in weaker fields on Fe-terminated side of the multilayer.",
"Unfortunately, in the region of low temperatures the increasing hysteresis smears the phase transitions and prevents accurate determination of the critical fields.",
"As a result, the experimental error is increasing.",
"Nevertheless, the observed behaviour is in agreement with the theoretical prediction that the surface twist phase arises on the side of the superlattice when the magnetization of the outermost layer is directed opposite to the applied field.",
"In the phase diagram Fig.",
"REF , the experimental stability regions for different phases are compared with the result of mean-field calculations (see [11] for details).",
"We note a good agreement between the experiment and the model.",
"Figure: Resulting H-TH-T phase diagram of the investigated Fe/Gd superlattice.",
"Points are obtained from MOKE data on two sides of the multilayer.",
"Lines are calculations within the mean-field approach .",
"The dashed line corresponds to a situation when Gd magnetization vanishes in the middle of Gd layer.Figure: Experimental resonance spectra at different temperatures (shown in the plot) obtained at f=7.65f=7.65 GHz (a), f=17.2f=17.2 GHz (b), and f=25.9f=25.9 GHz (c).Figure: Experimental (a) and calculated (b) frequency vs field dependencies at different temperatures (shown in the plots in Kelvins) and examples of calculated depth profiles of magnetization precession in the Gd layer for LT and HT modes (c)."
],
[
"Ferromagnetic resonance",
"Fig.",
"REF demonstrates the temperature evolution of experimental resonance spectra at several different frequencies.",
"At room temperature, one relatively narrow ($\\Delta H\\sim 100$ Oe) absorption peak is observed.",
"As temperature decreases, this \"high-temperature\" (HT) peak broadens and its position changes.",
"We note that the direction of the line shift depends on frequency.",
"At high frequencies ($f\\gtrsim 12$ GHz) the HT peak shifts towards lower fields (Fig.",
"REF b,c).",
"The same behaviour was observed earlier for different types of TM/Gd multilayers [21–23].",
"This effect can be qualitatively described, considering a strongly coupled layered ferrimagnet (see Appendix in the end of this paper), so this behaviour can be considered as \"normal\".",
"In our case, however, another situation takes place at low frequencies ($f\\lesssim 12$ GHz).",
"Here we observe the shift of the HT peak towards higher fields (Fig.",
"REF a).",
"This result is opposite to the behaviour reported in the previous works [21–23] and clearly contradicts to the simple approximation of strongly coupled FM layers.",
"At all frequencies under study, the HT peak disappears below $T\\approx 160$ K. At the same time a second \"low-temperature\" (LT) peak arises in the region of high fields.",
"As temperature decreases, this peak shifts towards lower fields and becomes more pronounced.",
"At high frequencies it can be clearly detected down to lowest temperature, however, at $f=7.65$ GHz it again disappears below $T\\approx 60$ K (Fig.",
"REF ).",
"Fig.",
"REF a demonstrates the resulting experimental frequency-vs-field $f(H)$ dependencies at different temperatures.",
"At room temperature the $f(H)$ curve for HT-mode can be qualitatively described by Kittel-like equation for FM film (see Appendix, Eq.",
"(REF )).",
"However at lower temperatures the shape of the $f(H)$ curve changes strongly and the simple Kittel's formula clearly becomes inapplicable.",
"This means that the approximation of uniform magnetization precession within the structure is not valid.",
"Taking into account a large exchange stiffness of Fe layers and a strong coupling at Fe-Gd interface, we can suppose that inhomogeneous precession occurs inside the Gd layers.",
"To describe such inhomogeneous resonance modes theoretically we use the approach of the work [11].",
"To model the non-uniform magnetization precession inside Gd layers they are divided into elementary \"atomic\" sublayers coupled with each other.",
"The static magnetization in each sublayer is calculated using the mean-field model while the dynamics is described by Landau-Lifshitz equations (LLE) with relaxation terms.",
"For relaxation terms, we consider Gilbert damping in Fe and Gd layers as well as diffusion-type damping in Gd (see [11] for details and model parameters).",
"As a result, we calculate the complex eigenfrequencies of the system $\\omega =\\omega ^\\prime +i\\omega ^{\\prime \\prime }$ .",
"The corresponding eigenvectors represent the depth profiles of magnetization precession in the superlattice.",
"The damping of the calculated resonance modes can be characterized by quality factor (Q-factor) $Q=\\omega ^\\prime /2\\omega ^{\\prime \\prime }$ .",
"The larger Q-factor is, the more intensive resonance peak is expected.",
"Following our previous work [11], we consider only the modes with in-phase precession of Fe layers and perform modelling for one period of the superlattice.",
"Fig.",
"REF b demonstrates the resulting calculated dependencies $f(H)$ at different temperatures (for illustrative purposes, only the modes with $Q>0.5$ , are shown).",
"The model predicts the existence of two spectral branches with different types of magnetic precession inside Gd layers (Fig.",
"REF c).",
"The HT-mode has a gap in the spectrum at low temperatures and corresponds to strongly non-uniform precession inside Gd layers.",
"The LT-mode is quasi-uniform.",
"Its frequency vanishes at $H=H_\\mathrm {b}$ , i.e.",
"at phase transition from field-aligned to twisted magnetic state.",
"In general, the behaviour of calculated curves $f(H)$ repeats qualitatively the experimental dependencies, except the temperature region $\\approx 200-225$ K (i.e.",
"slightly above $T_\\mathrm {C}^\\mathrm {Gd}$ ) and in weak magnetic fields $H\\lesssim 1.5$ kOe.",
"Above $T=225$ K the model predicts the crossing of two spectral branches.",
"One branch with increasing dependence $f(H)$ corresponds to preferable precession of Fe layers.",
"This branch has large Q-factor and is observed experimentally.",
"The second branch with decreasing dependence $f(H)$ corresponds to preferable precession of inner part of Gd layers.",
"This branch has small Q-factor and is not observed experimentally.",
"Below $T=225$ K the model predicts the repulsion of these two crossing modes.",
"As a consequence, a gap in the spectrum opens.",
"Experimentally, however, such a gap arises only at $T\\lesssim 180$ K (Fig.",
"REF a,b).",
"Despite this discrepancy, Fig.",
"REF b helps to understand different behaviour of the HT peak at frequencies below and above $f\\approx 12$ GHz, i.e different direction of the line shift at cooling the system below room temperature (Fig.",
"REF ).",
"The critical value 12 GHz corresponds to the frequency where the effect of modes repulsion arises.",
"Fig.",
"REF shows the resulting experimental and calculated temperature dependencies of the resonance fields $H_\\mathrm {res}(T)$ at different frequencies.",
"It can be seen that the experimental and theoretical curves demonstrate not only qualitative but also a certain quantitative agreement.",
"The noticeable discrepancy observed for HT-mode at 17.2 GHz below $T\\approx 230$ K is connected with the above-discussed inadequate description of the mode-repulsion region.",
"Figure: Temperature dependencies of the resonance field at different frequencies.",
"Points are experimental data, lines are calculations.",
"Solid, dashed, and dotted lines correspond to different Q-factor of resonance modes.It is interesting to note that at low frequency ($f=7.65$ GHz) the model predicts the existence of minimum in the $H_\\mathrm {res}(T)$ dependence for the LT-mode.",
"This minimum is connected with the fact that the LT-mode frequency vanishes at $H=H_\\mathrm {b}$ (i.e.",
"$H_\\mathrm {res}\\rightarrow H_\\mathrm {b}$ when $f\\rightarrow 0$ ).",
"Since $H_\\mathrm {b}$ turns to zero at $T_\\mathrm {comp}$ , we could expect the minimum of $H_\\mathrm {res}(T)$ at this temperature.",
"Experimentally, however, we did not manage to detect the absorption line below $T_\\mathrm {comp}$ .",
"The reason for this can be the large damping of the corresponding resonance mode.",
"Indeed, our calculations show that the Q-factor of the LT-mode increases below $T_\\mathrm {comp}$ at $f=7.65$ GHz (see Fig.",
"REF ).",
"To summarize, we achieved a reasonable agreement between the experiment and model calculations.",
"The model describes many features of the experimental spectra and helps to identify the types of the observed resonance modes.",
"The main discrepancy between the experiment and model arises in the vicinity of $T_\\mathrm {C}^\\mathrm {Gd}$ where the calculated spectra are very sensitive to magnetic parameters of the system and can be strongly influenced by structural inhomogeneities of the real superlattice."
],
[
"Conclusion",
"In this work we demonstrated the realization of non-collinear magnetic states and inhomogeneous magnetization dynamics in a Fe/Gd artificial layered ferrimagnet.",
"We have shown that both static and dynamic properties of the system are described taking into account essentially non-uniform magnetization distribution inside Gd layers.",
"Using the magneto-optical Kerr effect, we defined the regions of stability for surface and bulk twisted states of the investigated multilayer.",
"The resulting experimental $H-T$ phase diagram is in a good agreement with calculations based on the mean-field model.",
"Ferromagnetic resonance spectra obtained in this work reveal a complex temperature evolution with two spectral branches that can not be explained in terms of uniform magnetic precession within the superlattice.",
"The performed theoretical simulations of magnetization dynamics in the system show that the observed resonance modes correspond to different types of inhomogeneous precession inside Gd layers.",
"In the end we would like to emphasize that the nanostructured ferrimagnets provide possibility to study such complex magnetic phenomena under easily achievable experimental conditions: in magnetic fields up to 1 T and at microwave frequencies.",
"The traditional ferrimagnetic crystals would require magnetic fields and frequencies that are several orders of magnitude larger.",
"In this respect, the artificial structures can be considered as suitable model objects for experimental investigations of non-collinear magnetic phases and inhomogeneous magnetization dynamics in ferrimagnets."
],
[
"Acknowledgments",
"The work is partially supported by the Russian Foundation for Basic Research (grants No.",
"16-02-00061, No.",
"18-37-00182), by the Ministry of Education and Science of the Russian Federation (grant No.",
"14-Z-50.31.0025), and by the Basic Research Program of the Presidium of Russian Academy of Sciences.",
"Research in Ekaterinburg was performed in terms of the State assignment of Federal Agency of Scientific Organizations of the Russian Federation (theme “Spin” No.",
"AAAA-A18-188 020290104-2)."
],
[
"FMR frequency of a strongly coupled layered ferrimagnet",
"Let us consider two FM layers with different magnetic moments $\\mu _1 > \\mu _2$ .",
"We suppose that these layers are strongly AFM coupled (the exchange energy is infinity).",
"In this case, the magnetic field H applied in the film plane aligns $\\mu _1$ and $\\mu _2$ parallel and antiparallel to the field direction respectively.",
"Considering Zeeman and demagnetizing energy of both layers, the total energy of the system can be written as $E=-\\mathbf {H}\\left(\\mu _1+\\mu _2\\right) + 2\\pi \\left[ \\frac{\\left(\\mu _1 \\cdot \\mathbf {z}\\right)^2}{V_1} + \\frac{\\left(\\mu _2 \\cdot \\mathbf {z}\\right)^2}{V_2} \\right],$ where z is a unit vector normal to the film plane, $V_1$ and $V_2$ are volumes of layers.",
"Taking into account that $-\\mu _2\\upuparrows \\mu _1$ , the energy expression can be rewritten in the form $E=-\\mathbf {H}\\mu + 2\\pi \\frac{\\mu _1^2V/V_1+\\mu _2^2V/V_2}{(\\mu _1-\\mu _2)^2} \\cdot \\frac{\\left(\\mu \\cdot \\mathbf {z}\\right)^2}{V},$ where $\\mu =\\mu _1+\\mu _2$ and $V=V_1+V_2$ .",
"Now it has the form of magnetic energy for a single FM film with modified demagnetizing factor.",
"Thus, the FMR frequency of the system is defined by modified Kittel's formula $\\omega =\\gamma _\\mathrm {eff} \\sqrt{H\\left(H+4\\pi M_\\mathrm {eff} \\right)},$ where $4\\pi M_\\mathrm {eff}=4\\pi \\frac{\\mu _1^2/V_1+\\mu _2^2/V_2}{\\mu _1-\\mu _2},$ and $\\gamma _\\mathrm {eff}$ is a net gyromagnetic ratio of two coupled layers [27] $\\gamma _\\mathrm {eff}=\\frac{\\mu _1-\\mu _2}{\\mu _1/\\gamma _1-\\mu _2/\\gamma _2},$ where $\\gamma _1$ and $\\gamma _2$ are gyromagnetic ratios of individual layers.",
"If $\\gamma _1\\approx \\gamma _2$ , Eqs.",
"(REF ), (REF ) predict increasing FMR frequency when $\\mu _2$ is increasing.",
"This behaviour is opposite to the case of amorphous or crystal ferrimagnetic film when the effective demagnetizing field is defined by simple expression $4\\pi M_\\mathrm {eff}=4\\pi (M_1-M_2)$ , where $M_{1,2}$ are magnetizations of FM sublattices [27].",
"In this situation FMR frequency is decreasing with $M_2$ increase.",
"It is important to note that the approximation (REF )–(REF ) is valid only when the exchange fields $H_{\\mathrm {ex},i}$ acting on layers $i=1,2$ are much stronger than the corresponding demagnetizing fields $H_{\\mathrm {ex},i}\\gg 4\\pi M_i$ and the external field is far below the transition to the canted state: $H\\ll |H_{\\mathrm {ex},1}-H_{\\mathrm {ex},2}|$ [28]."
]
] | 1808.08466 |
[
[
"Meta-Learning for Low-Resource Neural Machine Translation"
],
[
"Abstract In this paper, we propose to extend the recently introduced model-agnostic meta-learning algorithm (MAML) for low-resource neural machine translation (NMT).",
"We frame low-resource translation as a meta-learning problem, and we learn to adapt to low-resource languages based on multilingual high-resource language tasks.",
"We use the universal lexical representation~\\citep{gu2018universal} to overcome the input-output mismatch across different languages.",
"We evaluate the proposed meta-learning strategy using eighteen European languages (Bg, Cs, Da, De, El, Es, Et, Fr, Hu, It, Lt, Nl, Pl, Pt, Sk, Sl, Sv and Ru) as source tasks and five diverse languages (Ro, Lv, Fi, Tr and Ko) as target tasks.",
"We show that the proposed approach significantly outperforms the multilingual, transfer learning based approach~\\citep{zoph2016transfer} and enables us to train a competitive NMT system with only a fraction of training examples.",
"For instance, the proposed approach can achieve as high as 22.04 BLEU on Romanian-English WMT'16 by seeing only 16,000 translated words (~600 parallel sentences)."
],
[
"Introduction",
"Despite the massive success brought by neural machine translation [37], [4], [38], it has been noticed that the vanilla NMT often lags behind conventional machine translation systems, such as statistical phrase-based translation systems [25], for low-resource language pairs [24].",
"In the past few years, various approaches have been proposed to address this issue.",
"The first attempts at tackling this problem exploited the availability of monolingual corpora [18], [33], [41].",
"It was later followed by approaches based on multilingual translation, in which the goal was to exploit knowledge from high-resource language pairs by training a single NMT system on a mix of high-resource and low-resource language pairs [12], [13], [28], [22], [20].",
"Its variant, transfer learning, was also proposed by [43], in which an NMT system is pretrained on a high-resource language pair before being finetuned on a target low-resource language pair.",
"In this paper, we follow up on these latest approaches based on multilingual NMT and propose a meta-learning algorithm for low-resource neural machine translation.",
"We start by arguing that the recently proposed model-agnostic meta-learning algorithm [11] could be applied to low-resource machine translation by viewing language pairs as separate tasks.",
"This view enables us to use MAML to find the initialization of model parameters that facilitate fast adaptation for a new language pair with a minimal amount of training examples (§).",
"Furthermore, the vanilla MAML however cannot handle tasks with mismatched input and output.",
"We overcome this limitation by incorporating the universal lexical representation [16] and adapting it for the meta-learning scenario (§REF ).",
"We extensively evaluate the effectiveness and generalizing ability of the proposed meta-learning algorithm on low-resource neural machine translation.",
"We utilize 17 languages from Europarl and Russian from WMT as the source tasks and test the meta-learned parameter initialization against five target languages (Ro, Lv, Fi, Tr and Ko), in all cases translating to English.",
"Our experiments using only up to 160k tokens in each of the target task reveal that the proposed meta-learning approach outperforms the multilingual translation approach across all the target language pairs, and the gap grows as the number of training examples decreases."
],
[
"Neural Machine Translation (NMT)",
"Given a source sentence $X=\\lbrace x_1, ..., x_{T^{\\prime }}\\rbrace $ , a neural machine translation model factors the distribution over possible output sentences $Y=\\lbrace y_1, ..., y_T\\rbrace $ into a chain of conditional probabilities with a left-to-right causal structure: $p(Y|X; \\theta ) = \\prod _{t=1}^{T+1} p(y_t| y_{0:t-1}, x_{1:T^{\\prime }}; \\theta ),$ where special tokens $y_0$ ($\\langle \\mathrm {bos}\\rangle $ ) and $y_{T+1}$ ($\\langle \\mathrm {eos}\\rangle $ ) are used to represent the beginning and the end of a target sentence.",
"These conditional probabilities are parameterized using a neural network.",
"Typically, an encoder-decoder architecture [37], [9], [4] with a RNN-based decoder is used.",
"More recently, architectures without any recurrent structures [14], [38] have been proposed and shown to speed up training while achieving state-of-the-art performance."
],
[
"Low Resource Translation",
"NMT is known to easily over-fit and result in an inferior performance when the training data is limited [24].",
"In general, there are two ways for handling the problem of low resource translation: (1) utilizing the resource of unlabeled monolingual data, and (2) sharing the knowledge between low- and high-resource language pairs.",
"Many research efforts have been spent on incorporating the monolingual corpora into machine translation, such as multi-task learning [18], [41], back-translation [33], dual learning [21] and unsupervised machine translation with monolingual corpora only for both sides [3], [27], [40].",
"For the second approach, prior researches have worked on methods to exploit the knowledge of auxiliary translations, or even auxiliary tasks.",
"For instance, [8], [6], [29], [7] investigate the use of a pivot to build a translation path between two languages even without any directed resource.",
"The pivot can be a third language or even an image in multimodal domains.",
"When pivots are not easy to obtain, [12], [28], [22] have shown that the structure of NMT is suitable for multilingual machine translation.",
"[16] also showed that such a multilingual NMT system could improve the performance of low resource translation by using a universal lexical representation to share embedding information across languages.",
"All the previous work for multilingual NMT assume the joint training of multiple high-resource languages naturally results in a universal space (for both the input representation and the model) which, however, is not necessarily true, especially for very low resource cases."
],
[
"Meta Learning",
"In the machine learning community, meta-learning, or learning-to-learn, has recently received interests.",
"Meta-learning tries to solve the problem of “fast adaptation on new training data.” One of the most successful applications of meta-learning has been on few-shot (or one-shot) learning [26], where a neural network is trained to readily learn to classify inputs based on only one or a few training examples.",
"There are two categories of meta-learning: learning a meta-policy for updating model parameters [1], [19], [31] learning a good parameter initialization for fast adaptation [11], [39], [36].",
"In this paper, we propose to use a meta-learning algorithm for low-resource neural machine translation based on the second category.",
"More specifically, we extend the idea of model-agnostic meta-learning [11] in the multilingual scenario.",
"Figure: The graphical illustration of the training process of the proposed MetaNMT.",
"For each episode, one task (language pair) is sampled for meta-learning.",
"The boxes and arrows in blue are mainly involved in language-specific learning (§), and those in purple in meta-learning (§)."
],
[
"Meta Learning for Low-Resource Neural Machine Translation",
"The underlying idea of MAML is to use a set of source tasks $\\left\\lbrace \\mathcal {T}^1, \\ldots , \\mathcal {T}^K \\right\\rbrace $ to find the initialization of parameters $\\theta ^0$ from which learning a target task $\\mathcal {T}^0$ would require only a small number of training examples.",
"In the context of machine translation, this amounts to using many high-resource language pairs to find good initial parameters and training a new translation model on a low-resource language starting from the found initial parameters.",
"This process can be understood as $\\theta ^* = \\text{Learn}(\\mathcal {T}^0; \\text{MetaLearn}(\\mathcal {T}^1, \\ldots , \\mathcal {T}^K)).$ That is, we meta-learn the initialization from auxiliary tasks and continue to learn the target task.",
"We refer the proposed meta-learning method for NMT to MetaNMT.",
"See Fig.",
"REF for the overall illustration."
],
[
"Learn: language-specific learning",
"Given any initial parameters $\\theta ^0$ (which can be either random or meta-learned), the prior distribution of the parameters of a desired NMT model can be defined as an isotropic Guassian: $\\theta _i \\sim \\mathcal {N}(\\theta ^0_i, 1/\\beta ),$ where $1/\\beta $ is a variance.",
"With this prior distribution, we formulate the language-specific learning process $\\text{Learn}(D_\\mathcal {T}; \\theta ^0)$ as maximizing the log-posterior of the model parameters given data $D_{\\mathcal {T}}$ : $&\\text{Learn}(D_\\mathcal {T}; \\theta ^0) =\\arg \\max _{\\theta } \\mathcal {L}^{D_\\mathcal {T}} (\\theta )\\\\&=\\arg \\max _{\\theta }\\!\\!\\!\\!\\sum _{(X,Y) \\in D_{\\mathcal {T}}} \\!\\!",
"\\!\\!\\log p(Y | X, \\theta )- \\beta \\Vert \\theta - \\theta ^0 \\Vert ^2,$ where we assume $p(X|\\theta )$ to be uniform.",
"The first term above corresponds to the maximum likelihood criterion often used for training a usual NMT system.",
"The second term discourages the newly learned model from deviating too much from the initial parameters, alleviating the issue of over-fitting when there is not enough training data.",
"In practice, we solve the problem above by maximizing the first term with gradient-based optimization and early-stopping after only a few update steps.",
"Thus, in the low-resource scenario, finding a good initialization $\\theta ^0$ strongly correlates the final performance of the resulting model.",
"Figure: An intuitive illustration in which we use solid lines to represent the learning of initialization, and dashed lines to show the path of fine-tuning."
],
[
"MetaLearn",
"We find the initialization $\\theta ^0$ by repeatedly simulating low-resource translation scenarios using auxiliary, high-resource language pairs.",
"Following [11], we achieve this goal by defining the meta-objective function as $\\mathcal {L}(\\theta ) =&\\mathbb {E}_{k} \\mathbb {E}_{D_{\\mathcal {T}^{k}}, D^{\\prime }_{\\mathcal {T}^{k}}} \\\\&\\left[\\sum _{(X,Y) \\in D^{\\prime }_{\\mathcal {T}^{k}}} \\!\\!\\!\\!\\!\\!\\!\\log p(Y|X; \\text{Learn}(D_{\\mathcal {T}^{k}}; \\theta ))\\right], \\nonumber $ where $k \\!\\sim \\!\\mathcal {U}(\\left\\lbrace 1, \\ldots , K \\right\\rbrace )$ refers to one meta-learning episode, and $D_{\\mathcal {T}}$ , $D^{\\prime }_{\\mathcal {T}}$ follow the uniform distribution over $\\mathcal {T}$ 's data.",
"We maximize the meta-objective function using stochastic approximation [32] with gradient descent.",
"For each episode, we uniformly sample one source task at random, $\\mathcal {T}^{k}$ .",
"We then sample two subsets of training examples independently from the chosen task, $D_{\\mathcal {T}^{k}}$ and $D^{\\prime }_{\\mathcal {T}^{k}}$ .",
"We use the former to simulate language-specific learning and the latter to evaluate its outcome.",
"Assuming a single gradient step is taken only the with learning rate $\\eta $ , the simulation is: $\\theta ^{\\prime }_k = \\text{Learn}(D_{\\mathcal {T}^k}; \\theta ) =\\theta - \\eta \\nabla _{\\theta } \\mathcal {L}^{D_{\\mathcal {T}^k}}(\\theta ).$ Once the simulation of learning is done, we evaluate the updated parameters $\\theta ^{\\prime }_k$ on $D^{\\prime }_{\\mathcal {T}^{k}}$ , The gradient computed from this evaluation, which we refer to as meta-gradient, is used to update the meta model $\\theta $ .",
"It is possible to aggregate multiple episodes of source tasks before updating $\\theta $ : $\\theta \\leftarrow \\theta - \\eta ^{\\prime } \\sum _k \\nabla _\\theta \\mathcal {L}^{D^{\\prime }_{\\mathcal {T}^{k}}}(\\theta ^{\\prime }_k),$ where $\\eta ^{\\prime }$ is the meta learning rate.",
"Unlike a usual learning scenario, the resulting model $\\theta ^0$ from this meta-learning procedure is not necessarily a good model on its own.",
"It is however a good starting point for training a good model using only a few steps of learning.",
"In the context of machine translation, this procedure can be understood as finding the initialization of a neural machine translation system that could quickly adapt to a new language pair by simulating such a fast adaptation scenario using many high-resource language pairs."
],
[
"Meta-Gradient",
"We use the following approximation property $H(x)v \\approx \\frac{\\nabla (x+\\nu v) - \\nabla (x)}{\\nu }$ to approximate the meta-gradient:We omit the subscript $k$ for simplicity.",
"$&\\nabla _\\theta \\mathcal {L}^{D^{\\prime }}(\\theta ^{\\prime }) = \\nabla _{\\theta ^{\\prime }} \\mathcal {L}^{D^{\\prime }}(\\theta ^{\\prime })\\nabla _{\\theta }(\\theta - \\eta \\nabla _{\\theta } \\mathcal {L}^{D}(\\theta )) \\\\&= \\nabla _{\\theta ^{\\prime }} \\mathcal {L}^{D^{\\prime }}(\\theta ^{\\prime })- \\eta \\nabla _{\\theta ^{\\prime }} \\mathcal {L}^{D^{\\prime }}(\\theta ^{\\prime }) H_{\\theta }(\\mathcal {L}^{D}(\\theta )) \\\\&\\approx \\nabla _{\\theta ^{\\prime }} \\mathcal {L}^{D^{\\prime }}(\\theta ^{\\prime })- \\frac{\\eta }{\\nu } \\left[\\nabla _{\\theta }\\mathcal {L}^D(\\theta )\\bigg |_{\\hat{\\theta }}- \\nabla _{\\theta }\\mathcal {L}^D(\\theta )\\bigg |_{\\theta }\\right],$ where $\\nu $ is a small constant and $\\hat{\\theta } = \\theta + \\nu \\nabla _{\\theta ^{\\prime }}\\mathcal {L}^{D^{\\prime }}(\\theta ^{\\prime }).$ In practice, we find that it is also possible to ignore the second-order term, ending up with the following simplified update rule: $\\nabla _\\theta \\mathcal {L}^{D^{\\prime }}(\\theta ^{\\prime }) \\approx \\nabla _{\\theta ^{\\prime }} & \\mathcal {L}^{D^{\\prime }}(\\theta ^{\\prime }).$"
],
[
"Related Work: Multilingual Transfer Learning",
"The proposed MetaNMT differs from the existing framework of multilingual translation [28], [22], [16] or transfer learning [43].",
"The latter can be thought of as solving the following problem: $\\max _{\\theta } \\mathcal {L}^{\\text{multi}}(\\theta ) = \\mathbb {E}_k\\left[\\sum _{(X,Y) \\in D_k} \\log p(Y|X; \\theta )\\right],$ where $D_k$ is the training set of the $k$ -th task, or language pair.",
"The target low-resource language pair could either be a part of joint training or be trained separately starting from the solution $\\theta ^0$ found from solving the above problem.",
"The major difference between the proposed MetaNMT and these multilingual transfer approaches is that the latter do not consider how learning happens with the target, low-resource language pair.",
"The former explicitly incorporates the learning process within the framework by simulating it repeatedly in Eq.",
"(REF ).",
"As we will see later in the experiments, this results in a substantial gap in the final performance on the low-resource task."
],
[
"Illustration",
"In Fig.",
"REF , we contrast transfer learning, multilingual learning and meta-learning using three source language pairs (Fr-En, Es-En and Pt-En) and two target pairs (Ro-En and Lv-En).",
"Transfer learning trains an NMT system specifically for a source language pair (Es-En) and finetunes the system for each target language pair (Ro-En, Lv-En).",
"Multilingual learning often trains a single NMT system that can handle many different language pairs (Fr-En, Pt-En, Es-En), which may or may not include the target pairs (Ro-En, Lv-En).",
"If not, it finetunes the system for each target pair, similarly to transfer learning.",
"Both of these however aim at directly solving the source tasks.",
"On the other hand, meta-learning trains the NMT system to be useful for fine-tuning on various tasks including the source and target tasks.",
"This is done by repeatedly simulating the learning process on low-resource languages using many high-resource language pairs (Fr-En, Pt-En, Es-En)."
],
[
"I/O mismatch across language pairs",
"One major challenge that limits applying meta-learning for low resource machine translation is that the approach outlined above assumes the input and output spaces are shared across all the source and target tasks.",
"This, however, does not apply to machine translation in general due to the vocabulary mismatch across different languages.",
"In multilingual translation, this issue has been tackled by using a vocabulary of sub-words [33] or characters [28] shared across multiple languages.",
"This surface-level sharing is however limited, as it cannot be applied to languages exhibiting distinct orthography (e.g., Indo-Euroepan languages vs.",
"Korean.)"
],
[
"Universal Lexical Representation (ULR)",
"We tackle this issue by dynamically building a vocabulary specific to each language using a key-value memory network [30], [17], as was done successfully for low-resource machine translation recently by [16].",
"We start with multilingual word embedding matrices $\\epsilon ^k_{\\text{query}} \\in \\mathbb {R}^{|V_k| \\times d}$ pretrained on large monolingual corpora, where $V_k$ is the vocabulary of the $k$ -th language.",
"These embedding vectors can be obtained with small dictionaries of seed word pairs [2], [35] or in a fully unsupervised manner [42], [10].",
"We take one of these languages $k^{\\prime }$ to build universal lexical representation consisting of a universal embedding matrix $\\epsilon _u \\in \\mathbb {R}^{M \\times d}$ and a corresponding key matrix $\\epsilon _{\\text{key}} \\in \\mathbb {R}^{M \\times d}$ , where $M < |V_k^{\\prime }|$ .",
"Both $\\epsilon ^k_{\\text{query}}$ and $\\epsilon _{\\text{key}}$ are fixed during meta-learning.",
"We then compute the language-specific embedding of token $x$ from the language $k$ as the convex sum of the universal embedding vectors by $\\epsilon ^0[x] = \\sum _{i=1}^M \\alpha _i \\epsilon _u[i],$ where $\\alpha _i \\propto \\exp \\left\\lbrace -\\tfrac{1}{\\tau } \\epsilon _{\\text{key}}[i]^\\top A \\epsilon ^k_{\\text{query}} [x] \\right\\rbrace $ and $\\tau $ is set to $0.05$ .",
"This approach allows us to handle languages with different vocabularies using a fixed number of shared parameters ($\\epsilon _u$ , $\\epsilon _{\\text{key}}$ and $A$ .)"
],
[
"Learning of ULR",
"It is not desirable to update the universal embedding matrix $\\epsilon _u$ when fine-tuning on a small corpus which contains a limited set of unique tokens in the target language, as it could adversely influence the other tokens' embedding vectors.",
"We thus estimate the change to each embedding vector induced by language-specific learning by a separate parameter $\\Delta \\epsilon ^k[x]$ : $\\epsilon ^k[x] = \\epsilon ^0[x] + \\Delta \\epsilon ^k[x].$ During language-specific learning, the ULR $\\epsilon ^0[x]$ is held constant, while only $\\Delta \\epsilon ^k[x]$ is updated, starting from an all-zero vector.",
"On the other hand, we hold $\\Delta \\epsilon ^k[x]$ 's constant while updating $\\epsilon _u$ and $A$ during the meta-learning stage."
],
[
"Target Tasks",
"We show the effectiveness of the proposed meta-learning method for low resource NMT with extremely limited training examples on five diverse target languages: Romanian (Ro) from WMT'16,http://www.statmt.org/wmt16/translation-task.html Latvian (Lv), Finnish (Fi), Turkish (Tr) from WMT'17,http://www.statmt.org/wmt17/translation-task.html and Korean (Ko) from Korean Parallel Dataset.https://sites.google.com/site/koreanparalleldata/ We use the officially provided train, dev and test splits for all these languages.",
"The statistics of these languages are presented in Table REF .",
"We simulate the low-resource translation scenarios by randomly sub-sampling the training set with different sizes."
],
[
"Source Tasks",
"We use the following languages from Europarlhttp://www.statmt.org/europarl/: Bulgarian (Bg), Czech (Cs), Danish (Da), German (De), Greek (El), Spanish (Es), Estonian (Et), French (Fr), Hungarian (Hu), Italian (It), Lithuanian (Lt), Dutch (Nl), Polish (Pl), Portuguese (Pt), Slovak (Sk), Slovene (Sl) and Swedish (Sv), in addition to Russian (Ru)A subsample of approximately 2M pairs from WMT'17.",
"to learn the intilization for fine-tuning.",
"In our experiments, different combinations of source tasks are explored to see the effects from the source tasks."
],
[
"Validation",
"We pick either Ro-En or Lv-En as a validation set for meta-learning and test the generalization capability on the remaining target tasks.",
"This allows us to study the strict form of meta-learning, in which target tasks are unknown during both training and model selection."
],
[
"Preprocessing and ULR Initialization",
"As described in §REF , we initialize the query embedding vectors $\\epsilon _{\\text{query}}^k$ of all the languages.",
"For each language, we use the monolingual corpora built from WikipediaWe use the most recent Wikipedia dump (2018.5) from https://dumps.wikimedia.org/backup-index.html.",
"and the parallel corpus.",
"The concatenated corpus is first tokenized and segmented using byte-pair encoding [34], resulting in $40,000$ subwords for each language.",
"We then estimate word vectors using fastText [5] and align them across all the languages in an unsupervised way using MUSE [10] to get multilingual word vectors.",
"We use the multilingual word vectors of the 20,000 most frequent words in English to form the universal embedding matrix $\\epsilon _u$ .",
"Table: BLEU Scores w.r.t.",
"the source task set for all five target tasks."
],
[
"Model",
"We utilize the recently proposed Transformer [38] as an underlying NMT system.",
"We implement Transformer in this paper based on [15]https://github.com/salesforce/nonauto-nmt and modify it to use the universal lexical representation from §REF .",
"We use the default set of hyperparameters ($d_\\text{model} = d_{\\text{hidden}} = 512$ , $n_\\text{layer}=6$ , $n_\\text{head}=8$ , $n_\\text{batch}=4000$ , $t_\\text{warmup} = 16000$ ) for all the language pairs and across all the experimental settings.",
"We refer the readers to [38], [15] for the details of the model.",
"However, since the proposed meta-learning method is model-agnostic, it can be easily extended to any other NMT architectures, e.g.",
"RNN-based sequence-to-sequence models with attention [4]."
],
[
"Learning",
"We meta-learn using various sets of source languages to investigate the effect of source task choice.",
"For each episode, by default, we use a single gradient step of language-specific learning with Adam [23] per computing the meta-gradient, which is computed by the first-order approximation in Eq.",
"(REF ).",
"For each target task, we sample training examples to form a low-resource task.",
"We build tasks of 4k, 16k, 40k and 160k English tokens for each language.",
"We randomly sample the training set five times for each experiment and report the average score and its standard deviation.",
"Each fine-tuning is done on a training set, early-stopped on a validation set and evaluated on a test set.",
"In default without notation, datasets of 16k tokens are used.",
"Figure: BLEU Scores w.r.t.",
"the size of the target task's training set."
],
[
"Fine-tuning Strategies",
"The transformer consists of three modules; embedding, encoder and decoder.",
"We update all three modules during meta-learning, but during fine-tuning, we can selectively tune only a subset of these modules.",
"Following [43], we consider three fine-tuning strategies; (1) fine-tuning all the modules (all), (2) fine-tuning the embedding and encoder, but freezing the parameters of the decoder (emb+enc) and (3) fine-tuning the embedding only (emb)."
],
[
"vs. Multilingual Transfer Learning",
"We meta-learn the initial models on all the source tasks using either Ro-En or Lv-En as a validation task.",
"We also train the initial models to be multilingual translation systems.",
"We fine-tune them using the four target tasks (Ro-En, Lv-En, Fi-En and Tr-En; 16k tokens each) and compare the proposed meta-learning strategy and the multilingual, transfer learning strategy.",
"As presented in Fig.",
"REF , the proposed learning approach significantly outperforms the multilingual, transfer learning strategy across all the target tasks regardless of which target task was used for early stopping.",
"We also notice that the emb+enc strategy is most effective for both meta-learning and transfer learning approaches.",
"With the proposed meta-learning and emb+enc fine-tuning, the final NMT systems trained using only a fraction of all available training examples achieve 2/3 (Ro-En) and 1/2 (Lv-En, Fi-En and Tr-En) of the BLEU score achieved by the models trained with full training sets."
],
[
"vs. Statistical Machine Translation",
"We also test the same Ro-En datasets with $16,000$ target tokens using the default setting of Phrase-based MT (Moses) with the dev set for adjusting the parameters and the test set for calculating the final performance.",
"We obtain $4.79 (\\pm 0.234)$ BLEU point, which is higher than the standard NMT performance (0 BLEU).",
"It is however still lower than both the multi-NMT and meta-NMT."
],
[
"Impact of Validation Tasks",
"Similarly to training any other neural network, meta-learning still requires early-stopping to avoid overfitting to a specific set of source tasks.",
"In doing so, we observe that the choice of a validation task has non-negligible impact on the final performance.",
"For instance, as shown in Fig.",
"REF , Fi-En benefits more when Ro-En is used for validation, while the opposite happens with Tr-En.",
"The relationship between the task similarity and the impact of a validation task must be investigated further in the future.",
"Table: Sample translations for Tr-En and Ko-En highlight the impact of fine-tuning which results in syntactically better formed translations.",
"We highlight tokens of interest in terms of reordering."
],
[
"Training Set Size",
"We vary the size of the target task's training set and compare the proposed meta-learning strategy and multilingual, transfer learning strategy.",
"We use the emb+enc fine-tuning on Ro-En and Fi-En.",
"Fig.",
"REF demonstrates that the meta-learning approach is more robust to the drop in the size of the target task's training set.",
"The gap between the meta-learning and transfer learning grows as the size shrinks, confirming the effectiveness of the proposed approach on extremely low-resource language pairs.",
"Figure: The learning curves of BLEU scores on the validation task (Ro-En)."
],
[
"Impact of Source Tasks",
"In Table REF , we present the results on all five target tasks obtained while varying the source task set.",
"We first see that it is always beneficial to use more source tasks.",
"Although the impact of adding more source tasks varies from one language to another, there is up to 2$\\times $ improvement going from one source task to 18 source tasks (Lv-En, Fi-En, Tr-En and Ko-En).",
"The same trend can be observed even without any fine-tuning (i.e., unsupervised translation, [27], [3]).",
"In addition, the choice of source languages has different implications for different target languages.",
"For instance, Ro-En benefits more from {Es, Fr, It, Pt} than from {De, Ru}, while the opposite effect is observed with all the other target tasks."
],
[
"Training Curves",
"The benefit of meta-learning over multilingual translation is clearly demonstrated when we look at the training curves in Fig.",
"REF .",
"With the multilingual, transfer learning approach, we observe that training rapidly saturates and eventually degrades, as the model overfits to the source tasks.",
"MetaNMT on the other hand continues to improve and never degrades, as the meta-objective ensures that the model is adequate for fine-tuning on target tasks rather than for solving the source tasks."
],
[
"Sample Translations",
"We present some sample translations from the tested models in Table REF .",
"Inspecting these examples provides the insight into the proposed meta-learning algorithm.",
"For instance, we observe that the meta-learned model without any fine-tuning produces a word-by-word translation in the first example (Tr-En), which is due to the successful use of the universal lexcial representation and the meta-learned initialization.",
"The system however cannot reorder tokens from Turkish to English, as it has not seen any training example of Tr-En.",
"After seeing around 600 sentence pairs (16K English tokens), the model rapidly learns to correctly reorder tokens to form a better translation.",
"A similar phenomenon is observed in the Ko-En example.",
"These cases could be found across different language pairs."
],
[
"Conclusion",
"In this paper, we proposed a meta-learning algorithm for low-resource neural machine translation that exploits the availability of high-resource languages pairs.",
"We based the proposed algorithm on the recently proposed model-agnostic meta-learning and adapted it to work with multiple languages that do not share a common vocabulary using the technique of universal lexcal representation, resulting in MetaNMT.",
"Our extensive evaluation, using 18 high-resource source tasks and 5 low-resource target tasks, has shown that the proposed MetaNMT significantly outperforms the existing approach of multilingual, transfer learning in low-resource neural machine translation across all the language pairs considered.",
"The proposed approach opens new opportunities for neural machine translation.",
"First, it is a principled framework for incorporating various extra sources of data, such as source- and target-side monolingual corpora.",
"Second, it is a generic framework that can easily accommodate existing and future neural machine translation systems."
],
[
"Acknowledgement",
"This research was supported in part by the Facebook Low Resource Neural Machine Translation Award.",
"This work was also partly supported by Samsung Advanced Institute of Technology (Next Generation Deep Learning: from pattern recognition to AI) and Samsung Electronics (Improving Deep Learning using Latent Structure).",
"KC thanks support by eBay, TenCent, NVIDIA and CIFAR."
]
] | 1808.08437 |
[
[
"Multiobjective Optimization Training of PLDA for Speaker Verification"
],
[
"Abstract Most current state-of-the-art text-independent speaker verification systems take probabilistic linear discriminant analysis (PLDA) as their backend classifiers.",
"The parameters of PLDA are often estimated by maximizing the objective function, which focuses on increasing the value of log-likelihood function, but ignoring the distinction between speakers.",
"In order to better distinguish speakers, we propose a multi-objective optimization training for PLDA.",
"Experiment results show that the proposed method has more than 10% relative performance improvement in both EER and MinDCF on the NIST SRE14 i-vector challenge dataset, and about 20% relative performance improvement in EER on the MCE18 dataset."
],
[
"Introduction",
"Although there are many kinds of frontends, such as GMM/DNN i-vector [1], [2], TDNN Xvector [3], and DNN embedding [4], [5], probabilistic linear discriminant analysis (PLDA) [6], [7] is still the most popular backend for a text-independent speaker verification system.",
"Many researchers aim at improving the performance of PLDA [8], [9], [10], [11].",
"Cumani proposes a new PLDA based on i-vector's posterior distribution, where an utterance is not mapped into a single i-vector, but into a posterior distribution to improve the performance for short utterances [8].",
"Burget refines the PLDA scoring by adopting discriminative models, e.g.",
"support vector machines or logistic regression [9].",
"Following his work, Rohdin introduces more constrains on PLDA parameters to boost system performance [10].",
"Inspired by the nonparametric discriminant analysis [12], Krosravani proposes a nonparametrically trained PLDA which achieves excellent performance on NIST SRE 2010 core c5 condition [11].",
"However, none of the above methods utilizes a discriminant way to train space matrix, which plays a crucial role in the PLDA modeling.",
"We adopt the advantages of discriminant and nonparametric methods, and propose a multi-objective optimization training for PLDA.",
"Experiment results on the NIST SRE14 [13] and MCE18 [14] demonstrate the effectiveness of proposed methods.",
"The remainder of this paper is organized as follows.",
"Section reviews the simplified Gaussian probabilistic linear discriminant analysis (sGPLDA).",
"Section proposes a multi-objective optimization training for sGPLDA.",
"Section analyzes and discusses the experiment results.",
"A conclusion is drawn in Section ."
],
[
"Simplified Gaussian probabilistic linear discriminant analysis",
"There are many variants of PLDA, the most widely used PLDA in the field of speaker verification is the simplified Gaussian PLDA (sGPLDA) [7] for its simplicity and proven performance on the recent NIST SREs.",
"The sGPLDA assumes that a length normalized i-vector $\\mathbf {x}$ is decomposed into three parts: a global mean vector $\\mathbf {\\mu }$ , a speaker space $F$ and factor $\\mathbf {h}_s$ , and a $\\varepsilon $ which consists of within-class variability and residual noise.",
"$ \\mathbf {x}_{si} = \\mathbf {\\mu } + F \\mathbf {h}_{s} + \\varepsilon _{si}$ where $s$ is the speaker index and $si$ is the segment index of speaker $s$ .",
"Under the sGPLDA assumption, the speaker factor $\\mathbf {h}_{s}$ has a standard normal prior, and $\\varepsilon _{si} \\sim \\mathcal {N}(0, \\Sigma _{w})$ .",
"The model parameters $\\lbrace F, \\Sigma _{w}\\rbrace $ and speaker factor $\\mathbf {h}_s$ are iteratively optimized by maximizing the log-likelihood function $f$ , $\\begin{aligned} \\mathop {\\arg \\max }_{F, \\Sigma _{w}, \\mathbf {h}} f = & \\frac{1}{\\sum _{s=1}^S sI} \\sum _{s=1}^S \\sum _{si=1}^{sI} \\\\& \\quad \\log \\mathcal {N}(\\mathbf {x}_{si}; \\mathbf {\\mu } + F\\mathbf {h}_{s}, \\Sigma _{w}) \\mathcal {N}(\\mathbf {h}_{s}; 0, I)\\end{aligned}$ via Expectation - Maximization (EM) algorithm [15], [7].",
"$\\mathcal {N}$ is a Gaussian distribution, $S$ is the total speaker number and $sI$ is the total segment number of speaker $s$ ."
],
[
"Motivation",
"Our motivation originates from the linear discriminant analysis (LDA) [16].",
"The LDA is to perform dimensionality reduction by analyzing within-class scatter matrix and between-class scatter matrix.",
"The within-class scatter matrix is $\\mathbf {S}_w = \\sum _{s=1}^S \\sum _{si=1}^{sI} (\\mathbf {x}_{si} - \\mathbf {\\mu }_{s}) (\\mathbf {x}_{si} - \\mathbf {\\mu }_{s})^t$ and the between-class scatter matrix is $\\mathbf {S}_b = \\sum _{s=1}^S (\\mathbf {\\mu }_{s} - \\mathbf {\\mu }) (\\mathbf {\\mu }_{s} - \\mathbf {\\mu })^t$ where $\\mathbf {\\mu }_{s} = \\frac{1}{sI} \\sum _{si=1}^{sI} \\mathbf {x}_{si}$ is the class mean, and $\\mathbf {\\mu } = \\frac{1}{S}\\sum _{s=1}^{S} \\mathbf {\\mu }_{s}$ is the global mean.",
"If we analogize sGPLDA and LDA, we will find that $\\mathbf {\\mu } + F \\mathbf {h}_{s}$ and $\\Sigma _{w}$ are equivalent to $\\mathbf {\\mu }_{s}$ and $\\mathbf {S}_w$ .",
"Finding a space $F$ which maximizes $f$ is similar to finding a space $V$ which maximizes $\\det {(V^t \\mathbf {S}_w V)^{-1}} $ .",
"By these comparison, we find that the objective function $f$ just focuses on within-class vectors, but ignores between-class vectors.",
"Here, the within-class vectors mean that the vectors are all from the same class and the between-class vectors mean that these vectors are not from the same class.",
"Take an extreme case for example, we only have one speaker's vectors for training.",
"We can compute $F$ by maximizing $f$ and $\\mathbf {S}_w$ because they only need within-class vectors, but failed to compute $\\mathbf {S}_b$ because it needs between-class vectors.",
"Clearly, effective use of between-class statistics can further enhance the discriminant ability of designed algorithm, e.g.",
"the space $V$ is obtained by maximizing $\\frac{\\det {(V^t\\mathbf {S}_b V)}}{\\det {(V^t\\mathbf {S}_w V)}}$ in LDA.",
"To achieve this goal, we try to integrate between-class statistics into the sGPLDA training."
],
[
"sGPLDA model for between-class vectors",
"For a speaker $s$ , $\\mathbf {x}_{si}$ denotes his/her $i$ -th i-vector.",
"Let $\\bar{\\mathbf {x}}_{sj}$ denote the $j$ -th i-vector that does not belong to speaker $s$ , $1 \\le j \\le sJ$ .",
"$\\lbrace \\mathbf {x}_{si}\\rbrace $ and $\\lbrace \\bar{\\mathbf {x}}_{sj}\\rbrace $ constitute between-class vectors of speaker $s$ , and we use $\\mathbf {y}$ to denote them for convenience.",
"Similar to (REF ) and (REF ), $\\mathbf {y}_{sk}$ is also decomposed into three parts: a global mean vector $\\mathbf {\\mu }$ , a speaker space $F$ and factor $\\mathbf {g}_{s}$ , and a $\\zeta $ which contains within-class variability of between-class vectors and residual noise.",
"$ \\mathbf {y}_{sk} = \\mathbf {\\mu } + F \\mathbf {g}_{s} + \\zeta _{sk}$ And the log-likelihood function $g$ is $\\begin{aligned} g = & \\frac{1}{\\sum _{s=1}^S sK} \\sum _{s=1}^S \\sum _{sk=1}^{sK} \\\\& \\quad \\log \\mathcal {N}(\\mathbf {y}_{sk}; \\mathbf {\\mu } + F\\mathbf {g}_{s}, \\Sigma _{b}) \\mathcal {N}(\\mathbf {g}_{s}; 0, I)\\end{aligned}$ Here, $sK = sI + sJ$ , the factor $\\mathbf {g}_{s}$ also has a standard normal prior, and $\\zeta _{sk} \\sim \\mathcal {N}(0, \\Sigma _{b})$ .",
"We name (REF ) and (REF ) as the sGPLDA model for within-class vectors and (REF ) and (REF ) as the sGPLDA model for between-class vectors."
],
[
"Multi-objective Optimization Training",
"The joint model parameters $\\lbrace F, \\Sigma _{w}, \\Sigma _{b}\\rbrace $ , $\\mathbf {h}_s$ , and $\\mathbf {g}_s$ are obtained by multi-objective optimization training, see Fig.",
"REF .",
"Our considerations are as follows: The sGPLDA model for within-class and between-class vectors share the same speaker space $F$ .",
"Intuitively, the desired $F$ is to maximize $f$ and to minimize $g$ at the same time.",
"Therefore, the objective function is $\\mathop {\\arg \\max }_{F} (\\alpha f - g)$ , where $\\alpha $ is an introduced factor which balances $f$ and $g$ , and will be examined in the experiment section.",
"$\\Sigma _{w}$ and $\\mathbf {h}_s$ are only related to $f$ , and the objective function is $\\mathop {\\arg \\max }_{\\Sigma _{w}, \\mathbf {h}_s} (f)$ .",
"$\\Sigma _{b}$ and $\\mathbf {g}_s$ are only related to $g$ , and the objective function is $\\mathop {\\arg \\max }_{\\Sigma _{b}, \\mathbf {g}_s} (g)$ .",
"The parameters are obtained by EM algorithm.",
"During the E-step, $\\mathbf {h}_s$ and $\\mathbf {g}_s$ are obtained by taking the derivation with $f$ and $g$ , respectively.",
"$\\begin{aligned}\\mathbf {h}_s & = [(sI) F^t \\Sigma _{w}^{-1} F + I]^{-1} F^t \\Sigma _{w}^{-1} \\sum _{si=1}^{sI} (\\mathbf {x}_{si} - \\mu ) \\\\\\mathbf {g}_s & = [(sK) F^t \\Sigma _{b}^{-1} F + I]^{-1} F^t \\Sigma _{b}^{-1} \\sum _{sk=1}^{sK} (\\mathbf {y}_{sk} - \\mu )\\end{aligned}$ During the M-step, $F$$, \\Sigma _{w}$ , and $\\Sigma _{b}$ are obtained by taking the derivation with $\\alpha f - g$ , $f$ , and $g$ , respectively.",
"$\\begin{aligned}F = & \\bigg (\\frac{\\alpha }{\\sum _{s=1}^{S} sI} \\sum _{s=1}^S \\sum _{si=1}^{sI} (\\mathbf {x}_{si} - \\mathbf {\\mu }) \\mathbf {h_s}^t \\\\& - \\frac{1}{\\sum _{s=1}^{S} sK} \\sum _{s=1}^S \\sum _{sk=1}^{sK} (\\mathbf {y}_{sk} - \\mathbf {\\mu }) \\mathbf {g_s}^t \\bigg ) \\\\& \\bigg ( \\frac{\\alpha }{\\sum _{s=1}^{S} sI}\\sum _{s=1}^S (sI) (\\mathbf {h}_s \\mathbf {h}_s^t) \\\\& - \\frac{1}{\\sum _{s=1}^{S} sK}\\sum _{s=1}^S (sK) (\\mathbf {g}_s \\mathbf {g}_s^t) \\bigg )^{-1}\\end{aligned}$ and $\\begin{aligned}\\Sigma _{w} &=\\frac{1}{\\sum _{s=1}^{S} sI} \\sum _{s=1}^S \\sum _{si=1}^{sI} (\\mathbf {x}_{si}-\\mathbf {\\mu }-F \\mathbf {h}_s) (\\mathbf {x}_{si}-\\mathbf {\\mu }-F \\mathbf {h}_s)^t \\\\\\Sigma _{b} &=\\frac{1}{\\sum _{s=1}^{S} sK} \\sum _{s=1}^S \\sum _{sk=1}^{sK} (\\mathbf {y}_{sk}-\\mathbf {\\mu }-F \\mathbf {g}_s) (\\mathbf {y}_{sk}-\\mathbf {\\mu }-F \\mathbf {g}_s)^t\\end{aligned}$ The E-step and M-step are iteratively performed.",
"Figure: A demo of multi-objective optimization training of sGPLDA"
],
[
"Selection of $\\bar{\\mathbf {x}}_{sj}$",
"As stated in [17], the most challenging task in text-independent speaker verification is to discriminate easily confusable speakers.",
"Krosravani also proposes a nonparametrically trained PLDA, in which the core idea is selecting nearest neighbor vectors during scoring [11].",
"Therefore, we adopt random and nearest neighbor selections to pick up $\\bar{\\mathbf {x}}_{sj}$ .",
"The former is used for comparison and we believe the latter is effective.",
"The nearest neighbor selection is that for a speaker $s$ , we calculate inner products between $\\mathbf {x}_s$ and $\\bar{\\mathbf {x}}_{sj}$ , sort them in a descending order and select the top $sI$ $\\bar{\\mathbf {x}}_{sj}$ ."
],
[
"Verification score",
"The scoring is also based on two-covariance model [7] and the log-likelihood ratio is $\\begin{aligned}score(\\mathbf {x}_1, \\mathbf {x}_2) = &\\log \\frac{p(\\mathbf {x}_1,\\mathbf {x}_2|\\text{same speaker})}{p(\\mathbf {x}_1,\\mathbf {x}_2|\\text{different speakers})} \\\\=& \\mathbf {x}_{1}^t Q \\mathbf {x}_{1} + \\mathbf {x}_{2}^t Q \\mathbf {x}_{2} +2 \\mathbf {x}_{1}^t P \\mathbf {x}_{2} + \\text{const}\\end{aligned}$ where $\\begin{aligned}Q &= \\Sigma _{tot,b}^{-1} -(\\Sigma _{tot,w} - \\Sigma _{ac}\\Sigma _{tot,w}^{-1}\\Sigma _{ac})^{-1} \\\\P &= \\Sigma _{tot,w}^{-1} \\Sigma _{ac} (\\Sigma _{tot,w} - \\Sigma _{ac}\\Sigma _{tot,w}^{-1}\\Sigma _{ac})^{-1} \\\\\\Sigma _{tot,w} &= FF^t + \\Sigma _{w}, \\Sigma _{tot,b} = FF^t + \\Sigma _{b}, \\text{and } \\Sigma _{ac} = FF^t\\end{aligned}$ Different from [7], we use $\\Sigma _{tot,b}$ instead of $\\Sigma _{tot,w}$ to compute $Q$ , because $\\Sigma _{tot,b}$ is a more reasonable choice under the different speakers assumption."
],
[
"NIST i-vector Machine Learning Challenge, SRE14",
"NIST i-vector machine learning challenge (SRE14) takes i-vectors instead of speech as input to examine the backend of speaker verification system [13].",
"It is gender independent, contains 1306 speaker models, 9634 test segments and 12582004 trials.",
"Each speaker model has 5 i-vectors.",
"The trials are randomly divided into a progress subset ($40\\%$ ) and an evaluation subset ($60\\%$ ).",
"In addition, NIST provided a development set, containing 36572 i-vectors.",
"All the i-vectors are 600-dimensional.",
"We study the backend learning algorithms with development labels known.",
"After applying LDA, traditional single objective function (SO) sGPLDA and multi-objective optimization training of sGPDLA (MO) are comparatively studied under the same condition.",
"The dimension of LDA, SO sGPLDA, and MO sGPLDA are 250, 150, and 150, respectively.",
"Unless otherwise specified, $\\bar{\\mathbf {x}}_{sj}$ is nearest selected.",
"Figure: The EER and MDCF14 vary with the α\\alpha Figure: The EER and MDCF14 vary with the dimensionFig.REF shows that the EER and MDCF14 of SO/MO sGPLDA vary with the $\\alpha $ .",
"It can be seen that as $\\alpha $ changes from $1.1$ to 2 with a step of $0.1$ , both EER and MDCF14 decrease first and then increase, which means a well balance between $f$ and $g$ is important for MO sGPLDA.",
"We choose $\\alpha = 1.7$ in the following experiments.",
"Fig.REF shows that the EER and MDCF14 of SO/MO sGPLDA vary with the sGPLDA dimension.",
"In most cases (150, 200, and 250), MO sGPLDA outperforms SO sGPLDA.",
"At a low dimension (50), the performance of MO sGPLDA is decreased.",
"Figure: The DET curves of sGPLDA on NIST SRE14From Table REF , we can see that nearest neighbor selection is better than random selection, which is in line with our expectation.",
"The nearest selected $\\bar{\\mathbf {x}}_{sj}$ are easily confusable i-vectors.",
"Taking them into considerations can boost system performance.",
"Compared with the SO sGPLDA, both EER and MDCF of proposed MO sGPLDA with nearest selection have more than $10\\%$ relative improvement on the progress set and the evaluation set of NIST SRE14, see Table REF and Fig.REF .",
"Table: Experiment results of sGPLDA on NIST SRE14."
],
[
"MCE18",
"The 1st Multi-target speaker detection and identification Challenge Evaluation [14] provides three i-vector sets: training, development and test sets.",
"Each set consists of blacklist and non-blacklist (background) speakers.",
"For the training set, there are 3,631 blacklist speakers and 5,000 background speakers.",
"Each blacklist speaker has 3 i-vectors, and there are 10,893 i-vectors for blacklist speakers in total.",
"For the development set, there also 3,631 blacklist speakers and 5,000 background speakers.",
"Each speaker has only one i-vector.",
"The blacklist speakers of the training and development sets are the same while the background speakers are not.",
"No information is provided about the distribution of speakers in the test set.",
"All the i-vectors are 600 dimension.",
"The MCE18 evaluation dataset includes the Fixed and Open conditions.",
"In the Fixed condition, we can only use data provided by the MCE18.",
"This limitation is removed in the Open condition.",
"We examined the Mot sGPLDA on the Fixed condition test.",
"Our procedure is classical, includes length normalization [15], LDA, PLDA and score normalization in turn.",
"We use both training and development sets to train these parameters.",
"The dimension of both LDA and PLDA is 350.",
"From the Table REF , we can see that, compared with the SO sGPLDA, the proposed MO sGPLDA has $19.8\\%$ and $22.0\\%$ relative improvement in the Top S and Top 1 conditions on the MCE18 evaluation dataset, which further proves our assert that the parameters trained by multi-objective optimization training not only have the ability to maximize the log-likelihood function on the within vector sets, but also have the ability to distinguish the vectors which are easily mis-judged.",
"Table: Experiment results of sGPLDA on MCE18."
],
[
"Conclusion",
"We propose a multi-objective optimization training for the sGPLDA.",
"It not only focuses on increasing the log-likelihood function, but also improves the distinction ability between easily mis-judged speakers.",
"Compared with the traditional method, the EER and MDCF of multi-objective optimized sGPLDA have $10.5\\%$ and $11.1\\%$ relative performance improvements on SRE14 progress set, and $16.2\\%$ and $12.1\\%$ relative performance improvements on SRE14 evaluation set, and the EER of multi-objective optimized sGPLDA have $19.8\\%$ and $22.0\\%$ relative performance improvements in the Top S and Top 1 conditions on the MCE18 evaluation set.",
"This method can also be extended to other types of PLDA with proper modification.",
"The python and matlab code for this paper can be downloaded from Github: git clone https://github.com/sanphiee/MOT-sGPLDA-SRE14 and git clone https://github.com/sanphiee/MOT-sGPLDA-MCE18."
]
] | 1808.08344 |
[
[
"Direct Observation of Cross-Polarized Excitons in Aligned\n Single-Chirality Single-Wall Carbon Nanotubes"
],
[
"Abstract Optical properties of single-wall carbon nanotubes (SWCNTs) for light polarized parallel to the nanotube axis have been extensively studied, whereas their response to light polarized perpendicular to the nanotube axis has not been well explored.",
"Here, by using a macroscopic film of highly aligned single-chirality (6,5) SWCNTs, we performed a systematic polarization-dependent optical absorption spectroscopy study.",
"In addition to the commonly observed angular-momentum-conserving interband absorption of parallel-polarized light, which generates $E_{11}$ and $E_{22}$ excitons, we observed a small but unambiguous absorption peak whose intensity is maximum for perpendicular-polarized light.",
"We attribute this feature to the lowest-energy cross-polarized interband absorption processes that change the angular momentum along the nanotube axis by $\\pm 1$, generating $E_{12}$ and $E_{21}$ excitons.",
"The energy difference between the $E_{12}$ and $E_{21}$ exciton peaks, expected from asymmetry between the conduction and valence bands, was smaller than the observed linewidth.",
"Unlike previous observations of cross-polarized excitons in polarization-dependent photoluminescence and circular dichroism spectroscopy experiments, our direct observation using absorption spectroscopy allowed us to quantitatively analyze this resonance.",
"Specifically, we determined the energy and oscillator strength of this resonance to be 1.54 and 0.05, respectively, compared with the values for the $E_{11}$ exciton peak.",
"These values, in combination with comparison with theoretical calculations, in turn led to an assessment of the environmental effect on the strength of Coulomb interactions in this aligned single-chirality SWCNT film."
],
[
"Introduction",
"Semiconducting single-wall carbon nanotubes (SWCNTs) possess rich optical properties arising from one-dimensional excitons with extremely large binding energies [1], [2], [3], [4], [5], [6], [7], [8], [9].",
"Although much has been understood about the properties of excitons that are active for parallel-polarized light, excitons excited by perpendicular-polarized light have not been explored experimentally.",
"Such cross-polarized excitons are predicted to exhibit strong many-body effects due to a subtle interplay of quantum confinement and Coulomb interactions [10], [11], [12], [13].",
"Figure REF schematically shows the lowest-energy allowed interband optical transitions in a semiconducting SWCNT [14].",
"For absorption of light polarized parallel to the nanotube axis, the band index is preserved in an allowed optical transition (the $E_{11}$ and $E_{22}$ transitions).",
"For light polarized perpendicular to the nanotube axis, a transition can occur when the subband index changes by 1 (the $E_{12}$ and $E_{21}$ transitions).",
"As first pointed out by Ajiki and Ando [10], the $E_{12}$ and $E_{21}$ absorption peaks are expected to be suppressed because of the depolarization effect.",
"However, subsequent theoretical studies [11], [12], [13] taking into account the electron-hole Coulomb interactions indicated that a small absorption peak due to cross-polarized excitons should still appear.",
"The $E_{12}$ /$E_{21}$ transitions were first observed in polarized photoluminescence excitation spectroscopy studies on aqueous suspensions of SWCNTs [15], [16].",
"By crossing the polarization of the excitation beam with respect to that of the collection beam, $E_{11}$ photoluminescence due to resonant absorption at the $E_{12}$ /$E_{21}$ transition was observed.",
"More recently, in circular dichroism (CD) studies [17], [18], [19], chirality-sorted nanotubes were further separated into enantiomers based on their “handedness,” i.e., (6,5) and (5,6) SWCNTs.",
"CD spectra for enantiomer-sorted nanotubes showed peaks due to $E_{12}$ and $E_{21}$ excitons.",
"However, such cross-polarized exciton transitions have never been directly identified in optical absorption spectra.",
"Therefore, quantitative characterization of $E_{12}$ /$E_{21}$ excitons has remained elusive.",
"Here, we report the direct observation of cross-polarized excitons by absorption spectroscopy.",
"Specifically, we investigated the polarization dependence of optical absorption in a macroscopic film of aligned, single-chirality (6,5) SWCNTs.",
"As the angle between the polarization of the incident beam and the nanotube alignment direction was increased from 0$^\\circ $ to 90$^\\circ $ , a peak due to the $E_{12}$ /$E_{21}$ excitons appeared and grew in intensity at the expense of the usual parallel-polarized excitons ($E_{11}$ and $E_{22}$ ).",
"The energy of the $E_{12}$ /$E_{21}$ exciton peak was 660 meV higher than the $E_{11}$ exciton peak and 250 meV lower than the $E_{22}$ exciton peak.",
"Together with the nematic order parameter of the aligned SWCNT film determined in the same analysis, these polarization-dependent absorption measurements allowed us to determine the oscillator strength of the $E_{12}$ /$E_{21}$ peak quantitatively.",
"We first prepared an aqueous suspension of extremely pure (6,5) SWCNTs based on pH-controlled gel chromatography [20], [21].",
"SWCNTs purchased from Sigma-Aldrich (Signis SG65i) were suspended in an aqueous solution of sodium cholate (SC).",
"After ultracentrifugation, the supernatant was collected as an initial suspension.",
"Sodium dodecyl sulfate (SDS) was added to the suspension, which was used for a two-stage gel chromatography process.",
"In the first-stage of gel chromatography to separate the semiconducting SWCNTs by a difference in chiral angle, the suspension was loaded onto gel beads (GE Healthcare, Sephacryl S-200 HR) under surfactant environment of 2.0% SDS and 0.5% SC, and the nonadsorbed fraction containing (6,5) nanotubes was collected as a filtrate.",
"This filtrate was used for the second-stage process to separate the semiconducting SWCNTs by a difference in diameter and to remove residual metallic SWCNTs.",
"Before separation, the surfactant concentrations of the filtrate were adjusted to 0.5% SDS and 0.5% SC.",
"The pH of the solution strongly influences the adsorption of residual metallic SWCNTs [21], and thus, we used pH-adjusted surfactant solutions.",
"The pH-adjusted solutions were loaded on gel beads, and the adsorbed (6,5) SWCNTs were eluted with a stepwise increase of the concentration of sodium deoxycholate (DOC).",
"Figure REF shows an absorbance spectrum for a purified (6,5) suspension in a cuvette with a 10-mm path length.",
"The assigned peaks are $E_{11}$ (1.26 eV), $E_{11}$ phonon sideband (1.46 eV), $E_{22}$ (2.17 eV), $E_{22}$ phonon sideband (2.38 eV), and $E_{33}$ (3.58 eV).",
"Small unresolved peaks due to residual metallic nanotubes exist in the range of 2.6–3.1 eV.",
"We estimate the (6,5) chirality purity of the sample to be 99.3% from this spectrum.",
"See Appendix for more details about the method we used for chirality purity determination.",
"The obtained suspension after surfactant exchange was poured into a 1-inch vacuum filtration system with a 80-nm-pore filter membrane to obtain a wafer-scale film of aligned SWCNTs [22].",
"The prepared suspension contained several surfactants, including SC, sodium dodecylbenzenesulfonate (SDBS), and DOC.",
"In order to have a thick film of highly-aligned (6,5) SWCNTs, we needed to have a mono-surfactant suspension.",
"Therefore, we used ultrafiltration to exchange the mixed surfactants to 0.04% (wt./vol.)",
"DOC.",
"The surfactant concentration was also adjusted to below the critical micelle concentration of DOC through ultrafiltration, which is a necessary condition for the controlled vacuum filtration technique we used to prepare an aligned film [22].",
"The average length of SWCNTs in the prepared suspension before vacuum filtration was $\\sim $ 200 nm.",
"The suspension was poured into a funnel with a polycarbonate filter membrane (Nuclepore track-etched polycarbonate hydrophilic membrane).",
"The pressure underneath the membrane was lowered by a mechanical vacuum pump connected to the side arm of a side-arm flask.",
"The filtration speed was adjusted to a rate of 1–2.5 mL/hour by controlling the valves in the vacuum line.",
"Near the end of the filtration process, the filtration speed was accelerated to $\\sim $ 10 mL/hour.",
"In this procedure, the filtration speed was also important to achieve spontaneous alignment [22].",
"The obtained circular film had a diameter of $\\sim $ 20 mm.",
"The thickness of the film gradually varied from the center ($\\sim $ 10 nm) to the circumference ($\\sim $ 1 nm).",
"This film was cut into 4 quadrants.",
"One of them was transferred onto a 1-mm-thick glass substrate by dissolving the filter membrane in chloroform."
],
[
"Polarization-dependent visible–near-infrared absorption spectroscopy",
"We performed optical transmission measurements on the prepared SWCNT film using linearly polarized light.",
"Our experimental setup consisted of a tungsten-halogen lamp (Thorlabs, SLS201L), a Glan-Thompson polarizer, and two spectrometers.",
"One of the spectrometers covered a spectral range of 520–1050 nm, utilizing a monochromator (Horiba/JY, Triax320) equipped with a liquid-nitrogen-cooled CCD camera (Princeton Instruments, Spec-10).",
"The other spectrometer, which covered a spectral range of 1050–1550 nm, consisted of a monochromator (Princeton Instruments, SP-2150) and a liquid-nitrogen-cooled 1D InGaAs detector array (Princeton Instruments, OMA V InGaAs System).",
"Polarization dependence was achieved through changing the polarization angle of the incident light beam by rotating the polarizer.",
"The light beam was focused down to 30 $\\mu $ m in diameter by a 50$\\times $ objective lens (Mitutoyo, M Plan NIR 50).",
"A schematic diagram of the experimental geometry is shown in Fig.",
"REF .",
"The incident beam was polarized along the horizontal direction.",
"The angle between the nanotube alignment direction and the light polarization direction is denoted by $\\beta $ throughout this manuscript.",
"Polarization-dependent transmittance ($T$ ) spectra were taken with a step size of 5 degrees.",
"The measured spot was $\\sim $ 1 mm away from the center of the film, and the film thickness was $\\sim $ 10 nm at that spot.",
"We calculated attenuation spectra through $A = -\\ln {(T)}$ .",
"Figure: Illustration of the geometry of the polarization-dependent transmission experiments performed on an aligned SWCNT film.",
"The incident beam is linearly polarized along the horizontal axis, and the nanotube alignment direction is rotated from the horizontal axis by angle β\\beta .Figure: (a) Polarization-dependent attenuation spectra for the aligned (6,5) SWCNT film for polarization angles (β\\beta ) of 0 ∘ 0^{\\circ }, 30 ∘ 30^{\\circ }, 45 ∘ 45^{\\circ }, 60 ∘ 60^{\\circ }, and 90 ∘ 90^{\\circ } with respect to the nanotube alignment direction.",
"(b) Comparison of attenuation spectra for 0 ∘ 0^{\\circ } (A ∥ A_\\parallel , black line) and 90 ∘ 90^{\\circ } (A ⊥ A_\\perp , red line).",
"A ⊥ A_\\perp is multiplied by 3.2.",
"They match except in the spectral region of E 12 /E 21 E_{12}/E_{21}.",
"(c) Comparison of the 0 ∘ 0^{\\circ } (A ∥ A_\\parallel ) and 90 ∘ 90^{\\circ } (A ⊥ A_\\perp ) spectra.",
"The blue line indicates 3.2A ⊥ -A ∥ 3.2A_\\perp - A_\\parallel .",
"(d) A normalized spectral difference (3.2A ⊥ -A ∥ 3.2A_\\perp -A_\\parallel )/A ∥ A_\\parallel , which shows a prominent peak due to the E 12 /E 21 E_{12}/E_{21} exciton.Figure: Detailed polarization dependence of the individual spectral components deduced from the fits: (a) E 11 E_{11} and E 22 E_{22}, (b) E 11 E_{11} phonon sideband, (c) E 12 /E 21 E_{12}/E_{21}, and, (d) polynomial baseline for polarization angles of 0, 30, 45, 60, 90 degrees.Figure: Integrated peak intensity as a function of polarization angle β\\beta extracted for (a) E 11 E_{11}, (b) E 22 E_{22}, (c) E 11 E_{11} phonon sideband, and (d) E 12 /E 21 E_{12}/E_{21}."
],
[
"EXPERIMENTAL RESULTS",
"Figure REF (a) displays representative attenuation spectra for polarization angles $\\beta $ = 0$^{\\circ }$ , 30$^{\\circ }$ , 45$^{\\circ }$ , 60$^{\\circ }$ , and 90$^{\\circ }$ .",
"The spectra are not intentionally offset.",
"The observed peaks at 1.22 eV and 2.13 eV are due to the $E_{11}$ and $E_{22}$ exciton transitions, respectively.",
"These peaks are red-shifted compared with the suspension spectrum in Fig.",
"REF by $\\sim $ 40 meV .",
"The peak at 1.44 eV is the phonon sideband of the $E_{11}$ exciton peak.",
"No other peaks are observed due to any residual semiconducting chiralities within this energy range.",
"As the polarization angle $\\beta $ increases from 0$^{\\circ }$ (parallel) to 90$^{\\circ }$ (perpendicular), these absorption peaks decrease in intensity.",
"The spectrum for perpendicular polarization ($\\beta $ = 90$^{\\circ }$ ) shows a new peak around 1.9 eV, which we assign to the $E_{12}/E_{21}$ transition.",
"As stated above, this transition is expected for light polarized perpendicular to the nanotube axis (Fig.",
"REF ).",
"A closer look at the polarization-dependent spectra allowed us to identify this peak in all spectra for polarization angles equal to or larger than 60$^{\\circ }$ .",
"Furthermore, it should be noted that this peak exists even in the suspension spectrum shown in Fig.",
"REF , although peak assignment was impossible since the nanotubes in the suspension are randomly oriented.",
"Figures REF (b)–(d) compare the 0$^{\\circ }$ ($A_\\parallel $ ) and 90$^{\\circ }$ ($A_\\perp $ ) spectra in more detail.",
"In these figures, a polynomial baseline was subtracted from each spectrum; see Sec.",
"IV for more details about this procedure.",
"In Fig.",
"REF (b), the red and black curves represent $A_\\parallel $ and $A_\\perp $ , respectively, where the $A_\\perp $ spectrum is multiplied by 3.2 so that the $E_{11}$ peak coincides in intensity between the two spectra.",
"As a result, the two spectra deviate from each other only in the spectral region of the $E_{12}/E_{21}$ peak.",
"In Fig.",
"REF (c), $A_\\perp $ multiplied by 3.2 is plotted in the upper ($y > 0$ ) plane, whereas $A_\\parallel $ is plotted in the lower ($y < 0$ ) plane.",
"The vertical dashed lines indicate the positions of the $E_{11}$ peak, the $E_{11}$ phonon sideband peak, the $E_{12}/E_{21}$ peak, and the $E_{22}$ peak, respectively.",
"The blue curve is $3.2A_\\perp -A_\\parallel $ , which is essentially zero everywhere except for the $E_{12}/E_{21}$ feature since the $E_{12}/E_{21}$ feature only appears in $A_\\perp $ .",
"Finally, Fig.",
"REF (d) shows a spectral difference ($3.2A_\\perp -A_\\parallel $ ) normalized by $A_\\parallel $ .",
"In this spectrum, the effects of the $E_{11}$ peak, the $E_{11}$ phonon sideband peak, and the $E_{22}$ peak are nearly eliminated, leaving a pronounced single peak due to the $E_{12}/E_{21}$ exciton."
],
[
"SPECTRAL ANALYSIS",
"To extract quantitative information from the obtained polarization-dependent spectra, we performed spectral analysis.",
"We fit each spectrum with a function consisting of Lorentzians representing the absorption peaks and a polynomial function representing the baseline: $A\\equiv -\\ln (T)=\\sum ^{3\\,\\text{or}\\,4}_{n=1}a_n\\frac{(b_n/2)^2}{(E_\\text{ph}-c_n)^2+(b_n/2)^2}+\\sum _{m=0}^{4}d_mE_\\text{ph}^m,$ where $E_\\text{ph}$ is the photon energy, acting as the independent variable, and $a_n$ , $b_n$ , $c_n$ , and, $d_m$ are the fitting parameters.",
"$a_n$ , $b_n$ , and $c_n$ are the peak amplitude, full width at half maximum, and peak position, respectively, of the $n$ -th peak, while $d_m$ is the $m$ -th polynomial coefficient.",
"We considered polynomials of order up to $m = 4$ .",
"We performed fitting on all spectra with polarization angles from $ -5^{\\circ }$ to 90$^{\\circ }$ with a step size of $5^{\\circ }$ .",
"The spectra from $ -5^{\\circ }$ to 30$^{\\circ }$ were fit with a polynomial function and three Lorentzians, to take account of the $E_{11}$ peak, the $E_{11}$ phonon sideband peak, and the $E_{22}$ peak.",
"The spectra from 35$^{\\circ }$ to 90$^{\\circ }$ were fit with four Lorentzians to take into account the $E_{12}/E_{21}$ peak as well.",
"Figure REF shows fitting results for the spectra for $\\beta =$ 0$^{\\circ }$ , 45$^{\\circ }$ , 60$^{\\circ }$ , and 90$^{\\circ }$ .",
"The solid black lines are experimental data.",
"The dashed red lines indicate the overall fit functions.",
"The blue curves indicate the individual components of the fit function.",
"Note that the spectrum for 0$^{\\circ }$ shown in Fig.",
"REF (a) does not contain the $E_{12}/E_{21}$ peak.",
"Figures REF (a)-REF (d) plot the extracted polarization-dependent spectra for the $E_{11}$ peak, the $E_{11}$ phonon sideband peak, the $E_{12}/E_{21}$ peak, the $E_{22}$ peak, and the polynomial baseline, respectively.",
"The shape of the baseline slightly changes with the polarization angle.",
"As the angle increases, the overall intensities of the baseline, the $E_{11}$ peak, the $E_{11}$ phonon sideband, and the $E_{22}$ peak decrease, while the $E_{12}/E_{21}$ peak grows in intensity.",
"The peak widths of the $E_{11}$ and $E_{22}$ peaks are $\\sim $ 120 meV and $\\sim $ 190 meV, respectively.",
"Finally, Figs.",
"REF (a)-REF (d) plot the integrated peak intensities of the $E_{11}$ peak, the $E_{11}$ phonon sideband, the $E_{22}$ peak, the $E_{12}/E_{21}$ peak, respectively, as a function of polarization angle $\\beta $ .",
"While the integrated intensities of the $E_{11}$ peak, the $E_{11}$ phonon sideband, and the $E_{22}$ peak decrease as the polarization angle $\\beta $ increases, the integrated intensity of the $E_{12}/E_{21}$ peak increases."
],
[
"Nematic order parameter",
"Since the average length of SWCNTs ($\\sim $ 200 nm) is much larger than the film thickness ($<$ 10 nm) in our sample, we use the two-dimensional (2D) theory of the optical absorption by an ensemble of anisotropic molecules, described in Appendix Section 2, to discuss our experimental data.",
"We assume that the nanotubes' angular distribution $f(\\theta )$ can be represented by the following Gaussian function with $\\theta =0$ as the alignment direction: $f(\\theta )=\\frac{1}{\\text{erf}\\left( \\pi /\\sqrt{2}\\sigma \\right) \\sqrt{2\\pi \\sigma ^2}}\\left( e^{-\\frac{\\theta ^2}{2\\sigma ^{2}}}+e^{-\\frac{(\\theta -\\pi )^2}{2\\sigma ^{2}}}\\right),$ where $\\theta $ is the angle between the macroscopic alignment direction and an individual nanotube and $\\sigma $ is the standard deviation.",
"Note that the nanotubes are distributed in an angular range of $0 \\le \\theta \\le \\pi $ , and $f(\\theta )$ is normalized in this range, i.e., $\\int _{0}^{\\pi }f(\\theta )d\\theta =1$ .",
"Figure REF shows three examples of $f(\\theta )$ for the cases of $\\sigma $ = 25$^\\circ $ , 32$^\\circ $ , and $\\infty $ .",
"When $\\sigma =25^\\circ $ (shown as a black dashed line), clear alignment along $\\theta =0$ is observed.",
"As $\\sigma $ increases, the distribution function $f\\left( \\theta \\right) $ becomes flatter.",
"Finally, when $\\sigma \\rightarrow \\infty $ , $f\\left( \\theta \\right) \\rightarrow 1/\\pi $ as indicated by the black solid line.",
"With the distribution function $f\\left( \\theta \\right)$ given by Eq.",
"(REF ), the 2D order parameter $S$ , defined by Eq.",
"(REF ), can be calculated as Figure: Simulated nanotubes' angular distribution f(θ)f(\\theta ), based on Eq. ().",
"The three traces correspond to σ=25 ∘ \\sigma = 25^\\circ , σ=32 ∘ \\sigma = 32^\\circ , and σ→∞\\sigma \\rightarrow \\infty , respectively.$S & =\\int _{0}^{\\pi }f(\\theta )\\left( 2\\cos ^2\\theta -1\\right) d\\theta \\nonumber \\\\& =\\frac{e^{-2\\sigma ^2}}{2\\text{erf}(\\pi /\\sqrt{2}\\sigma )} \\left[ \\text{erf}\\left( \\frac{\\pi }{\\sqrt{2}\\sigma }-i\\sqrt{2}\\sigma \\right) \\right.\\nonumber \\\\& \\left.",
"+ \\text{erf}\\left( \\frac{\\pi }{\\sqrt{2}\\sigma }+i\\sqrt{2}\\sigma \\right) \\right] .$ Since $S$ and $\\sigma $ have one-to-one correspondence, we can plot $S$ as a function of $\\sigma $ , as shown in Fig.",
"REF (a).",
"When $\\sigma \\rightarrow 0$ , $S \\rightarrow 1$ , as expected.",
"As $\\sigma $ increases, $S$ monotonically decreases, and finally, when $\\sigma \\rightarrow \\infty $ , $S \\rightarrow 0$ .",
"When the polarization angle is $\\beta $ with respect to the nanotube alignment direction (see Fig.",
"REF ), the absorption coefficient for incident light with photon energy $E_\\text{ph}$ is given by $\\alpha _\\text{abs}(\\beta )& = \\frac{NE_\\text{ph}}{\\hbar c n_0}\\left( \\alpha _1^{\\prime \\prime }\\frac{\\int _{0}^{\\pi }f(\\theta )\\cos ^2(\\theta -\\beta )d\\theta }{\\int _{0}^{\\pi }f(\\theta )d\\theta } \\right.",
"\\nonumber \\\\&+ \\left.",
"\\alpha _2^{\\prime \\prime }\\frac{\\int _{0}^{\\pi }f(\\theta )\\sin ^2(\\theta -\\beta )d\\theta }{\\int _{0}^{\\pi }f(\\theta )d\\theta }\\right) \\nonumber \\\\& = \\frac{NE_\\text{ph}}{\\hbar c n_0}\\left( \\alpha _1^{\\prime \\prime }\\int _{0}^{\\pi }f(\\theta -\\beta )\\cos ^2(\\theta -\\beta ) d\\theta \\right.",
"\\nonumber \\\\&\\left.",
"+\\alpha _2^{\\prime \\prime }\\int _{0}^{\\pi }f(\\theta -\\beta )\\sin ^2\\left(\\theta -\\beta \\right) d\\theta \\right),$ where $N$ is the total number of SWCNTs, $\\hbar $ is the reduced Planck constant, $c$ is the speed of light, $n_0$ is the refractive index, and $\\alpha _1^{\\prime \\prime }$ ($\\alpha _2^{\\prime \\prime }$ ) is the imaginary part of the molecular polarizability, $\\alpha $ , of an individual SWCNT parallel (perpendicular) to the tube axis.",
"See Appendix for more details.",
"We assume that the polarizability of an $E_{ii}$ transition is parallel to the nanotube axis ($\\xi _\\text{2D} = 0^\\circ $ ) whereas that of an $E_{ij}$ ($i\\ne j$ ) transition is perpendicular to the nanotube axis ($\\xi _\\text{2D} = 90^\\circ $ ), where $\\xi _\\text{2D} = \\tan ^{-1}\\left( \\sqrt{\\alpha _1^{^{\\prime \\prime }}/\\alpha _2^{^{\\prime \\prime }}} \\right)$ (see Appendix Section 2).",
"Namely, to consider the $E_{11}$ transition, we assume $\\xi _\\text{2D}=0^\\circ $ , i.e., $\\alpha _1^{\\prime \\prime } \\ne 0$ and $\\alpha _2^{\\prime \\prime } = 0$ .",
"With the distribution $f(\\theta )$ given by Eq.",
"(REF ), the absorption coefficient for the $E_{11}$ transition becomes $\\alpha _{\\text{abs},E_{11}}(\\beta )& = \\frac{NE_{11}}{\\hbar c n_0}\\alpha _1^{\\prime \\prime }\\int _{0}^{\\pi }f(\\theta -\\beta )\\cos ^2(\\theta -\\beta ) d\\theta \\nonumber \\\\& = \\frac{NE_{11}}{\\hbar c n_0}\\alpha _1^{\\prime \\prime }\\left[ \\frac{1}{2} +\\right.\\nonumber \\\\& \\frac{e^{-2\\sigma ^2}}{4\\text{erf}(\\pi /\\sqrt{2}\\sigma )}\\left\\lbrace \\text{erf}\\left( \\frac{\\pi }{\\sqrt{2}\\sigma }-i\\sqrt{2}\\sigma \\right) \\right.",
"\\nonumber \\\\& \\left.",
"\\left.",
"+\\text{erf}\\left( \\frac{\\pi }{\\sqrt{2}\\sigma }+i\\sqrt{2}\\sigma \\right) \\right\\rbrace \\cos 2\\beta \\right].$ Similarly, by assuming that $\\xi _\\text{2D}=90^\\circ $ , we obtain the absorption coefficient for the $E_{12}/E_{21}$ transition as $\\alpha _{\\text{abs},E_{12}}(\\beta )& = \\frac{NE_{12}}{\\hbar c n_0}\\alpha _2^{\\prime \\prime }\\int _{0}^{\\pi }f(\\theta -\\beta )\\sin ^2(\\theta -\\beta ) d\\theta .",
"\\nonumber \\\\& = \\frac{NE_{12}}{\\hbar c n_0}\\alpha _2^{\\prime \\prime }\\left[ \\frac{1}{2} + \\right.\\nonumber \\\\& \\frac{e^{-2\\sigma ^2}}{4\\text{erf}(\\pi /\\sqrt{2}\\sigma )}\\left\\lbrace \\text{erf}\\left( \\frac{\\pi }{\\sqrt{2}\\sigma }-i\\sqrt{2}\\sigma \\right) \\right.",
"\\nonumber \\\\& \\left.",
"\\left.",
"-\\text{erf}\\left( \\frac{\\pi }{\\sqrt{2}\\sigma }+i\\sqrt{2}\\sigma \\right) \\right\\rbrace \\cos 2\\beta \\right].$ Therefore, when the light polarization is parallel to the macroscopic alignment direction of the film, the absorption coefficient of the $E_{11}$ transition is given by $\\alpha _{\\text{abs},E_{11}}(0^\\circ ) & = \\frac{NE_{11}}{\\hbar c n_0}\\alpha _1^{\\prime \\prime }\\int _{0}^{\\pi }f(\\theta )\\cos ^2(\\theta ) d\\theta \\nonumber \\\\& = \\frac{NE_{11}}{\\hbar c n_0}\\frac{1+S}{2}\\alpha _1^{\\prime \\prime }.$ On the other hand, when the light polarization is perpendicular to the alignment direction, the absorption coefficient of the $E_{11}$ transition is given by $\\alpha _{\\text{abs},E_{11}}(90^\\circ ) & = \\frac{NE_{11}}{\\hbar c n_0}\\alpha _1^{\\prime \\prime }\\int _{0}^{\\pi }f\\left(\\theta -\\frac{\\pi }{2}\\right)\\cos ^2\\left(\\theta -\\frac{\\pi }{2} \\right) d\\theta \\nonumber \\\\& = \\frac{NE_{11}}{\\hbar c n_0}\\frac{1-S}{2}\\alpha _1^{\\prime \\prime }.$ Hence, the absorption coefficient ratio between parallel and perpendicular polarization is given by $\\frac{\\alpha _{\\text{abs},E_{11}}(0^\\circ )}{\\alpha _{\\text{abs},E_{11}}(90^\\circ )} =\\frac{1+S}{1-S}.$ By reversing Eq.",
"(REF ), we can express $S$ in terms of the absorption coefficient ratio as $S =\\frac{\\alpha _{\\text{abs},E_{11}}(0^\\circ )/\\alpha _{\\text{abs},E_{11}}(90^\\circ ) -1}{\\alpha _{\\text{abs},E_{11}}(0^\\circ )/\\alpha _{\\text{abs},E_{11}}(90^\\circ )+1}.$ In Fig.",
"REF (b), $S$ is plotted as a function of $\\alpha _{\\text{abs},E_{11}}(0^\\circ )/\\alpha _{\\text{abs},E_{11}}(90^\\circ )$ .",
"When $\\alpha _{\\text{abs},E_{11}}(0^\\circ )/\\alpha _{\\text{abs},E_{11}}(90^\\circ )=1$ , there is no anisotropy, meaning that $S = 0$ .",
"As the absorption ratio increases, $S$ increases.",
"As $\\alpha _{\\text{abs},E_{11}}(0^\\circ )/\\alpha _{\\text{abs},E_{11}}(90^\\circ ) \\rightarrow \\infty $ , $S$ asymptotically approaches 1.",
"Figure: (a) Nematic order parameter SS as a function of standard deviation angle σ\\sigma based on Eq. ().",
"(b) Nematic order parameter SS as a function of absorption ratio between parallel and perpendicular polarization based on Eq.",
"().Figure: Polarization angle dependence of the integrated intensity of the E 11 E_{11} peak.",
"Black dashed curve: theoretical calculation assuming S=0.52S = 0.52.",
"Red open circles: experimental data.",
"The experimental observation is well reproduced by the theoretical curve.",
"Blue dash-dotted curve: theoretical calculation assuming perfect alignment, i.e., S=1S = 1."
],
[
"Angular dependence of $E_{11}$ and {{formula:d9eb6962-8ddb-4095-890c-3874d9ded855}} absorption intensities",
"When the reflection loss can be neglected, the quantity we measured experimentally, i.e., the attenuation $A = -\\ln {(T)}$ is directly proportional to the absorption coefficient.",
"Namely, $A = \\alpha _\\text{abs}l$ , where $l$ is the film thickness.",
"Therefore, the experimentally determined $E_{11}$ integrated peak intensity ratio ($A_\\parallel / A_\\perp $ ) can be assumed to be equal to $\\alpha _{e,E_{11}}(0^\\circ )/\\alpha _{e,E_{11}}(90^\\circ )$ .",
"From Fig.",
"REF (a), $A_\\parallel / A_\\perp $ is determined to be 3.2, which, according to the plot in Fig.",
"REF (b), corresponds to $S =$ 0.52.",
"Accordingly, from Fig.",
"REF (a) and Eq.",
"(REF ), $\\sigma $ is determined to be 32$^\\circ $ .",
"Figure REF plots the angular distribution of nanotubes for this case as a red solid curve.",
"Furthermore, we calculated the integrated intensity of the $E_{11}$ peak in absorption coefficient as a function of polarization angle $\\beta $ for $S = 0.52$ , as shown in Fig.",
"REF as a black dashed line together with the experimental data (red open circles).",
"The calculated values are normalized by the experimental value for 0$^\\circ $ .",
"The observed angular dependence is accurately reproduced by the theoretical curve, confirming the overall correctness of our theoretical analysis.",
"Finally, the blue dash dotted line in Fig.",
"REF represents the angular dependence of the $E_{11}$ integrated absorption intensity calculated assuming perfect alignment, i.e., $S = 1$ ."
],
[
"Energy and oscillator strength of the $E_{12}/E_{21}$ transition",
"Figure REF shows a parallel-polarization spectrum ($\\beta = 0^\\circ $ ) exhibiting the $E_{11}$ and $E_{22}$ peaks, together with a perpendicular-polarization spectrum ($\\beta = 90^\\circ $ ) exhibiting the $E_{12}/E_{21}$ peak, which were extracted from the raw experimental data through the spectral analysis described in Section IV.",
"The perpendicular-polarization spectrum was multiplied by 10.",
"The energy position of the $E_{12}/E_{21}$ peak is 1.88 eV, which is 1.54 times that of the $E_{11}$ peak (1.22 eV) and 0.88 times that of the $E_{22}$ peak (2.13 eV).",
"Previously, the $E_{12}/E_{21}$ transition was observed through cross-polarized photoluminescence excitation experiments [16] and circular dichroism measurements [18], [19].",
"The reported energies range from 1.88 to 1.93 eV.",
"These fluctuations can be attributed to the different dielectric constants of the surrounding of the nanotubes studied under different conditions [11], [23].",
"Uryu and Ando calculated the energies of the $E_{11}$ , $E_{12}/E_{21}$ , and $E_{22}$ peaks for SWCNTs as a function of dielectric constant $\\kappa $ and diameter [11].",
"While we found no single value of $\\kappa $ that simultaneously makes the three calculated energies match the experimental values, we found reasonable overall agreement when $1.8 < \\kappa < 3.5$ .",
"Figure: Fit peak comparison of E 11 E_{11} and E 22 E_{22} for 0 ∘ 0^\\circ , and E 12 /E 21 E_{12}/E_{21} for 90 ∘ 90^\\circ .",
"The E 12 /E 21 E_{12}/E_{21} peak is multiplied by 10.We next discuss the oscillator strength ratio of the $E_{12}/E_{21}$ and $E_{11}$ transitions.",
"Directly from the traces presented in Fig.",
"REF , we can determine this ratio to be $I_{12}/I_{11}$ = 0.05.",
"Here, $I_{11}$ ($I_{12}$ ) is the integrated intensity of the $E_{11}$ ($E_{12}/E_{21}$ ) peak in the parallel-polarization (perpendicular-polarization) spectrum.",
"It is important to note that this ratio is independent of $S$ .",
"This can be easily seen by comparing Eq.",
"(REF ) and $\\alpha _{\\text{abs},E_{12}}(90^\\circ ) & = \\frac{NE_{12}}{\\hbar c n_0}\\alpha _2^{\\prime \\prime }\\int _{0}^{\\pi }f \\left(\\theta -\\frac{\\pi }{2}\\right)\\sin ^2\\left(\\theta -\\frac{\\pi }{2} \\right) d\\theta \\nonumber \\\\& = \\frac{NE_{12}}{\\hbar c n_0}\\frac{1+S}{2}\\alpha _2^{\\prime \\prime }.$ Namely, $\\frac{\\alpha _{\\text{abs},E_{11}}(0^\\circ )}{\\alpha _{\\text{abs},E_{12}}(90^\\circ )} = \\frac{E_{11}\\alpha _1^{\\prime \\prime }}{E_{12}\\alpha _2^{\\prime \\prime }}.$ By equating this ratio to $I_{11}/I_{12}$ , we can also obtain the ratio of the imaginary part of the molecular polarizability for perpendicular polarization at $E_{12}$ to that for parallel polarization at $E_{11}$ $\\frac{\\alpha _2^{\\prime \\prime }}{\\alpha _1^{\\prime \\prime }} = \\frac{E_{11}}{E_{12}} \\times 0.05 = 0.03.$ Finally, we can also use the obtained value of $I_{12}/I_{11}$ = 0.05 to get a value for the dielectric constant, $\\kappa $ , through comparison with the theoretical calculations of this ratio by Uryu and Ando [11].",
"The radiation power absorbed by a nanotube can be expressed as $P_\\parallel = \\frac{1}{2} \\sigma _{11}^{\\prime }D^2 \\\\P_\\perp = \\frac{1}{4} \\sigma _{12}^{\\prime }D^2$ for parallel and perpendicular polarizations, respectively.",
"Here, $\\sigma _{11}^{\\prime }$ ($\\sigma _{12}^{\\prime }$ ) is the real part of the optical conductivity parallel (perpendicular) to the nanotube axis at $E_\\text{ph} = E_{11}$ ($E_\\text{ph} = E_{12}$ ) and $D$ is the amplitude of the electric field of light.",
"Note that these expressions take into account the fact that only the wavenumber components $\\pm 2\\pi /L$ (where $L$ is the nanotube circumference) of the incident light can excite the $E_{12}/E_{21}$ transition whereas only the zero-wavenumber component of the incident light can excite the $E_{11}$ transition; the inclusion of the $\\pm 2\\pi /L$ components corresponds to the simultaneous excitation of the $E_{12}$ and $E_{21}$ transitions [10].",
"Spectrally integrated and properly normalized values of $\\sigma _{12}^{\\prime }$ and $\\sigma _{11}^{\\prime }$ (and thus those of $2P_\\perp $ and $P_\\parallel $ ) can be found in Fig.",
"7 of Ref. [11].",
"Hence, we compared the calculated ratio $2P_\\perp /P_\\parallel $ with our experimental value $2I_{12}/I_{11}$ = 0.10 and obtained $\\kappa $ = 1.52.",
"This value is slightly outside the range we deduced from the peak energy consideration above ($1.8 < \\kappa < 3.5$ ).",
"A better treatment of the surrounding dielectrics [23] as well as inclusion of higher-order terms in the band structure calculation are needed to fully explain the experimental results quantitatively."
],
[
"Summary",
"We prepared a macroscopic film of highly aligned single-chirality (6,5) SWCNTs and performed a polarization-dependent optical absorption spectroscopy study.",
"In addition to the usual $E_{11}$ and $E_{22}$ exciton peaks for parallel-polarized light, we observed a clear absorption peak due to the $E_{12}$ /$E_{21}$ exciton peak for perpendicular-polarized light.",
"Unlike previous observations of cross-polarized excitons in polarization-dependent photoluminescence and circular dichroism spectroscopy experiments, our direct absorption observation allowed us to quantitatively analyze this resonance.",
"We determined the energy of this peak to be 1.54 times that of the $E_{11}$ peak and the oscillator strength of this resonance to be 0.05 times that of the $E_{11}$ peak.",
"These values, in light of theoretical calculations available in the literature, led to an assessment of the environmental effect on the strength of Coulomb interactions in this aligned single-chirality SWCNT film."
],
[
"Acknowledgements",
"We thank Seiji Uryu, Tsuneya Ando, and Katsumasa Yoshioka for useful discussions.",
"This work was supported by the U.S. Department of Energy Basic Energy Sciences through grant no.",
"DEFG02-06ER46308 (optical spectroscopy experiments), the U.S. National Science Foundation through award no.",
"ECCS-1708315 (modeling), and the Robert A. Welch Foundation through grant no.",
"C-1509 (sample preparation).",
"K.Y.",
"acknowledges support by JSPS KAKENHI through Grant Numbers JP16H00919, JP17K14088, JP25107003, JP17H01069, JP17H06124, and JP15K21722, JST CREST through Grant Number JPMJCR17I5, Japan, and the Yamada Science Foundation."
],
[
"Chirality Purity Determination",
"To assess the chirality purity of our sample quantitatively, we analyzed the absorption spectrum shown in Fig.",
"REF using the method described in Ref. [24].",
"The spectrum is reproduced in Fig.",
"REF with two spectral regions of interest expanded.",
"In Region (i), we observe a shoulder, which we attribute to the $E_{11}$ peak of residual (9,1) SWCNTs.",
"In Region (ii), there are three small peaks, which can be attributed to the $E_{11}$ peaks of metallic SWCNTs.",
"Through line-fitting analysis shown in Fig.",
"REF , we determined the relative peak intensities of the observed peaks, as summarized in Table REF .",
"From these values, neglecting any ($n$ ,$m$ ) dependence of oscillator strength, we can calculate the relative population of (6,5) SWCNTs to be (41.658/41.941)$\\times $ 100 = 99.3%.",
"Table: Relative integrated peak intensities of the E 11 E_{11} peaks of (6,5), (9,1), and metallic SWCNTs in the sample.Figure: Absorbance spectrum for the SWCNT suspension used for making the film studied in this study.",
"Two spectral regions of interest – (i) and (ii) – are expanded in the bottom two panels.Figure: Spectral fitting analysis performed to determine the relative peak intensities of the E 11 E_{11} peaks of (6,5), (9,1), and metallic SWCNTs in the sample."
],
[
"Three-dimensional (3D) case",
"Let us cosnider an ensemble of spheroidal molecules and their anisotropic optical absorption properties.",
"As shown in Fig.",
"REF (a), we define the molecular polarizability along the long axis as $\\alpha _1$ and the molecular polarizability along the short axis as $\\alpha _2$ .",
"$\\theta $ is the angle between the alignment direction of the ensemble and the long axis of the particular individual molecule that we examine.",
"When an electric field is applied parallel to the alignment direction (which is the $z$ -direction in Fig.",
"REF (a)), the expectation value (i.e., the ensemble average) of the molecular polarizability $\\langle \\alpha \\rangle _{\\text{3D}}$ is given by $\\langle \\alpha \\rangle _{\\parallel ,\\text{3D}} & =\\alpha _1\\langle \\cos ^2\\theta \\rangle +\\alpha _2\\langle \\sin ^2\\theta \\rangle \\nonumber \\\\& =\\alpha _2+(\\alpha _1-\\alpha _2)\\langle \\cos ^2\\theta \\rangle ,$ where $\\langle \\cos ^2\\theta \\rangle $ and $\\langle \\sin ^2\\theta \\rangle $ are the expectation values of $\\cos ^2\\theta $ and $\\sin ^2\\theta $ , respectively.",
"On the other hand, when the applied electric field is parallel to the $y$ -axis in Fig.",
"REF (a), that is to say, the electric field is perpendicular to the alignment direction, the average molecular polarizability $\\langle \\alpha \\rangle _\\perp $ is given by $\\langle \\alpha \\rangle _{\\perp ,\\text{3D}} & =\\alpha _1\\langle \\cos ^2\\gamma \\rangle +\\alpha _2\\langle \\sin ^2\\gamma \\rangle \\nonumber \\\\& =\\alpha _2+(\\alpha _1-\\alpha _2)\\langle \\cos ^2\\gamma \\rangle ,$ where $\\gamma $ is the angle between the electric field, which is parallel to the $y$ -axis in Fig.",
"REF (a), and the long axis of the spheroidal molecule.",
"Here, $\\cos \\gamma $ can be written as $\\cos \\gamma =\\sin \\theta \\sin \\phi ,$ where $\\phi $ is the angle between the $x$ -axis and the direction of $\\alpha _1$ projected onto the $xy$ -plane.",
"Now, $\\langle \\cos ^2\\theta \\rangle _0$ , which is the expectation value of $\\cos ^2\\theta $ when the molecules are randomly oriented, is given by $\\langle \\cos ^2\\theta \\rangle _0=\\frac{\\int _{0}^{\\pi }\\cos ^2\\theta d\\Omega }{\\int _{0}^{\\pi } d\\Omega },$ where $d\\Omega $ is an infinitesimal solid angle, which is expressed as $2\\pi \\sin \\theta d\\theta d\\phi $ .",
"Hence, by substituting $d\\Omega = 2\\pi \\sin \\theta d\\theta d\\phi $ into Eq.",
"(REF ), we obtain $\\langle \\cos ^2\\theta \\rangle _{0,\\text{3D}}=\\frac{\\int _{-\\pi }^{\\pi }\\int _{0}^{\\pi }2\\pi \\cos ^2\\theta \\sin \\theta d\\theta d\\phi }{\\int _{-\\pi }^{\\pi }\\int _{0}^{\\pi }2\\pi \\sin \\theta d\\theta d\\phi }=\\frac{1}{3}.$ Similarly, $\\langle \\cos ^2\\gamma \\rangle _0$ , which is the expectation value of $\\cos ^2\\gamma $ when the molecules are randomly oriented, is given by $\\langle \\cos ^2\\gamma \\rangle _{0} &=\\frac{\\int _{-\\pi }^{\\pi }\\int _{0}^{\\pi }2\\pi \\cos ^2\\gamma \\sin \\theta d\\theta d\\phi }{\\int _{-\\pi }^{\\pi }\\int _{0}^{\\pi }2\\pi \\sin \\theta d\\theta d\\phi }=\\frac{1}{3}.$ The mean polarizability of randomly oriented spheroidal molecules can thus be obtained, through substitution of Eq.",
"(REF ) into Eq.",
"(REF ) or substitution of Eq.",
"(REF ) into Eq.",
"(REF ), as $\\langle \\alpha \\rangle _{0,\\text{3D}}=\\frac{1}{3}\\alpha _1+\\frac{2}{3}\\alpha _2.$ Figure: (a) An ensemble of spheroidal molecules in 3D space.",
"(b) Detailed illustration of a molecule in a 3D global coordinate system.",
"The alignment direction is along the zz-axis.",
"(c) Schematic of a 2D ensemble of carbon nanotubes.",
"The alignment direction is along the yy-axis.When the system is uniaxial, the distribution depends only on $\\theta $ .",
"Since $\\langle \\cos ^2\\gamma \\rangle $ does not depend on $\\phi $ in this case, $\\langle \\cos ^2\\gamma \\rangle $ is expressed as $\\langle \\cos ^2\\gamma \\rangle &=\\frac{1}{2}\\left( 1-\\langle \\cos ^2\\theta \\rangle \\right) .$ As a result, Eq.",
"(REF ) becomes $\\langle \\alpha \\rangle _{\\perp ,\\text{3D}} & =\\frac{1}{2}\\left( \\alpha _1+\\alpha _2-\\left( \\alpha _1-\\alpha _2\\right) \\langle \\cos ^2\\theta \\rangle \\right) .$ Therefore, the average polarizability for an ensemble of randomly orientated molecules in Eq.",
"(REF ) can be expressed in terms of $\\langle \\alpha \\rangle _\\parallel $ and $\\langle \\alpha \\rangle _\\perp $ as $\\langle \\alpha \\rangle _{0,\\text{3D}}=\\frac{1}{3}\\langle \\alpha \\rangle _{\\parallel ,\\text{3D}}+\\frac{2}{3}\\langle \\alpha \\rangle _{\\perp ,\\text{3D}}.$ Here, we introduce the nematic order parameter, $S$ , as a normalized degree of alignment.",
"Namely, we require that $S =1$ for a perfectly aligned ensemble and $S =0$ for a randomly oriented ensemble.",
"$S$ can be expressed as an average of the long axis distribution of the angle $\\theta $ , which is the angle between a nanotube and the macroscopic alignment direction.",
"For a 3D system [25], $S_{\\text{3D}}&=\\frac{1}{2}(3\\langle \\cos ^2\\theta \\rangle -1)$ satisfies the requirements above.",
"By reversing Eq.",
"(REF ), we obtain.",
"$\\langle \\cos ^2\\theta \\rangle &=\\frac{1}{3}(2S_{\\text{3D}}+1).$ The average polarizabilities for parallel and perpendicular electric fields, i.e., Eq.",
"(REF ) and Eq.",
"(REF ), can then be written in terms of $S$ : $\\langle \\alpha \\rangle _{\\parallel ,\\text{3D}} =\\frac{1}{3}\\left\\lbrace \\alpha _1+2\\alpha _2+2S_\\text{3D}\\left( \\alpha _1-\\alpha _2\\right) \\right\\rbrace .$ $\\langle \\alpha \\rangle _{\\perp ,\\text{3D}} =\\frac{1}{3}\\left\\lbrace \\alpha _1+2\\alpha _2-S_\\text{3D}\\left( \\alpha _1-\\alpha _2\\right) \\right\\rbrace .$ Given the average molecular polarizability, we can now obtain the susceptibility $\\chi $ of the molecular ensemble as $\\chi = N\\langle \\alpha \\rangle ,$ where $N$ is the number of molecules.",
"The absorption coefficient $\\alpha _\\text{abs}$ for incident light with angular frequency $\\omega $ is then obtained by $\\alpha _\\text{abs} & = \\frac{\\omega }{cn_0}\\chi ^{\\prime \\prime } = \\frac{E_\\text{ph}}{\\hbar c n_0}\\chi ^{\\prime \\prime } \\nonumber \\\\& = \\frac{NE_\\text{ph}}{\\hbar c n_0}\\langle \\alpha ^{\\prime \\prime }\\rangle ,$ where $\\chi ^{\\prime \\prime }$ is the imaginary part of $\\chi $ , $E_\\text{ph} = \\hbar \\omega $ is the photon energy of the incident light, $c$ is the speed of light, $\\hbar $ is the reduced Planck constant, $n_0$ is the refractive index, and $\\alpha ^{\\prime \\prime }$ is the imaginary part of the molecular polarizability, $\\alpha $ .",
"When the molecules are randomly oriented, $\\alpha _\\text{abs}$ can be obtained by substituting Eq.",
"(REF ) into Eq.",
"(REF ), i.e., $\\alpha _{\\text{abs},0,\\text{3D}} = \\frac{NE_\\text{ph}}{3\\hbar c n_0} \\left( \\alpha _1^{\\prime \\prime } + 2\\alpha _2^{\\prime \\prime } \\right),$ where $\\alpha _1^{\\prime \\prime }$ ($\\alpha _2^{\\prime \\prime }$ ) is the imaginary part of $\\alpha _1$ ($\\alpha _2$ ).",
"Using Eq.",
"(REF ) and Eq.",
"(REF ), $\\alpha _{\\text{abs},\\parallel ,\\text{3D}}$ and $\\alpha _{\\text{abs},\\perp ,\\text{3D}}$ , which are the absorption coefficients for parallel polarization and perpendicular polarization, respectively, can then be written as $\\alpha _{\\text{abs},\\parallel ,\\text{3D}} =\\frac{NE_\\text{ph}}{3\\hbar c n_0}\\left\\lbrace \\alpha _1^{\\prime \\prime }+2\\alpha _2^{\\prime \\prime }+2S_{\\text{3D}}\\left( \\alpha _1^{\\prime \\prime }-\\alpha _2^{\\prime \\prime }\\right) \\right\\rbrace .$ $\\alpha _{\\text{abs},\\perp ,\\text{3D}} =\\frac{NE_\\text{ph}}{3\\hbar c n_0}\\left\\lbrace \\alpha _1^{\\prime \\prime }+2\\alpha _2^{\\prime \\prime }-S_{\\text{3D}}\\left( \\alpha _1^{\\prime \\prime }-\\alpha _2^{\\prime \\prime }\\right) \\right\\rbrace .$ respectively.",
"From Eqs.",
"(REF ), (REF ), and (REF ), the following relation can also be derived: $\\alpha _{\\text{abs},0,\\text{3D}} = \\frac{1}{3}\\alpha _{\\text{abs},\\parallel ,\\text{3D}}+\\frac{2}{3}\\alpha _{\\text{abs},\\perp ,\\text{3D}}.$ When the reflection loss is negligible, the absorbance is given as $\\alpha _\\text{abs}l/\\ln (10)$ , where $l$ is the sample thickness.",
"Therefore, the linear dichroism $LD$ is written as $LD_{\\text{3D}} &= \\frac{l}{\\ln (10)} (\\alpha _{\\text{abs},\\parallel ,\\text{3D}} - \\alpha _{\\text{abs},\\perp ,\\text{3D}})\\\\ &= \\frac{NlE_\\text{ph}}{\\hbar c n_0 \\ln (10)}S_{\\text{3D}}\\left( \\alpha _1^{\\prime \\prime }-\\alpha _2^{\\prime \\prime }\\right).$ The reduced linear dichroism $LD^r$ , which is the linear dichroism normalized by $\\alpha _{\\text{abs},0,\\text{3D}}l/\\ln (10)$ , where $\\alpha _{\\text{abs},0,\\text{3D}}$ is given by Eq.",
"(REF ) or Eq.",
"(REF ).",
"Thus, $LD_{\\text{3D}}^r &= \\frac{3\\left(\\alpha _{\\text{abs},\\parallel ,\\text{3D}} - \\alpha _{\\text{abs},\\perp ,\\text{3D}} \\right) }{\\alpha _{\\text{abs},\\parallel ,\\text{3D}}+2\\alpha _{\\text{abs},\\perp ,\\text{3D}}}.$ Substituting Eq.",
"(REF ) and Eq.",
"(REF ) here, we obtain $LD_{\\text{3D}}^r &= \\frac{3S_{3\\text{D}}(\\alpha _1^{\\prime \\prime }-\\alpha _2^{\\prime \\prime })}{\\alpha _1^{\\prime \\prime }+2\\alpha _2^{\\prime \\prime }}.$ Defining an angle $\\xi _\\text{3D} \\equiv \\tan ^{-1} \\left( \\sqrt{\\alpha _1^{\\prime \\prime }/2\\alpha _2^{\\prime \\prime }} \\right)$ , $LD_{\\text{3D}}^r &= \\frac{1}{2}S\\left( 3\\cos ^2\\xi _\\text{3D}-1\\right) .$"
],
[
"Two-dimensional (2D) case",
"We apply the above-developed 3D theory to an ensemble of planar or 2D aligned nanotubes.",
"As shown in Fig.",
"REF (c), we define the polarizability along the tube axis as $\\alpha _1$ and the polarizability perpendicular to the tube axis as $\\alpha _2$ .",
"As before, $\\theta $ is the angle between the macroscopic alignment direction and the individual nanotube under question.",
"The expectation value of the polarizability of this 2D ensemble $\\langle \\alpha \\rangle _\\text{2D}$ for an electric field parallel to the alignment direction is given by $\\langle \\alpha \\rangle _{\\parallel ,\\text{2D}} & =\\alpha _1\\langle \\cos ^2\\theta \\rangle +\\alpha _2\\langle \\sin ^2\\theta \\rangle \\nonumber \\\\& =\\alpha _2+(\\alpha _1-\\alpha _2)\\langle \\cos ^2\\theta \\rangle ,$ and that for an electric field perpendicular to the alignment direction is given by $\\langle \\alpha \\rangle _{\\perp ,\\text{2D}} & =\\alpha _1\\langle \\sin ^2\\theta \\rangle +\\alpha _2\\langle \\cos ^2\\theta \\rangle \\nonumber \\\\& =\\alpha _1+(\\alpha _2-\\alpha _1)\\langle \\cos ^2\\theta \\rangle .$ When the nanotubes are randomly oriented, the expectation value of $\\cos ^2\\theta $ is given by $\\langle \\cos ^2\\theta \\rangle _{0,\\text{2D}}=\\frac{\\int _{0}^{\\pi }\\cos ^2\\theta d\\theta }{\\int _{0}^{\\pi } d\\theta }=\\frac{1}{2}.$ The mean polarizability of randomly oriented nanotubes can then be obtained by substituting Eq.",
"(REF ) into Eq.",
"(REF ) or Eq.",
"(REF ) as $\\langle \\alpha \\rangle _{0,\\text{2D}}=\\frac{1}{2}\\alpha _1+\\frac{1}{2}\\alpha _2.$ The order parameter $S$ in 2D is expressed as [26], [27], [28], $S_{\\text{2D}}=\\langle 2\\cos ^2\\theta -1\\rangle .$ By reversing this equation, we obtain $\\langle \\cos ^2\\theta \\rangle =\\frac{1}{2}\\left( S_{\\text{2D}}+1\\right) .$ The average polarizabilities for parallel and perpendicular electric fields, obtained as Eq.",
"(REF ) and Eq.",
"(REF ), respectively, can then be expressed in terms of $S_\\text{2D}$ as $\\langle \\alpha \\rangle _{\\parallel ,\\text{2D}} =\\frac{1}{2}\\left\\lbrace \\alpha _1+\\alpha _2+S_{\\text{2D}}\\left( \\alpha _1-\\alpha _2\\right) \\right\\rbrace $ and $\\langle \\alpha \\rangle _{\\perp ,\\text{2D}} =\\frac{1}{2}\\left\\lbrace \\alpha _1+\\alpha _2-S_{\\text{2D}}\\left( \\alpha _1-\\alpha _2\\right) \\right\\rbrace .$ respectively.",
"When the nanotubes are randomly oriented, the absorption coefficient $\\alpha _\\text{abs}$ can be obtained, by substituting Eq.",
"(REF ) into Eq.",
"(REF ), as $\\alpha _{\\text{abs},0,\\text{2D}} = \\frac{NE_\\text{ph}}{2\\hbar c n_0}\\left(\\alpha _1^{\\prime \\prime }+\\alpha _2^{\\prime \\prime }\\right),$ where $\\alpha _1^{\\prime \\prime }$ ($\\alpha _2^{\\prime \\prime }$ ) is the imaginary part of $\\alpha _1$ ($\\alpha _2$ ).",
"From Eq.",
"(REF ) and Eq.",
"(REF ), the absorption coefficients for parallel and perpendicular polarizations are given, respectively, by $\\alpha _{\\text{abs},\\parallel ,\\text{2D}} =\\frac{NE_\\text{ph}}{2\\hbar c n_0}\\left\\lbrace \\alpha _1^{\\prime \\prime }+\\alpha _2^{\\prime \\prime }+S_{\\text{2D}}\\left( \\alpha _1^{\\prime \\prime }-\\alpha _2^{\\prime \\prime }\\right) \\right\\rbrace ,$ and $\\alpha _{\\text{abs},\\perp ,\\text{2D}} =\\frac{NE_\\text{ph}}{2\\hbar c n_0}\\left\\lbrace \\alpha _1^{\\prime \\prime }+\\alpha _2^{\\prime \\prime }-S_{\\text{2D}}\\left( \\alpha _1^{\\prime \\prime }-\\alpha _2^{\\prime \\prime }\\right)\\right\\rbrace .$ The absorption coefficient for randomly orientated nanotubes is also expressed by $\\alpha _{\\text{abs},0,\\text{2D}} = \\frac{1}{2}\\alpha _{\\text{abs},\\parallel ,\\text{2D}} + \\frac{1}{2}\\alpha _{\\text{abs},\\perp ,\\text{2D}}.$ and $\\frac{\\alpha _{\\text{abs},\\parallel ,\\text{2D}}}{\\alpha _{\\text{abs},\\perp ,\\text{2D}}} =\\frac{\\alpha _1^{\\prime \\prime }+\\alpha _2^{\\prime \\prime }+S_{\\text{2D}}\\left( \\alpha _1^{\\prime \\prime }-\\alpha _2^{\\prime \\prime }\\right)}{\\alpha _1^{\\prime \\prime }+\\alpha _2^{\\prime \\prime }-S_{\\text{2D}}\\left( \\alpha _1^{\\prime \\prime }-\\alpha _2^{\\prime \\prime }\\right)}.$ In a manner similar to the 3D case, the linear dichroism, $LD$ , is expressed as $LD_{\\text{2D}} & = \\frac{l}{\\ln (10)} (\\alpha _{\\text{abs},\\parallel ,\\text{2D}} - \\alpha _{\\text{abs},\\perp ,\\text{2D}}) \\\\ &= \\frac{NlE_\\text{ph}}{2\\hbar c n_0 \\ln (10)}S_{\\text{2D}}\\left(\\alpha _1^{\\prime \\prime }-\\alpha _2^{\\prime \\prime }\\right).$ The reduced linear dichroism $LD^r$ is given by $LD_{\\text{2D}}^r &= \\frac{2 (\\alpha _{\\text{abs},\\parallel ,\\text{2D}} - \\alpha _{\\text{abs},\\perp ,\\text{2D}} ) }{\\alpha _{\\text{abs},\\parallel ,\\text{2D}}+\\alpha _{\\text{abs},\\perp ,\\text{2D}}} .$ Substituting Eq.",
"(REF ) and Eq.",
"(REF ) here, we obtain $LD_{\\text{2D}}^r &= \\frac{2S_{\\text{2D}}(\\alpha _1^{\\prime \\prime }-\\alpha _2^{\\prime \\prime })}{\\alpha _1^{\\prime \\prime }+\\alpha _2^{\\prime \\prime }}.$ Defining an angle $\\xi _\\text{2D} \\equiv \\tan ^{-1} \\left( \\sqrt{\\alpha _1^{\\prime \\prime }/\\alpha _2^{\\prime \\prime }} \\right)$ , $LD_{\\text{2D}}^r &= 2S_\\text{2D}\\left( \\cos ^2\\xi _\\text{2D}-1\\right) .$ Finally, we consider absorption coefficients for two cases: (i) $\\xi _\\text{2D}=0^\\circ $ ($\\alpha _1^{\\prime \\prime } \\ne 0$ , $\\alpha _2^{\\prime \\prime } = 0$ ), and (ii) $\\xi _\\text{2D}=90^\\circ $ ($\\alpha _1^{\\prime \\prime } = 0$ , $\\alpha _2^{\\prime \\prime } \\ne 0$ ).",
"In these cases, $\\alpha _{\\text{abs},\\parallel }$ , $\\alpha _{\\text{abs},\\perp }$ , $\\alpha _{\\text{abs},\\parallel }/\\alpha _{\\text{abs},\\perp }$ , and $LD^r$ are expressed as follows: (i) $\\xi _\\text{2D}=0^\\circ $ ($\\alpha _1^{\\prime \\prime } \\ne 0$ , $\\alpha _2^{\\prime \\prime } = 0$ ) $\\alpha _{\\text{abs},\\parallel ,\\text{2D}} = \\frac{NE_\\text{ph}\\alpha _1^{\\prime \\prime }}{2\\hbar c n_0}\\left( 1+S_{\\text{2D}} \\right)\\\\\\alpha _{\\text{abs},\\perp ,\\text{2D}} = \\frac{NE_\\text{ph}\\alpha _1^{\\prime \\prime }}{2\\hbar c n_0}\\left( 1-S_{\\text{2D}} \\right)\\\\\\frac{\\alpha _{\\text{abs},\\parallel ,\\text{2D}}}{\\alpha _{\\text{abs},\\perp ,\\text{2D}}} = \\frac{1+S_{\\text{2D}}}{1-S_{\\text{2D}}}\\\\LD_{\\text{2D}}^r = 2S_{\\text{2D}}.$ (ii) $\\xi _\\text{2D}=90^\\circ $ ($\\alpha _1^{\\prime \\prime } = 0$ , $\\alpha _2^{\\prime \\prime } \\ne 0$ ) $\\alpha _{\\text{abs},\\parallel ,\\text{2D}} =\\frac{NE_\\text{ph}\\alpha _2^{\\prime \\prime }}{2\\hbar c n_0}\\left( 1-S_{\\text{2D}} \\right)\\\\\\alpha _{\\text{abs},\\perp ,\\text{2D}} =\\frac{NE_\\text{ph}\\alpha _2^{\\prime \\prime }}{2\\hbar c n_0}\\left( 1+S_{\\text{2D}} \\right)\\\\\\frac{\\alpha _{\\text{abs},\\parallel ,\\text{2D}}}{\\alpha _{\\text{abs},\\perp ,\\text{2D}}} =\\frac{1-S_{\\text{2D}}}{1+S_{\\text{2D}}}\\\\LD_{\\text{2D}}^r =-2S_{\\text{2D}}.$"
]
] | 1808.08602 |
[
[
"Mallows Ranking Models: Maximum Likelihood Estimate and Regeneration"
],
[
"Abstract This paper is concerned with various Mallows ranking models.",
"We study the statistical properties of the MLE of Mallows' $\\phi$ model.",
"We also make connections of various Mallows ranking models, encompassing recent progress in mathematics.",
"Motivated by the infinite top-$t$ ranking model, we propose an algorithm to select the model size $t$ automatically.",
"The key idea relies on the renewal property of such an infinite random permutation.",
"Our algorithm shows good performance on several data sets."
],
[
"Introduction",
"Ranked data appear in many problems of social choice, user recommendation and information retrieval.",
"Examples include ranking candidates by a large number of voters in an election (e.g.",
"instant-runoff voting), and the document retrieval problem where one aims to design a meta-search engine according to a ranked list of web pages output by various search algorithms.",
"In the sequel, we use the words ranking and permutation interchangeably.",
"A ranking model is given by a collection of items, and an unknown total ordering of these items.",
"There is a rich body of literature on probabilistic ranking models.",
"The earliest work dates back to [51], [52] where items are ranked according to the order statistics of a Gaussian random vector.",
"[5] introduced an exponential family model by pairwise comparisons, which was extended by [36], [47] with comparisons of multiple items.",
"See also [27], [8], [9], [49], [26] for algorithms and statistical analysis of the Bradley-Terry model and its variants.",
"A more tractable subclass of the Bradley-Terry model was proposed by [37] as follows.",
"For $n \\ge 1$ , let $\\mathfrak {S}_n$ be the set of permutations of $[n]: = \\lbrace 1, \\ldots , n\\rbrace $ .",
"The parametric model $\\mathbb {P}_{\\theta , \\pi _0, d}(\\pi ) = \\frac{1}{\\Psi (\\theta , d)} e^{- \\theta d (\\pi ,\\pi _0)} \\quad \\mbox{for } \\pi \\in \\mathfrak {S}_n,$ is referred to as the Mallows model.",
"Here $\\theta > 0$ is the dispersion parameter, $\\pi _0$ is the central ranking, $d(\\cdot , \\cdot ): \\mathfrak {S}_n \\times \\mathfrak {S}_n \\rightarrow \\mathbb {R}_{+}$ is a discrepancy function which is right invariant: $d(\\pi , \\sigma ) = d(\\pi \\circ \\sigma ^{-1}, id) \\quad \\mbox{for } \\pi , \\sigma \\in \\mathfrak {S}_n$ , and $\\Psi (\\theta , d): = \\sum _{\\pi \\in \\mathfrak {S}_n} e^{- \\theta d (\\pi ,\\pi _0)}$ is the normalizing constant.",
"Mallows primarily considered two special cases of (REF ): [itemsep = 3 pt] Mallows' $\\theta $ model, where $d(\\pi , \\sigma ) = \\sum _{i = 1}^n (\\pi (i) - \\sigma (i))^2$ is the Spearman's rho, Mallows' $\\phi $ model, where $d(\\pi , \\sigma ) = \\operatorname{inv}(\\pi \\circ \\sigma ^{-1})$ , called the Kendall's tau, where $\\operatorname{inv}(\\pi ):= \\# \\lbrace (i,j) \\in [n]^2: i < j \\mbox{ and } \\pi (i) > \\pi (j)\\rbrace $ is the number of inversions of $\\pi $ .",
"The general form (REF ) was suggested by [17] along with other discrepancy functions.",
"[17], [18] and [14] also pioneered the group representation approach to ranked, and partially ranked data.",
"In this paper we are primarily concerned with the statistical properties of the maximum likelihood estimate (MLE) of the Mallows' $\\phi $ model and its infinite counterpart.",
"Specializing (REF ) with the Kendall's tau, the Mallows' $\\phi $ model is expressed as $\\mathbb {P}_{\\theta , \\pi _0}( \\pi ) = \\frac{1}{\\Psi (\\theta )} e^{- \\theta \\operatorname{inv}(\\pi \\circ \\pi _0^{-1})} \\quad \\mbox{for } \\pi \\in \\mathfrak {S}_n.$ It is easily seen that $\\mathbb {P}_{\\theta , \\pi _0}$ has a unique mode $\\pi _0$ if $\\theta > 0$ .",
"The Mallows' $\\phi $ model (REF ) is of particular interest, since it is an instance of two large classes of ranking models: distance-based ranking models [20] and multistage ranking models [21].",
"The one parameter model (REF ) also has an $n-1$ parameter extension where $\\theta $ is replaced with $\\vec{\\theta }:= (\\theta _1, \\ldots \\theta _{n-1})$ by factorizing the inversion.",
"This $n-1$ parameter model, called the Generalized Mallows (GM$_{\\vec{\\theta }, \\pi _0}$ ) model, will be discussed in Section .",
"See [15], [39] for a review of these ranking models.",
"[21] showed that if the central ranking $\\pi _0$ is known, the MLE of $\\theta $ (or $\\vec{\\theta }$ ) can be easily found by convex optimization.",
"But it is harder to compute the MLE of $\\pi _0$ , and only a few heuristic algorithms are available.",
"As pointed out in [43], the problem of finding the MLE of $\\pi _0$ for the Mallows' $\\phi $ model is the Kemeny's consensus ranking problem which is known to be NP-hard [54], [4].",
"They also gave a branch and bound search algorithm to estimate simultaneously $\\theta $ (or $\\vec{\\theta }$ ) and $\\pi _0$ .",
"See also [12], [1], [38] for approximation algorithms for consensus ranking problem, [6], [28], [29] for efficient sampling and learning of Mallows models, and [32], [10] for various generalizations of Mallows ranking models.",
"Though there have been efforts in developing algorithms to estimate $\\theta $ (or $\\vec{\\theta }$ ) and $\\pi _0$ , not much is known about the statistical properties of the MLE $\\widehat{\\theta }$ and $\\widehat{\\pi }_0$ even for the simplest model (REF ).",
"We provide statistical analysis to the MLE of the model (REF ), and answer the following questions in Section : Are the MLEs $\\widehat{\\theta }$ , $\\widehat{\\pi }_0$ consistent ?",
"Is the MLE $\\widehat{\\theta }$ unbiased ?",
"How fast does the MLE $\\widehat{\\pi }_0$ converge to $\\pi _0$ ?",
"When the number of items $n$ is large, learning a complete ranking model becomes impracticable.",
"A line of work by [20], [7], [41], [42] focused on the top-$t$ orderings for the GM$_{\\vec{\\theta }, \\pi _0}$ model.",
"Among these work, [41] proposed a probability model over the top-$t$ orderings of infinite permutations, called the Infinite Generalized Mallows (IGM$_{\\vec{\\theta }, \\pi _0}$ ) model: $\\mathbb {P}_{\\vec{\\theta }, \\pi _0}(\\pi ) = \\frac{1}{\\Psi (\\vec{\\theta })} \\cdot \\exp \\left(-\\sum _{j = 1}^t \\theta _j s_j (\\pi \\circ \\pi _0^{-1})\\right),$ where $s(\\pi ) := (s_1(\\pi ), s_2(\\pi ), \\ldots )$ is the inversion table of $\\pi $ defined by $s_j(\\pi ):= \\pi ^{-1}(j) - 1 - \\sum _{j^{\\prime } < j} 1_{\\lbrace \\pi ^{-1}(j^{\\prime }) < \\pi ^{-1}(j)\\rbrace },$ and $\\Psi (\\vec{\\theta }) = \\prod _{j = 1}^{t} (1-e^{-\\theta _j})^{-1}$ is the normalizing constant.",
"By convention, $\\theta _j = 0$ for any $j > t$ .",
"The integer `$t$ ' is referred to as the model size of the IGM model.",
"If $\\theta _ 1 = \\cdots = \\theta _t = \\theta $ , the IGM$_{\\vec{\\theta }, \\pi _0}$ model is called the single parameter IGM model.",
"As explained in [41], the IGM$_{\\vec{\\theta }, \\pi _0}$ model (REF ) is the marginal distribution of a random permutation of positive integers.",
"The single parameter IGM was called the infinite $q$ -shuffle in [22] with parameterization $q = e^{-\\theta }$ .",
"They provided a nice construction of the infinite $q$ -shuffle, which is reminiscent of absorption sampling [48], [30] and the repeated insertion model [19].",
"The infinite $q$ -shuffle was further extended by [46] to the $p$ -shifted permutations of positive integers.",
"But the link between Meilǎ-Bao's infinite ranking model and infinite $q$ -shuffle or $p$ -shifted permutations does not seem to have been previously noticed.",
"So we point out this connection which will be detailed in Section .",
"One disadvantage of the aforementioned top-$t$ ranking models is that they all require choosing `$t$ ' manually.",
"Small model size `$t$ ' often leads to poor accuracy, and large model size `$t$ ' needs a considerable amount of training time.",
"It was observed in [46] that the random infinite limit of the single parameter IGM model has a remarkable renewal property.",
"This suggests a heuristic procedure to select the model size `$t$ ' automatically based on Meilǎ-Bao's search algorithms.",
"We will discuss such an approach in Section .",
"To summarize, the main contributions of this paper are: Provide statistical analysis to the MLE of the Mallows' $\\phi $ model as well as the single parameter IGM.",
"Propose a selection algorithm for the model size `$t$ ' of the top-$t$ Mallows ranking models.",
"See also [31], [34], [35], [3], [53], [33] for the approximate Baysian inference of Mallows' mixture models for clustering heterogeneous ranked data, and [24], [25], [40] for hierarchical ranking models."
],
[
"Mallows Models: `$t$ ' Selection Algorithm",
"In this section we provide background on various Mallows ranking models, encompassing the closely related $q$ -shuffles and $p$ -shifted permutations.",
"We also give an algorithm to select the model size `$t$ ' for the IGM model (REF ).",
"We follow closely [21], [41], [46]."
],
[
"Finite Mallows Models",
"Given $n$ items labelled by $[n]$ , a ranking $\\pi \\in \\mathfrak {S}_n$ is represented by [itemsep = 3 pt] the word list $(\\pi (1), \\pi (2), \\ldots , \\pi (n))$ , the ranked list $(\\pi ^{-1}(1)| \\pi ^{-1}(2) |\\ldots |\\pi ^{-1}(n))$ .",
"Here $\\pi (i) = j$ means that the item $i$ has rank $j$ , and conversely $\\pi ^{-1}(j) = i$ means that the $j^{th}$ most preferred is item $i$ .",
"The idea of multistage ranking is to decompose the ranking procedure into independent stages.",
"The most preferred item is selected at the first stage, the best of the remaining at the second stage and so on until the least preferred item is selected.",
"The correctness of the choice at any stage is accessed through a central ranking $\\pi _0$ .",
"For example, if $\\pi _0 = (3|1|2)$ , then the ranking $\\pi = (3|2|1)$ gives a correct choice at the first stage, since item 3 is the most preferred in both $\\pi $ and $\\pi _0$ .",
"But at the second stage, among the two remaining items 1 and 2, item 2 is selected by $\\pi $ while the right choice is item 1 according to $\\pi _0$ .",
"For any ranking $\\pi \\in \\mathfrak {S}_n$ and $j \\in [n-1]$ , let $(s_1(\\pi ), \\ldots , s_{n-1}(\\pi ))$ be the inversion table of $\\pi $ defined by (REF ).",
"It is easy to see that $s_j(\\pi ) \\in \\lbrace 0, \\ldots , n -j\\rbrace $ , and there is a bijection between a ranking $\\pi $ and the inversion table $(s_1(\\pi ), \\ldots , s_{n-1}(\\pi ))$ .",
"The quantity $s_j(\\pi \\circ \\pi _0^{-1})$ measures the correctness of the choice at stage $j$ : $s_j(\\pi \\circ \\pi _0^{-1}) = k$ means that at stage $j$ the $(k+1)^{th}$ best of the remaining items is selected.",
"In the example with $\\pi _0 = (3|1|2)$ and $\\pi = (3|2|1)$ , we have $s_1(\\pi \\circ \\pi _0^{-1}) = 0$ and $s_2(\\pi \\circ \\pi _0^{-1}) = 1$ .",
"[20] introduced the multistage ranking models of the form: $\\mathbb {P}_{p, \\pi _0}(\\pi ) = \\prod _{j = 1}^{n-1} p_{j}\\left(s_j(\\pi \\circ \\pi _0^{-1})\\right),$ where $p_{j}(\\cdot )$ is a probability distribution on $\\lbrace 0, \\ldots , n - j\\rbrace $ at stage $j$ .",
"The choice of $p_j(k) = (1 - e^{-\\theta }) e^{-k \\theta }/(1 - e^{-(n-j+1)\\theta })$ for $k \\in \\lbrace 0, \\ldots , n-j\\rbrace $ in (REF ), and the remarkable identity $\\sum _{j = 1}^{n-1} s_j(\\pi ) = \\operatorname{inv}(\\pi )$ for any $\\pi \\in \\mathfrak {S}_n$ yield the Mallows' $\\phi $ model (REF ).",
"This one parameter model has a natural $n-1$ parameter extension by simply taking $p_j(k) = (1 - e^{-\\theta _j})e^{- \\theta _j k}/(1 - e^{-(n-j+1) \\theta _j})$ for $k \\in \\lbrace 0, \\ldots , n-j\\rbrace $ .",
"The $n - 1$ parameter model, called the Generalized Mallows (GM$_{\\vec{\\theta }, \\pi _0}$ ) model, is then defined by $\\mathbb {P}_{\\vec{\\theta }, \\pi _0}(\\pi ) = \\frac{1}{\\Psi (\\vec{\\theta })} \\exp \\left( - \\sum _{j = 1}^{n-1}\\theta _j s_j(\\pi \\circ \\pi _0^{-1}) \\right),$ where $\\Psi (\\vec{\\theta }) = \\prod _{j = 1}^{n-1} (1 - e^{-(n-j+1) \\theta _j})(1 - e^{-\\theta _j})^{-1}$ is the normalizing constant.",
"The GM$_{\\vec{\\theta }, \\pi _0}$ model is also called the Mallows' $\\phi $ -component model.",
"The model (REF ) can also be expressed in the form $e^{-d_{\\vec{\\theta }}}(\\pi , \\pi _0)/\\Psi (\\vec{\\theta })$ , with $d_{\\vec{\\theta }}(\\pi , \\pi _0): = \\sum _{j = 1}^{n-1} \\theta _j s_j(\\pi \\circ \\pi _0^{-1})$ ."
],
[
"Infinite Mallows Models",
"Given a countably infinite items labelled by $\\mathbb {N}_{+}: = \\lbrace 1, 2, \\ldots \\rbrace $ , a ranking $\\pi $ over $\\mathbb {N}_{+}$ is a bijection from $\\mathbb {N}_{+}$ onto itself represented by the word list $(\\pi (1), \\pi (2), \\ldots )$ or the ranked list $(\\pi ^{-1}(1)| \\pi ^{-1}(2) | \\ldots )$ .",
"A top-$t$ ordering of $\\pi $ is the prefix $(\\pi ^{-1}(1)| \\ldots |\\pi ^{-1}(t))$ .",
"Motivated by the GM$_{\\vec{\\theta }, \\pi _0}$ model (REF ), [41] proposed the Infinite Generalized Mallow (IGM$_{\\vec{\\theta }, \\pi _0}$ ) model (REF ), which can also be put in the form $e^{-d_{\\vec{\\theta }}(\\pi , \\pi _0)}/\\Psi (\\vec{\\theta })$ , with $d_{\\vec{\\theta }}(\\pi , \\pi _0): = \\sum _{j = 1}^t \\theta _j s_j(\\pi \\circ \\pi _0^{-1})$ .",
"In particular, $s_j$ is distributed as Geo$(1- e^{-\\theta _j})$ on $\\lbrace 0,1,\\ldots \\rbrace $ .",
"As explained in [41], one can regard $\\pi $ as a top-$t$ ordering, and $\\pi _0$ as an ordering over $\\mathbb {N}_{+}$ .",
"If $\\theta _1 = \\cdots =\\theta _t = \\theta $ , then the model (REF ) simplifies to $\\mathbb {P}_{\\theta , \\pi _0}(\\pi ) = \\frac{1}{\\Psi (\\theta )} \\exp \\left(-\\theta \\sum _{j = 1}^t s_j(\\pi \\circ \\pi _0^{-1}) \\right),$ called the single parameter IGM model.",
"It is easily seen that the single parameter IGM model (REF ) is the marginal distribution of a random permutation of positive integers.",
"Formally, this random infinite permutation is distributed as $\\mathbb {P}_{\\theta , \\pi _0}(\\pi ) = \\frac{1}{\\Psi (\\theta )} \\exp \\left( - \\theta \\sum _{j = 1}^{\\infty } s_j(\\pi \\circ \\pi _0^{-1}) \\right).$ In the terminology of [22], for $\\pi $ defined by (REF ), $\\pi \\circ \\pi _0^{-1}$ is the infinite $e^{-\\theta }$ -shuffle.",
"The latter was generalized by [46] to $p$ -shifted permutations, where $p = (p_1, p_2, \\ldots )$ is a discrete distribution on $\\mathbb {N}_{+}$ with $p_1 > 0$ .",
"Here we present a further extension of $p$ -shifted permutations.",
"Let $P = (p_{ij})_{i,j \\in \\mathbb {N}_{+}}$ be a stochastic matrix on $\\mathbb {N}_{+}$ with $p^i = (p_{ij})_{j \\in \\mathbb {N}_{+}}$ being the $i^{th}$ row of $P$ .",
"Assume that $\\lim _{n \\rightarrow \\infty } \\prod _{i = 1}^n \\left(1 - p_{i1} \\right) = 0$ .",
"Call a random permutation $\\Pi $ of $\\mathbb {N}_{+}$ a $P$ -shifted permutation of $\\mathbb {N}_{+}$ if $\\Pi $ has the distribution defined by the following construction from the independent sample $(X_i)_{i \\ge 1}$ , with $X_i$ distributed as $p^i$ .",
"Inductively, let $\\Pi _1 := X_1$ , and for $i \\ge 2$ , let $\\Pi _i := \\psi (X_i)$ where $\\psi $ is the increasing bijection from $\\mathbb {N}_{+}$ to $\\mathbb {N}_{+} \\setminus \\lbrace \\Pi _1, \\cdots , \\Pi _{i-1} \\rbrace $ .",
"For example, if $X_1 = 2$ , $X_2 = 1$ , $X_3 = 2$ , $X_4 = 3$ , $X_5 = 4$ , $X_6 = 1 \\ldots $ , then the associated permutation is $(2,1,4,6,8,3,\\ldots )$ .",
"Now the aforementioned infinite ranking models are subcases of the $P$ -shifted permutations.",
"[itemsep = 3 pt] If $p^i = p$ with $p_1 > 0$ for all $i$ , then we get the $p$ -shifted permutation.",
"If $p^i = \\mbox{Geo}(1 - e^{-\\theta })$ on $\\mathbb {N}_{+}$ for all $i$ , then we get the single parameter IGM model, or infinite $e^{-\\theta }$ -shuffle, i.e.",
"$\\Pi \\stackrel{d}{=} \\pi \\circ \\pi _0^{-1}$ for $\\pi $ distributed according to (REF ).",
"If $p^i = \\mbox{Geo}(1 - e^{-\\theta _i})$ on $\\mathbb {N}_{+}$ for each $i$ , then $\\Pi \\stackrel{d}{=} \\pi \\circ \\pi _0^{-1}$ for $\\pi $ an infinite version of the IGM model (REF )."
],
[
"`$t$ ' Selection Algorithm",
"We present an algorithm to select the model size `$t$ ' automatically for the top-$t$ IGM models.",
"The heuristic comes from the renewal property of the single parameter IGM model (REF ).",
"We need the following vocabulary.",
"Let $\\Pi $ be a permutation of $\\mathbb {N}_{+}$ .",
"Call $n \\in \\mathbb {N}_{+}$ a splitting time of $\\Pi $ if $\\Pi $ maps $[1,n]$ onto itself.",
"The set of splitting times of $\\Pi $ is the collection of finite right endpoints of some finite or infinite family of components of $\\Pi $ , say $\\lbrace I_j\\rbrace $ .",
"So $\\Pi $ does not act as a permutation on any proper subinterval of $I_j$ .",
"These components $I_j$ form a partition of $\\mathbb {N}_{+}$ , which is coarser than the partition by cycles of $\\Pi $ .",
"For example, the permutation $\\pi = (1)(2,4)(3) \\in \\mathfrak {S}_4$ induces the partition by components $[1] [2,3,4]$ .",
"The idea is to use the single parameter IGM model to preselect `$t$ ', which hinges on the renewal property of the latter.",
"Then we proceed to train a top-$t$ ranking model.",
"[46] proved that for $p = (p_1, p_2, \\ldots )$ a discrete distribution with $p_1 > 0$ and $\\sum _{i \\ge 1} ip_i < \\infty $ , a $p$ -shifted permutation $\\Pi $ is a concatenation of independent and identically distributed (i.i.d.)",
"components.",
"That is, $\\Pi $ is characterized by $L$ a distribution on $\\mathbb {N}_{+}$ , and $(Q_n)_{n \\ge 1}$ a sequence of distributions on indecomposable permutations such that [itemsep = 3 pt] the lengths of components $(L_i)_{i \\ge 1}$ are i.i.d.",
"as $L$ , given the length of a component $L_i = n_i$ , the reduced component defined via conjugation of $\\Pi $ by the shift from the component to $[n_i]$ is distributed as $Q_{n_i}$ .",
"To illustrate, $(\\, \\underbrace{2, \\, 3, \\, 4,\\, 1}_{L_1 = 4}, \\, \\underbrace{6,\\, 8, \\, 7, \\,10, \\,5, \\, 9}_{L_2 = 6}, \\, \\underbrace{12, \\, 13, \\, 11}_{L_3 = 3}, \\ldots )$ Moreover, the probability generating function of $L$ is given by $F(z) = 1 - \\frac{1}{1 + \\sum _{n = 1}^{\\infty } u_n z^n}$ , with $u_n: = \\prod _{i = 1}^n \\sum _{j = 1}^i p_i$ .",
"Specializing this renewal construction to the single parameter IGM model, we get the following proposition which is the foundation of Algorithm REF described right after.",
"Proposition 2.1 Let $\\Pi $ be a random permutation of $\\mathbb {N}_{+}$ distributed as $\\mathbb {P}_{\\theta , id}$ defined by (REF ).",
"Let $L$ be the common distribution of lengths of components of $\\Pi $ .",
"Then $\\mathbb {E}L = \\frac{1}{(e^{-\\theta }; e^{-\\theta })_{\\infty }},$ where $(a; q)_{\\infty }: = \\prod _{k = 0}^{\\infty } (1 - a q^{k})$ is the Q-Pochhammer function.",
"Figure: Plot of θ→1/(e -θ ;e -θ ) ∞ \\theta \\rightarrow 1/(e^{-\\theta }; e^{-\\theta })_{\\infty } for θ∈[0,5]\\theta \\in [0,5].Pitman-Tang's theory indicates that for the single parameter model (REF ), the first component of $\\pi \\circ \\pi _0^{-1}$ has length $L$ whose expectation is $1/(e^{-\\theta }; e^{-\\theta })_{\\infty }$ .",
"Given the dispersion parameter $\\theta $ , a complete permutation is expected to occur at some place close to $1/(e^{-\\theta }; e^{-\\theta })_{\\infty }$ .",
"The latter can be regarded as the effective length of the random infinite permutation, which suggests a natural candidate for `$t$ ' in top-$t$ ranking models.",
"Since $\\theta $ is unknown, we would like to find the `$t$ ' closest to the effective length.",
"Given $t$ in a suitable range $\\mathbb {T}$ , we fit the single parameter IGM model (REF ) to get the MLE $\\widehat{\\theta }(t)$ by the algorithms in [41].",
"Those algorithms also work for partially ranked data.",
"Then we search for $t : = \\min \\left\\lbrace \\operatorname{argmin}_{s \\in \\mathbb {T}} \\left|s - \\frac{1}{(e^{-\\widehat{\\theta }(s)}; e^{-\\widehat{\\theta }(s)})_{\\infty }}\\right|, \\lambda t_{\\max } \\right\\rbrace ,$ where $\\mathbb {T}$ is the range of search for model sizes, $\\lambda \\in (0,1)$ is a user-defined cutoff fraction to avoid overfitting, and $t_{\\max }$ is the maximum length of permutations in the data.",
"Practically, one starts with `$s$ ' of small values to narrow down the choices for the effective length.",
"That way, one only needs to search for a small proportion, not the full range of $[t_{\\min }, t_{\\max }]$ .",
"With `$t$ ' selected according to (REF ), we can then fit the IGM model (REF ) by means of MLE.",
"[tb] `$t$ ' selection algorithm $\\theta _0 \\leftarrow \\mbox{MB}(1)$ (Run Meilǎ-Bao's algorithm) Choose $\\mathbb {T} \\ni 1/(e^{-\\theta _0}; e^{-\\theta _0})_{\\infty }$ of a small range Initialize $Err \\leftarrow \\infty , \\,\\, t\\_\\mbox{SEL} \\leftarrow 0$ $t$ in $\\mathbb {T}$ $\\theta \\leftarrow \\mbox{MB}(t)$ (Run Meilǎ-Bao's algorithm) $|t - 1/(e^{-\\theta }; e^{-\\theta })_{\\infty }| < Err$ $Err \\leftarrow |t - 1/(e^{-\\theta }; e^{-\\theta })_{\\infty }|$ $ t\\_\\mbox{SEL} \\leftarrow t$ return $\\min (t\\_\\mbox{SEL}, \\lambda t_{\\max })$"
],
[
"Statistical Properties of the MLE",
"In this section we provide statistical analysis to the MLE of Mallows models."
],
[
"Main Theorems",
"[44] proved that the MLE $\\widehat{\\theta }$ for the general Mallows model (REF ) is consistent if $\\pi _0$ is known.",
"His approach relies on the concept of permutons [23].",
"The following result shows that the MLE $\\widehat{\\theta }$ is always biased upwards for the Mallows' $\\phi $ model.",
"Theorem 3.1 (Bias of $\\widehat{\\theta }$ ) Let $\\mathbb {P}_{\\theta , \\pi _0}$ be defined by (REF ), and $\\widehat{\\theta }$ be the MLE of $\\theta $ with $N$ samples.",
"Then for each $N \\ge 1$ , $\\mathbb {E}_{\\theta , \\pi _0} \\widehat{\\theta } > \\theta .$ The analysis of the MLE $\\widehat{\\pi }_0$ is more subtle since it lives in a discrete space.",
"By general results of [45], [11], one can prove the consistency of $\\widehat{\\pi }_0$ .",
"Here we establish a concentration bound of $\\widehat{\\pi }_0$ at $\\pi _0$ from which the consistency is straightforward.",
"This concentration bound also gives a confidence interval for the central ranking in the Mallows' $\\phi $ model.",
"Theorem 3.2 (Convergence rate of $\\widehat{\\pi }_0$ ) Let $\\mathbb {P}_{\\theta , \\pi _0}$ be defined by (REF ), and $\\widehat{\\pi }_0$ be the MLE of $\\pi _0$ with $N$ samples.",
"Then for $N$ large enough, $\\mathbb {P}_{\\theta , \\pi _0}(\\widehat{\\pi }_0 \\ne \\pi _0) \\ge \\frac{1}{1-e^{-\\theta }} \\sqrt{\\frac{2}{\\pi N}} \\left(\\cosh \\frac{\\theta }{2} \\right)^{-N},$ and $\\mathbb {P}_{\\theta , \\pi _0}(\\widehat{\\pi }_0 \\ne \\pi _0) \\le ( n - H_n) n!",
"\\left(\\cosh \\frac{\\theta }{2} \\right)^{-N},$ where $H_n : = \\sum _{i = 1}^n \\frac{1}{i}$ is the harmonic sum.",
"To the best of our knowledge, Theorems REF and REF are new.",
"Their proof will be given in the next two subsections.",
"Similar to Theorem REF , the MLE of $\\theta $ is biased upwards for the single parameter IGM model, confirming an observation in [41].",
"Theorem 3.3 (Bias of $\\widehat{\\theta }$ ) Let $\\mathbb {P}_{\\theta , \\pi _0}$ be the distribution of the single parameter IGM model, and $\\widehat{\\theta }$ be the MLE of $\\theta $ with $N$ samples.",
"Then for each $N \\ge 1$ , $\\mathbb {E}_{\\theta , \\pi _0} \\widehat{\\theta } > \\theta .$ Let us mention a few open problems.",
"The rate of $\\widehat{\\theta }$ is open for both the Mallows' $\\phi $ model and the single parameter IGM model.",
"We believe that the rate is of order $1/\\sqrt{N}$ for the Mallows' $\\phi $ model by applying the delta method with a finer variance analysis.",
"We can ask the same questions for the generalized Mallows model (GM$_{\\vec{\\theta }, \\pi _0}$ ).",
"It is expected that the rate of $\\widehat{\\pi }$ is of order $e^{-N \\beta (\\vec{\\theta })}$ but the explicit formula of $\\beta (\\vec{\\theta })$ is still missing.",
"Deriving a bound of $\\beta (\\vec{\\theta })$ would also be interesting.",
"The convergence rate of $\\widehat{\\pi }_0$ for the single parameter IGM model seems to be difficult, since the size of permutations goes to infinity.",
"We leave the analog of Theorem REF open."
],
[
"Proof of Theorem ",
"We first consider the case where the central ranking $\\pi _0$ is known.",
"Assume w.l.o.g.",
"that $\\pi _0 = id$ by suitably relabelling the items.",
"Then the model (REF ) simplifies to $\\mathbb {P}_{\\theta ,id}(\\pi ) = \\exp \\left(-\\theta \\operatorname{inv}(\\pi ) - \\ln \\Psi (\\theta )\\right).$ Given $N$ samples $(\\pi _i)_{1 \\le i \\le N}$ , the MLE $\\widehat{\\theta }$ is the solution to the following equation: $-\\frac{\\Psi ^{\\prime }(\\theta )}{\\Psi (\\theta )} = \\frac{1}{N} \\sum _{i=1}^N \\operatorname{inv}(\\pi _i).$ Now by writing $\\Psi (\\theta ) = f(e^{-\\theta })$ with $f(q):=\\sum _{\\pi \\in \\mathfrak {S}_n} q^{\\operatorname{inv}(\\pi )}$ , we have $-\\frac{\\Psi ^{\\prime }(\\theta )}{\\Psi (\\theta )} = g(e^{-\\theta }), \\quad \\mbox{with} \\quad g(q):=\\frac{q f^{\\prime }(q)}{f(q)}.$ So $\\widehat{\\theta } = - \\log g^{-1}(\\frac{1}{N} \\sum _{i=1}^N \\operatorname{inv}(\\pi _i))$ .",
"The function $f(q)$ is known as the $q$ -factorial [50].",
"We have $g(q) = q \\sum _{k = 1}^{n-1} \\frac{1-(k+1)q^k + kq^{k+1}}{(1-q)(1-q^{k+1})}.$ As observed by [37], [22], for $\\pi $ distributed according to (REF ), the number of inversions has the same distribution as a sum of independent truncated geometric random variables.",
"That is, $\\operatorname{inv}(\\pi ) \\stackrel{d}{=} G_{e^{-\\theta },1} + \\cdots + G_{e^{-\\theta },n} \\quad \\mbox{for } \\pi \\sim \\mathbb {P}_{\\theta , id},$ where $\\mathbb {P}(G_{q,k} = i) = q^i(1-q)/(1-q^k)$ for $i \\in \\lbrace 0,1,\\ldots , k-1\\rbrace $ .",
"Thus, $\\mathbb {E}\\operatorname{inv}(\\pi ) = g(e^{-\\theta })$ .",
"By elementary analysis, $\\theta \\mapsto g(e^{-\\theta })$ is strictly convex and decreasing.",
"So its inverse function $q \\mapsto -\\log g^{-1}(q)$ is strictly convex.",
"By Jensen's inequality, $- \\mathbb {E} \\log g^{-1}\\left( \\frac{1}{N} \\sum _{i=1}^N \\operatorname{inv}(\\pi _i) \\right)> - \\log g^{-1}\\left( \\mathbb {E}\\left( \\frac{1}{N} \\sum _{i=1}^N \\operatorname{inv}(\\pi _i) \\right) \\right)$ , which implies that $\\mathbb {E} \\widehat{\\theta } > \\theta $ .",
"Now consider the case where the central ranking $\\pi _0$ is unknown.",
"The MLE $\\widehat{\\theta }$ is given by $\\widehat{\\theta } = - \\log g^{-1} \\left(\\frac{1}{N} \\sum _{i = 1}^N \\operatorname{inv}(\\pi _i \\circ \\widehat{\\pi }_0^{-1}) \\right),$ where $g$ is defined as in (REF ), and $\\widehat{\\pi }_0$ is the MLE of $\\pi _0$ .",
"By the definition of $\\widehat{\\pi }_0$ , $\\sum _{i = 1}^N \\operatorname{inv}(\\pi _i \\circ \\widehat{\\pi }_0^{-1}) \\le \\sum _{i = 1}^N \\operatorname{inv}(\\pi _i \\circ \\pi _0^{-1}).$ Since $q \\mapsto -\\log g^{-1}(q)$ is strictly convex and decreasing, we get $\\mathbb {E}\\widehat{\\theta } & = - \\mathbb {E} \\log g^{-1}\\left(\\frac{1}{N} \\sum _{i = 1}^N \\operatorname{inv}(\\pi _i \\circ \\widehat{\\pi }_0^{-1}) \\right) \\\\&\\ge - \\mathbb {E} \\log g^{-1}\\left(\\frac{1}{N} \\sum _{i = 1}^N \\operatorname{inv}(\\pi _i \\circ \\pi _0^{-1}) \\right) \\\\& > - \\log g^{-1}\\left( \\mathbb {E}\\left( \\frac{1}{N} \\sum _{i=1}^N \\operatorname{inv}(\\pi _i \\circ \\pi _0^{-1}) \\right) \\right) = \\theta ,$ where the last equality follows from the fact that $\\pi _i \\circ \\pi _0^{-1}$ is distributed according to (REF ) for each $i$ ."
],
[
"Proof of Theorem ",
"Assume w.l.o.g.",
"that the true central ranking $\\pi _0 = id$ .",
"We aim to find the bounds of $\\mathbb {P}_{\\theta ,id}(\\widehat{\\pi }_0 \\ne id)$ given the dispersion parameter $\\theta $ .",
"For $\\pi , \\pi ^{\\prime } \\in \\mathfrak {S}_n$ , we say that $\\pi $ is more likely than $\\pi ^{\\prime }$ , denoted $\\pi \\succeq \\pi ^{\\prime }$ , if $\\sum _{i = 1}^N \\operatorname{inv}(\\pi _i \\circ \\pi ^{-1}) \\le \\sum _{i = 1}^N \\operatorname{inv}(\\pi _i \\circ \\pi ^{\\prime -1})$ .",
"For $1 \\le j < k \\le n$ , let $(j,k)$ be the transposition of $j$ and $k$ .",
"By the union bound, we get $\\mathbb {P}_{\\theta ,id}((1,2) \\succeq id) \\le \\mathbb {P}_{\\theta ,id}(\\widehat{\\pi }_0 \\ne id) \\le \\sum _{\\pi \\in \\mathfrak {S}_n} \\mathbb {P}_{\\theta ,id}(\\pi \\succeq id).$ Lower bound: For any permutation $\\pi \\in \\mathfrak {S}_n$ , if $\\pi (1) > \\pi (2)$ , then $\\operatorname{inv}(\\pi \\circ (1,2)) = \\operatorname{inv}(\\pi ) - 1$ , and if $\\pi (1) < \\pi (2)$ , then $\\operatorname{inv}(\\pi \\circ (1,2)) = \\operatorname{inv}(\\pi ) + 1$ .",
"Thus, $\\sum _{i = 1}^N \\operatorname{inv}(\\pi _i \\circ (1,2))$ equals to $\\sum _{i = 1}^N \\operatorname{inv}(\\pi _i) + \\#\\lbrace i: \\pi _i(1) < \\pi _i(2) \\rbrace - \\#\\lbrace i: \\pi _i(1) >\\pi _i(2) \\rbrace .$ Consequently, $\\mathbb {P}_{\\theta ,id}((1,2) \\succeq id)$ is given by $& \\quad ~ \\mathbb {P}_{\\theta ,id}\\Bigg ( \\#\\lbrace i: \\pi _i(1) < \\pi _i(2) \\rbrace \\le \\#\\lbrace i: \\pi _i(1) >\\pi _i(2) \\rbrace \\Bigg ) \\\\& = \\mathbb {P}_{\\theta ,id}\\left( \\#\\lbrace i: \\pi _i(1) >\\pi _i(2)\\rbrace \\ge \\frac{N}{2} \\right).$ Note that $\\mathbb {P}_{\\theta ,id}(\\pi _i(1) > \\pi _i(2)) = e^{-\\theta } \\mathbb {P}_{\\theta ,id}(\\pi _i(1) < \\pi _i(2))$ which implies that $\\mathbb {P}_{\\theta ,id}(\\pi _i(1) > \\pi _i(2)) = \\frac{1}{1 + e^{\\theta }}.$ Combining (REF ) and (REF ) yields $\\begin{aligned}\\mathbb {P}_{\\theta ,id}((1,2) \\succeq id) &= \\mathbb {P}\\left(\\operatorname{Bin}\\left(N, \\frac{1}{1+e^{\\theta }}\\right) \\ge \\frac{N}{2} \\right) \\\\& \\sim \\frac{1}{1- e^{-\\theta }} \\sqrt{\\frac{2}{\\pi N}} \\left( \\cosh \\frac{\\theta }{2}\\right)^{-N},\\end{aligned}$ where $\\operatorname{Bin}(N, p)$ is a binomial random variable with parameters $(N, p)$ , and the last estimate follows from the large deviation bound [2] that for $p<a$ , $\\mathbb {P}(\\operatorname{Bin}(N, p) > a N) \\sim \\frac{(1-p)\\sqrt{a}}{(a-p)\\sqrt{2 \\pi (1-a)N}} e^{-N H(a, p)},$ where $H(a,p) := a \\log \\left(\\frac{a}{p} \\right) + (1- a) \\log \\left(\\frac{1- a}{1- p} \\right)$ is the relative entropy between $\\operatorname{Bin}(N, p)$ and $\\operatorname{Bin}(N, a)$ .",
"Upper bound: We need the following comparison result.",
"Proposition 3.4 For $j,k \\in \\lbrace 1, \\ldots , n\\rbrace $ and $j < k$ , we have $\\mathbb {P}_{\\theta ,id}((j,k) \\succeq id) \\le \\left(\\cosh \\frac{\\theta }{2} \\right)^{-N}.$ Now decompose each permutation $\\pi \\in \\mathfrak {S}_n$ into a product of transpositions, say $\\pi = \\sigma _1 \\circ \\cdots \\circ \\sigma _k$ .",
"We have $& \\mathbb {P}_{\\theta , id}(\\pi \\succeq id) \\\\& \\qquad \\le \\mathbb {P}_{\\theta , id} (\\exists i \\in [k]: \\sigma _1 \\circ \\cdots \\circ \\sigma _i \\succeq \\sigma _1 \\circ \\cdots \\circ \\sigma _{i-1}) \\\\&\\qquad \\le \\#_{trans}(\\pi ) \\left(\\cosh \\frac{\\theta }{2} \\right)^{-N}, $ where $\\#_{trans}(\\pi )$ is the number of transpositions in $\\pi $ .",
"For any permutation $\\pi \\in \\mathfrak {S}_n$ , $\\#_{trans}(\\pi ) = n - \\#_{cyc}(\\pi )$ with $\\#_{cyc}(\\pi )$ the number of cycles in $\\pi $ .",
"The upper bound (REF ) follows from (REF ), the union bound and the fact that $\\sum _{\\pi \\in \\mathfrak {S}_n} \\#_{cyc}(\\pi ) = n!",
"H_n$ .",
"[Proof of Proposition REF ] A similar argument as before shows that $\\mathbb {P}((j, k) \\succeq id) = \\mathbb {P}((1,2) \\succeq id) \\quad \\mbox{for } k - j =1.$ Now let $\\ell := k - j \\in \\lbrace 1, \\ldots , n-1\\rbrace $ .",
"It is not hard to see that for any permutation $\\pi \\in \\mathfrak {S}_n$ , the number of inversions $\\operatorname{inv}(\\pi \\circ (j, k))$ can take $2 \\ell $ values: $\\operatorname{inv}(\\pi \\circ (j, k)) \\in \\lbrace \\operatorname{inv}(\\pi ) \\pm m: m = 1,3, \\ldots , 2 \\ell -1 ) \\rbrace .$ For $m \\in \\lbrace -2 \\ell + 1, -2 \\ell + 3, \\ldots , 2 \\ell -3, 2 \\ell -1\\rbrace $ , let $p_m: = \\mathbb {P}_{\\theta ,id}\\Bigg (\\operatorname{inv}(\\pi \\circ (j, k)) = \\operatorname{inv}(\\pi ) + m\\Bigg ).$ Observe that $\\mathbb {P}_{\\theta ,id}((j,k) \\succeq id)$ is given by $& \\quad ~ \\mathbb {P}_{\\theta ,id}\\left( \\sum _{i=1}^N \\Bigg ( \\operatorname{inv}(\\pi _i \\circ (j,k)) - \\operatorname{inv}(\\pi _i) \\Bigg )\\le 0 \\right) \\\\& = \\mathbb {P}\\left(\\sum _{i = 1}^N Z_i \\le 0 \\right),$ where $Z_i$ are i.i.d.",
"categorical random variables such that $Z_i = m$ with probability $p_m$ for $m \\in \\lbrace -2 \\ell + 1, -2 \\ell + 3, \\ldots , 2 \\ell -3, 2 \\ell -1\\rbrace $ .",
"By Cramer's theorem [16], $\\mathbb {P}\\left(\\sum _{i = 1}^N Z_i \\le 0 \\right) \\le \\exp \\left(-N \\sup _{\\lambda } \\lbrace - \\log F(\\lambda )\\rbrace \\right),$ where $F(\\lambda ): = \\sum _{m = -2 \\ell + 1, \\, m \\, \\mbox{\\tiny odd}}^{2 \\ell -1} p_m e^{\\lambda m},$ is the moment generating function of $Z_i$ .",
"Note that for $m > 0$ , we have $p_{-m} = e^{\\theta m} p_m$ .",
"Therefore, $\\sum _{m = 1, \\, m \\, \\mbox{\\tiny odd}}^{2 \\ell -1} p_m (1+e^{\\theta m}) = 1$ .",
"Moreover, $\\begin{aligned}\\sum _{m = 1, \\, m \\, \\mbox{\\tiny odd}}^{2 \\ell -1} p_m (1+e^{\\theta m}) &= \\sum _{m = 1, \\, m \\, \\mbox{\\tiny odd}}^{2 \\ell -1} p_m e^{\\frac{\\theta m}{2}}(e^{\\frac{\\theta m}{2}}+e^{-\\frac{\\theta m}{2}}) \\\\& \\ge \\sum _{m = 1, \\, m \\, \\mbox{\\tiny odd}}^{2 \\ell -1} p_m e^{\\frac{\\theta m}{2}}(e^{\\frac{\\theta }{2}}+e^{-\\frac{\\theta }{2}}),\\end{aligned}$ which yields $\\sum _{m = 1, \\, m \\, \\mbox{\\tiny odd}}^{2 \\ell -1} p_m e^{\\frac{\\theta m}{2}} \\le \\frac{1}{2} (\\cosh \\frac{\\theta }{2})^{-1}$ .",
"As a consequence, $\\begin{aligned}\\sup _{\\lambda } \\lbrace - \\log F(\\lambda )\\rbrace & \\ge - \\log F\\left(\\frac{\\theta }{2} \\right) \\\\& = - \\log \\sum _{m = 1, \\, m \\, \\mbox{\\tiny odd}}^{2 \\ell -1} 2 p_m e^{\\frac{\\theta m}{2}} \\\\& \\ge - \\log \\left(\\cosh \\frac{\\theta }{2} \\right)^{-1}.\\end{aligned}$ By (REF ), (REF ) and (REF ), we get the estimate (REF )."
],
[
"Experimental Results",
"In this section we apply Algorithm REF to provide experimental results on synthetic data and two real-world data: APA election data (large $N$ , small $t_{max}$ ), and University's homepage search (small $N$ , large $t_{max}$ )."
],
[
"Synthetic Data",
"We generate 50 sets of $N = 1000$ rankings from the IGM model (REF ) with $\\vec{\\theta } = (1,0.9,0.8,0.7,0.6,0.5, 0, \\ldots )$ and $\\pi _0 = id$ .",
"We restrict all observed rankings to the first $t_{\\max } = 6$ components.",
"Table REF displays the percentage of the model sizes selected by Algorithm REF .",
"Table: Percentage of model sizes selected for 50 simulated data sets.Now we fit the IGM model by MLE with the preselected model size $t$ .",
"The estimate $\\widehat{\\theta }_1$ lies in the range $[0.94, 1.08]$ with mean $1.0$ and standard deviation $0.03$ , and $\\widehat{\\theta }_2$ in the range $[0.84, 0.95]$ with mean $0.9$ and standard deviation $0.02$ .",
"Moreover, the estimated central rankings restricted to the top 6 ranks are always $(1|2|3|4|5|6)$ .",
"We also generate 50 sets of $N = 1000$ rankings from the IGM model restricted to the first $t_{\\max } = 10, 20$ and 40 components.",
"Table REF shows the accuracy of estimated ranking and average training time by the IGM model of model size $t = 1$ , $t = 10$ and Algorithm REF .",
"For rankings of small length, the IGM model of small size gives good accuracy and uses less training time.",
"Algorithm REF is more appealing for rankings of large length.",
"It has better accuracy than the IGM model of small size, and is less time-consuming than the IGM model of large size.",
"Table: Accuracy of estimated rank &\\& average training time for 50 simulated data with t max =10t_{max} = 10 (resp.",
"t max =20t_{max} = 20, t max =40t_{max} = 40) and θ →=(1,0.975,...,0.775,0,...)\\vec{\\theta } = (1, 0.975, \\ldots , 0.775, 0, \\ldots ) (resp.",
"θ →=(1,0.975,...,0.525,0,...)\\vec{\\theta } = (1, 0.975, \\ldots , 0.525, 0 , \\ldots ), θ →=(1,0.975,...,0.025,0,...)\\vec{\\theta } = (1, 0.975, \\ldots , 0.025, 0 , \\ldots ))by the IGM model of model size t=1t = 1, t=10t = 10 and Algorithm ."
],
[
"APA Data",
"We consider the problem of ranking with a small number of items and large sample size.",
"The data consists of $N = 15449$ rankings over $t_{max} = 5$ candidates during the American Psychological Association's presidential election in 1980.",
"Among these rankings, there are only 5738 complete rankings and the number of distinct rankings is $n = 207$ .",
"See [13], [18] for further background.",
"Table REF displays the estimate of $\\theta $ for the single parameter IGM with different model sizes.",
"Applying Algorithm 1 yields the selection of $t = 2$ .",
"Then by fitting the IGM model with $t = 2$ , we get $\\widehat{\\theta }_1 = 0.46$ , $\\widehat{\\theta }_2 = 0.54$ and $\\widehat{\\pi }_0 = (3|1|5|4|2)$ .",
"By the second order analysis, [18] argued that there is a strong effect of choosing candidates $\\lbrace 1,3\\rbrace $ and $\\lbrace 4,5\\rbrace $ , with candidate $\\lbrace 2\\rbrace $ in the middle.",
"Our result suggests that the pair of candidates $\\lbrace 1,3\\rbrace $ be in a more favorable position.",
"Table: Estimates of θ\\theta for the single parameter IGM model with t∈[1,5]t \\in [1,5]."
],
[
"University's Homepage Search",
"We consider the problem of learning a domain-specific search engine with the data collected by [12].",
"The data consists of 157 universities, the queries, and 21 search engines, the experts.",
"Each expert search engine outputs a ranked list of up to $t_{max} = 30$ URLs when queried with the university's name.",
"The target output is the university's homepage.",
"There are 10 universities without data, and some search engines return empty list.",
"So there are 147 ranking problems with sample size $N \\le 21$ , and each sample has length ranging from 1 to 30.",
"For each query, we fit the IGM model by MLE with Algorithm REF to calculate the rank of the university's homepage under the estimated central ranking $\\widehat{\\pi }_0$ .",
"This rank measures the correctness of the model.",
"If the target homepage is not among the URLs returned by the search engines, we put it to the end of the list.",
"The central ranking is estimated by the SORTR heuristic [41].",
"Table REF is an extract of the estimated rank of the target homepages.",
"Table: A list of 10 universities and their estimated ranks.Our training model is slightly different from that in [41], since they used the parameterization $\\vec{\\theta }= (\\theta _1, \\ldots , \\theta _{t-1}, \\theta _t, \\ldots .",
"\\theta _t, 0 , \\ldots )$ for the top-$t$ IGM model.",
"Figure REF provides the estimates of $\\theta $ for the single parameter IGM model with different model sizes.",
"By Algorithm REF , $68 \\%$ select $t = 5$ , $20 \\%$ select $t = 1$ , and $11 \\%$ select $t = 2$ over all 147 queries.",
"We also computed the rank of the homepage for each query and each model size $t$ .",
"Table REF summarizes the mean and the median of the target rank for these models.",
"Figure: Estimates of θ\\theta for the single parameter IGM model with t∈{1,2,3,6,10,30}t \\in \\lbrace 1,2,3,6,10,30\\rbrace .Table: Mean and median rank of the target homepage under the IGM models.",
"[41] got a mean rank around 15 and a median rank around 10.",
"Compared to their results, our experiments give better target rank for small model sizes.",
"This is reasonable because for each query there are only $N \\le 21$ samples and large model sizes may cause overfitting.",
"Small median ranks in Table REF also supports the validity of the IGM model."
],
[
"Conclusion",
"In this paper we study various Mallows ranking models.",
"The parameters of interest are the dispersion $\\theta $ (or $\\vec{\\theta }$ ), and the central ranking $\\pi _0$ .",
"Though there have been efficient algorithms to estimate the MLE $(\\widehat{\\theta }, \\widehat{\\pi }_0)$ , not much is known about the statistical properties of $(\\widehat{\\theta }, \\widehat{\\pi }_0)$ except the consistency of $\\widehat{\\theta }$ .",
"Aiming to fill this gap, we prove the biasedness and convergence rate of the MLE of the Mallows' $\\phi $ model and the single parameter IGM model.",
"To compare a large number of items, an infinite ranking model is often useful.",
"A natural infinite generalization of the finite Mallows model appeared in several contexts [22], [41].",
"But neither of them are aware of the other.",
"In this work we make a clear connection between these infinite rankings with further analysis.",
"Relying on a renewal property of the single parameter IGM model, we propose an algorithm to choose the model size `$t$ ' automatically.",
"The `$t$ ' selection algorithm is tested over synthetic and real data, and shows good performance."
]
] | 1808.08507 |
[
[
"Explicit solutions of certain orientable quadratic equations in free\n groups"
],
[
"Abstract For $g\\geq1$ denote by $F_{2g}=\\langle x_1, y_1,\\dots,x_g,y_g\\rangle$ the free group on $2g$ generators and by $B_g=[x_1,y_1]\\dots[x_g,y_g]$.",
"For $l,c\\geq 1$ and elements $w_1,\\dots,w_l\\in F_{2g}$ we study orientable quadratic equations of the form $[u_1,v_1]\\dots[u_h,v_h]=(B_g^{w_1})^c(B_g^{w_2})^c\\dots(B_g^{w_l})^c$ with unknowns $u_1,v_1,\\dots,u_h,v_h$ and provide explicit solutions for them for the minimal possible number $h$.",
"In the particular case when $g=1$, $w_i=y_1^{i-1}$ for $i=1,\\dots,l$ and $h$ the minimal number which satisfies $h \\geq l(c-1)/2+1$ we provide two types of solutions depending on the image of the subgroup $H=\\langle u_1,v_1,\\dots,u_h,v_h\\rangle$ generated by the solution under the natural homomorphism $p:F_2\\to F_2/[F_2,F_2]$: the first solution, which is called a primitive solution, satisfies $p(H)=F_2/[F_2,F_2]$, the second solution satisfies $p(H) = \\big\\langle p(x_1),p(y_1^l)\\big\\rangle$.",
"We also provide an explicit solution of the equation $[u_1,v_1]\\dots[u_k, v_k] = \\big(B_1\\big)^{k+l} \\big({B_1}^{y}\\big)^{k-l}$ for $k>l\\geq0$ in $F_2$, and prove that if $l\\neq0$, then every solution of this equation is primitive.",
"As a geometrical consequence, for every solution we obtain a map $f:S_h\\to T$ from the orientable surface $S_h$ of genus $h$ to the torus $T=S_1$ which has the minimal number of roots among all maps from the homotopy class of $f$.",
"Depending on the number $|p(F_2):p(H)|$ such maps have fundamentally different geometric properties: in some cases they satisfy the Wecken property and in other cases not."
],
[
"Introduction and preliminaries",
"Let $G$ be a group and $S$ be a symmetric subset of $G$ , i. e. a subset such that $1\\notin S$ and $S=S^{-1}$ .",
"For an element $a$ from $\\langle S\\rangle $ denote by $l_S(a)$ the minimal number $k$ such that $a$ is a product of $k$ elements from $S$ .",
"The number $l_S(a)$ is called the length of $a$ with respect to $S$ .",
"The numbers $l_S(a)$ and especially the value ${\\rm sup}\\big (l_S(a)~|~a\\in \\langle S\\rangle \\big )$ for different sets $S$ in different groups $G$ has been studied by various authors (see, for example, [22] and references therein).",
"If $S$ is the set of all nontrivial commutators in $G$ , then $l_S(a)$ is called the commutator length (or the genus) of an element $a\\in [G,G]$ .",
"The problem of determining the commutator length of an element $a\\in [G,G]$ corresponds to the problem of finding the minimal $h$ for which the equation $[u_1,v_1] \\dots [u_h, v_h]= a$ with unknowns $u_1,v_1,\\dots ,u_h,v_h$ admits a solution in $G$ .",
"Such equation is called an orientable quadratic equation.",
"The word “quadratic” means that every variable in the left side of the equation appears exactly twice.",
"The word “orientable” means that every unknown variable $x$ appears once in the form $x$ and once in the form $x^{-1}$ .",
"If there exists a variable $x$ in a quadratic equation which appears twice with exponent 1, then this equation is called non-orientable.",
"The problem of finding a solution of equation (REF ) in the free group is closely related with coincidence theory of maps between orientable surfaces, which includes the case of the study of roots.",
"See, for example, [8] and [3], [5], [12].",
"Many works concerning the problem of finding solutions of quadratic equations and especially of equation (REF ) in different groups have been done from both algebraic and geometric points of view.",
"For many years great attention was paid to equations in free groups [7], [8], [3], [5], [12], [13], [14], [15], [17], [18], [19], [20], [23], [24], [25].",
"The particular case of equation (REF ) when $h=1$ was studied in [25] (see also [11]).",
"For a given quadratic equation with any number of unknown variables in any free group with the right-hand side an arbitrary element an algorithm for solving the problem of the existence of a solution was given by Culler [7] using the surface method and generalizing the result of Wicks [25].",
"Based on different techniques, the problem has been studied by the first named author with coauthors [9], [10], [11] for parametric families of quadratic equations arising from continuous maps between closed surfaces.",
"The question about the existence of a solution of equation (REF ) can be solved in many cases.",
"However, the majority of results are either about non-existence of a solution, or about existence only and they do not provide an algorithm how to find an explicit solution.",
"The following result from [12] gives one simple necessary condition for solvability of equation (REF ) with the right-hand side of the special form motivated by geometry.",
"Proposition 1.1 Let $w_1,\\dots ,w_l$ be distinct elements of the free group $F_{2g}=\\langle x_1, y_1, \\dots ,x_g,y_g\\rangle $ and let $c_1,\\dots ,c_l$ be integers which are all positive or all negative.",
"Denote by $B_g=[x_1,y_1]\\dots [x_g,y_g]$ .",
"If the equation $[u_1,v_1] \\dots [u_h,v_h]= \\Big (B_g^{w_1}\\Big )^{c_1} \\Big (B_g^{w_2}\\Big )^{c_2} \\dots \\Big (B_g^{w_l}\\Big )^{c_l}$ with unknowns $u_1,v_1,\\dots ,u_h,v_h$ is solvable in $F_{2g}$ , then $(|c_1|+\\dots +|c_l|)(2g-1)\\le 2h-2+l$ .",
"Here and throughout the paper for elements $a,b$ we denote by $a^b=bab^{-1}$ the conjugate of $a$ by $b$ , and by $[a,b]=aba^{-1}b^{-1}$ the commutator of elements $a,b$ .",
"Orientable quadratic equations with the right-hand side as in (REF ) (not necessary for $c_1,\\dots ,c_l$ all positive or all negative) are of special interest in geometry.",
"If $f: S_h\\rightarrow S_g$ is a continuous map between orientable surfaces, then it induces a map $f_{\\#}: \\pi _1(S_h)\\rightarrow \\pi _1(S_g)$ between fundamental groups.",
"Denoting by $\\pi _1(S_h)=\\langle x_1,y_1,\\dots ,x_h,y_h~|~[x_1,y_1]\\dots [x_h,y_h]=1\\rangle $ and $f_{\\#}(x_i)=u_i$ , $f_{\\#}(y_i)=v_i$ we must have $[u_1,v_1]\\dots [u_h,v_h]=1$ in $\\pi _1(S_g)$ , i. e. $[u_1,v_1]\\dots [u_h,v_h]$ must be expressible as a right-side of (REF ).",
"So, there is a strong connection between maps between orientable surfaces and orientable quadratic equations with the right hand side as in (REF ).",
"Note that the result of Proposition REF says that if $(|c_1|+\\dots +|c_l|)(2g-1)> 2h-2+l$ , then equation (REF ) does not have a solution independently of elements $w_1,...,w_{l}$ .",
"However there is no guarantee that a solution exists for $(|c_1|+\\dots +|c_l|)(2g-1)\\le 2h-2+l$ .",
"Moreover a solution can exist for some elements $w_1,\\dots ,w_{l}$ but not for others.",
"For example, if $g=1$ , $l=2$ , $c_1=c_2=1$ , then $h=1$ .",
"Using Wicks criterion [25] it is easy to show that the equation $[u,v] = B_1 B_1^{y_1^2}$ has no solutions in $F_2=\\langle x_1,y_1\\rangle $ .",
"However, the equation $[u,v]=B_1 B_1^{y_1}$ has a solution $u=x_1$ , $v=y_1^2$ .",
"So, it is reasonable to ask for which integers $c_1,\\dots ,c_{l}$ and elements $w_1,\\dots ,w_{l}$ equation $(\\ref {eqpart})$ has a solution, and when it has, provide this solution.",
"Not much is known about this problem.",
"In the present work we consider equation (REF ) in the free group $F_{2g}$ with right parts of special forms and our goal is to provide explicit solutions for them.",
"In turn, for $g=1$ this will provide existence of maps from the orientable surface of genus $h$ into the torus which have some features about root theory.",
"For some cases we will find two types of solutions depending on the index of the image of the subgroup $H=\\langle u_1,v_1,\\dots ,u_h,v_h\\rangle $ generated by the solution under the natural homomorphism $p:F_{2g}\\rightarrow \\pi _1(S_g)=F_{2g}/\\langle B_g \\rangle ^{F_{2g}}$ in $\\pi _1(S_g)$ .",
"In order to explain the importance of the value $|\\pi _1(S_g):p(H)|$ let us recall some facts from Nielsen root theory.",
"Let $f:M_1\\rightarrow M_2$ be a continuous map between closed manifolds $M_1$ , $M_2$ and let $c\\in M_2$ .",
"Every element from $f^{-1}(c)$ is called a root.",
"The minimal number of roots in the homotopy class of a map $f$ is the number $MR[f]={\\rm min}_{g\\simeq f}\\big (|g^{-1}(c)|\\big )$ , where $\\simeq $ denotes the homotopy equivalence.",
"This number does not depend on $c$ .",
"Two roots $x,y\\in M_1$ are said to belong to the same Nielsen root class if there exists a path $\\gamma $ in $M_1$ connecting $x,y$ such that $f(\\gamma )$ is contractible.",
"For a map $f$ between two manifolds of the same dimension an index for a Nielsen root class is defined in [16].",
"A Nielsen root class is called essential if its index is not equal to zero.",
"The indices of all essential Nielsen root classes coincide.",
"The Nielsen root number $NR[f]$ is the number of essential Nielsen root classes, this number is always finite and it satisfies the inequality $NR[f]\\le MR[f]$ .",
"If $NR[f]=MR[f]$ , then $f$ is said to possess the Wecken property.",
"The map $f$ induces the map $f_{\\#}:\\pi _1(M_1)\\rightarrow \\pi _1(M_2)$ between fundamental groups.",
"Denote by $l(f)=|\\pi _1(M_2):f_{\\#}(\\pi _1(M_1))|$ if $|\\pi _1(M_2):f_{\\#}(\\pi _1(M_1))|$ is finite, and $l(f)=0$ otherwise.",
"If $M_1=S_h$ , $M_2=S_g$ are closed orientable surfaces of genus $h,g$ respectively, then the map $f$ induces a homomorphism beween second homology groups $\\mathbb {Z}=H_2(S_h)\\rightarrow H_2(S_g)=\\mathbb {Z}$ .",
"This map acts as a multiplication by some number $n$ .",
"This number is called the degree of $f$ and is denoted by ${\\rm deg}(f)$ .",
"The absolute value $|{\\rm deg}(f)|$ is denoted by $A(f)$ .",
"In [3] it is proved that, if $A(f)\\ne 0$ , then $MR[f]={\\rm max}\\Big (l(f), \\chi (M_1)+(1-\\chi (M_2))A(f)\\Big )&&NR[f]=l(f),$ where $\\chi $ denotes the Euler characteristic of the surface.",
"If $u_1,v_1,\\dots ,u_h,v_h$ is some solution of equation (REF ), then one can construct a continuous map $f:S_h\\rightarrow S_g$ which satisfies the following conditions: ${\\rm deg}(f)=c_1+\\dots +c_l$ , $|f^{-1}(y)|=l$ for some point $y\\in S_g$ , the index of every Nielsen root class of $f$ is equal to $c_{i_1}+\\dots +c_{i_k}$ for some indices $i_1,\\dots ,i_k$ and if $\\pi _1(S_h)=\\langle x_1,y_1,\\dots ,x_h,y_h~|~[x_1,y_1]\\dots [x_h,y_h]=1\\rangle $ , then $f_{\\#}(x_i)=u_i$ , $f_{\\#}(y_i)=v_i$ .",
"If ${\\rm deg}(f)=0$ , then $NR[f]=0$ and $|p(F_{2g}):p(H)|$ can be either finite or infinite.",
"If ${\\rm deg}(f)\\ne 0$ , then $|p(F_{2g}):p(H)|$ is finite and $NR[f]=|p(F_{2g}):p(H)|$ , where $p:F_{2g}\\rightarrow \\pi _1(S_g)$ is a natural homomorphism and $H=\\langle u_1,v_1,\\dots ,u_h,v_h\\rangle $ .",
"See details about the construction of $f$ in [12], here we are going to use only the properties of the constructed map.",
"The following result gives some information about the index of $p(H)$ in $\\pi _1(S_g)$ .",
"Proposition 1.2 Let $p:F_{2g} \\rightarrow \\pi _1(S_g)$ be the homomorphism which sends the free generators of $F_{2g}$ to the canonical system of generators of the fundamental group $\\pi _1(S_g)$ of an orientable surface $S_g$ of genus $g$ .",
"If $u_1,v_1,\\dots ,u_h,v_h$ is a solution of equation (REF ) with $c_1+c_2+...+c_{l}\\ne 0$ , then the index of $p\\big (\\langle u_1,v_1,\\dots ,u_h,v_h\\rangle \\big )$ in $\\pi _1(S_g)$ is less than or equal to $l$ .",
"Proof.",
"Let $f:S_h \\rightarrow S_g$ be the described before the proposition map constructed by the solution $u_1,v_1,\\dots ,u_h,v_h$ .",
"For some point $y\\in S_g$ the number of elements in the preimage of $y$ under $f$ is $l$ , therefore $MR[f]\\le l$ .",
"Since ${\\rm deg}(f)=c_1+\\dots +c_l\\ne 0$ , we have $|p(F_{2g}):p(H)|=NR[f]\\le MR[f]\\le l$ .$\\Box $ In the present paper we will find explicit solutions for particular cases of equation (REF ) which are in some sense “critical” from the point of view of Proposition REF : the first solution which is called a primitive solution satisfies the equality $p(H)=\\pi _1(S_g)$ (the word “primitive” appears here naturally from the notion of primitives in free groups [21]), and the second solution satisfies $|\\pi _1(S_g):p(H)|=l$ .",
"In Section we study the equation (REF ) for $c_1=c_2=\\dots =c_l=c$ and prove that if for $c=1$ this equation has a solution which generates a subgroup $H_1$ of $F_{2g}$ , then for every $c$ it has a solution which generates a subgroup $H_2$ such that $p(H_1)=p(H_2)$ (Theorem REF ).",
"In Section we consider the particular case of this equation in the free group $F_2=\\langle x,y\\rangle $ with the right part of the form $\\big ([x,y]\\big )^c \\big ({[x,y]}^y\\big )^c\\dots \\big ({[x,y]}^{y^{l-1}}\\big )^c$ for integers $c, l\\ge 1$ .",
"We provide an explicit algebraic algorithm for finding the solution in a minimal subgroup (Corollary REF ), and an algorithm for finding a primitive solution (Theorem REF ) for such equation.",
"In Section we consider equation (REF ) in the free group $F_2=\\langle x,y\\rangle $ with the right part of the form $\\big ([x,y]\\big )^{k+l} \\big ({[x,y]}^y\\big )^{k-l}$ for integers $k>l\\ge 0$ .",
"We construct an explicit primitive solution for such equation (formulas (REF ), (REF )) and prove that for $l\\ne 0$ every solution of such equation is primitive (Theorem REF ).",
"Some geometrical consequences derived from this algebraic results are formulated in Corollaries REF , REF .",
"Acknowledment: The first author would like to thank Prof. Richard Weidmann for many helpful discussions about the subject of the content of this work."
],
[
"The right part has the form $\\Big (B_g^{w_1}\\Big )^c\\Big (B_g^{w_2}\\Big )^c\\dots \\Big (B_g^{w_l}\\Big )^c$",
"The purpose of this section is to give an explicit solution for equation (REF ) in the particular case when $c_1=c_2=\\dots =c_l=c$ $[u_1,v_1]\\dots [u_h,v_h] = \\Big (B_g^{w_1}\\Big )^c \\Big ({B_g}^{w_2}\\Big )^c\\dots \\Big ({B_g}^{w_l}\\Big )^c$ for the minimal integer $h$ which satisfies the inequality $h \\ge l(c(2g-1)-1)/2+1$ .",
"We can assume that $c>0$ since if $c<0$ , then denoting by $x_i^{\\prime }=y_{g+1-i}$ , $y_i^{\\prime }=x_{g+1-i}$ for $i=1,\\dots ,g$ we have $F_{2g}=\\langle x_1^{\\prime }, y_1^{\\prime }, \\dots , x_g^{\\prime }, y_g^{\\prime }\\rangle $ , and in these generators equation (REF ) has the same form where $c$ is changed by $-c$ .",
"In the case when $h < l(c(2g-1)-1)/2+1$ by Proposition REF equation (REF ) does not have solutions.",
"At first, we need the following simple lemma.",
"Lemma 2.1 The word $w=a \\xi _1 b \\xi _2 c$ is a product of the commutator $[a \\xi _1 a^{-1}, aba^{-1}]$ and the element $ab\\xi _1\\xi _2 c$ .",
"Proof.",
"Straightforward calculation.",
"$\\Box $ The main result of this section is the following theorem.",
"Theorem 2.2 Let $l,c,g\\ge 1$ be integers, $h$ be the minimal integer which satisfies the inequality $h\\ge l\\big (c(2g-1)-1\\big )/2+1$ , $w_1,\\dots ,w_l$ be elements of the free group $F_{2g}=\\langle x_1, y_1, \\dots ,x_g,y_g\\rangle $ and $B_g=[x_1,y_1]\\dots [x_g,y_g]$ .",
"If for $c=1$ the equation $[u_1,v_1] \\dots [u_h,v_h]= \\Big (B_g^{w_1}\\Big )^{c} \\Big (B_g^{w_2}\\Big )^{c} \\dots \\Big (B_g^{w_l}\\Big )^{c}$ has a solution which generates the subgroup $H_1$ of $F_{2g}$ , then for an arbitrary $c\\ge 1$ it has a solution (explicitly constructed from the given solution for $c=1$ ) which generates the subgroup $H_2$ of $F_{2g}$ such that if $p:F_{2g}\\rightarrow \\pi _1(S_g)$ is the natural homomorphism, then $p(H_1)=p(H_2)$ .",
"Proof.",
"We will construct the solution inductively on the variable $c$ .",
"The basis of induction $c=1$ is given as the condition of the theorem.",
"Suppose that the statement is proved for $c=n$ and let us prove that it holds for $c=n+1$ .",
"We will consider two cases depending on the parity of $l$ .",
"Case 1: $l$ is even.",
"Rewrite the right-hand side of equation (REF ) for $c=n+1$ in the following form.",
"$\\big (B_g^{w_1}\\big )^{n+1} \\big (B_g^{w_2}\\big )^{n+1} \\dots \\big (B_g^{w_l}\\big )^{n+1}&=\\big (B_g^{w_1}\\big )\\Big (\\big (B_g^{w_1}\\big )^{n} \\big (B_g^{w_2}\\big )^{n} \\dots \\big (B_g^{w_l}\\big )^{n}\\Big )\\big (B_g^{w_1}\\big )^{-1}\\\\&\\cdot \\big (B_g^{w_1}\\big )\\big (B_g^{w_l}\\big )^{-n} \\big (B_g^{w_{l-1}}\\big )^{-n} \\dots \\big (B_g^{w_2}\\big )^{-n}\\\\&\\cdot \\big (B_g^{w_2}\\big )^{n+1} \\big (B_g^{w_3}\\big )^{n+1} \\dots \\big (B_g^{w_l}\\big )^{n+1}$ By the induction hypothesis there exist $u_1,v_1,\\dots ,u_h,v_h$ for $h=l\\big (n(2g-1)-1\\big )/2+1$ such that $[u_1,v_1] \\dots [u_h,v_h]= \\big (B_g^{w_1}\\big )^n\\big (B_g^{w_2}\\big )^n \\dots \\big (B_g^{w_l}\\big )^n$ and $p\\big (\\langle u_1,v_1,\\dots ,u_h,v_h\\rangle \\big )=p(H_1)$ .",
"So, it is enough to prove that the product of two last lines of equation (REF ) $\\big (B_g^{w_1}\\big )\\big (B_g^{w_l}\\big )^{-n} \\big (B_g^{w_{l-1}}\\big )^{-n} \\dots \\big (B_g^{w_3}\\big )^{-n}\\big (B_g^{w_2}\\big )\\big (B_g^{w_3}\\big )^{n+1} \\dots \\big (B_g^{w_l}\\big )^{n+1}$ is the product of $l\\big ((n+1)(2g-1)-1\\big )/2+1-l\\big (n(2g-1)-1\\big )/2-1=l(2g-1)/2$ commutators of elements images of which under $p$ belong to $p(H_1)$ .",
"In equation (REF ) denoting by $a=B_g^{w_1}$ , $\\xi _1=\\big (B_g^{w_l}\\big )^{-n}\\big (B_g^{w_{l-1}}\\big )^{-n}$ , $b=\\big (B_g^{w_{l-2}}\\big )^{-n}\\dots \\big (B_g^{w_3}\\big )^{-n}\\big (B_g^{w_2}\\big )\\big (B_g^{w_3}\\big )^{n+1}\\dots \\big (B_g^{w_{l-2}}\\big )^{n+1}\\big (B_g^{w_{l-1}}\\big )$ , $\\xi _2=\\big (B_g^{w_{l-1}}\\big )^{n}\\big (B_g^{w_l}\\big )^{n}$ , $c=\\big (B_g^{w_l}\\big )$ and applying Lemma REF we conclude that expression (REF ) is a product of the commutator of elements which belong to the kernel of $p$ times the element $\\big (B_g^{w_1}\\big )\\big (B_g^{w_{l-2}}\\big )^{-n} \\big (B_g^{w_{l-3}}\\big )^{-n} \\dots \\big (B_g^{w_3}\\big )^{-n}\\big (B_g^{w_2}\\big )\\big (B_g^{w_3}\\big )^{n+1} \\dots \\big (B_g^{w_{l-2}}\\big )^{n+1}\\big (B_g^{w_{l-1}}\\big )\\big (B_g^{w_l}\\big ).$ Repeating this idea denoting by $\\xi _1=\\big (B_g^{w_{l-2}}\\big )^{-n}\\big (B_g^{w_{l-3}}\\big )^{-n}$ , $\\xi _2=\\big (B_g^{w_{l-3}}\\big )^{n}\\big (B_g^{w_{l-2}}\\big )^{n}$ , we conclude that expression (REF ) is a product of two commutators of elements which belong to the kernel of $p$ times the element $\\big (B_g^{w_1}\\big )\\big (B_g^{w_{l-4}}\\big )^{-n} \\big (B_g^{w_{l-5}}\\big )^{-n} \\dots \\big (B_g^{w_3}\\big )^{-n}\\big (B_g^{w_2}\\big )\\big (B_g^{w_3}\\big )^{n+1} \\dots \\big (B_g^{w_{l-4}}\\big )^{n+1}\\big (B_g^{w_{l-3}}\\big )\\dots \\big (B_g^{w_l}\\big ).$ Repeating this procedure $(l-2)/2$ times we conclude that expression (REF ) is the product of $(l-2)/2$ commutators of elements which belong to the kernel of $p$ times the element $B_g^{w_1}B_g^{w_2} \\dots B_g^{w_l}$ which (by the induction hypothesis for $c=1$ ) is the product of $l(g-1)+1$ commutators of elements images of which under $p$ belong to $p(H_1)$ .",
"Therefore expression (REF ) is the product of $(l-2)/2+l(g-1)+1=l(2g-1)/2$ commutators of elements images of which under $p$ belong to $p(H_1)$ .",
"Case 2: $l$ is odd.",
"Rewrite the right-hand side of equation (REF ) for $c=n+1$ in the following form.",
"$\\big (B_g^{w_1}\\big )^{n+1} &\\big ({B_g}^{w_2}\\big )^{n+1}\\dots \\big ({B}_g^{w_l}\\big )^{n+1}=\\\\=&\\Big (\\big (B_g^{w_1}\\big ) \\big ({B_g}^{w_2}\\big )\\dots \\big ({B}_g^{w_l}\\big )\\Big )\\\\\\cdot &\\Big (\\big (B_g^{w_2}\\big ) \\big ({B_g}^{w_3}\\big )\\dots \\big ({B}_g^{w_l}\\big )\\Big )^{-1}\\Big (\\big (B_g^{w_1}\\big ) \\big ({B_g}^{w_2}\\big )\\dots \\big ({B}_g^{w_l}\\big )\\Big )\\Big (\\big (B_g^{w_2}\\big ) \\big ({B_g}^{w_3}\\big )\\dots \\big ({B}_g^{w_l}\\big )\\Big )\\\\\\cdot &\\Big (\\big (B_g^{w_2}\\big ) \\big ({B_g}^{w_3}\\big )\\dots \\big ({B}_g^{w_l}\\big )\\Big )^{-2}\\Big (\\big (B_g^{w_1}\\big )^{n-1}\\big ({B_g}^{w_2}\\big )^{n-1}\\dots \\big ({B}_g^{w_l}\\big )^{n-1}\\Big )\\Big (\\big (B_g^{w_2}\\big ) \\big ({B_g}^{w_3}\\big )\\dots \\big ({B}_g^{w_l}\\big )\\Big )^{2}\\\\\\cdot & \\big ({B_g}^{w_l}\\big )^{-1}\\dots \\big ({B_g}^{w_2}\\big )^{-1}\\big ({B_g}^{w_l}\\big )^{-1}\\dots \\big ({B_g}^{w_2}\\big )^{-1}\\\\\\cdot &\\big ({B_g}^{w_l}\\big )^{-n+1}\\dots \\big ({B_g}^{w_3}\\big )^{-n+1}\\big ({B_g}^{w_2}\\big )^{2}\\big ({B_g}^{w_3}\\big )^{n+1}\\dots \\big ({B_g}^{w_l}\\big )^{n+1}$ At first, we consider the particular case $n+1=2$ .",
"In this case equality (REF ) implies $\\big (B_g^{w_1}\\big )^{2}& \\big ({B_g}^{w_2}\\big )^{2}\\dots \\big ({B}_g^{w_l}\\big )^{2}=\\\\=&\\Big (\\big (B_g^{w_1}\\big ) \\big ({B_g}^{w_2}\\big )\\dots \\big ({B}_g^{w_l}\\big )\\Big )\\\\\\cdot &\\Big (\\big (B_g^{w_2}\\big ) \\big ({B_g}^{w_3}\\big )\\dots \\big ({B}_g^{w_l}\\big )\\Big )^{-1}\\Big (\\big (B_g^{w_1}\\big ) \\big ({B_g}^{w_2}\\big )\\dots \\big ({B}_g^{w_l}\\big )\\Big )\\Big (\\big (B_g^{w_2}\\big ) \\big ({B_g}^{w_3}\\big )\\dots \\big ({B}_g^{w_l}\\big )\\Big )\\\\\\cdot & \\big ({B_g}^{w_l}\\big )^{-1}\\dots \\big ({B_g}^{w_2}\\big )^{-1}\\big ({B_g}^{w_l}\\big )^{-1}\\dots \\big ({B_g}^{w_2}\\big )^{-1}\\\\\\cdot &\\big ({B_g}^{w_2}\\big )^{2}\\big ({B_g}^{w_3}\\big )^{2}\\dots \\big ({B_g}^{w_l}\\big )^{2}$ So, if $n+1=2$ , then since by induction hypothesis for $c=1$ the product of the first two lines of equation (REF ) is a product of $2l(g-1)+2$ commutators, it is enough to prove that the product of two last lines in (REF ) $ \\big ({B_g}^{w_l}\\big )^{-1}\\dots \\big ({B_g}^{w_2}\\big )^{-1}\\big ({B_g}^{w_l}\\big )^{-1}\\dots \\big ({B_g}^{w_3}\\big )^{-1}\\big ({B_g}^{w_2}\\big )\\big ({B_g}^{w_3}\\big )^{2}\\dots \\big ({B_g}^{w_l}\\big )^{2}$ is a product of $\\lceil l\\big (2(2g-1)-1\\big )/2\\rceil +1-2l(g-1)-2=(l-1)/2$ commutators.",
"Similarly to the first case (when $l$ is even) applying Lemma REF to expression (REF ) for $\\xi _1=\\big (B^{w_3}\\big )^{-1}\\big (B^{w_2}\\big )^{-1}$ , $\\xi _2=\\big (B^{w_2}\\big )\\big (B^{w_3}\\big )$ we conclude that (REF ) is a product of a commutator of elements which belong to $\\langle B_g\\rangle ^{F_{2g}}$ times the same expression without elements $B^{w_2}, B^{w_3}$ $\\big ({B_g}^{w_l}\\big )^{-1}\\dots \\big ({B_g}^{w_4}\\big )^{-1}\\big ({B_g}^{w_l}\\big )^{-1}\\dots \\big ({B_g}^{w_5}\\big )^{-1}\\big ({B_g}^{w_4}\\big )\\big ({B_g}^{w_5}\\big )^{2}\\dots \\big ({B_g}^{w_l}\\big )^{2}.$ Repeating this procedure $(l-1)/2$ times we conclude that expression (REF ) is a product of $(l-1)/2$ commutators of elements which belong to the kernel of $p$ , i. e. the case $n+1=2$ is proved.",
"For the general case $n>1$ by the induction hypothesis for $c=1$ the product of the first two lines of (REF ) is the product of $2l(g-1)+2$ commutators, by the induction hypothesis for $c=n-1$ the third line of (REF ) is the product of $\\lceil l\\big ((n-1)(2g-1)-1\\big )/2+1\\rceil $ commutators of elements such that their images under $p$ generate $H_1$ (the case $n+1=2$ was necessary for making this inductive step).",
"So, it is enough to prove that the product of two last lines in expression (REF ) $ \\big ({B_g}^{w_l}\\big )^{-1}\\dots \\big ({B_g}^{w_2}\\big )^{-1}\\big ({B_g}^{w_l}\\big )^{-1}\\dots \\big ({B_g}^{w_2}\\big )^{-1}\\cdot \\\\\\cdot \\big ({B_g}^{w_l}\\big )^{-n+1}\\dots \\big ({B_g}^{w_3}\\big )^{-n+1}\\big ({B_g}^{w_2}\\big )^{2}\\big ({B_g}^{w_3}\\big )^{n+1}\\dots \\big ({B_g}^{w_l}\\big )^{n+1}$ is a product of $\\lceil l\\big ((n+1)(2g-1)-1\\big )/2+1\\rceil -\\lceil l\\big ((n-1)(2g-1)-1\\big )/2+1\\rceil -2l(g-1)-2=l-2$ commutators of elements images of which under $p$ belong to $p(H_1)$ .",
"If we apply Lemma REF twice to expression (REF ) for $\\xi _1=\\big (B_g^{w_3}\\big )^{-1}\\big (B_g^{w_2}\\big )^{-1}$ , $\\xi _2=\\big (B_g^{w_2}\\big )\\big (B_g^{w_3}\\big )$ , then we conclude that expression (REF ) is a product of two commutators (of elements which belong to the kernel of $p$ ) times expression (REF ) without elements $\\big (B_g^{w_2}\\big ), \\big (B_g^{w_3}\\big )$ .",
"If we repeat this procedure $(l-3)/2$ times, we conclude that expression (REF ) is a product of $2(l-3)/2=l-3$ commutators times the expression $\\big (B_g^{w_l}\\big )^{-1}\\big (B_g^{w_{l-1}}\\big )^{-1}\\big (B_g^{w_l}\\big )^{-1}\\big (B_g^{w_{l-1}}\\big )^{-1}\\big (B_g^{w_l}\\big )^{-n+1}\\big (B_g^{w_{l-1}}\\big )^2\\big (B_g^{w_l}\\big )^{n+1}$ which is equal to the commutator $\\big (B_g^{w_l}\\big )^{-1}\\Big [\\big (B_g^{w_{l-1}}\\big )^{-1}\\big (B_g^{w_l}\\big )^{n-1},\\big (B_g^{w_l}\\big )^{-n}\\big (B_g^{w_{l-1}}\\big )^{-1}\\Big ]\\big (B_g^{w_l}\\big )$ , i. e. expression (REF ) is a product of $l-2$ commutators.",
"$\\Box $ Remark 2.3 We can suppose that if $u_1,v_1,\\dots ,u_h,v_h$ is a solution of equation (REF ) constructed in the proof of Theorem REF for an arbitrary $c$ , then $u_1,v_1,\\dots ,u_{l(g-1)+1}, v_{l(g-1)+1}$ is the solution of equation (REF ) for $c=1$ .",
"Remark 2.4 In particular case when $g=1$ , $l=1$ , $w_1=1$ , $c=2h-1$ the explicit solution of equation (REF ) is given in [12].",
"The following result, which is a consequence of the remark above, is a corollary of Theorem REF .",
"Corollary 2.5 Let $G$ be a group, $a,b\\in G$ , and $n$ be an integer.",
"Then $[a,b]^n$ can be expressed as the product of at most $\\lceil (n+1)/2\\rceil $ commutators.",
"Proof.",
"The equation $[u,v]=[x,y]$ has a solution $u=x,v=y$ in $F_2=\\langle x,y\\rangle $ .",
"Therefore the equation $[u_1,v_1]\\dots [u_h,v_h]=[x,y]^{n}$ has a solution in $F_2=\\langle x,y\\rangle $ for $h=\\lceil (n+1)/2\\rceil $ .",
"Acting on the equality $[u_1,v_1]\\dots [u_h,v_h]=[x,y]^{n}$ by the homomorphism $\\varphi :F_2\\rightarrow G$ which is induced by $\\varphi (x)=a$ , $\\varphi (y)=b$ we get the result.$\\square $"
],
[
"The right part has the form $\\Big ([x,y]\\Big )^c \\Big ({[x,y]}^y\\Big )^c\\dots \\Big ({[x,y]}^{y^{l-1}}\\Big )^c$",
"In this section we will consider a particular case of equation (REF ) in $F_2=\\langle x,y\\rangle $ .",
"In this case $p:F_2\\rightarrow \\pi _1(S_1)=F_2/[F_2,F_2]$ is the abelianization map.",
"Denote by $B=B_1=[x,y]$ .",
"The following statement gives a stronger version of Proposition REF for $c=g=h=1$ .",
"Proposition 3.1 Let $w_1,\\dots ,w_l$ be $l$ distinct elements from $F_2$ .",
"If $u,v$ is a solution of the equation $[u,v]=B^{w_1}\\dots B^{w_l},$ and $p:F_2\\rightarrow F_2/[F_2,F_2]$ is the natural homomorphism, then $|p(F_2):p\\big (\\langle u,v\\rangle \\big )|=l$ .",
"Proof.",
"By the solution $u,v$ we can construct a map $f:T\\rightarrow T$ such that ${\\rm deg}(f)=l\\ne 0$ .",
"Since ${\\rm deg}(f)\\ne 0$ , we have $NR[f]=|p(F_2):p\\big (\\langle u,v\\rangle \\big )|$ .",
"From the other side, since $f$ is a map from torus to torus, $NR[f]=|{\\rm deg}(f)|$ .",
"This follows promptly from the main result in [6] once one can identify the roots of $f$ with the fixed points of the map $g$ given by $g(x)=f(x)x$ , using the multiplication of the torus.",
"$\\square $ The purpose of this section is to give explicit solutions for the particular case of equation (REF ) $[u_1,v_1]\\dots [u_h,v_h] = \\Big ([x,y]\\Big )^c \\Big ({[x,y]}^y\\Big )^c\\dots \\Big ({[x,y]}^{y^{l-1}}\\Big )^c$ in $F_2=\\langle x,y\\rangle $ for the minimal integer $h$ which satisfies the inequality $h \\ge l(c-1)/2+1$ .",
"If $h < l(c-1)/2+1$ , then by Proposition REF equation (REF ) does not have solutions.",
"In contrast to Proposition REF , for $c>1$ (and therefore $h>1$ ) the index $|p(F_2):p(H)|$ , where $H=\\langle u_1,v_1,\\dots ,u_h,v_h\\rangle $ , can be different.",
"By Proposition REF this index is at most $l$ .",
"We are going to introduce two types of solution of equation (REF ): the first solution has the maximal possible index $|p(F_2):p(H)|=l$ , and the second solution is primitive, i. e. it has a minimal possible index $|p(F_2):p(H)|=1$ .",
"Since for $c=1$ equation (REF ) has a solution $u=x$ , $v=y^l$ , the case $|p(F_2):p(H)|=l$ is easy and it follows from Theorem REF in the following way.",
"Corollary 3.2 Let $c,l \\ge 1$ be integers and $h$ be the minimal number which satisfies the inequality $h \\ge l(c-1)/2+1$ .",
"Then the equation $[u_1,v_1]\\dots [u_h,v_h] = \\Big ([x,y]\\Big )^c \\Big ({[x,y]}^y\\Big )^c\\dots \\Big ({[x,y]}^{y^{l-1}}\\Big )^c$ with unknowns $u_1,v_1,\\dots ,u_h, v_h$ has an explicit solution in $F_2$ given by recurrence in $c$ which satisfies the equality $p\\Big (\\big \\langle u_1, v_1,\\dots ,u_h,v_h\\big \\rangle \\Big ) = \\big \\langle p(x),p(y^l)\\big \\rangle $ , where $p:F_2\\rightarrow F_2/[F_2,F_2]$ is the natural homomorphism.",
"Using equality (REF ) and construction of the map $f$ (obtained from the solution) described in Section and in details in [12] we have the following corollary.",
"Corollary 3.3 Let $c>1$ , $l\\ge 1$ be integers and $h$ be a minimal number which satisfies the inequality $h \\ge l(c-1)/2+1$ .",
"Then there exists a map $f: S_h \\rightarrow T$ with $A(f)=lc$ , $MR[f]=l$ , $NR[f]=l$ and each Nielsen root class has index $c$ .",
"So, the Wecken property holds for $f$ .",
"Now we are going to construct an explicit primitive solution of equation (REF ).",
"Results [2], [4] about primitive branching coverings give some evidence that such solutions might exist.",
"At first, we consider one simple particular case when $l=2$ , $c=2$ .",
"Lemma 3.4 The equation $[u_1,v_1][u_2,v_2]=[x,y]^2([x,y]^y)^2$ has as solution $u_1=x, v_1=y^3, u_2=y^3xy^{-2}x^{-1}y^{-1}xy^2x^{-1}y^{-3}, v_2=y^2x^2y^2x^{-1}y^{-3}$ which is primitive.",
"Proof.",
"Straightforward calculation.$\\Box $ The general case follows.",
"Theorem 3.5 Let $c>1$ , $l\\ge 1$ be integers and $h$ be a minimal number which satisfies the inequality $h \\ge l(c-1)/2+1$ .",
"Then the equation $[u_1,v_1]\\dots [u_h,v_h] = \\Big ([x,y]\\Big )^c \\Big ({[x,y]}^y\\Big )^c\\dots \\Big ({[x,y]}^{y^{l-1}}\\Big )^c$ with unknowns $u_1,v_1,\\dots ,u_h,v_h$ has an explicit primitive solution in $F_2$ given by recurrence in $l$ .",
"Proof.",
"We will use induction on $l$ .",
"For the basis of induction we consider two cases $l=1$ and $l=2$ .",
"The result for $l=1$ follows from Corollary REF .",
"If $l=2$ , then $h=c$ .",
"For $c=2$ the result follows from Lemma REF .",
"Suppose that we found a solution $u_1,v_1,\\dots ,u_h,v_h$ for an integer $c$ such that $u_1,v_1,u_2,v_2$ is the solution for $c=2$ .",
"Denoting by $u_{h+1}=\\big (B^y\\big )^{-c}x\\big (B^y\\big )^c$ , $v_{h+1}=\\big (B^y\\big )^{-c}y^2\\big (B^y\\big )^c$ we have $[u_{h+1},v_{h+1}]=\\big (B^y\\big )^{-c}[x, y^2]\\big (B^y\\big )^c$ and therefore $[u_1,v_1]\\dots [u_{h+1},v_{h+1}]=\\big (B\\big )^c\\big (B^y\\big )^c\\big (B^y\\big )^{-c}[x, y^2]\\big (B^y\\big )^c=\\big (B\\big )^{c+1}\\big (B^y\\big )^{c+1}.$ This solution is obviously primitive since $p(u_1), p(v_1), p(u_2), p(v_2)$ generate $F_2/[F_2,F_2]$ .",
"The basis is proved.",
"For the induction step we consider two similar cases depending on the parity of $l$ .",
"Case 1: $l=2n$ is even.",
"In this case $h=n(c-1)+1$ .",
"We will construct a primitive solution which satisfies the condition $u_1=x$ , $v_1=y^{l+1}$ .",
"If $l=2$ , then the statement follows from the basis of induction.",
"By the induction hypothesis we have a primitive solution $a_1=x$ , $b_1=y^{l+1},a_2,b_2,\\dots ,a_{h_1},b_{h_1}$ of equation (REF ) for $l=2n$ , $h_1=n(c-1)+1$ .",
"Also by induction hypothesis we have a primitive solution $r_1=x,s_1=y^2, r_2, s_2, \\dots , r_{h_2}, v_{h_2}$ of equation (REF ) for $l=2$ , $h_2=c$ .",
"If we denote by $u_1&=x&&\\\\v_1&=y^{2n+3}&&\\\\u_{j}&=\\big (B^{y^{2n+2}}\\big )^{-1}\\big (B^{y^{2n+1}}\\big )^{-1}a_j\\big (B^{y^{2n+1}}\\big )\\big (B^{y^{2n+2}}\\big )&&j=2,\\dots , h_1\\\\v_{j}&=\\big (B^{y^{2n+2}}\\big )^{-1}\\big (B^{y^{2n+1}}\\big )^{-1}b_j\\big (B^{y^{2n+1}}\\big )\\big (B^{y^{2n+2}}\\big )&&j=2,\\dots , h_1\\\\u_{h_1+j}&=y^{2n+1}r_{j+1}y^{-(2n+1)}&&j=1,\\dots , h_2-1\\\\ v_{h_1+j}&=y^{2n+1}s_{j+1}y^{-(2n+1)}&&j=1,\\dots , h_2-1$ and by $h=h_1+h_2-1=n(c-1)+1+c-1=(n+1)(c-1)+1$ , then we have $[u_1,v_1]\\dots [u_h,v_h]&=\\Big ([x,y^{2n+3}]\\big (B^{y^{2n+2}}\\big )^{-1}\\big (B^{y^{2n+1}}\\big )^{-1}[a_2,b_2]\\dots [a_{h_1}, b_{h_1}]\\Big )\\\\&\\cdot \\Big (\\big (B^{y^{2n+1}}\\big )\\big (B^{y^{2n+2}}\\big ) y^{2n+1}[r_2,s_2]\\dots [r_{h_2},s_{h_2}]y^{-(2n+1)}\\Big )\\\\&=\\Big ([x,y^{2n+1}][a_2,b_2]\\dots [a_{h_1}, b_{h_1}]\\Big )\\Big (y^{2n+1}BB^y[r_2,s_2]\\dots [r_{h_2},s_{h_2}]y^{-(2n+1)}\\Big )\\\\&=\\Big (\\big (B\\big )^c\\big (B^y\\big )^c\\dots \\big (B^{y^{2n}}\\big )^c\\Big )\\Big (y^{2n+1}\\big (B\\big )^c\\big (B^y\\big )^cy^{-(2n+1)}\\Big )\\\\&=\\big (B\\big )^c\\big (B^y\\big )^c\\dots \\big (B^{y^{2n+2}}\\big )^c.$ and the statement is proved for $l=2n+2$ .",
"The solution provided in (REF ) is primitive since the subgroup $p\\Big (\\big \\langle u_1,v_1,\\dots ,u_h,v_h\\big \\rangle \\Big )$ contains $p(u_1)=p(x)$ and $p(u_{h_1+1})=p(r_2)=p(y)^{-1}$ .",
"Case 2: $l=2n-1$ is odd.",
"In this case $h=\\lceil (2n-1)(c-1)/2+1\\rceil $ .",
"Similarly to the first case, we will show that there exists a primitive solution such that $u_1=x$ , $v_1=y^l$ .",
"For $l=1$ the result follows from Corollary REF and Remark REF .",
"By induction hypothesis we can suppose that we have a primitive solution $a_1=x, b_1=y^l,a_2,b_2,\\dots ,a_{h_1},b_{h_1}$ of equation (REF ) for $l=2n-1$ and $h_1=\\lceil (2n-1)(c-1)/2+1\\rceil $ .",
"Also by induction hypothesis we can suppose that we have a solution $r_1=x,s_1=y^2, r_2, s_2, \\dots , r_{h_2}, v_{h_2}$ of equation (REF ) for $l=2$ and $h_2=c$ .",
"If we denote by $u_1&=x&&\\\\v_1&=y^{2n+1}&&\\\\u_{j}&=\\big (B^{y^{2n}}\\big )^{-1}\\big (B^{y^{2n-1}}\\big )^{-1}a_j\\big (B^{y^{2n-1}}\\big )\\big (B^{y^{2n}}\\big )&&j=2,\\dots , h_1\\\\v_{j}&=\\big (B^{y^{2n}}\\big )^{-1}\\big (B^{y^{2n-1}}\\big )^{-1}b_j\\big (B^{y^{2n-1}}\\big )\\big (B^{y^{2n}}\\big )&&j=2,\\dots , h_1\\\\u_{h_1+j}&=y^{2n-1}r_{j+1}y^{1-2n}&&j=1,\\dots , h_2-1\\\\ v_{h_1+j}&=y^{2n-1}s_{j+1}y^{1-2n}&&j=1,\\dots , h_2-1$ and by $h=h_1+h_2-1=\\lceil (2n-1)(c-1)/2+1\\rceil +c-1=\\lceil (2n+1)(c-1)/2+1\\rceil $ , then we have $[u_1,v_1]\\dots [u_h,v_h]&=\\Big ([x,y^{2n+1}]\\big (B^{y^{2n}}\\big )^{-1}\\big (B^{y^{2n-1}}\\big )^{-1}[a_2,b_2]\\dots [a_{h_1}, b_{h_1}]\\Big )\\\\&\\cdot \\Big (\\big (B^{y^{2n-1}}\\big )\\big (B^{y^{2n}}\\big ) y^{2n-1}[r_2,s_2]\\dots [r_{h_2},s_{h_2}]y^{1-2n}\\Big )\\\\&=\\Big ([x,y^{2n-1}][a_2,b_2]\\dots [a_{h_1}, b_{h_1}]\\Big )\\Big (y^{2n-1}BB^y[r_2,s_2]\\dots [r_{h_2},s_{h_2}]y^{1-2n}\\Big )\\\\&=\\Big (\\big (B\\big )^c\\big (B^y\\big )^c\\dots \\big (B^{y^{2n-2}}\\big )^c\\Big )\\Big (y^{2n-1}\\big (B\\big )^c\\big (B^y\\big )^cy^{1-2n}\\Big )\\\\&=\\big (B\\big )^c\\big (B^y\\big )^c\\dots \\big (B^{y^{2n}}\\big )^c$ and the statement is proved for $l=2n+1$ .",
"The solution provided in (REF ) is primitive since the subgroup $p\\Big (\\big \\langle u_1,v_1,\\dots ,u_h,v_h\\big \\rangle \\Big )$ contains $p(u_1)=p(x)$ and $p(u_{h_1+1})=p(r_2)=p(y)^{-1}$ .$\\Box $ Remark 3.6 If $c=1$ , then by Proposition REF the result of Theorem REF holds only for $l=1$ .",
"Remark 3.7 An old problem in the geometric group theory is the problem of determining the genus and the number $f(g)$ of Nielsen classes for a given element $g\\in [F_n,F_n]$ (see [1] for the definition of Nielsen classes and [1] for the related question).",
"If $n=2$ , then Corollary REF and Theorem REF guarantee that for $g=([x,y])^c ({[x,y]}^y)^c\\dots ({[x,y]}^{y^{l-1}})^c$ the number of Nielsen classes is at least 2.",
"Using equality (REF ) and construction of the map $f$ (obtained from the solution) described in Section and in details in [12] we have the following corollary.",
"Corollary 3.8 Let $c>1$ , $l\\ge 1$ be integers and $h$ be a minimal number which satisfies the inequality $h \\ge l(c-1)/2+1$ .",
"Then there exists a map $f: S_h \\rightarrow T$ with $A(f)=lc$ , $MR[f]=l$ , $NR[f]=1$ and the only root class has index $lc$ .",
"So, the Wecken property does not hold for $f$ ."
],
[
"The right part has the form $\\Big ([x,y]\\Big )^{k+l} \\Big ({[x,y]}^{y}\\Big )^{k-l}$",
"The purpose of this section is to give an explicit solution for the equation $[u_1,v_1]\\dots [u_h,v_h] = \\Big ([x,y]\\Big )^{k+l} \\Big ({[x,y]}^{y}\\Big )^{k-l}$ for $h=k$ .",
"If $h < k$ , then by Proposition REF equation (REF ) does not have solutions.",
"The main result of this section is the following theorem Theorem 4.1 Let $k>l\\ge 0$ be integers.",
"Then the equation $[u_1,v_1]...[u_h, v_h]= \\Big ([x,y]\\Big )^{k+l} \\Big ({[x,y]}^{y}\\Big )^{k-l}$ with unknowns $u_1,v_1,\\dots ,u_h, v_h$ for $h=k$ has an explicit primitive solution.",
"Moreover, if $l\\ne 0$ , then every solution of this equation is primitive.",
"Proof.",
"At first, we will prove the moreover part of the theorem.",
"Let $u_1,v_1,\\dots ,u_k,v_k$ be a solution of equation (REF ) for $l\\ne 0$ and let $H=\\langle u_1,v_1,\\dots ,u_k,v_k\\rangle $ .",
"By Proposition REF , the index $|p(F_2):p(H)|$ is equal to 1 or 2, and we need to prove that this index is equal to 1.",
"By contrary, suppose that $|p(F_2):p(H)|=2$ .",
"From equality (REF ) follows that the map $f:S_k\\rightarrow T$ (obtained from the solution $u_1,v_1,\\dots ,u_k,v_k$ ) described in Section has two essential Nielsen root classes, one of this classes has the index $k+l$ and another one has the index $k-l$ .",
"If $l\\ne 0$ , then $k-l\\ne k+l$ , but the indices of all essential Nielsen root classes must coincide .",
"We have a contradiction.",
"In order to introduce the solution of equation (REF ) denote by $r_{i}&=yxy^{i-1}x^{-1}y^{-1}xy^{-i+1}x^{-1}y^{-1}&&i=1,\\dots ,l\\\\s_{i} &= yx y^{i-1}x^{-1}B^{l-i+1}yB^{k-l}y^{-i}x^2y^{-i+1}x^{-1}y^{-1}&&i=1,\\dots ,l\\\\r_{l+j}&=yxy^{(l+j)}x^{-1}y^{-1}xy^{(-l-j)}x^{-1}y^{-1}&&j=1,\\dots ,k-l-1\\\\s_{l+j} &= yxy^{(l+j)}x^{-1}B^{k-l-j}y^{-l-j}x^2y^{(-l-j)}x^{-1}y^{-1}&&j=1,\\dots ,k-l-1\\\\r_{k}&=yxy^{(k+1)}x^{-1}y^{-1}&&\\\\ s_{k}&=yxy^{-1}x^{-1}yx^{-1}y^{-1}&&$ and let us, at first, prove some auxiliary equalities involving $r_1,s_1,\\dots ,r_k,s_k$ .",
"Using induction on the number $t=1,\\dots , l$ let us prove that $[r_1,s_1]\\dots [r_{t},s_{t}]=B^lyB^{k-l}y^{-1}B^ty^{t+1}B^{l-k}y^{-1}B^{t-1-l}xy^{-t+1}x^{-1}y^{-1}$ The basis of induction ($t=1$ ) is proved in the following equality $[r_1,s_1]&=[y^{-1},yB^{l}yB^{k-l}y^{-1}xy^{-1}]\\\\&=y^{-1}yB^{l}yB^{k-l}y^{-1}xy^{-1}yyx^{-1}yB^{l-k}y^{-1}B^{-l}y^{-1}\\\\&=B^{l}yB^{k-l}y^{-1}By^2B^{l-k}y^{-1}B^{-l}y^{-1}$ The step of induction (omitting some detailed calculations) follows from the following equality $[r_1,s_1]\\dots [r_{t+1},s_{t+1}]&=\\Big ([r_1,s_1]\\dots [r_{t},s_{t}]\\Big )[r_{t+1},s_{t+1}]\\\\&=B^lyB^{k-l}y^{-1}B^ty^{t+1}B^{l-k}y^{-1}B^{t-1-l}xy^{-t+1}x^{-1}y^{-1}\\\\&\\cdot yxy^tx^{-1}\\Big [y^{-1}, B^{l-t}yB^{k-l}y^{-t-1}x\\Big ]xy^{-t}x^{-1}y^{-1}\\\\&=B^lyB^{k-l}y^{-1}B^{t+1}y^{t+2}B^{l-k}y^{-1}B^{t-l}xy^{-t}x^{-1}y^{-1}$ Similarly to equation (REF ) using induction on the number $t=1,\\dots , k-l-1$ we can prove the following equality.",
"$[r_{l+1},s_{l+1}]\\dots [r_{l+t},s_{l+t}]=yxy^{l+1}x^{-1}\\Big (y^{-1}B^{k-l-1}y^{-l-1}B^ty^{l+t+1}B^{l+t-k}\\Big )xy^{-l-t}x^{-1}y^{-1}$ We will not show the proof of (REF ) here since it repeats the proof of (REF ) almost completely.",
"Multiplying equality (REF ) for $t=l$ , equality (REF ) for $t=k-l-1$ and the value $[r_k,s_k]$ using formula (REF ) after some simple calculations we conclude that $[r_1,s_1]\\dots [r_k,s_k]=B^l\\big (B^y\\big )^{k-l}B^k$ .",
"From this equality follows that if for $i=1,\\dots ,k$ we denote by $u_u=B^{k}r_iB^{-k}, v_i=B^{k}s_iB^{-k},$ then $[u_1,v_1]\\dots [u_k,v_k]=B^{k+l}\\big (B^y\\big )^{k-l}$ , i. e. $u_1,v_1,\\dots ,u_k,v_k$ is the solution of equation (REF ).",
"The images of elements $u_1,v_1,\\dots ,u_k,v_k$ under the homomorphism $p:F_2\\rightarrow F_2/[F_2,F_2]$ generate $p(F_2)$ since $p(u_1)=p(y)^{-1}$ , $p(v_1)=p(x)$ .",
"Therefore $u_1,v_1,\\dots ,u_k,v_k$ is a primitive solution of equation (REF ).",
"$\\Box $ Remark 4.2 The same result for $k=l$ follows from Theorem REF .",
"Daciberg Lima Gonçalves Department of Mathematics-IME, University of São Paulo 05508-090, Rua do Matão 1010, Butanta-São Paulo-SP, Brazil email: dlgoncal@ime.usp.br Timur Nasybullov Department of Mathematics, KU Leuven KULAK 8500, Etienne Sabbelaan 53, Kortrijk, Belgium email: timur.nasybullov@mail.ru"
]
] | 1808.08456 |
[
[
"One-dimensional topological metal"
],
[
"Abstract We propose a new type of topological states of matter exhibiting topologically nontrivial edge states (ESs) within gapless bulk states (GBSs) protected by both spin rotational and reflection symmetries.",
"A model presenting such states is simply comprised of a one-dimensional reflection symmetric superlattice in the presence of spin-orbit coupling containing odd number of sublattices per unit cell.",
"We show that the system has a rich phase diagram including a topological metal (TM) phase where nontrivial ESs coexist with nontrivial GBSs at Fermi level.",
"Topologically distinct phases can be reached through subband gap closing-reopening transition depending on the relative strength of inter and intra unit cell spin-orbit couplings.",
"Moreover, topological class of the system is AI with an integer topological invariant called $\\mathbb{Z}$ index.",
"The stability of TM states is also analyzed against Zeeman magnetic fields and on-site potentials resulting in that the spin rotational symmetry around the lattice direction is a key requirement for the appearance of such states.",
"Also, possible experimental realizations are discussed."
],
[
"Acknowledgment",
"This work is partially supported by Iran Science Elites Federation under Grant No.",
"11/66332."
]
] | 1808.08530 |
[
[
"A numerical approximation for the standard one pressure system of two\n fluid flows with energy equations"
],
[
"Abstract We study numerically the standard one pressure model of two fluid flows with energy equations.",
"This system is not solved in time derivative.",
"It has been transformed into an equivalent system solved in time derivative.",
"We show that the scheme in this paper applies to both solved and nonsolved systems and gives same results.",
"One usually adds a nonphysical term to render the system hyperbolic.",
"However, explicit solutions and well posedness of the Cauchy problem for some nonlinear nonhyperbolic systems of physics have been obtained in some events by [B. Keyfitz et al.].",
"We also show that our scheme applies equally well to both versions, with and without the additional term, whether solved in time derivative or not, which provides four versions of the system.",
"We observe that the nonhyperbolic and the hyperbolic systems give very close but slightly different results: the step values are always the same but peaks in gas and liquid velocities are observed in the nonhyperbolic model, which is typically observed in experimental results concerning the gas kick phenomenon though we are unable to say if this result is related or not.",
"The numerical quality of the hyperbolic solved in time derivative system is better, therefore our results (same pressure and temperatures, and same step values in volume fraction and velocities besides the isolated peaks) provide a justification of the additional term that renders it hyperbolic.",
"Another difficulty lies in that these systems are in nonconservative form and therefore its discontinuous solutions cannot make sense in the theory of distributions and it has been observed that different numerical schemes can lead to different discontinuous solutions.",
"For the hyperbolic system this solution is identical to the main one in [S.T.",
"Munkejord, S. Evje, T. Flatten] obtained from completely different methods."
],
[
"We study numerically the standard one pressure model of two fluid flows with energy equations.",
"This system is not solved in time derivative.",
"It has been transformed into an equivalent system solved in time derivative [S.T.",
"Munkejord, S. Evje, T. Flatten, SIAM J. Sci.",
"Comput.",
"31,4,2009,2587-2622].",
"We show that the scheme in this paper applies to both solved and nonsolved systems and gives same results.",
"One usually adds a nonphysical term to render the system hyperbolic.",
"However, explicit solutions and well posedness of the Cauchy problem for some nonlinear nonhyperbolic systems of physics have been obtained in some events by [B. Keyfitz et al.].",
"We also show that our scheme applies equally well to both versions, with and without the additional term, whether solved in time derivative or not, which provides four versions of the system.",
"We observe that the nonhyperbolic and the hyperbolic systems give very close but slightly different results: the step values are always the same but peaks in gas and liquid velocities are observed in the nonhyperbolic model, which is typically observed in experimental results concerning the gas kick phenomenon though we are unable to say if this result is related or not.",
"The numerical quality of the hyperbolic solved in time derivative system is better, therefore our results (same pressure and temperatures, and same step values in volume fraction and velocities besides the isolated peaks) provide a justification of the additional term that renders it hyperbolic.",
"It has also been observed that additional viscosity does not modify significantly the results for all four versions.",
"Another difficulty lies in that these systems are in nonconservative form and therefore its discontinuous solutions cannot make sense in the theory of distributions and it has been observed that different numerical schemes can lead to different discontinuous solutions.",
"For the hyperbolic system this solution is identical to the main one in [S.T.",
"Munkejord, S. Evje, T. Flatten] obtained from completely different methods.",
"Finally, we improve the order one scheme in use above by transforming it explicitly into order three in space.",
"AMS classification: 65M08, 35D30, 35F25, 76T10.",
"Keywords: numerical analysis, partial differential equations, fluid dynamics, two phase flow models.",
"e-mail: m.colombeau@orange.fr *this research has been done thanks to financial support of FAPESP, processo 2012/15780-9, when the author was doing her postdoctoral studies at the University of São Paulo, USP, São Paulo, Brazil.",
"1.",
"Introduction.",
"We study numerically the basic system used to model a mixture of two immiscible fluids from the conservation laws of mass, momentum and energy with the natural assumption that the pressures at the same point are equal inside the two fluids [54] p. 373, [48] p. 2589 and 2590, [46] p. 413, [47] p. 479, [25] p. 179, [13] p. 465,....",
"This paper contributes to the research on this system in five ways.",
"$\\bullet $ First, since the system is in nonconservative form one should state precisely inside the schemes the formulas of discretization of the nonconservative products since various slight changes could lead to different weak solutions [48] p. 2620.",
"We observe that in the case we admit that the jumps between successive cells are \"`small\"' (which permits large jumps in the numerical solution provided a number of cells are involved inside these jumps) the ambiguity can be solved in a natural way by assuming that the step functions (constant on each cell) at times $n\\Delta t$ satisfy the equations.",
"One recovers the simplest formulas proposed by various authors but we have a mathematical justification of these formulas.",
"$\\bullet $ Second, a sequence of approximate solutions with full mathematical proof that they tend to satisfy the system has been constructed in [12] to play the role of a (lacking) explicit exact solution.",
"It has been checked in [12] that the scheme in this paper produces exactly this theoretical solution in both cases ($\\delta =0$ and $\\delta =2$ , see below).",
"This shows that the scheme in sections 3 and 4 gives a solution of the system but, in absence of a theoretical uniqueness result, one cannot claim this is the physically correct solution, although there is an accumulation of indications at least for the Toumi shock tube problem.",
"$\\bullet $ Our scheme permits to treat as well the original system [54] p. 373, [48] pp.2589-2590 and p. 2592, not solved in time derivatives and the system solved in time derivatives obtained in [48] p. 2595-2596.",
"We check that the two systems give exactly the same numerical solution, which is not trivial in the case of shock waves since nonlinear calculations that preserve the smooth solutions do not in general preserve the shock waves: it is well known that the equations $u_t+uu_x=0$ and $uu_t+u^2u_x=0$ have different jump conditions.",
"$\\bullet $ The authors on multifluid flows consider an additional term to render the system hyperbolic, then study the hyperbolic system so obtained.",
"Motivated by the fact that B. Keyfitz et al [32], [33], [34], [35], [41], [42], [31], [36], [37], [38] have discovered that, in certain cases, nonlinear nonhyperbolic systems could have realistic physical solutions we study also numerically the original nonhyperbolic system.",
"We observe same step values but a significant difference in form of a well defined peak in liquid flow velocity and gas flow velocity present in the original system and absent after the additional term.",
"Comparison with some experimental data suggest that these peaks could be the gas kick phenomenon.",
"$\\bullet $ Finally we transform our scheme into higher order schemes in view of an extension to multidimension.",
"Now we recall the equations.",
"The two fluids are denoted by the indices 1 and 2, for instance mixture of oil and natural gas in extraction tubes of oil exploitation for the gas kick simulation [2], [3], [43].",
"The system, as it stems from physics, called canonical nonconservative form, is stated as $\\frac{\\partial }{\\partial t}(\\rho _1 \\alpha _1)+\\frac{\\partial }{\\partial x}(\\rho _1\\alpha _1 v_1)=0,$ $\\frac{\\partial }{\\partial t}(\\rho _2 \\alpha _2)+\\frac{\\partial }{\\partial x}(\\rho _2\\alpha _2 v_2)=0,$ $\\frac{\\partial }{\\partial t}(\\rho _1 \\alpha _1v_1)+\\frac{\\partial }{\\partial x}(\\rho _1\\alpha _1( v_1)^2)+\\alpha _1\\frac{\\partial p}{\\partial x}+\\tau _i=g\\alpha _1\\rho _1,$ $\\frac{\\partial }{\\partial t}(\\rho _2 \\alpha _2v_2)+\\frac{\\partial }{\\partial x}(\\rho _2\\alpha _2( v_2)^2)+\\alpha _2\\frac{\\partial p}{\\partial x}-\\tau _i=g\\alpha _2\\rho _2,$ $\\frac{\\partial }{\\partial t}(E_1)+p\\frac{\\partial }{\\partial t}(\\alpha _1) +\\frac{\\partial }{\\partial x}(E_1v_1+p\\alpha _1v_1)+v_{\\tau }\\tau _i=g\\rho _1\\alpha _1v_1, $ $\\frac{\\partial }{\\partial t}(E_2)+p\\frac{\\partial }{\\partial t}(\\alpha _2) +\\frac{\\partial }{\\partial x}(E_2v_2+p\\alpha _2v_2)-v_{\\tau }\\tau _i=g\\rho _2\\alpha _2v_2, $ which are the conservation laws of mass, momentum and energy for fluids 1,2.",
"They are complemented by $\\alpha _1+\\alpha _2=1,$ $ \\tau _i=\\Delta p\\frac{\\partial \\alpha _1}{\\partial x}, \\ \\ \\Delta p=\\delta \\frac{\\alpha _1\\alpha _2\\rho _1\\rho _2}{\\rho _1\\alpha _2+\\rho _2\\alpha _1}(v_1-v_2)^2,$ $ v_\\tau =\\frac{\\alpha _2\\gamma _1v_1+\\alpha _1\\gamma _2v_2}{\\alpha _2\\gamma _1+\\alpha _1\\gamma _2},$ $\\delta \\ge 0, \\gamma _i\\ge 0$ and the state laws, $p_1=(K_1-1)(\\frac{E_1}{\\alpha _1}-\\frac{\\rho _1 (v_1)^2}{2})-K_1p_{\\infty ,1},$ $p_2=(K_2-1)(\\frac{E_2}{\\alpha _2}-\\frac{\\rho _2 (v_2)^2}{2})-K_2p_{\\infty ,2},$ in which we postulate $p_1=p_2$ , denoted by $p$ , from the equal pressure assumption.",
"The physical variables are the densities $\\rho _i(x,t)$ , the velocities $v_i(x,t)$ , the volumic proportions $\\alpha _i(x,t)$ , the equal pressures $p_i(x,t)$ , the total energy densities $E_i(x,t)$ , $i=1,2$ .",
"The constant $g$ is the component of the gravitational acceleration in the direction of the tube, $\\tau _i$ is the interphasic momentum exchange term, $\\Delta p$ is an interface pressure correction term and $v_\\tau $ is an operator with the dimension of a velocity which stems from the laws of thermodynamics as exposed in [48] where the choice of the formulas (8,9) is justified.",
"The letters $\\delta , \\gamma _1, \\gamma _2, K_1, K_2, p_{\\infty ,1}$ and $p_{\\infty ,2}$ represent real values which are chosen and explained in [48], [46].",
"We will adopt in the numerical calculations the values used in [48] p.2613 and [46] p.346 for the Toumi shock tube problem.",
"The system presents three basic peculiarities: obviously it is not solved in the time derivative and it is not in conservation form; non obviously it is not hyperbolic [48] p. 2595.",
"Each of these three peculiarities already causes serious problems for the elaboration of numerical schemes.",
"In [48] pp.",
"2592-2596 these authors, by a deep analysis involving the laws of thermodynamics, have produced a formally equivalent formulation which is resolved in the time derivative and has served to construct numerical methods, [48], [46].",
"The nonconservative character implies a nonuniqueness of stable (entropic) solutions [48] pp.",
"2587,2620, with the problem of searching the correct ones.",
"The absence of hyperbolicity generally suggests absence of well posedness.",
"On the other hand this system is of basic importance in industry such as oil extraction in deep sea [2], [3], [43] or cooling of nuclear power stations [4], [53].",
"In this paper we will address these three peculiarities.",
"Concerning nonhyperbolicity, somewhat unexpectedly, B. Keyfitz has pointed out various instances in which nonlinear nonhyperbolic systems modeling physics have physically meaningful solutions with applications in various domains, for instance in traffic flow, multifluid flows and porous medium, besides the fact the corresponding linear systems could be ill posed, see [32] pp.",
"150-151, the articles [33], [34], [35], [41], [42] and the review papers [31], [36], [37], [38].",
"This could be explained by the fact that some positive nonlinear effects could be lost in the linearization.",
"This motivates an attempt for a numerical study of the nonhyperbolic system, which has been shown realistic in the four equations model [11].",
"The scheme that we propose has been constructed from a scheme proposed in [5], [6] for pressureless fluids, [7] for ideal gases and shallow water, and [8] for various systems.",
"A version for the four equations model of two fluid flows is given in [11] section 7.",
"We begin to expose a scheme which is of order one in time and space.",
"Then we extend it to a scheme of order three in space that gives more accurate results.",
"The scheme applies to system (1-11), called canonical nonconservative form, although it is not solved in the time derivative, not conservative and not hyperbolic.",
"Applying it with the additional term (8) used in [48] ($\\delta =2$ ) to render the system hyperbolic one observes (figure 1) that one obtains exactly the numerical results in [48] p. 2615,2617 and [46] p. 437.",
"Using the systems solved and nonsolved in the time derivatives [48] we obtain (figures 1 and 2) with the scheme in this paper the same result for the discontinuous solution of the Toumi problem, which constitutes a numerical verification of the scheme and a numerical verification of the transformation of the equations in the case of discontinuous solutions [48] pp.",
"2592-2596.",
"Another interesting point is that the possibility to use the scheme with and without the additional term ($\\delta =2$ and $\\delta =0$ respectively in (8), figure 3) will permit to observe the influence of additional terms on the numerical solution and also to study numerically the original nonhyperbolic model [54] p.373.",
"These observations will be discussed in the paper and will finally provide a justification of the relevance of the additional term.",
"A theorem in section 4 shows, both for the hyperbolic and the nonhyperbolic model, that the approximate solutions from the scheme tend to satisfy the equations somewhat independently of a reasonable arbitrariness in a definition of a nonconservative product needed to give a sense to the equations.",
"2.",
"A choice of the nonconservative products inside the formulas of the scheme.",
"A basic problem lies in the nonconservative form of the formulas of the equations for example the single term $\\alpha _i\\partial _xp$ in (3,4).",
"It is known that different numerical schemes can converge to different weak solutions [48] p. 2620.",
"Therefore it is of basic importance to decide a precise choice of the formulas inside a numerical scheme for the discretization of the nonconservative products.",
"In this section we propose such a choice under a \"`small jump assumption\"' commonly used in numerical schemes: we postulate that the jumps $s_i$ between a cell $[(i-\\frac{1}{2})h, (i+\\frac{1}{2})h]$ and its neighbour cells are such that the higher order powers $(s_i)^n, \\ n=2,3$ can be neglected in the calculations, in which we retain therefore only the first order terms $s_i$ .",
"Then the reasoning that leads to a precise formula inside the scheme for the nonconservative products is as follows.",
"At times $t_n=n\\Delta t$ one would like to have a numerical solution in form of step functions on the cells that would satisfy the equations as much as possible.",
"How are these step functions in the case of a nonconservative product under the small jumps assumption?",
"Consider a general nonlinear equation $ \\partial _t u+A(u,v)\\partial _x u+B(u,v)\\partial _x v=0$ where $A$ and $B$ are polynomials in $u,v$ .",
"With standard notation, at the interface $(i+\\frac{1}{2})h$ , we set $u,v$ of the form $ u(x,t,h)=u_i+(u_{i+1}-u_i)H_u(x-ct), \\ v(x,t,h)=v_i+(v_{i+1}-v_i)H_v(x-ct), $ where $H_u$ and $H_v$ are regularizations of the Heaviside functions $H$ : here $H$ is discontinuous at $(i+\\frac{1}{2})h$ and $H_u, H_v$ are smooth functions that jump from 0 to 1 in the interval $]ih,(i+1)h[$ .",
"Then with this mollification the nonconservative products make sense.",
"What is their value in the mollified case, to be adopted at the limit of absence of mollification?",
"To this end plug the formulas (13) into the equation (12).",
"One obtains $ -c(u_{i+1}-u_i)H_u^{\\prime }+A(u,v)(u_{i+1}-u_i)H_u^{\\prime }+B(u,v)(v_{i+1}-v_i)H_v^{\\prime }=0.$ From the small jump assumption one replaces $A(u,v) $ and $B(u,v)$ by $A(u_i,v_i) $ and $B(u_i,v_i)$ respectively.",
"From (14) by integration one deduces that $H_u=H_v$ since both jump from 0 to 1 and are proportional.",
"Denoting $\\overline{H}=H_u=H_v$ a product such as $uv_x$ gives $\\int _ih^{(i+1)h}uv_x dx=\\int [u_i+(u_{i+1}-u_i)\\overline{H}](v_{i+1}-v_i)\\overline{H}^{\\prime }dx=u_i(v_{i+1}-v_i)+(u_{i+1}-u_i)(v_{i+1}-v_i)\\frac{1}{2}$ $=\\frac{u_{i+1}-u_i}{2}(v_{i+1}-v_i).$ Therefore one is led, under the small jump assumption, to discretize $uv_x$ by $\\frac{1}{h}\\frac{u_{i+1}+u_i}{2}(v_{i+1}-v_i)$ , or equivalently, seeking the terms in factor of $u_i$ for convenience, by $u_i( \\frac{v_{i+1}-v_{i-1}}{2h})$ which is simply a familiar centered discretization (formulas (15) and (16) are equivalent since we will understand the right-hand side of the equations in the sense of distributions).",
"This discretization will be applied to all nonconservative products in the scheme below.",
"The unique point would be to check that there will appear only small values of jumps between successive cells even inside physical discontinuities such as shock waves.",
"According to [48] p. 2597-2598 the above formula for nonconservative products consists in stating that the paths are same for $u$ and $v$ in the small jumps between two successive cells.",
"Indeed the mollifications $H_u$ and $H_v$ create a path for $H_u$ and a path for $H_v$ , not necessarily equal a priori.",
"Then we prove under the small jumps assumption that these paths should be equal in order that the numerical solutions at times $t_n=n\\Delta t$ stick to the equations.",
"Since the jumps in $u$ and in $v$ between the various cells do not necessarily remain proportional throughout a shock wave, the global paths in $u$ and $v$ for a physical jump involving a number of cells inside the jump can be very different, even when the paths between successive cells are same (an axample of this fact is given in [1]).",
"Note that due to their simplicity formulas (15) and (16) are widely used by various authors to discretize nonconservative products: for instance formula (16) is used in [1] p.2761 between formulas (7) and (8) there.",
"The novelty is that we provide a justification: under the small jump assumption other choices would give step functions (constant on each cell) at times $n\\Delta t$ that would not be solutions of the equations at times $t_n$ .",
"The authors of [1] give an example of a nonconservative system for which application of this formula for the jumps between successive cells does not produce the same jump formulas for the global jumps in the shock waves obtained from the scheme.",
"The scheme in this paper can also produce this natural phenomenon since the jumps between successive are in general not proportional for all interfaces of cells inside a physical jump.",
"3.",
"A method of sequences of smooth approximate solutions with full mathematical proof.",
"The system (1-11) cannot have discontinuous solutions in the sense of distribution theory in case of shock waves because of the terms $\\alpha _i \\partial _x p$ and $p\\partial _t\\alpha _i$ which do not permit a transfer of the derivatives to smooth test functions.",
"Therefore one has no other choice than approximation of the values of $\\alpha _i$ and $p$ by smooth functions.",
"Indeed this method has been widely developed in mathematics for systems without solutions in the sense of distributions and it has given very neat results for conservative systems, such as studies of $\\delta $ waves and $\\delta ^{(n)}$ waves, in [14], [15], [16], [17], [18], [19], [20], [21], [49], [50], [51] among other papers.",
"In this section we recall results in [12].",
"For the nonconservative system (1-11) a sequence of smooth approximate solutions with a full proof that the sequence tends to satisfy the equations has been constructed in [12].",
"This construction is based on a system of 6 ODEs in Banach space relevant of the Lipschitz theory of ODEs, therefore they can be solved numerically by convergent classical schemes for ODEs.",
"The defect of this approach lies in the present absence of a uniqueness result of a \"`limit\" (that could be simply a limit from observation) that \"`admissible sequences\"' could produce.",
"The sequence constructed in [12] has given exactly the results obtained in figure 2 from the schemes in sections 4.",
"Of course one could argue that the construction of this sequence of smooth solutions has been inspired by this scheme-although noticeably different, with intervention of physical arguments such as the convolution in pressure.",
"4.",
"A numerical scheme for the canonical nonconservative form (1-11).",
"The initial scheme in [5] has been extended to the more complicated system of collisional self gravitating fluids [6], [9], to the systems of ideal gases and shallow water equations [7], [10], to the singular shocks and the $\\delta ^{(n)}$ shocks of the sytems of Keyfitz-Kranzer [39], [40] and Panov-Shelkovich [49] in [8] and to the four equations model of two fluid flows in [11] section 7.",
"The basic method of the scheme consists in a splitting of the equations (1-6) into a transport step: transport of the fluid i with velocity $v_i$ , i=1,2 (though (16,17) are not exactly a transport).",
"$\\frac{\\partial }{\\partial t}(\\rho _1 \\alpha _1)+\\frac{\\partial }{\\partial x}(\\rho _1\\alpha _1 v_1)=0,$ $\\frac{\\partial }{\\partial t}(\\rho _2 \\alpha _2)+\\frac{\\partial }{\\partial x}(\\rho _2\\alpha _2 v_2)=0,$ $\\frac{\\partial }{\\partial t}(\\rho _1 \\alpha _1v_1)+\\frac{\\partial }{\\partial x}(\\rho _1\\alpha _1( v_1)^2)=0,$ $\\frac{\\partial }{\\partial t}(\\rho _2 \\alpha _2v_2)+\\frac{\\partial }{\\partial x}(\\rho _2\\alpha _2( v_2)^2)=0,$ $\\frac{\\partial }{\\partial t}(E_1)+p\\frac{\\partial }{\\partial t}(\\alpha _1) +\\frac{\\partial }{\\partial x}((E_1+p\\alpha _1)v_1)=0, $ $\\frac{\\partial }{\\partial t}(E_2)+p\\frac{\\partial }{\\partial t}(\\alpha _2) +\\frac{\\partial }{\\partial x}((E_2+p\\alpha _2)v_2)=0, $ and a pressure correction step $\\frac{\\partial }{\\partial t}(\\rho _1 \\alpha _1)=0,$ $\\frac{\\partial }{\\partial t}(\\rho _2 \\alpha _2)=0,$ $\\frac{\\partial }{\\partial t}(\\rho _1 \\alpha _1v_1)+\\alpha _1\\frac{\\partial p}{\\partial x}+\\tau _i=g\\alpha _1\\rho _1,$ $\\frac{\\partial }{\\partial t}(\\rho _2 \\alpha _2v_2)+\\alpha _2\\frac{\\partial p}{\\partial x}-\\tau _i=g\\alpha _2\\rho _2,$ $\\frac{\\partial }{\\partial t}(E_1)+p\\frac{\\partial }{\\partial t}(\\alpha _1) +v_{\\tau }\\tau _i=g\\rho _1\\alpha _1v_1, $ $\\frac{\\partial }{\\partial t}(E_2)+p\\frac{\\partial }{\\partial t}(\\alpha _2) -v_{\\tau }\\tau _i=g\\rho _2\\alpha _2v_2, $ with, in between, an averaging step and, in the case of system (1-11), auxiliary algebraic calculations not needed in the case of a single fluid.",
"We notice the groups $\\frac{\\partial }{\\partial t}(E_j)+p\\frac{\\partial }{\\partial t}(\\alpha _j), \\ j=1,2$ , which are not time derivatives.",
"We simply discretize them as $\\frac{(E_j)(t+\\Delta t)-(E_j)(t)+p(t) ((\\alpha _j)(t+\\Delta t)-(\\alpha _j)(t))}{\\Delta t}.$ Now we give the formulas of the scheme.",
"We set $r_j=\\rho _j\\alpha _j$ and $F_j=E_j+p\\alpha _j, \\ j=1,2$ .",
"The notation $F$ serves to contain $E$ and $\\alpha $ as a whole for time evolution according to formula (29), them to compute $E_j(t+\\Delta t)$ and $\\alpha _j(t+\\Delta t)$ from algebraic calculations.",
"We assume we have computed at time $t_n=n\\Delta t$ the family $\\lbrace (r_1)_i^n,(r_2)_i^n,(r_1v_1)_i^n, (r_2v_2)_i^n, (F_1)_i^n, (F_2)_i^n,p_i^n\\rbrace _{i\\in \\mathbb {Z}}.$ We seek the values at $t_{n+1}=(n+1)\\Delta t$ .",
"We set as usual $r=\\frac{\\Delta t}{h}$ where $h$ is the space step.",
"We set $(v_j)_i^n=\\frac{(r_jv_j)_i^n}{(r_j)_i^n}$ , and $ ((v_j)_i^n)^+=max((v_j)_i^n,0), ((v_j)_i^n)^-=max(-(v_j)_i^n,0), j=1,2.$ $\\bullet $ first step: transport.",
"$(\\widetilde{r_1})_i=(r_1)_i^n+r[(r_1)_{i-1}^n((v_1)_{i-1}^n)^+-(r_1)_{i}^n(|v_1|_{i}^n)+(r_1)_{i+1}^n((v_1)_{i+1}^n)^-],$ $(\\widetilde{r_2})_i=(r_2)_i^n+r[(r_2)_{i-1}^n((v_2)_{i-1}^n)^+-(r_2)_{i}^n(|v_2|_{i}^n)+(r_2)_{i+1}^n((v_2)_{i+1}^n)^-],$ $(\\widetilde{r_1v_1})_i=(r_1v_1)_i^n+r[(r_1v_1)_{i-1}^n((v_1)_{i-1}^n)^+-(r_1v_1)_{i}^n(|v_1|_{i}^n)+(r_1v_1)_{i+1}^n((v_1)_{i+1}^n)^-],$ $(\\widetilde{r_2v_2})_i=(r_2v_2)_i^n+r[(r_2v_2)_{i-1}^n((v_2)_{i-1}^n)^+-(r_2v_2)_{i}^n(|v_2|_{i}^n)+(r_2v_2)_{i+1}^n((v_2)_{i+1}^n)^-],$ $(\\widetilde{F_1})_i=(F_1)_i^n+r[(F_1)_{i-1}^n((v_1)_{i-1}^n)^+-(F_1)_{i}^n(|v_1|_{i}^n)+(F_1)_{i+1}^n((v_1)_{i+1}^n)^-],$ $(\\widetilde{F_2})_i=(F_2)_i^n+r[(F_2)_{i-1}^n((v_2)_{i-1}^n)^+-(F_2)_{i}^n(|v_2|_{i}^n)+(F_2)_{i+1}^n((v_2)_{i+1}^n)^-],$ For $\\omega =r_1,r_2,r_1v_1$ and $r_2v_2$ $\\widetilde{\\omega }$ stands for $\\omega (t+\\Delta t)$ while $\\widetilde{F_j}, j=1,2$ will be treated as $E_j(t+\\Delta t) +p(t)\\alpha _j(t+\\Delta t)$ from which one will extract $E_j(t+\\Delta t)$ and $\\alpha _j (t+\\Delta t)$ by algebraic calculations.",
"Now we do an averaging of the quantities calculated above.",
"This averaging is indispensable in most cases as exposed in [6], [7], [8].",
"$\\bullet $ second step: averaging of the quantities calculated above.",
"Let $a, 0<a<\\frac{1}{2}$ be a given real number.",
"We replace each of the values generically denoted $(\\widetilde{q})_i$ calculated in the first step (31-36) by $a(\\widetilde{q})_{i-1}+(1-2 a)(\\widetilde{q})_i+a(\\widetilde{q})_{i+1}$ .",
"This last quantity is denoted by $\\overline{q}_i$ .",
"The choice of the value $a, \\ 0<a<\\frac{1}{2}$ , is done from numerical tests as explained in [7] p.17.",
"Therefore after this averaging we have the six families of values $\\lbrace (\\overline{r_1})_i, (\\overline{r_2})_i, (\\overline{r_1v_1})_i, (\\overline{r_2v_2})_i,(\\overline{F_1})_i,(\\overline{F_2})_i\\rbrace _{i\\in \\mathbb {Z}}.$ Further, we set $(\\overline{v_1})_i=\\frac{(\\overline{r_1v_1})_i}{ (\\overline{r_1})_i}, (\\overline{v_2})_i=\\frac{(\\overline{r_2v_2})_i}{ (\\overline{r_2})_i}.$ $\\bullet $ third step: auxiliary algebraic calculations.",
"We compute values denoted by $\\overline{q} $ for $q=E_1, E_2, \\alpha _1, \\alpha _2, p_1$ and $p_2$ from the values (37,38).",
"This could also be done from the $\\widetilde{q}$ values in (31-36), defining $v_1$ and $v_2$ by the usual quotient.",
"We exploit the four algebraic formulas (10,11,7,$p_1=p_2$ ) and the two values $[(\\overline{E_1})_i+p_i^n(\\overline{\\alpha _1})_i]=(\\overline{F_1})_i$ and $[(\\overline{E_2})_i+p_i^n(\\overline{\\alpha _2})_i]=(\\overline{F_2})_i$ obtained above.",
"This gives 6 equations for the 6 unknown values $(\\overline{E_1})_i, (\\overline{E_2})_i, (\\overline{\\alpha _1})_i, (\\overline{\\alpha _2})_i, (\\overline{p_1})_i $ and $(\\overline{p_2})_i$ .",
"After some easy algebraic calculations one obtains that $(\\overline{\\alpha _1})_i$ is the root located in $]0,1[$ of the equation $AX^2+BX+C=0$ where $A=(K_1-1)p_i^n+K_1p_{\\infty ,1}-(K_2-1)p_i^n-K_2p_{\\infty ,2},$ $ B=-(K_1-1)(\\overline{F_1})_i-(K_2-1)(\\overline{F_2})_i-K_1p_{\\infty ,1}+K_2p_{\\infty ,2}+\\frac{1}{2}(K_1-1)(\\overline{r_1v_1})_i(\\overline{v_1})_i+$ $\\frac{1}{2}(K_2-1)(\\overline{r_2v_2})_i(\\overline{v_2})_i+(K_2-K_1)p_i^n,$ $C=(K_1-1)(\\overline{F_1})_i-\\frac{1}{2}(K_1-1)(\\overline{r_1v_1})_i(\\overline{v_1})_i.$ Once one has calculated $(\\overline{\\alpha _1})_i$ one sets $(\\overline{\\alpha _2})_i=1-(\\overline{\\alpha _1})_i,$ $(\\overline{E_1})_i=(\\overline{F_1})_i-p_i^n(\\overline{\\alpha _1})_i,(\\overline{E_2})_i=(\\overline{F_2})_i-p_i^n(\\overline{\\alpha _2})_i,$ $ (\\overline{\\rho _1})_i=\\frac{(\\overline{r_1})_i}{(\\overline{\\alpha _1})_i},(\\overline{\\rho _2})_i=\\frac{(\\overline{r_2})_i}{(\\overline{\\alpha _2})_i},$ then the values of $\\overline{p}_i$ obtained from (10,11) (by construction they are the same for $p_1$ and $p_2$ ).",
"From the formulas $\\gamma _1=K_1-1,\\gamma _2=K_2-1$ [48], [46] and from (8,9) one has the values $(\\overline{v_\\tau })_i$ and $(\\overline{\\Delta p})_i$ for $v_\\tau $ and $\\Delta p$ respectively.",
"$\\bullet $ fourth step: pressure correction.",
"Finally, we set $ (r_1v_1)_i^{n+1}=(\\overline{r_1v_1})_i-\\frac{r}{2}(\\overline{\\Delta p})_{i}((\\overline{\\alpha _1})_{i+1}-(\\overline{\\alpha _1})_{i-1})-\\frac{r}{2}(\\overline{\\alpha _1})_{i}((\\overline{p})_{i+1}-(\\overline{p})_{i-1})+rhg(\\overline{r_1})_i,$ $ (r_2v_2)_i^{n+1}=(\\overline{r_2v_2})_i+\\frac{r}{2}(\\overline{\\Delta p})_i((\\overline{\\alpha _1})_{i+1}-(\\overline{\\alpha _1})_{i-1})-\\frac{r}{2}(\\overline{\\alpha _2})_i((\\overline{p})_{i+1}-(\\overline{p})_{i-1})+rhg(\\overline{r_2})_i,$ $ (F_1^*)_i=(\\overline{F}_1)_i-\\frac{r}{2}(\\overline{v_\\tau })_i(\\overline{\\Delta p})_i((\\overline{\\alpha _1})_{i+1}-(\\overline{\\alpha _1})_{i-1})+rhg(\\overline{r_1v_1})_i,$ $ (F_2^*)_i=(\\overline{F}_2)_i+\\frac{r}{2}(\\overline{v_\\tau })_{i}(\\overline{\\Delta p})_{i}((\\overline{\\alpha _1})_{i+1}-(\\overline{\\alpha _1})_{i-1})+rhg(\\overline{r_2v_2})_{i}.$ We set $(r_1)_i^{n+1}=(\\overline{r_1})_i,\\ (r_2)_i^{n+1}=(\\overline{r_2})_i, (v_1)_i^{n+1}=\\frac{(r_1v_1)_i^{n+1}}{(r_1)_i^{n+1}}$ and $(v_2)_i^{n+1}=\\frac{(r_2v_2)_i^{n+1}}{(r_2)_i^{n+1}}$ .",
"$\\bullet $ fifth step: auxiliary algebraic calculations.",
"One has the values $(r_1)_i^{n+1}, (r_2)_i^{n+1}, (r_1v_1)_i^{n+1}, (r_2v_2)_i^{n+1}, (F_1)_i^*$ and $(F_2)_i^*$ as in situation (37), as well as the values $\\overline{p}_i$ obtained in the third step, that play here the role played by $p_i^n$ in the third step.",
"The formulas (39-42) with these values give values of $(\\alpha _j)_i^{n+1}$ , then $ (E_j)_i^{n+1}$ and $(p_j)_i^{n+1}$ with $(p_1)_i^{n+1}=(p_2)_i^{n+1}$ from (42,43,10,11).",
"We denote this value $p_i^{n+1}=(p_1)_i^{n+1}=(p_2)_i^{n+1}$ and we set $(F_j)_i^{n+1}=(E_j)_i^{n+1}+p_i^{n+1}(\\alpha _j)_i^{n+1}, j=1,2$ .",
"5.",
"A numerical scheme for the system solved in time derivatives.",
"A nicer formulation that involves time derivatives in the energy equations only in the variables $E_j, \\ j=1,2$ has been obtained in [48] p 2595-2596: the nonconservative temporal derivatives in (5,6) (formula (1) in [48]) are eliminated (formulas (2,3) in [48]).",
"Equations (1-4) are unchanged.",
"In [46] p. 414 equations (5,6) are stated as $\\frac{\\partial }{\\partial t}(E_1)+\\frac{\\partial }{\\partial x}(E_1v_1)+(\\alpha _1v_1-\\eta \\alpha _1\\alpha _2(v_1-v_2))\\frac{\\partial p}{\\partial x}+\\eta \\rho _2\\alpha _1(c_2)^2\\frac{\\partial }{\\partial x}(\\alpha _1v_1+\\alpha _2v_2))+v_{\\tau }\\tau _i=g\\rho _1\\alpha _1v_1, $ $\\frac{\\partial }{\\partial t}(E_2)+\\frac{\\partial }{\\partial x}(E_2v_2)+(\\alpha _2v_2+\\eta \\alpha _1\\alpha _2(v_1-v_2))\\frac{\\partial p}{\\partial x}+\\eta \\rho _1\\alpha _2(c_1)^2\\frac{\\partial }{\\partial x}(\\alpha _1v_1+\\alpha _2v_2))-v_{\\tau }\\tau _i=g\\rho _2\\alpha _2v_2, $ where $c_j, \\ j=1,2$ is the sound velocity $ (c_j)^2=\\frac{p+K_jp_{\\infty ,j}}{\\rho _j}$ and where $ \\eta =\\frac{p}{\\rho _1\\alpha _2(c_1)^2+\\rho _2\\alpha _1(c_2)^2}.$ It is proved in [48] section 2.3.3 that the equations (48-51) are mathematically equivalent to (5-9) in the case of smooth solutions.",
"Therefore it is important to check this transformation in the case of discontinuous solutions: a numerical verification will be done below in figures 1 and 2 in the hyperbolic and in the nonhyperbolic case respectively.",
"The numerical scheme is obtained as in section 2 from the three steps of transport, averaging and pressure correction, with auxiliary algebraic calculations.",
"The transport step is made of equations (17-20) for $r_1, r_2, r_1v_1$ and $r_2v_2$ and the following simple transport equations $ \\frac{\\partial E_j}{\\partial t}+\\frac{\\partial (E_jv_j)}{\\partial x}=0 $ for $E_1$ and $E_2$ , that replace (21,22).",
"Now we give the scheme in more details.",
"One starts with the values $\\lbrace (r_1)_i^n,(r_2)_i^n,(r_1v_1)_i^n,(r_2v_2)_i^n,(v_1)_i^n,(v_2)_i^n,(E_1)_i^n,(E_2)_i^n\\rbrace _{i\\in \\mathbb {Z}}.$ $\\bullet $ first step: transport.",
"The values $\\widetilde{r_1}, \\widetilde{r_2}, \\widetilde{r_1v_1}$ and $\\widetilde{r_2v_2}$ are given by formulas (31-34).",
"From (52) we obtain $\\widetilde{E_j}, \\ j=1,2$ : $(\\widetilde{E_j})_i=(E_j)_i^n+r[(E_j)_{i-1}^n((v_j)_{i-1}^n)^+-(E_j)_{i}^n(|v_j|_{i}^n)+(E_j)_{i+1}^n((v_j)_{i+1}^n)^-].$ $\\bullet $ second step: averaging.",
"The averaging step is identical to the one in section 2 for $r_1, r_2, r_1v_1$ and $r_2v_2$ ; it is also done for $E_1$ and $E_2$ .",
"Then one states $(\\overline{v_j})_i=\\frac{(\\overline{r_jv_j})_i}{(\\overline{r_j})_i}, \\ j=1,2$ .",
"At this point we have values $(\\overline{r_1})_i, (\\overline{r_2})_i, (\\overline{r_1v_1})_i, (\\overline{r_2v_2})_i, (\\overline{v_1})_i, (\\overline{v_2})_i, (\\overline{E_1})_i$ and $(\\overline{E_2})_i$ .",
"For the pressure correction term we need the values of $(\\alpha _1)_i^n, (\\alpha _2)_i^n, (\\rho _1)_i^n, (\\rho _2)_i^n$ and $p_i^n$ .",
"We obtain them by auxiliary algebraic calculations in the third step.",
"$\\bullet $ third step: auxiliary algebraic calculations.",
"Algebraic calculations with the values obtained in the second step, as this is done in section 2, would be more natural and would permit a better CFL condition ($r=0.002$ in the test of figure 1 instead of r=0.0012).",
"Nevertheless we perform the algebraic calculations with the values (53) because this is more convenient for the proof of the theorem, and at the same time gives same numerical result.",
"The value $\\alpha _i^n=(\\alpha _1)_i^n$ is obtained as the solution of the equation $AX^2+BX+C=0$ which lies in $]0,1[$ , with $A=K_1p_{\\infty ,1}-K_2p_{\\infty ,2},$ $ B=-(K_1-1)(E_1)_i^n-(K_2-1)(E_2)_i^n-K_1p_{\\infty ,1}+K_2p_{\\infty ,2}+\\frac{1}{2}(K_1-1)(r_1v_1)_i^n(v_1)_i^n+$ $\\frac{1}{2}(K_2-1)(r_2v_2)_i^n(v_2)_i^n,$ $C=(K_1-1)(E_1)_i^n-\\frac{1}{2}(K_1-1)(r_1v_1)_i^n(v_1)_i^n.$ Then we set $(\\alpha _2)_i^n=1-\\alpha _i^n, (\\rho _j)_i^n=\\frac{(r_j)_i^n}{(\\alpha _j)_i^n},$ $(p_j)_i^n=(K_j-1)(\\frac{(E_j)_i^n}{(\\alpha _j)_i^n}-\\frac{1}{2}(\\rho _j)_i^n((v_j)_i^n)^2)-K_j p_{\\infty ,j}, \\ p_i^n=(p_1)_i^n=(p_2)_i^n,$ $(\\eta )_i^n=\\frac{p_i^n}{(c_2)^2(\\alpha _1)_i^n(\\rho _2)_i^n+(c_1)^2(\\alpha _2)_i^n(\\rho _1)_i^n}.$ We calculate the quantities $(v_\\tau )_i^n$ and $(\\Delta p)_i^n$ from (8,9).",
"Then, dropping the superscript $n$ for convenience in second members of (60,61), we set (formulas 2.10,2.11 in [46] p. 414) $ (T_1)_i^n=(\\alpha _1)_i(v_1)_i(p_{i+1}-p_{i-1})-\\eta _i(\\alpha _1)_i(\\alpha _2)_i((v_1)_{i}-(v_2)_{i})(p_{i+1}-p_{i-1})+$ $\\eta _i(\\rho _2)_i(\\alpha _1)_i((c_2)^2)_i((\\alpha _1)_{i+1}(v_1)_{i+1}-(\\alpha _1)_{i-1}(v_1)_{i-1})+\\eta _i(\\rho _2)_i(\\alpha _1)_i((c_2)^2)_i$ $((\\alpha _2)_{i+1}(v_2)_{i+1}-(\\alpha _2)_{i-1}(v_2)_{i-1})+(v_\\tau )_i(\\Delta p)_i((\\alpha _1)_{i+1}-(\\alpha _1)_{i-1}),$ $ (T_2)_i^n=(\\alpha _2)_i(v_2)_i(p_{i+1}-p_{i-1})+\\eta _i(\\alpha _1)_i(\\alpha _2)_i((v_1)_{i}-(v_2)_{i})(p_{i+1}-p_{i-1})+$ $\\eta _i(\\rho _1)_i(\\alpha _2)_i((c_1)^2)_i((\\alpha _1)_{i+1}(v_1)_{i+1}-(\\alpha _1)_{i-1}(v_1)_{i-1})+\\eta _i(\\rho _1)_i(\\alpha _2)_i((c_1)^2)_i$ $((\\alpha _2)_{i+1}(v_2)_{i+1}-(\\alpha _2)_{i-1}(v_2)_{i-1})-(v_\\tau )_i(\\Delta p)_i((\\alpha _1)_{i+1}-(\\alpha _1)_{i-1}).$ $\\bullet $ fourth step: pressure correction.",
"For $r_1v_1$ and $r_2v_2$ one uses formulas (44) and (45) with the values of $(\\Delta p)_i^n,\\alpha _i^n,p_i^n$ obtained in the third step and the values $(r_j)_i^{n}$ .",
"For $E_1$ and $E_2$ , we set $ (E_j)_i^{n+1}=(\\overline{E_j})_i^n-\\frac{1}{2}r (T_j)_i^n+rhg(r_jv_j)_i^n, \\ j=1,2.$ At this point we have values $(r_1)_i^{n+1}, (r_2)_i^{n+1}, (r_1v_1)_i^{n+1}, (r_2v_2)_i^{n+1}$ , $(v_1)_i^{n+1}=\\frac{(r_1v_1)_i^{n+1}}{(r_1)_i^{n+1}}, (v_2)_i^{n+1}=\\frac{(r_2v_2)_i^{n+1}}{(r_2)_i^{n+1}}, (E_1)_i^{n+1}$ and $(E_2)_i^{n+1}$ , which permits to continue the induction on $n$ .",
"6.",
"Numerical Results on the Toumi shock tube problem.",
"INSERT FIGURE 1 Figure 1.",
"Comparison of the hyperbolic models ($\\delta =2$ ) solved (black) and nonsolved (red, nearly completely hidden under the black) in time derivative with the scheme of sections 3 and 4: exact coincidence.",
"The canonical nonconservative form (not solved in time derivative) and the rewritten model (solved in time derivative) have same solution in presence of the additional term that ensures hyperbolicity [48] p. 2595 (i.e.",
"$\\delta =2$ for both models).",
"Here $t= 0.06, 10.000$ space steps, $r=0.0012$ and $a=0.3$ for both systems.",
"Further, this solution is identical to the one presented in [48] pp.",
"2615,2617 and in [46] p. 437.",
"INSERT FIGURE 2 Figure 2.",
"Comparison of the nonhyperbolic models ($\\delta =0)$ solved (black) and nonsolved (red, nearly completely hidden under the black) in time derivative with the scheme of sections 3 and 4: exact coincidence.",
"The canonical nonconservative form (not solved in time derivative) and the rewritten model (solved in time derivative) have same solution in absence of the additional term that ensures hyperbolicity (i.e.",
"$\\delta =0$ for both models).",
"Here $t=0.06, 4000$ space steps, $r=0.0009$ and $a=0.3$ for both systems.",
"The results observed in figures 1 and 2 constitute a numerical verification of our scheme and of the validity of the transformation of equations in case of discontinuous solutions [48].",
"Figure 2 also brings a numerical confirmation that some nonlinear nonhyperbolic systems could have well posed solutions as noticed by B. Keyfitz et al.",
"in [31], [36], [37], [38], [32], [33], [34], [35], [41], [42].",
"INSERT FIGURE 3 Figure 3.",
"Comparison of the hyperbolic model ($\\delta =2$ , black) and the nonhyperbolic model ($\\delta =0$ , red) on 2000 space steps with the scheme of section 4.",
"Figure 3 shows numerical solutions of the system solved in time derivative with and without additional term at $t=0.06, 2000$ space steps, $r=0.0012$ and $a=0.3$ for both systems.",
"One observes that the results are really close, in particular the step values are always the same, with an irregularity in gas volume fraction and peaks in gas velocity and liquid velocity in absence of the additional term (i.e.",
"$\\delta =0$ ).",
"After a longer time-with correspondingly a longer tube-we have observed that the aspects of both solutions did not change significantly.",
"The first observation on figures 1 and 2 is that the two forms of the system, solved and nonsolved in time derivative, do give exactly the same numerical solutions even in presence of shocks and other irregularities.",
"Then we observe that the hyperbolic form of the system gives the solution obtained in [48] pp.",
"2615,2617 and [46] p. 437 by completely different numerical methods.",
"We observe on figure 3 differences due to the additional term that renders the system hyperbolic, [48] p. 2595, but also there are strong similarities such as all step values and three physical variables practically identical: pressure, gas temperature and liquid temperature.",
"What is the status of the irregularity observed in gas volume fraction, and of the peaks in gas velocity and liquid velocity in the nonhyperbolic model?",
"They appear as a small region with large gas and liquid velocities together with a liquid volume fraction close to one.",
"First, one observes that these results are to a large extent independent of the values of the space steps, of the number of iterations and of the averaging.",
"The top values and width of the peaks are well defined; they grow slowly with time.",
"They are reproduced from the higher order scheme even with a small number of space steps as expected for a genuine solution (figure 5).",
"Therefore they do not look like numerical artefacts.",
"The point to decide if these peaks, which could be easily observed by engineers, are really present in the physical solution depends on experimental measurements of the gas kick phenomenon.",
"Experimental reproductions of the gas kick and measurements have been done using a vertical 1240 meters deep test well [2].",
"First we note that when we reproduce all numerical tests in this paper in a vertical (100 meters long) tube (instead of the horizontal 100 meters long tube under consideration) one observes that all results are practically unchanged, and also in presence of viscosity.",
"Therefore, the results of the nonhyperbolic model in a vertical tube still show the same peak in liquid velocity conjointly with a liquid volume fraction close to one (top left figure on the gas volume fraction), which permits some attempts of comparison with the experimental results.",
"Experimental measurements of the liquid flow rate ($m^3/s$ ) are shown in [2], figure 4 p. 17.",
"Experimentalists observed a peak in liquid flow rate far higher than the surrounding step values (about 2 or 3 times higher: [2] figure 4 p. 17), which was underestimated by the hyperbolic model they used, [2] p.15.",
"Although these experimental conditions are somewhat different from the Toumi numerical test presented in this paper, even in a vertical tube and taking viscosity into account, this experimental observation and the analogy with the results in figure 3 suggest that the peak in liquid flow rate observed in the nonhyperbolic model could perhaps be physically realistic and, perhaps, be, in part, interpreted in the gas kick phenomenon.",
"Viscosity is present in the physical problem ([22], [23], [24], [26], [27], [28], [29], [30]).",
"An easy modification of the scheme by introduction of the viscosity into the transport step with the usual discretization of second order derivatives in the two momentum equations and the two energy equations with various relevant viscosity coefficients for gas (about $10^{-3}$ ) and liquid (from $10^{-3}$ to $10^2$ ) did not show a significant influence of the presence of viscosity both for the hyperbolic and the nonhyperbolic versions.",
"Now we compare the scheme of order 3 in space exposed in section 6 with the order 1 scheme used up to now.",
"INSERT FIGURE 4 Figure 4.",
"Hyperbolic model ($\\delta =2$ ).",
"Comparison of the order 3 scheme in space (black) and of the order 1 scheme of section 4 on 2000 space steps.",
"We observe that the $p=3$ scheme (black color, section 6) gives exactly the same result as the previous (i.e.",
"$p=1$ ) scheme (red color) but with a better accuracy since one observes details in figure 1 and $\\cite {Munk}$ which do not appear from the previous scheme (red) with same value of the space step; 2000 space steps, $r=0.001$ and $a=0.2$ for the previous scheme, $r=0.0002$ and $a=0.025$ for the $p=3$ scheme INSERT FIGURE 5 Figure 5.",
"Nonhyperbolic model ($\\delta =0$ ).",
"Comparison of the order 3 scheme in space (black) and of the order 1 scheme of section 3 (red) on 500 space steps.",
"The result from this last scheme is far from the final results in figures 2 and 3 while the scheme of section 7 gives this final result.",
"We observe that with 500 space steps the $p= 3$ scheme (black) gives practically the result of the $p=1$ scheme with 2000 cells (figure 3), with further some minor parasite oscillations; $r=0.001$ and $a=0.2$ for the $p=1$ scheme, $r=10^{-5}$ and $a=0.06$ for the $p=3$ scheme.",
"With 500 space steps the $p=1$ scheme of section 3 does not give a correct result since the peak in gas velocity is not apparent.",
"The observation that the peculiarities of the nonhyperbolic model in figures 2 and 3 appear with a better accuracy with the higher order $p=3$ scheme suggests again that they are really part of a solution of the nonhyperbolic model.",
"7.",
"Higher order versions of the scheme.",
"Since the system is demanding from the numerical viewpoint one faces the need of a more efficient scheme.",
"In this section the scheme is easily transformed into higher order schemes.",
"The scheme in use up to now is of order 1 in the time increment $\\Delta t=rh$ and of order 1 in the space increment $\\Delta x=h$ .",
"Since the value of $r$ is small the main point is to increase the order in space.",
"The scheme is made of 3 main steps: transport, averaging and pressure correction.",
"We introduce a $p$ -discretization in space , $p\\in \\mathbb {N}, \\ p$ odd, which consists in performing all of these 3 steps using values $\\omega _j, i-p\\le j\\le i+p$ to calculate the next value $\\omega _i$ , which is very easy since the needed coefficients can be obviously calculated before the iterations by resolving two small $(p+1)\\times (p+1)$ and $p\\times p$ linear systems whose entries are powers of integers from Taylor's formula so as to maximize the order in $h$ in the transport step and in the pressure correction step.",
"Explicit coefficients are given below for the $p=3$ scheme; for $p=5$ the coefficients were numerically computed.",
"The scheme used up to now corresponds to $p=1$ .",
"Such a $p$ -scheme will be of order $ p$ in the space increment: the transport step is discretized at order $p$ in $\\Delta x$ and the other steps are discretized at order $>p$ .",
"Since the stability appeared bad already for $p=3$ (too small value of $r=\\frac{\\Delta t}{\\Delta x}$ ), a second averaging step was used after the pressure correction step.",
"Since this improvement was still unsatisfactory the two averaging steps and the transport step were performed implicitly.",
"Then, in the case of ideal gases, the results became satisfactory with a value $r$ often comparable to the one of the $p=1$ scheme in use up to now (an exception in figure 5 where $r$ is very small since the nonhyperbolic model is very demanding).",
"In some cases when $p=3$ and 5 the tests show parasite oscillations on a few meshes at the vertical of the discontinuities.",
"They are essentially eliminated by a post-treatment on the final solution by replacing the final values by mean values on a few cells on the left and on the right.",
"The $p=3$ scheme and the similar $p=5$ scheme were tested on the 1-D system of ideal gases and for the system in this paper.",
"The $p=3$ scheme is used in figures 4 and 5 in this paper.",
"Now we give the detailed formulas of the $p=3$ scheme.",
"$\\bullet $ The transport step.",
"The values $\\lbrace \\omega _i^n,v_i^n\\rbrace _{i\\in \\mathbb {Z}}$ are known and we compute values improperly denoted $\\lbrace \\omega _i^{n+1}\\rbrace _{i\\in \\mathbb {Z}}$ at the end of the transport step to be inserted into the next step of averaging.",
"The implicit formula is: $ \\omega _i^{n+1}=\\omega _i^{n}-r\\lbrace -\\frac{1}{3}\\omega _{i-3}^{n+1}v_{i-3}^{n,+}+\\frac{3}{2}\\omega _{i-2}^{n+1}v_{i-2}^{n,+}-3\\omega _{i-1}^{n+1}v_{i-1}^{n,+}+\\frac{11}{6}\\omega _{i}^{n+1}v_{i}^{n,+}+$ $\\frac{11}{6}\\omega _{i}^{n+1}v_{i}^{n,-}-3\\omega _{i+1}^{n+1}v_{i+1}^{n,-}+\\frac{3}{2}\\omega _{i+2}^{n+1}v_{i+2}^{n,-}-\\frac{1}{3}\\omega _{i+3}^{n+1}v_{i+3}^{n,-}\\rbrace .$ Calculation is done with a sparse matrix having 7 nonzero diagonals.",
"The sum of the $v^+$ terms (respectively of the $v^-$ terms) above replaces the expression $\\omega _{i-1}^{n}v_{i-1}^{n,+}-\\omega _{i}^{n}v_{i}^{n,+}$ (resp.",
"$-\\omega _{i}^{n}v_{i}^{n,-}+\\omega _{i+1}^{n}v_{i+1}^{n,-}$ ) of the explicit $p=1$ scheme used up to now.",
"Semi-implicit RK2 and RK4 methods in time can be used easily in place of the implicit Euler order one method (63), due to the independence in time and linearity in $\\omega $ of the transport formula, and have given far better results.",
"$\\bullet $ The two averaging steps.",
"Starting from values improperly denoted $\\lbrace \\omega _i^n\\rbrace _{i\\in \\mathbb {Z}}$ the implicit formula is (composition of two Laplacians): $ \\omega _i^{n+1}=\\omega _i^{n}+a\\lbrace -\\omega _{i-3}^{n+1}+6\\omega _{i-2}^{n+1}-15\\omega _{i-1}^{n+1}+20\\omega _{i}^{n+1}-15\\omega _{i+1}^{n+1}+6\\omega _{i+2}^{n+1}-\\omega _{i+3}^{n+1}\\rbrace .$ The value $a$ is chosen from numerical tests as exposed for the order one scheme in [7] p.17.",
"One averaging step is done after the transport step and another averaging after the pressure correction step.",
"$\\bullet $ The pressure correction step.",
"We use a centered discretization in space: $\\partial _x \\omega $ is discretized by $ \\frac{1}{303}\\omega _{i+3}-\\frac{39}{404}\\omega _{i+2}+\\frac{69}{101}\\omega _{i+1}-\\frac{69}{101}\\omega _{i-1}+\\frac{39}{404}\\omega _{i-2}-\\frac{1}{303}\\omega _{i-3}.$ This can be done with the explicit Euler order one scheme in time (figures 4 and 5).",
"It has been observed that the explicit RK2 and RK4 schemes in time (using the above discretization in space) give better results.",
"The RK2 and RK4 discretizations in time have not been presented in figures 4 and 5 (where we used only the Euler order one methods in time, implicit in transport and explicit in pressure correction) to stress the very significant improvements already obtained only from the space discretization which is original, since the various possible time discretizations are standard.",
"Extension to 2-D and 3-D can be done using the transport formulas in [6] pp.",
"96-100.",
"In 1-D we considered separately terms in $v^+$ and in $v^-$ in (63) that were discretized up to indices $k, \\ i-p\\le k\\le i+p$ .",
"The $p=1$ expression involving $v^+$ , respectively $v^-$ , was transformed into the left, respectively right, part of the $\\lbrace \\dots \\rbrace $ in (63) for $p=3$ .",
"Similarly, in 2-D if the velocity vector is denoted by $(u,v)$ one considers separately terms in $(u^+,v^+), (u^+,v^-), (u^-,v^+)$ and $ (u^-,v^-)$ in the $p=1$ transport formulas in [6] pp.",
"96-100, that we replace for the calculation of $\\omega _{i,j}^{n+1}$ in the $p$ -scheme by expressions involving $\\omega _{k,q}^n$ for $i-p\\le k\\le i+p, j-p\\le q\\le j+p$ chosen from Taylor's formula in 2-D so as to maximize the order in the space step $\\Delta x=\\Delta y$ .",
"As an application one could study multifluid flows in multidimension as done in 1-D in this paper.",
"The 2-D $p=1$ scheme was sufficient in [7] to obtain exactly the results presented in [44], [45] (obtained there by order two schemes) but in the more demanding context of multifluid flows the higher order schemes could presumably be useful.",
"8.",
"Conclusion.",
"We have checked that a numerical scheme used in [5], [6], [7], [8], [11] applies also for the standard system of one pressure model of two fluid flows with energy equations.",
"The novelty is that our scheme works in the hyperbolic and nonhyperbolic cases, whether solved and nonsolved in time derivative, that we obtain well defined approximate solutions that tend to satisfy the equations somewhat independently on a choice of the nonconservative product both in the hyperbolic case and in the nonhyperbolic case.",
"Comparisons with previous works [48], [46] in the hyperbolic case show perfect agreement.",
"In the nonhyperbolic case we observe the appearance of a peak in liquid flow rate, whose possible physical relevance in relation with the gas kick phenomenon remains to be elucidated.",
"Our scheme has been explicitely stated at order three in space and possible extensions to higher order in multidimension are sketched.",
"Acknowledgements.",
"Figure: NO_CAPTION Figure: NO_CAPTION Figure: NO_CAPTION Figure: NO_CAPTION Figure: NO_CAPTION"
]
] | 1808.08467 |
[
[
"Composable block solvers for the four-field double porosity/permeability\n model"
],
[
"Abstract The objective of this paper is twofold.",
"First, we propose two composable block solver methodologies to solve the discrete systems that arise from finite element discretizations of the double porosity/permeability (DPP) model.",
"The DPP model, which is a four-field mathematical model, describes the flow of a single-phase incompressible fluid in a porous medium with two distinct pore-networks and with a possibility of mass transfer between them.",
"Using the composable solvers feature available in PETSc and the finite element libraries available under the Firedrake Project, we illustrate two different ways by which one can effectively precondition these large systems of equations.",
"Second, we employ the recently developed performance model called the Time-Accuracy-Size (TAS) spectrum to demonstrate that the proposed composable block solvers are scalable in both the parallel and algorithmic sense.",
"Moreover, we utilize this spectrum analysis to compare the performance of three different finite element discretizations (classical mixed formulation with H(div) elements, stabilized continuous Galerkin mixed formulation, and stabilized discontinuous Galerkin mixed formulation) for the DPP model.",
"Our performance spectrum analysis demonstrates that the composable block solvers are fine choices for any of these three finite element discretizations.",
"Sample computer codes are provided to illustrate how one can easily implement the proposed block solver methodologies through PETSc command line options."
],
[
"INTRODUCTION",
"Due to recent growth in the exploration of hydrocarbons from unconventional sources (i.e., oil and gas from tight shale), there has been a growing interest to understand and to model flows in porous media with complex pore-networks [52].",
"This interest has been amplified due to recent advances in additive manufacturing, which allow for creating materials with complex pore-networks for various applications ranging from water purification filters to composite manufacturing.",
"The porous materials in the aforementioned applications typically exhibit two or more dominant pore-networks with each pore-network displaying distinctive hydro-mechanical properties [51], [23].",
"Moreover, due to presence of fissures, there could be mass transfer across the pore-networks [6].",
"To address flows in these types of porous materials, several mathematical models have been proposed in the literature (see [52] and references therein).",
"A class of models, which is commonly referred to as double porosity/permeability (DPP) models, have been found to be particularly attractive in modeling flows in porous media with two pore-networks (e.g., see [6], [53], [24], [9], [44], [20]).",
"Recently, a DPP mathematical model with strong continuum thermomechanics underpinning has been derived in [44].",
"This model, which will be central to this paper and will be referred to as the DPP model from here on, describes the flow of a single-phase incompressible fluid in a rigid porous medium with two distinct pore-networks, with possible mass transfer across the pore-networks.",
"The governing equations form a boundary value problem in terms of four-fields and the nature of the PDE is elliptic under steady-state responses.",
"Except for some academic problems, it is not possible to obtain analytical solutions for the governing equations under the DPP model.",
"Hence, there is a need to resort to numerical solutions.",
"Recently, several numerical formulations have been developed to solve the governing equations under the DPP model; which include [19], [34], [35].",
"However, these works addressed small-scale problems.",
"The problems that arise in subsurface modeling and other applications involving flow through porous media are typically large-scale in nature.",
"These large-scale problems cannot be solved on a standard desktop or by employing direct solvers; as such a computation will be prohibitively expensive.",
"But large-scale problems from subsurface modeling are routinely tackled using parallel computing tools and by employing iterative linear solvers.",
"The current iterative solver methodologies have been developed and successfully employed for either single-field problems (e.g., Poisson's equation, linear elasticity) or for two-field problems (e.g., Darcy equations, Stokes equations) using two-field composable solvers [49], [47], [13].",
"However, there is a gap in knowledge when one wants to solve large-scale problems under the DPP model, which involves four independent field variables.",
"Unlike Darcy equations, the governing equations under the DPP model cannot be written as a single-field Poisson equation solely in terms of pressures [34], or even as a two-field problem.",
"To facilitate solving large-scale problems under the DPP model, we present two four-field composable block solver methodologies.",
"Appealing to PETSc's composable solver features [4], [13] and Firedrake Project's finite element libraries [47], we will show that the proposed composable block solvers can be effectively implemented in a parallel setting.",
"The two salient features of the proposed block solvers are: they are scalable in both the algorithmic and parallel senses.",
"They can be employed under a wide variety of finite element discretizations.",
"Both these features will be illustrated in this paper using representative two- and three-dimensional problems.",
"In order to illustrate that the proposed composable solvers can be used under a wide variety of finite element discretizations, we will employ three popular finite element discretizations – the classical mixed formulation (which is based on the Galerkin formalism) using H(div) elements, the CG-VMS stabilized formulation [34] and the DG-VMS stabilized formulation [35].",
"We will consider H(div) discretizations for simplicial elements (triangle [TRI] and tetrahedron [TET]) and non-simplicial elements (quadrilateral [QUAD] and hexahedron [HEX]).",
"In particular, we employ the lowest-order Raviart-Thomas spaces for simplicial elements [48], [8] The classical mixed formulation using the lowest-order Raviart-Thomas spaces for simplicial elements is commonly referred to as the RT0 formulation.. For non-simplicial elements, the velocity spaces for QUAD and HEX elements are, respectively, $\\mathrm {RCTF_1}$ and $\\mathrm {NCF_1}$ [43], [1] See Figure REF and Table REF for a description of these discretizations..",
"The CG-VMS formulation is based on the variational multi-scale (VMS) formalism [32] and is stable under any arbitrary interpolation order for velocity and pressure fields.",
"The DG-VMS formulation is a discontinuous version of the CG-VMS formulation and is built by combining the VMS formalism and discontinuous Galerkin techniques.",
"The DG-VMS formulation has been shown to accurately capture physical jumps in flow profiles across highly heterogeneous porous media [35].",
"Recently, [16] have proposed the Time-Accuracy-Size (TAS) performance spectrum model, which is an enhanced version of the original spectrum model proposed in [17] obtained by incorporating accuracy into the spectrum model.",
"The TAS spectrum model can be used to study performance of numerical formulations in a parallel setting.",
"Herein, we will utilize the TAS model specifically to achieve the following: (i) We show that the proposed composable solvers are algorithmically scalable.",
"(ii) We compare the performance of the two proposed composable solvers on a particular hardware.",
"(iii) We discuss how the choice of finite element mesh type could affect the solver performance.",
"(iv) We compare the performance of the chosen three finite element discretizations (the classical mixed formulation with H(div) elements, the CG-VMS stabilized formulation and the DG-VMS stabilized formulation) for solving the governing equations under the DPP model.",
"The work reported in this paper will be valuable to subsurface modelers on three fronts.",
"First and the obvious one is that the proposed composable block solver methodologies facilitate solving large-scale problems involving flow through porous media with multiple pore-networks.",
"Second, our work can guide an application scientist to choose a finite element discretization among several choices.",
"Third, our work illustrates on how to utilize performance metrics other than the commonly used metric – the total time to solution – in subsurface modeling.",
"A couple of these other metrics include Digits of Efficacy (DoE) and the total Degrees-of-Freedom (DoF) processed per second (DoF/s).",
"The rest of this paper is organized as follows.",
"The governing equations under the DPP model and convenient grouping of the field variables are presented in Section .",
"The weak forms of the three finite element formulations that are employed in this paper are presented in Section .",
"The proposed two block solver methodologies are discussed in detail in Section .",
"The framework of performance spectrum model along with the guidelines on how to interpret the resulting diagrams are presented in Section .",
"The performance of the proposed block solvers is illustrated using numerical simulations in Section .",
"In the same section, we also compare the performance of the chosen finite element discretizations using the TAS performance spectrum model, which provides guidance to an application scientist with respect to several metrics (e.g., time-to-solution, digits-of-efficacy).",
"Finally, conclusions are drawn in Section ."
],
[
"GOVERNING EQUATIONS AND GROUPING OF FIELD VARIABLES",
"We now document the most important equations under the DPP mathematical model.",
"In this paper, we restrict our treatment of the model to a steady-state response; however, the proposed composable block solvers are equally applicable in a transient setting.",
"We refer the two dominant pore-networks as macro-pore and micro-pore, and the quantities associated with the two pore-networks are, respectively, identified by subscripts 1 and 2.",
"The porous medium is denoted by $\\Omega $ .",
"Mathematically, $\\Omega \\subset \\mathbb {R}^{nd}$ is assumed to be open and bounded, where $nd$ denotes the number of spatial dimensions.",
"In this paper, $nd = 2 \\; \\mathrm {or} \\; 3$ .",
"The gradient and divergence operators with respect to a spatial point $\\mathbf {x} \\in \\Omega $ are, respectively, denoted by $\\mathrm {grad}[\\cdot ]$ and $\\mathrm {div}[\\cdot ]$ .",
"The pressure and the discharge (or Darcy) velocity fields in the macro-pore network are, respectively, denoted by $p_{1}(\\mathbf {x})$ and $\\mathbf {u}_{1}(\\mathbf {x})$ , and the corresponding fields in the micro-pore network are denoted by $p_{2}(\\mathbf {x})$ and $\\mathbf {u}_{2}(\\mathbf {x})$ .",
"The viscosity and true density of the fluid are denoted by $\\mu $ and $\\gamma $ , respectively.",
"The governing equations for a steady-response under the DPP mathematical model take the following form: 2 k1-1 u1(x) + grad[p1] = b(x) in k2-1 u2(x) + grad[p2] = b(x) in div[u1] = -(p1(x) - p2(x)) in div[u2] = +(p1(x) - p2(x)) in where $k_1(\\mathbf {x})$ and $k_2(\\mathbf {x})$ , respectively, denote the (isotropic) permeabilities of the macro-pore and micro-pore networks, $\\beta $ is a dimensionless characteristic of the porous medium, and $\\mathbf {b}(\\mathbf {x})$ denotes the specific body force.",
"In the above equations, the mass transfer per unit volume, $\\chi (\\mathbf {x})$ , from the macro-pore network to the micro-pore network is modeled as follows: (x) = -(p1 - p2) As one can see from equations ()–(), the flow in one pore-network is coupled with its counterpart in the other pore-network through the inter-pore mass transfer.",
"Unlike Darcy equations, it is not possible to rewrite the governing equations under the DPP model in form of a single-field Poisson's equation in terms of pressures.",
"One has to deal with the governing equations in their mixed form."
],
[
"Grouping of field variables in continuum setting",
"We now discuss two ways of grouping the field variables, which form the basis for the proposed composable block solvers.",
"Under the first approach, the field variables are grouped based on the scale of the pore-network.",
"That is, all the field variables (i.e., velocity and pressure) pertaining to the macro-pore network are placed in one group, and the field variables of the micro-pore network are placed into another.",
"We refer to this splitting of field variables as the scale-split and the associated grouping takes the following form: 1 = {c u1(x) p1(x) u2(x) p2(x) } The governing equations of the DPP model under the scale-split can be compactly written as follows: L1[1] = F1 In the above equation, the differential operator takes the following form: L1 := [cc|cc k1-1 I grad[] O 0 div[] O - O 0 k2-1 I grad[] O - div[] ] where $\\mathbf {I}$ denotes the identity tensor, $\\mathbf {O}$ denotes the zero tensor, and the forcing function takes the following form: F1 = {c b(x) 0 b(x) 0 } Under the second approach, the field variables are grouped based on the nature of the fields.",
"That is, field variables of a similar kind are placed in the same group.",
"We refer to this splitting of field variables as the field-split and the associated grouping takes the following form: 2 = {c u1(x) u2(x) p1(x) p2(x) } The governing equations of the DPP model under the field-split can be compactly written as follows: L2[2] = F2 where the differential operator takes the following form: L2 := [cc|cc k1-1 I O grad[] 0 O k2-1 I 0 grad[] div[] O - O div[] - ] and the forcing function can be written as follows: F2 = {c b(x) b(x) 0 0 }"
],
[
"Boundary conditions",
"The boundary of the domain will be denoted by $\\partial \\Omega $ .",
"The unit outward normal to the boundary at $\\mathbf {x} \\in \\partial \\Omega $ is denoted by $\\widehat{\\mathbf {n}}(\\mathbf {x})$ .",
"The velocity boundary condition (i.e., no penetration boundary condition) for each pore-network takes the following form: 2 u1(x) n(x) = un1(x) on u1 u2(x) n(x) = un2(x) on u2 where $\\Gamma _{i}^{u}$ denotes that part of the boundary on which the normal component of the velocity is prescribed in the macro-pore ($i=1$ ) and micro-pore ($i=2$ ) networks, and $u_{n1}(\\mathbf {x})$ and $u_{n2}(\\mathbf {x})$ denote the prescribed normal components of the velocities on $\\Gamma ^{u}_{1}$ and $\\Gamma ^{u}_{2}$ , respectively.",
"The pressure boundary condition for each pore-network reads: 2 p1(x) = p01 (x) on p1 p2(x) = p02 (x) on p2 in which $\\Gamma _{i}^{p}$ is that part of the boundary on which the pressure is prescribed in the macro-pore ($i=1$ ) and micro-pore ($i=2$ ) networks, and $p_{01}(\\mathbf {x})$ and $p_{02}(\\mathbf {x})$ denote the prescribed pressures on $\\Gamma _{1}^{p}$ and $\\Gamma _{2}^{p}$ , respectively.",
"For mathematical well-posedness, we assume that 1u 1p = , 1u 1p = , 2u 2p = and 2u 2p ="
],
[
"CLASSICAL AND STABILIZED MIXED WEAK FORMULATIONS",
"As mentioned earlier, we will employ three different mixed formulations – the classical mixed formulation using H(div) discretizations, a continuous stabilized mixed formulation and a discontinuous stabilized mixed formulation.",
"Under a mixed formulation, velocities and pressures are taken to be the primary variables.",
"However, for numerical stability, a mixed formulation should either satisfy or circumvent the Ladyzhenskaya-Babuška-Brezzi (LBB) inf-sup stability condition [10].",
"This naturally places all the mixed formulations into either of two categories.",
"A mixed formulation in the first category is built on the classical mixed formulation (which is based on the Galerkin formalism) but places restrictions on the interpolation functions for the independent field variables to satisfy the LBB condition.",
"To put it differently, not all combinations of interpolation functions for the field variables satisfy the LBB condition under the classical mixed formulation.",
"A mixed formulation in the second category augments the classical mixed formulation with stabilization terms so as to circumvent the LBB condition and to render a stable formulation.",
"In this paper, we consider one mixed formulation from the first category and two from the second category.",
"We denote the set of all square-integrable functions on $\\Omega $ by $L_{2}(\\Omega )$ .",
"The standard $L_2$ inner-product over a set $K$ is denoted as follows: (a;b)K K a(x)b (x) d K For convenience, the subscript $K$ will be dropped if $K = \\Omega $ in the case of classical mixed formulation and the stabilized mixed continuous Galerkin formulation.",
"In the case of the stabilized mixed discontinuous Galerkin formulation, the subscript $K$ will be dropped if $K = \\widetilde{\\Omega }$ , which will be defined later in equation (REF ).",
"We now provide details of the three chosen mixed finite element discretizations.",
"Let us define the following function spaces for the velocities and pressures fields as follows: U1 := {u1(x) (L2())nd div[u1] L2(), u1(x) n(x) = un1(x) H-1/2(1u)} U2 := {u2(x) (L2())nd div[u2] L2(), u2(x) n(x) = un2(x) H-1/2(2u)} W1 := {w1(x) (L2())nd div[w1] L2(), w1(x) n(x) = 0 on 1u } W2 := {w2(x) (L2())nd div[w2] L2(), w2(x) n(x) = 0 on 2u} P := {(p1(x),p2(x)) L2() L2() ( p1(x) d ) ( p2(x) d ) = 0 } Q := {(p1(x),p2(x)) H1() H1() ( p1(x) d ) ( p2(x) d ) = 0 } where $H^{1}(\\Omega )$ is a standard Sobolev space, and $H^{-1/2}(\\cdot )$ is the dual space corresponding to $H^{1/2}(\\cdot )$ .",
"Rigorous discussion of Sobolev spaces are accessible in [37]; and further discussion of function spaces are provided by [10]."
],
[
"Classical mixed formulation\nusing H(div) elements",
"The classical mixed formulation can be written as follows: Find $\\left(\\mathbf {u}_1(\\mathbf {x}),\\mathbf {u}_2(\\mathbf {x})\\right) \\in \\mathcal {U}_1 \\times \\mathcal {U}_2$ and $\\left(p_1(\\mathbf {x}),p_2(\\mathbf {x})\\right) \\in \\mathcal {P}$ such that we have BGal(w1,w2,q1,q2; u1,u2,p1,p2) = LGal (w1,w2,q1,q2) (w1(x), w2(x)) W1 W2, (q1(x), q2(x)) P where the bilinear form and the linear functional are, respectively, defined as follows: BGal := (w1;k1-1u1) - (div[w1];p1) + (q1;div[u1]) + (w2;k2-1u2) - (div[w2];p2) + (q2;div[u2]) + (q1 - q2;/(p1 - p2)) LGal := (w1;b) - (w1 n;p01 )p1 + (w2;b) - (w2 n;p02 )p2 Classes of H(div) finite element discretizations such as Raviart-Thomas (RT) [48], generalized RTN [45], BDM [11], and BDFM [12] have been shown to satisfy the LBB condition.",
"Moreover, these finite element discretizations satisfy element-wise mass balance property [10].",
"The classical mixed formulation based on discretizations from the lowest-order Raviart-Thomas spaces is commonly referred to as the RT0 formulation; which is frequently used in subsurface modeling [18].",
"The unknowns under the RT0 formulation on a triangle are fluxes at the midpoints of edges of the element and element-wise constant pressures.",
"The finite dimensional subspaces for each velocity and pressure fields under the lowest-order Raviart-Thomas discretization on a triangle, which are collectively denoted by $\\mathrm {RTF}_1\\oplus \\mathrm {DP}_0$ , take the following form: Uh:={u = (u,v) | uK = aK+bKx,vK = cK+bKy; aK, bK, cK R; K Th} Ph:= { p | p = constant on each triangle K Th } where $\\mathcal {T}_{h}$ is a triangulation on $\\Omega $ .",
"These subspaces on a tetrahedron, which are denoted by $\\mathrm {N1F}_1\\oplus \\mathrm {DP}_0$ , take the following form: Uh:={u = (u,v,w) | uK = aK + bK x, vK = cK + bK y, wK = dK + bK z; aK, bK, cK, dK R; K Th} Ph:= { p | p = constant on each tetrahedron K Th } where $\\mathcal {T}_{h}$ , in this case, is a tetrahedralization on $\\Omega $ .",
"In addition to H(div) discretizations on simplicial meshes, we also consider the corresponding discretizations on non-simplicial element – QUAD and HEX.",
"The velocity spaces for QUAD and HEX elements are, respectively, $\\mathrm {RCTF_1}$ and $\\mathrm {NCF_1}$ [43], [1].",
"The (macro- and micro-) pressures are element-wise constants, and $\\mathrm {DG}_0$ is commonly used to denote element-wise constant discretization on non-simplicial elements.",
"See Figure REF and Table REF for a description of these discretizations.",
"The finite dimensional subspaces for the $\\mathrm {RCTF_1}$ and $\\mathrm {NCF}_1$ discretizations can be written precisely using the language of finite element exterior calculus.",
"But such a description needs introduction of additional jargon and notation, which is beyond the scope of this paper.",
"We, therefore, refer the reader to [2], [3], [1].",
"However, to guide the reader, the degrees-of-freedom for these discretizations are shown in Figure REF and Table REF .",
"Table: The element-level discretizationfor different mesh types and the chosenthree formulations.",
"⨁\\bigoplus denotesthe direct sum operator between two finiteelement spaces.",
"The notation used in thistable is based on the Periodic Tableof the Finite Elements ."
],
[
"Stabilized mixed continuous\nGalerkin formulation (CG-VMS)",
"The weak form of the CG-VMS formulation can be written as follows: Find $\\left(\\mathbf {u}_1(\\mathbf {x}),\\mathbf {u}_2(\\mathbf {x}) \\right)\\in \\mathcal {U}_1 \\times \\mathcal {U}_2$ and $\\left(p_1(\\mathbf {x}),p_2(\\mathbf {x})\\right)\\in \\mathcal {Q}$ such that we have BCGstab(w1,w2,q1,q2; u1,u2,p1,p2) = LCGstab(w1,w2,q1,q2) (w1(x), w2(x)) W1 W2, (q1(x),q2(x)) Q where the bilinear form and the linear functional are defined, respectively, as follows: BCGstab := BGal(w1,w2,q1,q2; u1,u2,p1,p2) -12 (k1-1 w1 - grad[q1];1 k1 (k1-1 u1 + grad[p1])) -12 (k2-1 w2 - grad[q2]; 1 k2 (k2-1 u2 + grad[p2])) LstabCG := LGal(w1,w2,q1,q2) -12 (k1-1 w1 - grad[q1]; 1 k1 b) -12 (k2-1 w2 - grad[q2]; 1 k2 b) An attractive feature of the CG-VMS formulation is that nodal-based equal-order interpolation for all the field variables (micro- and macro- velocities and pressures) is stable, which is not the case with the classical mixed formulation.",
"The stability is achieved by the addition of stabilization terms, which circumvent the LBB condition."
],
[
"Stabilized mixed discontinuous\nGalerkin formulation (DG-VMS)",
"Formulations under the discontinuous Galerkin (DG) method inherit attractive features of both finite element and finite volume methods by allowing discontinuous basis functions (e.g., in the form of piecewise polynomials) [28].",
"The DG method supports non-matching grids and hanging nodes, and hence ideal for $hp$ adaptivity [21].",
"Moreover, the method can naturally handle jumps in the profiles of the solution variables [33], [35].",
"Since the DG method offers several attractive features, we choose the DG formulation that is recently proposed by [35] for the DPP model, which will be referred to as the DG-VMS formulation, as one of the three representative formulations to illustrate the performance of the proposed composable block solvers.",
"We now document the weak form under the DG-VMS formulation.",
"To this end, we decompose the domain into $Nele$ open subdomains such that = i=1Nele i where $\\omega ^{i}$ denotes the $i$ -th subdomain.",
"The union of all open subdomains will be denoted by = i=1Nele i We denote the two adjacent subdomains sharing a given interior edge by $\\omega ^{+}$ and $\\omega ^{-}$ .",
"The unit normal vectors on the shared interface $\\Gamma ^{\\pm }$ pointing outwards to $\\omega ^{+}$ and $\\omega ^{-}$ are, respectively, denoted by $\\widehat{\\mathbf {n}}^{+}$ and $\\widehat{\\mathbf {n}}^{-}$ .",
"The jump and average operators on an interior facet for a scalar field $\\varphi (\\mathbf {x})$ are, respectively, defined as follows: := + n+ + - n- and {} := + + -2 where + = |+ and - = |- For a vector field ${\\tau }(\\mathbf {x})$ these operators are defined as follows: := + n+ + - n- and { } := + + - 2 on int where ${\\tau }^{+}$ and ${\\tau }^{-}$ are defined similar to equation (REF ).",
"We denote the set of all square-integrable functions on $\\omega ^{i}$ by $L_2(\\omega ^{i})$ .",
"We denote the set of all functions that belong to $L_2(\\omega ^{i})$ and are continuously differentiable by $H^{1}(\\omega ^{i})$ .",
"We then introduce the following broken Sobolev spaces (which are piece-wise discontinuous spaces): Udg := {u(x) | u(x)|i (L2(i) )nd; div[u] L2(i); i = 1, , Nele } Qdg := {(p1(x),p2(x)) | p1(x)|i H1(i), p2(x)|i H1(i), .",
".",
"( p1(x) d ) ( p2(x) d ) = 0 } The weak form under the DG-VMS formulation reads as follows: Find $\\left(\\mathbf {u}_1(\\mathbf {x}),\\mathbf {u}_2(\\mathbf {x})\\right)\\in \\mathcal {U}^{\\mathrm {dg}} \\times \\mathcal {U}^{\\mathrm {dg}}$ , $\\left(p_1(\\mathbf {x}),p_2(\\mathbf {x})\\right)\\in \\mathcal {Q}^{\\mathrm {dg}}$ such that we have BstabDG(w1,w2,q1,q2; u1,u2,p1,p2) = LstabDG(w1,w2,q1,q2) (w1(x), w2(x)) Udg Udg, (q1(x),q2(x)) Qdg where the bilinear form and the linear functional are defined, respectively, as follows: BDGstab := ( w1 ; k1-1 u1 ) - (div[w1] ; p1 ) +( w2 ; k2-1 u2 ) - (div[w2] ; p2 ) +(w1 ; {{ p1 }} )int - ({{q1 }} ; u1 )int + (q1;div[u1]) + (q2;div[u2]) + (w2 ; {{p2 }} )int - ({{q2 }} ; u2 )int + (q1 - q2; (p1 - p2) ) + (w1 n; p1 )1u + (w2 n; p2 )2u -(q1;u1 n)u1 -(q2;u2 n)u2 -12 (k1-1 w1 - grad[q1]; -1 k1 (k-11 u1 + grad[p1])) -12 (k2-1 w2 - grad[q2]; -1 k2(k-12 u2 + grad[p2])) + u h ( {{k1-1}} w1 ; u1 )int + u h ( {{k2-1}} w2 ; u2 )int + ph ( {{-1 k1}} q1 ; p1 )int + ph ( {{-1 k2}} q2 ; p2 )int LstabDG := ( w1 ; b1 ) + ( w2 ; b2 ) -(w1 n; p01)1p -(w2 n; p02)2p - (q1;un1 )u1 - (q2;un2)u2 -12 (k1-1 w1 - grad[q1]; -1 k1 b1) -12 (k2-1 w2 - grad[q2]; -1 k2 b2) where $\\eta _{u}$ and $\\eta _{p}$ are non-negative, non-dimensional numbers.",
"The DG-VMS formulation also circumvents the LBB condition and is stable under arbitrary combinations of interpolation functions for the field variables."
],
[
"PROPOSED FOUR-FIELD SOLVERS",
"The fully discrete formulations for the DPP model can be assembled into the following linear problem: Ku=f where ${K}$ is the stiffness matrix, ${u}$ is the vector of unknown velocities and pressure, and ${f}$ is the corresponding forcing or RHS vector.",
"Solving the system of equations () in a fast and scalable way requires careful composition and manipulation of the four different physical fields.",
"In this section, we demonstrate how this can be done through PETSc [4], [5], [22] and its composable solver capabilities [13].",
"The individual block components of the stiffness matrix ${K}$ for the mixed Galerkin formulation using H(div) elements can be categorized into the following: K1uu (w1;k1-1u1) K1up - (div[w1];p1) K1pu (q1;div[v1]) K1pp (q1;p1) K2uu ( w2;k-12u2) K2up - ( div[w2]; p2) K2pu ( q2;div[v2]) K2pp (q2;p2) K12pp -(q1;p2) K21pp -(q2;p1) For the CG-VMS formulation, the individual block components of the stiffness matrix can be categorized into the following: K1uu 12(w1;k1-1u1) K1up - (div[w1];p1) - 12(w1;grad[p1]) K1pu (q1;div[v1]) + 12(grad[q1];u1) K1pp 12(grad[q1];1k1grad[p1]) + (q1;p1) K2uu 12( w2;k-12u2) K2up - ( div[w2]; p2) - 12( w2;grad[p2]) K2pu ( q2;div[v2]) + 12(grad[q2];u2) K2pp 12(grad[q2];1k2grad[p2]) + (q2;p2) K12pp -(q1;p2) K21pp -(q2;p1) Likewise, the block components of the stiffness matrix for the DG-VMS formulation read: K1uu 12( w1;k-11u1) + u h ( {{k1-1}}w1 ;u1 )int K1up - ( div[w1]; p1) - 12( w1;grad[p1]) + (w1 ;{{ p1 }})int + (w1 n;p1)1u K1pu ( q1;div[v1]) + 12(grad[q1];u1) - ({{ q1 }};v1 )int - (q1;v1 n)1u K1pp 12(grad[q1];1k1grad[p1]) + (q1;p1) + ph ( {{-1 k1}}q1 ; p1 )int K2uu 12( w2;k-12u2) + u h ( {{k2-1}}w2 ;u2 )int K2up - ( div[w2]; p2) - 12( w2;grad[p2]) + (w2 ;{{ p2 }})int + (w2 n;p2)2u K2pu ( q2;div[v2]) + 12(grad[q2];u2) - ({{ q2 }};v2 )int - (q2;v2 n)2u K2pp 12(grad[q2];1k2grad[p2]) + (q2;p2) + ph ( {{-1 k2}}q2 ; p2 )int K12pp -(q1;p2) K21pp -(q2;p1) The components of the corresponding RHS vector ${f}$ for equations (), () and () are f1u (w1;b) - (w1n;p01)p1 f1p 0 f2u (w2;b) - (w2n;p02)p2 f2p 0 and f1u 12(w1;b) - (w1n;p01)p1 f1p 12(grad[q1];1k1b) f2u 12(w2;b) - (w2n;p02)p2 f2p 12(grad[q2];1k2b) and f1u 12(w1;b) - (w1n;p01)p1 f1p 12(grad[q1];1k1b) - (q1;un1)u1 f2u 12(w2;b) - (w2n;p02)p2 f2p 12(grad[q2];1k2b) - (q2;un2)u2 respectively.",
"Specifically, we employ PETSc's block solver capabilities, in the PCFIEDLSPLIT class, taking two fields at a time.",
"However, the global DPP model is a four field problem so we subdivide our problem recursively such that we end up with 2$\\times $ 2 blocks.",
"Conceptually, PETSc can employ a wide variety of block solver methodologies on a 2$\\times $ 2 matrix: K = A B C D , where ${A}$ , ${B}$ , ${C}$ , and ${D}$ are individual block matrices which also consist of 2$\\times $ 2 blocks.",
"Although equation () is conceptually a 4$\\times $ 4 block matrix, PETSc's field-splitting capabilities enables us to break the system down dynamically at runtime into two levels of 2$\\times $ 2 blocks.",
"We now propose two different ways one can compose scalable and efficient solvers and preconditioners for blocks ${A}$ , ${B}$ , ${C}$ , and ${D}$ with the individual components shown in equations (), (), and ()."
],
[
"Method 1: splitting by scales",
"One option is to split the global problem by scales.",
"That is, each macro- or micro- scale 2$\\times $ 2 block will contain its corresponding velocity and pressure fields.",
"Under this solver strategy, equation () is then rewritten as: K1uu K1up 0 0 K1pu K1pp 0 K12pp 0 0 K2uu K2up 0 K21pp K2pu K2pp u1 p1 u2 p2 = fu1 fp1 fu2 fp2 where 0 is a zero matrix, ${u}_1$ and ${p}_1$ are the respective macro-scale velocity and pressure vectors, ${v}_2$ and ${p}_2$ are the respective micro-scale velocity and pressure vectors.",
"The individual 2$\\times $ 2 blocks from equation () would be A := K1uu K1up K1pu K1pp , B := 0 0 0 K12pp , C := 0 0 0 K21pp , D := K2uu K2up K2pu K2pp Although the off diagonal blocks ${B}$ and ${C}$ contain the inter-scale pressure coupling terms, they are very sparse so we will ignore these blocks for now.",
"The composition of the ${A}$ and ${D}$ blocks are similar to the classical mixed Poisson problem so the Schur complement approach outlined in [14], [42] and the references within can be applied.",
"The task is to individually precondition the decoupled ${A}$ and ${D}$ blocks.",
"We note that they admit factorizations of A = I 0 K1pu(Kuu1)-1 I Kuu1 0 0 S1 I (K1uu)-1K1up 0 I D = I 0 K2pu(Kuu2)-1 I Kuu2 0 0 S2 I (K2uu)-1K2up 0 I where ${I}$ is the identity matrix and S1=K1pp-K1pu(K1uu)-1K1up S2=K2pp-K2pu(K2uu)-1K2up are the Schur complements for the ${A}$ and ${D}$ blocks, respectively.",
"The inverses can therefore be written as A-1 = I -(K1uu)-1K1up 0 I (K1uu)-1 0 0 (S1)-1 I 0 -Kpu1(K1uu)-1 I D-1 = I -(K2uu)-1K2up 0 I (K2uu)-1 0 0 (S2)-1 I 0 -Kpu2(K2uu)-1 I The task at hand is to approximate the inverses of the ${K}^{1}_{vv}$ , ${K}^2_{uu}$ , ${S}^1$ , and ${S}^2$ blocks.",
"The first two blocks are simply mass matrices so we can invert them using the ILU$(0)$ (incomplete lower upper) solver.",
"For the Schur complement blocks, we employ a diagonal mass-lumping of ${K}^1_{uu}$ and ${K}^2_{uu}$ to estimate $\\left({K}^1_{uu}\\right)^{-1}$ and $\\left({K}^2_{uu}\\right)^{-1}$ because they are spectrally equivalent to the identity.",
"That is, S1p = K1pp-K1pudiag( K1uu)-1K1up S2p = K2pp-K2pudiag( K2uu)-1K2up to precondition the inner solvers responsible for inverting ${S}^1$ and ${S}^2$ .",
"For these blocks we employ the multigrid V-cycle on ${S}^1_p$ and ${S}^2_p$ from the HYPRE BoomerAMG package [25].",
"We expect these to work because the ${S}$ blocks are spectrally equivalent to the Laplacian, modulo the penalty terms.",
"In [42] it turns out the presence of the VMS stabilization terms in the ${K}^1_{pp}$ and ${K}^2_{pp}$ blocks do not drastically affect the performance or scalability of this solver strategy.",
"Instead of completely solving for the ${K}_{uu}^{-1}$ and ${S}_p$ of both scales, we apply only a single sweep of ILU(0)/block Jacobi and V-cycle, respectively, and rely on GMRES [50] to solve the entire 4$\\times $ 4 block system.",
"Thus this outer GMRES is able to pick up the inter-scale pressure coupling blocks ${B}$ and ${C}$ .",
"The PETSc command-line options for this solver methodology is given in listing .",
"-ksp_type gmres -pc_type fieldsplit -pc_fieldsplit_0_fields 0,1 -pc_fieldsplit_1_fields 2,3 -pc_fieldsplit_type additive -fieldsplit_0_ksp_type preonly -fieldsplit_0_pc_type fieldsplit -fieldsplit_0_pc_fieldsplit_type schur -fieldsplit_0_pc_fieldsplit_schur_fact_type full -fieldsplit_0_pc_fieldsplit_schur_precondition selfp -fieldsplit_0_fieldsplit_0_ksp_type preonly -fieldsplit_0_fieldsplit_0_pc_type bjacobi -fieldsplit_0_fieldsplit_1_ksp_type preonly -fieldsplit_0_fieldsplit_1_pc_type hypre -fieldsplit_1_ksp_type preonly -fieldsplit_1_pc_type fieldsplit -fieldsplit_1_pc_fieldsplit_type schur -fieldsplit_1_pc_fieldsplit_schur_fact_type full -fieldsplit_1_pc_fieldsplit_schur_precondition selfp -fieldsplit_1_fieldsplit_0_ksp_type preonly -fieldsplit_1_fieldsplit_0_pc_type bjacobi -fieldsplit_1_fieldsplit_1_ksp_type preonly -fieldsplit_1_fieldsplit_1_pc_type hypre where we assume that the global ordering of the mixed function space is macro-scale velocity (0), macro-scale pressure (1), micro-scale velocity (2), and micro-scale pressure (3)."
],
[
"Method 2: splitting by fields",
"Another option is to group the velocities and pressures of both scales into two different blocks.",
"If this approach is taken, equation () is then rewritten as: K1uu 0 K1up 0 0 K2uu 0 K2up K1pu 0 K1pp K12pp 0 K2pu K21pp K2pp u1 u2 p1 p2 = fu1 fu2 fp1 fp2 and the individual blocks in equation () would now look like A := K1uu 0 0 K2uu , B := K1up 0 0 K2up , C := K1pu 0 0 K2pu , D := K1pp K12pp K21pp K2pp Unlike the previous methodology, we can work directly with the above stiffness matrix, which admits a factorization of K = I 0 CA-1 I A 0 0 S I A-1B 0 I , where the Schur complement ${S}$ is S=D-CA-1B.",
"The inverse can therefore be written as K-1 = I -A-1B 0 I A-1 0 0 S-1 I 0 -CA-1 I .",
"Although ${A}$ is a 2$\\times $ 2 block containing velocities spanning across two different scales, we can still approximate ${A}^{-1}$ by inverting the entire ${A}$ block using ILU$(0)$ because the off-diagonal blocks are zero and the diagonal blocks consist of only mass matrices.",
"Approximating ${S}^{-1}$ is a little trickier because equation (REF ) is a dense 2$\\times $ 2 block with off-diagonal terms.",
"However, we can still employ a diagonal mass-lumping of ${A}$ to estimate ${A}^{-1}$ because it is again spectrally equivalent to the identity.",
"The preconditioner needed for ${S}^{-1}$ is: Sp = D-Cdiag( A)-1B = K1pp K12pp K21pp K2pp - K1pu 0 0 K2pu diag( K1uu 0 0 K2uu )-1 K1up 0 0 K2up = K1pp - K1pudiag(K1uu)K1up K12pp K21pp K2pp - K2pudiag(K2uu)K2up The off-diagonal blocks only consist of mass-matrix terms but the decoupled diagonal blocks are identical to equations (REF ) and (REF ).",
"Thus, we individually employ multigrid V-cycle on each of the diagonal blocks.",
"As in the previous solver methodology, only a single sweep of ILU(0) and the two multigrid V-cycles are needed for the ${A}^{-1}$ matrix and the two diagonal terms within the ${S}_p$ matrix, respectively, and the GMRES method is employed to solve the entire block system.",
"The PETSc implementation is shown in listing .",
"-ksp_type gmres -pc_type fieldsplit -pc_fieldsplit_0_fields 0,2 -pc_fieldsplit_1_fields 1,3 -pc_fieldsplit_type schur -pc_fieldsplit_schur_fact_type full -pc_fieldsplit_schur_precondition selfp -fieldsplit_0_ksp_type preonly -fieldsplit_0_pc_type bjacobi -fieldsplit_1_ksp_type preonly -fieldsplit_1_pc_type fieldsplit -fieldsplit_1_pc_fieldsplit_type additive -fieldsplit_1_fieldsplit_0_ksp_type preonly -fieldsplit_1_fieldsplit_0_pc_type hypre -fieldsplit_1_fieldsplit_1_ksp_type preonly -fieldsplit_1_fieldsplit_1_pc_type hypre where we again assume that the global ordering of the mixed function space is macro-scale velocity (0), macro-scale pressure (1), micro-scale velocity (2), and micro-scale pressure (3)."
],
[
"Computer implementation",
"The finite element capabilities are provided by the Firedrake Project package [47], [40], [29], [43], [7], [41], [31], [30] with GNU compilers.",
"This sophisticate finite element simulation package and its software dependencies can be found at [57], [60], [61], [54], [62], [56], [55], [58], [59].",
"The computational meshes are built on top of the DMPlex unstructured grid format [36], [38], [39] and partitioned through the Chaco package [27].",
"Pictorial descriptions of the specific elements represented by this mesh format and utilized in this paper are illustrated in Figure REF .",
"The DMPlex data structure interfaces very nicely with PETSc's suite of parallel solvers and provide excellent scalability across thousands of MPI processes [15], [17].",
"Sample Firedrake codes for some of these benchmark problems can be found in Appendix .",
"In our PETSc implementation, the same global matrix will be assembled for both solvers.",
"The preconditioners differ by the subblocks which are extracted.",
"The different sparsity pattern of the subblocks contributes to the performance differences seen in the solvers, but the overall assembly time remains unchanged for either solvers."
],
[
"PERFORMANCE SPECTRUM MODELING",
"To understand the parallel performance and algorithmic scalability of the proposed DPP composable block solver methodologies for the three finite element formulations, a performance model is needed.",
"The performance model based on the Time-Accuracy-Size (TAS) spectrum analysis outlined in [16] shall be used as the basis for understanding the quality of these finite element formulations with the proposed block solvers.",
"We now briefly highlight the performance metrics used in this section and why they are each important in each of their own ways."
],
[
"Mesh convergence",
"This criterion uses the convergence notion to account for numerical accuracy of a solution in the performance spectrum.",
"In this paper, we are adopting $L_2$ norm of the error defined as: L2norm= uh-uL2 where $u$ is the exact solution, $u_h$ is the the finite element solution, and $h$ is measure of element size.",
"Based on theory, most finite element discretizations will have an upper-bound for $L_2$ error norm as follows: L2normCh where $\\alpha $ is known as convergence rate and $C$ is some constant.",
"When reporting and comparing how much accuracy is attained for each discretization, we use the notation of Digits of Accuracy (DoA) defined as: DoA := -log10 ( L2norm) and plot DoA against Digits of Size (DoS), which is defined as: DoS := -log10 (DoF) Noting that for most formulations $DoF=Dh^{-d}$ , where $d$ is the spatial dimension and $D$ is some constant, the slope of DoA vs DoS plot is in the order of $\\frac{\\alpha }{d}$ .",
"Any tailing off from the line plot is an indicator of incorrect implementation or solver convergence tolerances being too relaxed.",
"Furthermore, the ratio DoA/DoS can be a good indicator of how much accuracy is achieved per DoF."
],
[
"Strong-scaling",
"In this basic parallel scaling while the size of problem remains unchanged, the number of processes increases.",
"In general, this metric comments on the marginal efficiency of each additional processes assigned to a problem.",
"It is conventional to plot number of processes against the parallel efficiency defined as: Parallel eff.",
"$(\\%)$ = T1Tpproc100% where $proc$ is the number of MPI processes, $T_1$ is the total wall-clock time needed on a single MPI processes, and $T_p$ is the total wall-clock time needed with $proc$ MPI processes.",
"However, this metric must be interpreted carefully for the following reasons: Solver iteration counts: The number of solver iterations may fluctuate as the number of MPI processes changes.",
"This can happen for a number of reasons, whether it is algorithmic implementation or relaxed convergence criterion.",
"It is necessary to also report the number of KSP iterations required as the number of processes changes.",
"Problems too small: If the DoF count is too small for a particular MPI concurrency, communication time will swamp the computation time, thus reducing the parallel efficiency.",
"This issue may arise when making comparative studies between different finite element discretizations, as different formulations have different DoF counts for a given $h$ -size.",
"Furthermore, for Python-based simulation packages like Firedrake, overheads from just-in-time compilation and instantiation of objects can also affect the strong-scaling .",
"Problems too large: If the DoF count is too large for a particular MPI concurrency, the problems not only drop out of the various levels of cache in the memory hierarchy but also invoke several expensive cache misses which can slow down the overall performance.",
"This may result in superlinear speedups, like the BLMVM bound-constrained optimization solver in [15].",
"Lastly, the global problem size for this scaling analysis is fixed, so we need an additional parallel scaling metric which explains whether the performance of our solvers might degrade due to increased KSP iteration counts or memory contention as the problem size increases."
],
[
"Static-scaling",
"As described in [17], static-scaling is a scaling analysis where the MPI concurrency is fixed but the problem size is increased.",
"The essential metric for this analysis is the computation rate (DoF over Time).",
"In this paper, we run a series of problem sizes at a fixed parallelism and plot the computation rate against the wall-clock time.",
"Note that the time need not be the total time to solution, instead one could look at various phases like the finite element assembly or solver computation rates.",
"Static-scaling returns information on performance and scalability of software and solvers across different hardware architectures.",
"This scaling analysis also captures both strong-scaling and weak-scaling effects.",
"Assuming that the block solvers are of $\\mathcal {O}(N)$ scalability, where $N = DoF$ , optimal scaling is indicated by a horizontal curve.",
"Any tail offs at small problem sizes suggests strong-scaling effects whereas tail offs at large problem sizes indicate suboptimal algorithmic or memory effects.",
"The exact reasoning for the tail offs towards the right can be verified through arithmetic intensity, which is the measure of the total work over the total bytes transferred (see [17] and the references within)."
],
[
"Digits-of-Efficacy ($\\mathrm {DoE}$ )",
"The final metric needed for our performance spectrum study is the Digits of Efficacy (DoE).",
"This metric measures the accuracy production by a particular scheme in a given amount of time.",
"The DoE could be defined as: DoE:=-log10(L2normTime) Assuming that straight lines are captured in both the mesh convergence and static-scaling diagrams, the DoE has a linear dependence on problem size and returns a slope of $d-\\alpha $ (see [16] for details on the exact derivation).",
"This efficacy measure is analogous to the action of a mechanical system, that is the product of energy and time.",
"In the TAS spectrum analysis, the DoE represents an analogous action for computation, and we speculate that an optimal algorithm minimizes this product over its runtime.",
"Since the DoE takes the negative logarithm of action, a higher DoE is desirable."
],
[
"REPRESENTATIVE NUMERICAL RESULTS ",
"In this section, after clarifying the terminology and framework adopted for the performance spectrum model, we solve the four-field DPP model in two- and three-dimensional settings in order to demonstrate the implementation of the proposed composable block solvers and gauge their performances.",
"The two-dimensional problem will be conducted in serial (one MPI process) on a dual socket Intel Xeon E5-2609v3 server node.",
"The three-dimensional problems will be conducted on a dual socket Intel Xeon E5-2698v3 server node and will utilize up to 16 MPI processes (8 MPI processes per socket).",
"On different performance metrics, H(div), CG-VMS, and DG VMS formulations are compared for both simplicial (TRI, TET) and non-simplicial (QUAD, HEX) meshes.",
"Both two-dimensional and three-dimensional problems were adopted by [34] for the convergence analysis of continuous stabilized mixed formulation (CG-VMS) and by [35] for the convergence analysis of discontinuous stabilized mixed formulation (DG-VMS) for the DPP model.",
"We are generating three series of outputs for first-order CG-VMS, DG-VMS with $\\eta _p=\\eta _u=10$ , and H(div) formulations."
],
[
"Two-dimensional study",
"For this first problem, let us consider a two-dimensional DPP boundary value problem with governing equations stated in equations ()–() and (REF )–(REF ).",
"The homogeneous (i.e., constant macro and micro-permeabilities) bi-unit square computational domain and boundary conditions for this study are shown in Figure REF , and the corresponding parameters are described in Table REF .",
"Table: Parameters for two-dimensional problem.The analytical solution for the pressure and velocity fields takes the following form: u1(x,y) = -k1 ex(y) ex(y)-k1 ey p1(x,y)= ex(y) -k1ey u2(x,y) = -k2 ex(y) ex(y)+k2 ey p2(x,y)= ex(y) +k2ey where $\\eta $ is defined as: :=k1+k2k1 k2 For two-dimensional performance spectrum analysis, all three finite element formulations will start off with the same $h$ -sizes and will be refined up to 6 times.",
"The initial TRI and QUAD coarse meshes are shown in Figures REF and REF and the corresponding DoF counts for each formulation is shown in Table REF .",
"Table: This table illustrates degrees-of-freedom for two-dimensional h-size refinement study.The mesh convergence results with respect to DoA and DoS are performed under field-splitting solver and are shown in Figures REF and REF respectively for TRI and QUAD meshes.",
"It should be noted that by applying scale-splitting solver, very same results could be obtained and for brevity, we decided not to plot them in figures.",
"It can be seen in these diagrams that the CG-VMS and DG-VMS lines exhibit a slope $\\frac{\\alpha }{d}=1$ , which verifies that our Firedrake implementation of these discretizations is correct.",
"The H(div) lines exhibit a slope of 0.5 for TRI meshes but appear to have superlinear convergence for the QUAD meshes, which has also been observed in other Firedrake endeavors [26].",
"It can also be seen that if the solver tolerances are not strict enough, the mesh convergence lines will tail off.",
"Nonetheless, the CG-VMS and DG-VMS have the highest ratios of DoA over DoS in most of these diagrams which suggests that each DoF in VMS formulations has a greater level of contribution to the overall numerical accuracy than their H(div) counterparts.",
"Static-scaling results for both block solver strategies are shown in Figure REF , and we see that the total wall clock time is almost equally distributed among the assemble and solve phases.",
"The field-splitting methodologies are slightly worse than their scale-splitting counterparts for the VMS formulations.",
"However, the difference in performance is almost negligible when we look at the total time.",
"The DoF counts are too small as the line curves for both the assembly and solve phases flatten out when all three formulations have roughly 10K DoF or more.",
"No matter which mesh is utilized, the H(div) formulation processes its DoF count faster than either VMS formulations.",
"Figures REF and REF contain DoE diagrams for TRI and QUAD meshes, respectively.",
"Although H(div) appears to have the highest computation rates, it has a lower DoA than its VMS counterparts which results in a much smaller DoE.",
"The QUAD mesh on the other hand has a very high DoA and it beats out its VMS counterparts for all the fields."
],
[
"Three-dimensional study",
"In this section, we are solving a three-dimensional problem which is constructed by the Method of Manufactured Solutions (MMS) [46].",
"The homogeneous computational domain and boundary conditions for this problem are illustrated in Figure REF , and related parameters are listed in Table REF .",
"Also, a representative TET and HEX coarse meshes are shown in Figures REF and REF , respectively.",
"Table: Parameters for three-dimensional problem.The analytical solution for the pressure and velocity fields in the two pore-networks takes the following form: u1(x,y,z) = -k1 ex( (y)+(z) ) ex(y)-k1 ey ex(z)-k1 ez p1(x,y,z)= ex((y)+(z)) -k1(ey+ez) u2(x,y,z) = -k2 ex( (y)+(z) ) ex(y)+k2 ey ex(z)+k2 ez p2(x,y,z)= ex((y)+(z)) +k1(ey+ez)"
],
[
"Test 1: Strong-scaling results",
"First we investigate the strong-scaling performance of the proposed block solvers when applied to different finite element formulations.",
"Two case studies are shown: first we fix the $h$ -size for all finite element formulations, and second we modify each formulation's $h$ -size such that they all have roughly matching DoF counts.",
"Table REF contains the corresponding $h$ -sizes and DoF counts needed for both case studies.",
"First, we consider when all discretizations have an $h$ -size = 1/16.",
"Strong-scaling results for both field-splitting and scale-splitting block solver methodologies for H(div), CG-VMS, and DG-VMS can be found in Tables REF , REF , and REF , respectively.",
"All three Tables indicate that the KSP iteration counts between field-splitting and scale-splitting are identical whereas the wall-clock time for scale-splitting is slightly smaller.",
"The KSP counts for H(div) and CG-VMS do not change much when the number of MPI processes increases, whereas DG-VMS's KSP counts increase drastically.",
"This increase in KSP iteration counts will affect the parallel efficiency so one has to be careful when interpreting these results.",
"Nonetheless, we see that the DG-VMS parallel efficiency is the highest, even with its proliferated KSP counts.",
"This is attributed to the fact that the DoF count for DG-VMS is larger than CG-VMS and H(div).",
"All three tables indicate that higher DoF counts bring in more efficiency in the parallel sense.",
"Second, we consider the case when all discretizations contain approximately 200K degrees-of-freedom.",
"Strong-scaling results for both field-splitting and scale-splitting block solver methodologies for H(div), CG-VMS, and DG-VMS can be found in Tables REF , REF , and REF , respectively.",
"Like with the same $h$ -size case, the scale-splitting method appears to be more efficient in terms of wall-clock time needed despite having the same KSP counts as the field-splitting method.",
"It can also be seen that the H(div) and CG-VMS KSP counts do not fluctuate much with MPI processes and that DG-VMS KSP counts still increase dramatically.",
"However, tuning the mesh sizes such that all finite element discretizations have the same DoF count enables us to have better understanding of the parallel performance, especially for three-dimensional problems.",
"It can be seen that H(div) requires the least amount of wall-clock time resulting in the lowest parallel efficiency, but that does not mean this is a bad formulation.",
"In order to understand the quality of the H(div) discretizations, we need to take into consideration the numerical accuracy and perform a TAS spectrum analysis.",
"Table: This table shows h-size and corresponding degrees-of-freedom for three-dimensional strong-scaling studies.",
"Table: 3D problem: Strong-scaling results for H(div) formulation with same h-size.Table: 3D problem: Strong-scaling results for CG-VMS formulation with same h-size.Table: 3D problem: Strong-scaling results for DG-VMS formulation with same h-size."
],
[
"Test 2: TAS Spectrum Analysis",
"For the TAS spectrum analysis, we consider a range of problems, shown in Table REF , such that all finite element formulations in each refinement step have roughly the same DoF count.",
"The mesh convergence results with respect to DoA and DoS, for both TET and HEX meshes, are shown in Figure REF .",
"CG-VMS and DG-VMS lines indicate a slope of $\\frac{2}{3}$ , which again corroborates that our Firedrake implementation of these formulations are correct.",
"The H(div) lines exhibit a slope of $\\frac{1}{3}$ for TET mesh.",
"However, similar to the two-dimensional problem for non-simplicial element QUAD, H(div) exhibits super linear convergence for the HEX meshes.",
"We are not observing any tail-offs in these results as the solver relative convergence tolerance of $1e-7$ was strict enough.",
"The observation that both CG- and DG-VMS have the highest DoA over DoS ratio for almost all velocity and pressure fields implies that they have greater levels of contribution to the overall numerical accuracy than the H(div) schemes.",
"Static-scaling results for both block solver strategies are presented in Figure REF .",
"Flat lines appear in all six subfigures, indicating that the proposed block-solver methodologies are scalable under the chosen $h$ -sizes and hardware environment.",
"It is a common belief among application scientists that a solver exhibits worse scaling than an assembly procedure, since assembly is almost entirely local.",
"However, the results show that for all the chosen discretizations—no matter what solver methodology is employed—time to assemble stiffness matrix is higher than the solver time.",
"This infers that we have successfully optimized solvers to such an extent that the assembly procedure is more dominant.",
"Analogous to the two-dimensional problem, the scale-splitting methodologies are slightly better than their field-splitting counterparts for the all formulations.",
"Evidently, this disparity is more clear for VMS formulations at the solve time level.",
"However, the difference in performance is almost inconsequential when we look at the total time.",
"It can be seen that the DoF counts are sufficient to level out the curves when all three formulations have roughly 20K DoFs or more.",
"Regardless of the mesh type, the H(div) formulation processes its DoF count faster than either VMS formulations.",
"Figures REF and REF contain DoE diagrams for TET and HEX meshes, respectively.",
"For the case of TET mesh type, in spite of H(div) having the fastest computation rates, it has a lower DoA than its VMS counterparts which in turns lead to a much smaller DoE with steep declining curve.",
"On the contrary, for the case of the HEX mesh, H(div) surpasses its VMS counterparts due to its high DoA values.",
"These diagrams demonstrate how numerical accuracy can have a drastic effect on the overall computational performance of these various finite element formulations.",
"Table: 3D problem: Strong-scaling results for H(div) formulation with same DoF count.Table: 3D problem: Strong-scaling results for CG-VMS formulation with same DoF count.Table: 3D problem: Strong-scaling results for DG-VMS formulation with same DoF count.Table: This table illustrates three-dimensional h-size refinement for each discretization such that at each step DoF approximately doubles."
],
[
"CLOSURE",
"We have developed two block solver methodologies which are capable of solving large-scale problems under the four-field DPP mathematical model.",
"We have also presented a systematic performance analysis of various finite element discretizations for the DPP model using the recently proposed Time-Accuracy-Size (TAS) spectrum model, which takes into consideration important metrics such as mesh convergence, static-scaling, and Digits of Efficacy (DoE).",
"We have also identified strong-scaling issues one needs to be cognizant of when the block solvers are applied to various finite elements.",
"In our numerical studies, two- and three-dimensional problems had analogous performance trends, despite their marked discrepancy in time to solution.",
"Some of the salient features of the proposed composable block solver methodologies are as follows: Both composable solvers are compatible with different kinds of mixed finite element formulations: H(div) and non-H(div) elements, simplicial and non-simplicial elements, node- and edge-based discretizations, and continuous and discontinuous approximations.",
"Both composable solvers are scalable in both parallel and algorithmic senses.",
"The solvers can be implemented seamlessly using the existing PETSc's composable solver options.",
"Hence, one can leverage on the existing parallel computing tools to implement these composable solvers into existing simulators.",
"Some of the main conclusions from the performance analysis based on the TAS spectrum model are as follows: Scale-split vs. field-split.",
"For a fixed problem size, the scale-splitting methodology tends to be slightly more efficient in terms of wall-clock time needed despite having the same KSP counts as the field-splitting method.",
"However, selecting either solver methodologies will be left to the programmer's convenience and limitations as switching from one strategy to another exerts negligible overall effects on performance metrics.",
"H(div) vs. VMS formulations.",
"(a) No matter what mesh type is chosen, DoFs are processed the fastest under the H(div) formulation compared to the CG-VMS or DG-VMS formulations.",
"(b) The VMS formulations yield much higher overall numerical accuracy for all velocity and pressure fields than their H(div) counterparts.",
"The exception is for non-simplicial meshes, where the H(div) formulation exhibits super linear convergence."
],
[
"Computer codes",
"In the following, we have provided Firedrake-based computer codes for CG-VMS formulation (listing ), DG-VMS formulation (listing ), and H(div) formulation (listing ), which was earlier discussed in Section .",
"language=Python,caption=Firedrake code for 3D problem with TET mesh using H(div) formulation, label=Code:ex3, frame=single]Sections/Samplecode/3DcomposablesolversRT0.py language=Python,caption=Firedrake code for 3D problem with TET mesh using CG-VMS formulation , label=Code:ex1,frame=single]Sections/Samplecode/3DcomposablesolversCG.py language=Python,caption=Firedrake code for 3D problem with TET mesh using DG-VMS formulation, label=Code:ex2, frame=single]Sections/Samplecode/3DcomposablesolversDG.py"
],
[
"ACKNOWLEDGMENTS",
"KBN acknowledges the support through the High Priority Area Research Seed Grant from the Division of Research, University of Houston."
]
] | 1808.08328 |
[
[
"Paraphrases as Foreign Languages in Multilingual Neural Machine\n Translation"
],
[
"Abstract Paraphrases, the rewordings of the same semantic meaning, are useful for improving generalization and translation.",
"However, prior works only explore paraphrases at the word or phrase level, not at the sentence or corpus level.",
"Unlike previous works that only explore paraphrases at the word or phrase level, we use different translations of the whole training data that are consistent in structure as paraphrases at the corpus level.",
"We train on parallel paraphrases in multiple languages from various sources.",
"We treat paraphrases as foreign languages, tag source sentences with paraphrase labels, and train on parallel paraphrases in the style of multilingual Neural Machine Translation (NMT).",
"Our multi-paraphrase NMT that trains only on two languages outperforms the multilingual baselines.",
"Adding paraphrases improves the rare word translation and increases entropy and diversity in lexical choice.",
"Adding the source paraphrases boosts performance better than adding the target ones.",
"Combining both the source and the target paraphrases lifts performance further; combining paraphrases with multilingual data helps but has mixed performance.",
"We achieve a BLEU score of 57.2 for French-to-English translation using 24 corpus-level paraphrases of the Bible, which outperforms the multilingual baselines and is +34.7 above the single-source single-target NMT baseline."
],
[
"Introduction",
"Paraphrases, rewordings of texts with preserved semantics, are often used to improve generalization and the sparsity issue in translation [5], [9], [11], [23], [27].",
"Unlike previous works that use paraphrases at the word/phrase level, we research on different translations of the whole corpus that are consistent in structure as paraphrases at the corpus level; we refer to paraphrases as the different translation versions of the same corpus.",
"We train paraphrases in the style of multilingual NMT [16], [14] .",
"Implicit parameter sharing enables multilingual NMT to learn across languages and achieve better generalization [16].",
"Training on closely related languages are shown to improve translation [32].",
"We view paraphrases as an extreme case of closely related languages and view multilingual data as paraphrases in different languages.",
"Paraphrases can differ randomly or systematically as each carries the translator's unique style.",
"We treat paraphrases as foreign languages, and train a unified NMT model on paraphrase-labeled data with a shared attention in the style of multilingual NMT.",
"Similar to multilingual NMT's objective of translating from any of the $N$ input languages to any of the $M$ output languages [10], multi-paraphrase NMT aims to translate from any of the $N$ input paraphrases to any of the $M$ output paraphrases in Figure REF .",
"In Figure REF , we see different expressions of a host showing courtesy to a guest to ask whether sake (a type of alcohol drink that is normally served warm in Asia) needs to be warmed.",
"In Table REF , we show a few examples of parallel paraphrasing data in the Bible corpus.",
"Different translators' styles give rise to rich parallel paraphrasing data, covering wide range of domains.",
"In Table REF , we also show some paraphrasing examples from the modern poetry dataset, which we are considering for future research.",
"Indeed, we go beyond the traditional NMT learning of one-to-one mapping between the source and the target text; instead, we exploit the many-to-many mappings between the source and target text through training on paraphrases that are consistent to each other at the corpus level.",
"Our method achieves high translation performance and gives interesting findings.",
"The differences between our work and the prior works are mainly the following.",
"Unlike previous works that use paraphrases at the word or phrase level, we use paraphrases at the entire corpus level to improve translation performance.",
"We use different translations of the whole training data consistent in structure as paraphrases of the full training data.",
"Unlike most of the multilingual NMT works that uses data from multiple languages, we use paraphrases as foreign languages in a single-source single-target NMT system training only on data from the source and the target languages.",
"Our main findings in harnessing paraphrases in NMT are the following.",
"Our multi-paraphrase NMT results show significant improvements in BLEU scores over all baselines.",
"Our paraphrase-exploiting NMT uses only two languages, the source and the target languages, and achieves higher BLEUs than the multi-source and multi-target NMT that incorporates more languages.",
"We find that adding the source paraphrases helps better than adding the target paraphrases.",
"We find that adding paraphrases at both the source and the target sides is better than adding at either side.",
"We also find that adding paraphrases with additional multilingual data yields mixed performance; its performance is better than training on language families alone, but is worse than training on both the source and target paraphrases without language families.",
"Adding paraphrases improves the sparsity issue of rare word translation and diversity in lexical choice.",
"In this paper, we begin with introduction and related work in Section and .",
"We introduce our models in Section .",
"Finally, we present our results in Section and conclude in Section ."
],
[
"Paraphrasing",
"Many works generate and harness paraphrases [2], [24], [4], [21], [11], [3], [26], [20], [29], [15].",
"Some are on question and answer [9], [7], evaluation of translation [31] and more recently NMT [23], [27].",
"Past research includes paraphrasing unknown words/phrases/sub-sentences [5], [23], [27], [8].",
"These approaches are similar in transforming the difficult sparsity problem of rare words prediction and long sentence translation into a simpler problem with known words and short sentence translation.",
"It is worthwhile to contrast paraphrasing that diversifies data, with knowledge distillation that benefits from making data more consistent [13].",
"Our work is different in that we exploit paraphrases at the corpus level, rather than at the word or phrase level."
],
[
"Multilingual Attentional NMT",
"Machine polyglotism which trains machines to translate any of the $N$ input languages to any of the $M$ output languages from many languages to many languages, many languages is a new paradigm in multilingual NMT [10], [33], [6], [12], [1], [30].",
"The objective is to translate from any of the $N$ input languages to any of the $M$ output languages [10].",
"Many multilingual NMT systems involve multiple encoders and decoders [14], and it is hard to combine attention for quadratic language pairs bypassing quadratic attention mechanisms [10].",
"An interesting work is training a universal model with a shared attention mechanism with the source and target language labels and Byte-Pair Encoding (BPE) [16], [14].",
"This method is elegant in its simplicity and its advancement in low-resource language translation and zero-shot translation using pivot-based translation mechanism [16], [10].",
"Unlike previous works, our parallelism is across paraphrases, not across languages.",
"In other words, we achieve higher translation performance in the single-source single-target paraphrase-exploiting NMT than that of the multilingual NMT.",
"Table: Comparison of adding source paraphrases and adding target paraphrases.All acronyms including data are explained in Section .Table: Comparison of adding a mix of the source paraphrases and the target paraphrases against the baselines.",
"All acronyms including data are explained in Section ."
],
[
"Models",
"We have four baseline models.",
"Two are single-source single-target attentional NMT models, the other two are multilingual NMT models with a shared attention [16], [14].",
"In Figure REF , we show an example of multilingual attentional NMT.",
"Translating from all 4 languages to each other, we have 12 translation paths.",
"For each translation path, we label the source sentence with the source and target language tags.",
"Translating from UTF8gbsn“你的清酒凉了吗?” to “Has your sake turned cold?”, we label the source sentence with __opt_src_zh __opt_tgt_en.",
"More details are in Section .",
"In multi-paraphrase model, all source sentences are labeled with the paraphrase tags.",
"For example, in French-to-English translation, a source sentence may be tagged with __opt_src_f1 __opt_tgt_e0, denoting that it is translating from version “f1” of French data to version “e0” of English data.",
"In Figure REF , we show 2 Japanese and 2 English paraphrases.",
"Translating from all 4 paraphrases to each other ($N=M=4$ ), we have 12 translation paths as $N\\times (N-1)=12$ .",
"For each translation path, we label the source sentence with the source and target paraphrase tags.",
"For the translation path from UTF8min“お酒冷めましたよね?” to “Has your sake turned cold?”, we label the source sentence with __opt_src_j1 __opt_tgt_e0 in Figure REF .",
"Paraphrases of the same translation path carry the same labels.",
"Our paraphrasing data is at the corpus level, and we train a unified NMT model with a shared attention.",
"Unlike the paraphrasing sentences in Figure REF , We show this example with only one sentence, it is similar when the training data contains many sentences.",
"All sentences in the same paraphrase path share the same labels.",
"Table: Examples of French-to-English translation trained using 12 French paraphrases and 12 English paraphrases."
],
[
"Data",
"Our main data is the French-to-English Bible corpus [22], containing 12 versions of the English Bible and 12 versions of the French Bible We considered the open subtitles with different scripts of the same movie in the same language; they covers many topics, but they are noisy and only differ in interjections.",
"We also considered the poetry dataset where a poem like “If” by Rudyard Kipling is translated many times, by various people into the same language, but the data is small.. We translate from French to English.",
"Since these 24 translation versions are consistent in structure, we refer to them as paraphrases at corpus level.",
"In our paper, each paraphrase refers to each translation version of whole Bible corpus.",
"To understand our setup, if we use all 12 French paraphrases and all 12 English paraphrases so there are 24 paraphrases in total, i.e., $N=M=24$ , we have 552 translation paths because $N\\times (N-1)=552$ .",
"The original corpus contains missing or extra verses for different paraphrases; we clean and align 24 paraphrases of the Bible corpus and randomly sample the training, validation and test sets according to the 0.75, 0.15, 0.10 ratio.",
"Our training set contains only 23K verses, but is massively parallel across paraphrases.",
"For all experiments, we choose a specific English corpus as e0 and a specific French corpus as f0 which we evaluate across all experiments to ensure consistency in comparison, and we evaluate all translation performance from f0 to e0."
],
[
"Training Parameters",
"In all our experiments, we use a minibatch size of 64, dropout rate of 0.3, 4 RNN layers of size 1000, a word vector size of 600, number of epochs of 13, a learning rate of 0.8 that decays at the rate of 0.7 if the validation score is not improving or it is past epoch 9 across all LSTM-based experiments.",
"Byte-Pair Encoding (BPE) is used at preprocessing stage [14].",
"Our code is built on OpenNMT [17] and we evaluate our models using BLEU scores [25], entropy [28], F-measure and qualitative evaluation."
],
[
"Baselines",
"We introduce a few acronyms for our four baselines to describe the experiments in Table REF , Table REF and Figure REF .",
"Firstly, we have two single-source single-target attentional NMT models, Single and WMT.",
"Single trains on f0 and e0 and gives a BLEU of 22.5, the starting point for all curves in Figure REF .",
"WMT adds the out-domain WMT'14 French-to-English data on top of f0 and e0; it serves as a weak baseline that helps us to evaluate all experiments' performance discounting the effect of increasing data.",
"Moreover, we have two multilingual baselinesFor multilingual baselines, we use the additional Bible corpus in 22 European languages that are cleaned and aligned to each other.",
"built on multilingual attentional NMT, Family and Span [32].",
"Family refers to the multilingual baseline by adding one language family at a time, where on top of the French corpus f0 and the English corpus e0, we add up to 20 other European languages.",
"Span refers to the multilingual baseline by adding one span at a time, where a span is a set of languages that contains at least one language from all the families in the data; in other words, span is a sparse representation of all the families.",
"Both Family and Span trains on the Bible in 22 Europeans languages trained using multilingual NMT.",
"Since Span is always suboptimal to Family in our results, we only show numerical results for Family in Table REF and REF , and we plot both Family and Span in Figure REF .",
"The two multilingual baselines are strong baselines while the fWMT baseline is a weak baseline that helps us to evaluate all experiments' performance discounting the effect of increasing data.",
"All baseline results are taken from a research work which uses the grid of (1, 6, 11, 16, 22) for the number of languages or equivalent number of unique sentences and we follow the same in Figure REF [32].",
"All experiments for each grid point carry the same number of unique sentences.",
"Table: Entropy increases with the number of paraphrase corpora in Vmix.",
"The 95% confidence interval is calculated via bootstrap resampling with replacement.Table: F1 score of frequency 1 bucket increases with the number of paraphrase corpora in Vmix, showing training on paraphrases improves the sparsity at tail and the rare word problem.Furthermore, Vsrc refers to adding more source (English) paraphrases, and Vtgt refers to adding more target (French) paraphrases.",
"Vmix refers to adding both the source and the target paraphrases.",
"Vmf refers to combining Vmix with additional multilingual data; note that only Vmf, Family and Span use languages other than French and English, all other experiments use only English and French.",
"For the x-axis, data refers to the number of paraphrase corpora for Vsrc, Vtgt, Vmix; data refers to the number of languages for Family; data refers to and the equivalent number of unique training sentences compared to other training curves for WMT and Vmf.",
"Figure: BLEU plots showing the effects of different ways of adding training data in French-to-English Translation.",
"All acronyms including data are explained in Section ."
],
[
"Results",
"Training on paraphrases gives better performance than all baselines: The translation performance of training on 22 paraphrases, i.e., 11 English paraphrases and 11 French paraphrases, achieves a BLEU score of 55.4, which is +32.9 above the Single baseline, +8.8 above the Family baseline, and +26.1 above the WMT baseline.",
"Note that the Family baseline uses the grid of (1, 6, 11, 16, 22) for number of languages, we continue to use this grid for our results on number of paraphrases, which explains why we pick 22 as an example here.",
"The highest BLEU 57.2 is achieved when we train on 24 paraphrases, i.e., 12 English paraphrases and 12 French paraphrases.",
"Adding the source paraphrases boosts translation performance more than adding the target paraphrases: The translation performance of adding the source paraphrases is higher than that of adding the target paraphrases.",
"Adding the source paraphrases diversifies the data, exposes the model to more rare words, and enables better generalization.",
"Take the experiments training on 13 paraphrases for example, training on the source (i.e., 12 French paraphrases and the English paraphrase e0) gives a BLEU score of 48.8, which has a gain of +1.4 over 47.4, the BLEU score of training on the target (i.e., 12 English paraphrases and the French paraphrase f0).",
"This suggests that adding the source paraphrases is more effective than adding the target paraphrases.",
"Adding paraphrases from both sides is better than adding paraphrases from either side: The curve of adding paraphrases from both the source and the target sides is higher than both the curve of adding the target paraphrases and the curve of adding the source paraphrases.",
"Training on 11 paraphrases from both sides, i.e., a total of 22 paraphrases achieves a BLEU score of 50.8, which is +3.8 higher than that of training on the target side only and +1.9 higher than that of training on the source side only.",
"The advantage of combining both sides is that we can combine paraphrases from both the source and the target to reach 24 paraphrases in total to achieve a BLEU score of 57.2.",
"Adding both paraphrases and language families yields mixed performance: We conduct one more experiment combining the source and target paraphrases together with additional multilingual data.",
"This is the only experiment on paraphrases where we use multilingual data other than only French and English data.",
"The BLEU score is 49.3, higher than training on families alone, in fact, it is higher than training on eight European families altogether.",
"However, it is lower than training on English and French paraphrases alone.",
"Indeed, adding paraphrases as foreign languages is effective, however, when there is a lack of data, mixing the paraphrases with multilingual data is helpful.",
"Adding paraphrases increases entropy and diversity in lexical choice, and improves the sparsity issue of rare words: We use bootstrap resampling and construct 95% confidence intervals for entropies [28] of all models of Vmix, i.e., models adding paraphrases at both the source and the target sides.",
"We find that the more paraphrases, the higher the entropy, the more diversity in lexical choice as shown in Table REF .",
"From the word F-measure shown in Table REF , we find that the more paraphrases, the better the model handles the sparsity of rare words issue.",
"Adding paraphrases not only achieves much higher BLEU score than the WMT baseline, but also handles the sparsity issue much better than the WMT baseline.",
"Adding paraphrases helps rhetoric translation and increases expressiveness: Qualitative evaluation shows many cases where rhetoric translation is improved by training on diverse sets of paraphrases.",
"In Table REF , Paraphrases help NMT to use a more contemporary synonym of “silver”, “money”, which is more direct and easier to understand.",
"Paraphrases simplifies the rhetorical or subtle expressions, for example, our model uses “rejoice” to replace “break out into song”, a personification device of mountains to describe joy, which captures the essence of the meaning being conveyed.",
"However, we also observe that NMT wrongly translates “clap the palm” to “strike”.",
"We find the quality of rhetorical translation ties closely with the diversity of parallel paraphrases data.",
"Indeed, the use of paraphrases to improve rhetoric translation is a good future research question.",
"Please refer to the Table REF for more qualitative examples."
],
[
"Conclusion",
"We train on paraphrases as foreign languages in the style of multilingual NMT.",
"Adding paraphrases improves translation quality, the rare word issue, and diversity in lexical choice.",
"Adding the source paraphrases helps more than adding the target ones, while combining both boosts performance further.",
"Adding multilingual data to paraphrases yields mixed performance.",
"We would like to explore the common structure and terminology consistency across different paraphrases.",
"Since structure and terminology are shared across paraphrases, we are interested in a building an explicit representation of the paraphrases and extend our work for better translation, or translation with more explicit and more explainable hidden states, which is very important in all neural systems.",
"We are interested in broadening our dataset in our future experiments.",
"We hope to use other parallel paraphrasing corpora like the poetry dataset as shown in Table REF .",
"There are very few poems that are translated multiple times into the same language, we therefore need to train on extremely small dataset.",
"Rhetoric in paraphrasing is important in poetry dataset, which again depends on the training paraphrases.",
"The limited data issue is also relevant to the low-resource setting.",
"We would like to effectively train on extremely small low-resource paraphrasing data.",
"As discussed above about the potential research poetry dataset, dataset with multiple paraphrases is typically small and yet valuable.",
"If we can train using extremely small amount of data, especially in the low-resource scenario, we would exploit the power of multi-paraphrase NMT further.",
"Cultural-aware paraphrasing and subtle expressions are vital [19], [18].",
"Rhetoric in paraphrasing is a very important too.",
"In Figure REF , “is your sake warm enough?” in Asian culture is an implicit way of saying “would you like me to warm the sake for you?”.",
"We would like to model the culture-specific subtlety through multi-paraphrase training."
],
[
"Supplemental Materials",
"We show a few examples of parallel paraphrasing data in the Bible corpus.",
"We also show some paraphrasing examples from the modern poetry dataset, which we are considering for future research.",
"Table: Examples of parallel paraphrasing data with English, French, Tagalog and Spanish paraphrases in Bible translation.Table: Examples of parallel paraphrasing data with German, Chinese, and Portuguese paraphrases of the English poem “If” by Rudyard Kipling."
]
] | 1808.08438 |
[
[
"Similarity model for corner roll in turbulent Rayleigh-Benard convection"
],
[
"Abstract The corner roll (CR) in the Rayleigh-B\\'{e}nard (RB) convection accounts for the behaviors of heat transport and convection flow at the corner.",
"Streamlines of the three-dimensional direct numerical simulations for $10^8<Ra<5\\times10^9$ show that CR presents well-defined similarity and multi-layer structure.",
"A stream function for CR is developed by homotopy and the structure ensemble dynamics.",
"The model presents the scaling of Reynolds number of corner roll $Re_{cr}\\sim Ra^{1/4}$.",
"Scaling of CR scale $r = 0.77 Ra^{-0.085}$ indicates strong near-wall shearing induced by wind and provides a probability of the `ultimate regime' at high $Ra$."
],
[
"Similarity model for corner roll in turbulent Rayleigh-Bénard convection Wen-Feng Zhou Jun Chen jun@pku.edu.cn State Key Laboratory for Turbulence and Complex Systems Department of Mechanics, College of Engineering, Peking University, Beijing 100871, China The corner roll (CR) in the Rayleigh-Bénard (RB) convection accounts for the behaviors of heat transport and convection flow at the corner.",
"Streamlines of the three-dimensional direct numerical simulations for $10^8<Ra<5\\times 10^9$ show that CR presents well-defined similarity and multi-layer structure.",
"A stream function for CR is developed by homotopy and the structure ensemble dynamics.",
"The model presents the scaling of Reynolds number of corner roll $Re_{cr}\\sim Ra^{1/4}$ .",
"Scaling of CR scale $r = 0.77 Ra^{-0.085}$ indicates strong near-wall shearing induced by wind and provides a probability of the `ultimate regime' at high $Ra$ .",
"47.55.pb, 47.27.-i, 44.25.+f, 47.27.E- The Rayleigh-Bénard (RB) convection is generated in a cell filled with fluid which is cooled from the top plate and heated from the bottom plate.",
"The control parameters of an RB system are the Rayleigh number $Ra = \\alpha g\\Delta {L_z^3}/(\\kappa \\nu )$ and the Prandtl number $Pr=\\nu /\\kappa $ of the fluid, and the aspect ratio $\\Gamma =L_x/L_z$ , where $g$ is the gravitational acceleration, $\\alpha $ the thermal expansion coefficient, $\\nu $ the kinematic viscosity, and $\\kappa $ the thermal diffusivity, $L_x$ and $L_z$ are the dimensions of the cell in width and height directions, respectively.",
"The response parameters are the Nusselt number, $Nu=q/\\left[-\\kappa (\\partial T/\\partial z)\\right]$ and the Reynolds number, $Re = UL/\\nu $ .",
"The corner roll (CR) is the secondary flow at the corners of an RB cell, induced by the large-scale-circulation (LSC).",
"Usually found in experiments [1], [2], [3] and numerical simulations [4], [5], [6], the CR as a quasi-steady structure at the corner persistently contributes to heat transfer for high $Ra$ number, despite $Pr$ [7].",
"Some authors suggested that the local heat transfer coefficient (or the local $Nu$ number) for the CR is larger than that in the shear region [8], [9], being explained as the result of strong fluctuation and strong vorticity carried by CR with cool jet from cooling plate impinging on the heated wall [10].",
"In this sense, it is worthwhile to quantify the spatial structure of the CR and its heat transfer performance with a mathematical model.",
"Some evidence indicates that the scale of CR decreases as increasing $Ra$ .",
"Thus investigating the behavior of CR also provide a way to explore the physics of the ultimate regime predicted by [11] in 1962.",
"Some researches have attempted to study the CR-type flow based on its properties.",
"For high $Re$ number cases, [12] first proved that, under the steady Euler limit, for two-dimensional enclosed by vortex sheets, the vorticity away from the sheets is constant.",
"[13] generalized this work to fluid with body force.",
"For moderate $Re$ number, [14] found that when $Re > 100$ , the vortex develops from completely viscous to inviscid rotational core.",
"They also derived that under Euler limit, the temperature in the core is uniform.",
"For small $Re$ number, [15] found a series of similarity solutions using separation variable method solving the Stokes flow.",
"He found that for the right angle, any flow sufficiently near the corner must consist of a sequence of eddies decreasing their size and intensity.",
"In mathematical modelling of CR, [12] and [14] proposed the analytical solutions by assuming the circular shape of the CR.",
"However, the conditions for the solutions limit their application to the flow with complex configurations, e.g.",
"the corner region in the RB convection.",
"A solution for small $Re$ number was suggested by [15], but it still cannot be directly applied to an RB cell either in the corner region of relatively high $Re$ number or high $Ra$ number.",
"Therefore, it is necessary to develop a model to describe the CR in turbulent convection.",
"The SED theory claims that there are three kinds of basic ansatz in turbulent boundary layer (BL), i.e.",
"power law, defect power law and generalized invariant relation by Lie group analysis [16].",
"Having been examined by canonical wall-bounded turbulence [17], [18], the SED also unifies the temperature profile and the $Ra$ -scaling of coefficient of the log-law [19].",
"In applications, the SED has been extended to develop turbulent transition model and RANS model for the flow around foils considering the effects of pressure gradient and finite $Re$ number [20].",
"Being invoked by the geometric similarity of the corner flow and the SED theory, we here developed a self-similarity model for CR based on SED and homotopy analysis.",
"We performed the 3D DNS for $Ra=10^{8}, 5\\times 10^{8}, 10^{9}, 5\\times 10^{9}$ with $Pr=0.7$ of the Boussinesq equations using a second-order staggered finite difference scheme from [21].",
"The flow is confined in a narrow rectangular cell with aspect ratio of $L_{x} \\text{(width)}: L_{y} \\text{(depth)}: L_{z} \\text{(height)} = 1:1/6:1$ .",
"The periodic boundary condition is employed in the depth ($y$ ) direction, and the large-scale convection flow is thus held on the $xz$ -plane.",
"The resolution of the simulations is up to $1024\\times 256 \\times 800$ by clustering grid points near the boundaries.",
"The largest grid scale is still smaller than the Kolmogorov and Batchelor scales [22], [23], ensuring that both momentum and thermal energy are well resolved.",
"The scaling of $Nu$ of the DNS is $Nu = 3.27 Ra^{0.294}$ , consistent with other DNS and experiments [24], [25].",
"Since no reversal of LSC occurs through the computational time and the quasi-steady CRs are held at the corners of the cell, the statistical properties of CR are obtained by time-average of flow field, which are shown in Fig.",
"REF .",
"The triangular CRs are wedged at the corners along the diagonal direction of the cell.",
"The slip line as the interface between CR and LSC playing a key role in transporting kinetic energy between the large-scale structures.",
"In statistical sense, the energy sustaining rotating of a CR is supported both by shearing of slip line from LSC and the buoyancy from horizontal conducting plate.",
"Figure: (a) Time-average field colored by temperature for Ra=1×10 8 Ra=1\\times 10^8.",
"The arrowed-lines represent the streamline.",
"The purple dashed lines mark the slip lines.",
"(b)The RaRa-scaling of ReRe of CR, slip line, represented by symbols.",
"The blue line is the CR scaling given by the model.",
"The black line is the empirical fitting of the slip line scaling.",
"The red line is the correlation of CR and the slip line.It is seen that the $Ra$ number has limited effect on the triangle-shaped CR, the scale of which diminishes with increasing $Ra$ .",
"The velocity on the slip line reaches it maximum in the middle with zero velocity at both stagnation ends.",
"Particularly, the maximum velocity in the middle of the slip line implies local intensive momentum and energy transport between CR and LSC, which decreases with increasing $Ra$ .",
"To quantify the momentum transfer ability of the slip line, we define the Reynolds number of the slip line as $Re_{sl}=\\int \\frac{v_{sl}}{\\upsilon }\\mathrm {d}l$ , where $v_{sl}$ is the average velocity on slip line and $l$ the length of slip line, representing a scaling law of $Ra$ , $Re_{sl} = 3.15 Ra^{0.295}$ ; see Fig.",
"REF .",
"A model based on the circumferential similarity of the stremlines inside CR is developed.",
"An algorithmic expression of the streamlines is obtained by homotopy, giving the transformation expression of two formula continuously deforming from one to another [26].",
"We here construct a homotopy $H: X\\times I \\rightarrow Y, I=[0,1]$ for any $f \\in X, H(f,0)=f_{2}$ and $H(f,1) = f_{1}$ , where $f_{2}(x,z) = 0$ represents a small vicinity of the CR center, and $f_{1}(x,z)=0$ is the function of the CR boundary.",
"When $\\xi \\in I$ as the similarity variable changes from 0 to 1, $f(x,z)=0$ for $H(f,\\xi )=0$ deforms continuously from $f_{2}(x,z)=0$ to $f_{1}(x,z)=0$ .",
"The implicit homotopy expression is simplified by linearly combining $f_{1}$ and $f_{2}$ , $H=\\xi f_{1} + (1-\\xi )\\Lambda _{0}f_{2}$ .",
"If $H=0$ , we get the explicit form of $\\xi $ as $\\xi = {\\Lambda _{0}f_{2}}/ ({-f_{1}+\\Lambda _{0}f_{2}}) $ where $\\Lambda _{0}$ is considered as the constraint strength between the inner and outer layer.",
"The least square procedure is used to obtain $f_{1}$ , $f_{2}$ and the magnitude $\\Lambda _{0}$ .",
"The boundary of CR, $f_{1}$ , is composed of two walls and the slip line.",
"To keep accuracy, the slip line is expressed as a cubic function $z-(a + bx + cx^{2} + dx^{3})=0$ .",
"The walls are respectively written as $x=0$ and $z=0$ .",
"Thus the boundary of CR is given as $f_{1}=-xz(z-(a + bx + cx^{2} + dx^{3}))=0$ .",
"On the other hand, the inner boundary as a small core at the central region of CR is expressed as $f_{2} = (x-s_{0})^{2} + (z-t_{0})^{2} - \\epsilon ^2 = 0$ , where $(s_{0}, t_{0})$ is the coordinates of the center, and $\\epsilon $ the radius of the core.",
"In the case of $Ra = 10^{8}$ , they are $a=0.36, b=-1,955, c=10.07, d=-29.58, s_{0}=0.11, t_{0}=0.11, \\epsilon =0.0001$ .",
"The strength factor $\\Lambda _{0}$ in the range of $[0.05, 0.2]$ (e.g.",
"$\\Lambda _{0}=0.12$ for $Ra=1\\times 10^{8}$ ) indicates that, for the present cases, the constraint of the rigid wall is always stronger than that of the center.",
"We consider independent similarity variable $\\xi $ (Eq.",
"(REF )) and the stream function $\\psi $ as the dependent similarity variable.",
"The dimensionless Reynolds-averaged Naiver-Stokes (RANS) equations can be written as $u_i\\partial _i u_j = -\\partial _j p+ Ra^{-1/2} Pr^{1/2} \\partial _{ii} u_j - \\partial _i\\overline{u_j u_i} + \\theta \\delta _{j3},$ where $u_j, p, \\theta $ represent the Reynolds-average quantities.",
"The velocity component is expressed in the stream-function form, using $u_1 = {{\\partial \\psi }}/{{\\partial z}}$ , $u_3 = -{{\\partial \\psi }}/{{\\partial x}}$ .",
"Then we have $ A=V+\\Pi ,$ where $A=\\partial _3\\psi \\partial _{ii}\\partial _1\\psi - \\partial _1\\psi \\partial _{ii}\\partial _3\\psi $ is advection term, $V=Ra^{-1/2} Pr^{1/2} \\partial _{ii}\\partial _{jj} \\psi $ is viscous term and $\\Pi = \\partial _{13}\\left(\\overline{u^{\\prime }_3u^{\\prime }_3-u^{\\prime }_1u^{\\prime }_1}\\right)+\\partial _{11}\\overline{u^{\\prime }_1u^{\\prime }_3} - \\partial _{33}\\overline{u^{\\prime }_1u^{\\prime }_3} -\\partial _1 \\theta $ is fluctuation term.",
"As discussed above, the viscosity effect on CR is relatively small and negligible.",
"Thus, the balance is $A \\simeq \\Pi $ .",
"Based on the geometry similarity, we assume that stream function has a similarity solution under the similarity variable $\\xi $ , i.e.",
"$\\psi = \\psi [\\xi ] = \\psi [x,y;a_0]$ , and have $\\psi _\\xi \\left(\\psi _{\\xi \\xi } h_1 +\\psi _\\xi h_2\\right) = \\Pi ,$ where $h_1 = 2[\\partial _1\\xi \\partial _3 \\xi \\partial _{11}\\xi +\\left(\\partial _3 \\xi \\right)^2\\partial _{13}\\xi -\\left(\\partial _1 \\xi \\right)^2\\partial _{13}\\xi - \\partial _1\\xi \\partial _3\\xi \\partial _{33}\\xi ]$ and $h_2 = \\partial _{111}\\xi \\partial _3\\xi +\\partial _{133}\\xi \\partial _3\\xi - \\partial _{113}\\xi \\partial _1\\xi -\\partial _{333}\\xi \\partial _1\\xi $ , representing the coordinate transformation functions.",
"A similarity solution requires $\\Pi $ to be a function of $\\xi $ , i.e.",
"$\\Pi =\\Pi (\\xi )$ , and the ratio of $h_2/h_1 = c_2$ is a constant or a function of $\\xi $ .",
"The DNS show that the viscosity effect on CR is so weak that negligible.",
"Thus a relationship between the fluctuation terms and the steam-function-differential term $\\psi _\\xi $ is established as $\\Pi /\\psi _\\xi ^2 = c_1$ .",
"We have $\\psi _{\\xi } \\left[\\psi _{\\xi \\xi } + \\psi _\\xi \\left(c_1 + c_2\\right) \\right] = 0$ with its solution $\\psi = \\psi _{0}{\\mathrm {exp}(- \\xi /\\xi _0)} + B$ , where $\\xi _{0}=1/({c_1} + {c_2})$ .",
"Considering the inner boundary condition $\\psi (\\xi = 0) = 0$ , then the stream-function follows $\\psi = \\psi _{0}\\left(1 - {e^{ -\\xi /\\xi _{0} }}\\right),$ as an exact similarity solution.",
"The coefficients $\\psi _{0}$ and $\\xi _{0}$ can be measured from DNS (e.g.",
"$\\psi _{0}=0.028, \\xi _{0}=1.9429$ for $Ra=1\\times 10^8$ ).",
"By analyzing the stream function profiles along the similarity coordinate $\\xi $ at different central angles, one can see that the stream function collapses well to the DNS, except for the profile of angle of $\\pi /6$ .",
"Eq.",
"(REF ) is the solution of the flow under the solid wall condition, thus the profile near the slip line has a perceptive deviation.",
"Figure: The CR and its stream function profiles at five angles for the case of Ra=10 8 Ra=10^8.The stream function for the bulk is no longer proper for the near-wall flow — viscosity should be taken into account.",
"Canonical turbulent BL theory cannot successfully describe the undeveloped flow in CR.",
"Moreover, the fluid near the wall being advected downstream undergoes the varying pressure gradient along with emission of thermal plumes.",
"To obtain a function of boundary layer (BL) in CR, we apply the structure ensemble dynamics (SED) theory [16], by introducing the stress length and the symmetry of the wall taking into account the constraint of solid wall, pressure gradient, and thermal effects [19].",
"Space average of the momentum equation in the steamwise direction and integration from 0 to $z$ gives $\\frac{{d{u^ + }}}{{d{z^ + }}} - \\frac{{\\overline{u^{\\prime }w^{\\prime }} }}{{u_\\tau ^2}} = 1 + \\frac{1}{{u_{\\tau }^{2}L_{z}}}\\int _0^z {({p_r} - {p_l})dz},$ where $u$ and $u^{\\prime }w^{\\prime }$ are steamwise-average variable and $p_r$ and $p_l$ are the right and left side wall pressure, respectively.",
"Velocity and length scale are normalized by friction velocity ${u_\\tau } = \\sqrt{{{\\left.",
"{\\nu du/dz} \\right|}_{z = 0}}} $ and viscous length ${l_\\nu } = \\nu /{u_\\tau }$ .",
"By introducing ${S^ + } = \\frac{du^{+}}{dz^{+}}$ ,${W^ + } = - \\frac{{\\overline{u^{\\prime }w^{\\prime }} }}{u_{\\tau }^{2}}$ , ${\\tau ^ + } = 1 + \\frac{1}{{u_{\\tau }^{2}L_{z}}}\\int _0^z {({p_r} - {p_l})dz} $ , we have ${S^ + } + {W^ + } = {\\tau ^ + }$ .",
"In the region near the wall, $\\tau ^{+} \\simeq 1$ .",
"Substituting the stress length function $\\ell _M^ + = \\sqrt{{W^ + }} /{S^ + }$ as the multilayer similarity function of the BL gives ${S^ + } = \\left({{ - 1 + \\sqrt{1 + 4\\ell _M^{ + 2}{\\tau ^ + }} }}\\right)/\\left({{2\\ell _M^{ + 2}}}\\right).$ The velocity profile is eventually obtained by integrating from 0 to $z$ , $u = {u_\\tau }{u^ + } = {u_\\tau }\\int _0^z {{S^ + }dz}$ .",
"We apply SED to quantify the pressure gradient effect on BL in CR.",
"The stress length along $x$ direction is $\\ell _{M,x}^ + = \\sqrt{{W_x}^ + } /{S_x}^ + $ for the horizontal BL and the stress length in $z$ direction $\\ell _{M,z}^ + = \\sqrt{{W_z}^ + } /{S_z}^ + $ for the vertical BL.",
"We apply Taylor expansion to the shear rate $S^+$ and the Reynolds stress $W^+$ .",
"The no-slip boundary and continuity condition follow $u \\sim z$ , $w\\sim z^{2}$ for horizontal BL.",
"We have $\\ell _{M,x}^ + = \\sqrt{{W_x}^ + } /{S_x}^ + = {\\ell _{0,x}}{z^{3/2}}$ for the BL.",
"Similarly, the stress length of vertical BL is $\\ell _{M,z}^ + = \\sqrt{{W_z}^ + } /{S_z}^ + = {\\ell _{0,z}}{(x^+)^{3/2}}$ , where $\\ell _{0,x}$ and $\\ell _{0,z}$ are the coefficient and the function of $x$ or $z$ , respectively.",
"The parameter $\\ell _{0,x}$ is expressed as follows ${\\ell _{0,x}} = \\left\\lbrace \\begin{array}{l}a_1(x - x_0),{\\rm { }}0 < x < x_c\\\\b_1/(x_{ap} - x),{\\rm { }} x_c < x < x_{ap}\\end{array} \\right.$ where $x_0$ ($0.01<x_0<0.0533$ for the present study) is the critical point defining the two ranges; $x_c$ ($0.0747<x_c<0.11$ ) the position of the core of CR for different $Ra$ numbers.",
"The parameters, $a_1$ ($2.35<a_1<4.5$ ), $b_1(=0.4Ra^{-0.125})$ , $x_{ap}$ ($0.19<x_{ap}<0.26$ ) are related to $Ra$ number.",
"Similarly, $\\ell _{0,z}$ can be expressed as ${\\ell _{0,z}} = \\left\\lbrace \\begin{array}{l}b_3/z, 0< z < z_c\\\\a_3(z_0-z), z_c < z < z_{0},\\end{array} \\right.$ where $z_0$ ($0.206<z_0<0.2692$ for the present study) is the position of the separation point, $z_c$ ($0.0819<z_c<0.11$ ) is the position of the core of CR.",
"The parameters, $a_3(=500Ra^{-0.28})$ and $b_3(=1.3Ra^{-0.2})$ are determined by $Ra$ number.",
"It is noteworthy that the relations between the parameters and $Ra$ number, i.e.",
"$x_0$ , $x_c$ , $a_1$ , $x_{ap}$ , $z_0$ and $z_c$ as the functions of $Ra$ , cannot be determined with the present simulated cases.",
"Further simulations with wider range of $Ra$ are necessary to obtain the parameters in the stress lengths.",
"Combining the stream function for the bulk and the multi-layer structure for the BL, we have a unified model for CR.",
"$\\ell _{0,x}$ and $\\ell _{0,z}$ indicate that the eddy scale is inversely proportional to the upstream distance with favorable pressure gradient and decreases linearly with adverse pressure gradient.",
"Comparison of the model-reconstructed CR and the DNS, we find that the streamlines and velocity distribution are almost the same except for those very close to the core and the slip line.",
"At the core, low Reynolds effect might cause the elliptical core.",
"The slip line is expressed in Eq.",
"(REF ), which does not specify the difference between the solid wall and the slip line, though the solid wall has stronger constraint than the slip line.",
"As an assessment of the model, the relative error is examined as $10.93\\%$ for $Ra = 1\\times 10^8$ and $10.45\\%$ for $Ra = 10^9$ — all below $11\\%$ .",
"Figure: Comparison of DNS with reconstruction of the similarity solution.",
"Velocity magnitude nephogram and vector arrow represent the DNS and the steam-line is obtained from Eq.",
"() (a) Ra=1×10 8 Ra=1\\times 10^8; (b) Ra=10 9 Ra=10^9The downwelling flow around the slip line is fully turbulence, and the CR is confined in the corner.",
"As the heated bottom plate pumps energy into the corner region, enforcing the vorticity of CR.",
"Thus, the scale of the CR varying with $Ra$ is detemined by these effects.",
"The CR is characterized both by $|\\psi _{0}/\\xi _{0}|$ , which is $\\psi _{0}/\\xi _{0} = 1.45Ra^{ - 1/4}$ , and the scale of CR, $r$ .",
"The scale of CR, $r$ is defined as the average distance of the core to the intersection point of walls, which is found to obey the scaling law as $r = 0.77 Ra^{-0.085}$ .",
"The negative power exponent indicates that high $Ra$ would suppress the CR and thus elongate the wind shearing region.",
"Recalling the conjecture of realization of `ultimate region' induced by the fully turbulent BL [11], [27], [28], a relatively wilder shear region would improve turbulence of the BL [29], [30].",
"Therefore, the Reynolds number of CR is given as ${Re_{cr,mod }} = \\frac{{{u_{cr,mod }}r\\cos \\theta }}{\\nu } \\simeq \\frac{{\\left| \\psi _{0}/\\xi _{0} \\right|}\\sqrt{Ra/Pr}}{{C_{1} }}$ Using $r/L_{z}=0.77Ra^{-0.085}$ , $|\\psi _{0}/\\xi _{0}|=1.45Ra^{ - 1/4}$ and $C_{1}=1/\\sqrt{2}$ , we have $Re_{cr,mod} = 1.45\\sqrt{2}Ra^{1/4}Pr^{1/2}$ .",
"It is noted that although the characteristic velocity and scale become smaller at higher $Ra$ , the solution of Moffatt for the Stokes flow of the corner cannot be applied either.",
"The heat transport in the corner roll region is investigated.",
"Being invoked by the hypothesis of mixing zone [31], we obtained the correlation between $Re_{cr}$ and $Nu_{cr}$ .",
"The assumptions are listed as follows: (a) The characteristic velocity of CR fulfils an anomalous scaling as $u_{cr} \\sim (\\alpha g \\Delta _{cr} L)^{1/2} Ra^{\\chi }$ ; (b) temperature scale of the bulk of CR is proportional to the temperature difference of the top and bottom plate, i.e.",
"$\\Delta _{cr}/\\Delta \\sim const.",
"$ ; (c) heat flux is determined by the inner convection heat transport, $H \\sim u_{cr} \\Delta _{cr}$ ; (d) $w_{h}$ is of the same order of the CR's characteristic velocity, i.e.",
"$u_{cr} \\sim w_{h}$ ; (e) during the emission of plumes, viscous force and buoyancy are balanced as $\\alpha g \\Delta \\sim \\nu w_{h}/\\lambda ^{2}_{cr}$ ; (f) defining a number of scaling indexes through the relations $Re_{cr}=\\frac{u_{cr}r}{\\nu } \\sim Ra^{\\epsilon }, Nu_{cr}^{-1}= \\lambda _{cr}/L_{z} \\sim \\frac{\\kappa \\Delta /L_{z}}{H} \\sim Ra^{-\\beta }, \\frac{r}{L_{z}} \\sim Ra^{\\eta } $ .",
"Based on relations of (a), (b) and (f), we have $\\frac{u_{cr}r}{\\nu } \\sim (\\frac{\\alpha g \\Delta L_{z}^{3}}{\\kappa \\nu })^{1/2} (\\frac{\\kappa }{\\nu })^{1/2} Ra^{\\chi } \\frac{r}{L_{z}}.$ Relations of (c) and (f) follow $\\frac{u_{cr}r}{\\nu } \\frac{\\nu }{\\kappa } \\frac{L_{z}}{r} \\sim Ra^{\\beta }.$ Relations of (d), (e) and (f) give $\\frac{u_{cr}r}{\\nu } \\sim \\frac{\\alpha g \\Delta L_{z}^{3}}{\\kappa \\nu } \\frac{\\lambda _{cr}^{2}}{L_{z}^2} \\frac{\\kappa }{\\nu } \\frac{r}{L_{z}}.$ The power exponents in Eq.",
"(REF ) – (REF ) give $ \\left\\lbrace \\begin{array}{l}\\varepsilon = 1/2 + \\chi + \\eta \\\\\\beta = \\varepsilon - \\eta \\\\\\varepsilon = 1 - 2\\beta + \\eta \\end{array} \\right..$ With the scaling of CR scale $\\eta =-0.085$ , we get the rest exponents, which are $\\beta = 1/3$ , $\\epsilon =1/3 - \\eta = 0.248$ , and $\\chi = 1/3 -1/2 = -1/6 \\approx -0.167$ .",
"The DNS show that the scalings of the characteristic velocity $u_{cr}/U_{f} \\sim Ra^{-0.165}$ , and that of Reynolds number $Re_{cr}=\\frac{u_{cr}r}{\\nu } \\sim Ra^{0.250}$ , close to the scalings from the assumptions.",
"More importantly, the thermal boundary thickness described by the approximation is $\\lambda _{cr}/L \\sim Ra^{-1/3}$ , in agreement with the DNS, $\\lambda _{cr,dns}/L_{z} \\sim Ra^{-0.331}$ .",
"It is noteworthy that the assumption of the constant CR temperature has only been examined with the unit aspect ratio.",
"The validity of the assumption at different aspect ratios needs further investigation.",
"The present results can be extended to the large scale circulation.",
"While we established the stream function for corner roll, it is likely that the stream-function for the boundary for the triangular shape of the corner roll but also applicable for the hexagon boundary of the bulk flow; see Fig.",
"REF .",
"Another important aspect is the multilayer structure of the vertical and horizontal profiles near the walls.",
"We anticipate that, depending on the boundary type, the boundary layer of the large-scale circulation may be described by the similar multilayer function, in particular for the solid wall boundary, and for the slip line.",
"Another intriguing question relates to the assumptions specified for the CR, in relation to the $Re$ number.",
"The positive scaling of $Re_{cr}$ ($\\epsilon = 0.248$ ) indicates that the corner roll is enforced by increasing the $Ra$ number, though the size of CR decreases with the $Ra$ number with a negative exponent $\\eta = -0.085$ .",
"The enhancement of convection leads intensive heat transport at higher $Ra$ number, i.e.",
"$Nu\\sim Ra^{0.331}$ , where the power exponent even higher than in the wind shearing region ($Nu\\sim Ra^{0.295}$ ).",
"From the hypothesis building perspective, our work also motivates investigating separation scenarios leading to enhanced heat transport.",
"In the future, we also plan to explore parameterization of the streamline functions of CR and bulk flow.",
"Finally, it may prove fruitful for more detailed simulations to study the corner roll under various control parameters, such as the $Pr$ R number, $\\Gamma $ .",
"We thank Zhen-Su She for helpful comments and suggestions.",
"This work is supported by National Nature Science (China) Fund 11452002, 11521091, and 11372362, and by MOST (China) 973 project 2009CB724100."
]
] | 1808.08510 |
[
[
"Title-Guided Encoding for Keyphrase Generation"
],
[
"Abstract Keyphrase generation (KG) aims to generate a set of keyphrases given a document, which is a fundamental task in natural language processing (NLP).",
"Most previous methods solve this problem in an extractive manner, while recently, several attempts are made under the generative setting using deep neural networks.",
"However, the state-of-the-art generative methods simply treat the document title and the document main body equally, ignoring the leading role of the title to the overall document.",
"To solve this problem, we introduce a new model called Title-Guided Network (TG-Net) for automatic keyphrase generation task based on the encoder-decoder architecture with two new features: (i) the title is additionally employed as a query-like input, and (ii) a title-guided encoder gathers the relevant information from the title to each word in the document.",
"Experiments on a range of KG datasets demonstrate that our model outperforms the state-of-the-art models with a large margin, especially for documents with either very low or very high title length ratios."
],
[
"Introduction",
"Keyphrases are short phrases that can quickly provide the main information of a given document (the terms “document”, “source text” and “context” are interchangeable in this study, and all of them represent the concatenation of the title and the main body.).",
"Because of the succinct and accurate expression, keyphrases are widely used in information retrieval [14], document categorizing [12], opinion mining [2], etc.",
"Due to the huge potential value, various automatic keyphrase extraction and generation methods have been developed.",
"As shown in Figure REF , the input usually consists of the title and the main body, and the output is a set of keyphrases.",
"Figure: An example of keyphrase generation.",
"The present keyphrases are bold and italic in the source text.Most typical automatic keyphrase extraction methods [37], [25], [27] focus on extracting present keyphrases like “relevance profiling” in Figure REF , which are the exact phrases appearing in the source text.",
"The main ideas among them are identifying candidate phrases first and then ranking algorithms.",
"However, these methods ignore the semantic meaning underlying the context content and cannot generate absent keyphrases like “interactive information retrieval”, which do not appear in the source text.",
"To overcome the above drawbacks, several encoder-decoder based keyphrase generation methods have been proposed including CopyRNN [26] and CopyCNN [39].",
"First, these methods treat the title and the main body equally and concatenate them as the only source text input.",
"Then, the encoder maps each source text word into a hidden state vector which is regarded as the contextual representation.",
"Finally, based on these representations, the decoder generates keyphrases from a predefined vocabulary regardless of the presence or absence of the keyphrases.",
"A serious drawback of these models is that they ignore the leading role of the title and consequently fail to sufficiently utilize the already summarized information in it.",
"It is a widely agreed fact that the title can be viewed as a high-level summary of a document and the keyphrases provide more details of the key topics introduced in the document [20].",
"They play a similar and complementary role with each other.",
"Therefore, keyphrases should have close semantic meaning to the title [20].",
"For example, as shown in Figure REF , the title contains most of the salient points reflected by these keyphrases including “retrieval”, “profiling”, and “evaluation”.",
"Statistically, we study the proportion of keyphrases related to the title on the largest KG dataset and show the results in Table REF .",
"For simplicity, we define a TitleRelated keyphrase as the keyphrase containing at least one common non-stop-word with the title.",
"From Table REF , we find that about 33% absent keyphrases are TitleRelated.",
"For present keyphrases, the TitleRelated percentage is up to around 60%.",
"By considering the fact that the length of a title is usually only 3%-6% of the corresponding source text, we can conclude that the title, indeed, contains highly summative and valuable information for generating keyphrases.",
"Moreover, information in the title is also helpful in reflecting which part of the main body is essential, such as the part containing the same or related information with the title.",
"For instance, in Figure REF , the point “evaluation” in the title can guide us to focus on the part “... task-oriented, comparative evaluation ...” of the main body, which is highly related to the absent keyphrase “task-oriented evaluation”.",
"To sufficiently leverage the title content, we introduce a new title-guided network by taking the above fact into the keyphrase generation scenario.",
"In our model, the title is additionally treated as a query-like input in the encoding stage.",
"First, two bi-directional Gated Recurrent Unit (GRU) [3] layers are adopted to separately encode the context and the title into corresponding contextual representations.",
"Then, an attention-based matching layer is used to gather the relevant title information for each context word according to the semantic relatedness.",
"Since the context is the concatenation of the title and the main body, this layer implicitly contains two parts.",
"The former part is the “title to title” self-matching, which aims to make the salient information in the title more important.",
"The latter part is the “main body to title” matching wherein the title information is employed to reflect the importance of information in the main body.",
"Next, an extra bi-directional GRU layer is used to merge the original contextual information and the corresponding gathered title information into the final title-guided representation for each context word.",
"Finally, the decoder equipped with attention and copy mechanisms utilizes the final title-guided context representation to predict keyphrases.",
"We evaluate our model on five real-world benchmarks, which test the ability of our model to predict present and absent keyphrases.",
"Using these benchmarks, we demonstrate that our model can effectively exploit the title information and it outperforms the relevant baselines by a significant margin: for present (absent) keyphrase prediction, the improvement gain of F1-measure at 10 (Recall at 50) score is up to 9.4% (19.1%) compared to the best baseline on the largest dataset.",
"Besides, we probe the performance of our model and a strong baseline CopyRNN on documents with different title length ratios (i.e., the title length over the context length).",
"Experimental results show that our model consistently improves the performance with large gains, especially for documents with either very low or very high title length ratios.",
"Table: The statistics of TitleRelated keyphrases on the validation set of KP20k.Our main contributions consist of three parts: A new perspective on keyphrase generation is explored, which sufficiently employs the title to guide the keyphrase prediction process.",
"A novel TG-Net model is proposed, which can effectively leverage the useful information in the title.",
"The overall empirical results on five real-world benchmarks show that our model outperforms the state-of-the-art models significantly on both present and absent keyphrase prediction, especially for documents with either very low or very high title length ratios."
],
[
"Automatic Keyphrase Extraction",
"Most of the automatic keyphrase extraction methods consist of two steps.",
"Firstly, the candidate identification step obtains a set of candidate phrases such as phrases with some specific part-of-speech (POS) tags [25], [37].",
"Secondly, in the ranking step, all the candidates are ranked based on the importance computed by either unsupervised ranking approaches [35], [27], [5] or supervised machine learning approaches [25], [37], [30], [6].",
"Finally, the top-ranked candidates are selected as the keyphrases.",
"Besides these widely developed two-step approaches, there are also some methods using a sequence labeling operation to extract keyphrases [38], [22], [9].",
"But they still cannot generate absent keyphrases.",
"Some extraction approaches [20], [21] also consider the influence of the title.",
"[20] li2010semititle proposes a graph-based ranking algorithm which initializes the importance score of title phrases as one and the others as zero and then propagates the influence of title phrases iteratively.",
"The biggest difference between [20] li2010semititle and our method is that our method utilizes the contextual information of the title to guide the context encoding, while their model only considers the phrase occurrence in the title.",
"[21] liu2011automatictitle models keyphrase extraction process as a translation operation from a document to keyphrases.",
"The title is used as the target output to train the translator.",
"Compared with our model, one difference is that this method still cannot handle semantic meaning of the context.",
"The other is that our model regards the title as an extra query-like input instead of a target output."
],
[
"Automatic Keyphrase Generation",
"Keyphrase generation is an extension of keyphrase extraction which explicitly considers the absent keyphrase prediction.",
"CopyRNN [26] first frames the generation process as a sequence-to-sequence learning task and employs a widely used encoder-decoder framework [34] with attention [1] and copy [10] mechanisms.",
"Based on CopyRNN, various extensions [11], [15] are recently proposed.",
"However, these recurrent neural network (RNN) based models may suffer the low-efficiency issues because of the computation dependency between the current time step and the preceding time steps in RNN.",
"To overcome this shortcoming, CopyCNN [39] applies a convolutional neural network (CNN) based encoder-decoder model [8].",
"CopyCNN employs position embedding for obtaining a sense of order in the input sequence and adopts gated linear units (GLU) [4] as the non-linearity function.",
"CopyCNN not only achieves much faster keyphrase generation speed but also outperforms CopyRNN on five real-world benchmark datasets.",
"Nevertheless, both CopyRNN and CopyCNN treat the title and the main body equally, which ignores the semantic similarity between the title and the keyphrases.",
"Motivated by the success of query-based encoding in various natural language processing tasks [7], [33], [28], [36], we regard the title as an extra query-like input to guide the source context encoding.",
"Consequently, we propose a TG-Net model to explicitly explore the useful information in the title.",
"In this paper, we focus on how to incorporate a title-guided encoding into the RNN-based model, but it is also convenient to apply this idea to the CNN-based model in a similar way."
],
[
"Problem Definition",
"We denote vectors with bold lowercase letters, matrices with bold uppercase letters and sets with calligraphy letters.",
"We denote $\\Theta $ as a set of parameters and $\\mathbf {W}$ as a parameter matrix.",
"Keyphrase generation (KG) is usually formulated as follows: given a context $\\mathbf {x}$ , which is the concatenation of the title and the main body, output a set of keyphrases $\\mathcal {Y}=\\lbrace \\mathbf {y}^i\\rbrace _{i=1,\\dots ,M}$ where $M$ is the keyphrase number of $\\mathbf {x}$ .",
"Here, the context $\\mathbf {x}=[x_1,\\dots ,x_{L_{\\mathbf {x}}}]$ and each keyphrase $\\mathbf {y}^i=[y_1^i,\\dots ,y^i_{L_{\\mathbf {y}^i}}]$ are both word sequences, where $L_{\\mathbf {x}}$ is the length (i.e., the total word number) of the context and $L_{\\mathbf {y}^i}$ is the length of the $i$ -th produced keyphrase $\\mathbf {y}^i$ .",
"To adapt the encoder-decoder framework, $M$ context-keyphrase pairs $\\lbrace (\\mathbf {x}, \\mathbf {y}^i)\\rbrace _{i=1,\\dots ,M}$ are usually split.",
"Since we additionally use the title $\\mathbf {t}=[t_1,\\dots ,t_{L_{\\mathbf {t}}}]$ with length $L_{\\mathbf {t}}$ as an extra query-like input, we split $M$ context-title-keyphrase triplets $\\lbrace (\\mathbf {x}, \\mathbf {t}, \\mathbf {y}^i)\\rbrace _{i=1,\\dots ,M}$ instead of context-keyphrase pairs to feed our model.",
"For conciseness, we use $(\\mathbf {x}, \\mathbf {t}, \\mathbf {y})$ to represent such a triplet, where $\\mathbf {y}$ is one of its target keyphrases."
],
[
"Title-Guided Encoder Module",
"As shown in Figure REF , the title-guided encoder module consists of a sequence encoding layer, a matching layer, and a merging layer.",
"First, the sequence encoding layer reads the context input and the title input and learns their contextual representations separately.",
"Then the matching layer gathers the relevant title information for each context word reflecting the important parts of the context.",
"Finally, the merging layer merges the aggregated title information into each context word producing the final title-guided context representation.",
"Figure: The title-guided encoder module.",
"(Best viewed in color.)"
],
[
"Sequence Encoding Layer",
"At first, an embedding look-up table is applied to map each word within the context and the title into a dense vector with a fixed size $d_e$ .",
"To incorporate the contextual information into the representation of each word, two bi-directional GRUs [3] are used to encode the context and the title respectively: $\\overrightarrow{\\textbf {u}}_i &= \\text{GRU}_{11}(\\textbf {x}_i, \\overrightarrow{\\textbf {u}}_{i-1}), \\\\\\overleftarrow{\\textbf {u}}_i &= \\text{GRU}_{12}(\\textbf {x}_i, \\overleftarrow{\\textbf {u}}_{i+1}), \\\\\\overrightarrow{\\textbf {v}}_j &= \\text{GRU}_{21}(\\textbf {t}_j, \\overrightarrow{\\textbf {v}}_{j-1}),\\\\\\overleftarrow{\\textbf {v}}_j &= \\text{GRU}_{22}(\\textbf {t}_j, \\overleftarrow{\\textbf {v}}_{j+1}),$ where $i=1,2,\\dots ,L_{\\mathbf {x}}$ and $j=1,2,\\dots ,L_{\\mathbf {t}}$ .",
"$\\mathbf {x}_i$ and $\\mathbf {t}_j$ are the $d_e$ -dimensional embedding vectors of the $i$ -th context word and the $j$ -th title word separately.",
"$\\overrightarrow{\\textbf {u}}_i$ , $\\overleftarrow{\\textbf {u}}_i$ , $\\overrightarrow{\\textbf {v}}_j$ , and $\\overleftarrow{\\textbf {v}}_j$ are $d/2$ -dimensional hidden vectors where $d$ is the hidden dimension of the bi-directional GRUs.",
"The concatenations $\\mathbf {u}_i = [\\overrightarrow{\\mathbf {u}}_i; \\overleftarrow{\\mathbf {u}}_i] \\in \\mathbb {R}^d$ and $\\mathbf {v}_j = [\\overrightarrow{\\mathbf {v}}_j; \\overleftarrow{\\mathbf {v}}_j] \\in \\mathbb {R}^d$ are used as the contextual vectors for the $i$ -th context word and the $j$ -th title word respectively."
],
[
"Matching Layer",
"The attention-based matching layer is engaged to aggregate the relevant information from the title for each word within the context.",
"The aggregation operation $\\mathbf {c}_i = attn(\\mathbf {u}_i, [\\mathbf {v}_1, \\mathbf {v}_2,\\dots , \\mathbf {v}_{L_{\\mathbf {t}}}]; \\mathbf {W}_1)$ is as follows: $\\mathbf {c}_i &= \\sum _{j=1}^{L_{\\mathbf {t}}} \\alpha _{i,j} \\mathbf {v}_j, \\\\\\alpha _{i,j} &= \\exp (s_{i,j}) \\slash \\sum _{k=1}^{L_{\\mathbf {t}}} \\exp (s_{i,k}),\\\\s_{i,j} &= (\\mathbf {u}_i)^T \\mathbf {W}_1 \\mathbf {v}_j,$ where $\\mathbf {c}_i \\in \\mathbb {R}^d$ is the aggregated information vector for the $i$ -th word of $\\mathbf {x}$ .",
"$\\alpha _{i,j}$ ($s_{i,j}$ ) is the normalized (unnormalized) attention score between $\\mathbf {u}_i$ and $\\mathbf {v}_j$ .",
"Here, the matching layer is implicitly composed of two parts because the context is a concatenation of the title and the main body.",
"The first part is the “title to title” self-matching part, wherein each title word attends the whole title itself and gathers the relevant title information.",
"This part is used to strengthen the important information in the title itself, which is essential to capture the core information because the title already contains much highly summative information.",
"The other part is the “main body to title” matching part wherein each main body word also aggregates the relevant title information based on semantic relatedness.",
"In this part, the title information is employed to reflect the importance of information in the main body based on the fact that the highly title-related information in the main body should contain core information.",
"Through these two parts, this matching layer can utilize the title information much more sufficiently than any of the previous sequence to sequence methods."
],
[
"Merging Layer",
"Finally, the original contextual vector $\\mathbf {u}_i$ and the aggregated information vector $\\mathbf {c}_i$ are used as the inputs to another information merging layer: $\\overrightarrow{\\mathbf {m}}_i &= \\text{GRU}_{31}([\\mathbf {u}_i; \\mathbf {c}_i], \\overrightarrow{\\mathbf {m}}_{i-1}),\\\\\\overleftarrow{\\mathbf {m}}_i &= \\text{GRU}_{32}([\\mathbf {u}_i; \\mathbf {c}_i], \\overleftarrow{\\mathbf {m}}_{i+1}),\\\\\\widetilde{\\mathbf {m}}_i &= \\lambda \\mathbf {u}_i + (1-\\lambda ) [\\overrightarrow{\\mathbf {m}}_i; \\overleftarrow{\\mathbf {m}}_i], $ where $[\\mathbf {u}_i; \\mathbf {c}_i] \\in \\mathbb {R}^{2d}$ , $\\overrightarrow{\\mathbf {m}}_i \\in \\mathbb {R}^{d/2}$ , $\\overleftarrow{\\mathbf {m}}_i \\in \\mathbb {R}^{d/2}$ , $[\\overrightarrow{\\mathbf {m}}_i, \\overleftarrow{\\mathbf {m}}_i] \\in \\mathbb {R}^d$ , and $\\widetilde{\\mathbf {m}}_i \\in \\mathbb {R}^d$ .",
"The $\\mathbf {u}_i$ in Eq.",
"(REF ) is a residual connection, and $\\lambda \\in (0,1)$ is the corresponding hyperparameter.",
"Eventually, we obtain the title-guided contextual representation of the context (i.e., $[\\widetilde{\\mathbf {m}}_1, \\widetilde{\\mathbf {m}}_2,\\dots , \\widetilde{\\mathbf {m}}_{L_{\\mathbf {x}}}]$ ), which is regarded as a memory bank for the later decoding process."
],
[
"Decoder Module",
"After encoding the context into the title-guided representation, we engage an attention-based decoder [23] incorporating with copy mechanism [32] to produce keyphrases.",
"Only one foward GRU is used in this module: $\\mathbf {h}_t &= \\text{GRU}_4([\\mathbf {e}_{t-1}; \\tilde{\\mathbf {h}}_{t-1}], \\mathbf {h}_{t-1}), \\\\\\hat{\\mathbf {c}}_t &= attn(\\mathbf {h}_t, [\\widetilde{\\mathbf {m}}_1, \\widetilde{\\mathbf {m}}_2,\\dots , \\widetilde{\\mathbf {m}}_{L_{\\mathbf {x}}}]; \\mathbf {W}_2), \\\\\\tilde{\\mathbf {h}}_{t} &= \\text{tanh}(\\mathbf {W}_3[\\hat{\\mathbf {c}}_t; \\mathbf {h}_t]),$ where $t=1,2,\\dots ,L_{\\mathbf {y}}$ , $\\mathbf {e}_{t-1} \\in \\mathbb {R}^{d_e}$ is the embedding of the $(t-1)$ -th predicted word wherein $\\mathbf {e}_0$ is the embedding of the start token, $\\hat{\\mathbf {c}}_t \\in \\mathbb {R}^d$ is the aggregated vector for $\\mathbf {h}_t \\in \\mathbb {R}^d$ from the memory bank $[\\widetilde{\\mathbf {m}}_1, \\widetilde{\\mathbf {m}}_2,\\dots , \\widetilde{\\mathbf {m}}_{L_{\\mathbf {x}}}]$ , and $\\tilde{\\mathbf {h}}_t \\in \\mathbb {R}^d$ is the attentional vector at time step $t$ .",
"Consequently, the predicted probability distribution over the predefined vocabulary $\\mathcal {V}$ for current step is computed by: $P_{v}(y_{t}|\\mathbf {y}_{t-1}, \\mathbf {x}, \\mathbf {t}) = \\text{softmax}(\\mathbf {W}_v\\tilde{\\mathbf {h}}_{t} + \\mathbf {b}_v),$ where $\\mathbf {y}_{t-1} = [y_1,\\dots , y_{t-1}]$ is the previous predicted word sequence, and $\\mathbf {b}_v \\in \\mathbb {R}^{|\\mathcal {V}|}$ is a learnable parameter vector.",
"Before generating the predicted word, a copy mechanism is adopted to efficiently exploit the in-text information and to strengthen the extraction capability of our model.",
"We follow [32] see2017gettothepoint and first calculate a soft switch between generating from the vocabulary and copying from the source context $\\mathbf {x}$ at time step $t$ : $g_t = \\sigma (\\mathbf {w}^T_g\\tilde{\\mathbf {h}}_{t} + b_g),$ where $\\mathbf {w}_g \\in \\mathbb {R}^d$ is a learnable parameter vector and $b_g$ is a learnable parameter scalar.",
"Eventually, we get the final predicted probability distribution over the dynamic vocabulary $\\mathcal {V}\\cup \\mathcal {X}$ , where $\\mathcal {X}$ are all words appearing in the source context.",
"For simplicity, we use $P_v(y_t)$ and $P_{final}(y_t)$ to denote $P_v(y_{t}|\\mathbf {y}_{t-1}, \\mathbf {x}, \\mathbf {t})$ and $P_{final}(y_{t}|\\mathbf {y}_{t-1}, \\mathbf {x}, \\mathbf {t})$ respectively: $P_{final}(y_t) = (1-g_t) P_v(y_t) + g_t\\sum _{i:x_i=y_t}\\hat{\\alpha }_{t,i},$ where $\\hat{\\alpha }_{t,i}$ is the normalized attention score between $\\mathbf {h}_t$ and $\\widetilde{\\mathbf {m}}_i$ .",
"For all out-of-vocabulary (OOV) words (i.e., $y_t\\notin \\mathcal {V}$ ), we set $P_{v}(y_t)$ as zero.",
"Similarly, if word $y_t$ does not appear in the source context $\\mathbf {x}$ (i.e., $y_t\\notin \\mathcal {X}$ ), the copy probability $\\sum _{i:x_i=y_t}\\hat{\\alpha }_{t,i}$ is set as zero."
],
[
"Training",
"We use the negative log likelihood loss to train our model: $\\mathcal {L} = -\\sum _{t=1}^{L_{\\mathbf {y}}} log P_{final}(y_t|\\mathbf {y}_{t-1}, \\mathbf {x}, \\mathbf {t}; \\Theta ),$ where $L_{\\mathbf {y}}$ is the length of target keyphrase $\\mathbf {y}$ and $y_t$ is the $t$ -th target word in $\\mathbf {y}$ , and $\\Theta $ represents all the learnable parameters."
],
[
"Experiment Settings",
"The keyphrase prediction performance is first evaluated by comparing our model with the popular extractive methods and the state-of-the-art generative methods on five real-world benchmarks.",
"Then, comparative experiments of different title length ratios are performed on our model and CopyRNN for further model exploration.",
"Finally, an ablation study and a case study are conducted to better understand and interpret our model.",
"The experiment results lead to the following findings: Our model outperforms the state-of-the-art models on all the five benchmark datasets for both present and absent keyphrase prediction.",
"Our model consistently improves the performance on various title length ratios and obtains relative higher improvement gains for both very low and very high title length ratios.",
"The title-guided encoding part and the copy part are consistently effective in both present and absent keyphrase prediction tasks.",
"We implement the models using PyTorch [31] on the basis of the OpenNMT-py system [18]."
],
[
"Training Dataset",
"Because of the public accessibility, many commonly-used scientific publication datasets are used to evaluate the explored KG methods.",
"This study also focuses on generating keyphrases from scientific publications.",
"For all the generative models (i.e.",
"our TG-Net model as well as all the encoder-decoder baselines), we choose the largest publicly available keyphrase generation dataset KP20k constructed by [26] meng2017dkg as the training dataset.",
"KP20k consists of a large amount of high-quality scientific publications from various computer science domains.",
"Totally 567,830 articles are collected in this dataset, where 527,830 for training, 20,000 for validation, and 20,000 for testing.",
"Both the validation set and testing set are randomly selected.",
"Since the other commonly-used datasets are too small to train a reliable generative model, we only train these generative models on KP20k and then test the trained models on all the testing part of the datasets listed in Table REF .",
"As for the traditional supervised extractive baseline, we follow [26] meng2017dkg and use the dataset configuration shown in Table REF .",
"To avoid the out-of-memory problem, for KP20k, we use the validation set to train the traditional supervised extractive baseline."
],
[
"Testing Datasets",
"Besides KP20k, we also adopt other four widely-used scientific datasets for comprehensive testing, including Inspec [13], Krapivin [19], NUS [29], and SemEval-2010 [16].",
"Table REF summarizes the statistics of each testing dataset.",
"Table: The statistics of testing datasets.",
"The “Training” means the training part for the traditional supervised extractive baseline.",
"The “FFCV” represents five-fold cross validation.",
"The “Testing” means the testing part for all models."
],
[
"Implementation Details",
"For all datasets, the main body is the abstract, and the context is the concatenation of the title and the abstract.",
"During preprocessing, various operations are performed including lowercasing, tokenizing by CoreNLP [24], and replacing all the digits with the symbol $\\langle digit \\rangle $ .",
"We define the vocabulary $\\mathcal {V}$ as the 50,000 most frequent words.",
"We set the embedding dimension $d_e$ to 100, the hidden size $d$ to 256, and $\\lambda $ to 0.5.",
"All the initial states of GRU cells are set as zero vectors except that $\\mathbf {h}_0$ is initialized as $[\\overrightarrow{m}_{L_{\\mathbf {x}}}; \\overleftarrow{m}_1]$ .",
"We share the embedding matrix among the context words, the title words, and the target keyphrase words.",
"All the trainable variables including the embedding matrix are initialized randomly with uniform distribution in [-0.1, 0.1].",
"The model is optimized by Adam [17] with batch size = 64, initial learning rate = 0.001, gradient clipping = 1, and dropout rate = 0.1.",
"We decay the learning rate into the half when the evaluation perplexity stops dropping.",
"Early stopping is applied when the validation perplexity stops dropping for three continuous evaluations.",
"During testing, we set the maximum depth of beam search as 6 and the beam size as 200.",
"We repeat the experiments of our model three times using different random seeds and report the averaged results.",
"We do not remove any predicted single-word phrase in the post-processing for KP20k during testing, which is different from [26] meng2017dkg, since our model is trained on this dataset and it can effectively learn the distribution of the single-word keyphrases.",
"But for other testing datasets, we only keep the first predicted single-word phrase following [26] meng2017dkg."
],
[
"Baseline Models and Evaluation Metric",
"For present keyphrase predicting experiment, we use two unsupervised models including TF-IDF and TextRank [27], and one supervised model Maui [25] as our traditional extraction baselines.",
"Besides, we also select CopyRNN [26] and CopyCNN [39], the two state-of-the-art encoder-decoder models with copy mechanism [10], as the baselines for present keyphrase prediction task.",
"As for absent keyphrase prediction, since traditional extraction baselines cannot generate such keyphrases, we only choose CopyRNN and CopyCNN as the baseline models.",
"For all baselines, we use the same setups as [26] meng2017dkg and [39] zhang2017dkgconv.",
"The recall and F-measure ($\\text{F}_1$ ) are employed as our metrics for evaluating these algorithms.",
"Recall is the number of correctly predicted keyphrases over the total number of target kayphrases.",
"$\\text{F}_1$ score is computed based on the Recall and the Precision, wherein Precision is defined as the number of correctly predicted keyphrases over the total predicted keyphrase number.",
"Following [26] meng2017dkg and [39] zhang2017dkgconv, we also employ Porter Stemmer for preprocessing when determining whether two keyphrases are matched.",
"Table: Present keyphrase predicting results on all test datasets.",
"“% gain” is the improvement gain over CopyCNN.Table: Absent keyphrase predicting results on all test datasets.",
"“% gain” is the improvement gain over CopyCNN."
],
[
"Present Keyphrase Predicting",
"In this section, we compare present keyphrase prediction ability of these models on the five real-world benchmark datasets.",
"The F-measures at top 5 and top 10 predictions of each model are shown in Table REF .",
"From this table, we find that all the generative models significantly outperforms all the traditional extraction baselines.",
"Besides, we also note that our TG-Net model achieves the best performance on all the datasets with significant margins.",
"For example, on KP20k dataset, our model improves 9.4% ($\\text{F}_1$ @10 score) than the best generative model CopyCNN.",
"Compared to CopyRNN which also applies an RNN-based framework, our model improves about 20.2%.",
"The results show that our model obtains much stronger keyphrase extraction ability than CopyRNN and CopyCNN."
],
[
"Absent Keyphrase Predicting",
"In this setting, we consider the absent keyphrase predicting ability which requires the understanding of the semantic meaning of the context.",
"Only the absent target keyphrases and the absent predictions are preserved for this evaluation.",
"Generally, recalls at top 10 and top 50 predictions are engaged as the metrics to evaluate how many absent target keyphrases are correctly predicted.",
"The performance of all models is listed in Table REF .",
"It is observed that our TG-Net model consistently outperforms the previous sequence-to-sequence models on all the datasets.",
"For instance, our model exceeds 19.1% (R@50 score) on KP20k than the state-of-the-art model CopyCNN.",
"Overall, the results indicate that our model is able to capture the underlying semantic meaning of the context content much better than these baselines, as we have anticipated.",
"Figure: Present keyphrase predicting ability (F1@5 measure) on various title length ratios.Figure: A prediction example of CopyRNN and TG-Net.",
"The top 5 predictions are compared and the correct predictions are highlighted in bold."
],
[
"Keyphrase Predicting on Various Title Length Ratios",
"To find out how our title incorporation influences the prediction ability, we compare the keyphrase predicting ability of two RNN-based models (i.e., our model and CopyRNN) on different title length ratios.",
"The title length ratio is defined as the title length over the context length.",
"This analysis is based on the KP20k testing dataset.",
"In view of the title length ratio, we preprocess the testing set into five groups (i.e., $<$ 3%, 3%-6%, 6%-9%, 9%-12% and $>$ 12%).",
"Then, the present keyphrase prediction results (F1@5 measure) and the improvement gain on each group are depicted in Figure REF .",
"In Figure REF (a), we notice that both CopyRNN and our TG-Net model generally perform better when the title length ratio is higher.",
"One possible explanation is that when the title is long, it conveys substantial salient information of the abstract.",
"Therefore, the chance for the models to attend to the core information is enhanced, which leads to the observed situation.",
"This figure also shows that both TG-Net and CopyRNN get worse performance on $>$ 12% group than 9%-12% group.",
"The main reason is that there exist some data with a short abstract in $>$ 12% group, which leads to the lack of enough context information for correctly generating all keyphrases.",
"In Figure REF (b), we find that our TG-Net consistently improves the performance with a large margin on five testing groups, which again indicates the effectiveness of our model.",
"In a finer perspective, we note that the improvement gain is higher on the lowest (i.e., $<$ 3%) and the highest (i.e., $>$ 12%) title length ratio groups.",
"In $>$ 12% group, the title plays a more important role than in other groups, and consequently our model benefits more by not only explicitly emphasizing the title information itself, but also utilizing it to guide the encoding of information in the main body.",
"As for $<$ 3% group, the effect of such a short title is small on the latter part of the context in CopyRNN because of the long distance.",
"However, our model explicitly employs the title to guide the encoding of each context word regardless of the distance, which utilizes the title information much more sufficiently.",
"Consequently, our model achieves much higher improvement in this group.",
"While we only display the results of present keyphrase prediction, the absent keyphrase predicting task gets the similar results."
],
[
"Ablation Study",
"We also perform an ablation study on Krapivin for better understanding the contributions of the main parts of our model.",
"For a comprehensive comparison, we conduct this study on both present keyphrase prediction and absent keyphrase prediction.",
"As shown in Table REF , after we remove the title-guided part and only reserve the sequence encoding for the context (i.e., -title), both the present and absent keyphrase prediction performance become obviously worse, indicating that our title-guided context encoding is consistently critical for both present and absent keyphrase generation tasks.",
"We also investigate the effect of removing the copy mechanism (i.e., -copy) from our TG-Net.",
"From Table REF , we notice that the scores decrease dramatically on both present and absent keyphrase prediction, which demonstrates the effectiveness of the copy mechanism in finding important parts of the context."
],
[
"Case Study",
"A keyphrase prediction example for a paper about the exponential stability of uncertain switched stochastic delay systems is shown in Figure REF .",
"To be fair, we also only compare the RNN-based models (i.e., TG-Net and CopyRNN).",
"For present keyphrase, we find that a present keyphrase “non-linear uncertainties”, which is a title phrase, is correctly predicted by our TG-Net, while CopyRNN fails to do so.",
"As for absent keyphrase, we note that CopyRNN fails to predict the absent keyphrase “time-delay systems”.",
"But our TG-Net can effectively utilize the title information “stochastic delay systems” to locate the important abstract information “stochastic systems with time-delay” and then successfully generate this absent keyphrase.",
"These results exhibit that our model is capable of capturing the title-related core information more effectively and achieving better results in predicting present and absent keyphrases.",
"Table: Ablation study on Krapivin dataset."
],
[
"Conclusion",
"In this paper, we propose a novel TG-Net for keyphrase generation task, which explicitly considers the leading role of the title to the overall document main body.",
"Instead of simply concatenating the title and the main body as the only source input, our model explicitly treats the title as an extra query-like input to guide the encoding of the context.",
"The proposed TG-Net is able to sufficiently leverage the highly summative information in the title to guide keyphrase generation.",
"The empirical experiment results on five popular real-world datasets exhibit the effectiveness of our model for both present and absent keyphrase generation, especially for a document with very low or very high title length ratio.",
"One interesting future direction is to explore more appropriate evaluation metrics for the predicted keyphrases instead of only considering the exact match with the human labeled keyphrases as the current recall and F-measure do."
],
[
"Acknowledgments",
"The work described in this paper was partially supported by the Research Grants Council of the Hong Kong Special Administrative Region, China (No.",
"CUHK 14208815 and No.",
"CUHK 14210717 of the General Research Fund), and Microsoft Research Asia (2018 Microsoft Research Asia Collaborative Research Award).",
"We would like to thank Jingjing Li, Hou Pong Chan, Piji Li and Lidong Bing for their comments."
]
] | 1808.08575 |
[
[
"Magnetic-field-induced parity effect in insulating Josephson junction\n chains"
],
[
"Abstract We report the experimental manifestation of even-odd parity effects in the transport characteristics of insulating Josephson junction chains which occur as the superconducting gap is suppressed by applied magnetic fields at millikelvin temperatures.",
"The primary signature is a non-monotonic dependence of the critical voltage, $V_c$, for the onset of charge transport through the chain, with the parity crossover indicated by a maximum of $V_c$ at the parity field $B^*$.",
"We also observe a distinctive change in the transport characteristics across the parity transition, indicative of Cooper-pair dominated transport below $B^*$, giving way to single-electron dominated transport above $B^*$.",
"For fields applied in the plane of the superconducting aluminum films, the parity effect is found to occur at the field, $B^*_{||}$, such that the superconducting gap equals the single-electron charging energy, $\\Delta(B^*_{||})=E_C$.",
"On the contrary, the parity effect for perpendicularly applied fields can occur at relatively lower fields, $B^*_\\perp\\simeq 2\\Phi_0/A_I$, depending only on island area, $A_I$.",
"Our results suggest a novel explanation for the insulating peak observed in disordered superconducting films and one-dimensional strips patterned from such films."
],
[
"Magnetic-field-induced parity effect in insulating Josephson junction chains Timothy Duty [corresponding author, ]t.duty@unsw.edu.au.",
"School of Physics, University of New South Wales, Sydney, NSW 2052 Australia Karin Cedergren School of Physics, University of New South Wales, Sydney, NSW 2052 Australia Sergey Kafanov Physics Department, Lancaster University, Lancaster, UK LA1 4YB.",
"Roger Ackroyd School of Physics, University of New South Wales, Sydney, NSW 2052 Australia Jared H. Cole Chemical and Quantum Physics, School of Science, RMIT University, Melbourne,VIC 3001 Australia We report the experimental manifestation of even-odd parity effects in the transport characteristics of insulating Josephson junction chains which occur as the superconducting gap is suppressed by applied magnetic fields at millikelvin temperatures.",
"The primary signature is a non-monotonic dependence of the critical voltage, $V_c$ , for the onset of charge transport through the chain, with the parity crossover indicated by a maximum of $V_c$ at the parity field $B^*$ .",
"We also observe a distinctive change in the transport characteristics across the parity transition, indicative of Cooper-pair dominated transport below $B^*$ , giving way to single-electron dominated transport above $B^*$ .",
"For fields applied in the plane of the superconducting aluminum films, the parity effect is found to occur at the field, $B^*_{||}$ , such that the superconducting gap equals the single-electron charging energy, $\\Delta (B^*_{||})=E_C$ .",
"On the contrary, the parity effect for perpendicularly applied fields can occur at relatively lower fields, $B^*_\\perp \\simeq 2\\Phi _0/A_I$ , depending only on island area, $A_I$ .",
"Our results suggest a novel explanation for the insulating peak observed in disordered superconducting films and one-dimensional strips patterned from such films.",
"The ground state of a mesoscopic BCS superconductor has been shown to contain an even number of electrons, as long as the single-electron charging energy is less than the superconducting gap, $E_C<\\Delta $ , and the temperature is less than a characteristic temperature $T^*$ .",
"An even-odd parity effect occurs as $T$ exceeds $T^*$ , or at very low temperatures, if $\\Delta $ becomes lower than $E_C$ .",
"This has been demonstrated experimentally as a change from $2e$ to $e$ -periodicity in the gate-voltage modulation of superconducting aluminum single-charge transistors and Cooper-pair box qubits[1], [2], [3], [4], [5].",
"A quantitative analysis of the parity effect in hybrid superconductor-semiconductor islands has become an important experimental tool in identifying Majorana modes[6].",
"In this Letter, we show that even-odd parity effects strongly affect magnetotransport in insulating Josephson junction chains, and lead to a peak in the critical voltage with magnetic field.",
"Such an insulating peak is observed in homogeneously disordered superconducting films and strips[7], [8], [9], [10], [11], which are conjectured to behave as Josephson-coupled grains or droplets[12], [13].",
"Josephson junctions chains are also discrete versions of superconducting nanowires We recently reported results on the thermal parity effect observed in Josephson junction chains very deep in the insulating state, where the characteristic Josephson energy is much less than the Cooper-pair charging energy, $E_\\mathrm {J} \\ll E_{CP}$ , $(E_{CP}=4E_C$ )[14].",
"The hallmark of the insulating state—a voltage gap to conduction—was found to vanish sharply at the characteristic temperature $\\textrm {k}_\\textrm {B}T^* = \\Delta / \\ln N_\\mathrm {eff} \\simeq \\Delta /9$ , which coincides with the presence of $\\sim 1$ thermally excited BCS quasiparticle per island, where $N_\\mathrm {eff} (T)\\approx \\mathcal {V}\\rho (0)\\sqrt{2\\pi k_\\mathrm {B}T\\Delta (T)}$ is the effective number of states arising from integration over the quasiparticle density of states, $\\mathcal {V}$ is the volume of the island and $\\rho (0)$ is the density of states for the normal metal at the Fermi energy[1].",
"In a more recent Letter, we showed that insulating Josephson junction chains in zero magnetic field, with $E_\\mathrm {J}\\,\\sim \\,E_{CP}$ , behave as 1D Luttinger liquids, pinned by offset charge disorder, and therefore can be understood as a circuit implementation of the one-dimensional Bose glass[15].",
"The key result was that the voltage gap, $V_c,$ for the onset of conduction arises from collective depinning of Cooper-pair quasicharge, and is inversely related to the localisation length as calculated by Giamarchi and Schultz[16].",
"We also found $V_c$ to be proportional to the number of junctions in the chain, $N$ , and decreasing as a power law in bandwidth, $W$ .",
"The Bloch bandwidth $W$ is prescribed by the single-junction theory of quasicharge energy bands[17], [18], and can be envisioned as the amplitude for coherent tunneling of flux quanta.",
"$W$ decreases exponentially with $\\sqrt{8E_\\mathrm {J}/E_C}$ .",
"In this work, we examine the dependence of $V_c$ on both parallel and perpendicularly-applied magnetic fields, finding a non-monotonic dependence of the critical voltage with magnetic field.",
"Below an orientation-dependent cross over field, $B^*$ , the critical voltage $V_c$ increases with field in accordance with increasing $W$ , which occurs by suppression of the superconducting gap, $\\Delta $ , and hence decreasing $E_\\mathrm {J}$ .",
"Above $B^*$ , however, the critical voltage decreases with field, until finally becoming independent of magnetic field above the superconducting critical field $B_c$ .",
"We show that the peak behavior in the critical voltage, along with the change in current-voltage characteristics (IVC's) at $B^*$ reveal a parity crossover where collective depinning of Cooper pairs gives way to that of single-electron charges.",
"Moreover, when the field is applied perpendicular to the island films, the parity effect can be accompanied by a change in vorticity of the superconducting islands.",
"In other words, the parity effect occurs in sync with formation of the single-vortex state.",
"Charge transport above $B^*$ for both geometries is found to be markedly different from that below, lending additional support to our interpretation of the insulating peak as an experimental signature of an even-odd parity transition.",
"The interplay of parity and vortex dynamics has been discussed theoretically by Khaymovich et al.",
"[19] for single-island devices in the context of charge pumping, and single-vortex trapping was observed experimentally in hybrid normal-superconducting-normal transistors[20].",
"Although the parity effect in superconducting single-charge transistors and Cooper-pair boxes has been well known for some years, it has received very little attention in studies of Josephson junction arrays.",
"A theoretical paper by Feigel'man et al.",
"[21] pointed out the implications of the parity effect on the experimental search for the Kosterliz-Thouless charge-unbinding transition.",
"The parity effect for chains in the sequential tunneling limit, $E_\\mathrm {J} \\ll E_C$ , was studied theoretically by [22], inspired by the experimental results of Bylander et al.[23].",
"Detailed treatments of parity effects in junction arrays with $E_\\mathrm {J}$$\\,$$\\sim $$\\,$$E_C$ , two-dimensional disordered superconducting films, and one-dimensional superconducting nanowires, however, are conspicuously absent.",
"Table: DevicesFigure: Measured current on a logarithmic scale, (log 10 |I|\\log _{10}|I|), versus bias voltage VV and magnetic field for (a) device C5 in a parallel magnetic field B || B_{||}, and (b) for device B6 in a perpendicular field B ⊥ B_{\\perp }.",
"Critical voltage V c (B || V_c(B_{||} found for device C5 (c), and V c (B ⊥ V_c(B_\\perp for device B6 (d).",
"The parity transition occurs at B * B^*, the field where critical voltage, V c V_c, is maximum.",
"The critical field B c B_c is identified as the field at which V c V_c becomes independent of field.We have fabricated and measured a large ensemble of Al/AlO$_ {x}$ /Al single-junction chains[15].",
"Several families of devices were produced, where within each family, we varied the junction area, $A_J$ , in order to geometrically tune the ratio $g=E_\\mathrm {J0}/4E_C$ , where $E_\\mathrm {J0}$ is the Josephson tunneling energy at zero magnetic field (see SM[24]).",
"For each device, we first obtain an accurate measure of the average junction charging energy, $E_C$ , from the voltage offset, $V_\\mathrm {off}$ , of each device found from extrapolating its linear current-voltage characteristic (IVC) from large voltage biases, $V \\gg 2 N \\Delta _0/e$ , where $\\Delta _0$ is the superconducting gap in zero magnetic field, and $N$ is the number of junctions in the chain.",
"As noted in [14], [24], [25], the experimentally determined charging energy is found as$\\text{,}$ $E_C = e V_\\mathrm {off}/N$ , and the junction Josephson energy $E_\\mathrm {J0}$ across the chain is found from the normal state conductance using the Ambegaokar-Baratoff relation, $E_\\mathrm {J0}=\\Delta _0R_Q/2R_N$ , where $R_Q$ is the superconducting quantum of resistance, and $R_N$ is the junction resistance in the normal state The experimentally determined device parameters are listed in Table I.",
"Next we measured the critical voltage $V_\\mathrm {c}$ , deep in the subgap region, $V \\ll 2 N \\Delta _0/e$ .",
"In addition to the zero-field measurements, IVC's for some devices were also measured with varying parallel, or perpendicularly-applied magnetic fields.",
"Most of the measured devices have non-hysteretic IVC's in this region, for all values of applied field, however, a few devices exhibited hysteretic IVC's for some values of magnetic fields.",
"In this work, we are interested in using the voltage gap as a probe of the parity effect.",
"We therefore take $V_c$ as the return voltage, that is, the voltage magnitude at which the device returns to the zero-current state (current less than the noise floor), when stepping from the non-zero current state.",
"The return voltage is found in all cases to be characterized by an extremely narrow distribution (smaller than the experimental resolution).",
"We note that unlike the situation for nanowires and films based on disordered superconducting films[11], the voltage gaps we observe are very hard: we do not find an observable zero-bias resistance arising from a postulated parallel quasiparticle conductance channel, and therefore we have no need to subtract a finite current, $V/R_N$ , in order to observe a robust critical voltage.",
"This indicates that our junction chains are significantly more homogeneous than devices based on disordered films.",
"Figure: Transport data II vs. VV (upper plot), and dI/dVdI/dV vs. VV (lower plot) for device C5 at selected values of parallel magnetic field: below the parity field, B || B_{||} = 0.10.1 T, at the parity field, B || * =0.35B^*_{||}=0.35 (grey), and above the parity field, B || =0.47B_{||}=0.47 T. Although V c V_c is approximately equal at 0.1 T and 0.47 T (inset), the transport characteristics are found to be substantially different.",
"The conductance above V c V_c decreases with VV for fields below B * B^* as expected for Cooper-pair dominated transport, and increases with VV for B || B_{||} above B * B^*, indicative of single-electron transport.",
"Similar behavior is observed for perpendicularly-applied fields (see SM).Figure 1, (a) and (b), show logarithmic scale ($\\log _{10} |I|$ ) image plots of the IVC's for stepped magnetic fields for devices C5 and B6.",
"Device C5 was measured in a parallel magnetic field, while device B6 was measured in a perpendicular field.",
"Both devices were fabricated during the same fabrication run in neighboring squares on the same chip.",
"The critical voltage for each value of magnetic field is identified as the voltage magnitude for which the measured current becomes less that the noise floor for the measurement, which is $\\lesssim 1$ pA. One can readily identify a peak in the critical voltages, as shown in Fig.",
"1, (c) and (d).",
"We estimate the precise field for the peak, $B^*_{||}$ (or $B^*_\\perp $ ), by fitting to few points of the experimentally determined $V_c$ around it's maximum.",
"From Figure 2, one notes that the transport data for both parallel and perpendicular fields are remarkably different above and below their respective parity fields.",
"This can be seen more directly in the individual IVC's, as shown for example for Fig.",
"2, where we plot data for device C5 at magnetic fields both above and below $B^*_{||}$ , fields where $V_c$ is approximately equal.",
"We note that all measured devices show qualitatively different transport characteristics above compared to below their $B^*$ 's (see Supplemental Material[24] for more examples).",
"In Figure 2 (lower plot), it is evident that for $B_{||}$ = $0.1$ T ($< B^*$ ), the conductance $dI/dV$ is strongly peaked just above $\\pm V_c$ , and strictly decreasing with $|V|$ outside the voltage gap.",
"This supercurrent-like feature is indicative of Cooper-pair dominated transport.",
"Conversely, for $B_{||}=0.47$ T ($> B^*$ ), the conductance increases monotonically outside of the voltage gap.",
"We note that both sets of data asymptotically approach each other.",
"This can be understood qualitatively, as below $B^*$ , increasing charge transport invariably populates higher bands in quasicharge via Landau-Zener tunneling.",
"The ensuing relaxation from higher bands produces BCS quasiparticles, which eventually suppress the even-odd free energy difference that permits Cooper-pair tunneling to dominate.",
"Figure: Measured parity field B * B^* versus experimentally determined single-electron charging energy, E C E_C, for devices measured in parallel (squares, left axis), and perpendicular (diamonds, right axis), magnetic fields.",
"The solid black line follows from suppression of the homogeneous superconducting gap, Δ(B)\\Delta (B) = Δ 0 [1-(B/B 0 ) 2 ]\\Delta _0[1-(B/B_0)^2], and setting, Δ(B * )\\Delta (B^*) = E C E_C, to find B * B^*.",
"The dashed line is a fit to B * B^* = fΦ 0 /A I f\\Phi _0/A_I (using E C E_C as a proxy for 1/A I 1/A_I), where A I A_I = wlwl, ww is the island film width, and ll, the between-junction island length.",
"A fit to perpendicular field devices having E C <110μE_C<110\\,\\mu eV, yields f 1 f_1 = 2.00±0.052.00\\pm 0.05.",
"The inset shows B ⊥ * B^*_\\perp vs. A I -1 A_I^{-1} for these devices and the fit directly.In Figure 3, we plot the experimentally determined parity fields for six devices in parallel magnetic fields (squares, left axis), and seven devices in perpendicular magnetic fields (diamonds, right axis), as a function of their experimentally determined single-electron charging energy.",
"First we focus on the parallel field data.",
"The solid line follows from suppression of the superconducting gap according to $\\Delta =\\Delta _0\\left(1-B^2/B_0^2\\right)$ , which we previously found adequate to describe suppression of the superconducting gap in comparably-sized aluminum islands in a parallel magnetic fields[14].",
"Setting $E_C=\\Delta $ , at $B^*$ , one finds $B^*=B_0\\sqrt{1-E_C/\\Delta _0}$ .",
"$B_0$ is known as the pair-breaking parameter, which is expected to be of the same order as the critical field $B_c$ .",
"The solid line represents the best fit to the parallel field data, resulting in $B_0=0.42$ T, which is comparable to the average critical field found for these devices, $B_c=0.48$ T. An alternative method to estimate the depairing parameter, is to fit $V_c(B)$ at fields below $B^*$ to the scaling law behaviour of $V_c(W)$ , as detailed in Ref.[15].",
"Since $V_c$ depends on $E_\\mathrm {J}$ through $W$ , the dependence of $V_c$ on $\\Delta $ and hence $B$ can be found assuming $E_\\mathrm {J}$ to be given by the Ambegaokar-Baratoff formula, resulting a field dependence $E_\\mathrm {J}(B)=E_\\mathrm {J0}\\left(1-B^2/B_0^2\\right)$ (see Supplemental Material[24]).",
"Averaging over the parallel field devices (see Table I), we find $B_0=0.43\\pm 0.02$ T, which agrees extremely well with $B_0=0.42$ T found above.",
"Considering now the devices measured in a perpendicular field, it is clear that $B^*_\\perp (E_C)$ does not follow the dependence, $B_0\\sqrt{1-E_C/\\Delta _0}$ , in particular, for the five devices with $E_C<110\\, \\mu $ eV.",
"We can, however, estimate the depairing parameter, $B_0$ , as discussed in the preceding paragraph.",
"Averaging over the perpendicular field devices, we arrive at $B_0=0.086\\pm 0.003$ T, which agrees with the observed critical perpendicular field, $B_c = 0.086$ T. For perpendicular field devices, the solid line of Figure 3 corresponds to $B_0=0.086$ T. The experimental dependence of $B^*_\\perp $ on $E_C$ for devices with $E_C \\lesssim 110$ $\\mu $ eV is nearly orthogonal to that predicted by the black line.",
"This suggests a different mechanism driving the even-odd parity effect: in a perpendicular field, the parity effect can occur in sync with trapping of a single flux quanta.",
"Concomitant with formation of the vortex state, the average superconducting gap across the island becomes nearly zero, destroying the free energy difference between even and odd parity.",
"Extensive studies, both theoretical and experimental, have been reported detailing the vortex states of mesoscopic superconducting islands, typically within the context of finding eigenvalues of the linearised Ginzburg-Landau equations[26], [27], [28], [29], [30], [31].",
"For a thin superconducting disc of radius, $R$ , it is found, [28], [29], that the single vortex state becomes energetically favourable for an applied flux $\\Phi \\simeq f_1$ $\\Phi _0$ , or $B=f_1$ $\\Phi _0/\\pi R^2$ , with $f_1$ $=$ $1.924$ .",
"For a square, the transition is found to occur at $f_1\\simeq 2.0$[30], Our chains are composed of thin films structured approximately as rectangular islands having a length $l\\simeq 890$ nm, thickness $d$ $=$ 30 nm, and varying width (we varied the junction area across each family to modulate the ratio, $g=E_\\mathrm {J}/E_C$ , while using the same oxidation parameters[15]).",
"We have analysed SEM images of our devices finding device-averaged junction widths, $w$ , ranging from 80 to 120 nm.",
"We find that in accordance with a parallel plate capacitor model of the junctions, the charging energy indeed scales inversely with device-averaged junction area, $A_J=w^2$ .",
"For the family of devices measured in perpendicular field we find, $E_C=0.96/A_J$ , with $A_J$ in $\\mu $ m$^2$[24].",
"This also gives us a relation between charging energy and average island area, since the island widths equal those of the junctions, $A_I=w (0.89-2w)$ $\\mu $ m$^2$ .",
"If we neglect the region of the islands making up the junction, i.e.",
"we take for the island area, $A_I$ , the area between the junctions, remarkably, we find that we can fit the larger area devices ($E_C \\lesssim 110$ $\\mu $ eV), measured in perpendicular fields, to a single curve, $B^*_\\perp =f_1 \\Phi _0/A_I$ , with$f_1=2.00\\pm 0.05$ , as show by the dashed lines in Figure 3 (main plot and inset).",
"In the dirty limit, which applies to our films, the coherence length is given by $\\xi _0=\\sqrt{\\hbar D/\\Delta _0}$ , where $D$ is the diffusion coefficient, which can be deduced from the conductivity, $\\sigma $ , using Einstein's relation, $\\sigma =N(0)e^2D$ , where $N(0)$ = $2.15 \\times 10^{47}$ J$^{-1}$ m$^{-3}$ is the density of states at the Fermi level for aluminum, and $e$ is the electron charge.",
"From the measured resistivity of our 30 nm films, we estimate a coherence length, $\\xi _0\\sim 60$ nm.",
"The critical field for a large film, $B_{c2}=\\Phi _0/2\\pi \\xi _0^2$ , for $\\xi _0\\sim 60$ nm gives an estimate $B_{c2}\\simeq 90$ mT, which roughly agrees with the observed critical fields of our devices in perpendicular fields.",
"For thin films in parallel applied fields, calculations based on Ginzburg-Landau theory find that the vortex state can only occur when the thickness, $d$ , is greater than $1.84\\,\\xi _0$[32], [12].",
"This corresponds well to our parallel field results.",
"Since $d<\\xi _0$ , we see only homogenous suppression of the superconducting gap up to the parity field where $\\Delta (B^*) = E_C$ .",
"For perpendicular fields, analyses of the vortex states of mesoscopic islands based on the linearized Ginzburg-Landau equations show that the single vortex state is stable only for film widths greater than a critical value $w_c \\sim 2\\xi _0$[29], [30].",
"The measured widths of our devices appear to cross this borderline width.",
"For such strong confinement, ($w\\gtrsim \\xi _0$ ), however, an analysis based on the nonlinear Ginzburg-Landau equations, or the Bogoliubov-de Gennes equations, may be required for a more quantitative comparison.",
"In conclusion, we find that the non-monotonic dependence and peak in the voltage gap of insulating Josephson junction chains with magnetic field arises from an even-odd parity effect.",
"Below the parity field, the ground state of our insulating junctions chains is that of a one-dimensional Bose glass of localized Cooper-pairs, with the onset of transport arising from depinning of the compressible Cooper-pair quasicharge[15].",
"Above $B^*$ , odd parity quasicharge bands become accessible, so that the nature of the ground state becomes that of a Fermi glass, and depinning involves single-electron charges rather than Cooper pairs.",
"Moreover, in a perpendicular field the transition can occur simultaneously with trapping of a single flux quantum in the thin-film superconducting islands.",
"Our results are relevant for the insulating peak observed in disordered superconducting films, and strips patterned from such films, which are postulated to form a network of Josephson-coupled superconducting droplets[7], [8], [9], [10], [11].",
"A current explanation for such data is the formation of random SQUID loops in the network[13].",
"Our results suggest such a peak arises from an even-odd parity effect, and may occur with formation of the vortex state in the effective superconducting droplets.",
"We suggest that even-odd parity effects could also be observed in superconducting nanowires.",
"TD acknowledges useful discussions with Alexander Shnirman.",
"This work was supported by the ARC Centre of Excellence for Engineered Quantum Systems, CE11000101.",
"Devices were fabricated at the UNSW Node of the Australian National Fabrication Facility.",
"JHC is supported by the Australian Government's NCI National Facility through the National Computational Merit Allocation Scheme, and the ARC Centre of Excellence in Future Low-Energy Electronics Technologies (FLEET), CE170100039."
]
] | 1808.08552 |
[
[
"A short exposition of S. Parsa's theorems on intrinsic linking and\n non-realizability"
],
[
"Abstract We present a short exposition of the following results by S. Parsa.",
"Let $L$ be a graph such that the join $L*\\{1,2,3\\}$ (i.e.",
"the union of three cones over $L$ along their common bases) piecewise linearly (PL) embeds into $\\mathbb R^4$.",
"Then $L$ admits a PL embedding into $\\mathbb R^3$ such that any two disjoint cycles have zero linking number.",
"There is $C$ such that every 2-dimensional simplicial complex having $n$ vertices and embeddable into $\\mathbb R^4$ contains less than $Cn^{8/3}$ simplices of dimension 2.",
"We also present the analogue of the second result for intrinsic linking."
],
[
"all This paper provides short proofs of Theorems REF , REF and REF below.",
"Let $[k]:=\\lbrace 1,\\ldots ,k\\rbrace $ .",
"Theorem 1 (see Remark REF ) Let $L$ be a graph such that the join $L*[3]$ (i.e.",
"the union of three cones over $L$ along their common bases) piecewise linearly (PL) embeds into ${\\mathbb {R}}^4$ .",
"Then $L$ admits a PL embedding into ${\\mathbb {R}}^3$ such that any two disjoint cycles have zero linking number.",
"Consider $L*[3]$ as a subcomplex of some triangulation of ${\\mathbb {R}}^4$ .",
"Then there is a small general position 4-dimensional PL ball $\\Delta ^4$ containing the point $\\emptyset *1\\in {\\mathbb {R}}^4$ .",
"Hence the intersection $\\partial \\Delta ^4\\cap (L*[3])$ is PL homeomorphic to $L$ .",
"Let us prove that this very embedding of $L$ into the 3-dimensional sphere $\\partial \\Delta ^4$ satisfies the required property.",
"Take any two disjoint oriented closed polygonal lines $\\beta ,\\gamma \\subset \\partial \\Delta ^4\\cap (L*[3])\\cong L$ .",
"Then $(\\beta * \\lbrace 1,2\\rbrace )-\\mathop {Int}\\Delta ^4$ and $(\\gamma *\\lbrace 1,3\\rbrace ) -\\mathop {Int}\\Delta ^4$ are two disjoint 2-dimensional PL disks in ${\\mathbb {R}}^4-\\Delta ^4$ whose boundaries are $\\beta $ and $\\gamma $ .",
"Hence $\\beta $ and $\\gamma $ have zero linking number in the 3-dimensional sphere $\\partial \\Delta ^4$ (by the following well-known lemma applied to the 4-dimensional ball $S^4-\\mathop {Int}\\Delta ^4$ ).",
"Consider $L*[3]$ as a subcomplex of some triangulation of ${\\mathbb {R}}^4$ .",
"Then there is a small general position 4-dimensional PL ball $\\Delta ^4$ containing the point $\\emptyset *1\\in {\\mathbb {R}}^4$ .",
"Hence the intersection $\\partial \\Delta ^4\\cap (L*[3])$ is PL homeomorphic to $L$ .",
"Let us prove that this very embedding of $L$ into the 3-dimensional sphere $\\partial \\Delta ^4$ satisfies the required property.",
"Take any two disjoint oriented closed polygonal lines $\\beta ,\\gamma \\subset \\partial \\Delta ^4\\cap (L*[3])\\cong L$ .",
"Then $(\\beta * \\lbrace 1,2\\rbrace )-\\mathop {Int}\\Delta ^4$ and $(\\gamma *\\lbrace 1,3\\rbrace ) -\\mathop {Int}\\Delta ^4$ are two disjoint 2-dimensional PL disks in ${\\mathbb {R}}^4-\\Delta ^4$ whose boundaries are $\\beta $ and $\\gamma $ .",
"Hence $\\beta $ and $\\gamma $ have zero linking number in the 3-dimensional sphere $\\partial \\Delta ^4$ (by the following well-known lemma applied to the 4-dimensional ball $S^4-\\mathop {Int}\\Delta ^4$ ).",
"Lemma 2 If two disjoint oriented closed polygonal lines in the 3-dimensional sphere $\\partial D^4$ bound two disjoint 2-dimensional PL disks in the 4-dimensional ball $D^4$ , then the polygonal lines have zero integer linking number in $\\partial D^4$ .",
"This proof is well-known.",
"For elementary versions of this well-known lemma see e.g.",
"[14].",
"Denote by $B,\\Gamma \\subset D^4$ the two disjoint oriented disks bounded by the polygonal lines $\\beta ,\\gamma \\subset \\partial D^4$ .",
"Denote by $B^{\\prime }\\subset {\\mathbb {R}}^4-D^4$ an oriented disk (e.g.",
"a cone) bounded by $\\beta $ .",
"Denote by $\\Gamma ^{\\prime }\\subset \\partial D^4$ a general position oriented singular disk (e.g.",
"singular cone) bounded by $\\gamma $ .",
"We have $\\beta \\cap \\Gamma ^{\\prime }=(B\\cup B^{\\prime })\\cap (\\Gamma \\cup \\Gamma ^{\\prime })$ .",
"Denote by the same letters integer chains carried by $\\beta ,\\gamma ,B,\\Gamma $ .",
"Denote by $\\cdot _M$ the algebraic intersection of integer chains in $M$ .",
"Then by general position the linking number of $\\beta $ and $\\gamma $ is $\\beta \\cdot _{\\partial D^4}\\Gamma ^{\\prime }=(B-B^{\\prime })\\cdot _{{\\mathbb {R}}^4}(\\Gamma -\\Gamma ^{\\prime })=0$ .",
"This proof is well-known.",
"For elementary versions of this well-known lemma see e.g.",
"[14].",
"Denote by $B,\\Gamma \\subset D^4$ the two disjoint oriented disks bounded by the polygonal lines $\\beta ,\\gamma \\subset \\partial D^4$ .",
"Denote by $B^{\\prime }\\subset {\\mathbb {R}}^4-D^4$ an oriented disk (e.g.",
"a cone) bounded by $\\beta $ .",
"Denote by $\\Gamma ^{\\prime }\\subset \\partial D^4$ a general position oriented singular disk (e.g.",
"singular cone) bounded by $\\gamma $ .",
"We have $\\beta \\cap \\Gamma ^{\\prime }=(B\\cup B^{\\prime })\\cap (\\Gamma \\cup \\Gamma ^{\\prime })$ .",
"Denote by the same letters integer chains carried by $\\beta ,\\gamma ,B,\\Gamma $ .",
"Denote by $\\cdot _M$ the algebraic intersection of integer chains in $M$ .",
"Then by general position the linking number of $\\beta $ and $\\gamma $ is $\\beta \\cdot _{\\partial D^4}\\Gamma ^{\\prime }=(B-B^{\\prime })\\cdot _{{\\mathbb {R}}^4}(\\Gamma -\\Gamma ^{\\prime })=0$ .",
"Remark 3 (a) Theorem REF trivially generalizes to a $d$ -dimensional finite simplicial complex $L$ and embeddings $L*[3]\\rightarrow {\\mathbb {R}}^{2d+2}$ , $L\\rightarrow {\\mathbb {R}}^{2d+1}$ .",
"For $d\\ne 2$ and a $d$ -complex $L$ embeddability of $L*[3]$ into ${\\mathbb {R}}^{2d+2}$ even implies embeddability of $L$ into ${\\mathbb {R}}^{2d}$ .",
"For the case $d=1$ considered in Theorem REF this improvement follows from a theorem of Grünbaum [4] (whose proof is more complicated).",
"For the case $d\\ge 3$ this improvement is proved in [7], [11], [12].",
"(b) Theorem REF is formally a corollary of [9] but is essentially a restatement of [9] accessible to non-specialists.",
"In spite of being much shorter, the above proof of Theorem REF is not an alternative proof comparatively to [9] but is just a different exposition avoiding sophisticated language.",
"The above proof of Theorem REF is analogous to [13] where a relation between intrinsic linking in 3-space and non-realizability in 4-space was found and used.",
"Although the proof is simple, it easily generalizes to non-trivial results like a simple solution of the Menger 1929 conjecture and its generalizations [13], see survey [14].",
"The following Theorem REF .a is a higher-dimensional generalization of upper estimation on the number of edges in a planar graph.",
"An embedding of a simplicial complex into ${\\mathbb {R}}^{2d+1}$ is called linkless if the images of any two $d$ -dimensional spheres have zero linking number.",
"Theorem 4 (S. Parsa) (a) For every $d$ there is $C$ such that for every $n$ every $d$ -dimensional simplicial complex having $n$ vertices and embeddable into ${\\mathbb {R}}^{2d}$ contains less than $Cn^{d+1-3^{1-d}}$ simplices of dimension $d$ .",
"(b) For every $d$ there is $C$ such that for every $n$ every $d$ -dimensional simplicial complex having $n$ vertices and linklessly embeddable into ${\\mathbb {R}}^{2d+1}$ contains less than $Cn^{d+1-4^{1-d}}$ simplices of dimension $d$ .",
"The result (a) improves analogous result with $Cn^{d+1-3^{-d}}$ [3] and is covered by the Grünbaum-Kalai-Sarkaria conjecture (whose proof is announced in [1]; I did not check that proof).",
"See [9] and [10].",
"The Flores 1934 Theorem states that the $d+1$ join power $[3]^{*(d+1)}=[3]*\\ldots *[3]$ ($d+1$ copies of $[3]$ ) is not (PL or topologically) embeddable into $\\mathbb {R}^{2d}$ [6].",
"(We have $[3]^{*2}=K_{3,3}$ , so the case $d=1$ is even more classical.)",
"The $d+1$ join power $[4]^{*(d+1)}$ is not linklessly embeddable into $\\mathbb {R}^{2d+1}$ [13].",
"(We have $[4]^{*2}=K_{4,4}$ , so the case $d=1$ is due to Sachs.)",
"Theorem REF is implied by these results and the following theorem.",
"Theorem 5 For every $d,r$ there is $C$ such that for every $n$ every $d$ -dimensional simplicial complex having $n$ vertices and not containing a subcomplex homeomorphic to $[r]^{*(d+1)}$ contains less than $Cn^{d+1-r^{1-d}}$ simplices of dimension $d$ .",
"Proof of Theorem REF is based on the following lemma similar to the estimation of the number of edges in a graph not containing $K_{s+1,a}$ (Kovari-Sos-Turań Theorem).",
"Lemma 6 For every integers $r,m,a,s$ and subsets $S_1,\\ldots ,S_m\\subset [a]$ every whose $r$ -tuple intersection contains at most $s$ elements we have $|S_1|+\\ldots +|S_m|\\le r(ma^{1-1/r}s^{1/r}+a).$ The case $r=3$ of Theorem REF and of Lemma REF is essentially proved in [9], see [10].",
"The case of arbitrary $r$ is analogous.",
"Denote by $d_q$ the number of subsets among $S_1,\\ldots ,S_m$ containing element $q\\in [a]$ .",
"We may assume that there is $\\nu \\le a$ such that $d_q\\ge r$ when $q\\le \\nu $ and $d_q<r$ when $q>\\nu $ .",
"Then the required inequality follows by $\\sum _{j=1}^m|S_j| = \\sum _{q=1}^a d_q < ra+\\sum _{q=1}^\\nu d_q\\quad \\text{and}$ $\\left(\\sum _{q=1}^\\nu d_q\\right)^r\\ \\overset{(1)}{\\le }\\ \\nu ^{r-1}\\sum _{q=1}^\\nu d_q^r\\ \\overset{(2)}{\\le }\\ r^r\\nu ^{r-1} \\sum _{q=1}^\\nu {d_q\\atopwithdelims ()r}\\ \\le \\ r^ra^{r-1} \\sum _{q=1}^a{d_q\\atopwithdelims ()r}\\ \\overset{(3)}{=}$ $\\overset{(3)}{=}\\ r^ra^{r-1}\\sum \\limits _{\\lbrace j_1,\\ldots ,j_r\\rbrace }|S_{j_1}\\cap \\ldots \\cap S_{j_r}|\\ \\overset{(4)}{\\le }\\ r^ra^{r-1}s{m\\atopwithdelims ()r}\\ <\\ r^rm^ra^{r-1}s.$ Here $\\bullet $ the inequality (1) is the inequality between the arithmetic mean and the degree $r$ mean; $\\bullet $ since $d_q\\ge r$ when $q\\le \\nu $ , the inequality (2) follows by $r^r{d_q\\atopwithdelims ()r} = \\frac{r^rd_q^r}{r!",
"}\\left(1-\\frac{1}{d_q}\\right)\\left(1-\\frac{2}{d_q}\\right)\\ldots \\left(1-\\frac{r-1}{d_q}\\right) \\ge \\frac{r^rd_q^r}{r!",
"}\\frac{r-1}{r}\\frac{r-2}{r}\\ldots \\frac{1}{r}=d_q^r;$ $\\bullet $ the (in)equalities (3) and (4) are obtained by double counting the number of pairs $(\\lbrace j_1,\\ldots ,j_r\\rbrace ,q)$ of an $r$ -element subset of $[m]$ and $q\\in S_{j_1}\\cap \\ldots \\cap S_{j_r}$ .",
"Denote by $d_q$ the number of subsets among $S_1,\\ldots ,S_m$ containing element $q\\in [a]$ .",
"We may assume that there is $\\nu \\le a$ such that $d_q\\ge r$ when $q\\le \\nu $ and $d_q<r$ when $q>\\nu $ .",
"Then the required inequality follows by $\\sum _{j=1}^m|S_j| = \\sum _{q=1}^a d_q < ra+\\sum _{q=1}^\\nu d_q\\quad \\text{and}$ $\\left(\\sum _{q=1}^\\nu d_q\\right)^r\\ \\overset{(1)}{\\le }\\ \\nu ^{r-1}\\sum _{q=1}^\\nu d_q^r\\ \\overset{(2)}{\\le }\\ r^r\\nu ^{r-1} \\sum _{q=1}^\\nu {d_q\\atopwithdelims ()r}\\ \\le \\ r^ra^{r-1} \\sum _{q=1}^a{d_q\\atopwithdelims ()r}\\ \\overset{(3)}{=}$ $\\overset{(3)}{=}\\ r^ra^{r-1}\\sum \\limits _{\\lbrace j_1,\\ldots ,j_r\\rbrace }|S_{j_1}\\cap \\ldots \\cap S_{j_r}|\\ \\overset{(4)}{\\le }\\ r^ra^{r-1}s{m\\atopwithdelims ()r}\\ <\\ r^rm^ra^{r-1}s.$ Here $\\bullet $ the inequality (1) is the inequality between the arithmetic mean and the degree $r$ mean; $\\bullet $ since $d_q\\ge r$ when $q\\le \\nu $ , the inequality (2) follows by $r^r{d_q\\atopwithdelims ()r} = \\frac{r^rd_q^r}{r!",
"}\\left(1-\\frac{1}{d_q}\\right)\\left(1-\\frac{2}{d_q}\\right)\\ldots \\left(1-\\frac{r-1}{d_q}\\right) \\ge \\frac{r^rd_q^r}{r!",
"}\\frac{r-1}{r}\\frac{r-2}{r}\\ldots \\frac{1}{r}=d_q^r;$ $\\bullet $ the (in)equalities (3) and (4) are obtained by double counting the number of pairs $(\\lbrace j_1,\\ldots ,j_r\\rbrace ,q)$ of an $r$ -element subset of $[m]$ and $q\\in S_{j_1}\\cap \\ldots \\cap S_{j_r}$ .",
"Induction on $d$ .",
"The base $d=1$ follows because if a graph on $n$ vertices does not contain a subgraph homeomorphic to $K_{r,r}$ , then the graph does not contain a subgraph homeomorphic to $K_{2r}$ and hence has $O(n)$ vertices [2] (this was apparently proved in the paper [5] which is not easily available to me).",
"Let us prove the inductive step.",
"If a $d$ -dimensional simplicial complex $K$ having $n$ vertices does not contain a subcomplex homeomorphic to $[r]^{*(d+1)}$ , then any $r$ -tuple intersection of the links of vertices from $K$ does not contain a subcomplex homeomorphic to $[r]^{*d}$ .",
"Apply Lemma REF to the set of $a\\le {n\\atopwithdelims ()d}<n^d$ simplices of $K$ having dimension $d-1$ , and to $m=n$ subsets defined by links of the vertices.",
"By the inductive hypothesis $s\\le Cn^{d-r^{2-d}}$ .",
"Hence the number of $d$ -simplices of $K$ does not exceed $rnn^{(r-1)d/r}\\left(Cn^{d-r^{2-d}}\\right)^{1/r}=C^{\\prime }n^{d+1-r^{1-d}}$ .",
"Induction on $d$ .",
"The base $d=1$ follows because if a graph on $n$ vertices does not contain a subgraph homeomorphic to $K_{r,r}$ , then the graph does not contain a subgraph homeomorphic to $K_{2r}$ and hence has $O(n)$ vertices [2] (this was apparently proved in the paper [5] which is not easily available to me).",
"Let us prove the inductive step.",
"If a $d$ -dimensional simplicial complex $K$ having $n$ vertices does not contain a subcomplex homeomorphic to $[r]^{*(d+1)}$ , then any $r$ -tuple intersection of the links of vertices from $K$ does not contain a subcomplex homeomorphic to $[r]^{*d}$ .",
"Apply Lemma REF to the set of $a\\le {n\\atopwithdelims ()d}<n^d$ simplices of $K$ having dimension $d-1$ , and to $m=n$ subsets defined by links of the vertices.",
"By the inductive hypothesis $s\\le Cn^{d-r^{2-d}}$ .",
"Hence the number of $d$ -simplices of $K$ does not exceed $rnn^{(r-1)d/r}\\left(Cn^{d-r^{2-d}}\\right)^{1/r}=C^{\\prime }n^{d+1-r^{1-d}}$ .",
"The following version of Lemma REF is also possibly known.",
"It was (re)invented by I. Mitrofanov and the author in discussions of the $r$ -fold Khintchine recurrence theorem, see [8].",
"Lemma 7 For every integers $r,m,a$ and subsets $S_1,\\ldots ,S_m\\subset [a]$ we have $a^{r-1}\\sum _{j_1,\\ldots ,j_r=1}^m |S_{j_1}\\cap \\ldots \\cap S_{j_r}| \\ge \\left(\\sum _{j=1}^m|S_j|\\right)^r.$ Consider the decomposition of $[a]$ by the sets $S_j$ and their complements.",
"The sets of this decomposition correspond to subsets of $[m]$ .",
"Denote by $\\mu _A$ the number of elements in the set of this decomposition corresponding to a subset $A\\subset [m]$ .",
"To every pair $(A,j)$ of a subset $A\\subset [m]$ and a number $j\\in [m]$ assign 0 if $j\\notin A$ and assign $\\mu _A$ if $j\\in A$ .",
"Let us double count the sum $\\Sigma $ of the obtained $2^m\\cdot m$ numbers.",
"We obtain $\\sum \\limits _{j=1}^m|S_j|=\\Sigma =\\sum \\limits _{A\\subset [m]}|A|\\mu _A.$ To every pair $(A,(j_1,\\ldots ,j_r))$ of a subset $A\\subset [m]$ and a vector $(j_1,\\ldots ,j_r)\\in [m]^r$ assign 0 if $\\lbrace j_1,\\ldots ,j_r\\rbrace \\lnot \\subset A$ and assign $\\mu _A$ if $\\lbrace j_1,\\ldots ,j_r\\rbrace \\subset A$ .",
"Let us double count the sum $\\Sigma _r$ of the obtained $2^m\\cdot m^r$ numbers.",
"We obtain $\\sum \\limits _{j_1,\\ldots ,j_r=1}^m|S_{j_1}\\cap \\ldots \\cap S_{j_r}|=\\Sigma _r=\\sum \\limits _{A\\subset [m]}|A|^r\\mu _A.$ Hence by the inequality between the weighted arithmetic mean and the weighted degree $r$ mean, and using $\\sum _{A\\subset [m]}\\mu _A=a$ , we obtain $a^{r-1}\\Sigma _r \\ge \\left(\\sum \\limits _{A\\subset [m]}|A|\\mu _A\\right)^r = \\left(\\sum \\limits _{j=1}^n|S_j|\\right)^r.$ Consider the decomposition of $[a]$ by the sets $S_j$ and their complements.",
"The sets of this decomposition correspond to subsets of $[m]$ .",
"Denote by $\\mu _A$ the number of elements in the set of this decomposition corresponding to a subset $A\\subset [m]$ .",
"To every pair $(A,j)$ of a subset $A\\subset [m]$ and a number $j\\in [m]$ assign 0 if $j\\notin A$ and assign $\\mu _A$ if $j\\in A$ .",
"Let us double count the sum $\\Sigma $ of the obtained $2^m\\cdot m$ numbers.",
"We obtain $\\sum \\limits _{j=1}^m|S_j|=\\Sigma =\\sum \\limits _{A\\subset [m]}|A|\\mu _A.$ To every pair $(A,(j_1,\\ldots ,j_r))$ of a subset $A\\subset [m]$ and a vector $(j_1,\\ldots ,j_r)\\in [m]^r$ assign 0 if $\\lbrace j_1,\\ldots ,j_r\\rbrace \\lnot \\subset A$ and assign $\\mu _A$ if $\\lbrace j_1,\\ldots ,j_r\\rbrace \\subset A$ .",
"Let us double count the sum $\\Sigma _r$ of the obtained $2^m\\cdot m^r$ numbers.",
"We obtain $\\sum \\limits _{j_1,\\ldots ,j_r=1}^m|S_{j_1}\\cap \\ldots \\cap S_{j_r}|=\\Sigma _r=\\sum \\limits _{A\\subset [m]}|A|^r\\mu _A.$ Hence by the inequality between the weighted arithmetic mean and the weighted degree $r$ mean, and using $\\sum _{A\\subset [m]}\\mu _A=a$ , we obtain $a^{r-1}\\Sigma _r \\ge \\left(\\sum \\limits _{A\\subset [m]}|A|\\mu _A\\right)^r = \\left(\\sum \\limits _{j=1}^n|S_j|\\right)^r.$ Books, surveys and expository papers in this list are marked by the stars."
]
] | 1808.08363 |
[
[
"A Novel Deep Neural Network Architecture for Mars Visual Navigation"
],
[
"Abstract In this paper, emerging deep learning techniques are leveraged to deal with Mars visual navigation problem.",
"Specifically, to achieve precise landing and autonomous navigation, a novel deep neural network architecture with double branches and non-recurrent structure is designed, which can represent both global and local deep features of Martian environment images effectively.",
"By employing this architecture, Mars rover can determine the optimal navigation policy to the target point directly from original Martian environment images.",
"Moreover, compared with the existing state-of-the-art algorithm, the training time is reduced by 45.8%.",
"Finally, experiment results demonstrate that the proposed deep neural network architecture achieves better performance and faster convergence than the existing ones and generalizes well to unknown environment."
],
[
"Introduction",
"Generally, there are three basic phases during Mars exploration missions—entry, descent and landing (EDL) [1], where the landing phase finally determines whether Mars rovers land on Martian surface safely and precisely.",
"Due to the large uncertainties and dispersions derived from Martian environment, existing algorithms used for EDL phase cannot guarantee the precision of the Mars rovers' landing on the target point.",
"Moreover, after landing, Mars rovers are usually required to move to new target points constantly in order to carry out new exploration tasks.",
"Hence, in future Mars missions, autonomous navigation algorithms are essential for Mars rovers to avoid risky areas (such as craters, high mountains and rocks) and reach target points precisely and efficiently (Fig.",
"REF ).",
"Figure: Visual navigation phase after landing.Currently, one of the most significant methods for Mars navigation is visual navigation [2].",
"Two main methods for Mars visual navigation are blind drive and autonomous navigation with hazard avoidance (AutoNav) [3].",
"In blind drive, all commands for Mars rovers are determined by engineers from the earth before starting missions.",
"This method promptly reduces the efficiency and flexibility of exploration missions.",
"By contrast, AutoNav can lender Mars rover execute missions unmannedly.",
"Thus, it is more in coincidence with the increasing future demand on Mars rovers' autonomy and intelligence.",
"Classical algorithms for AutoNav such as Dijkstra [4], [5], $A^*$ [6], [7] and $D^*$ [8], [9] have been widely researched in the past decades.",
"It is noteworthy that these algorithms have to search the optimal path iteratively on cellular grip maps, which are both time consuming and memory consuming [10].",
"When dimensions of maps become large and computation resources are limited, these algorithms may fail to offer the optimal navigation policy.",
"To overcome the dimension explosion problem, intelligent algorithms such as neural network [11], genetic algorithm [12] and particle swarm algorithm [13] were extended into planetary navigation problem.",
"However, prior knowledge about the obstacles in maps is prerequisite for these algorithms to work.",
"To provide the optimal navigation policy directly from natural Martian scenes, effective feature representation algorithms are required.",
"That is, these algorithms have to understand deep features of input image such as the shape and location of obstacles firstly and then determine the navigation policy according to these deep features.",
"In recent years, Deep Convolutional Neural Networks (DCNNs) have received wide attention in computer vision field for their superior feature representation capability [14].",
"Notably, although the training process of DCNNs consumes massive time and computation resource, it is completed offline.",
"When applying DCNNs to represent deep features of images online after training, it costs little time and computation resource.",
"Therefore, DCNNs have been widely applied in varieties of visual tasks such as image classification [15], object detection [16], visual navigation [17] and robotic manipulation [18].",
"Inspired by the state-of-art performance of DCNNs in computer vision field, planetary visual navigation algorithms based on deep neural network have been researched.",
"In [19], a 3 dimensional DCNN was designed to create a safety map for autonomous landing zone detection from terrain image.",
"In [20], a DCNN was trained to predict rover's position from terrain images for Lunar navigation.",
"Though these algorithms are capable of extracting deep features of raw images, they are unable to provide the optimal policy for navigation directly.",
"To solve this probelm, Value Iteration Network (VIN) was firstly proposed in [21] to plan path directly from images and applied to Mars visual navigation problem successfully.",
"Then, in [22], Memory Augmented Control Network was proposed to find the optimal path for rovers in partially observable environment.",
"Both of these networks for visual navigation employed Value Iteration Module.",
"However, it takes massive time to train them.",
"In this paper, an efficient algorithm to determine the optimal navigation policy directly from original Martian images is investigated.",
"Specifically, a novel DCNN architecture with double branches is designed.",
"It can represent both global and local deep features of input images and then achieve precise navigation efficiently.",
"The main contributions of this paper are summarized as follows: Emerging deep learning techniques (deep neural networks) are leveraged to deal with Mars visual navigation problem.",
"The proposed DCNN architecture with double branches and non-recurrent structure can find the optimal path to target point directly from global Martian environment images and prior knowledge about risky areas in images are not required.",
"Compared with acknowledged (VIN), the proposed DCNN architecture achieves better performance on Mars visual navigation and the training time is reduced by 45.8%.",
"The accuracy and efficiency of this novel architecture are demonstrated through experiment results and analysis.",
"The rest paper is organized as follows.",
"Section II provides preliminaries of this paper.",
"Section III describes the novel DCNN architecture for Mars visual navigation.",
"Experimental results and analysis are illustrated in Section IV, followed by discussion and conclusions in Section V. Mars visual navigation can be formulated as a Markov Decision Process (MDP), since the next state of Mars rover can be determined by its current state and action completely.",
"A standard MDP for sequential decision making is composed of action space $\\mathcal {A}$ , state space $\\mathcal {S}$ , reward $r:\\mathcal {S}\\times \\mathcal {A}\\rightarrow \\mathcal {R}$ , transition probability distribution $P:\\mathcal {S}\\times \\mathcal {A} \\times \\mathcal {S}\\rightarrow \\mathcal {R}$ and policy $\\pi _{\\theta }: \\mathcal {S}\\times \\mathcal {A}\\rightarrow \\mathcal {R}$ .",
"At time step $t$ , the agent can obtain its state $s_t \\in \\mathcal {S}$ from environment and then choose its action $a_t$ satisfying distribution $a_t \\sim \\pi _{\\theta }(a|s_t), a \\in \\mathcal {A}$ .",
"After that, its state will transit into $s_{t+1}\\in \\mathcal {S}$ and the agent will then receive reward $r_t=r(s_t,a_t) \\in \\mathcal {R}$ from environment, where $s_{t+1} \\in \\mathcal {S}$ satisfies the transition probability distribution $s_{t+1}\\sim P(s|s_t,a_t), s \\in \\mathcal {S}$ .",
"The whole process is shown in Fig.",
"REF .",
"Denote the discount factor of reward by $\\gamma \\in \\mathcal {R}$ .",
"A policy is defined as optimal if and only if its parameter $\\theta $ satisfies $\\theta ^{*} = \\arg \\max _{\\theta } E_{s_0,a_0,s_1,a_1,...}[r_0+\\gamma r_1+\\gamma ^{2} r_2+...],$ Figure: Markov decision process.To measure the expected accumulative reward of $s_t$ and $(s_t,a_t)$ , the state value function and the action value function are defined respectively as $V^{\\pi _{\\theta }}(s_t)=E_{a_t,s_{t+1},a_{t+1}...}[\\sum _{i=t}^{+\\infty }\\gamma ^{i-t}r(s_{i},a_{i})],$ $\\begin{aligned}Q^{\\pi _{\\theta }}(s_t,a_t)&=r_{t}+E_{s_{t+1},a_{t+1}...}[\\sum _{i=t+1}^{+\\infty }\\gamma ^{i-t}r(s_{i},a_{i})]\\\\&=r_{t} + E_{s_{t+1}}[V^{\\pi _{\\theta }}(s_{t+1})].\\end{aligned}$ By substituting Eq.",
"(REF ) into Eq.",
"(REF ), the following equation is derived as $\\theta ^{*} = \\arg \\max _{\\theta } E_{s_0}[V^{\\pi _{\\theta }}(s_0)].$ By solving Eq.",
"(REF ), the optimal policy is determined such that the objective of MDP is achieved.",
"However, since both state value function and action value function are unknown before, Eq.",
"(REF ) cannot be solved directly.",
"Therefore, state value function and action value function have to be estimated in order to solving MDP problem."
],
[
"Value Function Estimation",
"Value iteration is an typical method for value function estimation and then addressing MDP problem [25].",
"Denote the estimated state value function at step $k$ by $V_{k}(s)$ , and the estimated action value function for each state at step $k$ by $Q_{k}(s,a)$ .",
"$\\pi _{k}$ is utilized to represent the policy at step $k$ .",
"Then, the value iteration process can be expressed as $\\pi _{k}(s_i) = \\arg \\max _{a_i}Q_{k}(s_i,a_i) \\quad (i=0,1,\\cdots ),$ $\\begin{aligned}V_{k+1}(s_i) &= Q_{k+1}(s_i,\\pi _{k}(s_i)) \\\\&= r_i + E_{s_{i+1}}[V_{k}(s_{i+1})] \\quad (i=0,1,\\cdots ).\\end{aligned}$ Through iteration, the policy and value functions will converge to optimum $\\pi ^{*}$ , $Q^{*}$ and $V^{*}$ simultaneously.",
"However, since it is difficult to determine the explicit representation of $\\pi _{k}$ , $Q_{k}$ and $V_{k}$ (especially when the dimension of $s_t$ is high), VIN is applied to approximate this process successfully.",
"Specifically, VIN is designed with Value Iteration Module, which consists of recurrent convolutional layers [21].",
"As illustrated in Fig.",
"REF , the value function layer $V_k$ is stacked with the reward layer $R_k$ and then filtered by a convolutional layer and a max-pooling layer recurrently.",
"Furthermore, through VIN, navigation information including global environment and target point can be conveyed to each state in the final value function layer.",
"Experiments demonstrate that this architecture performs well in navigation tasks.",
"However, it takes lots of time and computation resource to train such a recurrent convolutional neural network when the value of $K$ becomes large.",
"Therefore, replacing Value Iteration Module with a more efficient and non-recurrent architecture without losing its excellent navigation performance becomes the focus of this paper.",
"Figure: Value iteration module."
],
[
"Learning-Based Algorithms",
"Typically, there exist two learning-based algorithms for training DCNNs in value function estimation—Reinforcement learning [25] and Imitation learning [26].",
"In Reinforcement learning, no prior knowledge is required and the agent can find the optimal policy in complex environment by trial and error [27].",
"However, the training process of Reinforcement learning is computationally inefficient.",
"In Imitation learning, when the expert dataset is given $\\lbrace (s_i, y_i)\\rbrace _{i=1}^{i=N}$ , the training process transforms into supervised learning with higher data-efficiency and fitting accuracy.",
"Considering that expert dataset $\\lbrace (s_i, y_i)\\rbrace _{i=1}^{i=N}$ for global visual navigation is available ($y_i \\in \\lbrace 0,1,\\cdots ,N\\rbrace $ is the optimal action at state $s_t$ and $N$ is the number of samples), in this paper, Imitation learning method is applied to find the optimal navigation policy."
],
[
"Mars Visual Navigation Model",
"In this subsection, the process of formulating Mars visual navigation into MDP is presented.",
"More precisely, state $s_t =\\lbrace I_t,(g_{1t},g_{2t}),(x_{1t},x_{2t})\\rbrace $ is composed of the Martian environment image $I_t\\in \\mathcal {R}^{M\\times M\\times 3}$ , target point $(g_{1t},g_{2t}) \\in \\mathcal {Z}\\times \\mathcal {Z}$ and the current position of Mars rover $(x_{1t},x_{2t})\\in \\lbrace 0,1,\\cdots ,M-1\\rbrace \\times \\lbrace 0,1,\\cdots ,M-1\\rbrace $ at time step $t$ .",
"The action $a_t \\in \\lbrace 0,1,\\cdots ,7\\rbrace $ represents the moving direction of the Mars rover at time step $t$ (0:east, 1:south, 2:west, 3:north, 4:southeast, 5:northeast, 6:southwest, 7:northwest).",
"After taking action $a_t$ , the current location of the Mars rover will change and the state $s_t$ will transit into $s_{t+1}$ .",
"If the Mars rover reaches the target point precisely at time $t+1$ , a positive reward will be obtained (such as $r_t=1.0$ ).",
"Otherwise, the Mars rover will get a negative reward (such as $r_t=-1.0$ ).",
"Furthermore, the output vector of the proposed DCNN is defined as $\\Pi _{\\theta }(s_t)=[Q^{\\pi _{\\theta }}(s_t,a=0), \\cdots , Q^{\\pi _{\\theta }}(s_t,a=7)]^{T}$ ($s_t \\in \\mathcal {S}$ ).",
"Then the training loss is defined in cross entropy form with $L_2$ norm as [28] $\\begin{aligned}L(\\theta ) = -\\frac{1}{N}\\sum _{i=1}^{N}Y_i log( \\Pi _{\\theta }(s_i)) + \\lambda ||\\theta ||_2,\\end{aligned}$ where $N$ is the number of training samples, $Y_i$ is the one-hot vector [29] of $y_i$ and $\\lambda $ is the hyperparameter adjusting the effect of $L_2$ norm on the loss function.",
"By minimizing the loss function $L(\\theta )$ , the optimal parameter of navigation policy is determined as follows $\\begin{aligned}\\theta ^* = \\arg \\min _{\\theta }L(\\theta ).\\end{aligned}$"
],
[
"The Novel Deep Neural Network Architecture",
"In this subsection, the novel deep neural network architecture—DB-Net with double branches for deep feature representations and value function estimation is illuminated.",
"The principle design idea of DB-Net is to replace Value Iteration Module of VIN with a non-recurrent convolutional network structure.",
"Firstly, the reprocessing layers of DB-Net compresses the input Martian environment image $I_t$ into feature map $I_{t}^{^{\\prime }}(I_t,g_{1t},g_{2t}) \\in \\mathcal {R}^{N\\times N\\times A}$ ($M = lN, l\\in \\mathcal {Z}, A \\in \\mathcal {Z}$ and $N\\in \\mathcal {Z}$ ).",
"Then, the global deep feature $f_1(I_t^{^{\\prime }}) \\in \\mathcal {R}^{B}$ ($B\\in \\mathcal {Z}$ ) and the local deep feature $f_2(I_{t}^{^{\\prime }},x_{1t},x_{2t}) \\in \\mathcal {R}^{C}$ ($C\\in \\mathcal {Z}$ ) are extracted from feature map $I_{t}^{^{\\prime }}$ by branch one and branch two respectively.",
"By fusing $f_1(I_{t}^{^{\\prime }})$ and $f_2(I_{t}^{^{\\prime }},x_{1t},x_{2t})$ , the final deep feature (value function estimation) $\\Pi _{\\theta }(s_t)=\\Pi _{\\theta }(I_t,g_{1t},g_{2t},x_{1t},x_{2t}) \\in \\mathcal {R}^{8}$ of Martian environment image is derived.",
"Then, the optimal navigation policy can be determined through Eq.",
"(REF ).",
"The diagram of DB-Net is illustrated in Fig.",
"REF , where Conv, Pool, Res, Fc and S are short for convolutional layer, max-pooling layer, residual convolution layer, fully-connected layer and softmax layer respectively.",
"More specific explanations of DB-Net are given as follows.",
"Figure: The diagram of DB-Net.",
"(1) The reprocessing layers comprises of two convolutional layers (Conv-00, Conv-01) and two max-pooling layers (Pool-00, Pool-01).",
"After compressing the original image $I_t$ , the navigation policy becomes area by area instead of point by point (each area has size $l\\times l$ ).",
"Thus, the efficiency of visual navigation is promptly enhanced.",
"(2) Branch one consists of one convolutional layer (Conv-10), three residual convolutional layers (Res-11, Res-12, Res-13), four max-pooling layers (Pool-10, Pool-11, Pool-12, Pool-13) and two fully connected layers (Fc-1, Fc-2).",
"Notably, residual convolutional layer (Fig.",
"REF ) is one kind of convolutional layer proposed in [30], which not only increases the training accuracy of convolutional neural networks with deep feature representations, but also makes them generalize well to testing data.",
"Considering that DB-Net is required to represent deep features of Martian image and achieves high-precision in unknown Martian environment images, residual convolutional layers are employed on DB-Net.",
"The deep feature $f_1(I_t^{^{\\prime }})$ represented by this branch is a global guidance to the Mars rover, containing abstract information about global Martian environment $I_t$ and target point $(g_{1t},g_{2t})$ .",
"(3) Branch two is composed of two convolutional layers (Conv-20, Conv-21) and four residual convolutional layers (Res-21, Res-22, Res-23, Res-24).",
"The deep feature $f_2(I_t^{^{\\prime }},x_{1t},x_{2t})$ represented by this branch depicts the local value distribution of Martian environment image $I_t$ with target $(g_{1t},g_{2t})$ , which acts as a local guidance to Mars rover.",
"Figure: Residual convolutional layer(4) The final deep feature $\\Pi _{\\theta }(I_t,g_{1t},g_{2t},x_{1t},x_{2t})=\\Pi _{\\theta }(s_t)$ is fully connected with $f_1$ and $f_2$ through Fc-3, corresponding to the $Q^{\\pi _{\\theta }}$ value of one action $a_t$ at current state $s_t$ .",
"Hence, following Eq.",
"(REF ), the optimal visual navigation policy is determined.",
"Compared with VIN, not only the depth of DB-Net is reduced significantly (since it is non-recurrent), but also both global and local information of the image is kept and represented effectively.",
"Detailed parameters of DB-Net are demonstrated in TABLE REF .",
"Table: Detailed parameters of DB-Net"
],
[
"Experiments and Analysis",
"In this section, DB-Net and VIN are firstly trained and tested on Martian image dataset derived from HiRISE [31].",
"The dataset consists of 10000 high-resolution Martian images, each of which has 7 optimal trajectory samples (generated randomly).",
"The training set and the testing set consist of 6/7 and 1/7 dataset respectively.",
"Then, navigation accuracy and training efficiency of DB-Net and VIN are compared.",
"Finally, detailed analysis of DB-Net is made through model ablation experiments.",
"More precisely, the following questions will be investigated: Could DB-Net provide the optimal navigation policy directly from original Martian environment images?",
"Could DB-Net outperform the best framework—VIN in accuracy and efficiency?",
"Could DB-Net keep its performance after ablating some of its components?"
],
[
"Experiment Results on Martian Images",
"In this subsection, the process of training and testing DB-Net and VIN on Martian image dataset is described.",
"The input image has a size of $128\\times 128$ with 3 channels (i.g.",
"$M=128$ ), consisting of the gray image of original Martian environment, the edge image of original Martian environment generated by Canny algorithm [32] and the target image (Fig.",
"REF ).",
"Then, training accuracy and testing accuracy of DB-Net and VIN are counted to contrast the proportion of the optimal action they take each step.",
"To compare the navigation performance of DB-Net and VIN, success rate on both training images and testing images are counted.",
"It is worth noting that a navigation process is considered successful if and only if the Mars rover reaches target point from start point without running into any risky areas.",
"Figure: The input image of DB-Net.",
"(Channel1 is the gray image.",
"Channel2 is the edge image.",
"Channel3 is the target image.",
")As illustrated in Fig.",
"REF , both training loss and training error of DB-Net converge faster that VIN.",
"After 200 training epoches, DB-Net achieves 96.4% training accuracy and 95.4% testing accuracy, outperforming VIN significantly in precision (as shown in TABLE REF ).",
"Moreover, compared with VIN, average time cost of DB-Net in one training epoch is reduced by 45.8%, exceeding VIN in efficiency promptly.",
"Finally, DB-Net achieves high success rate both in training data and testing data.",
"Remarkably, Martian environment images in testing data are totally unknown to DB-Net, since training data differs from testing data.",
"Therefore, even if the environment is unknown before, DB-Net can still achieve high-precision visual navigation.",
"By contrast, VIN exhibits poor performance on success rate, which is less than 80% in testing data.",
"Examples of successful navigation process are demonstrated in Fig.",
"REF .",
"It can be seen that the rover avoid craters with varying size precisely under the guidance of DB-Net.",
"Furthermore, the trajectories are nearly optimal.",
"It is worth noting that prior knowledge of craters are unknown and DB-Net has to understand deep representations of original Martian images intuitively.",
"Therefore, the performance of DB-Net is marvellous.",
"Figure: Training results of DB-Net and VINTable: Results on 128x128 Martian imageFigure: Experiments on 128×128128\\times 128 Martian Images.",
"(Green points are landing points.",
"Blue points are the target points.",
"Navigation trajectories are red.)"
],
[
"Model Ablation Analysis of ",
"In this subsection, to test whether DB-Net could keep its performance after ablating some of its components, model ablation experiments are conducted.",
"Define DB-Net without branch one as B1-Net.",
"Then, derive B2-net by replacing residual convolutional layers of B1-Net with normal convolutional layers.",
"As illustrated in Fig.",
"REF and TABLE REF , without global deep features, the navigation accuracy and success rate of B1-Net drop promptly compared with DB-Net.",
"Moreover, with only normal convolutional layers, training cost and error of B2-Net remain at high levels, unable to provide reliable navigation policy for the Mars rover.",
"Therefore, both of the two-branch architecture and the residual convolutional layers make indispensable contributions to the final performance of DB-Net.",
"Figure: Training results of DB-Net, B1-Net and B2-NetTable: Results of model ablation experimentsMoreover, to explore the inner mechanism of DB-Net, the final value function layers ($\\Pi _{\\theta }$ ) of DB-Net, B1-Net and VIN are contrasted in a visualized way.",
"The value function layers estimates the action value distribution of current Martian images and target point.",
"After being visualized, locations $(x_{1t},x_{2t})$ close to target point should be lighter (larger value) while location far from target point or near risky areas should be darker (smaller value).",
"As demonstrated in Fig.",
"REF , the value functions estimated by by DB-Net are more in coincidence with the original Martian images compared with B2-Net.",
"It is clear that risky areas are darker and the lighter locations are around target points in value function layers generated by DB-Net from Fig.",
"REF .",
"By contrast, B1-Net without global deep features cannot estimate the value function as precisely as DB-Net.",
"VIN also fails to recognize risky areas of Martian images evidently.",
"Therefore, DB-Net indeed has a remarkable capability of representing deep features and estimating the value distribution of current Martian environment.",
"Figure: Visualization of value function layers."
],
[
"Conclusions",
"In this paper, a novel deep neural network architecture—DB-Net with double branches and non-recurrent structure is designed for dealing with Martian visual navigation problem.",
"DB-Net is able to determine the optimal navigation policy to target point directly from original Martian environment images without any prior knowledge.",
"Moreover, compared with the existing best architecture—VIN, DB-Net achieves higher precision and efficiency.",
"Most significantly, the average training time of DB-Net is reduced by 45.8%.",
"In future research, more effective deep neural network architecture will be explored and the robustness of the architecture will be researched further."
],
[
"Acknowledgement",
"This work was supported by the National Key Research and Development Program of China under Grant 2018YFB1003700, the Beijing Natural Science Foundation under Grant 4161001, the National Natural Science Foundation Projects of International Cooperation and Exchanges under Grant 61720106010, and by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China under Grant 61621063."
]
] | 1808.08395 |
[
[
"An Incremental Construction of Deep Neuro Fuzzy System for Continual\n Learning of Non-stationary Data Streams"
],
[
"Abstract Existing FNNs are mostly developed under a shallow network configuration having lower generalization power than those of deep structures.",
"This paper proposes a novel self-organizing deep FNN, namely DEVFNN.",
"Fuzzy rules can be automatically extracted from data streams or removed if they play limited role during their lifespan.",
"The structure of the network can be deepened on demand by stacking additional layers using a drift detection method which not only detects the covariate drift, variations of input space, but also accurately identifies the real drift, dynamic changes of both feature space and target space.",
"DEVFNN is developed under the stacked generalization principle via the feature augmentation concept where a recently developed algorithm, namely gClass, drives the hidden layer.",
"It is equipped by an automatic feature selection method which controls activation and deactivation of input attributes to induce varying subsets of input features.",
"A deep network simplification procedure is put forward using the concept of hidden layer merging to prevent uncontrollable growth of dimensionality of input space due to the nature of feature augmentation approach in building a deep network structure.",
"DEVFNN works in the sample-wise fashion and is compatible for data stream applications.",
"The efficacy of DEVFNN has been thoroughly evaluated using seven datasets with non-stationary properties under the prequential test-then-train protocol.",
"It has been compared with four popular continual learning algorithms and its shallow counterpart where DEVFNN demonstrates improvement of classification accuracy.",
"Moreover, it is also shown that the concept drift detection method is an effective tool to control the depth of network structure while the hidden layer merging scenario is capable of simplifying the network complexity of a deep network with negligible compromise of generalization performance."
],
[
"Introduction",
"Deep neural network (DNN) has gained tremendous success in many real-world problems because its deep network structure enables to learn complex feature representations [1].",
"Its structure is constructed by stacking multiple hidden layers or classifiers to produce a high level abstraction of input features which brings improvement of model's generalization.",
"There exists a certain point where introduction of extra hidden nodes or models in a wide configuration has negligible effect toward enhancement of generalization power.",
"The power of depth has been theoretically proven [2] with examples that there are simple functions in $d$ -dimensional feature space that can be modelled with ease by a simple three-layer feedforward neural network but cannot be approximated by a two-layer feedforward neural network up to a certain accuracy level unless the number of hidden nodes is exponential in the dimension.",
"Despite of these aforementioned working advantages, the success of DNN relies on its static network structure blindly selected by trial-error approaches or search methods [3].",
"Although a very deep network architecture is capable of delivering satisfactory predictive accuracy, this approach incurs excessive computational cost.",
"An over-complex DNN is also prone to the so-called vanishing gradients [4] and diminishing feature reuse [5].",
"In addition, it calls for a high number of training samples to ensure convergence of all network parameters.",
"To address the issue of a fixed and static model, model selection of DNN has attracted growing research interest.",
"It aims to develop DNN with elastic structure featuring stochastic depth which can be adjusted to suit the complexity of given problems [6].",
"In addition, a flexible structure paradigm also eases gradient computation and addresses the well-known issues: diminishing feature reuse and vanishing gradient.",
"This approach starts from an over-complex network structure followed by complexity reduction scenario via dropout, bypass, highway [5], hedging [7], merging [8], regularizer [9], etc.",
"Another approach utilizes the idea of knowledge transfer or distillation [10].",
"That is, the training process is carried out using a very deep network structure while deploying a shallow network structure during the testing phase.",
"Nonetheless, this approach is not scalable for data stream applications because most of which are built upon iterative training process.",
"Notwithstanding that [7] characterizes an online working principle, it starts its training process with a very deep network architecture and utilizes the hedging concept which opens direct link between hidden layer and output layer.",
"The final output is determined from aggregation of each layer output.",
"This strategy has strong relationship to the weighted voting concept.",
"The concept of DNN is introduced into fuzzy system in [11], [12], [13] making use of the stacked generalization principle [14].",
"Two different architectures, namely random shift [11] and feature augmentation [12], [13], are adopted to create deep neuro fuzzy structure.",
"It is also claimed that fuzzy rule interpretability is not compromised under these two structures because intermediate features still have the same physical meaning as original input attributes.",
"Similar work is done in [15] but the difference exists in the parameter learning aspect adopting the online learning procedure instead of the batch learning module.",
"These works, however, rely on a fixed and static network configuration which calls for prior domain knowledge to determine its network architecture.",
"In [16], a deep fuzzy rule-based system is proposed for image classification.",
"It is built upon a four-layered network structure where the first three layer consists of normalization layer, scaling layer and feature descriptor layer while the final layer is a fuzzy rule layer.",
"Unlike [11], [12], [13], [7], this algorithm is capable of self-organizing its fuzzy rules in the fuzzy rule layer but the network structure still has a fixed depth (4 layer).",
"Although the area of deep learning has grown at a high pace, the issue of data stream processing or continual learning remains an open issue in the existing deep learning literature.",
"A novel incremental DNN, namely Deep Evolving Fuzzy Neural Network (DEVFNN), is proposed in this paper.",
"DEVFNN features a fully elastic structure where not only its fuzzy rule can be autonomously evolved but also the depth of network structure can be adapted in the fully automatic manner [17].",
"This property is capable of handling dynamic variations of data streams but also delivering continuous improvement of predictive performance.",
"The deep structure of DEVFNN is built upon the stacked generalization principle via the augmented feature space where each layer consists of a local learner and is inter-connected through augmentation of feature space [12], [13].",
"That is, the output of previous layer is fed as new input information to the next layer.",
"A meta-cognitive Scaffolding learner, namely Generic Classifier (gCLass), is deployed as a local learner, the main driving force of hidden layer, because it not only has an open structure and works in the single-pass fashion but also answers two key issues: what-to-learn and when-to-learn [18].",
"The what-to-learn module is driven by an active learning scenario which estimates contribution of data points and selects important samples for model updates while the when-to-learn module controls when the rule premise of multivariate Gaussian rule is updated.",
"gClass is structured under a generalized Takagi Sugeno Kang (TSK) fuzzy system which incorporates the multivariate Gaussian function as the rule premise and the concept of functional link neural network (FLANN) [19] as the rule consequent.",
"The multivariate Gaussian function generates non axis-parallel ellipsoidal clusters while the FLANN expands the degree of freedom (DoF) of the rule consequent via the up to second order Chebyshev series rectifying the mapping capability[20].",
"The major contribution of this paper is elaborated as follows: Elastic Deep Neural Network Structure: DEVFNN is structured by a deep stacked network architecture inspired by the stacked generalization principle.",
"Unlike the original stacked generalization principle having two layers, DEVFNN’s network structure is capable of being very deep with the use of feature augmentation concept.",
"This approach adopts the stacked deep fuzzy neural network concept in [12] where the feature space of the bottom layer to the top one is growing incorporating the outputs of previous layers as extra input information.",
"DEVFNN differs itself from [12], [13], [15] where it characterizes a fully flexible network structure targeted to address the requirement of continual learning [1].",
"This property is capable of expanding the depth of the DNN whenever the drift is identified to adapt to rapidly changing environments.",
"The use of drift detection method for the layer growing mechanism generates different concepts across each layer supposed to induce continuous refinement of generalization power.",
"The elastic characteristic of DEVFNN is borne out with the introduction of a hidden layer merging mechanism as a deep structure simplification approach which shrinks the depth of network structure on the fly.",
"This mechanism focuses on redundant layers having high mutual information to be coalesced with minor cost of predictive accuracy.",
"Dynamic Feature Space Paradigm: an online feature selection scenario is integrated in the DEVFNN learning procedure and enables the use of different input combinations for each sample.",
"This scenario enables flexible activation and deactivation of input attributes across different layers which prevents exponential increase of input dimension due to the main drawback of feature augmentation approach.",
"As with [21], [22], the feature selection process is carried out with binary weights (0 or 1) determined from the relevance of input features to the target concept.",
"Such feature selection method opens likelihood of previously deactivated features to be active again whenever its relevance is substantiated with current data trend.",
"Moreover, the dynamic feature space paradigm is realized using the concept of hidden layer merging method which functions as a complexity reduction approach.",
"The use of hidden layer merging approach has minor compression loss because one layer can be completely represented by another hidden layer.",
"An Evolving Base Building Unit: DEVFNN is constructed from a collection of Generic Classifier (gClass) [18] hierarchically connected in tandem.",
"gClass functions as the underlying component of DEVFNN and operates in every layer of DEVFNN.",
"gClass features an evolving and adaptive trait where its structural construction process is fully automated.",
"That is, its fuzzy rules can be automatically generated and pruned on the fly.",
"This property handles local drift better than a non-evolving base building unit because its network structure can expand on the fly.",
"The prominent trait of gClass lies in the use of online active learning scenario as the what-to-learn part of metacognitive learner which supports reduction of training samples and labeling cost.",
"This strategy differs from [23] since the sample selection process is undertaken in a decentralized manner and not in the main training process of DEVFNN.",
"Moreover, the how-to-learn scenario is designed in accordance with the three learning pillars of Scaffolding theory [24]: fading, problematizing and complexity reduction which processes data streams more efficiently than conventional self-evolving FNNs due to additional learning modules: local forgetting mechanism, rule pruning and recall mechanism, etc.",
"Real Drift Detection Approach: a concept drift detection method is adopted to control the depth of network structure.",
"This idea is confirmed by the fact that the hidden layer should produce intermediate representation which reveals hidden structure of data samples through multiple linear mapping.",
"Based on the recent study [25], it is outlined from the hyperplane perspective that the number of response region has a direct correlation to model$^{\\prime }$ s generalization power and DNN is more expressive than a shallow network simply because it has much higher number of response region.",
"In realm of deep stacked network, we interpret response region as the amount of unique information a base building unit carries.",
"In other words, the drift detection method paves a way to come up with a diverse collection of hidden layers or building units.",
"Our drift detection method is a derivation of the Hoeffding bound based drift detection method in [26] but differs from the fact that the accuracy matrix which corresponds to prequential error is used in lieu of sample statistics [27].",
"This modification targets detection of real drift which moves the shape of decision boundary.",
"Another salient aspect of DEVFNN$^{\\prime }$ s drift detector exists in the confidence level of Hoeffding$^{\\prime }$ s bound which takes into account sample$^{\\prime }$ s availability via an exponentially decreasing confidence level.",
"The exponentially decreasing confidence parameter links the depth of network structure and sample’s availability.",
"This strategy reflects the fact that the depth of DNN should be adjusted to the number of training samples as well-known from the deep learning literature.",
"A shallow network is generally preferred for small datasets because it ensures fast model convergence whereas a deep structure is well-suited for large datasets.",
"Adaptation of Voting Weight: the dynamic weighting scenario is implemented in the adaptive voting mechanism of DEVFNN and generates unique decaying factors of every hidden layer.",
"Our innovation in DEVFNN lies in the use of dynamic penalty and reward factor which enables the voting weight to rise and decline with different rates.",
"It is inspired by the fact that the voting weight of a relevant building unit should decrease slowly when making misclassification whereas that of a poor building unit should increase slowly when returning correct prediction.",
"This scenario improves a static decreasing factor as actualized in pENsemble [28], pENsemble+ [23], DWM [29] which imposes too violent fluctuations of voting weights.",
"The novelty of this paper is summed up in five facets: 1) contributes methodology in building the structure of DNNs which to the best of our knowledge remains a challenging and open issue; 2) offers a DNN variant which can be applied for data stream processing; 3) puts forward online complexity reduction mechanism of DNNs based on the hidden layer merging strategy and the online feature selection method; 4) introduces a dynamic weighting strategy with the dynamic decaying factor.",
"The efficacy of DEVFNN has been numerically validated using seven synthetic and real-world datasets possessing non-stationary characteristics under the prequential test-then-train approach - a standard evaluation procedure of data stream algorithms.",
"DEVFNN has been also compared by its static version and other online learning algorithms where DEVFNN delivers more encouraging performance in term of accuracy and sample consumption than its counterparts while imposing comparable computational and memory burden.",
"This paper is organized as follows: Section 2 outlines the network architecture of DEVFNN and its hidden layer, gClass; Section 3 elaborates the learning policy of DEVFNN encompassing the hidden layer growing strategy, the hidden layer merging strategy and the online feature selection strategy; Section 4 offers brief summary of gClass learning policy; Section 5 describes numerical study and comparison of DEVFNN; some concluding remarks are drawn in the last section of this paper."
],
[
"Problem Formulation",
"DEVFNN is deployed in the continual learning environment to handle data streams $C_t=[C_1,C_2,...,C_T]$ which continuously arrives in a form of data batch in the $T$ time stamps.",
"In practise, the number of time stamps is unknown and is possible to be infinite.",
"The size of data batch is $C_t=[X_1,X_2,...,X_P]\\in \\Re ^{P\\times n}$ and is set to be equal-sized in every time stamp.",
"Note that the size of data batches may vary as well across different time stamps.",
"$n$ stands for the number of input attributes.",
"In realm of online learning environments, it is impractical to assume the direct access to the true class label vector $Y_t=[y_1,y_2,...,y_P]\\in \\Re ^{P\\times m}$ and labelling process often draws some costs subject to the existence of ground truth or expert knowledge.",
"$m$ here stands for the number of target classes.",
"This fact confirms the suitability of prequential test-then-train scenario as evaluation procedure of the continual learning.",
"The $1-0$ encoding scenario is applied to form a multi-output target matrix.",
"For instance, if a data samples lies in a class 1 and there exists in total three classes, the $1-0$ encoding scheme produces a target vector $[1,0,0]$ , while a class 2 leads to $[0,1,0]$ .",
"The issue of labelling cost is tackled with the use of metacognitive learner as a driving force of hidden layer having the online active learning scenario as a part of the what-to-learn phase.",
"Data stream is inherent to the problem of concept drift where data batches do not follow static and predictable data distributions $P(Y|X)_t\\ne P(Y|X)_{t-1}$ well-known to be in two typical types: real and covariate.",
"The real drift is generally more challenging to handle than the covariate drift because variations of input data induce the shape of decision boundary usually resulting in classification error.",
"This problem cannot be solved by inspecting the statistic of input data as proposed in [26].",
"One approach to capture the presence of concept drift is by constructing accuracy metric which summarizes predictive performance of a classifier.",
"A decreasing trend of classifier$^{\\prime }$ s accuracy is a strong indicator of concept drift [27].",
"This issue is resolved here by the use of accuracy matrix which advances the idea of accuracy matrix in [27] with an adaptive windowing scheme based on the cutting point concept.",
"The presence of concept drift also calls for innovation in the construction of deep network structure because every layer is supposed to represent different concepts rather than different levels of data abstraction.",
"DEVFNN handles this issue with direct connection of hidden layer to output layer under the deep stacked network structure with the feature augmentation approach in which it allows every layer to contribute toward the final classification decision.",
"Specifically, the weighting voting scheme is implemented where the voting weight is derived from the reward and punishment approach with dynamic decaying rates.",
"The dynamic decaying rate sets unique reward and penalty factors giving added flexibility according to relevance of every hidden layer.",
"On the other hand, the feature augmentation method in building deep network structure risks on the curse of dimensionality due to the expansion of input dimension as the network depth.",
"The network simplification procedure is realized with the hidden layer merging procedure and the online feature selection scenario.",
"Fig.",
"1 delineates the working principle of DEVFNN."
],
[
"Network Architecture of DEVFNN",
"This section elaborates on the network architecture of DEVFNN including both the deep network structure and the hidden layer structure.",
"Figure: Learning Architecture of DEVFNN"
],
[
"Architecture of Hidden Layer",
"The hidden layer of DEVFNN is constructed by Generic Classifier (gClass) [18] working cooperatively in tandem to produce the final output.",
"gClass realizes the generalized TSK fuzzy inference system procedure where the multivariate Gaussian function with non diagonal covariance matrix is used in the rule layer while the consequent layer adopts the concept of expansion block of the FLANN via a nonlinear mapping of the up-to second order Chebyshev series [20].",
"The multivariate Gaussian rule generates better input space partition than those classical rules using the dot product t-norm operator because it enables rotation of ellipsoidal cluster [30].",
"This trait allows a reduction of fuzzy rule demand especially when data samples are not spanned in the main axes.",
"The use of FLANN idea in the consequent layer [19] aims to improve the approximation power because it has a higher DoF than the first order TSK rule consequent.",
"Several nonlinear mapping functions such as polynomial power function, trigonometric function, etc.",
"can be applied as the expansion block but the up-to second order Chebyshev function is utilized here because it incurs less free parameters than trigonometric function but exhibits better approximation power than other polynomial functions of the same order [31].",
"The fuzzy rule of gClass is formally expressed as follows: $\\Re _i$ : IF $X_k$ is $N_i(X_k;C_i,A_{i}^{-1})$ Then $\\tilde{y_i}=\\Phi W_i=W_{0,i}T_0+W_{1,i}T_1(x_1)+W_{2,i}T_2(x_2)+...+W_{2n-1,i}T_1(x_n)+W_{2n,i}T_2(x_n)$ where $X_k\\in \\Re ^{1\\times n}$ denotes the input vector at the $k-th$ time instant and $N_i(X_k;C_i,A_{i}^{-1})$ stands for the multivariate Gaussian function which corresponds to the rule premise of $i-th$ fuzzy rule.",
"$\\tilde{y_i}$ labels the rule consequent of the $i-th$ rule constructed by the up to second order Chebyshev polynomial expansion where $T_0(.),T_1(.),T_2(.",
")$ refer to the zero, first and second order Chebyshev polynomial expansion respectively.",
"The multivariate Gaussian function generates the firing strength of $i-th$ fuzzy rule which reveals its degree of compatibility as follows: $\\phi _i=exp(X-C_i)A_i^{-1}(X-C_i);$ where $C_i\\in \\Re ^{1\\times n}$ denotes the center of multivariate Gaussian function while $A_i^{-1}\\in \\Re ^{n \\times n}$ stands for the non-diagonal inverse covariance matrix while $R,n$ are respectively the number of fuzzy rules and input features.",
"The non-diagonal covariance matrix makes possible to describe inter-relations among input variables which vanishes in the case of diagonal covariance matrix and steers the orientation of ellipsoidal cluster.",
"On the other hand, the rule consequent of gClass is built upon the non-linear mapping of the up-to second order Chebyshev series defined as $\\tilde{y_i}=\\Phi W_i$ where $\\Phi \\in \\Re ^{1 \\times (2n+1)}$ is the output of functional expansion block while $W_i\\in \\Re ^{(2n+1)\\times 1}$ is the weight vector of the $i-th$ rule.",
"$\\Phi $ is produced by the up-to second order Chebyshev function expressed as follows: $T_{n+1}(x_j)=2x_jT_n(x_j)-T_{n-1}(x_j)$ Since the Chebyshev series is expanded up to the second order, $T_0(x_j)=1,T_1(x_j)=x_j,T_2(x_j)=2x_j^{2}-1$ .",
"Suppose that a predictive task is navigated by two input attributes, the functional expansion block $\\Phi $ is crafted as follows: $\\Phi =[T_0(x_j),T_1(x_1),T_2(x_j),T_1(x_2),T_2(x_2)]\\\\\\Phi =[1,x_1,2x_1^{2}-1,x_2,2x_2^{2}-1]$ The concept of FLANN in the framework of TSK fuzzy system improves approximation capability of zero or first order TSK rule consequent because it enhances the degree of freedom of rule consequent, although it does draw around extra $n$ network parameters.",
"Under the MIMO structure, the rule consequent incurs $(2n+1)\\times R \\times m$ parameters while the rule premise of gClass imposes $(n\\times n)\\times R+(n\\times R)$ .",
"The output of gClass is inferred using the weighted average operation of the rule firing strength and the output of functional expansion block as follows $y_o=\\frac{\\sum _i^R\\phi _i\\tilde{y_i}}{\\sum _i^R\\phi _i}=\\frac{\\sum _i^R exp(X-C_i)A_i^{-1}(X-C_i)^{T}\\tilde{y_i}}{\\sum _i^R exp(X-C_i)A_i^{-1}(X-C_i)^{T}}$ gClass is formed in the MIMO architecture possessing the class-specific rule consequent [32], [33] addressing the issue of class overlapping better than the popular one-versus-rest architecture because each class is handled separately.",
"Because gClass generates multiple output, the final classification decision is obtained by applying the maximum operation to find the most confident prediction as follows: $y=\\max _{1 \\le o \\le m} y_o$ where $m$ is the number of target classes.",
"The MIMO structure is also more stable than the direct regression approach hampered by the class shifting issue [32], [33].",
"This problem is caused by the smooth transition of the TSK fuzzy system which cannot cope with a dramatic class change.",
"Figure: Deep Stacked Network Structure with Feature Augmentation"
],
[
"Deep Stacked Network",
"DEVFNN is realized in the deep stacked network structure where each hidden layer is connected in series.",
"Hidden layers except the first layer accept predictive outputs of previous layers as input attributes plus original input patterns [12].",
"This trait retains the physical meaning of original input variables which often loses in conventional DNN architectures due to multiple nonlinear mappings.",
"This aspect also implies three exists a growth of input space across each layer.",
"That is, the deeper the hidden layer position the higher the input dimension is, since it is supposed to receive all predictive outputs of preceding layers as extra input features.",
"This structure is inspired by [12], [13] where the concept of feature augmentation is introduced in building a DNN architecture.",
"Our approach extends this approach with the introduction of structural learning process which fully automates the construction of deep stacked network via online analysis of dynamic data streams.",
"Moreover, this approach is equipped by the online feature selection and hidden layer merging scenarios which dynamically compress the input space of each layer.",
"This mechanism replaces the random feature selection of the original work in [12], [13].",
"Our approach also operates in the one-pass learning scenario being compatible with online learning setting whereas [12], [13] still applies an offline working scenario.",
"The network architecture of DEVFNN is shown in Fig.",
"2.",
"DEVFNN works in the chunk-by-chunk basis where each chunk is discarded once learned and the chunk size may vary in different time stamps.",
"Suppose that a data chunk with the size of $P$ $C_t=(X_k,Y_k)_{k=1,...,P}$ is received at the $k-th$ time stamp while DEVFNN is composed of $D$ hidden layers built upon gClass, the first hidden layer receives the original input representation with no feature augmentation while the $2-nd$ hidden layer is presented with the following input attributes: $\\widehat{X}_k^{2}=[X_k,\\widehat{Y}_k^{1}]$ where $\\widehat{X}_k^{2}\\in \\Re ^{P\\times (n+m)}$ consists of the input feature vector and the predictive output of the first layer.",
"Additional $m$ input attributes are incurred because gClass is structured under the MIMO architecture deploying a class-specific rule consequent.",
"For $d-th$ hidden layer, its input features are expressed as $\\widehat{X}_k^{d}=[X_k,\\widehat{Y}_k^{1},...,\\widehat{Y}_k^{d-1}]_{k=1,...,P}\\in \\Re ^{P\\times (n+(m\\times (d-1)))}$ .",
"This situation also implies that dimensionality of input space linearly increases as the depth of network structure.",
"This issue is coped with the self-organizing property of DEVFNN featuring a controlled growth of network depth by means of the drift detection method which assures the right network complexity of a given problem.",
"In addition, the hidden layer merging and online feature selection mechanisms further contribute to compression of input dimension to handle possible curse of dimensionality.",
"The deep stacked network through feature augmentation is consistent to the stacked generalization principle [14] which is capable of achieving an improved generalization performance since the manifold structure of the original feature space is constantly opened.",
"In addition, each hidden layer functions as a discriminative layer producing predicted class label.",
"The predicted class label delivers an effective avenue to open the manifold structure of the original input space to the next hidden layer supposed to improve generalization power.",
"The end output of DEVFNN is produced by the weighted voting scheme where the voting weight of hidden layer increases and decreases dynamically in respect to their predictive accuracy.",
"This strategy reflects the fact that each hidden layer is constructed with different concepts navigated by the drift detection scenario.",
"That is, a hidden layer is rewarded when returning correct prediction by enhancing its voting weight while being punished by lowering its voting weight when incurring misclassification.",
"It reduces the influence of irrelevant classifier where it is supposed to carry low voting power due to the penalty as a result of misclassification [28].",
"It is worth noting that the predictive performance of hidden layers is examined against the prequential error when true class labels are gathered.",
"The prequential test-then-train scenario is simulated in our numerical study.",
"It enables creation of real data stream environment where data streams are predicted first and used for model updates once the true class label becomes available.",
"This scenario is more plausible to capture real data stream environments in practise in which true class labels often cannot be immediately fed by the operator.",
"Furthermore, DEVFNN is composed of independent building blocks interconnected with the feature augmentation approach meaning that each hidden layer is trained independently with the same cost function and outputs the same target variables.",
"The final output of DEVFNN is formalized as follows: $Class_{final}=\\max _{1 \\le o \\le m} \\sigma _{o}$ where $\\sigma _o$ denotes the accumulated weight of a target class resulted from predicted class label of hidden layers $\\sigma _{o}=\\chi _{d}^{class}+\\sigma _{o}$ .",
"$\\chi _{d}^{o}$ stands for the voting weight of $d-th$ hidden layer predicting $o-th$ class label.",
"This procedure aims to extract the most relevant subsets of hidden layer to the $C_t$ data batch.",
"That is, each hidden layer is set to carry different concepts and is triggered when the closest concept comes into the picture.",
"In addition, the tuning phase is restricted to the $win$ hidden layer in order to induce a stable concept for every hidden layer.",
"That is, irrelevant hidden layers are frozen from the tuning phase when they represent different concepts.",
"The $win$ hidden layer can be determined from its voting weight where a high voting weight indicates strong relevance of a hidden layer to the current concept."
],
[
"Learning Policy of DEVFNN",
"This section elaborates the learning policy of DEVFNN in the deep structure level.",
"An overview of DEVFNN learning procedure is outlined in Algorithm 1.",
"DEVFNN initiates its learning process with the adaptation of voting weight with dynamic decaying factors.",
"The tuning of voting weight results in a unique voting weight of each hidden layer with respect to their relevance to the current concept measured from their prequential error.",
"This method can be also interpreted as a soft structural simplification method since it is capable of ruling out hidden layers in the training process by suppressing their voting weights to a small value.",
"The training procedure continues with the online feature selection procedure which aims to cope with the curse of dimensionality due to the growing feature space bottleneck of the deep stacked network structure.",
"The feature selection is performed based on the idea of feature relevance examining the sensitivity of input variables in respect to the target classes.",
"The hidden layer merging mechanism is put forward to capture redundant hidden layers and reduces the depth of network structure by coalescing highly correlated layers.",
"Furthermore, the drift detection mechanism is applied to deepen the deep stacked network if changing system dynamics are observed in data streams.",
"The drift detection module is designed using the concept of Hoeffding bound [26], [27] which determines a conflict level to flag a drift.",
"There exist three conditions resulted from the drift detector: stable, warning and drift.",
"The drift phase leads to addition of a hidden layer, while the warning phase accumulates data samples into a buffer later used to create a new hidden layer if a drift is detected.",
"The stable phase simply updates the winning hidden layer determined from its voting weight.",
"In what follows, we elaborate on the working principle of DEVFNN:"
],
[
"Adaptation of Voting Weight",
"The voting weight is dynamically adjusted by a unique decaying factor for each layer $\\rho _d \\in [0,1]$ which plays a major role to adapt to the concept drift.",
"A low value of $\\rho _d$ leads to slow adaptation to rapidly changing conditions but works very well in the presence of gradual or incremental drift where the drift is not too obvious initially.",
"A high value of $\\rho _d$ adapts to sudden drift more timely than the low value but compromises the stability in the gradual drift where data samples are drawn from the mixture between two distributions during the transition period.",
"Considering this issue, $\\rho _d$ is not kept fixed rather it continuously adjusted to reflect the learning performance of a hidden layer.",
"The decaying factor $\\rho _d$ is tuned using a step size, $\\pi $ , as follows: $\\rho _d=\\rho _d \\pm \\pi ;$ This mechanism underpins a flexible voting scenario where the decaying factor mirrors hidden layer compatibility to existing data streams.",
"In other words, the voting weight of hidden layers augments and diminishes with different intensities.",
"It is also evident that the voting weight of a strong hidden layer should be confirmed whereas the influence of poor building units should be minimized in the voting phase.",
"That is, $\\rho _d=\\rho _d+\\pi $ occurs if a hidden layer returns a correct prequential prediction whereas $\\rho _d=\\rho _d-\\pi $ takes place if a sample leads to a prequential error.",
"The step size $\\pi $ sets the rate of change where the higher the value increases its sensitivity to the hidden layer performance.",
"The penalty and reward scenario is undertaken by increasing and decreasing the voting weight of a hidden layer.",
"A penalty is imposed if a wrong prediction is produced by hidden layer as follows: $\\chi _d=\\chi _d*\\rho _d;$ On the other hand, the reward scenario is performed if correct prediction is returned as follows: $\\chi _d=min(\\chi _d(1+\\rho _d),1)$ the reward scenario also functions to handle the cyclic drift because it reactivates inactive hidden layer with a minor voting weight.",
"Because it is also observed from (REF ) and the range of the decaying factor $\\rho _d$ is $[0,1]$ , the voting weight is bounded in the range of $[0,1]$ .",
"Compared to similar approaches in [28], [23], this penalty and reward scenario emphasizes rewards to a strong hidden layer while discouraging a penalty to such layers whereas a poor hidden layer should be penalized with a high intensity while receiving little reward when returning correct prediction.",
"This strategy is done since every hidden layer has direct connection to output layer outputting its own predicted class label.",
"The final predicted class label should take into account relevance of each hidden layer reflected from its prequential error.",
"Furthermore, this mechanism also aligns with the use of drift detection method as an avenue to deepen the depth of network structure since the drift detection method evolves different concepts of each hidden layer."
],
[
"Online Feature Weighting Mechanism",
"DEVFNN implements the feature weighting strategy based on the concept of feature relevance.",
"That is, a relevant feature is defined as that showing high sensitivity to the target concept whereas low sensitivity feature signifies poor input attributes which should be assigned a low weight to reduce its influence to the final predictive outcomes [34].",
"There exist several avenues to check the sensitivity of input attributes in respect to the target concept: Fisher Separability Criterion [35], statistical contribution [30], etc.",
"The correlation measure is considered as the most plausible strategy here because it reveals the mutual information of input and target features [22] which signals the presence of changing system dynamics.",
"The correlation between two variables, $x_1,x_2$ , can be estimated using the Pearson correlation index (PCI) as follows: $\\zeta (x_1,x_2)=\\frac{cov(x_1,x_2)}{\\sqrt{var(x_1,x_2)}}$ where $cov(x_1,x_2),var(x_1,x_2)$ respectively stand for covariance and variance of $x_1,x_2$ which can be calculated in recursive manners.",
"Although the PCI can be directly integrated into the feature weighting scope without any transformation because the highest correlation is attained when it returns either $-1$ or 1, the PCI method is sensitive to the rotation and translation of data samples [36].",
"The mutual information compression index (MICI) method [36] is applied to achieve trade-off between accuracy of correlation measure and computational simplicity.",
"The MICI method works by estimating the amount of information loss if one of the two variables is discarded.",
"It is expressed as follows: $\\gamma (x_{1},x_{2})=\\frac{1}{2}(vr_1+vr_2-\\nonumber \\\\\\sqrt{vr_1^{2}+vr_2^{2}-4vr_1vr_2(1-\\zeta (x_{1},x_{2})^{2})})$ where $\\zeta {(x_1,x_2)}$ denotes the Pearson correlation index of two variables and $vr_1,vr_2$ represent $var(x_1),var(x_2)$ , respectively.",
"Unlike the PCI method where -1 and 1 signify the maximum correlation, the maximum similarity is attained at $\\gamma (x_{1},x_{2})=0$ .",
"This method is also insensitive to rotation and translation of data points[36].",
"Once the correlation of the $j-th$ input variable and all $m$ target classes are calculated, the score of the $j-th$ input feature is defined by its relevance to all $m-th$ target classes.",
"This aspect is realized by taking average correlation across $m$ target classes as follows: $Score_j=mean_{o=1,..,m}\\gamma (x_j,y_o)$ where $\\gamma (x_j,y_o)$ denotes the maximum information compression index between the $j-th$ input feature and the $o-th$ target class.",
"The use of average operator is to assign equal importance of each target class and to embrace the fact that an input feature is highly needed to only identify one target class.",
"This strategy adapts to the characteristic of gClass formed in the MIMO architecture [33].",
"The feature selection mechanism is carried out by assigning binary weights $\\lambda _j$ , either 0 or 1, which changes on demand in every time stamp.",
"This strategy is applied to induce flexible activation and deactivation of input variables during the whole course of training process which avoids loss of information due to complete forgetting of particular input information.",
"An input feature is switched off by assigning 0 weight if its score falls above a given threshold $\\delta _1$ meaning that it shows low relationship to any target classes as follows: $Score_j>=\\delta _1$ where $\\delta _1\\in [0,1]$ stands for the predefined threshold.",
"The higher the value of this threshold the less aggressive the hidden layer merging is performed and vice versa.",
"The feature selection process is carried out by setting the input weight to zero with likelihood being set to one again in the future whenever the input attribute becomes relevant again.",
"Moreover, the input weighting strategy is committed in the centralized manner where the similarity of all input attributes are analyzed at once.",
"That is, all input attributes are put together.",
"The online feature weighting scenario only analyses the $n$ dimensional original feature space rather than the $n+m\\times (D-1)$ dimensional augmented feature space.",
"Extra input feature is subject to the hidden layer merging mechanism which studies the redundancy level of $D$ hidden layers."
],
[
"Hidden Layer Merging Mechanism",
"DEVFNN implements the hidden layer merging mechanism to cope with the redundancy across different base building units.",
"This scenario is realized by analyzing the correlation of the outputs of different layers [23].",
"From manifold learning viewpoint, a redundant layer containing similar concept is expected not to inform salient structure of the given problem because it does not open manifold of learning problem to unique representation - at least already covered by previous layers.",
"Suppose that the MICI method is applied to explore the correlation of two hidden layers and $\\gamma (y_i,y_j),i\\ne j$ , the hidden layer merging condition is formulated as follows: $\\gamma (y_i,y_j)<\\delta _2$ where $\\delta _2\\in [0,1]$ is a user-defined threshold.",
"This threshold is linearly proportional to the maximum correlation index where the lower the value the less merging process is undertaken.",
"The merging procedure is carried out by setting its voting weight as zero, thus it is considered as a \"don$^{\\prime }$ t care\" input attribute of the next layers.",
"This strategy expedites the model updates because redundant layers can be bypassed without being revisited for both inference and training procedures.",
"The hard pruning mechanism is not implemented in the merging process because it causes a reduction of input dimension which undermines the stability of next layers unless a retraining mechanism from scratch is carried out.",
"It causes dimensional reduction of the output covariance matrix of next layers.",
"This strategy is deemed similar to the dropout scenario [37] in the deep learning literature but the weight setting is analyzed from the similarity analysis rather than purely probabilistic approach.",
"The dominance of two hidden layers is simply determined from its voting weight.",
"The voting weight is deemed a proper indicator of hidden layer dominance because it is derived from a dynamic penalty and reward scenario with unique and adaptive decaying factors.",
"The redundancy-based approach such as merging scenario is more stable than relevance-based approach because information of one layer can be perfectly represented by another layer.",
"In addition, parameters of two hidden layers are not fused because similarity of two hidden layers is observed in the output level rather than fuzzy rule level."
],
[
"Hidden Layer Growing Mechanism",
"DEVFNN realizes an evolving deep stacked network structure which is capable of introducing new hidden layers on demand.",
"This strategy aims to embrace changing training patterns of data streams and to enhance generalization performance by increasing the level of abstraction of training data.",
"This is done by utilizing a drift detection module which vets the status of data streams whether the concept change is present.",
"Notwithstanding that the idea of adding a new component to handle concept drift is well-established in the literature as presented in the ensemble learning literature, each ensemble member has no interaction at all to their neighbors.",
"The drift detection scenario makes use of the Hoeffding$^{\\prime }$ s bound drift detection mechanism proposed in [26] where the evaluation window is determined from the switching point rather than a fixed window size.",
"Nevertheless, the original method is developed based on the increase of population mean which ignores the change of data distribution in the target space.",
"In other words, it is only capable of detecting covariate drift.",
"Instead of looking at the increase of population mean, the error index is applied.",
"This modification is based on the fact that the change of feature space does necessarily induce the concept drift in the target domain $P(Y|X)_{k}\\ne P(Y|X)_{k-1}$ .",
"This concept is implemented by constructing a binary accuracy vector where 0 elements presents correct prediction while 1 elements are inserted for false predictions.",
"This scenario is inspired by the fast Hoeffding drift detection method (FDDM) [27] but our approach differs from [26] in which the window size is set fully adaptive according to the switching point.",
"Moreover, each data sample is treated with equal importance with the absence of any weights which detects sudden drift rapidly although it is rather inaccurate to pick up gradual drift [26].",
"The advantage of the Hoeffding$^{\\prime }$ s method is free of normal data distribution assumption - too restrictive in many applications.",
"Moreover, it is statistically sound because a Hoeffding bound corresponds to a particular confidence level.",
"Assuming that $P$ denotes the chunk size, the data chunk is partitioned into three groups $F\\in \\Re ^{P\\times (n+m)},G\\in \\Re ^{cut\\times (m+n)},H\\in \\Re ^{(P-cut)\\times (m+n)}$ where $cut$ is the switching point.",
"$F,G,H$ records the error index instead of the original data points in which only two values, namely 0 or 1, is present - 0 for true prediction, 1 for false prediction.",
"Note that $cut$ is elicited by evaluating data samples - the first sample up to the switching sample.",
"Each data partition $F,G,H$ is assigned with the error bounds $\\epsilon _F,\\epsilon _G,\\epsilon _H$ calculated as follows: $\\epsilon _{F,G,H}=(b-a)\\sqrt{\\frac{size}{2 (size*cut)}\\ln ({\\frac{1}{\\alpha }})}$ where $size$ denotes the size of data partition and $\\alpha $ labels the significance level of Hoeffding's bound.",
"$a,b$ denote the maximum and minimum values in the data partition.",
"The significance level has a clear statistical interpretation because it corresponds to the confidence level of Hoeffding's bound $1-\\alpha $ .",
"In realm of DNN, the model$^{\\prime }$ s complexity must consider the availability of training samples to ensure parameter convergence especially in the context of continual learning situation where a retraining process over a number of epochs is prohibited.",
"A shallow model is generally preferred over a deep model to handle a small dataset.",
"Considering the aspect of sample$^{\\prime }$ s availability, a dynamic significance level is put forward where the significance level exponentially rises as the number of time stamps with a limit.",
"A limit is required here to avoid loss of detection accuracy because the significance level is inversely proportional to the confidence level: $\\alpha _{D}=min(1-e^{\\frac{-k}{T}},\\alpha _{min}^{D}),\\alpha _{W}=min(1-e^{\\frac{-k}{T}},\\alpha _{min}^{W}) $ where $k,T$ respectively denote the number of time stamps seen thus far and the total number of time stamps while $\\alpha _{min}^{D},\\alpha _{min}^{W}$ stand for the minimum significance level of the drift and warning phases.",
"The minimum significance level has to be capped at 0.1 to induce above 90% confidence level.",
"Once the confidence level is calculated, the next step is to find the switching point, $cut$ , indicating the horizon of the drift detection problem or the time window in which a drift is likely to be present.",
"The switching point is found if the following condition is met as follows: $\\hat{F}+\\epsilon _{F}\\le \\hat{G}+\\epsilon _{G}$ where $\\hat{F},\\hat{G},\\hat{H}$ denote the statistics of the three data partitions.",
"It is observed that the switching point targets a transition point between two concepts where the statistic of $G$ is larger than $H$ .",
"The switching point portrays a time index where a drift starts to come into picture.",
"It is worth noting that the statistics of the data partition $\\hat{G}$ is expected to be constant or to reduce during the stable phase.",
"The drift phase, therefore, refers to the opposite case where the empirical mean of the accuracy vector $\\hat{G}$ increases.",
"The condition (REF ) aims to find the cutting point $cut$ where the accuracy vector is no longer in the decreasing trend.",
"Once $cut$ is located, the three error index vectors, $F,G,H$ , can be formed.",
"Our drift detector returns two conditions: warning and drift tailored from the two significance levels $\\alpha _{warning},\\alpha _{drift}$ .",
"The two significance levels $\\alpha _{warning},\\alpha _{drift}$ can be set to the confidence level of the drift detection.",
"The smaller the value of the significance level implies the more accurate the drift detection method delivers.",
"The warning and drift conditions are signaled if two following situations are come across as follows: $|\\hat{H}-\\hat{G}|> \\alpha _{drift}$ $|\\hat{H}-\\hat{G}|> \\alpha _{warning}$ These two conditions present a case where the null hypothesis $H_{0}:E[G]\\le E[H]$ is rejected.",
"The warning condition is meant to capture the gradual drift.",
"It pinpoints a situation where a drift is not obvious enough to be declared or next instances are called for to confirm a drift.",
"No action is taken during the warning phase, only data samples of the warning condition are accumulated in the data matrix $\\phi =[X_{warning},Y_{warning}]$ .",
"Once the drift condition is satisfied, the structure of deep network is deepened by appending a new hidden layer stacked at the top level.",
"The new hidden layer is created using the incoming data chunk and the accumulated data samples in the warning phase $\\Phi =[X_{new},Y_{new};\\phi ]$ .",
"The opposite case or the normal case portrays the alternative $H_{1}:E[G]> E[H]$ .",
"That is, the stable phase only activates an adjustment of the winning hidden layer to improve the generalization power of DEVFNN.",
"The winning hidden layer is selected for the adaptation scenario instead of all hidden layers to expedite the model update.",
"Moreover, DEVFNN adopts the concept of different-depth network structure which opens the room for each layer to produce the end-output of DEVFNN because each layer is trained to solve the same optimization problem.",
"Because the dynamic voting mechanism is implemented, the winning hidden layer is simply selected based on its voting weight."
],
[
"gClass learning procedure",
"This section provides brief recap of gClass working principle [18] constructed under three learning pillars of meta-cognitive learning: what-to-learn, how-to-learn and when-to-learn.",
"The what-to-learn component functions as the sample selection module implemented under the online active learning scenario, while the when-to-learn component sets the sample update condition.",
"The how-to-learn module realizes a self-adaptive learning principle developed under the framework of Scaffolding theory [24]: problematizing, fading and complexity reduction.",
"This procedure encompasses rule generation, pruning, forgetting and tuning mechanisms.",
"A flowchart of gClass learning policy is placed in the supplemental document.",
"Learning Policy of DEVFNN [1] Input: $(X_n,Y_n)\\in \\Re ^{P\\times (n+m)}$ , $\\pi $ , $\\delta _1$ , $\\delta _2$ , $Vig$ and $k_{prune}$ Output: $Class_{final}$ final predicted Class Testing phase: Predict: The class label of the current data batch.",
"Training phase: Step 1: Tuning of $\\chi $ : $k = 1$ to $P$ $d=1$ to $D$ ($\\hat{y}_k^d\\ne y_k$ ) Execute: $\\rho _d=\\rho _d-\\pi $ and $chi_d=\\chi _d*\\rho _d$ Update: the accuracy matrix $Acc(k)=1$ Execute: $\\rho _d=\\rho _d+\\pi $ , $\\chi _d=min(\\chi _d(1+\\rho _d),1)$ Update: the accuracy matrix $Acc(k)=0$ Step 2: Online Feature Selection Mechanism: $j = 1$ to $n$ $o=1$ to $m$ Calculate: the input target correlation (12) ($score>\\delta _1$ ) Set: the input weight to $\\lambda _j=0$ Step 3: Hidden Layer Merging Mechanism : $d_1 = 1$ to $D$ $d_2=1$ to $D$ Calculate: the correlation coefficient (14) ($\\gamma (y_{d_1},y_{d_2}) <\\delta _1, \\forall d_1 \\ne d_2$ ) Set: the voting weight of weaker layer to 0 Step 4: Drift Detection Mechanism: $k = 1$ to $P$ Construct: $F=Acc,G=Acc(1:k)$ Calculate: $\\epsilon _F,\\epsilon _G, \\hat{G},\\hat{H}$ (15) ($\\hat{G}+\\epsilon _G \\ge \\hat{F}+\\epsilon _F$ ) Set: $cut=k$ Construct: $H=Acc(P-cut+1:cut)$ and $\\hat{H}$ Calculate: the drift and warning level $\\epsilon _{drift},\\epsilon _{warning}$ ($|\\hat{G}-\\hat{F}|\\ge \\epsilon _{drift}$ ) Deepen: the network structure using $(X_n,Y_n)\\in \\Re ^{P\\times (n+m)}$ and $\\phi $ ($|\\hat{G}-\\hat{F}|\\ne \\epsilon _{drift} \\text{and} |\\hat{G}-\\hat{F}|\\ge \\epsilon _{warning}$ ) Create: $\\phi $ Update: The winning layer using $(X_n,Y_n)\\in \\Re ^{P\\times (n+m)}$ What-To-Learn Phase - Online Active Learning Scenario: The online active learning scenario of gClass is driven by the extension of extended conflict ignorance (ECI) principle [38] which relies on two sample contribution measures.",
"The first concept is developed from the idea of extended recursive density estimation (ERDE) applied as the rule growing scenario in [39].",
"Our approach distinguishes itself from [39] in which the RDE concept is modified for the multivariate Gaussian rule and integrates the sample weighting concept overcoming the outlier$^{\\prime }$ s bottleneck.",
"Unlike the rule growing concept finding salient samples as those having maximum and minimum densities, the sample of interest for deletion purpose is redundant samples defined as those violating the maximum and minimum conditions.",
"The second approach is designed using the distance-to-boundary concept.",
"A classifier is said to be confident if it safely classifies a sample to one of classes or is far from decision boundary.",
"The sample-to-boundary concept is set as a ratio of the first and second dominant classes examined by the classifier$^{\\prime }$ s outputs.",
"An uncertain sample is indicated if the ratio returns a value around 0.5 whereas a high ratio signifies a confident case.",
"How-To-Learn - Sample Learning Strategy: Once a sample is accepted by the online active learning scenario, it is passed to the how-to-learn scenario evolving parameter and structure of gClass.",
"The problematizing part concerns on the issue of concept drift handling and the complexity reduction part relieves the problem$^{\\prime }$ s complexity while the fading part is devised for reduction of model$^{\\prime }$ s complexity.",
"The three components are integrated in a single dedicated learning process and executed in the one-pass learning scenario.",
"The problematizing phase consists of two learning modules: the rule growing and forgetting scenarios.",
"The rule growing phase of gClass adopts the same criteria of pClass [32] in which the three rule growing conditions, namely data quality (DQ), datum significance (DS), volume check, are consolidated to pinpoint an ideal observation to expand the rule base size.",
"The data quality method is a derivation of the RDE method in [39] involving the sample weighting concept.",
"Moreover, it is tailored to accommodate the multivariate Gaussian rule.",
"The DS concept estimates the statistical contribution of a data sample whether it deserves to be a candidate of new rule [40], while the volume check is integrated to prevent the over-sized rule which undermines model$^{\\prime }$ s generalization.",
"The DS concept extends the concept of neuron significance [41] to be compatible with the non axis-parallel ellipsoidal rule while the volume check examines the volume of winning rule and a new rule is introduced given that its volume exceeds a pre-specified limit.",
"Once the three rule growing conditions are satisfied, a new fuzzy rule is initialized using the class overlapping situation.",
"The class overlapping method arranges the three initialization strategies in respect to spatial proximity of a data sample to inter-class and intra-class data samples.",
"This strategy is carried our using the quality per class method which studies relationship of current sample to target classes.",
"The rule forgetting scenario is carried out in the local mode which deploys unique forgetting levels of each rule [42].",
"It is derived using the local DQ (LDQ) method performing recursive local density estimation.",
"The gradient of LDQ method for each rule is calculated and used to define the forgetting level in the rule premise and consequent.",
"Moreover, the rule consequent tuning scenario is driven by the fuzzily weighted generalized recursive least square (FWGRLS) method inspired by the work of generalized recursive least square method (GRLS) method [43] putting forward the weight decay term in the cost function of RLS method.",
"This method can be also perceived as a derivation of FWRLS method in [39] in which it improves the tuning scenario with the weight decay term to improve model$^{\\prime }$ s generalization.",
"The fading phase lowers structural complexity of gClass to avoid the overfitting problem and to expedite the runtime.",
"The fading phase of gClass is crafted under the same strategy as pClass [32] where two rule pruning strategy, namely extended rule significance (ERS) and potential+ (P+) methods, are put forward.",
"The ERS method shares the same principle as the DS method where it approximates the statistical contribution of fuzzy rules.",
"This approach can be also seen as an estimator of expected outputs of gClass under uniform distribution.",
"The P+ method adopts the concept of rule potential [39] but converts this method to perform the rule pruning task.",
"Unlike the ERS method capturing superfluous rules playing little role during their lifespan, the P+ method discovers obsolete rule, no longer relevant to represent the current data distribution.",
"By extension, the P+ method is also applied in the rule recall scenario to cope with the cyclic drift.",
"That is, obsolete rules are only deactivated with possibility to be reactivated again in the future if it becomes relevant again.",
"gClass implements the online feature weighting mechanism based on the fisher seperability criteria (FSC) in the empirical feature space using the kernel concept as adopted in pClass [32] as a part of complexity reduction method.",
"Nevertheless, the online feature weighting scenario of gClass is switched into a sleep mode because DEVFNN is already equipped by an online feature selection and layer merging module in the top level.",
"When-To-Learn - Sample Reserved Strategy: The when-to-learn strategy of gClass is built upon the standard sample reserved strategy of the meta-cognitive learner [44].",
"Our approach, however, differs from that [44] where the sample learning condition is designed under different criteria.",
"The sample reserved strategy incorporates a condition for the tuning scenario of rule premise and data samples violating the rule growing and tuning procedures are accumulated in a data buffer reserved for the future training process.",
"Our approach simply exempts those samples and exploit them merely for the rule consequent adaptation scenario because such sample quickly become outdated in rapidly changing environments.",
"In addition, This procedure is designated to reduce the computational cost.",
"A flowchart of DEVFNN learning policy is attached in the supplemental document."
],
[
"Numerical Study",
"This section discusses experimental study of DEVFNN in seven popular real-world and synthetic data stream problems: electricity-pricing, weather, SEA, hyperplane, SUSY, kddCUP and indoor RFID localization problem from our own project.",
"DEVFNN is compared against prominent continual learning algorithms in the literature: PNN [45], HAT [46], DEN [4], DSSCN [17], staticDEVFNN.",
"PNN, HAT and DEN are popular continual learning algorithms in the deep learning literature also designed to prevent the catastrophic forgetting problems.",
"They feature self-evolution of hidden nodes but still adopt the static network depth.",
"DSSCN represents a deep algorithm having stochastic depth.",
"The key difference with our approach lies in the concept of random shift to form deep stacked network structure rather than the feature augmentation approach.",
"Comparison with staticDEVFNN is important to demonstrate the efficacy of flexible structure.",
"That is, the hidden layer expansion and merging modules are switched off in the static DEVFNN.",
"The MATLAB codes of DEVFNN are made publicly available in $https://bit.ly/2ZfnN5y$.",
"Because of the page limit, only SEA problem is discussed in the paper while the remainder of numerical study is outlined in the supplemental document.",
"Nonetheless, Table 1 displays numerical results of all problems.",
"Our numerical study follows the prequential test-then-train protocol.",
"That is, a dataset is divided into a number of equal-sized data batches.",
"Each data batch is streamed to DEVFNN in which it is used to first test DEVFNN's generalization performance before being used for the training process.",
"This scenario aims to simulate real data stream environment [47] where numerical evaluation is independently performed per data batch.",
"The numerical results in Table 1 are reported as the average of numerical results across all data batches.",
"Five evaluation criteria are used here: Classification Rate (CR), Fuzzy Rule (FR), Precision (P), Recall (R), Hidden Layer (HL).",
"Table: Numerical results of consolidated algorithms"
],
[
"SEA Problem",
"The SEA problem is one of the most prominent problems in the data stream area where the underlying goal is to categorize data samples into two classes based on the summation of two input attributes [48].",
"That is, a class 1 is returned if the condition $f_1+f_2<\\theta $ while the opposite case $f_1+f_2>\\theta $ indicates a class 2.",
"The concept drift is induced by shifting the class threshold three times $\\theta =4 \\rightarrow 7 \\rightarrow 4 \\rightarrow 7$ .",
"This shift results in the abrupt drift and furthermore the cyclic drift because the class boundary is changed in the recurring fashion.",
"The SEA problem is built upon three input attributes where the third input attribute is merely a noise.",
"We make use of the modified SEA problem in [49] which incorporates 5 to 25% minority class proportion.",
"Data points are generated from the range of $[0,10]$ and the concept drift takes place in the target domain due to the drifting class boundary.",
"The trace of classification rate, hidden layer and training sample are depicted in Fig.",
"4(a)-(d) of supplemental documents.",
"Numerical results are reported in Table 1 along with numerical results of other problems.",
"The advantage of DEVFNN is reported in Table 1 where it outperforms other algorithms in terms of classification rates, precision and recall.",
"This result highlights the efficacy of a deep stacked network structure compared to the static depth network in improving the generalization power.",
"The use of a deep stacked network structure allows continuous refinement of predictive power where the output of preceding layer is fed as extra input information of current layer supposed to guide the predictive error toward zeros.",
"The main bottleneck of the deep stacked network architecture through the feature augmentation approach lies in the linear increase of input dimensionality as the number of hidden layer.",
"Notwithstanding that the hidden layer merging mechanism is integrated in DEVFNN and is supposed to lower the input dimension due to reduction of extra input attributes, the soft dimensionality reduction is applied by zeroing the voting weight of a hidden layer and the hidden layer is excluded from any training and inference processes.",
"This strategy is to prevent instability issue due to the discontinuity of the training process which imposes a retraining process from scratch.",
"Another complexity reduction method is implemented in terms of a dynamic voting weight scenario which overrides the influence of hidden layers in the final classification decision.",
"This scenario generates unique reward and penalty weights giving heavy reward to a good hidden layer whereas strong penalty is imposed to that of poor hidden layers.",
"Because of the page limit, our numerical results of other datasets are presented in the supplemental document.",
"A brief summary of our numerical results is as follows: 1) the decentralized online active learning strategy in the layer level is less efficient than the centralistic variant because it is classifier-dependent; 2) the proposal of real drift detection scenario is more timely than the covariate drift detection scenario because it reacts when the classifier$^{\\prime }$ s performance is compromised; 3) the dynamic voting scenario with dynamic decaying factors lead to more stable voting scenario because the effect of penalty and reward can be controlled in respect to the classifier$^{\\prime }$ s performance; 4) The application self-evolving depth enhances learning performance of the static depth because it grows the network on demand considering variation and availability of data samples."
],
[
"Conclusions",
"A novel deep fuzzy neural network, namely dynamic evolving fuzzy neural network (DEVFNN), is proposed for mining evolving and dynamic data streams in the lifelong fashion.",
"DEVFNN is constructed with a deep stacked network structure via the feature space augmentation concept where a hidden layer receives original input features plus the output of preceding layers as input features.",
"This strategy generates continuous refinement of predictive power across a number of hidden layers.",
"DEVFNN features an autonomous working principle where its structure and parameters are self-configured on the fly from data streams.",
"A drift detection method is developed based on the principle of FDDM but it integrates an adaptive windowing scheme using the idea of cutting point.",
"The drift detection mechanism is not only designed to monitor the dynamic of input space, covariate drift, but also to identify the nature of output space, real drift.",
"Furthermore, DEVFNN is equipped by a hidden layer merging mechanism which measures correlation between two hidden layers and combines two redundant hidden layer.",
"This module plays a key role in the deep stacked network structure via the feature augmentation concept to address uncontrollable increase of input dimension in rapidly changing conditions.",
"DEVFNN also incorporates the online feature weighting method which assigns a crisp weight to input features in respect to their relevance to the target concept.",
"The dynmic voting concept is introduced with the underlying notion \"unique penalty and reward intensity\" examined by the relevance of hidden layers.",
"DEVFNN is created with a local learner as a hidden layer, termed gClass, interconnected in tandem.",
"gClass characterizes the meta-cognitive learning approach having three learning phases: what-to-learn, how-to-learn, when-to-learn.",
"The what-to-learn and when-to-learn schemes provide added flexibility to gClass putting forward the online active learning scenario and the sample reserved strategy while the how-to-learn scheme is devised according to the Scaffolding theory.",
"The online active learning method selects relevant samples to be labeled and to train a model which increases learning efficiency and mitigates the overfitting risk.",
"The sample reserved strategy sets conditions for rule premise update while the scaffolding theory is followed to enhance the learning performance with several learning modules tailored according to the problematizing, fading and complexity reduction concepts of the Scaffolding theory.",
"The efficacy of DEVFNN has been numerically validated using seven prominent data stream problems in the literature where it produces more accurate classification rates, precision and recall than other benchmarked algorithms while incurring minor increase of computational and memory demand.",
"It is found that the depth of network structure possesses a linear correlation with a generalization power given that every hidden layer is properly initialized and trained while the stochastic depth property improves learning performance compared to the static depth.",
"Moreover, the dynamic adjustment of voting weights makes possible to adapt the voting weight with the dynamic adjustment factor which dynamically augments and shrinks according to its prequential error.",
"It is perceived that the merging scenario dampens the network complexity without compromising classification accuracy.",
"The concept drift detection method is applied to grow the hidden layer of network structure which sets flexibility for direct access of hidden layer to output layer.",
"This implies that the final output is inferred by a combination of every layer output.",
"In other words, DEVFNN actualizes a different-depth network paradigm where each level puts forward unique aspects of data streams.",
"Our future work will investigate different approaches for self-generating the hidden layer of deep fuzzy neural network because it is admitted that the use of a concept drift detection method replaces the multiple nonlinear mapping of one concept generating a high level feature description with diverse concepts per layer."
]
] | 1808.08517 |
[
[
"Noiseprint: a CNN-based camera model fingerprint"
],
[
"Abstract Forensic analyses of digital images rely heavily on the traces of in-camera and out-camera processes left on the acquired images.",
"Such traces represent a sort of camera fingerprint.",
"If one is able to recover them, by suppressing the high-level scene content and other disturbances, a number of forensic tasks can be easily accomplished.",
"A notable example is the PRNU pattern, which can be regarded as a device fingerprint, and has received great attention in multimedia forensics.",
"In this paper we propose a method to extract a camera model fingerprint, called noiseprint, where the scene content is largely suppressed and model-related artifacts are enhanced.",
"This is obtained by means of a Siamese network, which is trained with pairs of image patches coming from the same (label +1) or different (label -1) cameras.",
"Although noiseprints can be used for a large variety of forensic tasks, here we focus on image forgery localization.",
"Experiments on several datasets widespread in the forensic community show noiseprint-based methods to provide state-of-the-art performance."
],
[
"Introduction",
"In the last few years, digital image forensics has been drawing an ever increasing attention in the scientific community and beyond.",
"With cheap and powerful cameras available to virtually anyone in the world, and the ubiquitous diffusion of social networks, images and videos have become a dominant source of information.",
"Unfortunately, they are used not only for innocent purposes, but more and more often to shape and and distort people's opinion for commercial, political or even criminal aims.",
"In this context, image and video manipulations are becoming very common, and increasingly dangerous for individuals and society as a whole.",
"Figure: Two forged images (left) with their noiseprints (right).",
"The inconsistencies caused by the manipulation are visiblein the extracted noiseprint.Driven by these phenomena, in the last decade, a large number of methods have been proposed for forgery detection and localization or camera identification [1], [2], [3].",
"Some of them rely on semantic or physical inconsistencies [4], [5], but statistical methods, based on pixel-level analyses of the data, are by far the most successful and widespread.",
"Mostly, they exploit the fact that any acquisition device leaves on each captured image distinctive traces, much like a gun barrel leaves peculiar striations on any bullet fired by it.",
"Statistical methods can follow both a model-based and a data-driven approach.",
"Methods of the first class try to build mathematical models of some specific features and exploit them for forensic purposes.",
"Popular targets of such analyses are lens aberration [6], [7], [8], camera response function [9], [10], [11], color filter array (CFA) [12], [13], [14] or JPEG artifacts [15], [16], [17], [18].",
"Having models to explain the available evidence has an obvious appeal, but also a number of shortcomings, first of all their usually narrow scope of application.",
"As an alternative, one can rely on data-driven methods, where models are mostly abandoned, and the algorithms are trained on a suitably large number of examples.",
"Most data-driven methods work on the so-called noise residual, that is, the noise-like signal which remains once the high-level semantic content has been removed.",
"A noise residual can be obtained by subtracting from the image its “clean” version estimated by means of denoising algorithms, or by applying some high-pass filters in the spatial or transform (Fourier, DCT, wavelet) domain [19], [20], [21], [22], [23], [24].",
"Noise residuals can be also used in a blind context (no external training) to reveal local anomalies that indicate possible image tampering [25], [26], [27], [28].",
"Figure: From left to right: the forged image, its noiseprint, the noise residual obtained using a Wavelet-based denoising filter (a tool commonly used for PRNU extraction) and the noise residual obtained through a 3rd order derivative filter (used in the Splicebuster algorithm ).Among all methods based on noise residuals, those relying on the photo-response non-uniformity noise (PRNU) deserve a special attention for their popularity and performance.",
"In the seminal paper by Lukas et al.",
"[30] it was observed that each individual device leaves a specific mark on all acquired images, the PRNU pattern, due to imperfections in the device manufacturing process.",
"Because of its uniqueness, and stability in time, the PRNU pattern can be regarded as a device fingerprint, and used to carry out multiple forensic tasks.",
"PRNU-based methods have shown excellent performance for source identification [31] and for image forgery detection and localization [30], [32], [33], [34], [35].",
"Note that they can find any type of forgeries, irrespective of their nature, since the lack of PRNU is seen as a possible clue of manipulation.",
"The main drawbacks of PRNU-based methods are i) the need of a large number of images taken from the camera to obtain good estimates and ii) the low power of the signal of interest with respect to noise, which impacts heavily on the performance.",
"In particular, the prevailing source of noise is the high-level image content, which leaks in the PRNU due to imperfect filtering.",
"The latter often overwhelms the information of interest, especially in the presence of saturated, dark or textured areas.",
"This latter is a typical problem of all the methods based on noise residuals.",
"In this work, to overcome these problems we propose a new method to extract a noise residual.",
"Our explicit goal is to improve the rejection of semantic content and, at the same time, emphasize all the camera-related artifacts, since they bear traces of the whole digital history of an image.",
"While doing this, we want to avoid any external dependency.",
"Therefore, we will rely neither on prior information of any type, nor on the availability of a labelled set of training data.",
"To this end, we follow a data driven approach and exploit deep learning.",
"A suitable architecture is designed, inspired to Siamese networks, and trained on a large dataset which includes images from many different camera models.",
"Once the training is over, the network is freezed, and can be used with no further supervision on images captured by any camera model, both inside and outside the training set.",
"In this way the approach is completely unsupervised.",
"To any single image the network associates a noise residual, called noiseprint from now on, which shows clear traces of camera artifacts.",
"Therefore, it can be regarded as a camera model fingerprint, much like the PRNU pattern represents a device fingerprint.",
"It can also happen that image manipulations leave traces very evident in the noiseprint, such to allow easy localization even by direct inspection.",
"As an example, Fig.1 shows two images subject to a splicing attack, which can be easily detected by visual inspection of their noiseprints.",
"It is worth to observe that these artifacts cannot be spotted so clearly using other noise residuals (see Fig.2).",
"In the rest of the paper, we first analyze related work on noise residuals to better contextualize our proposal (Section II), then describe the proposed architecture and its training (Section III), carry out a thorough comparative performance analysis of a noiseprint-based algorithm for forgery localization (Section IV), provide ideas and examples on possible uses of noiseprints for further forensic tasks (Section V), and eventually draw conclusions (Section VI).",
"The observation that the local noise level within an image may help revealing possible manipulations dates back at least to 2004, with the work of Popescu and Farid [25].",
"The underlying idea is that each image has an intrinsic uniform amount of noise introduced by the imaging process or by digital compression.",
"Therefore, if two images are spliced together, for example, or some local post-processing is carried out on part of the image to hide traces of tampering, inconsistencies in the noise level may occur, which can be used to reveal the manipulation.",
"In [25] the local noise variance is estimated over partially overlapping blocks based on the second and fourth moments of the data, assuming the kurtosis of signal and noise to be known.",
"Detection of inconsistencies is then left to visual inspection.",
"In [26] the same approach is adopted, but the noise variance is estimated through wavelet decomposition and a segmentation process is carried out to check for homogeneity.",
"In [27], instead, the local noise level is estimated based on a property of natural images, the projection kurtosis concentration, and estimation is formulated as an optimization problem with closed-form solution.",
"Further methods based on noise level inconsistencies have been recently proposed in [36] and [37].",
"A major appeal of all these unsupervised methods is their generality.",
"They require only a reliable estimator of noise variance to discover possible anomalies, and need no further hypotheses and no training.",
"On the down side, since the noise due to in-camera processing is certainly non-white, using only intensity as a descriptor neglects precious information.",
"This consideration justifies the quest for better noise descriptors and the use of machine learning in forensics.",
"One of the first methods in this class, proposed back in 2005 [38], exploits statistics extracted from the high-pass wavelet subbands of the image to train a suitable classifier.",
"In [20], the set of wavelet-based features of [38] is augmented with prediction error features computed on a noise residual extracted through denoising.",
"A more accurate discrimination is carried out in [21], [22] by computing both first-order and higher-order Markovian features on DCT or Wavelet coefficients and also on prediction errors.",
"Interestingly, these features were inspired by prior work carried out in steganalysis [39].",
"This is the same path followed by the popular rich models, which were proposed originally in steganalysis [40], and then applied successfully in image forensics for the detection and localization of various types of manipulations [41], [42], [28], [24].",
"Like in [20] the rich models rely on noise residuals, but multiple high-pass filters are used to extract them, and discriminative features are built based on the co-occurrence of small local patterns.",
"Even though these methods exhibit a very good performance, they need a large training set to work properly, a condition rarely met in the most challenging real-world cases.",
"To overcome this limitation, the methods proposed in [28] and [43] exploit rich-model features only to perform unsupervised anomaly detection.",
"In Splicebuster [28] the expectation-maximization algorithm is used to this end, while [43] resorts to an ad hoc autoencoder-based architecture.",
"Eventually, these methods are used for blind forgery detection and localization with no supervision or external training.",
"The papers by Swaminathan et al.",
"[44], [45] are conceptually related to our noiseprint proposal since they aim at identifying all the possible traces of in-camera and out-camera processing, called intrinsic fingerprints.",
"However, the proposed solution, strongly model-based, is completely different from ours.",
"The traces of interest are estimated on the basis of a suitable camera model.",
"Then, any further post-processing is regarded as a filtering operation whose coefficients can be estimated using blind deconvolution.",
"In the end, inconsistencies in the estimated model parameters suggest that the image has been manipulated in some way.",
"However, correctly modeling such processes is quite difficult, and this approach does not work well in realistic conditions.",
"It is also worth mentioning a recent paper [46] in which traces of camera model artifacts in the noise residual are preserved and collected in the so-called sensor linear pattern, and exploited to find inconsistencies."
],
[
"Using deep learning for image forensics",
"Recently, deep learning methods have been applied to image forensics.",
"Interestingly, the first proposed architectures, inspired again by work in steganalysis [47], all focus on suppressing the scene content, forcing the network to work on noise residuals.",
"This is obtained by adding a first layer of high-pass filters, either fixed [48], [49], or trainable [50], or else by recasting a conventional feature extractor as a convolutional neural network (CNN) [51].",
"A two-stream network is proposed in [52], [53] to exploit both low-level and high-level features, where a first network constrained to work on noise residuals is joined with a general purpose deep CNN (ResNet 101 in [53]).",
"Slightly different CNN-based architectures have been proposed in [54], [55].",
"All such solutions, however, rely on a training dataset strongly aligned with the test set, which limits their value for real-world problems.",
"Instead, to gain higher robustness, the training phase should be completely independent of the test phase.",
"This requirement inspires a group of recently proposed methods [56], [57], [58] which share some high-level ideas with our own proposal.",
"In [56] a CNN trained for camera model identification is used to analyze pairs of patches: different sources suggest possible image splicing.",
"Results look promising, but only a synthetic dataset is used for experiments, and the performance degrades sharply if the camera models are not present in the training set.",
"A similar approach, based on a similarity network, is followed in [57] for camera model identification.",
"First, the constrained network of [50] is trained to extract high-level camera model features, then, another network is trained to learn the similarity between pairs of such features, with a procedure similar to a Siamese network.",
"A Siamese network is also used in [58] to predict the probability that two image patches share the same value for each EXIF metadata attribute.",
"Once trained, the network can be used on any possible type of image without supervision.",
"This is a very important property that we also pursue in our work, Unlike in [58], however, we do not use metadata information in the training phase, but rely only on the image content and on the information about the camera model.",
"Figure: Using CNNs to extract noise residuals.Top: the target CNN processes the input image to generate its noiseprint, a suitable noise residual with enhanced model-based artifacts.Bottom: the CNN proposed in processes the input image to generate its AWGN pattern, a strict-sense noise residual."
],
[
"Proposed approach",
"A digital camera carries out a number of processes to convert the input light field into the desired output image.",
"Some of these processes, like data compression, interpolation, and gamma correction, are common to virtually all cameras, although with different implementations.",
"Others, included to offer more advanced functionalities and to attract customers, vary from model to model.",
"Due to all these internal processing steps, each camera model leaves on each acquired image a number of artifacts which are peculiar of the model itself, and hence can be used to perform forensic analyses.",
"However, such artifacts are very weak, certainly imperceptible to the eye, and their exploitation requires sophisticated statistical methods.",
"To this end, a typical approach consists in extracting a noise residual of the image, by means of a high-pass filter or a denoiser.",
"Our goal is to improve the noise residual extraction process, enhancing the camera model artifacts to the point of allowing their direct use for forensic analyses.",
"Accordingly, the product of our system will be an image-size noise residual, just like in PRNU-based methods, a noiseprint image that will bear traces of camera model artifacts, rather than of the individual device imperfections.",
"In the following two subsections we describe the noiseprint extraction process, based on the Siamese network concept, and provide implementation details on the network training."
],
[
"Extracting noiseprints",
"Our aim is to design a system, Fig.3(top), which takes a generic image as input and produces a suitable noise residual in output, the image noiseprint.",
"As said before, the noiseprint is desired to contain mostly camera model artifacts.",
"For sure, we would like to remove from it, or strongly attenuate, the high-level scene content, which acts as a disturbance for our purposes.",
"This latter is precisely the goal of the CNN-based denoiser proposed by Zhang et al.",
"[59].",
"In fact, rather than trying to generate the noiseless version of the image, this denoiser, Fig.3(bottom), aims at extracting the noise pattern affecting it (by removing the high-level content).",
"Eventually, this is subtracted from the input to obtain the desired clean image.",
"For this reason, this denoiser is obviously a good starting point to develop our own system, so we keep its architecture, and initialize it with the optimal parameters obtained in [59] for AWGN image denoising.",
"Then, we will update such parameters through a suitable training phase.",
"Figure: Training the CNN-based denoiser.To each clean patch y i y_i of the dataset a synthetic AWGN pattern w i w_i is added to create a noisy patch: x i =y i +w i x_i=y_i+w_i.The (x i ,w i )(x_i,w_i) pairs are used to train the CNN.The distance between the residual generated by the CNN, r i =f(x i )r_i=f(x_i), and the true noise pattern, w i w_i, is back-propagated to update the net weights.In [59] these parameters have been obtained by training the CNN with a large number of paired input-output patches, where the input is a noisy image patch and the output its noise content (see Fig.4).",
"We should therefore resume the training by submitting new paired patches, where the input is a generic image patch, and the output the corresponding noiseprint.",
"The only problem is that we have no model of the image noiseprint therefore we cannot produce the output patches necessary for this training procedure.",
"Nonetheless, we have precious information to rely upon.",
"In fact, we know that image patches coming from the same camera model should generate similar noiseprint patches, and image patches coming from different camera models dissimilar noiseprint patches.",
"Leveraging this knowledge, we can train the network to generate the desired noise residual where not only the scene content but all non-discriminative information is discarded, while discriminative features are enhanced.",
"Figure: Using a Siamese architecture for training.The output of one CNN takes the role of desired (same model and position) or undesired (different models or positions) reference for the other twin CNN.Consider the Siamese architecture of Fig.5, formed by the parallel of two identical CNNs, that is two CNNs which have both the same architecture and the same weights.",
"Two different input patches acquired with the same camera model are now fed to the two branches.",
"Since the outputs are expected to be similar, the output of net 1 can take the role of desired output for the input of net 2, and vice-versa, thus providing two reasonable input-output pairs.",
"For both nets, we can therefore compute the error between the real output and the desired output, and back-propagate it to update the network weights.",
"More in general, all pairs formed by the input to one net and the output to its sibling represent useful training data.",
"For positive examples (same model) weights are updated so as to reduce the distance between the outputs, while for negative examples (different models) weights are updated to increase this distance.",
"It is worth emphasizing that negative examples are no less important than positive ones.",
"Indeed, they teach the network to discard irrelevant information, common to all models, and keep in the noiseprint only the most discriminative features.",
"Until now, for the sake of simplicity, we have neglected the following important point.",
"In order for two input patches to merit a positive label, they must come not only from the same camera model but also from the same position in the image.",
"In fact, artifacts generated by in-camera processes are not spatially stationary, just think of JPEG compression with its regular 8$\\times $ 8 grid, or to the regular sampling pattern used for acquiring the three color channels.",
"Therefore, noiseprint patches corresponding to different positions are different themselves (unless the displacement is a multiple of all artifacts' periods), and input patches drawn from different positions must not be pooled during training, in order not to dilute the artifacts' strength.",
"An important consequence for forensic analyses is that any image shift, not to talk of rotation, will impact on the corresponding noiseprint, thereby allowing for the detection of many types of manipulations.",
"When the training process ends, the system is freezed.",
"Consequently, to each input image a noiseprint is deterministically associated, which enhances the camera model artifacts with their model-dependent spatial distribution.",
"Of course, the noiseprint will also contain random disturbances, including traces of the high-level scene.",
"Nonetheless, the enhanced artifacts appear to be much stronger than these disturbances, and such to provide a satisfactory basis for forensic tasks."
],
[
"Implementation",
"In the previous subsection our aim was to convey the main ideas about the nature of noiseprints and how to extract them.",
"Here we provide crucial implementation details, which allow a fast and accurate training of the system.",
"Figure: Detailed architecture of the CNN-based denoiser."
],
[
"Initialization",
"As already said, we start from the architecture of the denoiser proposed in [59], shown in some more detail in Fig.6.",
"Like in [59], for complexity issues, the network is trained using minibatches of $N$ patches of $K\\times K$ pixels, with $N$ =200 and $K$ =48.",
"It is worth underlining, however, that the system is fully convolutional and hence, once trained, it works on input images of any given size, not just 48$\\times $ 48 patches.",
"Therefore, there is no patch stitching issue."
],
[
"Boosting minibatch information",
"Concerning training, it should be realized that the Siamese architecture is only an abstraction, useful to understand how input-output pairs are generated.",
"In practice, there is only one CNN, which must be trained by submitting a large number of examples.",
"Each minibatch, as said before, includes $N$ =200 patches, which are used to form suitable input-output pairs.",
"However, this does not imply that only $N/2$ pairs can be formed: in fact, each individual patch can be meaningfully combined with all the others, as proposed in [17], lifting a batch of examples into a dense pairwise matrix.",
"When two patches come from the same model and image position, the pair will have a positive label, otherwise a negative label.",
"Therefore, each minibatch provides $O(N^2)$ examples rather than just $O(N)$ , with a significant speed-up of training.",
"In particular, to implement this strategy, our minibatches comprise 50 groups of 4 patches.",
"Each group is internally homogeneous (same model same position) but heterogeneous with respect to the other groups."
],
[
"Distance-based logistic loss",
"As for the loss function, ${\\cal L}$ , we adopt the distance based logistic (DBL) proposed in [18].",
"Let $\\lbrace x_1, \\ldots , {\\bf x}_n\\rbrace $ be a minibatch of input patches, and $\\lbrace r_1, \\ldots , r_n\\rbrace $ the corresponding residuals output by the net.",
"Then, let $d_{ij} = \\Vert r_i-r_j \\Vert ^2$ be the squared Euclidean distance between residuals $i$ and $j$ .",
"We require such distances to be small when $(i,j)$ belong to the same group, and large otherwise.",
"Now, for the generic $i$ -th patch, we can build a suitable probability distribution through softmax processing as $p_i(j) = \\frac{e^{-d_{ij}}}{\\sum _{j \\ne i} e^{-d_{ij}}}$ With this definition, our original requirement on distances is converted in the requirement that $p_i(j)$ be large whenever $(i,j)$ are in the same group, that is, $l_{ij}=+1$ , and small otherwise.",
"This leads us to define the $i$ -th patch loss as ${\\cal L}_i = -\\log \\!",
"\\sum _{j: l_{ij}=+1} p_i(j)$ When all the probability mass is concentrated in same-group patches, the sum is unitary and the loss is null, while all deviations from this condition cause an increase of the loss.",
"Accordingly, the minibatch loss is defined as the sum of all per-patch losses, hence ${\\cal L}_0 = \\sum _i \\left[ -\\log \\!",
"\\sum _{j: l_{ij}=+1} p_i(j) \\right]$"
],
[
"Regularization",
"To encourage diversity of noiseprints, we add a regularization term to the previous DBL loss.",
"Let $R_i(u,v) = {\\cal F}(r_i(m,n))$ be the 2D discrete Fourier transform of patch $r_i$ , where $(m,n)$ and $(u,v)$ indicate spatial and spectral discrete coordinates, respectively.",
"The quantity $S(u,v) = \\frac{1}{N} \\sum _{i=1}^N |R_i(u,v)|^2$ is therefore an estimate of the power spectral density (PSD) of the whole minibatch.",
"For a given camera model, the power spectrum will peak at the fundamental frequencies of artifacts and their combinations.",
"It is expected, however, that different camera models will have different spectral peaks.",
"Actually, since the locations of such peaks are powerful discriminative features, it is desirable that they be uniformly distributed over all frequencies.",
"To this end, we include in the loss function a regularization term given by the log-ratio between geometric and arithmetic means of the PSD components ${\\cal R}& = & \\log \\left[ \\frac{S_{GM}}{S_{AM}} \\right] \\\\& = & \\left[ \\frac{1}{K^2} \\sum _{u,v} \\log S(u,v) \\right] -\\log \\left[ \\frac{1}{K^2} \\sum _{u,v}S(u,v) \\right] \\nonumber $ In fact, the GM/AM ratio is maximized for uniform distribution, and therefore its inclusion encourages the maximum spread of frequency-related features across the model noiseprints.",
"Eventually, the complete loss function reads as ${\\cal L}= {\\cal L}_0 - \\lambda {\\cal R}$ with the weight $\\lambda $ to be determined by experiments.",
"We now provide some experimental evidence on the potential of noiseprints for forensic analyses.",
"Camera fingerprints can be used for a multiplicity of goals, as proven by the large body of literature on the applications of PRNU patterns.",
"In Section V we provide some insights into the possible uses of noiseprints.",
"However, we leave a detailed investigation of all these cases for future work, and focus, instead, on just one major forensic task, the localization of image manipulations, irrespective of their nature.",
"To analyze performance in depth, we carry out an extensive experimental analysis, considering 9 datasets of very different characteristics, and comparing results, under several performance criteria, with all the most promising reference techniques.",
"In the rest of this Section, we first present our noiseprint-based localization method, then describe the reference methods, the datasets, and the performance metrics, provide details on the training of the noiseprint extractor, and finally present and discuss experimental results."
],
[
"Forgery localization based on noiseprints",
"In the presence of localized image manipulations, the image noiseprint shows often clear traces of the attack, allowing direct visual detection and localization.",
"However, this is not always the case, and an automatic localization tool is necessary to support the analyst's work.",
"In particular, we look for a localization algorithm which takes the image and its noiseprint as input, and outputs a real-valued heatmap which, for each pixel, provides information on the likelihood that it has has been manipulated.",
"Here, we use the very same blind localization algorithm proposed for Splicebuster [28].",
"By so doing, we obtain an objective measure of the improvement granted by adopting the image noiseprint in place of the third-order image residual used in [28].",
"The algorithm assumes that the pristine and manipulated parts of the image are characterized by different models.",
"Accordingly, it looks for anomalies w.r.t.",
"the dominant pristine model to locate the manipulated part.",
"To each pixel of a regular sampling grid, a feature vector is associated, accounting for the spatial co-occurrences of residuals.",
"These vectors are then fed to the expectation-maximization (EM) algorithm, which learns the two models together with the corresponding segmentation map.",
"The interested reader is referred to [28] for a more detailed description.",
"However, it is worth emphasizing the blind nature of this localization algorithm, which relies only on the given image with no need of prior information."
],
[
"Reference methods",
"We consider only reference methods which are blind, like our proposal, that is, they do not need specific datasets for training or fine tuning, nor do they use metadata or other prior information on the test data.",
"Besides being more general, these methods are less sensitive to dataset-related polarizations, allowing a fair comparison.",
"Most of these methods can be considered state-of-the-art in the field, except for a few ones, like the error level analysis (ELA) included for their diffusion among practitioners.",
"They can be roughly classified in three classes according to the features they exploit: i) JPEG artifacts [60], [61], [62], [63], [16], [64], ii) CFA artifacts [14], [13], iii) inconsistencies in the spatial distribution of features [65], [26], [66], [27], [28], [58].",
"Tab.REF lists all methods under comparison together with a link to the available source or executable code.",
"To save space, we use compact acronyms, for example EXIF-SC to mean EXIF self-consistency algorithm [58].",
"Our own proposed noiseprint-based localization algorithm will be referred to simply as Noiseprint from now on."
],
[
"Datasets",
"To assess performance we use 9 datasets, which are listed in Tab.REF together with their main features.",
"Some of them focus only on splicing, like the DSO-1 dataset [5], the VIPP dataset [16], created to evaluate double JPEG compression artifacts, and the FaceSwap dataset [52], where only automatic face manipulation have been created using code available on-linehttps://github.com/MarekKowalski/FaceSwap/.",
"All other datasets, instead, present a wide variety of manipulations, sometimes cascaded on one another on the same image.",
"They also present very different characteristics in terms of number of cameras, resolution and format.",
"For example, the dataset proposed by Korus et al.",
"[67] comprises only raw images of the same resolution, acquired by only four different cameras.",
"This low variability can induce some polarizations of the results.",
"On the contrary, the NIMBLEhttps://www.nist.gov/itl/iad/mig/nimble-challenge-2017-evaluation and MFChttps://www.nist.gov/itl/iad/mig/media-forensics-challenge-2018 datasets designed by NIST for algorithm development and evaluation in the context of the Medifor program, are extremely variable, beyond what can be found in real practice.",
"Therefore, they can be considered very challenging benchmarks for all methods under test."
],
[
"Performance measures",
"Forgery localization can be regarded as a binary classification problem.",
"Pixels belong to one of two classes, pristine (background or negative) or forged (foreground or positive), and a decision must be made for each of them.",
"All performance metrics rely on four basic quantities TP (true positive): # positive pixels declared positive; TN (true negative): # negative pixels declared negative; FP (false positive): # negative pixels declared positive; FN (false negative): # positive pixels declared negative; Since the last two items correspond to errors, a natural performance measure is the overall accuracy $A = \\frac{TP+TN}{TP+TN+FP+FN}$ However, often there are many more negative than positive pixels, and errors on positive pixels impact very little on accuracy, which becomes a poor indicator of performance.",
"This is exactly the case of forgery localization, where the manipulated area is often much smaller than the background.",
"To address this problem a number of other metrics have been proposed.",
"Precision and recall, defined as $\\mbox{precision} = \\frac{TP}{TP+FP} \\hspace{17.07164pt} \\mbox{recall} = \\frac{TP}{TP+FN}$ put emphasis on the positive (forged) class, measuring, respectively, the method's ability to avoid false alarms and detect forged pixels.",
"These quantities are summarized in a single index by their harmonic mean, the F1 measure $\\mbox{F1} = \\frac{2}{\\frac{1}{\\mbox{precision}} + \\frac{1}{\\mbox{recall}}} = \\frac{2\\,TP}{2\\,TP + FN + FP}$ Another popular metric is the Matthews Correlation Coefficient (MCC), that is, the cross correlation coefficient between the decision map and and the ground truth, computed as $\\mbox{MCC} = \\frac{TP \\times TN - FP \\times FN}{\\sqrt{(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}$ which is robust to unbalanced classes.",
"Both F1 and MCC work on a binary decision map.",
"However, most methods provide a continuous-valued heatmap, which is converted to a binary map by thresholding.",
"Therefore, to free performance assessment from the threshold selection problem, for both F1 and MCC the maximum over all possible thresholds is taken.",
"An alternative approach is followed with the Average Precision (AP) which is computed as the area under the precision-recall curve, and therefore can be regarded as an average of performance measures over all thresholds.",
"In order to carry out a solid assessment of performance, we consider all three measures, F1, MCC, and AP.",
"Note that, in some cases, the output heatmap may have an inverted polarity w.r.t.",
"the ground truth (see Fig.7).",
"Since this is immaterial for the semantics of the map, but may disrupt performance, we always consider both the original and inverted heatmaps, and keep the best of the two.",
"Finally, we exclude from the evaluation pixels near the foreground-background boundary, where all methods are very unreliable due to limited resolution.",
"Figure: From left to right: forged images, ground truth, heatmaps from the method proposed in and in .The forged area can be dark or light, this does not really matter, since the approaches look for image inconsistencies."
],
[
"Training procedure",
"For the proposed method the network used to extract all noiseprints is trained on a large variety of models.",
"To this end, we formed a large dataset, including both cameras and smartphones, using various publicly available datasets, plus some other private cameras.",
"In detail, we used 44 cameras from the Dresden dataset [68], 32 from Socrates dataset [69], 32 from VISION [70], 17 from our private dataset, totaling 125 individual cameras from 70 different models and 19 brands.",
"For the experiments, this dataset is split, on a per-camera basis, in training and validation sets, comprising 100 and 25 cameras, respectively.",
"We note explicitly that the datasets used to form the training and validation sets are not used in the test phase.",
"All images are originally in JPEG format.",
"The network is initialized with the weights of the denoising network of [59].",
"During training, each minibatch contains 200 patches of 48$\\times $ 48 pixels extracted from 100 different images of 25 different cameras.",
"In each batch, there are 50 sets, each one formed by 4 patches with same camera and position.",
"Training is performed using the ADAM optimizer.",
"All hyper-parameters (learning rate, number of iterations, and weight of regularization term) are chosen using the validation set.",
"Considering the major impact of JPEG compression on performance, we use a different network for each JPEG quality factor, training it on images that are preliminary JPEG compressed with the same factor.",
"To ensure reproducibility of results, all trained nets will be made available on-line upon publication of this manuscript."
],
[
"Results",
"We now present and discuss experimental results for all 9 datasets, 15 methods under comparison, and 3 performance metrics.",
"Our choice is to consider one metric at a time, in order to allow a synoptic view of the performance of all methods over all datasets.",
"Under this respect, it is worth noting in advance that, although numbers change significantly from one metric to another, the relative ranking of methods remains pretty much the same.",
"Therefore, in the tables REF -REF we report results in terms of MCC, F1 and AP, respectively.",
"We complement each performance value with the corresponding rank on the dataset, in parentheses, using red for the three best methods, blue for the others.",
"The last two columns show the average performance and average ranking over all datasets.",
"We begin our analysis from these latter quantities which allow for a first large-scale assessment.",
"The proposed noiseprint-based method provides the best average performance, MCC=0.403, which is 10% better than the second best (Splicebuster) and much better than all the others, which go from 0.101 to 0.333.",
"This is the effect of a uniformly good performance over all datasets.",
"Noiseprint ranks always among the best three methods (red), with an average rank of 1.7, testifying of a remarkable robustness across datasets with wildly different characteristics.",
"The comparison with Splicebuster (average MCC=0.365, average ranking=2.7) is especially meaningful, since the two methods differ only in the input noise residual, obtained thorough high-pass filtering in Splicebuster and given by noiseprint here.",
"It is also worth noting that the third best technique, based on EXIF metadata inconsistencies, looks for similarity among patches and uses a deep CNN with intensive training (not influenced by the test set), further supporting the soundness of the proposed approach.",
"Some specific cases deserve a deeper analysis.",
"On the Korus dataset, for example, Noiseprint ranks only third, after CFA1 and Splicebuster.",
"However, this is a dataset of raw images (not JPEG compressed) while Noiseprint is trained on JPEG-compressed images.",
"On this dataset, CFA-based methods perform especially well, since two of the four cameras (the two Nikon) fit very well the model developed in [14].",
"All this said, Noiseprint keeps providing a good performance, while methods based on JPEG artifacts show a dramatic impairment.",
"Conversely, on the VIPP dataset, JPEG-based methods exhibit a boost in performance, especially ADQ1 and ADQ2 which look for double JPEG compression artifacts.",
"Indeed, the VIPP dataset was built originally to expose this very type of artifacts.",
"In this case as well, Noiseprint ranks among the best methods.",
"These examples ring an alarm bell on the use of polarized dataset.",
"In fact, these are precious tools to study a specific phenomenon but cannot be taken as reliable predictors of performance in uncontrolled scenarios.",
"For this latter task, datasets should be much more varied and, even better, multiple independent datasets should be considered at once.",
"Table: EXPERIMENTAL RESULTS: F1 (F-measure)Table: EXPERIMENTAL RESULTS: AP (Average Precision)The NIMBLE and MFC datasets, developed by NIST under the Medifor program, fit very well this latter profile.",
"They are characterized by a large variety of attacks (e.g., splicing, copy-move, removal through inpainting, local blurring, contrast enhancement) often cascaded on the same image.",
"Therefore, they represent very challenging testbeds, especially for robustness.",
"In fact, all methods exhibit a worse average performance on these datasets than on the first four.",
"Noteworthy, Noiseprint ranks always first or second on these datasets and, when second, just inches away from the best (0.324 vs. 0328, or 0.295 vs. 0.297).",
"Since also Splicebuster and EXIF-SC perform quite well, it seems safe to say that the spatial inconsistency of features is the key for good results.",
"A relatively good performance is also ensured by ADQ2, CAGI, and NOI2.",
"Tab.REF and Tab.REF report experimental results for the F1 and AP metrics, with the same structure as Tab.REF .",
"We will keep comments to a minimum here, since the numbers, and especially the relative ranking, change very little by replacing one metric with another.",
"The most remarkable variation is a slight improvement in the ranking of EXIF-SC on the AP metric, which is now the same as Splicebuster on the average.",
"Noiseprint keeps providing the best average performance, with a remarkable stability across all datasets.",
"Readers familiar with the F1 measure may notice the impressive 0.78 obtained on DSO-1, but this is a simple dataset, with large splicings and uncompressed images.",
"Still, this shows that in favourable conditions, near-perfect localization is possible.",
"A better insight on the actual quality of results can be gathered by the visual inspection of the examples of Fig.REF .",
"Note that these examples were cherry-picked from all datasets to show cases where Noiseprint provides a good results even when most of the best competitor fail.",
"It is worth noting that the correct localization in these examples comes together with an accurate delineation of contours and rare false alarms.",
"This is not always the case, of course, as is shown in Fig.REF .",
"In general, errors are due to the leakage of high-level content into the noiseprint.",
"This happens especially in the presence of strongly textured areas, since the leaked regular patterns, with almost periodic structures, are misinterpreted as traces of an alien noiseprint.",
"A further critical case is given by very small images, with data too scarce to allow correct interpretation.",
"Finally, global image processing, like compression, resizing, and blurring, tend to reduce the noiseprint strength and hence impair the performance of subsequent steps.",
"Figure: Examples from all the test datasets.From left to right: forged image, ground truth, heatmaps from the six best performing methods: ADQ2, CAGI, NOI2, EXIF-SC, Splicebuster, Noiseprint.Note that some heatmaps are color-inverted, for example, the NOI2 map for the Nim.16 image.Figure: Noiseprint failure examples.",
"Top: original image, middle: ground truth, bottom: heatmap.Problems are mostly due to strongly textured regions, whose leaks in the noise residual are interpreted as an alien noiseprint."
],
[
"Further noiseprint-based forensic analyses",
"The main goal of this work was to present the noiseprint idea and its implementation.",
"Forgery localization was selected to prove the potential of this approach on a well-studied forensic problem, where plenty of reference methods and datasets are available.",
"It should be clear, however, that a strong camera model fingerprint can be used for many other applications in the forensic field.",
"This Section is meant to highlight some possible applications, something more than a “future work” list but less than a set of functioning and well studied algorithms.",
"A first obvious application is camera model identification.",
"So, we carried out a very basic source identification experiment, comparing the conventional PRNU-based method of [30] with the corresponding noiseprint-based method.",
"We used 3 different camera models (Nikon D70, Nikon D200 and Smartphone OnePlus) with 2 devices per camera.",
"Only the central 128$\\times $ 128-pixel crop of test images was used for identification.",
"For each device, 100 training images were used to estimate the ideal PRNU/noiseprint reference pattern by sample averaging.",
"Then, each image was attributed to the device whose reference pattern had minimum Euclidean distance w.r.t.",
"the noise residual.",
"Tab.REF shows the confusion matrices obtained in the two cases.",
"In terms of model identification, the noiseprint-based method provides 100% accuracy, to be compared with the 77% accuracy ensured by PRNU.",
"Of course, PRNU allows one to perform also device identification, with 70% accuracy.",
"Interestingly, for this latter task noiseprint provides a 62% accuracy, that is, the choice between the two devices of the same model is not entirely random, a fact that deserves further investigation.",
"Table: Confusion Matrices for camera identificationLet us now move to some unconventional attacks.",
"In Fig.REF we show, on the left, some images subject to seam carving [71], horizontal, vertical, or both, and, on the right, the corresponding noiseprint heatmaps obtained as described in Section IV.A.",
"In the heatmaps, traces of the inserted seams are clearly visible, allowing easy detection of the attack by a human observer.",
"Figure: Seam-carved images (left part) and corresponding noiseprint heatmaps (right part).Top-left: original image; diagonal: vertical/horizontal seam carving; bottom-right: vertical and horizontal seam carving.Image inpainting has seen huge progresses with the advent of deep learning.",
"A recently proposed [72] method based on generative adversarial networks has proven to produce results with a remarkably natural appearance in the presence of quite complex scenes.",
"In Fig.REF we show two such examples, together with the noiseprints extracted from inpainted images.",
"A visual inspection of the noiseprints (with suitable zoom) reveals a clear textural change in correspondence of the inpainted area.",
"Converting such information into an automatic algorithm for inpainting detection should be at hand.",
"We conclude this short review going back to image splicing.",
"In this case, however, the target images shown in Fig.REF , were acquired by sensors mounted on board a satellite, which has quite different characteristics w.r.t.",
"common camera sensors, and its peculiar processing chain.",
"Nonetheless, the associated noiseprints reveal quite clearly the manipulations, which are captured with great accuracy in the heatmaps.",
"Figure: Noiseprints of images inpainted by an advanced GAN-based method.Figure: In first line, there is a image of Washington by WorldView2 satellite (https://www.digitalglobe.com/resources/product-samples/washington-dc-usa).",
"In the second line a image of Caserta by Ikonos satellite."
],
[
"Conclusions",
"In this paper we proposed a deep learning method to extract a noise residual, called noiseprint, where the scene content is largely suppressed and model-related artifacts are enhanced.",
"Therefore, a noiseprint bears traces of the ideal camera model fingerprint much like a PRNU residual bears traces of the ideal device fingerprint.",
"In noiseprints, however, the signal of interest is much stronger than in PRNU residuals, allowing for the reliable accomplishment of many forensic tasks.",
"Experiments on forgery localization provide support to this statement, but many more forensic applications can be envisioned, which certainly deserve thorough investigation.",
"Despite the promising results, one must keep in mind that no tool can solve all forensic tasks by itself.",
"As an example, noiseprints will probably allow excellent camera model identification, but cannot help for device identification.",
"Therefore, the fusion of noiseprint-based tools with other approaches is a further topic of interest for future research."
],
[
"Acknowledgment",
"This material is based on research sponsored by the Air Force Research Laboratory and the Defense Advanced Research Projects Agency under agreement number FA8750-16-2-0204.",
"The U.S.Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.",
"The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory and the Defense Advanced Research Projects Agency or the U.S. Government.",
"The authors are grateful to Prof. David Doermann for precious discussions and for suggesting the name “noiseprint”."
]
] | 1808.08396 |
[
[
"Cech cover of the complement of the discriminant variety. Part I:\n Goresky--MacPherson stratification"
],
[
"Abstract Interest in Conformal Field Theories and Quantum Field Theory lead physicists to consider configuration spaces of marked points on the complex projective line, $Conf_{0,d}(\\mathbb{P})$.",
"We investigate the moduli space $\\mathcal{M}_{0,[d]}(\\mathbb{C})$, of Riemann surfaces of genus 0, with $n$ unordered marked points, in the framework of real geometry.",
"A decomposition of this moduli space, relying on Gauss drawings - objects being reminiscent of Grothendieck's dessin d'enfants, is presented.",
"The main result of the paper is that this decomposition forms a (real) topological stratification, which can be subtly modified so as to form a good Aleksandrov-Cech cover.",
"This approach provides a new very rich and detailed geometric description of this space, and in the panorama of results concerning $\\overline{\\mathcal{M}}_{0,[d]}(\\mathbb{C})$ and $\\overline{\\mathcal{M}}_{0,[d]}(\\mathbb{R})$, this approach fills in the gap between the two different methods employed to study the real and complex part of this moduli space.",
"The results can be extended to calculate explicitly the Aleksandrov-Cech cohomology of braids with values of any sheaf."
],
[
"Introduction",
"The decomposition of the space of configurations of $n$ marked points on the complex plane has been considered for over more than fifty years [24] and has lead to many important works , , , , , , , .",
"The braid group $\\mathcal {B}_n$ with $n$ strands and the space $\\mathop {{}^{\\textsc {D}}\\text{Pol}}_{n}$ of degree $n$ complex monic polynomials with distinct roots are objects which are deeply connected: the space $\\mathop {{}^{\\textsc {D}}\\text{Pol}}_{n}$ is a $K (\\pi , 1)$ of fundamental group $\\mathcal {B}_n$ .",
"The richness of their interactions allows each object to provide information on the other.",
"In particular, one may use the space $\\mathop {{}^{\\textsc {D}}\\text{Pol}}_{n}$ to give a description of braids with $n$ strands.",
"By means of a spectral sequence V. Arnold [1] gives a method to calculate the integral cohomology of the braid group.",
"An other version was given by D.B.",
"Fuks [16], which allows to have the cohomology of the braid group with values in $\\mathbb {Z}_{2}$ .",
"This latter approach was further on developed in a more general way by F.V.",
"Vainshtein [34] using Bockstein homomorphisms.",
"Many other results followed more recently, namely by V. Lin, F. Cohen [7] and A. Goryunov [21].",
"In very close subjects we can also cite works of De Concini, D. Moroni, C. Procesi, M. Salvetti [11], [2].",
"In spite of an abundant literature concerning this subject, we give a new approach to the configuration space of $n$ marked points on $\\mathbb {C}$ and thus on the moduli space $\\overline{\\mathcal {M}}_{0,n}$ of $n$ marked points on the Riemann sphere $\\mathbb {P}^1$ .",
"We develop in this paper the real-geometry on the canonical stratifications of the spaces $\\overline{\\mathcal {M}}_{0,n}$ , which gives a new insight on [10], [12], [25], [26], [28].",
"To be more precise, the aim is to present a real-algebraic stratification of the parametrizing $\\mathbb {C}$ –scheme of $\\overline{\\mathcal {M}}_{0,n}$ .",
"We show that it is a Goresky–MacPherson stratification.",
"This highlights real-geometry properties of $\\overline{\\mathcal {M}}_{0,n}$ and thus gives a different approach to this object.",
"The notion of Goresky–MacPherson stratification can be found in [18], [19], [20].",
"Let us recall the notion of stratification.",
"A stratification of a topological space $X$ is a locally finite partition of this space $\\mathcal {S}=(X^{(\\sigma )})_{\\sigma \\in S}$ of $X$ into elements called strata, locally closed and verifying the condition: for all $\\tau ,\\sigma \\in S,$ $X^{(\\tau )}\\cap \\overline{X}^{(\\sigma )}\\ne \\emptyset \\iff X^{(r)}\\subset \\overline{X}^{(\\sigma )}.$ The boundary condition can be reformulated as follows: the closure of a stratum is a union of strata.",
"A different interpretation of this would be to define a stratification as a filtration $X=X_{(n)}\\supset X_{(n-1)}\\supset \\dots \\supset X_{(1)}\\supset X_{(0)}\\supset X_{-1}=\\emptyset ,$ where the $X_{(i)}$ are closed, and where by defining $X^{(i)}:=X_{(i)}\\setminus X_{(i-1)},$ we have $\\overline{X^{(i)}}=X_{(i)}$ .",
"A Goresky–MacPherson stratification is defined as follows.",
"An $n$ -dimensional topological stratification of a paracompact Hausdorff topological space $X$ is a filtration by closed subsets $X=X_{(n)}\\supset X_{(n-1)}\\supset \\dots \\supset X_{(1)}\\supset X_{(0)}\\supset X_{-1}=\\emptyset ,$ such that for each point $p\\in X_{(i)}-X_{(i-1)}$ there exists a distinguished neighborhood $N$ of $p$ in $X$ , a compact Hausdorff space $L$ with an $n-i-1$ dimensional topological stratification $L=L_{n-i-1}\\supset \\dots \\supset L_1\\supset L_0\\supset L_{-1}=\\emptyset $ and a homeomorphism $\\phi :\\mathbb {R}^i\\times cone^\\circ (L)\\rightarrow N,$ which takes $\\mathbb {R}^i \\times cone^{\\circ } (L_j) \\rightarrow N\\cap X_{i+j+1}$ .",
"The symbol $cone^{\\circ } (L_j)$ denotes the open cone, $L\\times [0,1)/(l,0)\\sim (l^{\\prime },0)$ for all $l, l^{\\prime }\\in L$ .",
"Focusing on the case where the marked points on $\\mathbb {C}$ are pairwise distinct, we show that from this stratification can be constructed a good Čech cover.",
"This paper includes the explicit construction of the open sets of the Čech cover.",
"Definition 1 (Čech cover [4]) Consider a cover of a topological space $X$ .",
"A good Čech cover is a cover such that its components are open and have contractible multiple intersections.",
"We prove the following theorems.",
"Theorem 1 Consider the configuration space of $n$ marked points on the complex plane, where points are pairwise disjoint.",
"Then, there exists a real algebraic stratification $\\mathcal {S}$ of this space, where strata are indexed by oriented and bicolored forests verifying the following properties: there exist $n$ vertices of valency 4, incident to edges of alternating color and orientation; there exist at most $n-1$ vertices of even valency, incident to edges of only one color and of alternating orientations; there are $4n$ leaves (i.e.",
"vertices incident to one edge), where the colors and orientations of the incident edges alternate.",
"Theorem 2 Let $A_\\sigma $ be a generic stratum (i.e.",
"of codimension 0).",
"Then, the topological closure $\\overline{A}_\\sigma $ defines a Goresky–MacPherson stratification.",
"Thickening these strata prepares the ground for defining a good Čech cover, which leads to the following statement: Theorem 3 Consider the configuration space of $n$ marked points on the complex plane, where points are pairwise disjoint, and the stratification $\\mathcal {S}$ defined above.",
"Then, the thickened strata $\\underline{A_{\\sigma }}^+$ form a good Čech cover.",
"In a more global way, what we have in mind is a new manner of defining the generators for $\\Gamma _{0,[n]}$ , where $\\Gamma _{0,[n]}$ is nothing but the orbifold fundamental group of the moduli space of smooth curves of genus 0 with $n$ unordered marked points [22], [27].",
"This new cell decomposition turns out to have many interesting symmetries (this is the subject of the paper [9]), where we aim at underlining the existence of polyhedral symmetries explicitly in the presentation.",
"In this paper we show the existence of a good cover in the sense of Čech which we will use, in further investigations, to calculate explicitly the cohomology groups [8].",
"Namely, to stratify the configuration space $\\mathop {{}^{\\textsc {D}}\\text{Pol}}_n$ , we consider drawings of polynomials—objects reminiscent of Grothendieck's dessin's d'enfant in the sense that we consider the inverse image of the real and imaginary axis under a complex polynomial.",
"The drawing associated to a complex polynomial is a system of blue and red curves properly embedded in the complex plane, being the inverse image under a polynomial $P$ of the union of the real axis (colored red) and the imaginary axis (colored blue) [8].",
"For a polynomial of degree $d$ , the drawing contains $d$ blue and $d$ red curves, each blue curve intersecting exactly one red curve exactly once.",
"The entire drawing forms a forest whose leaves (terminal edges) go to infinity in the asymptotic directions of angle $k \\pi /4d$ .",
"Polynomials belong to a given stratum if their drawings are isotopic, relatively to the $4d$ asymptotic directions.",
"An element of the stratification is indexed by an equivalence class of drawings which we call a signature $\\sigma $ .",
"A stratum shall be denoted by $A_{\\sigma }$ .",
"Each stratum of this decomposition is attributed to a decorated graph (i.e.",
"a graph where edges are oriented and colored in red or blue; complementary faces to the embedded graph are colored in colors ($A,B,C,D$ ).",
"This paper is organized as follows.",
"The section 2, we present the definition and construction of the real-algebraic stratification.",
"We introduce the notion of drawings, signatures and present some of its properties.",
"In section 3 we introduce Whitehead moves on the signatures which allow a definition of the combinatorial closure of a signature.",
"An important result which follows from this section is that the topological closure of a stratum is given by its combinatorial closure.",
"In particular, it is the union of all the incident signatures to $\\sigma $ (i.e the combinatorial closure).",
"Finally, we prove that this stratification is a Goresky–MacPherson stratification.",
"Section 4 shows the construction of the Čech cover.",
"In particular, we define the components of the cover and prove that multiple intersections between those components are contractible.",
"Finally, an appendix is introduced, where we discuss the multiple intersections of closures of generic strata and classify the possible diagrams in order to have non-empty multiple intersections."
],
[
"Moduli spaces of genus 0 curves with marked points",
"Let $\\mathbb {C}$ be fixed field and $\\bf {T}$ a finite set.",
"Consider the moduli space (generally a stack) $\\overline{\\mathcal {M}}_{0,\\bf {T}}$ of stable genus zero curves with a finite set of smooth pairwise distinct closed points defined over $\\mathbb {C}$ and bijectively marked (labeled) by $\\bf {T}$ .",
"For a $\\mathbb {C}$ –scheme $B$ , a $B$ –family of such curves is represented by a fibration $C_B \\rightarrow B$ with genus zero fibres of dimension 1, endowed by pairwise distinct sections $s_i : B \\rightarrow C$ labeled by $\\bf {T}$ as above (cf.",
"[28]).",
"From the known complex results, the combinatorial type of each such curve over a closed point is encoded by a stable tree $\\sigma $ .",
"Leaves (or else, tails) of such a tree are labeled by the elements of a subset $\\bf {T}_\\sigma \\subseteq \\bf {T}$ .",
"It has been considered in [26] and later in [28] and [10] the spaces $\\mathcal {M}_{0,\\sigma }$ and their natural closures $M_{0,\\sigma } \\subset \\overline{\\mathcal {M}}_{0,\\sigma }.$ One point of $\\overline{\\mathcal {M}}_{0,n+1}(\\mathbb {C})$ “is” a stable complex curve of genus 0 with $n + 1$ points labeled by $\\lbrace 1,2,...,n-2, 0, 1, \\infty \\rbrace $ .",
"These points are distributed along different locally closed strata that are naturally labeled by stable trees.",
"The complex codimension of a stratum is the number of edges of the respective tree.",
"We will refer to this number as level.",
"Each stratum is reduced and irreducible, two different strata do not intersect.",
"At the level zero, all these pairwise different points live on a fixed $\\mathbb {P}^1$ endowed with a fixed real structure, with respect to which $(0,1,\\infty )$ are real.",
"Equivalently, this projective line is endowed with a fixed homogeneous coordinate system $(y : z)$ such that $0=(0:1)$ , $1=(1:1)$ , $\\infty =(1:0)$ ."
],
[
"Definition-construction",
"Given a stratification of $\\mathcal {S}=(X^{(\\sigma )})_{\\sigma \\in S}$ one can define an incidence relation on the set $S$ of indices by: $\\tau \\dot{\\preceq } \\sigma \\quad \\text{if and only if} \\quad X_{(\\tau )}\\cap \\overline{X}^{(\\sigma )}\\ne \\emptyset .$ In other words, in a stratification the incidence relation gives a poset."
],
[
"For the construction of this new stratification, it is very important to have a fixed real structure on $\\mathbb {P}^1$ . The main idea of our construction is to take the inverse image, under a polynomial {{formula:8b2210c7-a076-40cb-ba3d-64f98ab01224}} in {{formula:b147281a-15d1-4f79-a8b5-39dbb17bdb9c}} , of the real and imaginary axis, i.e. {{formula:772bd0bd-2f9f-4cac-b094-04165ef80481}} . This inverse image forms a system of oriented curves in the complex plane. To distinguish {{formula:4075f9b7-7e6a-4bd1-b0cc-f389d2e79c37}} from {{formula:1774ce51-7159-4921-8c34-c76e6001f1e7}} , we color in red the pre-image of {{formula:b17fd6e8-0eba-4897-ad66-a0e8ef3dbfab}} and in blue the pre-image of {{formula:04673fdf-0d71-4e18-a909-94f4997fa7b1}} . The orientation of the curves is inherited from the natural orientation on the real and imaginary axis (see Figure ",
"Definition 2 An $n$ -drawing $\\mathcal {C}_{P}$ of a degree $n$ -polynomial $P\\in \\mathop {{}^{\\textsc {D}}\\text{Pol}}_n$ is a system of curves properly embedded in the complex plane given by $P^{-1}(\\mathbb {R}\\cup \\imath \\mathbb {R})$ .",
"Figure: Partition of the complex planeUsing the coloring convention of the curves above, we have three families of intersections in a drawing: The roots of the polynomials given by the intersection points of a red curve with a blue curve.",
"The critical points $z_0$ of $P$ such that the critical value associated $P(z_0)$ is an imaginary, $Re(P)(z_0)=0$ , are given by the intersection points between the blue curves.",
"The critical points $z_0$ of $P$ such that the critical value associated to $P(z_0)$ is a real, $Im(P)(z_0)=0$ , are given by the intersection points between the red curves.",
"We define isotopy classes of drawings.",
"Definition 3 Two $n$ -drawings $\\mathcal {C}_{P_{1}}$ and $\\mathcal {C}_{P_{2}}$ are equivalent if and only if there exists an isotopy $h$ of $\\mathbb {C}$ (a continuous family of homeomorphisms of $\\mathbb {C}$ ), such that $h:\\mathcal {C}_{P_{1}}\\rightarrow \\mathcal {C}_{P_{2}}$ ; $h$ preserves the $4n$ asymptotic directions, the colouring and orientation of curves.",
"We denote the equivalence class of isotopic drawings by $[\\mathcal {C}_{P}]$ .",
"This definition serves to construct the decomposition of the configuration space of marked points on the complex plane.",
"Definition 4 Consider the space of configurations of $n$ marked points on $\\mathbb {C}$ .",
"Let $A_{[\\mathcal {C}_{P}]}$ be the set of polynomials with drawings in the isotopy class $[\\mathcal {C}_{P}]$ .",
"The family $(A_{[\\mathcal {C}_{P}]})_{[\\mathcal {C}_{P}] \\in S}$ , where $S$ is the set of isotopy classes of $n$ -drawings, partitions the configuration space.",
"The component $A_{[\\mathcal {C}_{P}]}$ is called stratum.",
"Definition 5 (Codimension) Let $P$ be a degree $n$ complex polynomial in $\\mathop {{}^{\\textsc {D}}\\text{Pol}}_n$ .",
"A polynomial $P$ with no critical values on $\\mathbb {R}\\cup \\imath \\mathbb {R}$ is called generic, such a polynomial is of codimension 0.",
"The special critical points of $P$ are the critical points $z$ such that $P(z) \\in \\mathbb {R}\\cup \\imath \\mathbb {R}$ .",
"The local index at a special critical point $z\\in \\mathbb {R}$ (resp.",
"$ \\imath \\mathbb {R}$ ) is equal to $2m-3$ , where $m$ is the number of red (or blue) diagonals crossing at the point $z$ .",
"The real codimension of $P$ is the sum of the local indices of all the special critical points.",
"Example 1 Figure REF illustrates a generic (codimension 0) polynomial of degree 6.",
"Figure: P(z)=(z+1.54*(0.98+i*0.2))*(z+0.68*(0.98+i*0.2))*(z+0.4*(0.98+i*0.2))*(z-1.54*(0.98+i*0.2))*(z-0.68*(0.98+i*0.2))*(z-0.4*(0.98+i*0.2)).P(z)= (z+1.54*(0.98+i*0.2))*(z+0.68*(0.98+i*0.2))*(z+0.4*(0.98+i*0.2))*(z-1.54*(0.98+i*0.2))*(z-0.68*(0.98+i*0.2))*(z-0.4*(0.98+i*0.2))."
],
[
"Embedded forests",
"Consider the category $\\Gamma $ of graphs.",
"Objects of this category are graphs, i.e.",
"objects defined by a set $V$ of elements called vertices, equipped with a symmetric, reflexive, relation $E$ .",
"For given vertices $x,y\\in V$ there is an edge $(x,y)$ if and only if $(x,y)\\in E$ and $x\\ne y$ .",
"We define a morphism $(V,E)\\rightarrow (W,F)$ as a function $f: V\\rightarrow W$ such that $(x,y)\\in E$ implies $(f(x),f(y))\\in F$ .",
"We equip this category $\\Gamma $ with a coproduct $\\sqcup $ , being here the disjoint union of graphs.",
"In the category $\\Gamma $ , we are interested only in those graphs being trees, i.e.",
"acyclic connected graphs.",
"By convention, a tree has at least one edge.",
"Vertices of valency one are called leaves.",
"We equip the standard definition of trees with a colouring and an orientation on the set of edges.",
"Each edge is mapped to the set $\\lbrace R^{+},B^{-},B^{+},R^{-}\\rbrace $ , where the capital letter $R$ (resp.",
"$B$ ) corresponds to a red (resp.",
"blue) colouring and the sign \"+\" or \"-\" corresponds to the orientation of the edge.",
"We will write $R^{-}$ (or $B^{-}$ ) if at a given vertex, the direction points inwards, i.e.",
"towards the chosen vertex.",
"Similarly, if we have the opposite sign, then the direction points in the opposite direction i.e.",
"outwards.",
"Concerning the notation, we use the symbol $(i,j)_{R}$ (resp.",
"$(i,j)_{B}$ ) for a red (blue) diagonal in the forest, connecting the leaf $i$ to the leaf $j$ .",
"We define the following object.",
"Definition 6 An $n$ -signature is an object of the category $(\\Gamma ,\\sqcup )$ being a disjoint union of trees (i.e.",
"a forest) such that: There exist $k$ vertices of valency four.",
"The incident edges are in bijection with the set $\\lbrace R^{+},B^{-},B^{+},R^{-}\\rbrace $ .",
"Orientation and colors alternate.",
"There exist $n-k$ vertices of valency a multiple of four.",
"The incident edges are in bijection with the set $\\lbrace R^{+},B^{-},B^{+},R^{-}\\rbrace $ .",
"Orientation and colors alternate.",
"There exist at most $k-1$ vertices of even valency (at least 4), incident to edges of the same color.",
"Orientations alternate.",
"There exist $4n$ leaves.",
"The set of incident edges to those leaves are of alternating color and orientation.",
"Note that for the case of pairwise distinct points on the complex plane we have exactly $n$ vertices incident to four edges of alternating colors/orientations and at most $n-1$ vertices incident to monochromatic edges.",
"A geometric realisation of a signature is a (union of) 1-dimensional CW-complex, being contractible, and properly embedded in the complex plane.",
"It is important for the construction presented in this note, to insist on the difference between the geometric realisation of a given graph and the graph itself, which is a purely combinatorial notion.",
"An embedded forest is a subset of the plane which is the image of a proper embedding of a forest minus leaves to the plane."
],
[
"From real curves to forests",
"An equivalent way of describing the orientation on the edges on the graphs is to consider the complementary part to the real and imaginary axis in $\\mathbb {C}$ and to color them.",
"Let us label the quadrants in the colors $A,B,C,D$ as in the figure REF .",
"Theorem 4 The set of $n$ -signatures is in bijection with the set of isotopy classes of drawings relatively to the $4n$ asymptotic directions.",
"One direction is easy.",
"The other direction is proved as follows.",
"Let $\\gamma $ be the embedding in $\\mathbb {C}$ of a signature $\\sigma $ , which satisfies the asymptotic directions.",
"Then, we can construct a function $f$ such that: $f: \\mathbb {C}\\rightarrow \\mathbb {C}$ is a smooth function, $f^{-1}(\\mathbb {R}\\cup \\imath \\mathbb {R})=\\gamma $ The function $f$ is a bijection between each region of $\\gamma $ and a quadrant colored $A, B, C, D$ in the complex plane, $f$ is injective and regular on the edges of $\\gamma $ .",
"Let $J$ be the standard conformal structure on $\\mathbb {C}$ .",
"We have $J= f_{*}(J_{0})$ so $f:(\\mathbb {C},J)\\rightarrow (\\mathbb {C},J_{0})$ is holomorphic.",
"By Riemann’s mapping theorem there exists a biholomorphic function $\\rho : (\\mathbb {C}, J ) \\rightarrow (\\mathbb {C}, J_0)$ .",
"The classical theorem of complex analysis by Rouché implies that $f \\circ \\rho ^{-1}$ is a polynomial.",
"Lemma 1 Let $\\gamma \\in [\\mathcal {C}_{P}]$ , then $\\gamma $ is a forest.",
"Consider an embedded forest $F_R$ (resp.",
"$F_B$ ) with $2n$ leaves, whose edges are colored in $R$ (resp.",
"B) and such that all vertices in the plane are of even valencies.",
"Then, according to theorem 1.3 in [15], there exists a harmonic polynomial of degree $n$ whose zero set is equivalent to $F_R$ (resp.",
"$F_B$ ).",
"The class $[\\mathcal {C}_{P}]$ is a forest, since it is easy to see that the level set of the degree $d$ harmonic polynomials $\\lbrace (x,y)\\in \\mathbb {R}^2: Re P(x,y)=0\\rbrace $ (resp.",
"$\\lbrace (x,y)\\in \\mathbb {R}^2: ImP(x,y)=0\\rbrace $ ) is an embedded forest (the existence of a cycle would contradict the maximum principle).",
"Theorem 5 Let $\\sigma $ be an $n$ -signature.",
"Then, the set $A_{\\sigma }$ is contractible.",
"We need to use a theorem of J. Cerf, which we recall.",
"Let $V$ be a manifold with boundary, compact; $W$ is a \"target\" manifold and $Pl(V,W)$ is the space of smooth embeddings of $V$ into $W$ .",
"Theorem 6 (Cerf, theorem 3, [6]) Let $G$ be a subgroup of the group of all diffeomorphisms of $V$ , where $G$ acts on $Pl(V,W)$ and determines a structure of principal bundle.",
"If $G$ is open, then the canonical map: $Pl(V,W)\\rightarrow Pl(V,W)/G$ is a locally trivial fibration.",
"For an exposition on principal bundles, we refer to [3].",
"We recall a lemma (cited as lemma 1 in [6]), useful for the application of this result to our case.",
"Lemma 2 Let $(E,B, p)$ be a fiber bundle where $E$ ans $B$ are topological spaces, $p$ is a continuous map $p: E \\rightarrow B$ .",
"A sufficient condition to have a locally trivial fibration is that for $x_{0}\\in B$ there exists a topological group $G$ operating (left wise) on the total space $E$ and on the base space $B$ so that the following diagram is commutative: $\\begin{tikzcd}G \\times E {r}{\\phi } [swap]{d}{Id \\times p} & E {d}{p} \\\\G \\times B {r}{\\Phi } & B\\end{tikzcd}$ and the map $g \\rightarrow g x_{0}$ from $G$ into $B$ has a continuous section in the neighborhood of $x_{0}$ .",
"Remark 1 Let us comment on this result.",
"Let $\\gamma $ be the embedding in $\\mathbb {C}$ of a signature $\\sigma $ , i.e.",
"$\\gamma \\in Pl(\\sigma ,\\mathbb {C})$ which satisfies the asymptotic directions.",
"Let $T$ be a tubular neighbourhood closed of $\\gamma $ in $\\mathbb {C}$ and denote by $H$ the group of diffeomorphisms, inducing the identity on $\\mathbb {C}-T$ .",
"From REF , it is shown in [6] that there exists a neighborhood $\\mathcal {V}$ of $\\gamma $ in $Pl(\\sigma ,\\mathbb {C})$ and a continuous map $s: \\mathcal {V} \\rightarrow H$ such that : $s(\\gamma )=e;$ For all $\\gamma ^{\\prime }\\in \\mathcal {V}$ , $s(\\gamma ^{\\prime })\\gamma ^{\\prime }$ is identified with a diffeomorphism of $\\sigma $ in the neighborhood of the identity, which is of the form $\\gamma g$ where $g$ is a diffeomorphism of $\\sigma $ .",
"$\\forall \\gamma ^{\\prime }\\in \\mathcal {V}$ and any diffeomorphism $g$ of $\\sigma $ such that $\\gamma ^{\\prime } g\\in \\mathcal {V}$ , $s(\\gamma ^{\\prime } g)=s(\\gamma ^{\\prime })$ We will mainly use the Cerf theorem and the remark REF , above.",
"The embedded graphs of $\\sigma $ in $\\mathbb {C}$ correspond to a class of drawings.",
"We know that this class is contractible, by theorems 5.3 and 5.4 of Epstein in [14].",
"For each $\\gamma $ in the neighborhood $V$ , we can construct the function $f$ , as mentioned above in theorem REF .",
"Define $E_{\\gamma }$ to be the space of those functions corresponding to the embedded graph $\\gamma $ , and verifying the properties described in the poof of theorem REF .",
"This space is contractible.",
"Define the Cerf fibration $E_\\sigma =\\lbrace (\\gamma ,f) | \\gamma \\in \\sigma ,f \\in E_{\\gamma }\\rbrace \\rightarrow \\sigma .$ $(\\gamma ,f )\\mapsto \\gamma $ By the result of Cerf in Theorem REF , we have a structure of principal bundle on $E_\\sigma \\rightarrow E_\\sigma /G$ , where $G$ is the group of diffeomorphisms $G$ acting on $\\mathbb {C}$ , which are homotopic to the identity and preserving orientation.",
"This group is contractible, by the theorem of Earle and Eells [13].",
"Now, by a theorem of Cerf [5] the total space $E_{\\sigma }$ is contractible.",
"Note that $G$ acts without fixed points on $E$ and that orbits are closed.",
"Using the Riemann mapping theorem and the classical Rouché theorem, implies that each class modulo $G$ has a representent which is a complex polynomial."
],
[
"Stratification on $\\overline{\\mathcal {M}}_{0,n}$",
"Let $\\tau $ be the (labeled) graph of a stable connected curve with $n$ marked points $C_B$ .",
"Denote by $M_{0, {\\tau }} \\subset \\overline{\\mathcal {M}}_{0,n}$ the moduli submanifold (or generally, substack) parametrizing all curves having the same graph ${\\tau }$ .",
"Its closure will be denoted $\\overline{\\mathcal {M}}_{0,\\tau }$ .",
"A morphism $f:\\, {\\tau } \\rightarrow {\\sigma }$ is determined by its covariant surjective action upon vertices $f_v:\\, V_{{\\tau }}\\rightarrow V_{ {\\sigma }}$ and contravariant injective actions upon tails and edges: $f^t:\\, \\bf {T}_{{\\sigma }}\\rightarrow \\bf {T}_{\\tau }, \\quad f^e:\\, E_{\\sigma }\\rightarrow E_{\\tau }.$ Geometrically, $f$ contracts edges from $E_{{\\bf \\tau }}\\setminus f^e(E_{{\\sigma }})$ and tails from $\\bf {T}_{{\\tau }}\\setminus f^t(\\bf {T}_{{\\sigma }})$ , compatibly with its action upon vertices.",
"Similarly, given a stratum $\\overline{D}( {\\tau })$ in $B$ , its closure is formed from the union of subschemes $D({\\sigma })$ , such that ${\\tau }>{\\sigma }$ and, where ${\\tau }$ and ${\\sigma }$ , have the same set of tails.",
"In our case, where the genus of the curve is zero, the condition ${\\tau }> {\\sigma }$ is uniquely specified by the splitting data, which can be described as a certain type of Whitehead move.",
"The splitting data is as follows.",
"Choose a vertex $v$ of the set of vertices of $\\tau $ and a partition of the set of flags, incident to $v$ : $F_{{\\tau }}(v)=F^{\\prime }_{{\\tau }}(v)\\cup F^{\\prime \\prime }_{{\\tau }}(v)$ such that both subsets are invariant under the involution $j_{\\tau }: F_{{\\tau }}\\rightarrow F_{{\\tau }}$ .",
"To obtain $\\sigma $ , replace the vertex $v$ by two vertices $v^{\\prime }$ and $v^{\\prime \\prime }$ connected by an edge $e$ , where the flags verify $F^{\\prime }_{{\\tau }}(v^{\\prime })=F^{\\prime }_{{\\tau }}(v)\\cup \\lbrace e^{\\prime }\\rbrace $ , $F^{\\prime }_{{\\tau }}(v^{\\prime \\prime })=F^{\\prime \\prime }_{{\\tau }}(v)\\cup \\lbrace e^{\\prime \\prime }\\rbrace $ , where $e^{\\prime },e^{\\prime \\prime }$ are the two halves of the edge $e$ .",
"The remaining vertices, flags and incidence relations stay the same for ${\\tau }$ and $ {\\sigma }$ .",
"For more details, see [28] ch.III $§$ 2.7, p.90.",
"Finally, the scheme $\\overline{\\mathcal {M}}_{0,n}$ is decomposed into pairwise disjoined locally closed strata, indexed by the isomorphism classes of $n$ -graphs.",
"Such as depicted in [28] ch.III $§3$ , the stratification of the scheme is given by trees.",
"If the set $\\bf {T}$ is finite, then $\\mathcal {T}((\\bf {T}))$ is the set of isomorphism classes of trees ${\\tau }$ , whose external edges are labeled by the elements of $T$ .",
"The set of trees is graded by the number of edges: $\\mathcal {T}((\\bf {T}))=\\bigcup _{i=0}^{|T|-3}\\mathcal {T}_{i}(({\\bf T})), $ where $\\mathcal {T}_{i}((\\bf {T}))$ is a tree with $i$ edges.",
"The tree $\\mathcal {T}_{0}((\\bf {T}))$ is the tree with one vertex and the set of flags equals to $\\bf {T}$ ."
],
[
"Real stratification of the parametrizing space",
"Compared to the previous exposition, some additional material is necessary to define the real stratification of $\\overline{\\mathcal {M}}_{0,n}$ .",
"Bridging strata between each other requires the introduction of Whitehead-moves.",
"Let us discuss the case of a forest of one color.",
"This forest has $2d$ leaves and vertices of even valency.",
"Suppose that in this forest there exist $m\\ge 2$ one-edged trees, occurring as part of the boundary of a given cell.",
"Add one vertex (different from the leafs) to each of the trees and a polygon, joining these vertices.",
"Contract this polygon to a point.",
"This contracting morphism gives a new tree, locally star-like.",
"We call this a complete contracting Whitehead move.",
"This operation also holds in the case of a tree where two vertices are connected to each other by an edge.",
"In this situation, the edge can be contracted, leaving only one vertex, exactly as in the contracting morphism discussed above.",
"We call this a partial contracting Whitehead move.",
"The opposite of this operation exists too.",
"It is called a smoothing operation.",
"Geometrically speaking, a smoothing is applied to the vertex (not a leaf) of a tree having $m\\ge 2$ edges.",
"This smoothing is obtained by ungluing those edges, giving $m$ one-edged trees.",
"This is called a complete smoothing Whitehead move.",
"In this smoothing process, we include also the splitting operation above, i.e.",
"for a given tree with $m\\ge 3$ edges, replace one vertex by two vertices connected by an edge, such that the valency of the new vertices remains even, and that the condition for flags holds (where ”flags” are replaced here by incident half-edges).",
"This is called the partial smoothing Whitehead move.",
"Figure: Example of a complete contracting Whitehead moveFigure: Example of a partial contracting Whitehead moveWe now introduce the deformation retract lemma.",
"Lemma 3 (Deformation retract) Consider a signature $\\sigma $ .",
"Suppose that in $\\sigma $ there exist edges of $m \\ge 2$ trees, occurring in the boundary of a cell of the plane $\\mathbb {C}$ .",
"Applying a complete contraction Whitehead move onto these edges corresponds to a deformation retract of $\\sigma $ onto a new signature $\\tau $ .",
"To prove this lemma, we essentially need some notions from Morse theory (see [29]).",
"From the Morse lemma we know the following.",
"Let $p$ be a non-degenerate critical point of $f : M \\rightarrow \\mathbb {R}$ .",
"Then there exists a chart $(x_1, x_2, \\dots , x_n)$ in a neighborhood $U$ of $p$ such that $f(x)=f(p)-x_{1}^{2}-\\cdots -x_{\\alpha }^{2}+x_{\\alpha +1}^{2}+\\cdots +x_{n}^{2}$ throughout $U$ .",
"Here $\\alpha $ is equal to the index of $f$ at $p$ .",
"A corollary of the Morse lemma, is that non-degenerate critical points are isolated.",
"A smooth real-valued function on a manifold $M$ is a Morse function if it has no degenerate critical points.",
"Let $M^a=f^{-1}(-\\infty , a]=\\lbrace x\\in M: f(x)\\le a\\rbrace $ .",
"We recall two important results.",
"Theorem 7 (Milnor, Morse theory, Theorem 3.1, [29]) Let $M$ be a differentiable manifold.",
"Suppose $f$ is a smooth real-valued function on $M$ , $a < b$ , $f^{-1}[a, b]$ is compact, and there are no critical values between $a$ and $b$ .",
"Then $M^a$ is diffeomorphic to $M^b$ , and $M^b$ deformation retracts onto $M^a$ .",
"Theorem 8 (Milnor, Morse theory, Theorem 3.2, [29]) Suppose $f$ is a smooth real-valued function on $M$ and $p$ is a non-degenerate critical point of $f$ of index $k$ , and that $f(p) = q$ .",
"Suppose $f^{-1}[q - \\epsilon , q + \\epsilon ]$ is compact and contains no critical points besides $p$ .",
"Then $M^{q+\\epsilon }$ is homotopy equivalent to $M^{q-\\epsilon }$ with a $k$ -cell attached.",
"We are now able to prove that the lemma, in order to show that a Whitehead contraction step is a deformation retract.",
"Consider the simplest case i.e.",
"when we have two curves of the same color lying in a cell of $\\mathbb {C}$ and after the contracting Whitehead move, intersecting.",
"Applying the Morse lemma, we take a coordinate system $x,y$ in a neighborhood $U$ of a critical point $p$ , such that $f(b)=c$ and so that the identity $f=c-x^2+y^2$ holds throughout $U$ and the critical point $p$ will have coordinates $x(p)=y(p)=0$ .",
"Choose $\\epsilon >0$ , sufficiently small so that: the region $f^{-1}[c-\\epsilon , c+\\epsilon ]$ is compact and contains no critical point other than $p$ .",
"The image of $U$ under the diffeomorphic embedding $(x,y): U\\rightarrow \\mathbb {R}^2$ contains the closed ball $\\lbrace (x,y): x^2+y^2\\le 2\\epsilon \\rbrace $ .",
"Coordinate lines are $x=0, y=0$ ; the hyperbolas represent $f^{-1}(c-\\epsilon )$ and $f^{-1}(c+\\epsilon )$ .",
"A slight modification of the proof of theorem REF in [29], and using the Morse function defined above, shows that the set $M^c$ is a deformation retract of $M^{c+\\epsilon }$ (reciprocally, a deformation retract of $M^{c-\\epsilon }$ ).",
"The generalisation for more than two curves intersecting follows from a modification of the function $f$ to higher degrees than 2.",
"Remark 2 Using the Morse theory two previous results i.e.",
"Theorems REF and REF , and the fact that there exists a Morse function on any differentiable manifold, one can prove that any differentiable manifold is a CW complex with an $n$ -cell for each critical point of index $n$ .",
"Figure: Contracting Whitehead moves onto the middle figureLemma 4 Consider a tree (or a subtree of a tree) containing a vertex which is of valency greater or equal to 8; incident edges having of alternating colors.",
"The smoothing Whitehead move applied to this vertex forms a Milnor fiber.",
"This Milnor fiber is homotopy equivalent to a circle.",
"To prove this let us note the following.",
"Milnor introduced the Milnor fibration for any holomorphic germ $(X_0,0)$ in $\\mathbb {C}^N$ and proved that the Milnor fiber is always a CW-complex of dimension at most $(n-1)$ (see [30]).",
"This tool can be used when $X_0$ is smoothable.",
"Let us recall this notion.",
"Consider an open ball in $\\mathbb {C}^N$ with centre zero; a small disk $ in $ C$ with centre zero and a closed subspace $ X$ of $ B.",
"The flat holomorphic mapping $f:X \\rightarrow is a smoothing of $ X0$ (which is the restriction to $ X$ of the projection $ p: B) when the pre-image of 0 under $f$ is the germ ($X_0,0)$ and the pre-image $f^{-1}(t)$ is smooth for $t \\in different from zero.In the case in which $ f$ has an isolated singularity at the origin, Milnor proved that the Milnor fiber is homotopy equivalent to a bouquet of $ (n-1)$-spheres.",
"The number of spheres is equal to the Milnor number, which is given by $ dim(K[[x1,...,xN]]/jacob(f)),$ where $ jacob(f)$ is the ideal generated by the partials of $ f$.$ This case appears precisely when the polynomial in $\\mathop {{}^{\\textsc {D}}\\text{Pol}}_n$ has a root of multiplicity 2, i.e.",
"when we have a pair of colliding points in the configuration space.",
"This corresponds to a germ of the type $(z^2,0)$ .",
"Now, here $N=1$ and the Milnor number is 1.",
"Applying Milnor's result, the fiber, for $t\\ne 0$ , is equivalent to a circle.",
"Remark that, for this lemma the Milnor fiber is locally illustrated by the drawings in Figure REF ."
],
[
"Some properties of the stratification",
"Let $P\\in \\mathop {{}^{\\textsc {D}}\\text{Pol}}_{d}$ , explicitly: $P(z)=z^{n}+a_{n-1} z^{n-1}+\\ldots + a_0$ .",
"We denote the critical points by ${\\underline{r}}=(r_1,\\ldots ,r_n)$ ($P^{\\prime }(r_i)=0$ ) and the critical values by ${\\underline{v}}=(v_1,\\ldots ,v_{n-1})$ , so that for any $i$ there is a $j$ such that $P(r_i)=v_j$ .",
"There are no constraints on the $r_i$ 's and $v_j$ 's other than $v_j\\ne 0$ (for all $j$ ) because the roots of $P$ are assumed to be simple.",
"Let $\\mathcal {V}_n$ denote the (affine) space of the ${\\underline{v}}$ 's and recall that there is a ramified cover $\\pi _w :\\mathop {{}^{\\textsc {D}}\\text{Pol}}_{n} \\rightarrow \\mathcal {V}_n$ of degree $(n)^{n-1}$ [31].",
"Let $\\mathbb {C}_r^n$ denote the affine space of the critical points ${\\underline{r}}$ and let $p : \\mathbb {C}_r\\rightarrow \\mathbb {C}_v$ denote the natural map given by $P$ ($P(r)=v)$ .",
"Denote by $c(v_k)$ (or just $c(k)$ ) the number of distinct critical points above $v_k$ and by ${\\underline{m}}=(m_1,\\ldots ,m_{c(k)})$ their multiplicities ($m_i>1$ ).",
"Generically, above a point $v$ there is just one critical point of multiplicity 2 and $n-1$ simple non critical points, i.e.",
"$c(v)=1$ , $m_1=2$ .",
"We call such a point, a simple critical point.",
"Lemma 5 Suppose $P$ is a polynomial with signature $\\tau $ and a critical point of multiplicity $m$ at $z_0$ , and let $\\sigma $ be the signature obtained by smoothing $\\tau $ .",
"Then $A_\\tau \\subset \\overline{A}_\\sigma $ and there exists a neighborhood $U$ of $P$ in $A_\\sigma $ such that $U\\cong V \\times (\\mathbb {D})^m$ , where $V$ is a neighborhood of $p\\in A_\\sigma $ and the polydisk $(\\mathbb {D})^m$ corresponds to a canonical local perturbation of $(z - z_0)^m$ .",
"Let us work in the fiber given by $p_w$ .",
"Now, return to a given polynomial $P$ and its critical values $\\mathcal {V}_n$ .",
"It is generic if neither $v_j^2\\notin \\mathbb {R}$ nor $v_j^2\\notin \\imath \\mathbb {R}$ for all $j$ .",
"Assume it is not generic and let $v_0$ ($=v_j$ for some $j$ ) be such that $v_0^2\\in \\mathbb {R}$ ; purely for notational simplicity, we suppose that $v_0$ is real (rather than pure imaginary).",
"Let $c=c(v_0)$ be the number of distinct critical points above $v_0$ , with multiplicities ${\\underline{m}}=(m_1,\\ldots ,m_{c})$ and denote these points $(r_1, \\ldots , r_c)$ (here the $z_i$ 's are distinct).",
"So, choose one of the $r_i$ ($1\\ne i\\ne c$ ), and call it $r_0$ , with multiplicity $m=m_0$ .",
"First, this is shown via holomorphic surgery and we cannot explicitly write down a universal family in terms of the coefficients $a_i$ of the polynomial $P$ .",
"However, we do know that there exists such a local universal family and that it is biholomorphically equivalent to the one obtained by completing the polynomial $(z-r_0)^m$ into the generic polynomial in $(z-r_0)$ of degree $m$ near $r_0$ (for instance, see [35], chapter 2, paragraph 3 for more details).",
"In other words, let ${\\underline{\\varepsilon }}=({\\varepsilon }_1,\\ldots , {\\varepsilon }_m)$ be complex numbers with $\\vert {\\varepsilon }_i \\vert <{\\varepsilon }<\\!< 1$ for all $i$ (with some ${\\varepsilon }>0$ ) and let $p_{\\varepsilon }(z) = (z-r_0)^m+{\\varepsilon }_1(z-r_0)^{m-1}+\\ldots +{\\varepsilon }_m \\ .$ Then, there is a biholomorphic map between the set of the ${\\underline{\\varepsilon }}$ 's (i.e.",
"$(0,{\\varepsilon })^m$ ), or equivalently the family $p_{\\varepsilon }$ , and a universal family $P_{\\varepsilon }(z)$ with $P_0=P$ (where $0=(0,\\ldots , 0)$ ).",
"By varying ${\\underline{\\varepsilon }}$ , let the critical point vary into $r_0({\\varepsilon })$ , leading to a deformation $v_0({\\varepsilon })$ .",
"The $v_0({\\varepsilon })$ has non zero derivative at ${\\varepsilon }=0$ (otherwise, the polynomial would have multiple roots, which we exclude).",
"By the implicit function theorem $v_0({\\varepsilon })$ covers a neighborhood of $v_0$ in the $v$ -plane and a Whitehead move is nothing but the results of what happens when $v_0({\\varepsilon })$ moves into the upper or lower half-plane.",
"Lemma 6 Strata in the topological closure $\\overline{A_{\\sigma }}$ are indexed by signatures obtained from a contracting Whitehead move on $\\sigma $ .",
"Let us suppose first that we apply on a generic signature $\\sigma $ a complete contracting Whitehead.",
"Let $P$ be a polynomial in $A_{\\sigma }$ and $\\tilde{P}$ a polynomial in $A_{\\tilde{\\sigma }}$ , where $\\sigma \\prec \\tilde{\\sigma }$ .",
"We first use the Whitehead move of first type.",
"A contracting Whitehead move on $\\sigma $ corresponds to a path of the critical values of $P$ in the space of critical values.",
"We will show that by using a contracting Whitehead move on $\\sigma $ this corresponds to defining a convergent sequence of critical values in $ \\mathbb {C}_w^{n-1}$ .",
"In $\\tilde{\\sigma }$ the intersection point of a set of $m$ curves of the same color lies on a critical point $c_i$ of $\\tilde{P}$ .",
"Suppose that there exist $I$ (where $|I|<d$ ) such intersections.",
"Therefore, for $i\\in I$ we have $\\tilde{P}^{\\prime }(r_i)=0$ and $Re(\\tilde{P}(r_i))=0$ (resp $Im(\\tilde{P}(r_i))=0$ ) and the critical value $\\tilde{P}(r_i)=v_i \\in \\imath \\mathbb {R}$ (resp.",
"$\\tilde{P}(r_i)=v_i \\in \\mathbb {R}$ ).",
"So, this indicates a sequence of critical values converging to $(v_1,....,v_{n-1})$ , where the subset $v_i, i \\in \\lbrace 1,...,n-1\\rbrace $ lies on the imaginary (resp.",
"real) axes.",
"Hence it indicates a topological closure.",
"Consider the case of a partial contracting Whitehead move.",
"In this case, the initial signature $\\sigma $ has a set of critical values lying on the real or imaginary axis and the partial contracting Whitehead operation merges a subset of those critical values together.",
"A partial contracting Whitehead move corresponds to a converging sequence of critical values in the space $\\mathbb {C}_v^{n-1}$ and hence it indicates a topological closure of the stratum $A_{\\sigma }$ ."
],
[
"Critical values",
"Consider the space $V_{n}=(\\mathbb {C}^{n-1}\\setminus 0)/S_{n-1}$ , where $S_{n-1}$ is the group of permutations.",
"If $X$ denotes an equivalence class of points in $\\mathbb {C}^{n-1}$ , we can associate a unique $\\sigma $ -sequence $(a,b,c,d,e,f,g,h)$ of positive integers to $X$ enumerating the number of points in $X$ in the quadrants $A,B,C,D$ and on the semi-axes.",
"The set of points $X$ in $V_{n}$ having a given $\\sigma $ -sequence $(a,b,c,d,e,f,g,h)$ forms a polygonal cell in $V_{d}$ isomorphic to $A^{a} /S_{a} \\times B^{b}/ S_{b}\\times C^{c}/ S_{c} \\times D^{d}/ S_{d} \\times (\\mathbb {R}^{+})^{e} /S_{e} \\times (\\mathbb {R}^{-})/ S_{f}\\times (\\imath \\mathbb {R}^{+})^{g}/ S_{g} \\times (\\imath \\mathbb {R}^{-})^{h}/ S_{h}.$ The real dimension of this cell is equal to $2(a+b+c+d)+ (e+f+g+h)$ .",
"The cells are disjoint and thus form a stratification of $V_{n}$ .",
"Definition 7 A subset $V$ inside $\\mathbb {C}^{n-1}/S_{n-1}$ for $n>2$ is said to be a $\\mathbf {non-compact\\ \\ stratification}$ if it is equipped with a stratification by a finite number of open cells of varying dimensions having the following properties: the relative closure of a $k$ -dimensional cell of $V$ is a union of cells in the stratification.",
"the relative closure of a $k$ -dimensional cell of $V$ is a “semi-closed” polytope, i.e.",
"the union of the interior of a closed polytope in $\\mathbb {C}^{n-1}/S_{n-1}$ with a subset of its faces.",
"Lemma 7 Let $n>2$ , and let $\\mathcal {V}_{n}$ denote the space $V_{n}$ equipped with the stratification by $\\sigma $ -sequences.",
"Then $\\mathcal {V}_{n}$ is a non-compact stratification.",
"The closure of the region of $\\mathcal {V}_{n}$ described by REF is given by $\\overline{A}^{a} /S_{a} \\times \\overline{ B}^{b}/ S_{b}\\times \\overline{ C}^{c}/ S_{c} \\times \\overline{D}^{d}/ S_{d} \\times (\\mathbb {R}^{+})^{e} /S_{e} \\times (\\mathbb {R}^{-})/ S_{f}\\times (\\imath \\mathbb {R}^{+})^{g}/ S_{g} \\times (\\imath \\mathbb {R}^{-})^{h}/ S_{h},$ where $\\overline{A}$ denotes the closure in $V_1=\\mathbb {C}\\setminus 0$ of the quadrant $A$ , namely the union of $A$ with $\\mathbb {R}^{+}$ and $\\imath \\mathbb {R}^{+}$ , and similarly for $\\overline{B}, \\overline{C}, \\overline{D}$ .",
"The direct product of semi-closed polytopes is again a semi-closed polytope, as is the quotient of a semi-closed polytope by a sub-group of its symmetry group.",
"Theorem 9 The map $\\nu $ that sends a polynomial in $\\mathop {{}^{\\textsc {D}}\\text{Pol}}_{n}$ to its critical values realizes $\\mathop {{}^{\\textsc {D}}\\text{Pol}}_{n}$ as a finite ramified cover of $\\mathcal {V}_{n}$ .",
"The image of $\\nu $ contains only unordered tuples of $n-1$ complex numbers different from zero, since a polynomial can have 0 as a critical value if and only if it has multiple roots.",
"Therefore, the image of $\\nu $ lies in $\\mathcal {V}_{n}$ .",
"To show that $\\nu $ is surjective, we use a theorem of R. Thom [33] (1963), stating that given $n-1$ complex critical values, there exists a complex polynomial $P$ of degree $d$ such that $P(r_{i})=v_{i}$ for $1\\le i\\le d-1$ , where the $r_{i}$ are the critical points of $P$ , and $P(0)=0$ .",
"To find a Tschirnhausen representative polynomial of $\\mathop {{}^{\\textsc {D}}\\text{Pol}}_{n}$ having the same property it suffices to take $P(z-\\frac{a_{n-1}}{n-1})$ where, $a_{n-1}$ is the coefficient of $z^{n-1}$ in $P$ .",
"By a result of J. Mycielski [31], the map $\\nu $ is a finite ramified cover, of degree $\\frac{n^{n-1}}{n-1}$ , see [31]."
],
[
"The cases $n=2,3,4$",
"The exact nature of the ramified cover $\\mathop {{}^{\\textsc {D}}\\text{Pol}}_{n} \\rightarrow \\mathcal {V}_{n}$ is complicated and interesting, especially in terms of describing the ramification using the signatures.",
"In this section, we work out full details in the small dimensional cases, and for generic strata.",
"Let $n=2$ .",
"The spaces $\\mathop {{}^{\\textsc {D}}\\text{Pol}}_{2}$ and $\\mathcal {V}_{2}$ are one-dimensional.",
"The space $\\mathcal {V}_{2}$ is $\\mathbb {C}\\setminus 0$ equipped with the stratification given by the four quadrants $A,B,C,D$ and the four semi-axes.",
"The only Tschirnhausen polynomial of degree 2 having given critical value $v$ is $z^2 +v$ , therefore the covering map $\\nu $ is unramified of degree 1, an isomorphism.",
"The four signatures corresponding to the strata of real dimensional 2 and the four corresponding to the one dimensional strata are illustrated in figure REF .",
"Figure: Relations between strata for n=2n=2Let $n=3$ .",
"In this case the covering map $\\nu $ is of degree 3: explicitly, if $P(z)=z^3+az+b$ has critical values $v_1$ and $v_2$ then so do the polynomials $P(\\zeta z)$ and $P(\\zeta ^2 z)$ where $\\zeta ^3 =1$ .",
"The 10 open strata of real dimension 4 in $\\mathcal {V}_3$ are given by: $A\\times A/S_{2},\\quad B\\times B/S_{2},\\quad C\\times C/S_{2},\\quad D\\times D/S_{2}, $ $A\\times B,\\quad A\\times C,\\quad A\\times D, \\quad B\\times C,\\quad B\\times D, \\quad C\\times D.$ Ramification occurs only above the first four; in fact exactly when the two critical values are equal, i.e.",
"$P(z)=z^3+b$ .",
"Thus above each of the first four cells, there is only one stratum, corresponding to the four rotations of the left most signature (see figure REF ).",
"Figure: Diagrams d=3d=3In contrast, there are three distinct signatures above each of the remaining 6 strata.",
"The six signatures in the middle of the figure REF form two orbits under the $\\frac{2\\pi }{3}$ rotation which lie above the cells, $ A\\times C$ and $ B\\times D$ , whereas the twelve signatures on the right form four orbits lying over the strata $A\\times B; B\\times C; C\\times D; D\\times A$ .",
"Let $n=4$ .",
"The degree of the covering map $\\nu $ is 16.",
"There are 20 generic strata in $\\mathcal {V}_4$ .",
"Four generic strata correspond to taking three critical values in three different quadrants.",
"There is no ramification above these strata; each of these have 16 distinct strata in the preimage of $\\nu $ , corresponding to four rotations each of the fifth, sixth, eighth and eleventh signatures in the figure below.",
"Four more cells of $\\mathcal {V}_4$ correspond to taking three critical values in the same quadrant.",
"Only one stratum of $\\mathop {{}^{\\textsc {D}}\\text{Pol}}_{4}$ lies above each of these cells, namely, the first signature in the figure below, with ramification of order sixteen.",
"The remaining twelve cells correspond to two critical points in one quadrant and the third in a different quadrant.",
"When the quadrants are adjacent, only six cells lie above the corresponding regions of $\\mathcal {V}_4$ .",
"For example over the region $A,A,B$ lie the two rotations of the second signature in the figure below with ramification of order 2, and the four rotations of the tenth signature each of ramification of order 3.",
"The situation is analogous when the quadrants are opposed.",
"For example over the region $A,A,C$ , there are the two rotations of the third figure (below) each with ramification of order 2, and the four rotations of the ninth signature each of ramification order 3.",
"All 1 classes of size 4: [scale=1] (0,0) circle (1) ; [blue, name=B0B2] (0*180/6:1) .. controls(0*180/6:0.7) and (2*180/6:0.7) .. (2*180/6:1) ; [blue, name=B4B6] (4*180/6:1) .. controls(4*180/6:0.7) and (6*180/6:0.7) .. (6*180/6:1) ; [blue, name=B8B10] (8*180/6:1) .. controls(8*180/6:0.7) and (10*180/6:0.7) .. (10*180/6:1) ; [blue, name=B12B14] (12*180/6:1) .. controls(12*180/6:0.7) and (14*180/6:0.7) .. (14*180/6:1) ; [red, name=B1B3] (1*180/6:1) .. controls(1*180/6:0.7) and (3*180/6:0.7) .. (3*180/6:1) ; [red, name=B5B7](5*180/6:1) .. controls(5*180/6:0.7) and (7*180/6:0.7) .. (7*180/6:1) ; [red, name=B9B11] (9*180/6:1) .. controls(9*180/6:0.7) and (11*180/6:0.7) .. (11*180/6:1) ; All 3 classes of size 8: [scale=1] (0,0) circle (1) ; [blue, name=B0B2] (0*180/6:1) .. controls(0*180/6:0.7) and (2*180/6:0.7) .. (2*180/6:1) ; [blue, name=B4B6] (4*180/6:1) .. controls(4*180/6:0.7) and (6*180/6:0.7) .. (6*180/6:1) ; [blue, name=B8B10] (8*180/6:1) .. controls(8*180/6:0.7) and (10*180/6:0.7) .. (10*180/6:1) ; [blue, name=B12B14] (12*180/6:1) .. controls(12*180/6:0.7) and (14*180/6:0.7) .. (14*180/6:1) ; [red, name=B1B7] (1*180/6:1) .. controls(1*180/6:0.7) and (7*180/6:0.7) .. (7*180/6:1) ; [red, name=B3B5] (3*180/6:1) .. controls(3*180/6:0.7) and (5*180/6:0.7) .. (5*180/6:1) ; [red, name=B9B15] (9*180/6:1) .. controls(9*180/6:0.7) and (15*180/6:0.7) .. (15*180/6:1) ; [red, name=B11B13] (11*180/6:1) .. controls(11*180/6:0.7) and (13*180/6:0.7) .. (13*180/6:1) ; [scale=1] (0,0) circle (1) ; [blue, name=B0B2] (0*180/6:1) .. controls(0*180/6:0.7) and (2*180/6:0.7) .. (2*180/6:1) ; [blue, name=B4B14] (4*180/6:1) .. controls(4*180/6:0.7) and (14*180/6:0.7) .. (14*180/6:1) ; [blue, name=B6B12] (6*180/6:1) .. controls(6*180/6:0.7) and (12*180/6:0.7) .. (12*180/6:1) ; [blue, name=B8B10] (8*180/6:1) .. controls(8*180/6:0.7) and (10*180/6:0.7) .. (10*180/6:1) ; [red, name=B1B3] (1*180/6:1) .. controls(1*180/6:0.7) and (3*180/6:0.7) .. (3*180/6:1) ; [red, name=B5B15] (5*180/6:1) .. controls(5*180/6:0.7) and (15*180/6:0.7) .. (15*180/6:1) ; [red, name=B7B13] (7*180/6:1) .. controls(7*180/6:0.7) and (13*180/6:0.7) .. (13*180/6:1) ; [red, name=B9B11] (9*180/6:1) .. controls(9*180/6:0.7) and (11*180/6:0.7) .. (11*180/6:1) ; [scale=1] (0,0) circle (1) ; [blue, name=B0B2] (0*180/6:1) .. controls(0*180/6:0.7) and (2*180/6:0.7) .. (2*180/6:1) ; [blue, name=B4B6] (4*180/6:1) .. controls(4*180/6:0.7) and (6*180/6:0.7) .. (6*180/6:1) ; [blue, name=B8B10] (8*180/6:1) .. controls(8*180/6:0.7) and (10*180/6:0.7) .. (10*180/6:1) ; [blue, name=B12B14] (12*180/6:1) .. controls(12*180/6:0.7) and (14*180/6:0.7) .. (14*180/6:1) ; [red, name=B1B3] (1*180/6:1) .. controls(1*180/6:0.7) and (3*180/6:0.7) .. (3*180/6:1) ; [red, name=B5B15] (5*180/6:1) .. controls(5*180/6:0.7) and (15*180/6:0.7) .. (15*180/6:1) ; [red, name=B7B13] (7*180/6:1) .. controls(7*180/6:0.7) and (13*180/6:0.7) .. (13*180/6:1) ; [red, name=B9B11] (9*180/6:1) .. controls(9*180/6:0.7) and (11*180/6:0.7) .. (11*180/6:1) ; All 7 classes of size 16: [scale=1] (0,0) circle (1) ; [blue, name=B0B2] (0*180/6:1) .. controls(0*180/6:0.7) and (2*180/6:0.7) .. (2*180/6:1) ; [blue, name=B4B6] (4*180/6:1) .. controls(4*180/6:0.7) and (6*180/6:0.7) .. (6*180/6:1) ; [blue, name=B8B14] (8*180/6:1) .. controls(8*180/6:0.7) and (14*180/6:0.7) .. (14*180/6:1) ; [blue, name=B10B12] (10*180/6:1) .. controls(10*180/6:0.7) and (12*180/6:0.7) .. (12*180/6:1) ; [red, name=B1B3] (1*180/6:1) .. controls(1*180/6:0.7) and (3*180/6:0.7) .. (3*180/6:1) ; [red, name=B5B15] (5*180/6:1) .. controls(5*180/6:0.7) and (15*180/6:0.7) .. (15*180/6:1) ; [red, name=B7B13] (7*180/6:1) .. controls(7*180/6:0.7) and (13*180/6:0.7) .. (13*180/6:1) ; [red, name=B9B11] (9*180/6:1) .. controls(9*180/6:0.7) and (11*180/6:0.7) .. (11*180/6:1) ; [scale=1] (0,0) circle (1) ; [blue, name=B0B2] (0*180/6:1) .. controls(0*180/6:0.7) and (2*180/6:0.7) .. (2*180/6:1) ; [blue, name=B4B6] (4*180/6:1) .. controls(4*180/6:0.7) and (6*180/6:0.7) .. (6*180/6:1) ; [blue, name=B8B14] (8*180/6:1) .. controls(8*180/6:0.7) and (14*180/6:0.7) .. (14*180/6:1) ; [blue, name=B10B12] (10*180/6:1) .. controls(10*180/6:0.7) and (12*180/6:0.7) .. (12*180/6:1) ; [red, name=B3B9] (3*180/6:1) .. controls(3*180/6:0.7) and (9*180/6:0.7) .. (9*180/6:1) ; [red, name=B5B7] (5*180/6:1) .. controls(5*180/6:0.7) and (7*180/6:0.7) .. (7*180/6:1) ; [red, name=B11B13] (11*180/6:1) .. controls(11*180/6:0.7) and (13*180/6:0.7) .. (13*180/6:1) ; [red, name=B15B1] (15*180/6:1) .. controls(15*180/6:0.7) and (1*180/6:0.7) .. (1*180/6:1) ; [scale=1] (0,0) circle (1) ; [blue, name=B0B2] (0*180/6:1) .. controls(0*180/6:0.7) and (2*180/6:0.7) .. (2*180/6:1) ; [blue, name=B4B6] (4*180/6:1) .. controls(4*180/6:0.7) and (6*180/6:0.7) .. (6*180/6:1) ; [blue, name=B8B10] (8*180/6:1) .. controls(8*180/6:0.7) and (10*180/6:0.7) .. (10*180/6:1) ; [blue, name=B12B14] (12*180/6:1) .. controls(12*180/6:0.7) and (14*180/6:0.7) .. (14*180/6:1) ; [red, name=B1B3] (1*180/6:1) .. controls(1*180/6:0.7) and (3*180/6:0.7) .. (3*180/6:1) ; [red, name=B5B7](5*180/6:1) .. controls(5*180/6:0.7) and (7*180/6:0.7) .. (7*180/6:1) ; [red, name=B9B15] (9*180/6:1) .. controls(9*180/6:0.7) and (15*180/6:0.7) .. (15*180/6:1) ; [red, name=B11B13] (11*180/6:1) .. controls(11*180/6:0.7) and (13*180/6:0.7) .. (13*180/6:1) ; [scale=1] (0,0) circle (1) ; [blue, name=B0B2] (0*180/6:1) .. controls(0*180/6:0.7) and (2*180/6:0.7) .. (2*180/6:1) ; [blue, name=B4B6] (4*180/6:1) .. controls(4*180/6:0.7) and (6*180/6:0.7) .. (6*180/6:1) ; [blue, name=B8B14] (8*180/6:1) .. controls(8*180/6:0.7) and (14*180/6:0.7) .. (14*180/6:1) ; [blue, name=B10B12] (10*180/6:1) .. controls(10*180/6:0.7) and (12*180/6:0.7) .. (12*180/6:1) ; [red, name=B1B3] (1*180/6:1) .. controls(1*180/6:0.7) and (3*180/6:0.7) .. (3*180/6:1) ; [red, name=B5B15](5*180/6:1) .. controls(5*180/6:0.7) and (15*180/6:0.7) .. (15*180/6:1) ; [red, name=B7B9] (7*180/6:1) .. controls(7*180/6:0.7) and (9*180/6:0.7) .. (9*180/6:1) ; [red, name=B11B13] (11*180/6:1) .. controls(11*180/6:0.7) and (13*180/6:0.7) .. (13*180/6:1) ; [scale=1] (0,0) circle (1) ; [blue, name=B0B2] (0*180/6:1) .. controls(0*180/6:0.7) and (2*180/6:0.7) .. (2*180/6:1) ; [blue, name=B4B6] (4*180/6:1) .. controls(4*180/6:0.7) and (6*180/6:0.7) .. (6*180/6:1) ; [blue, name=B8B14] (8*180/6:1) .. controls(8*180/6:0.7) and (14*180/6:0.7) .. (14*180/6:1) ; [blue, name=B10B12] (10*180/6:1) .. controls(10*180/6:0.7) and (12*180/6:0.7) .. (12*180/6:1) ; [red, name=B1B3] (1*180/6:1) .. controls(1*180/6:0.7) and (3*180/6:0.7) .. (3*180/6:1) ; [red, name=B5B7] (5*180/6:1) .. controls(5*180/6:0.7) and (7*180/6:0.7) .. (7*180/6:1) ; [red, name=B9B15] (9*180/6:1) .. controls(9*180/6:0.7) and (15*180/6:0.7) .. (15*180/6:1) ; [red, name=B11B13] (11*180/6:1) .. controls(11*180/6:0.7) and (13*180/6:0.7) .. (13*180/6:1) ; [scale=1] (0,0) circle (1) ; [blue, name=B0B2] (0*180/6:1) .. controls(0*180/6:0.7) and (2*180/6:0.7) .. (2*180/6:1) ; [blue, name=B4B6] (4*180/6:1) .. controls(4*180/6:0.7) and (6*180/6:0.7) .. (6*180/6:1) ; [blue, name=B8B10] (8*180/6:1) .. controls(8*180/6:0.7) and (10*180/6:0.7) .. (10*180/6:1) ; [blue, name=B12B14] (12*180/6:1) .. controls(12*180/6:0.7) and (14*180/6:0.7) .. (14*180/6:1) ; [red, name=B1B3] (1*180/6:1) .. controls(1*180/6:0.7) and (3*180/6:0.7) .. (3*180/6:1) ; [red, name=B5B15](5*180/6:1) .. controls(5*180/6:0.7) and (15*180/6:0.7) .. (15*180/6:1) ; [red, name=B7B9] (7*180/6:1) .. controls(7*180/6:0.7) and (9*180/6:0.7) .. (9*180/6:1) ; [red, name=B11B13] (11*180/6:1) .. controls(11*180/6:0.7) and (13*180/6:0.7) .. (13*180/6:1) ; [scale=1] (0,0) circle (1) ; [blue, name=B0B2] (0*180/6:1) .. controls(0*180/6:0.7) and (2*180/6:0.7) .. (2*180/6:1) ; [blue, name=B4B6] (4*180/6:1) .. controls(4*180/6:0.7) and (6*180/6:0.7) .. (6*180/6:1) ; [blue, name=B8B14] (8*180/6:1) .. controls(8*180/6:0.7) and (14*180/6:0.7) .. (14*180/6:1) ; [blue, name=B10B12] (10*180/6:1) .. controls(10*180/6:0.7) and (12*180/6:0.7) .. (12*180/6:1) ; [red, name=B1B7] (1*180/6:1) .. controls(1*180/6:0.7) and (7*180/6:0.7) .. (7*180/6:1) ; [red, name=B9B15](9*180/6:1) .. controls(9*180/6:0.7) and (15*180/6:0.7) .. (15*180/6:1) ; [red, name=B3B5] (3*180/6:1) .. controls(3*180/6:0.7) and (5*180/6:0.7) .. (5*180/6:1) ; [red, name=B11B13] (11*180/6:1) .. controls(11*180/6:0.7) and (13*180/6:0.7) .. (13*180/6:1) ;"
],
[
"Combinatorial closure of a signature",
"The existence of the notion of incidence relation between strata, which leads to the notion of poset and chain.",
"A poset (partially ordered set) is a set $P$ together with a binary relation $\\prec $ which is transitive ($x \\prec y$ and $y \\prec z$ implies $x \\prec z$ ) and irreflexive ($x \\prec y$ and $y \\prec x$ cannot both hold).",
"The elements $x$ and $y$ are comparable if $x \\prec y$ and/or $y \\prec x$ hold.",
"A chain in a poset $P$ is a subset $C \\subseteq P$ such that any two elements in $C$ are comparable.",
"We write $\\sigma \\prec \\tau $ , when $\\tau $ can be obtained from $\\sigma $ by a sequence of repeated contracting Whitehead moves.",
"The signature $\\tau $ is incident to $\\sigma $ .",
"Definition 8 We call the union $\\overline{\\sigma }=\\lbrace \\tau : \\sigma \\prec \\tau \\rbrace $ the combinatorial closure of $\\sigma $ , and we define $A_{\\overline{\\sigma }}=\\cup _{\\sigma \\prec \\tau } A_{\\tau }.$ Definition 9 Let $S$ be the set of signatures.",
"An upper bound in a subset of $(S,\\prec )$ is a signature such that there exists no other signature $\\tau $ in this subset verifying $\\sigma \\prec \\tau $ .",
"Lemma 8 Let $\\tau $ be a signature with a given intersection of two red (or blue) diagonals.",
"There are exactly two ways to smooth the intersection, which give two non-isotopic signatures $\\sigma _1$ and $\\sigma _2$ such that $\\tau $ is incident to both $\\sigma _1$ and $\\sigma _2$ .",
"The proof follows from Morse theory.",
"As in the proof of lemma REF : let $x,y$ be in a neighbourhood $U$ of $p$ , so that the identity $f=f(p)-x^2+y^2$ holds throughout $U$ and the critical point $p$ will have coordinates $x(p)=y(p)=0$ , with $f(p)=c$ .",
"Choose $\\epsilon >0$ , sufficiently small so that the region $f^{-1}[c-\\epsilon , c+\\epsilon ]$ is compact and contains no critical point other than $p$ .",
"Then, at $c-\\epsilon $ and $c+\\epsilon $ we have two pairs of curves.",
"They give two signatures which are non-isotopic.",
"Similarly, we have the following for contracting Whitehead moves.",
"Lemma 9 Let $\\tau _1$ and $\\tau _2$ be obtained from a signature $\\sigma $ by two different complete contracting Whitehead moves.",
"Then, $\\tau _1$ and $\\tau _2$ are different.",
"Let $\\tau _1$ and $\\tau _2$ be obtained from a signature $\\sigma $ from two different contracting Whitehead moves.",
"Two contracting Whitehead moves are different if they operate on different sets of edges.",
"Let us consider $m_1$ (resp.",
"$m_2$ ) edges of $\\sigma $ lying in the boundary of cell of $\\mathbb {C}$ .",
"Suppose that their set of leaves is $\\lbrace i_1,...i_{2m_{1}}\\rbrace $ (resp.",
"$\\lbrace j_1,...j_{2m_{2}}\\rbrace $ ).",
"A contracting Whitehead move glues those edges at one point.",
"This gives a (star shaped) tree with leaves in the set $\\lbrace i_1,...i_{2m}\\rbrace $ (resp.$\\lbrace j_1,...j_{2m_{2}}\\rbrace $ ).",
"This gives the signature $\\tau _1$ (resp.",
"$\\tau _2$ ).",
"Clearly $\\tau _1$ can not be isotopic to $\\tau _2$ , with respect to the leaves, since $\\lbrace i_1,...i_{2m}\\rbrace $ is different from $\\lbrace j_1,...j_{2m_{2}}\\rbrace $ .",
"Lemma 10 Let $\\tau $ be a non generic signature.",
"Consider in $\\tau $ a vertex, incident to $m>2$ red (or blue) curves.",
"The signatures, obtained from $\\tau $ in one single smoothing Whitehead move, are all distinct.",
"Locally, around the intersection, the graph resembles a star shaped tree with one inner node and $2m$ leaves which we can number $1,..,2m$ .",
"After one partial Whitehead move, the graph is still connected.",
"However, the set of leaves are splitted into two disjoint sets $U_1\\sqcup U_2=\\lbrace 1,...,2m\\rbrace $ .",
"It is clear that for different splittings the graphs are not isotopic.",
"After a complete smoothing Whitehead move, the graph is disconnected: there exists a star-like tree and a tree with at least one edge.",
"There are at $2m$ possible of creating such graphs, if there are no symmetries in the graph.",
"There are $m$ possibilities if there are some symmetries in the graph.",
"These graphs are non-isotopic, with respect to the asymptotic directions of the leaves $2m$ .",
"Lemma 11 Let $\\tau $ be a signature of codimension $k$ with a given red (or blue) intersection of $m>2$ curves.",
"Then there exists $Cat(m)$ distinct signatures which are smoothings of $\\tau $ and of codimension $(k-(2m-3))$ obtained by complete smoothing Whitehead moves, where $Cat(m)$ is the $m^{th}$ -Catalan number.",
"Draw in the neighborhood of the critical point $p$ a $2m$ -gon.",
"The vertices are the boundaries of the $m$ curves.",
"Since we only consider the combinatorics, we may assume that the $2m$ -gon is regular.",
"Ungluing those $m$ curves gives $m$ disjoint curves in the regular $2m$ -gon and the number of such possibilities is given by the $m^{th}$ -Catalan number (see [32])."
],
[
"Topological closure of a stratum",
"Lemma 12 Let $\\tau $ be a non-generic signature and let $A_\\tau $ be the corresponding strata of $\\mathop {{}^{\\textsc {D}}\\text{Pol}}_{d}$ .",
"Let $P_0\\in A_\\tau $ .",
"Then for every generic signature $\\sigma $ such that $\\tau $ is incident to $\\sigma $ , every $2d$ -dimensional open ball $B_{\\epsilon }$ containing $P_0$ intersects the generic stratum $A_\\sigma $ .",
"Suppose that there exists a generic signature $\\sigma $ which does not verify that $B_{\\epsilon }$ containing $P_0$ intersects the generic stratum $A_\\sigma $ .",
"Then there does not exist any path from a point $y \\in A_{\\tau }$ to $x\\in A_{\\sigma }$ .",
"Therefore $A_{\\sigma }$ and $A_{\\tau }$ are disjoint.",
"In particular this implies that $A_{\\tau }$ is not in the closure of $A_{\\sigma }$ .",
"So, $\\tau $ is not incident to $\\sigma $ .",
"Lemma 13 Let $\\sigma $ be a generic signature and let $\\tau \\ne \\sigma $ .",
"Then $\\tau $ is incident to $\\sigma $ if and only if the following holds: for every pair of points $x,y\\in \\mathop {{}^{\\textsc {D}}\\text{Pol}}_{d}$ with $x\\in A_\\sigma $ and $y\\in A_\\tau $ , there exists a continuous path $\\gamma :[0,1]\\rightarrow \\mathop {{}^{\\textsc {D}}\\text{Pol}}_n$ such that $\\gamma (0)=x$ , $\\gamma (1)=y$ and $\\gamma (t)\\in A_\\sigma $ for all $t\\in [0,1)$ .",
"Any other such path $\\rho $ from a point $x^{\\prime }\\in A_\\sigma $ to a point $y^{\\prime }\\in A_\\tau $ is homotopic to $\\gamma $ .",
"The result follows from lemma REF .",
"Indeed, if $\\tau $ is incident to $\\sigma $ then, a ball around $y\\in A_\\tau $ necessarily intersects $A_\\sigma $ and therefore there is a path from $y$ to a point $x^{\\prime }\\in A_\\sigma $ .",
"Composing this with a path from $x^{\\prime }$ to $x$ in $A_\\sigma $ we obtain $\\gamma $ .",
"Conversely, if there is a path $\\gamma $ from $x\\in A_\\sigma $ to $y\\in A_\\tau $ , then it is impossible to have an open ball containing $y$ that does not intersect $A_\\sigma $ .",
"We return to the original polynomials in order to define Whitehead moves in an analytic and then topological fashion.",
"Lemma 14 A contracting Whitehead move which takes a signature $\\sigma $ to an incident signature $\\tau $ induces a retraction of the stratum $A_{\\sigma }$ onto the boundary stratum $A_{\\tau }$ .",
"From the above discussion it follows that we can construct the retraction locally near $A_{\\tau }$ ; then we extend it to the whole of $A_{\\sigma }$ using the contractibility of this stratum.",
"Proposition 1 If $\\sigma $ is a signature, the closure $\\overline{A}_\\sigma $ of the stratum $A_\\sigma $ in $\\mathop {{}^{\\textsc {D}}\\text{Pol}}_n$ is given by $\\overline{A}_\\sigma =\\cup _{\\tau \\in \\overline{\\sigma }} A_\\tau ,$ where the boundary of $\\sigma $ denoted $\\overline{\\sigma }$ consists of all incident signatures $\\tau $ .",
"One direction is easy.",
"Indeed, if $x\\in A_{\\tau }$ where $\\tau $ is incident to $\\sigma $ then every $ 2d$ -dimensional open set containing $x$ must intersect $A_{\\sigma }$ , so $\\cup _{\\tau \\in \\overline{\\sigma }} A_\\tau \\subset \\overline{A}_\\sigma $ .",
"For the other direction, let $x \\in \\overline{A}_\\sigma \\setminus A_{\\sigma }$ and let $\\tau $ be the signature of $x$ .",
"We first note that the dimension of $\\tau $ can not be equal to the dimension of $\\sigma $ because if they were equal, $A_{\\tau }$ would be an open stratum disjoint from $A_{\\sigma }$ .",
"Therefore the dimension of $\\tau $ is less than the dimension of $\\sigma $ .",
"Let $U$ be any small open neighborhood of $x$ .",
"Let $y\\in U\\cap A_\\sigma $ and let $\\gamma $ be a path from $y$ to $x$ such that $\\gamma \\setminus x\\subset A_{\\sigma }$ .",
"Then every point $z \\in \\gamma \\setminus x$ has the same signature $\\sigma $ .",
"Using theorem REF , any path from the interior to any point not in $A_{\\sigma }$ must pass through the boundary of the polytope.",
"Therefore any sequence of Whitehead moves and smoothings from $\\sigma $ to $\\tau $ must begin with Whitehead moves bringing $\\sigma $ to a signature that is incident to $\\sigma $ .",
"But $x$ is the first point on $\\gamma $ where the signature changes and therefore $\\tau $ must be incident to $\\sigma $ .",
"Lemma 15 Let $P^\\sigma _r$ (resp.",
"$P^\\sigma _r$ ) be a red (resp.",
"blue) forest in the signature $\\sigma $ ; let $P^{\\tau }_r$ (resp.",
"$P^\\sigma _b$ ) be a red (resp.",
"blue) forest in the signature $\\tau $ .",
"The set of signatures incident to a generic signature $\\sigma $ is equal to the set of signatures $\\tau $ such that (i) $P^\\sigma _r\\subset P^{\\tau }_r$ and $P^\\sigma _b\\subset P^{\\tau }_b$ (with at least one of the containments being strict), (ii) the set of edges $(i,j)$ in $\\tau $ for the pairs in $P^\\sigma _r$ (resp.",
"those for $(k,l)$ in $P^\\sigma _v$ ) intersect only at isolated points (no shared segments).",
"Performing a Whitehead move on a signature can never eliminate an edge but can only add edges, which shows that if $\\tau $ is incident to $\\sigma $ then (i) holds.",
"Furthermore, Whitehead moves cause the arcs of $\\sigma $ to cross only at isolated points.",
"Conversely, suppose that (i) and (ii) hold for $\\tau $ .",
"Consider the red forest of $\\tau $ .",
"We claim that the set $P^\\sigma _r$ provides a recipe for smoothing the red forest of $\\tau $ to obtain the red forest $\\sigma $ (and subsequently the blue using $P^\\sigma _b$ ), as follows.",
"Let us use the term “short edges” for edges joining two neighboring (even for red, odd for blue) leaves.",
"We start with the short edges, joining pairs of the form $(i,i+2)$ in $P^\\sigma _r$ (and with labels $\\mod {4}n$ ).",
"Because the edges cross only at points, smoothing the edge $(i,i+2)$ by separating it off from the rest of the red forest of $\\tau $ eliminates only edges from $i$ or $i+2$ to the other points; paths between all the other pairs of $P^{\\tau }_r$ remain.",
"We then erase the edges $(i,i+2)$ from the signature and now treat the “new” short edges (those from $P^\\sigma _r$ joining points $j$ , $j+6$ ) separating them from the rest of the tree, then erase those and continue in the same way.",
"The final result reduces the red forest of $P^{\\tau }_r$ entirely to a set of $d$ disjoint edges, endpoints of red edges, which is equal to $P^\\sigma _r$ since they all belong to $P^\\sigma _r$ .",
"This proves that any signature $\\tau $ satisfying (i) and (ii) can be smoothed to $\\sigma $ .",
"Theorem 10 The set of all $n-$ signatures determine a stratification of the configuration space $Conf_n(\\mathbb {C})$ in which a signature of dimension $k$ corresponds to a stratum of dimension $k$ in $Conf_n(\\mathbb {C})$ .",
"A signature corresponds uniquely to a class of drawings, a class of drawings describes a region of polynomials (i.e.",
"the set of polynomials having the same signature) and the statement from theorem REF shows that these regions are contractible.",
"Due to proposition REF , lemma REF and lemma REF , we have a filtration verifying the definition of a stratified space.",
"Theorem 11 Let $\\sigma $ be a generic signature.",
"Then, $\\overline{A}_{\\sigma }$ is a Goresky–MacPherson stratified space.",
"We have the following filtration: $\\overline{A}_{\\tau _0}\\subset \\overline{A}_{\\tau _{1}}\\subset \\dots \\subset \\overline{A}_{\\tau _{n-1}} \\subset \\overline{A}_{\\tau _{n}}=\\overline{A}_{\\sigma }$ .",
"Take a point $p$ in $\\overline{A}_{\\tau _{i}}\\setminus \\overline{A}_{\\tau _{i-1}}$ .",
"Then, by lemma REF , $A_\\sigma $ retracts onto $\\overline{A}_{\\tau _{i}}$ .",
"Therefore, we have a topological cone structure $cone(L)$ , where $L$ is a compact Hausdorff space.",
"From the stratification, it follows that $L$ is endowed with an $n-i-1$ topological stratification: $ L=L_{n-i-1}\\supset \\dots \\supset L_{1}\\supset L_{0} \\supset L_{-1}=\\emptyset .$ By lemma REF , the neighborhood of $p$ in $\\overline{A}_{\\tau _{i}}$ behaves as an $i$ -dimensional Euclidean space.",
"So, there exists a distinguished neighborhood $N$ of $p$ in $\\overline{A}_{\\sigma }$ and a homeomorphism $\\phi $ such that: $\\phi :\\mathbb {R}^i\\times cone^{\\circ }(L)\\rightarrow N,$ ($cone^{\\circ }(L_{j})$ denotes the open cone) which takes each $\\mathbb {R}^i\\times cone^{\\circ }(L_{j})$ homeomorphically to $N\\cap \\overline{A}_{\\tau _{i+j+1}}$ ."
],
[
"Multiple intersections of closures of strata",
"This section investigates a combinatorial method to study multiple intersections between closures of strata indexed by generic signatures.",
"This shows that a finite non-empty intersection of closures of generic strata has a lowest upper bound.",
"This part is an independent investigation of the properties of the stratification, and is not directly used for the construction of the Čech cover.",
"For simplicity, instead of signatures, we will use diagrams i.e.",
"embedded forests in the complex plane, such that their leaves lie on the boundary of a disc, on the $4n$ roots of the unity.",
"We label the leaves in the trigonometric sense.",
"We focus essentially on the combinatorics.",
"So, from now on, we can assume that all the discs are of the same canonical radius and that their leaves coincide on the $4n$ roots of the unity.",
"The core idea of the construction is to superimpose generic diagrams $\\sigma _0,...,\\sigma _p$ such that leaves $\\sqrt{1}^{4n}$ (and their labels) coincide.",
"Then, we apply contracting Whitehead moves only to those diagonals, of the generic signatures, which in the superimposition are non identical."
],
[
"Admissible superimposition of diagrams",
"Let $\\sigma _0,\\dots , \\sigma _p$ be generic signatures and let $\\Theta $ denote their superimposition.",
"This superimposition is not well-defined as the diagonals of the different signatures can be positioned differently with respect to each other, since the diagonals of each signature are given only up to isotopy, but we will consider only those having the following properties: all intersections are crossings (but not tangents) of at most two diagonals, the superimposition $\\Theta $ cuts the disk into polygonal regions; we require that no region is a bigon, the number of crossings of a given diagonal with other diagonals of $\\Theta $ must be minimal, with respect to a possible isotopy.",
"We take representatives of isotopy classes of arcs.",
"We call such superimpositions admissible.",
"Lemma 16 Let $\\sigma _0,\\dots , \\sigma _p$ be generic signatures.",
"If there exists no superimposition $\\Theta $ with the property that one diagonal has more than $p+1$ intersections with diagonals of the opposite color then there is no signature $\\tau $ incident to all the $\\sigma _{i}$ .",
"The key point is the following.",
"If $\\sigma _{0},...,\\sigma _{p}$ admit a signature $\\tau $ incident to all of them, then $\\tau $ has the following property: every segment of the tree $\\tau $ (a segment is the part of an edge contained between two vertices, including leaves) belongs to at most one diagonal $(i,j)$ of each $\\sigma _i$ .",
"Thus, in particular, each segment can be considered as belonging to at most $p+1$ diagonals, one from each $\\sigma _i$ .",
"Thus, if a red diagonal of $\\tau $ crosses $p+2$ or more blue diagonals in the superimposition, there is no one segment of $\\tau $ which can belong to all of them, so the red diagonal will necessarily cross more than one blue segment of $\\tau $ , which is impossible.",
"We will say that the set of signatures $\\sigma _0,\\dots , \\sigma _p$ is compatible if it admits an admissible superimposition $\\Theta $ with the property that no diagonal crosses more than $p+1$ diagonals of the opposite color.",
"Note that a red diagonal can never cross a blue diagonal in more than one point.",
"Compatible sets of generic signatures may potentially have non-empty intersection.",
"We will now show how to give a condition on $\\Theta $ to see whether or not this is the case."
],
[
"Graph associated to an intersection of generic signatures.",
"Let $\\sigma _0,\\dots , \\sigma _p$ be a set of compatible generic signatures and let $\\Theta $ be an admissible superimposition.",
"Then $\\Theta $ cuts the disk into polygonal regions.",
"Color a region red (resp.",
"blue) if all its edges are red (resp.",
"blue); the intersecting regions are purple.",
"Construct a graph from $\\Theta $ as follows : place a vertex in each red or blue region (but not purple) with number of sides greater than three.",
"If two vertices lie in blue (resp.",
"red) polygons that meet at a point, join them with a blue (resp.",
"red) edge (even if this edge crosses purple regions).",
"If two vertices lie in blue (resp.",
"red) polygons that intersect along an edge of the opposite color, connect them with a blue (resp.",
"red) edge.",
"If two vertices lying in the same red (resp.",
"blue) polygon can be connected by a segment inside the polygon which crosses only one purple region, add this segment.",
"Connect each vertex to every terminal vertex lying in the same red (resp.",
"blue) region, and also to any terminal vertex which can be reached by staying within the original red (resp.",
"blue) polygon but crossing through a purple region formed by two blue (resp.",
"red) diagonals emerging from that terminal vertex.",
"Finally, if any vertex of the graph has valency 2, we ignore this vertex and consider the two emerging edges as forming a single edge.",
"We call this graph the graph associated to the superimposition.",
"Lemma 17 The graph associated to $\\Theta $ is independent of the actual choice of admissible superimposition $\\Theta $ .",
"Let $\\Theta $ and $\\Theta ^{\\prime }$ be admissible superimpositions, and consider a given diagonal $D$ .",
"By the admissibility conditions the number of crossings of $D$ with diagonals of the other color is equal in $\\Theta $ and $\\Theta ^{\\prime }$ , and in fact the set of diagonals of the other color crossed by $D$ is identical in $\\Theta $ and $\\Theta ^{\\prime }$ .",
"Therefore, the only possible modifications of the $\\Theta $ is to move $D$ across an intersection of two diagonals of the other color.",
"But this does not change the associated graph."
],
[
"Compatible signatures",
"Definition 10 The canonical graph associated to a set of compatible signatures $\\sigma _0,\\dots ,\\sigma _p$ is the graph associated to any admissible superimposition $\\Theta $ .",
"Theorem 12 Let $\\sigma _0,\\dots , \\sigma _p$ be generic signatures.",
"Then, there exists a signature incident to all the $\\sigma _i$ if and only if the set $\\sigma _0,\\dots , \\sigma _p$ is compatible and the associated canonical graph is a signature.",
"Replacing a blue (or red) polygon by a graph having the shape of a star as in the construction above involves diagonals of the different $\\sigma _i$ which must be identified if we want to construct a common incident signature.",
"The contracting moves may be stronger than strictly necessary (i.e.",
"the signature $\\tau $ may not be the signature of maximal dimension in the intersection), but any signature in the intersection must either have the same connected components as $\\tau $ , i.e.",
"be obtained from $\\tau $ by applying only smoothing Whitehead moves which do not increase the number of connected components of $\\tau $ (these smoothings are the partial smoothings), or lie in the closures of these.",
"Thus, up to such smoothings, the moves constructing $\\tau $ are necessary in order to identify the diagonals of the $\\sigma _i$ .",
"The contracting moves in the construction of the graph associated to $\\Theta $ , restricted to just one of the signatures $\\sigma _i$ , has the effect of making a contracting Whitehead move on the blue (resp.",
"red) curves of this signature.",
"Thus, on each of the signatures, the graph construction reduces to a sequence of contracting Whitehead moves.",
"Thus the $\\sigma _i$ possess a common incident signature if and only if $\\tau $ is such a signature.",
"Remark 3 In the appendix we show the classification of intersections between polygons which are allowed in order to have a non-empty intersection between closures of strata.",
"At the end we show how to construct the canonical graph from the superimposition of a red and a blue polygon."
],
[
"Good Čech cover",
"This section shows how to construct the Čech cover, and includes a proof that multiple intersections are contractible.",
"We will introduce the notation $\\underline{A_{\\sigma }}$ , for the set of all elements $A_{\\tau }$ in the set of strata verifying $\\tau \\prec \\sigma $ i.e.",
"$\\underline{A_{\\sigma }}=\\cup _{\\tau \\prec \\sigma } A_{\\tau }$ and $codim(\\tau )<codim(\\sigma )$ .",
"The set of elements verifying $\\tau \\prec \\sigma $ , where $codim(\\tau )<codim(\\sigma )$ is denoted by $\\underline{\\sigma }$ ."
],
[
"Contractibility theorem",
"Theorem 13 Let $\\sigma $ be a non-generic signature .",
"Then, $\\underline{A_{\\sigma }}$ is a contractible set.",
"The proof goes as follows.",
"Using an induction on the codimension of the incident strata to $\\sigma $ , and the deformation retract lemma we show that $\\underline{A}_{\\sigma }$ is a deformation retract of $A_{\\sigma }$ .",
"Then, we use the theorem REF on contractibility of strata.",
"An element $\\tau $ lying in $\\underline{A_{\\sigma }}$ verifies $\\tau \\prec \\sigma $ .",
"In other words, there exists one unique contracting Whitehead move from $\\tau $ to $\\sigma $ (uniqueness follows from lemma REF and lemma REF ).",
"Suppose that $ codim(\\sigma )=codim(\\tau )+1$ .",
"Now by the deformation retract lemma REF , $A_{\\sigma }$ is a deformation retract of $A_{\\tau }$ .",
"Let $\\tau ^{\\prime }$ in $\\underline{A_{\\sigma }}$ be of codimension $k$ and suppose that $A_{\\sigma }$ is a deformation retract of $A_{\\tau ^{\\prime }}$ .",
"Let us show that for $ \\tau ^{\\prime \\prime }\\prec \\sigma $ , where $codim(A_{\\tau ^{\\prime \\prime }})=codim(A_{\\tau ^{\\prime }})+1$ , $A_{\\sigma }$ is a deformation retract of $A_{\\tau ^{\\prime \\prime }}$ .",
"Now, by the deformation retract lemma REF , $A_{\\tau ^{\\prime }}$ is a deformation retract of $A_{\\tau ^{\\prime \\prime }}$ .",
"So, by the induction hypothesis, we have that $A_{\\sigma }$ is a deformation retract of $A_{\\tau ^{\\prime \\prime }}$ .",
"Finally, using the fact that each stratum is contractible (c.f.",
"Theorem REF ), we have that $\\underline{A_{\\sigma }}$ is contractible.",
"This proof easily adapts to give the following result.",
"Corollary 1 Let $\\sigma $ be any (non-generic) signature and let $\\tilde{\\sigma }$ be a subset of $\\underline{\\sigma }$ .",
"Let $A_{\\tilde{\\sigma }}$ be the union of $A_{\\rho }$ with $\\rho $ in $\\tilde{\\sigma }$ .",
"Then, $A_{\\tilde{\\sigma }}$ is contractible."
],
[
"Thickening of strata",
"Let $(\\mathop {{}^{\\textsc {D}}\\text{Pol}}_n, \\mathcal {S})$ be the stratified smooth topological space.",
"A system of tubes is a family $T$ of triplets $(T_\\sigma ,\\pi _\\sigma ,\\rho _\\sigma )\\sigma \\in S$ , where $T_\\sigma $ is an open neighbourhood of the stratum $A_{\\sigma }$ , called its tubular neighbourhood; $\\pi _\\sigma $ is a deformation retract $T_\\sigma \\rightarrow A_{\\sigma }$ (i.e.",
"$\\pi _\\sigma $ is continuous and $\\pi _\\sigma (x) = x$ for all $x \\in A_{\\sigma }$ ); $\\rho _\\sigma $ is a continuous function $T_\\sigma \\rightarrow \\mathbb {R}_{+}$ , called the distance function of the stratum, such that $A_{\\sigma } = \\rho ^{-1}(0)$ ; For any pair of indices $\\tau \\prec \\sigma $ the restriction of $(\\pi _\\tau ,\\rho _\\tau )$ to $T_\\tau \\cap A_{\\sigma } \\rightarrow A_{\\tau ^{\\prime }} \\times \\mathbb {R}_+$ is a smooth submersion for all $x\\in T_{\\sigma } \\cap T_\\tau $ we have $\\pi _{\\sigma }(x)\\in T_\\tau $ ,$\\pi _r\\pi _\\sigma (x)=\\pi _\\tau (x)$ and $\\rho _\\tau (\\pi _\\sigma (x)) = \\rho _\\tau (x)$"
],
[
"The Čech cover",
"We consider only the red (resp.",
"blue) curves of a signature $\\sigma $ .",
"This is denoted by $\\sigma ^{R}$ (resp.",
"$\\sigma ^{B}$ ).",
"Definition 11 We call a connected component a tree in $\\sigma ^{R}$ (resp.",
"$\\sigma ^B$ ) .",
"A tree is in bijection with its set of leaves.",
"So, a connected component is in bijection with a subset of $\\lbrace 1,...,4n\\rbrace $ , corresponding to the labels of its leaves.",
"Such a set is of even cardinality.",
"A complete Whitehead move partitions the set into two subsets of even cardinality.",
"Lemma 18 Consider two non-generic signatures $\\sigma $ and $\\tau $ .",
"Then, $\\underline{\\sigma }\\cap \\underline{\\tau }\\ne \\emptyset $ if and only if for each connected component with leaves in $I$ of $\\sigma ^{R}$ (resp.",
"$\\sigma ^B$ ) there exists a connected component of leaves in $J$ of $\\tau ^{R}$ (resp.",
"$\\tau ^B$ ) verifying that $I\\setminus (I\\cap J)$ and $J\\setminus (I\\cap J)$ are of even cardinality.",
"Suppose that there exists one connected component of $\\sigma ^{R}$ which does not verify the property.",
"Without loss of generality we suppose that a connected component of $\\sigma ^{R}$ is not contained in a connected component of $\\tau ^{R}$ .",
"We suppose that this connected component of $\\sigma ^{R}$ is defined by the set $I=\\lbrace i_1,...,i_{2p}\\rbrace $ ; whereas the investigated component of $\\tau ^{R}$ is defined by the set $J=\\lbrace i_2,...,i_{2p+1}\\rbrace $ (a generalisation of this case can be easily obtained using $J=\\lbrace i_{2r},...,i_{2k+1}\\rbrace $ , for some positive integers $r,k$ and where $k\\ge p$ ).",
"Then, a sequence of complete Whitehead moves on both connected components induce partitions of $I$ and $J$ , into subsets of even cardinality.",
"The map between these partitions can never be the identity.",
"The same procedure applies to the color $B$ .",
"We proceed by the following algorithm.",
"Lemma 19 Let $\\sigma $ and $\\tau $ be two non-generic signatures such that $\\underline{\\sigma }\\cap \\underline{\\tau }\\ne \\emptyset $ .",
"If the lowest upper bound exists in $\\underline{\\sigma }\\cap \\underline{\\tau }$ then it is unique.",
"Consider the connected components of $\\sigma ^{R}$ (resp.",
"$\\sigma ^{B}$ ) and $\\tau ^{R}$ (resp.",
"$\\tau ^{B}$ ) as two partitions of the set of cardinality $2n$ : $\\sqcup _{i=1}^p A_i$ and $\\sqcup _{j=1}^r B_j$ .",
"Then, there exists a unique partition which is given by $\\cup _{j=1}^{r} (B_j \\cap (\\sqcup _{i=1}^p A_i))$ .",
"However, that this does not provide a unique signature: to one connected component of leaves $\\lbrace i_1,...,i_{2k}\\rbrace $ corresponds a set of non isomorphic trees, if $k>1$ .",
"Let us consider this connected component $\\lbrace i_1,...,i_{2k}\\rbrace $ , obtained by smoothing Whitehead moves in $\\sigma ^{R}$ and $\\tau ^{R}$ with the above method.",
"Take the maximum on the number of vertices (which are not leaves!)",
"in both connected components.",
"Modify the connected component having the minimal number of vertices, by a partial smoothing Whitehead move.",
"Repeat until the number of vertices in both connected components coincide and in such a way that edge relations $E$ between leaves and inner vertices are identical in both connected components.",
"Those trees are thus isomorphic and there exists one unique such graph.",
"Repeat this for $\\sigma ^{B}$ and $\\tau ^{B}$ .",
"Therefore, the greatest lower bound is unique.",
"Lemma 20 Let $\\sigma _i$ be an upper bound.",
"Each $\\underline{A_{\\sigma _i}}$ is a lattice.",
"In $\\underline{A_{\\sigma _i}}$ we have a poset, where we have a subset of the set of signatures and the relation is the incidence relation $\\prec $ .",
"There is a join $\\vee $ and meet $ \\wedge $ structure.",
"The join $\\vee $ is given by smoothing Whitehead moves and the meet $\\wedge $ is given by the contracting Whitehead moves on the elements of the set $\\underline{A_{\\sigma _i}}$ .",
"We can easily verify that the commutativity, associativity and absorption laws are verified.",
"We will use this construction to thicken all our strata.",
"Concerning notation, a thickened stratum $A$ is denoted by $A^+$ .",
"Theorem 14 Let $\\sigma _0,...,\\sigma _p$ be a set of non-generic signature, being upper bounds.",
"Then, the open sets of the Čech cover are formed by the thickened sets $\\underline{A_{\\sigma _i}}^{+}$ .",
"Indeed, to have a Čech cover, it is necessary to have open sets and that their multiple intersections are contractible.",
"We have shown that $\\underline{A_{\\sigma }}$ is contractible.",
"Applying the construction above, we thicken every stratum in $\\underline{A_{\\sigma }}$ .",
"Condition 4 of this construction implies that to cover our space we need the union of all thickened strata lying in $\\underline{A_{\\sigma _i}}$ , which we denote by $\\underline{A_{\\sigma }}^{+}$ .",
"Multiple intersections are contractible.",
"For simplicity we consider the case of two intersecting sets, where $A=\\underline{A_{\\sigma }}^{+}$ and $B= \\underline{A_{\\sigma ^{\\prime }}}^{+}$ .",
"We have shown in the lemma REF that if the intersection is non-empty then there exists a unique greatest lower bound.",
"Therefore, by the deformation retract lemma REF , all the signatures in $A\\cap B$ retract onto this greatest lower bound.",
"Theorem REF states that the stratum is contractible.",
"The generalisation can be easily obtained by induction.",
"Therefore, multiple intersections are contractible."
],
[
"Superimposition of signatures",
"Let $\\sigma _0 \\cup \\dots \\cup \\sigma _p$ be compatible generic signatures and $\\Theta $ denote an admissible superimposition.",
"In this subsection, we digress briefly in order to give a visual description of the conditions on the superimposition $\\Theta $ for the associated graph to be a signature.",
"In fact, it is quite rare for signatures to intersect.",
"Almost always the canonical graph will not be a signature.",
"Given a red polygon and a blue polygon of $\\Theta $ , they must either be disjoint or intersect in one of exactly four possible ways: the intersection is a three sided polygon with two red (resp.",
"blue) edges and one blue edge (resp.",
"red); [scale=1.2] [blue, name=B4B9] (4*180/6:1) .. controls(4*180/6:0.1) and (9*180/6:0.7) .. (9*180/6:1) ; [red, name=R2R6] (2*180/6:1) .. controls(2*180/6:0.7) and (6*180/6:0.7) .. (6*180/6:1) ; [red, name=R6R11] (6*180/6:1) .. controls(6*180/6:0.7) and (11*180/6:0.7) .. (11*180/6:1) ; The intersection is a four sided polygon with two red and two blue edges; [scale=1.2] [blue, name=B0B1] (0*180/6:1) .. controls(0*180/6:0.7) and (1*180/6:0.7) .. (1*180/6:1) ; [blue, name=B0B11] (0*180/6:1) .. controls(0*180/6:0.7) and (11*180/6:0.7) .. (11*180/6:1) ; [blue, name=B1B5] (1*180/6:1) .. controls(1*180/6:0.7) and (5*180/6:0.7) .. (5*180/6:1) ; [blue, name=B7B11] (7*180/6:1) .. controls(7*180/6:0.7) and (11*180/6:0.7) .. (11*180/6:1) ; [blue, name=B5B6] (5*180/6:1) .. controls(5*180/6:0.7) and (6*180/6:0.7) .. (6*180/6:1) ; [blue, name=B6B7] (6*180/6:1) .. controls(6*180/6:0.7) and (7*180/6:0.7) .. (7*180/6:1) ; [red, name=R3R4] (3*180/6:1) .. controls(3*180/6:0.7) and (4*180/6:0.7) .. (4*180/6:1) ; [red, name=R2R3] (2*180/6:1) .. controls(2*180/6:0.7) and (3*180/6:0.7) .. (3*180/6:1) ; [red, name=R4R8] (4*180/6:1) .. controls(4*180/6:0.7) and (8*180/6:0.7) .. (8*180/6:1) ; [red, name=R2R10] (2*180/6:1) .. controls(2*180/6:0.7) and (10*180/6:0.7) .. (10*180/6:1) ; [red, name=R9R8] (9*180/6:1) .. controls(9*180/6:0.7) and (8*180/6:0.7) .. (8*180/6:1) ; [red, name=R9R10] (9*180/6:1) .. controls(9*180/6:0.7) and (10*180/6:0.7) .. (10*180/6:1) ; The intersection is two triangles joined at a point, formed by two crossing blue (resp.",
"red) diagonals, cut transversally on either side of the intersection by two red (resp.",
"blue) diagonals; note that this means that two polygons of the same color meet at a point and both intersect the polygon of the other color; [scale=1.2] [blue, name=B0B1] (0*180/6:1) .. controls(0*180/6:0.7) and (1*180/6:0.7) .. (1*180/6:1) ; [blue, name=B0B11] (0*180/6:1) .. controls(0*180/6:0.7) and (11*180/6:0.7) .. (11*180/6:1) ; [blue, name=B1B7] (1*180/6:1) .. controls(1*180/6:0.7) and (7*180/6:0.7) .. (7*180/6:1) ; [blue, name=B5B11] (5*180/6:1) .. controls(5*180/6:0.7) and (11*180/6:0.7) .. (11*180/6:1) ; [blue, name=B5B6] (5*180/6:1) .. controls(5*180/6:0.7) and (6*180/6:0.7) .. (6*180/6:1) ; [blue, name=B6B7] (6*180/6:1) .. controls(6*180/6:0.7) and (7*180/6:0.7) .. (7*180/6:1) ; [red, name=R3R4] (3*180/6:1) .. controls(3*180/6:0.7) and (4*180/6:0.7) .. (4*180/6:1) ; [red, name=R2R3] (2*180/6:1) .. controls(2*180/6:0.7) and (3*180/6:0.7) .. (3*180/6:1) ; [red, name=R4R8] (4*180/6:1) .. controls(4*180/6:0.7) and (8*180/6:0.7) .. (8*180/6:1) ; [red, name=R2R10] (2*180/6:1) .. controls(2*180/6:0.7) and (10*180/6:0.7) .. (10*180/6:1) ; [red, name=R9R8] (9*180/6:1) .. controls(9*180/6:0.7) and (8*180/6:0.7) .. (8*180/6:1) ; [red, name=R9R10] (9*180/6:1) .. controls(9*180/6:0.7) and (10*180/6:0.7) .. (10*180/6:1) ; The intersection is a point at which two blue diagonals and two red diagonals all cross in the cyclic order red, red, blue, blue, red, red, blue, blue; note that this means that in fact four polygons meet at a point, each being the same color as the opposing one.",
"[scale=1.2] [blue, name=B0B1] (0*180/6:1) .. controls(0*180/6:0.7) and (1*180/6:0.7) .. (1*180/6:1) ; [blue, name=B0B11] (0*180/6:1) .. controls(0*180/6:0.7) and (11*180/6:0.7) .. (11*180/6:1) ; [blue, name=B1B7] (1*180/6:1) .. controls(1*180/6:0.7) and (7*180/6:0.7) .. (7*180/6:1) ; [blue, name=B5B11] (5*180/6:1) .. controls(5*180/6:0.7) and (11*180/6:0.7) .. (11*180/6:1) ; [blue, name=B5B6] (5*180/6:1) .. controls(5*180/6:0.7) and (6*180/6:0.7) .. (6*180/6:1) ; [blue, name=B6B7] (6*180/6:1) .. controls(6*180/6:0.7) and (7*180/6:0.7) .. (7*180/6:1) ; [red, name=R3R4] (3*180/6:1) .. controls(3*180/6:0.7) and (4*180/6:0.7) .. (4*180/6:1) ; [red, name=R2R3] (2*180/6:1) .. controls(2*180/6:0.7) and (3*180/6:0.7) .. (3*180/6:1) ; [red, name=R2R8] (2*180/6:1) .. controls(2*180/6:0.7) and (8*180/6:0.7) .. (8*180/6:1) ; [red, name=R4R10] (4*180/6:1) .. controls(4*180/6:0.7) and (10*180/6:0.7) .. (10*180/6:1) ; [red, name=R9R8] (9*180/6:1) .. controls(9*180/6:0.7) and (8*180/6:0.7) .. (8*180/6:1) ; [red, name=R9R10] (9*180/6:1) .. controls(9*180/6:0.7) and (10*180/6:0.7) .. (10*180/6:1) ; From such a disposition of diagonals, we obtain the graph described above.",
"The graph must be a forest with even valency at every non-terminal vertex.",
"In the following we show how to construct a canonical graph from the superimposition of a red and a blue polygon, using the second superimposition above.",
"Figure: Admissible superimpositionFigure: Canonical graph"
]
] | 1808.08411 |
[
[
"Ensemble Learning Applied to Classify GPS Trajectories of Birds into\n Male or Female"
],
[
"Abstract We describe our first-place solution to the Animal Behavior Challenge (ABC 2018) on predicting gender of bird from its GPS trajectory.",
"The task consisted in predicting the gender of shearwater based on how they navigate themselves across a big ocean.",
"The trajectories are collected from GPS loggers attached on shearwaters' body, and represented as a variable-length sequence of GPS points (latitude and longitude), and associated meta-information, such as the sun azimuth, the sun elevation, the daytime, the elapsed time on each GPS location after starting the trip, the local time (date is trimmed), and the indicator of the day starting the from the trip.",
"We used ensemble of several variants of Gradient Boosting Classifier along with Gaussian Process Classifier and Support Vector Classifier after extensive feature engineering and we ranked first out of 74 registered teams.",
"The variants of Gradient Boosting Classifier we tried are CatBoost (Developed by Yandex), LightGBM (Developed by Microsoft), XGBoost (Developed by Distributed Machine Learning Community).",
"Our approach could easily be adapted to other applications in which the goal is to predict a classification output from a variable-length sequence."
],
[
"Introduction",
"The Animal Behavior Challenge was organized by the 2018 Symposium on Systems Science of Bio-Navigation http://navi-science.org/2017/09/19/symposium-on-systems-science-of-bio-navigation-2018/, sponsored by Technosmart http://www.technosmart.eu/index.php and proposed as a CodaLab competition https://competitions.codalab.org/competitions/16283.",
"It consisted in classifying the gender of shearwater based on trajectories (latitude and longitude) and some meta-information associated to each shearwaters' trip.",
"Such prediction models could help to understand shearwater more efficiently and how they navigate themselves, like male and female shearwater could use different trajectories along the way of trip.",
"The training dataset is composed of all the GPS trajectories of 631 streaked shearwaters (326 male and 305 female) breeding on Awashima Island, Japan.",
"Each datapoints in the training dataset representing a complete bird trip and being composed of the following attributes https://competitions.codalab.org/competitions/16283#participate-get_data : longitude latitude sun azimuth: clockwise from the North sun elevation: upward from the horizon daytime: 1 being day, or 0 being night elapsed time: after starting the trip local time: only time with a format (hh:mm:ss) days: days after the trip starts In the competition setup, the testing dataset is composed of all the GPS trajectories of 275 streaked shearwaters.",
"In the Development Phase of the competition 10% of the submission labels are randomly modified to report score.",
"In the Final Phase of the competition, all the submission are recalculated for the final ranking.",
"Our approach uses extensive feature engineering prior to ensemble learning.",
"Section describes feature engineering techniques.",
"Section introduces our winning model, which is based on a ensemble learning architecture.",
"Section 4 and Section 5 compares and analyses our various models quantitatively and qualitatively on the competition dataset set.",
"The source code of our first-place solution can be found online https://github.com/dfayzur/Animal-Behavior-Challenge-ABC2018."
],
[
"Basic New Features",
"We first created velocity, acceleration, distance features for each of the GPS points from the gives dataset.",
"At this time we have 7 key features for each GPS points to work with, such features are: velocity, acceleration, distance, longitude, latitude, azimuth, and elevation.",
"We also created the differences of features at time t to the next point at time t+1.",
"We call these features as delta of velocity, longitude, latitude, azimuth, and elevation.",
"From these 12 features, we took quintiles at 0%, 5%, 10%, 20%, 25%, 30%, 40%, 50%, 60%, 70%, 75%, 80%, 90%, 95%, and 100% of them.",
"In addition, we also calculated average, minimum, and maximum of these 12 features.",
"All these operations are applied on each sequence of GPS trajectories for each bird individually.",
"We also calculated the number of times a velocity exceeds to values (average and quintiles at 5%, 10%, 15%, 25%, 50%, 75%, 80%, 85%, 90%, 95%, and 99%) calculated over all GPS trajectories combined.",
"Finally we took first 5 longitude and latitude values of each birds' trajectories.",
"We also included Principal component analysis (PCA) on longitude, latitude, azimus, elevation, and velocity of individual birds' and added them to the final features list."
],
[
"Preparing Dataset",
"We have created two training dataset based on the features generated from REF .",
"In the first dataset, we splitted the original dataset to day and night trajectories and apply features generation on them according to features described in REF .",
"So, this operation doubles the number of features, and we call this dataset as split.",
"In the second dataset, we consider all trajectories together apply features generation process.",
"We call this dataset as together.",
"The test dataset were created on similar ways."
],
[
"Modeling",
"We have trained 10 binary predictive models on each of the dataset created in REF .",
"All together we trained 20 models with 5 fold cross validations, and predicted on the test dataset.",
"The trained models are: Gradient boosted decision trees [1]: Gradient boosting is a machine learning technique for regression and classification problems, which produces a prediction model in the form of an ensemble of weak prediction models, typically decision trees.",
"We modeled variants of Gradient boosted decision trees: XGBoost [2]: a distributed gradient boosting library designed to be highly efficient, flexible and portable.",
"We modeled two version of XGBoost with two different loss functions: binary logistic: logistic regression for binary classification, output probability.",
"(model name which we call xgb_binary) pairwise rank: set XGBoost to do ranking task by minimizing the pairwise loss.",
"(model name which we call xgb_rank) LightGBM [3]: an effective gradient boosting decision tree which contains two novel techniques: Gradient-based One-Side Sampling and Exclusive Feature Bundling to deal with large number of data instances and large number of features respectively.",
"We modeled two version of LightGBM with two different boosting types: gbdt: traditional Gradient Boosting Decision Tree.",
"(model name which we call lgb_gbdt) rf: Random Forest.",
"(model name which we call lgb_rf) CatBoost [4]: a state-of-the-art gradient boosting on decision trees library, which support for both numerical and categorical features.",
"(model name which we call cat) GradientBoostingClassifier [1]: a Gradient Boosting for classification algorithm from scikit-learn [5] library.",
"It builds an additive model in a forward stage-wise fashion and it allows for the optimization of arbitrary differentiable loss functions.",
"(model name which we call sk_gbt) RandomForestClassifier [6]: a meta estimator that fits a number of decision tree classifiers on various sub-samples of the dataset to improve the predictive accuracy and control over-fitting.",
"This algorithm is from scikit-learn [5] library.",
"(model name which we call sk_rf) ExtraTreesClassifier [7]: a meta estimator that fits a number of randomized decision trees on various sub-samples of the dataset to improve the predictive accuracy and control over-fitting.",
"This algorithm is from scikit-learn [5] library.",
"(model name which we call sk_et) SVC [8]: a libsvm based Support Vector Machines estimator from scikit-learn [5] library.",
"(model name which we call sk_svc) GPC [9]: a probabilistic predictions with Gaussian process classification estimator from scikit-learn [5] library.",
"(model name which we call sk_gpc) The model parameters are identified by 5 fold cross validations on each of the algorithms on both together and split dataset.",
"Thus all the model name will be prefixed with together and split to distinguish.",
"Models estimated on together dataset are called: together_xgb_rank, together_xgb_binary, together_lgb_gbdt, together_lgb_rf, together_cat, together_sk_gbt, together_sk_rf, together_sk_et, together_sk_svc, and together_sk_gpc.",
"Models estimated on split dataset are called: split_xgb_rank, split_xgb_binary, split_lgb_gbdt, split_lgb_rf, split_cat, split_sk_gbt, split_sk_rf, split_sk_et, split_sk_svc, and split_sk_gpc.",
"The cross validation results are described in the section .",
"Finally, we report the simple majority vote ensemble of all predictions generated from 20 trained models."
],
[
"Experimental Results",
"In the competition setup, wee needed to submit directly target on the classification decision boundary without producing the probability estimation.",
"This kind of prediction is called hard classification problem, in contrast to soft classification problem where we could submit probability estimation.",
"The evaluation matric is Accuracy which is also hard to optimize if the prediction problem is unbalanced.",
"We decided to train all the models listed in REF to optimize for better F1 score [10].",
"The F1 score is the harmonic average of the precision and recall, often perform better in classification problem settings.",
"Thus, F1 score was the stoping criterion at the time of finding optimal parameters of each models."
],
[
"Cross Validation",
"As the competition testing dataset is small and submission of predicted classification is randomly modified and reported, we can not reliably compare models on leaderboard results.",
"Thus we decided stick on the cross validation F1 score to tune models and select models in the development phase.",
"A 5 fold cross validation [11] strategies was used throughout the development phase.",
"The 5 fold cross validation split is same for all models and for both custom together and split dataset.",
"Table REF shows the 5 fold cross validation scores of our various models on our custom dataset as well as on the ensemble ones.",
"The different hyperparameters of each model have been tuned and used in the final submission models in subsection REF .",
"Table: 5 fold cross validation F1 score"
],
[
"Final Submission",
"To reduce the effect of random seeds in the 20 models setting, we simulated each model settings for 10 times with 10 different random seeds.",
"For each model setting, we used the hyperparameters found in the previous subsection REF except number of iterations of a particular model, which the model determine dynamically.",
"Finally, we applied simple majority vote ensemble [12] on all 200 hard prediction on the test dataset.",
"Table REF shows the testing scores from the final phase of the competition https://competitions.codalab.org/competitions/16283#results.",
"Table: Models accuracy on test dataset (Final standing)"
],
[
"Conclusion",
"We introduced an ensemble learning approach to predict gender of shearwater based on trajectory and associated metadata.",
"The ensemble learning approach was able to model trajectories to output the predictions.",
"One potential limitation of our approach is not considering cluster based information, which might be valuable to increase accuracy.",
"As a future direction, we want to make cluster of grids from the trajectories and analyze the problem.",
"One more interesting feature to see how 3D (latitude, longitude, and elevation) clustering might help increase accuracy.",
"In aligned with this, it would be good to have date of the birds's trajectories, so that we might benefit from seasonal pattern of water level (elevation)."
]
] | 1808.08613 |
[
[
"The Ultra-Fast Outflow of WKK 4438: Suzaku and NuSTAR X-ray Spectral\n Analysis"
],
[
"Abstract Previous X-ray spectral analysis has revealed an increasing number of AGNs with high accretion rates where an outflow with a mildly relativistic velocity originates from the inner accretion disk.",
"Here we report the detection of a new ultra-fast outflow (UFO) with a velocity of $v_{\\rm out}=0.319^{+0.005}_{-0.008}c$ in addition to a relativistic disk reflection component in a poorly studied NLS1 WKK~4438, based on archival \\nustar and \\suzaku observations.",
"The spectra of both \\suzaku and \\nustar observations show an Fe~\\textsc{xxvi} absorption feature and the \\suzaku data also show evidence for an Ar~\\textsc{xviii} with the same blueshift.",
"A super-solar argon abundance ($Z^{\\prime}_{\\rm Ar}>6Z_{\\odot}$) and a slight iron over-abundance ($Z^{\\prime}_{\\rm Fe}=2.6^{+1.9}_{-2.0}Z_{\\odot}$) are found in our spectral modelling.",
"Based on Monte-Carlo simulations, the detection of the UFO is estimated to be around at 3$\\sigma$ significance.",
"The fast wind most likely arises from a radius of $\\geq20r_g$ away from the central black hole.",
"The disk is accreting at a high Eddington ratio ($L_{\\rm bol}=0.4-0.7L_{\\rm Edd}$).",
"The mass outflow rate of the UFO is comparable with the disk mass inflow rate ($\\dot M_{\\rm out}>30\\%\\dot M_{\\rm in}$), assuming a maximum covering factor.",
"The kinetic power of the wind might not be high enough to have influence in AGN feedback ($\\dot E_{\\rm wind}/L_{\\rm bol}\\approx 3-5\\%$) due to a relatively small column density ($12^{+9}_{-4}\\times10^{22}$~cm$^{-2}$).",
"However note that both the inferred velocity and the column density could be lower limits owing to the low viewing angle ($i=23^{+3}_{-2}$$^{\\circ}$)."
],
[
"Introduction",
"Recently there has been increasing evidence showing the presence of absorption line features above 7 keV in the X-ray band of various sources, including Active Galactic Nuclei (AGNs) , , [2], , , .",
"These absorption line features are commonly interpreted as blueshifted Fe xxv or Fe xxvi K absorption in a highly ionized environment ($\\log (\\xi /$ erg cm s$^{-1}$ )>3) and can occasionally correspond to a very large line-of-sight outflow velocity of up to $0.2-0.4c$ .",
"Such outflows are often referred to as ultra-fast outflows (UFOs), and lie in the mildly relativistic regime indicating that the outflow is driven from the inner accretion disk.",
"There are two key technical challenges when it comes to searching for ultra-fast outflows.",
"The first is that they are close to the upper limit of the instrumental effective area of most soft energy cameras, such as Suzaku XIS and XMM-Newton EPIC.",
"Elements lighter than iron are generally fully ionized and therefore show weak or absent absorption features in the soft energy band.",
"Only strong iron absorption feature remains above 7 keV where the signal-to-noise (S/N) and the spectral resolution are worse than in the soft band.",
"The second is that the broadband continuum needs to be correctly modeled in order robustly determine the key absorption parameters .",
"This is particularly critical for AGN sources, which often exhibit complex X-ray spectra with strong reflection.",
"One popular theory for the origin of these extreme outflows is that the radiation pressure due to a high accretion rate drives the UFO (e.g.",
"PDS 456; ).",
"It is therefore interesting to note the discovery of UFOs in ultraluminous X-ray sources , , , which appear to be sources accreting above their Eddington limit.",
"Another ideal population to test this theory are Narrow Line Seyfert 1 (NLS1) galaxies.",
"NLS1s are characterized by having low-mass, high-accretion-rate black holes in the center.",
"For example, IRAS 13224$-$ 3809 accretes around the Eddington limit and shows a flux-dependent blueshifted Fe absorption feature above 8 keV , , , which is interpreted as a UFO with velocity up to $0.236\\pm 0.006c$ .",
"WKK 4438 is a nearby (z=0.016) NLS1 galaxy hosting a low-mass supermassive black hole $M_{\\rm BH}=2\\times 10^{6}M_{\\odot }$ .",
"In this work, we analyse its archival X-ray spectra obtained by Suzaku and NuSTAR satellites, which show blueshifted Fe xxvi and Ar xviii absorption lines in addition to a relativistic reflection component."
],
[
"Data Reduction",
"WKK 4438 has not been studied well in the X-ray band.",
"The only long soft band observation is one 70 ks Suzaku observation (obsID 703011010), taken in 2012.",
"The only archival NuSTAR observation (obsID 60061259002) is a 20 ks snapshot in 2013."
],
[
"We reduced the Suzaku data with the latest HEASOFT software package.",
"We processed the raw event files for each XIS CCD and created filtered event lists for XIS0, XIS1 and XIS3 detectors by running the Suzaku pipeline.",
"The CALDB version we use for XIS detectors is 20160607.",
"XSELECT was used to extract the spectral products.",
"The size of the circular region we used to extract the source regions is 3.5$^{\\prime }$ in radius and the background regions were selected from the surrounding areas free from the target source and the calibration source.",
"XISRESP was used to generate the corresponding response files with ‘medium’ resolution.",
"The spectra and the response files of the front-illuminated CCDs (XIS0 and XIS3) were combined by using ADDSPEC.",
"We grouped the spectra to have a minimum count of 50 per bin with GRPPHA.",
"Hereafter, the combined spectrum of the front-illuminated CCD XIS0 and XIS3 is called FI spectrum (blue in figures) and the spectrum of the back-illuminated CCD XIS1 is called BI spectrum (green in figures).",
"We analysed the FI and BI data over the 0.7–10 keV and 0.7–9 keV energy range, respectively.",
"The 1.7–2.5 keV band is ignored in both spectra due to calibration issues around the Si K edge https://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/sical.html."
],
[
"The NuSTAR data were reduced using the NuSTAR data analysis software NuSTARDAS and CALDB version 20161021.",
"In addition to the standard science (mode 1) event files, we also extracted the products from the spacecraft science (mode 6) event files by following to maximize the exposure of the observation and thus increase the S/N of our FPM spectra.",
"The mode 6 events contribute an additional 4 ks good exposure in addition to the mode 1 events.",
"The total net exposure is 26 ks.",
"The source spectra were selected from circular regions with radii of 60$^{\\prime \\prime }$ , and the background was obtained from nearby circular regions with radii of 120$^{\\prime \\prime }$ .",
"Spectra were extracted from the cleaned event files using NUPRODUCTS for both FPMA and FPMB.",
"The FPMA and FPMB spectra were grouped to have a minimum count of 50 per bin by using GRPPHA.",
"The 3.0–50.0 keV band is considered as the background spectrum dominates above 50 keV.",
"Two spectra were analysed independently but we only show the combined spectrum in our plots (in red) for clarity."
],
[
"Spectral Analysis",
"For the spectral analysis, we use the XSPEC(12.9.1k) software package [1] to fit all the spectra discussed, and $\\chi ^2$ -statistics is considered in this work.",
"The Galactic column density towards WKK 4438 is fixed at the nominal value $4.3\\times 10^{21}$ cm$^{-2}$ if not specified.",
"The parameter values are reported in the source rest frame.",
"The tbnew model is used for calculating the Galactic absorption.",
"The solar abundance is taken from .",
"The cross section is taken from ."
],
[
"Spectral Modelling",
"First, we fitted the Suzaku and NuSTAR spectra independently with a Galactic absorbed powerlaw model.",
"The top panel of Fig.",
"REF shows the ratio plot against the best-fit powerlaw model ($\\Gamma =2.00, 1.76$ for Suzaku and NuSTAR respectively).",
"An Fe K$\\alpha $ emission line feature in two sets of spectra and a Compton hump above 15 keV in the FPM spectra indicate a strong reflection component.",
"By fitting the emission feature at the iron band with a simple zgauss line model, it improves the fit by $\\Delta \\chi ^2=71$ .",
"The best-fit line width ($\\sigma $ ) values are $0.41^{+0.13}_{-0.12}$ keV and $0.31^{+0.34}_{-0.16}$ keV for XIS and FPM spectra respectively, indicating a broad Fe emission line in both two sets of spectra.",
"Both lines are broader than the instrument energy resolution.",
"The rest frame energy of the zgauss line model is $6.3\\pm 0.1$ keV for the XIS spectra and $6.3^{+0.2}_{-0.3}$ keV for the FPM spectra.",
"A broad absorption line feature is visible around 10 keV in the FPM spectra.",
"The FI spectrum also shows some evidence of a broad absorption line feature at 10 keV but it lies too close to the upper limit of the observable energy range.",
"A second absorption line feature is visible at $\\approx 4.6$ keV in both the Suzaku FI and BI spectra (see Fig.",
"REF ).",
"As far as we know, there is no obvious calibration issue during this Suzaku observation (Yukikatsu Terada, Katja Pottschmidt priv.",
"comm.).",
"All three cameras of the Suzaku satellite show consistent absorption features at 4.6 keV (see Fig.REF ).",
"Second, we fitted both the NuSTAR and Suzaku spectra with the relativistic disk reflection model relxilllp .",
"This model calculates consistent emissivity profile for the relativistic blurred disk iron line according to the coronal height in the lamp-post scenario.",
"The spin parameter is fixed at its maximum value 0.998 and the inner radius of the disk is allowed to vary.",
"The disk inner radius $R_{\\rm in}$ , viewing angle $i$ and the disk iron abundance $Z_{\\rm Fe}$ are linked between two epochs.",
"relxilllp gives a best-fit with $\\chi ^2/\\nu = 1396.4/1438$ , where $\\nu $ is the number of the degrees of freedom.",
"The relativistic blurred disk reflection model can fit the soft excess below 1 keV, the emission line at the iron band, and the Compton hump.",
"It improves the fit by $\\Delta \\chi ^2=533$ compared to the simple powerlaw+zgauss modelling.",
"However, the broad absorption line features at 10 keV and 4.6 keV are still visible in the ratio plot (see the middle panel of Fig.",
"REF and Fig.",
"REF for absorption lines).",
"Based on the best-fit relxilllp model, we fitted the absorption features with two simple line models zgauss.",
"The redshift parameter $z$ is fixed at the source redshift and the line energy is allowed to vary to obtain the line position in the source rest frame.",
"The line parameters except the normalization are all linked between two observations.",
"The inclusion of two zgauss models can improve the $\\chi ^2$ by 33 with additional 8 free parameters ($\\Delta \\chi ^2=16$ for the $4.6$ keV absorption line and $\\Delta \\chi ^2=17$ for the 10 keV absorption line).",
"The other parameters can be found in Tab.REF .",
"In this section, we model the two absorption features with one ionized fast wind absorber.",
"If the absorption line at 10 keV is interpreted as the Fe xxvi Ly$\\alpha $ line (6.97 keV), it corresponds to a redshift value of $z_{\\rm Fe}=-0.28^{+0.03}_{-0.04}$ .",
"If the absorption line at $4.6$ keV is interpreted as the Ar xviii Ly$\\alpha $ line (3.32 keV), it corresponds to a redshift value of $z_{\\rm Ar}=-0.281^{+0.005}_{-0.004}$ .",
"The two redshift values are consistent within the uncertainty, corresponding to a relativistic outflow with a velocity of $v\\approx 0.3$ c. Table: The best-fit parameters of the simple line models.",
"The line energy is the value in the source rest frame.",
"The numbers in the brackets show the values obtained for the NuSTAR spectra.Finally, we modelled the absorption features in both the NuSTAR and Suzaku spectra with the same photoionized absorption model xstar .",
"The warmabs model, the alternative analytical version of xstar, is used first to estimate the line width of the absorption lines.",
"We fitted with the turbulent velocity as a free parameter and obtained a best-fit value of $v_{\\rm tur}>6000$ km s$^{-1}$ at the 2$\\sigma $ confidence level.",
"Then we constructed custom absorption models with xstar.",
"The grids are calculated assuming solar abundances except for that of iron and argon, a fixed turbulent velocity of 6000 km s$^{-1}$ and an ionizing luminosity of $10^{43}$ erg s$^{-1}$ .",
"Free parameters are the ionization of the plasma ($\\log \\xi ^{\\prime }$ ), the column density ($N^{\\prime }_{\\rm H}$ ), the iron abundance ($Z^{\\prime }_{\\rm Fe}$ ), the argon abundance ($Z^{\\prime }_{\\rm Ar}$ ) and the redshift (z).",
"The prime symbol is to distinguish the parameter of the outflow from that of the disk.",
"The iron and argon abundances of the UFO ($Z^{\\prime }_{\\rm Ar,Fe}$ ) are treated as independent free parameters during our fit.",
"The total model reads tbnew*xstar*relxilllp in XSPEC format.",
"The best-fit model parameters obtained by fitting the two sets of spectra with tbnew*xstar*relxilllp can be found in Table REF .",
"The ratio plot can be found in the bottom panel of Fig.REF .",
"One xstar model with a turbulent velocity of $v_{\\rm tur}=6000$ km s$^{-1}$ can fit two absorption lines well simultaneously and improve the fit by $\\Delta \\chi ^2=33$ with additional 5 free parameters after stepping the redshift parameter $z$ between -0.35 and 0 with 70 steps with all the other parameters free to vary.",
"See the left panel of Fig.REF for the constraint on the redshift parameter $z$ .",
"The STEPPAR function in XSPEC is used for the stepping purpose.",
"The best-fit xstar model is shown in the bottom panel of Fig.REF ."
],
[
"Alternative scenarios for the absorptions",
"In this section, we test other possible solutions for the two absorption features.",
"First, an alternative interpretation of the 10 keV absorption feature is the Fe K edge from a slower wind.",
"We fitted the 10 keV absorption line with a simple edge absorption model zedge and obtained an improved fit with $\\Delta \\chi ^2=7.4$ .",
"The edge model provides a worse fit of the 10 keV absorption feature than the line absorption model ($\\Delta \\chi ^2=17$ , see Table REF ).",
"Therefore, we exclude the edge interpretation of the 10 keV absorption feature.",
"Second, an alternative identification of the 4.6 keV absorption feature could be a blueshifted Ca xx absorption line, with a lower blueshift value compared to the Ar xviii interpretation.",
"The rest frame energy of the dominant Ca xx Ly$\\alpha $ 1 absorption line is around 4.11 keV.",
"A very high relativistic outflowing velocity of $v=0.166^{+0.008}_{-0.009}c$ is still required in addition to a fast wind component obtained in Section REF .",
"In this case, a highly super-solar Ca abundance ($Z^{\\prime }_{\\rm Ca}>70Z_{\\odot }$ ) is required to fit the 4.6 keV line, given the lack of the other absorption lines.",
"So although the fast wind model does require an unusual Ar abundance, the slower wind model requires an even more extreme abundance ratio.",
"Third, we explored the possibility of a warm absorber solution for the 4.6 keV absorption feature by limiting the velocity of the warmabs absorber within $v$ <1000 km s$^{-1}$ and assuming solar abundances.",
"Warm absorbers are partially ionzied, optically thin, circumnuclear materials causing a series of absorption features and are not uncommn among nearby AGNs , , .",
"However, the additional warmabs model fails to improve the fit of the 4.6 keV absorption line.",
"Because no strong absorption lines of other elements are produced around 4.6 keV at an ionization state of $\\log (\\xi )=0-4$ without any other absorption features being produced simultaneously.",
"For example, if the 4.6 keV absorption line is interpreted as the Ca xix absorption line at 4.58 keV at an ionization state of $\\log (\\xi )=2$ , a stronger Ca xiv absorption line at 3.76 keV with an optical depth 1000 times larger than Ca xix is expected at the same ionization state but not detected in the spectra.",
"Fourth, another possible origin of the 4.6 keV line feature is the Fe xxv/xxvi absorption from a relativistic inflow .",
"An inflowing velocity of $0.38c$ and $0.35c$ is required if the absorption feature corresponds to the Fe xxv and Fe xxvi line respectively.",
"However, absorption features from an inflow with such a high velocity ($>0.35c$ ) have not been confirmed in other sources.",
"In conclusion, we argue that the fast wind model that can account for both the 4.6 keV and 10 keV absorption features simultaneously is preferred.",
"According to the best-fit reflection model, the disk is truncated with an inner radius of $R_{\\rm in}=15^{+8}_{-5} r_{g}$ .",
"The disk viewing angle is well constrained at $i=23^{+3}_{-2}$$^{\\circ }$ with respect to the normal direction of the disk.",
"An additional distant reflection model xillver with the abundance linked to the disk reflection component only improves the fit by $\\Delta \\chi ^2=4$ and gives a disk viewing angle of $i=24^{+4}_{-5}$$^{\\circ }$ .",
"The absorber model xstar prefers the Fe xxvi interpretation of the broad absorption line at 10 keV, as it can fit the absorption lines at 4.6 keV and 10 keV simultaneously (see the left panel of Fig.REF ).",
"The outflow has a column density of $N^{\\prime }_{\\rm H} = 12^{+9}_{-4}\\times 10^{22}$ cm$^{-3}$ , an ionization of $\\log (\\xi ^{\\prime }$ /erg cm s$^{-1})$ = $3.9^{+0.4}_{-0.3}$ and a redshift of $z=-0.270\\pm 0.006$ , corresponding to a line-of-sight outflowing velocity $v=-0.319^{+0.005}_{-0.008}c$ accounting for the relativistic corrections and the cosmological redshift of WKK 4438.",
"When the UFO argon abundance $Z^\\prime _{\\rm Ar}$ and iron abundance $Z^\\prime _{\\rm Fe}$ are treated as independent free parameters, a lower limit on the argon abundance is obtained ($Z^\\prime _{\\rm Ar}>6$ ) and a slight iron over-abundance is required compared to solar ($Z^\\prime _{\\rm Fe}=2.6^{+1.9}_{-2.0}$ ).",
"Note that the abundances are probably degenerate with the column density.",
"More discussion about the uncommon argon abundance can be found in Section .",
"Table: The best-fit model parameters.",
"The model is tbnew*xstar*relxilllp.",
"The tbnew model is used to account for the Galactic absorption.",
"The 3–10 keV band flux of the Suzaku observation is (9.01±0.03)×10 -12 (9.01\\pm 0.03)\\times 10^{-12}erg cm -2 ^{-2} s -1 ^{-1}.",
"The same band flux of the NuSTAR observation is (5.15±0.08)×10 -12 (5.15\\pm 0.08)\\times 10^{-12}erg cm -2 ^{-2} s -1 ^{-1}.Figure: The ratio plots against the best-fit Galactic absorbed powerlaw (top), relxilllp (middle) and xstar*relxilllp model (bottom).",
"Blue: Suzaku FI spectrum; red: combined FPM spectrum.",
"Suzaku FI and BI spectra are both analysed in our work.",
"Only FI spectrum is shown here.",
"FPMA and FPMB are fitted independently and the combined spectrum is shown only for clarity.",
"All the spectra show a fairly broad Fe Kα\\alpha emission line at the iron band.",
"A Compton hump is shown above 15 keV.",
"The NuSTAR spectra show a broad absorption line feature around 10 keV in the rest frame which is consistent with the Suzaku FI spectrum.Figure: The ratio plot against the best-fit relxilllp model.",
"Blue: Suzaku FI spectrum; green: Suzaku BI spectrum.",
"Both two Suzaku spectra show an absorption line around 4.6 keV in the rest frame, which can be explained as a blueshifted Ar xviii absorption line.Figure: Top left: the constraint on the redshift parameter zz of the absorber model xstar.",
"Top right: the constraint on the Ar abundance Z Ar ' Z^{\\prime }_{\\rm Ar} of the absorber when being fitted as a free independent parameter as well as Z Fe ' Z^{\\prime }_{\\rm Fe}.",
"Bottom: The best-fit xstar model applied to a simple powerlaw continuum.",
"The detected lines are marked in red."
],
[
"Monte-Carlo Simulations",
"In this section, we present our estimation on significance of the outflow detection in the current archival data based on Monte-Carlo simulations.",
"We simulated 7000 sets of Suzaku (FI and BI) and NuSTAR (FPMA and FPMB) spectra using the best-fit relxilllp model discussed in Section .",
"The same exposures and the flux levels as the real observations are considered.",
"The key reflection model parameters used are the best-fit values obtained for the relxilllp-only model in Section REF (h=5.06 $r_g$ , $i=23$$^{\\circ }$ , $R_{\\rm in}=16r_{g}$ , and $Z_{\\rm Fe}=0.91$ ).",
"The reflection fraction is $R_{\\rm refl}=1.52$ for the Suzaku simulations and $R_{\\rm refl}=2.38$ for NuSTAR simulations.",
"The powerlaw continuum has a photon index of $\\Gamma =2.05$ for Suzaku simulations and $\\Gamma =1.99$ for NuSTAR simulations.",
"The fakeit command is used to simulate the spectra.",
"All the spectra are grouped to have a minimum count of 50 per bin using GRPPHA.",
"We analysed each set of simulated spectra with both relxilllp and xstar*relxilllp models and measure the distribution of the statistical improvements achieved by adding the xstar absorber.",
"All the parameters in Table REF are free to vary during our spectral fitting processes.",
"We conducted a search for any absorption features in each set of simulated spectra by stepping the redshift parameter $z$ of the xstar model in the same way as in the real spectral analysis.",
"The inclusion of the xstar model improves the fit statistic by $\\Delta \\chi ^2=33.1$ in the real spectral analysis.",
"The fit improvements in 17 out of the 7000 simulations exceed this threshold (a chance probability of 0.024), which means our detection of UFO in the current data is slightly higher than 3$\\sigma $ significance by combining two observations."
],
[
"Discussion",
"Archival Suzaku and NuSTAR data have revealed the NLS1 WKK 4438 shows evidence for both relativistic disk reflection and an ultra-fast outflow.",
"The best-fit UFO parameters are a column density of $N^{\\prime }_{\\rm H}=12^{+9}_{-4}\\times 10^{22}$ cm$^{-2}$ , an ionization state of $\\log (\\xi ^{\\prime }$ /erg cm s$^{-1})=3.9^{+0.4}_{-0.3}$ , a slight iron over-abundance ($2.6^{+1.9}_{-2.0}Z_{\\odot }$ ), an argon over-abundance (>6$Z_{\\odot }$ ), and an outflowing velocity of $v_{\\rm out}=0.319^{+0.005}_{-0.008}c$ .",
"The UFO absorption features are consistent when fitting the continuum only with the distant reflection model xillver.",
"The inferred line significance is however higher with a distant reflection modelling.",
"We discuss the physics properties of the outflow, other systematic uncertainties of the measurements, and future work in this section.",
"To calculate the disk accretion rate, we apply an average bolometric correction factor $\\kappa =20$ to the 2–10 keV flux in the Suzaku and NuSTAR observation.",
"The bolometric luminosity of WKK 4438 is estimated to be $L_{\\rm bol}=1.6-2.8\\times 10^{44}$ erg s$^{-1}=0.4-0.7L_{\\rm Edd}$ , assuming $M_{\\rm BH}=2\\times 10^6$ $M_{\\odot }$ .",
"A combination of high accretion rate and UFO is seen in other sources, such as rapidly accreting AGNs with high accretion rate (e.g.",
"IRAS 13224$-$ 3809: , , ; 1H0707$-$ 495: ; PDS 456 , ; PG 1211$+$ 143: , ), and ULXs , , .",
"The Eddington accretion rate for a black hole of $M_{\\rm BH}=2\\times 10^6$ $M_{\\odot }$ is $9\\times 10^{20}$ kg s$^{-1}$ .",
"Therefore the mass accretion rate of WKK 4438 is $\\dot{M}_{\\rm in}\\approx 0.006-0.01M_{\\odot }$ yr$^{-1}$ .",
"We estimate the mass outflow rate by following .",
"A lower limit on the location of the wind can be derived from $r=2GM_{\\rm BH}/v_{\\rm out}^2\\approx 5.8\\times 10^{10}$ m, which means that the wind is launched at $r\\ge 20r_g$ away from the central SMBH.",
"The mass outflow rate of the wind is $\\dot{M}_{\\rm out}=4 \\pi \\mu m_{p} r N_{\\rm H} C_{\\rm F} v_{\\rm out}$ , where $\\mu =1.4$ is the average atomic mass per proton, $C_{\\rm F}$ is the covering factor of the wind, and $m_{p}$ is the proton mass.",
"Therefore, the lower limit on the mass outflow rate is $\\dot{M}_{\\rm out}\\approx 0.003C_{\\rm F}M_{\\odot }$ yr$^{-1}$ .",
"The covering factor $C_{\\rm F}$ of the wind remains unknown.",
"The mass outflow rate is comparable with the mass accretion rate, assuming a maximum covering factor ($C_{\\rm F}\\approx 1$ for PDS 456 and IRAS F11119$+$ 3257 ).",
"The kinetic power of the wind is then $\\dot{E}_{\\rm wind}=(1/2)\\dot{M}_{\\rm out}v_{\\rm out}^2\\approx 9.59\\times 10^{42}$ erg s$^{-1}\\approx 3-5\\%L_{\\rm bol}$ , assuming maximum covering factor.",
"The low $\\dot{E}_{\\rm wind}/L_{\\rm bol}$ ratio is due to a small column density of the wind and indicates the energetics might not be high enough to have an influence in AGN feedback .",
"However, note that both the inferred velocity and the column density could be lower limits owing to the low inclination we infer, assuming the wind still has a roughly equatorial geometry.",
"In order to understand the high argon abundance required by the best-fit spectral model, we first investigated relative argon abundance against several other elements, such as silicon, sulfur, calcium, iron and oxygen, by fitting the spectra with the warmabs model.",
"It turns out that the spectral fitting requires a high $Z^{\\prime }_{\\rm {Ar}}/Z^{\\prime }_{\\rm {Fe}}>4.2$ in the outflow.",
"When the UFO and disk iron abundances are linked during the fit ($Z^\\prime _{\\rm Fe}=Z_{\\rm Fe}$ ), a solar iron abundance is required ($Z^{\\prime }_{\\rm Fe}=Z_{\\rm Fe}=0.96^{+0.66}_{-0.18}Z_{\\odot }$ ) with $\\chi ^2=1364.1$ , slightly worse than the fit with $Z^{\\prime }_{\\rm Fe}$ and $Z_{\\rm Fe}$ both as a free parameter.",
"The other parameter values do not change too much when the two iron abundances are linked.",
"Second, we also tried fitting the data with a single matallicity ($Z^{\\prime }$ ), with the abundance of all elements heavier than He linked together.",
"It offers a metallicity value of $Z^{\\prime }>0.6Z_{\\odot }$ with $\\chi ^2=1388.4$ .",
"Finally, if we allow for a free iron abundance in addition to a single metallicity for the rest of the heavy elements, we obtained a fit with $\\chi ^2=1381.2$ ($Z^\\prime _{\\rm Fe}<3Z_{\\odot }$ and $Z^{\\prime }>0.6Z_{\\odot }$ ) and only the 10 keV absorption feature being fitted.",
"We note that report a similar scenario, where absorption from iron and argon with a common blueshift, and no other lines detected, for the Seyfert galaxy NGC 7582. analysed the 1H0707-495 XMM-Newton spectra and found a possible P Cygni profile of H-like Ar in the 2008 observations, where the redshift of the Ar line feature is however different from other absorption lines.",
"In contrast, another NLS1 IRAS 13224$-$ 3809 shows a series of absorption lines in the middle energy band, including Ne x, S xvi and Si xiv absorption lines, but shows no evidence of Ar xviii absorption line in the spectra .",
"Although, the variability spectrum of IRAS 13224$-$ 3809 shows some evidence of Ar xviii feature .",
"By fitting the spectra with the relativistic reflection model relxilllp, we obtain a disk viewing angle of $i=23^{+3}_{-2}$$^{\\circ }$ .",
"Such a small viewing angle is unusual for a source with visible UFO absorption features due to the opening angle of the wind.",
"Other sources where UFO is detected show evidence of large disk viewing angle when being fitted with relativistic disk reflection model, such as 1H0707$-$ 495 , PDS 456 , IRAS 13224$-$ 3809 .",
"An alternative explanation of the absorption features is ionized materials corotating above the disk, where the relativistic velocities occur naturally .",
"Such a model has been successfully applied to PG 1211$+$ 143 and IRAS 13224$-$ 3809 (Fabian et al.",
"in prep).",
"However, this scenario seems to be unlikely for WKK 4438 due to its small viewing angle and truncated disk.",
"It is now becoming apparent that these extreme outflows are variable phenomena in various sources, such as IRAS 13224$-$ 3809 , , Mrk 509 [3], PDS 456 , , , and PG1211$+$ 143 , .",
"In many of these cases, the black hole masses are high (e.g.",
"$M_{\\rm BH}\\approx 10^9$ M$_{\\odot }$ for PDS 456 and $M_{\\rm BH}\\approx 10^8$ M$_{\\odot }$ for PG1211$+$ 143).",
"However, similar to IRAS 13224$-$ 3809, the mass of the black hole in WKK 4438 is rather low ($M_{\\rm BH}\\approx 2\\times 10^6$ $M_{\\odot }$ , ).",
"This means the outflow in WKK 4438 is potentially of particular interest, as variability timescales are generally expected to scale with black hole mass.",
"It may therefore be possible to study the variability of the outflow in WKK 4438 – and any potential response to intrinsic changes in the source – with a few deep observations in the future.",
"Unfortunately, the data currently available do not have sufficient signal-to-noise or broadband coverage to undertake more studies at the present time."
],
[
"Acknowledgements",
"J.J. acknowledges support by the Cambridge Trust and the Chinese Scholarship Council Joint Scholarship Programme (201604100032).",
"D.J.W.",
"acknowledges support from an STFC Ernest Rutherford fellowship.",
"M.L.P.",
"is supported by a European Space Agency (ESA) Research Fellowship.",
"A.C.F.",
"acknowledges support by the ERC Advanced Grant 340442.",
"This work made use of data from the NuSTAR mission, a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by NASA, and data obtained from the Suzaku satellite, a collaborative mission between the space agencies of Japan (JAXA) and the USA (NASA).",
"This research has made use of the NuSTAR Data Analysis Software (NuSTARDAS) jointly developed by the ASI Science Data Center and the California Institute of Technology."
]
] | 1808.08398 |
[
[
"Novel Time Asynchronous NOMA schemes for Downlink Transmissions"
],
[
"Abstract In this work, we investigate the effect of time asynchrony in non-orthogonal multiple access (NOMA) schemes for downlink transmissions.",
"First, we analyze the benefit of adding intentional timing offsets to the conventional power domain-NOMA (P-NOMA).",
"This method which is called Asynchronous-Power Domain-NOMA (AP-NOMA) introduces artificial symbol-offsets between packets destined for different users.",
"It reduces the mutual interference which results in enlarging the achievable rate-region of the conventional P-NOMA.",
"Then, we propose a precoding scheme which fully exploits the degrees of freedom provided by the time asynchrony.",
"We call this multiple access scheme T-NOMA which provides higher degrees of freedom for users compared to the conventional P-NOMA or even the modified AP-NOMA.",
"T-NOMA adopts a precoding at the base station and a linear preprocessing scheme at the receiving user which decomposes the broadcast channel into parallel channels circumventing the need for Successive Interference Cancellation (SIC).",
"The numerical results show that T-NOMA outperforms AP-NOMA and both outperform the conventional P-NOMA.",
"We also compare the maximum sum-rate and fairness provided by these methods.",
"Moreover, the impact of pulse shape and symbol offset on the performance of AP-NOMA and T-NOMA schemes are investigated."
],
[
"Introduction",
"For future radio access, significant gains in the system capacity/efficiency and quality of user experience are required.",
"In particular, the multiple access approach is a key part of radio access technology [1].",
"In [2], [3], and the references therein, non orthogonal multiple access (NOMA) is proposed as a candidate of future radio access to partially fulfill the requirements of future networks and the possibility of downlink NOMA for 5G is currently being examined by 3GPP.",
"The currently prevailing approach for multiple access lies in the category of orthogonal multiple access (OMA).",
"In 2G systems, time division multiple access (TDMA) is adopted .",
"In the 3G mobile communication systems such as W-CDMA and CDMA2000, direct sequence-code division multiple access (DS-CDMA) is used and the receiver is based on simple single-user detection using the Rake receiver.",
"OMA based on orthogonal frequency division multiple access (OFDMA) or single carrier-frequency division multiple access (SC-FDMA) is used in the 4th generation mobile communication systems such as LTE and LTE-Advanced.",
"These approaches first partition resources into orthogonal resource blocks and then assign each resource block exclusively to one user.",
"After this, the problem is reduced to a point-to-point (P2P) communication problem and then well developed single-user encoders/decoders can be applied.",
"The significant advantage of OMA methods is that their complexity is merely the complexity of single-user encoders/decoders.",
"On the other hand, assigning resource blocks exclusively can be very inefficient (in terms of achievable rate-regions) and may pose a serious fairness problem among users.",
"In contrast to OMA, NOMA allows users to utilize the same resource blocks for transmission simultaneously and therefore is potentially more efficient.",
"In fact, when evaluated under the LTE system characteristics, NOMA demonstrates significant gains over OMA systems [4], [5].",
"The problem of communicating with many receivers arises in many “downlink” scenarios such as communication from an access point to stations in WiFi or from a base station in cellular systems.",
"Although OMA approaches eliminates interference between transmissions, it does not in general achieve the highest possible transmission rates for a given packet error rate [6].",
"In fact, superposition Coding (SC) is a well-known non-orthogonal scheme that achieves the capacity on a scalar Gaussian broadcast channel [7].",
"Superposition coding is a technique of simultaneously communicating information to several receivers by a single source.",
"In other words, it allows the transmitter to send the information of multiple users at the same time and frequency.",
"At the receiver's side successive interference cancellation (SIC) is applied which exploits the differences in signal strength among the signals of interest [8].",
"The basic idea behind SIC is that user signals are successively decoded.",
"In fact, superposition coding and SIC are the optimal encoder/decoder methods for degraded broadcast channels where users can be ordered in terms of the quality of the received signals [9], [6].",
"This is of particular importance in cellular systems where the channel conditions vary significantly among users due to the near-far effect[3].",
"From an information-theoretic perspective, NOMA with a SIC is an optimal multiple access scheme from the viewpoint of the achievable multiuser capacity region, in the downlink [10], [11], [12], [13], [14] and in the uplink [15].",
"Applications of NOMA in the downlink scenario have been widely studied [16].",
"In [17] and [18], various power allocation and user scheduling algorithms were proposed to improve the sum-rate of the NOMA-based multi-user system.",
"In [19], cooperative NOMA scheme was investigated to improve the spectral efficiency and transmission reliability.",
"More recently, the study of the combination of multiple-input and multiple-output (MIMO) and NOMA has received considerable attention [20], [21].",
"However, most of the previous research on NOMA only considered symbol-synchronous transmission.",
"In fact, often in the literature, timing mismatch is considered as an impairment and different synchronization methods are applied to eliminate it [22].",
"However, in this work, we show that time asynchrony can indeed be beneficial.",
"By using proper transmission and receiver design, time asynchrony can decrease interference and also provide additional degrees of freedom which can be exploited to improve the performance.",
"The usefulness of timing offset, or time asynchrony, have been studied in the literature.",
"For example, the results in [23] show that time asynchrony can increase the capacity region in multiple-access channels.",
"Also, time asynchrony can improve the performance in other scenarios if the proper sampling and detection methods are used [24], [25], [26], [27], [28], [29], [30].",
"In this work, we introduce two schemes called AP-NOMA and T-NOMA which use time asynchrony to improve and enlarge the downlink rate-region.",
"We have the following specific contributions: We introduce two NOMA broadcast methods, AP-NOMA and T-NOMA which exploit the intentionally added timing offsets between the superimposed signals and improve the performance of the conventional P-NOMA We analytically prove that for a wide range of pulse shaping filters including, rectangular, sinc and raised cosine, AP-NOMA decreases the inter-user interference (IUI), thus improves the overall performance.",
"We demonstrate that T-NOMA can take advantage of limited time communication by appropriate precoding in order to provide higher degrees of freedom and hence improve the performance.",
"We derive the achievable rate-regions for the proposed schemes for arbitrary number of users and provide numerical results.",
"We analytically show that for the P-NOMA method to achieve the maximum sum-rate, it needs to assign all the power to the strongest user, thus violating the fairness.",
"However, AP-NOMA and T-NOMA methods can achieve the maximum sum-rate while maintaining fairness among the users.",
"The rest of the paper is organized as follows: In Section , we provide an overview of the concepts of NOMA including superposition coding and SIC.",
"Next, we provide some insights regarding the benefits of time asynchrony.",
"The concept behind the proposed method AP-NOMA is the reduction in IUI by using intentional time delays, and the concept behind T-NOMA is exploiting extra degrees of freedom provided by time asynchrony which are explained in Section .",
"Then, we present the system model and its characteristics in Section .",
"We present the achievable rate-region results for the conventional P-NOMA, modified AP-NOMA and T-NOMA in Section .",
"At the end, we provide the numerical results and the final remarks in Sections and , respectively."
],
[
"Concepts of P-NOMA",
"Simultaneous transmission of information from one source to several receivers has been studied under the title of broadcast channel[31].",
"Superposition coding at the transmitter and SIC at the receivers provide the capacity achieving performance and thus play important roles in the P-NOMA.",
"In this section, we briefly summarize the results in the literature and explain the concepts of superposition coding and SIC method.",
"Consider the Gaussian broadcast channel $\\nonumber Y_1&=X+Z_1\\\\\\nonumber Y_2&=X+Z_2$ where $Z_1$ and $Z_2$ follow Gaussian distributions, i.e., $Z_1\\sim N(0,N_1)$ and $Z_2\\sim N(0,N_2)$ , assuming that $N_2>N_1$ .",
"Theorem 1 The capacity region for the Gaussian broadcast channel, with signal power constraint $P$ , is given by: $\\nonumber R_1&\\le \\frac{1}{2}\\log \\left(1+\\frac{\\alpha P}{N_1}\\right)\\\\\\nonumber R_2&\\le \\frac{1}{2}\\log \\left(1+\\frac{(1-\\alpha ) P}{\\alpha P+N_2}\\right)$ This region is achieved by the superposition coding and SIC schemes described in [7].",
"To encode the messages, the transmitter generates two codebooks, one with power $\\alpha P$ at rate $R_1$ , and another codebook with power $(1-\\alpha )P$ at rate $R_2$ , where $R_1$ and $R_2$ lie in the capacity region above.",
"Then, to send an index $w_1\\in \\lbrace 1,2,\\cdots ,2^{nR_1}\\rbrace $ and $w_2 \\in \\lbrace 1,2,\\cdots ,2^{nR_2}\\rbrace $ to $Y_1$ and $Y_2$ , respectively, the transmitter takes the codeword $X(w_1)$ from the first codebook and codeword $X(w_2)$ from the second codebook and computes the sum.",
"The transmitter sends the sum over the channel [9].",
"An example of superposition Coding using 4-PSK and 8-QAM constellations for two users are shown in Fig.",
"REF .",
"The input bits of the user with weaker channel are modulated with 4-PSK modulation as shown by the coarse points in Fig.",
"REF and the input bits for the user with stronger channel are modulated with 8-QAM modulation and sum of the modulated symbols results in a 32-QAM constellation shown by the fine points in Fig.",
"REF .",
"Figure: An Example of Superposition CodingThe receivers must now decode the messages.",
"The weak receiver, $Y_2$ , merely looks through the second codebook to find the closest codeword to the received vector $Y_2$ .",
"His effective signal-to-noise ratio is $\\frac{(1-\\alpha )P}{\\alpha P +N_2}$ , since $Y_1$ 's message acts as noise to $Y_2$ .",
"The strong receiver, $Y_1$ , first decodes $Y_2$ 's codeword, which he can accomplish because of his lower noise $N_1$ .",
"After subtracting this codeword $\\hat{X}_2$ from $Y_1$ , the receiver looks for the codeword in the first codebook closest to $Y_1-\\hat{X}_2$ .",
"The resulting probability of error can be made as low as desired.",
"A nice dividend of optimal encoding for degraded broadcast channels is that the strong receiver $Y_1$ always knows the message intended for receiver $Y_2$ in addition to its own message [9].",
"Fig.",
"REF presents the technique for decoding the superposed signal (Fig.",
"REF ) at the receiving side.",
"As shown in Fig.",
"REF , the weak receiver only decodes the coarse points by mapping the received signal to the nearest point in the corresponding constellation (4-PSK).",
"The stronger user is also able to decode the coarse points and after subtracting the decoded symbol from the received signal, the resulting signal is decoded using the corresponding constellation (8-QAM) as shown in Fig.",
"REF .",
"Figure: An example of SIC decoding"
],
[
"Motivations Behind Asynchronous Transmission",
"It is shown in the literature that time asynchrony which is the intrinsic feature of most of the wireless networks, not only is not disruptive, but also can be beneficial if the proper processing is employed [24], [25], [26], [27], [28], [29], [30].",
"We introduce two NOMA schemes enjoying the benefits of time asynchrony, i.e., AP-NOMA and T-NOMA.",
"The details of these two methods will be elaborated later, but before that, let us briefly express the intuition and motivations behind each of the mentioned methods."
],
[
"Motivation Behind AP-NOMA",
"In conventional P-NOMA, the underlying assumption is the reception with perfect synchronization.",
"In the perfect synchronous scenario, the peak points of all users are aligned, however, by adding intentional time delays to each user, the peak points drift apart.",
"The comparison between synchronous and asynchronous reception for two users is shown in Fig.",
"REF .",
"Figure: Illustration of IUI for two cases of synchronous and asynchronous receptionDenoting the overall pulse shape, including the transmit pulse shape and the matched filter, as $g(t)$ , the interuser interference (IUI) power from interfering user can be calculated as: $IUI(\\tau )=\\sum _{i=-\\infty }^{\\infty }{|g(\\tau +iT)|^2}$ where $\\tau $ is the time delay and $T$ is the symbol interval.",
"In the next lemma, we show that adding intentional time delay will decrease the IUI power.",
"Lemma 1 For any pulse shape, denoted as g(t), with real spectrum in frequency domain (i.e., real and even in the time domain), we will have $IUI(\\tau )\\le IUI(0) \\ \\ \\ $ The proof is presented in Appendix .",
"Note that the conditions mentioned above encompasses almost all the the pulse shapes in the literature like rectangular and sinc pulse shapes and even practical pulse shapes like the raised cosine pulse shape.",
"Thus, adding time asynchrony can decrease the IUI which is the main degradation in NOMA schemes.",
"However, the benefits of time asynchrony is not limited to decreasing IUI power.",
"In fact, as we will explain later, adding time asynchrony provides additional degrees of freedom which can be exploited to serve more users over the same time and frequency resources."
],
[
"Motivation Behind T-NOMA",
"Here, we use Hilbert space formulation to show the potential of asynchronous transmission in providing additional degrees of freedom.",
"Hilbert space generalizes the Euclidean space of real numbers to finite energy signals.",
"Each finite energy signal can be represented by a vector in the Hilbert space with each coordinate given by an inner product with the corresponding orthonormal basis functions.",
"In more details, any finite energy signal like $x(t)$ can be written as a linear combination of the orthonormal basis functions as: $x(t)=\\sum _{n\\in \\Gamma }x[n]p_n(t)$ where $p_n(t)$ is an orthonormal basis function, i.e, $\\langle p_n(t).p_m(t)\\rangle =\\delta (n-m)$ and $x[n]$ is the corresponding coefficient in the direction of the basis function $p_n(t)$ which is obtained by the following inner product.",
"$x[n]=\\langle x(t).p_n(t)\\rangle $ If we further constrain our finite energy signals to strictly band-limited ones, then the Hilbert Space is called the Paley-Wiener space.",
"The Nyquist sampling theorem states that any signal in Paley-Wiener space whose Fourier transform is supported on $f \\in (-W \\ W)$ can be written as the linear combination of some sinc pulses, i.e, $x(t)=\\sum _{n=-\\infty }^{\\infty }x[n]\\left(\\sqrt{2W}sinc(2W(t-(n-1)T))\\right)$ where $T$ is the Nyquist interval, i.e., $T=\\frac{1}{2W}$ [32].",
"Due to completeness of the of sinc pulses, all band-limited signals, even sinc pulses that do not lie at integer multiples of $T$ , e.g., their shifted version, i.e., $sinc(2W(t-(n-1)T-\\tau ))$ , still lie completely in the Paley-Wiener space.",
"Assume that, in Eq.",
"(REF ), $x[n]$ is the transmitted symbol modulated on the sinc pulse $p_n(t)=\\sqrt{2W}sinc(2W(t-(n-1)T))$ .",
"In practice, pulses spanning an unlimited time domain are not feasible, hence, they are usually truncated within a desired interval.",
"Assume that the transmission interval is truncated into $NT$ seconds, then we are capable of transmitting approximately $2WNT$ symbols.",
"In other words, $2WNT$ dimensions is used in the case of finite-time transmissions [33].",
"However, due to the truncation, the finite set of sinc pulses, i.e.",
": $S\\equiv \\begin{Bmatrix}p_n(t)=\\sqrt{2W}sinc(2W(t-(n-1)T))\\\\n=1,\\cdots , 2WNT\\end{Bmatrix}$ is not complete anymore and does not span the whole signal space.",
"Therefore, we can insert additional pulses to exploit more signaling dimensions which leads to higher data throughput [34].",
"For example, defining $b_{2WNT}=\\sqrt{2W}sinc(2W(t-\\tau ))$ and applying the well-known Gram-Schmidt orthogonalization process, provide us an orthonormal basis function with size $2WNT+1$ .",
"The newly formed basis function exploits an additional signaling dimension.",
"We provide a numerical example next.",
"Example 1 Assume that $W=0.5,N=5,T=1$ , then $p_n(t)=sinc(t-(n-1))$ for $n=1,\\cdots , 5$ with truncation length of $5T$ .",
"The Gram matrix of the aforementioned set is equal to: $G_S=\\begin{pmatrix}0.959& 0.052 & -0.084 & 0.057 & -0.013\\\\0.052&0.959 & 0.052 & -0.084 & 0.057\\\\-0.084& 0.052 &0.959 & 0.052 & -0.084\\\\0.057& -0.084 & 0.052 & 0.959 & 0.052 \\\\-0.013 & 0.057 & -0.084 & 0.052 & 0.959\\end{pmatrix}$ If the pulse shapes were strictly band-limited, i.e., unlimited time support, matrix $G_S$ would be the identity matrix.",
"In addition, with unlimited time support, any other function like $p_6(t)=sinc(t-0.5)$ can be written as $\\sum _{n=-\\infty }^{\\infty }{a_n sinc(t-n)}$ where $a_n=sinc(n-0.5)$ .",
"On the other hand, in a time-limited scenario, $p_6(t)$ cannot be written as the weighted sum of truncated sinc functions.",
"Thus, performing the Gram-Schmidt process, we can get the following orthonormal set: $\\nonumber \\begin{Bmatrix}p^{\\prime }_1(t)=sinc(t), e_1(t)=\\frac{p^{\\prime }_1(t)}{|p^{\\prime }_1(t)|}\\\\p^{\\prime }_2(t)=p_2(t)-0.053e_1(t), e_2(t)=\\frac{p^{\\prime }_2(t)}{|p^{\\prime }_2(t)|}\\\\p^{\\prime }_3(t)=p_3(t)+0.086e_1(t)-0.058e_2(t), e_3(t)=\\frac{p^{\\prime }_3(t)}{|p^{\\prime }_3(t)|}\\\\p^{\\prime }_4(t)=p_4(t)-0.059e_1(t)+0.089e_2(t)-0.064e_3(t), e_4(t)=\\frac{p^{\\prime }_4(t)}{|p^{\\prime }_4(t)|}\\\\p^{\\prime }_5(t)=p_5(t)+0.014e_1(t)-0.06e_2(t)+0.091e_3(t)-0.066e_4(t), e_5(t)=\\frac{p^{\\prime }_5(t)}{|p^{\\prime }_5(t)|}\\\\p^{\\prime }_6(t)=p_6(t)-0.647e_1(t)-0.612e_2(t)+0.191e_3(t)-0.099e_4(t)+0.06e_5(t), e_6(t)=\\frac{p^{\\prime }_6(t)}{|p^{\\prime }_6(t)|}\\end{Bmatrix}$ Now we have a new set with six elements, i.e., $S^{\\prime }\\equiv \\lbrace e_n(t), n=1,\\cdots ,6\\rbrace $ , for which the Gram matrix is the identity matrix.",
"We can continue this procedure and take advantage of the rest of the available signaling dimensions.",
"In fact, it is shown in the literature that the available degree of freedom in a time-limited channel is unbounded [35].",
"We will use asynchronous transmission to fully exploit the available degrees of freedom in a NOMA framework.",
"In the next section, the general system model and its characteristics for the asynchronous transmission is explained."
],
[
"System Model",
"After performing coding and modulation, the modulated symbols intended for each user, namely, User $k$ , are shaped with appropriate waveforms suited to the communication channel, particularly its bandwidth ($W$ ).",
"We denote the block length by $N$ , then the intended signal for User k will be: $x_k(t)=\\sum _{n=1}^{N}\\sqrt{P_{kn}}{x_k[n]p(t-(n-1)T)}$ where $p(t)$ is the pulse shape, e.g., root raised cosine, which is truncated and its length is denoted by $T_p$ and $T$ is the symbol interval and is usually equal to $\\frac{1}{2W}$ .",
"The transmit power assigned to $x_k(t)$ is denoted by $P_k$ which is calculated as: $P_k=E\\left[\\int _{-\\infty }^{\\infty } x_k(t)x_k(t)^* dt\\right]$ The relation between the transmit power, $P_k$ and individual symbol magnitudes of symbols $P_{kn}$ will be examined later.",
"The transmitted signal from BS will be the super-position of signals from all users, i.e.",
": $x_{synch}(t)=\\sum _{k=1}^{K}{x_k(t)}$ where the transmit power, i.e., $P_{synch}=E\\left[\\int _{-\\infty }^{\\infty } x_{synch}(t)x_{synch}(t)^* dt\\right]$ , satisfies the total power constraint for $N$ time slots, i.e., $P_{synch}\\le NP$ .",
"To take advantage of asynchrony, assume that each sub-stream is shifted with a specific time delay $\\tau _k$ .",
"Then, the transmitted signal will be: $x_{asynch}(t)=\\sum _{k=1}^{K}{x_k(t-\\tau _k)}$ such that $P_{asynch}\\le NP$ .",
"By assuming flat fading and additive white Gaussian noise, the received signal at the $r$ th user is described as: $y^r(t)=h_r\\sum _{k=1}^{K}{x_k(t-\\tau _k)}+n^r(t)$ To detect the transmitted symbols $x_k[n]$ , instead of working with the continuous random process $y^r(t)$ , we use a set of statistics, i.e., $Z_j=r_j(y^r(t)), j=1,\\cdots , J$ that are sufficient for detection of transmitted symbols.",
"We use the well-known factorization theorem to find the sufficient statistics.",
"Theorem 2 Let $Y_1, Y_2,\\cdots ,Y_n$ be random variables with joint density $f(y_1,y_2,\\cdots ,y_n|\\theta )$ .",
"The statistics $Z_j=r_j(Y_1,Y_2,\\cdots ,Y_n), j=1,\\cdots , J$ are jointly sufficient to estimate $\\theta $ if and only if the joint density can be factored as follows: $\\nonumber &f(y_1,y_2,\\cdots ,y_n|\\theta )=u(y_1,y_2,\\cdots ,y_n).\\\\&v(r_1(y_1,y_2,\\cdots ,y_n),\\cdots ,r_J(y_1,y_2,\\cdots ,y_n),\\theta )$ where $u$ and $v$ are non-negative functions [36].",
"Assuming the knowledge of channel coefficients, power assignments and time delays, the density of $y^r(t)$ given the transmitted symbols is calculated as follows: $\\nonumber f(y^r(t)|\\lbrace x_k[n]\\rbrace )=\\nonumber c\\exp {\\left[\\int _{-\\infty }^{\\infty }{[z(t)]^2dt}\\right]}$ where $z(t)=y^r(t)-h_r\\sum _{k=1}^K\\sum _{n=1}^{N}{\\sqrt{P_{kn}}x_k[n]p(t-(n-1)T-\\tau _k)}$ and $c$ is a constant value independent of the transmitted symbols.",
"Expanding the likelihood function, we will have $\\nonumber f(y^r(t)|\\lbrace x_k[n]\\rbrace )=u_1(y^r(t)).u_2(x).Re\\left\\lbrace h^*_r\\sum _{k=1}^K\\sum _{n=1}^{N}{\\sqrt{P_{kn}}x^*_k[n]\\int _{-\\infty }^{\\infty }{y^r(t)p(t-(n-1)T-\\tau _k)dt}}\\right\\rbrace $ where $u_1(y^r(t))$ and $u_2(x)$ are only functions of the output signal and the input symbols, respectively.",
"Therefore, using the factorization theorem stated above, we can conclude that sufficient statistics for detecting the transmitted symbols are: $y^r_l[m]=&\\int _{-\\infty }^{\\infty }{y^r(t)p(t-(m-1)T-\\tau _l)dt}\\\\\\nonumber &l=1,\\cdots ,K \\ \\ \\ m=1,\\cdots ,N$ which is similar to the matched filter for the synchronous case, although it involves $K$ times more samples, and can also be implemented using convolution followed by a sampler, i.e.,: $y^r_l[m]=y^r(t)*p(t)|_{(m-1)T+\\tau _l}$ Denoting $p(t)*p(t)$ as $g(t)$ and $g((m-n)T+(\\tau _l-\\tau _k))$ as $g_{lk}(m-n)$ , the sufficient statistics can be represented as: $y_l^r[m]=h_r\\sum _{k=1}^K\\sum _{n=1}^{N}{\\sqrt{P_{kn}}g_{lk}(m-n)x_k[n]}+n^r_l[m]$ where $n^r_l[m]=n(t)*p(t)|_{(m-1)T+\\tau _l}$ .",
"Defining $y_l^r=\\left(y^r_l[1],\\cdots ,y^r_l[N]\\right)^T$ , $P_k=$ diag$(\\sqrt{P_{k1}},\\sqrt{P_{k2}},\\cdots , \\sqrt{P_{kN}})$ , and $x_k=\\left(x_k[1],\\cdots ,x_k[N]\\right)^T$ , the input-output system representation in matrix form is as follows: $y^r=h_rRPx+n^r$ where $y^r=\\left(y^r_1^T,\\cdots ,y^r_K^T\\right)^T$ , $x=\\left(x_1^T,\\cdots ,x_K^T\\right)^T$ , $n^r=\\left(n^r_1^T,\\cdots ,n^r_K^T\\right)^T$ and matrices $P$ and $R$ are defined as: $P=\\left(\\begin{matrix}P_{1}& 0& \\cdots & 0\\\\0& P_{2}& \\ddots & \\vdots \\\\\\vdots & \\ddots & \\ddots & 0 \\\\0& \\cdots & 0 & P_{K}\\end{matrix}\\right),R=\\left(\\begin{matrix}R_{11}& R_{12}& \\cdots & R_{1K}\\\\R_{21}& R_{22}& \\cdots & R_{2K}\\\\\\vdots & \\ddots & \\ddots & \\vdots \\\\R_{K1}& R_{K2} & \\cdots & R_{KK}\\end{matrix}\\right)$ where the elements of each sub-matrix $R_{lk}$ is defined as: $R_{lk}(m,n)=g_{lk}(m-n)$ Matrix $R$ is a Hermitian matrix whose sub-blocks, i.e., $R_{lk}$ are banded Toeplitz blocks of order $u$ , where $u=\\frac{T_p}{T}$ .",
"Due to additional signaling, the noise samples are not independent anymore and their covariance is equal to $R\\sigma ^2_n$ .",
"Note that, in the perfect synchronous case, sub-blocks $R_{lk}$ turn into identity matrices, i.e., $I_N$ .",
"However, for the asynchronous case, only diagonal blocks are identity matrices, i.e., $R_{ll}=I_N$ , and the other sub-blocks have non-zero off diagonal elements.",
"The question of whether this matrix is invertible or not and how it behaves asymptotically as the block length N tends to infinity will have important consequences in the performance of the system.",
"Hence, we will investigate this question next."
],
[
"Properties of Matrix $R$",
"To understand the asymptotic behavior of matrix $R$ , we will utilize the Szego theorem which states that[37]: Theorem 3 Let $T_N = [t_{k-j};k,j = 0,1,2,\\cdots ,N-1]$ be a sequence of Hermitian Toeplitz matrices whose generating functions is defined as $f(w)=\\sum _{k=-\\infty }^{\\infty }{t_ke^{ikw}}, w \\in [0,2\\pi ]$ .",
"Also, $\\lambda _{1}\\ge \\lambda _{2}\\ge \\cdots \\ge \\lambda _{N}$ are the sorted eigenvalues of matrix $T_N$ .",
"Then, for any function $F$ that is continuous on the range of $f$ , we have: $\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{k=0}^{N}F(\\lambda _{k})=\\frac{1}{2\\pi }\\int _0^{2\\pi }F(f(w))dw$ In addition, the largest and smallest eigenvalues asymptotically converge to: $\\nonumber \\lim _{N\\rightarrow \\infty } \\lambda _{1}=\\max _{w}f(w)\\\\\\nonumber \\lim _{N\\rightarrow \\infty }\\lambda _{N}=\\min _{w}f(w)$ The proof comes from asymptotic equivalence of sequences of Hermitian Toeplitz matrices and their corresponding circulant versions which results in asymptotic convergence of their eigenvalues.",
"Our system model is not Toeplitz in general except for the special case of equally spaced timing offsets.",
"However, luckily, the Szego Theorem can be extended to Toeplitz block matrices [38].",
"The generalized Szego Theorem relates the collective behavior of the eigenvalues to the generalized generating function, $R(w)$ , which is defined as: $R(w)=\\left(\\begin{matrix}f_{11}(w)& f_{12}(w)& \\cdots & f_{1K}(w)\\\\f_{21}(w)& f_{22}(w)& \\cdots & f_{2K}(w)\\\\\\vdots & \\ddots & \\ddots & \\vdots \\\\f_{K1}(w)& f_{K2}(w) & \\cdots & f_{KK}(w)\\end{matrix}\\right)$ where $f_{lk}(w)$ is the generating function for the corresponding Toeplitz block $R_{lk}$ .",
"The generalized Szego Theorem states that for any continuous function $F$ [39]: $\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{k=1}^{NK}F[\\lambda _{k}(R)]=\\frac{1}{2\\pi }\\int _0^{2\\pi }\\sum _{j=1}^{K}F[\\lambda _{j}(R(w))]dw$ In particular, for $F(x)=x$ , $\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{k=1}^{NK}\\lambda _{k}(R)=\\frac{1}{2\\pi }\\int _0^{2\\pi }\\sum _{j=1}^{K}\\lambda _{j}(R(w))dw$ Moreover, the same convergence results can be obtained for the largest and smallest eigenvalues, i.e,: $\\nonumber \\lim _{N\\rightarrow \\infty } \\lambda _{1}(R)=\\max _{w}\\lambda _{1}(R(w))\\\\\\nonumber \\lim _{N\\rightarrow \\infty } \\lambda _{NK}(R)=\\min _{w}\\lambda _{K}(R(w))$ Therefore, some of the properties of matrix $R$ can be deducted by properties of matrix $R(w)$ when the block length is sufficiently large.",
"For example, it is proved in the literature that for time limited transmission, i.e., finite value of u, matrix $R(w)$ is positive definite with bounded eigenvalues, thus, eigenvalues of matrix $R$ are nonzero and bounded.",
"On the other hand, when the pulse shapes are strictly band-limited, matrix $R(w)$ is singular which results in the singularity of $R$ [40]."
],
[
"Transmit Power Examination",
"In this section, we analyze the transmit power for synchronous and asynchronous scenarios.",
"We calculate $P_{asynch}$ in the next lemma and $P_{synch}$ will be found by assuming that the time delays are zero.",
"Lemma 2 The transmit power of the asynchronous transmission of $K$ superimposed sub-streams defined in Eq.",
"(REF ) can be calculated as: $\\nonumber P_{asynch}&=\\sum _{i=1}^{K}\\sum _{j=1}^{K}trace({R_{ij}Q_{ij}})\\\\&=trace(RQ)$ where $Q_{ij}=COV[x_i,x_j]$ and $Q=COV[x]$ .",
"The proof is presented in Appendix Note that the total power in Eq.",
"(REF ) is not the same as the sum of individual powers.",
"This is because the symbols are not independent in general and $E\\left[\\sum {|x_k|}^2\\right]$ is not equal to $E\\left[|\\sum {x_k}|^2\\right]$ , i.e., the cross-terms will not be zero.",
"For the synchronous transmission where the sub-blocks of matrix $R$ are identity matrices, the transmit power will be simplified to $P_{synch}=\\sum _{i=1}^{K}\\sum _{j=1}^{K}trace({Q_{ij}})$ .",
"If no precoding performed in the BS, i.e., $Q_{ij}=P_i^2\\delta [i-j]$ , then the transmit power for both synchronous and asynchronous transmission will be $\\sum _{k=1}^{K}{P_k}$ where $P_k=\\sum _{n=1}^{N}{P_{kn}}$ .",
"Because the channel is assumed to be fixed during the transmission of a packet, the index of $n$ can be discarded for P-NOMA and AP-NOMA schemes which use the same power for all time instants.",
"However, for T-NOMA method which uses precoding and exploits the variation of effective channel induced by asynchrony, the values of $P_{kn}$ will be assigned accordingly.",
"Thus, the power constraint for P-NOMA, AP-NOMA and T-NOMA methods can be stated as: $\\textnormal {P-NOMA, AP-NOMA}&:\\sum _{k=1}^{K}{P_k}\\le P\\\\\\textnormal {T-NOMA}&:trace(RQ) \\le NP$ Note that, from now on, $P_k$ is the transmit power assigned to the transmission of each symbol by User $k$ in P-NOMA and AP-NOMA methods and $P_{kn}$ is the magnitude assigned to the transmission of the $n$ th symbol by User $k$ in T-NOMA method.",
"In all methods, the transmit power constraint is $NP$ in $N$ symbol times."
],
[
"Achievable Rate-Region Analysis",
"This section analyzes the achievable rate-region of the proposed NOMA schemes, AP-NOMA and T-NOMA.",
"For comparison purposes, we also present the achievable rate-regions of the synchronous P-NOMA."
],
[
"Achievable Rates for Conventional P-NOMA",
"We begin by applying the SIC detection at each user in the synchronous NOMA scheme.",
"The optimal detection sequence is $x_K, \\cdots ,x_2, x_1$ assuming $\\frac{|h_1|^2}{\\sigma ^2_n} > \\cdots > \\frac{|h_K|^2}{\\sigma ^2_n}$ .",
"In particular, User 1 first decodes $x_2, \\cdots , x_K$ and subtracts their components from the received signal $y_1$ .",
"Then, User 1 decodes $x_1$ without interference from other users.",
"On the other hand, User $K$ can directly decode $x_K$ while considering other users as noise.",
"Assuming successful decoding and no error propagation, the achievable rate-region can be represented as: $R_{P-NOMA}\\triangleq \\left\\lbrace \\begin{matrix}0\\le R_1 \\le \\frac{1}{2}\\log _2\\left( 1+\\frac{P_1|h_1|^2}{\\sigma ^2_n}\\right)\\\\0\\le R_2 \\le \\frac{1}{2}\\log _2\\left( 1+\\frac{P_2|h_2|^2}{P_1|h_2|^2+\\sigma ^2_n}\\right)\\\\\\vdots \\\\0\\le R_K \\le \\frac{1}{2}\\log _2\\left( 1+\\frac{P_K|h_K|^2}{|h_K|^2\\sum _{k=1}^{K-1}{P_k}+\\sigma ^2_n}\\right)\\end{matrix}\\right.$"
],
[
"Achievable Rates for AP-NOMA",
"In AP-NOMA, the SIC detection is the same as P-NOMA, however, the set of samples that is used to decode symbols of each user is different.",
"In the synchronous case, there is only one set of samples at each user, namely, User $r$ , with no ISI and can be written as $y_r^r=h_r\\sum _{k=1}^{K}{I_Nx_k}+n_r^r$ .",
"Based on the sufficient statistics derived in Eq.",
"(REF ), in the asynchronous case, there are $K$ sets of samples at each user, each of them matched to the timing offset of one of the users represented as: $y_j^r=h_r\\sum _{\\begin{array}{c}j=1\\end{array}}^{K}{\\sqrt{P_k}R_{jk}x_k}+n_j^r, \\ \\ \\ j=1,\\cdots , K$ Based on the assumption of $\\frac{|h_1|^2}{\\sigma ^2_n} > \\cdots > \\frac{|h_K|^2}{\\sigma ^2_n}$ , User $r$ can decode signals of Users $\\left\\lbrace r+1, \\cdots ,K\\right\\rbrace $ using the sample sets of $\\left\\lbrace y_{r+1}^r, \\cdots ,y_K^r\\right\\rbrace $ and subtract them from the corresponding sample set of $y_r^r$ .",
"Note that based on the ordering of channel strengths, User $r$ only needs $K-r$ sets of samples.",
"In particular, the strongest user needs all $K$ sets of samples and the weakest user only needs its own corresponding set of samples.",
"The resulting set of samples after subtraction, at User $r$ , is calculated as: $\\hat{y}_r^r=h_r\\sqrt{P_r}I_Nx_r+h_r\\sum _{\\begin{array}{c}k=1\\end{array}}^{r-1}{\\sqrt{P_k}R_{rk}x_k}+n_r^r$ Due to the Toeplitz structure of the sub-blocks $R_{lk}$ , the resulting sample at each time instant $i$ can be written as: $\\nonumber \\hat{y}_r^r[i]=h_r\\sqrt{P_r}x_r[i]+h_r\\sum _{\\begin{array}{c}k=1\\end{array}}^{r-1}\\sqrt{P_k}\\sum _{j=-u}^{u}{g_{rk}(j)x_k[i-j]}+n_r^r[i], \\ \\ \\ \\ i=u+1,\\cdots ,N-u$ Treating the remaining interferers as noise will result in the following achievable rate for User $r$ : $0\\le R_r \\le \\frac{1}{2}\\log _2\\left( 1+\\frac{P_r|h_r|^2}{|h_r|^2\\sum _{k=1}^{r-1}{G_{rk}P_k}+\\sigma ^2_n}\\right)$ where $G_{rk}=\\sum _{i=-u}^{u}{g^2_{rk}(i)}$ .",
"In general, the overall rate-region depends on how we assign the timing offsets to users with different channel strengths.",
"For a specific assignment, namely $\\psi $ , the achievable rate-region, $R_{AP-NOMA}^{\\psi }$ can be defined as: $R_{AP-NOMA}^{\\psi }\\triangleq \\left\\lbrace \\begin{matrix}0\\le R_1 \\le \\frac{1}{2}\\log _2\\left( 1+\\frac{P_1|h_1|^2}{\\sigma ^2_n}\\right)\\\\0\\le R_2 \\le \\frac{1}{2}\\log _2\\left( 1+\\frac{P_2|h_2|^2}{G_{21}P_1|h_2|^2+\\sigma ^2_n}\\right)\\\\\\vdots \\\\0\\le R_K \\le \\frac{1}{2}\\log _2\\left( 1+\\frac{P_K|h_k|^2}{|h_K|^2\\sum _{k=1}^{K-1}{G_{Kk}P_k}+\\sigma ^2_n}\\right)\\end{matrix}\\right.$ Note that there are $K!$ different assignments of time delays to users.",
"Hence, the total rate-region is the convex hull of all possible assignments, i.e., $R_{AP-NOMA}\\triangleq \\bigcup _{\\psi =1}^{K!}",
"R_{AP-NOMA}^{\\psi }$ Using Lemma 1, it can be shown that $\\sum _{-u}^{u}{g^2_{rk}(i)}<IUI(\\tau )<IUI(0)=1$ where the last identity is valid for all pulse shapes that satisfy the Nyquist no-ISI condition, including, the rectangular, the sinc and the raised cosine pulse shapes.",
"As a result, the rate-region for each assignment and thus the total rate-region for AP-NOMA is larger than that of the conventional P-NOMA.",
"Note that for the 2-user scenario, the rate-region for both possible assignments is the same because $G_{21}=G_{12}$ .",
"However, this is not valid for more number of users unless the difference between time delays is equal which results in matrix $R$ to be Hermitian Toeplitz.",
"The numerical results are shown in Section ."
],
[
"Achievable Rates for T-NOMA",
"In this section, we derive the achievable rate-region for the T-NOMA method.",
"As mentioned before, asynchrony provides additional degrees of freedom which will be exploited in T-NOMA by using precoding at the BS.",
"T-NOMA method applies a simple precoding at the BS.",
"In more details, after power assignment to users' intended symbols, they are precoded by a unitary matrix, i.e., $x_T=U_TPx$ .",
"Then, the received signal at User $r$ is calculated by: $y^r=h_rRU_TPx+n^r$ where the power constraint is stated as $trace(RQ) \\le NP$ .",
"The covariance matrix of the transmitted vector and noise vector are $Q=U_TP^2U_T^H$ and $Q_n=R\\sigma ^2_n$ , respectively.",
"To find the proper precoding matrix, let us consider the eigen-decomposition of matrix $R$ .",
"Matrix $R$ is a Hermitian matrix, thus its eigen-decomposition can be written as: $\\nonumber R&={U}_{R} \\left(\\begin{matrix}\\lambda _1& 0& \\cdots & 0\\\\0& \\lambda _2& \\cdots & 0\\\\\\vdots & \\ddots & \\ddots & \\vdots \\\\0& 0 & \\cdots &\\lambda _{NK}\\end{matrix} \\right){U}_{R}^H\\\\&={U}_{R} \\Lambda _R {U}_{R}^H$ where $\\lambda _{1}\\ge \\lambda _{2}\\ge \\cdots \\ge \\lambda _{NK}$ are the eigenvalues of matrix $R$ and ${U}_R$ is a Unitary matrix.",
"Therefore, the received signal can be rewritten as: $y^r=h_rU_{R} \\Lambda _R U_{R}^HU_TPx+n^r$ Because the time delays are known at the transmitter, the matrix $R$ is known at the transmitter.",
"Hence, the transmitted symbols can be precoded in the direction of the eigen-vectors of matrix $R$ .",
"In addition, the sub-channels can be decomposed at each user by a post-processing matrix $U_U$ .",
"By choosing $U_T=U_R$ and $U_U=U_R^H$ , the received signal will be written as $y^r=h_r\\Lambda _RPx+\\hat{n}^r$ where $\\hat{n}^r=U^Hn^r$ is a white Gaussian noise with covariance matrix of $Q_{\\hat{n}}=\\Lambda _R\\sigma ^2_n$ .",
"By using proper precoding and post processing at the destination, the channel has turned into $NK$ independent sub-channels.",
"In other words, we have used the available degrees of freedoms offered by asynchrony to decompose the transmitted symbols and eliminate interference.",
"As we will see later, the way of assigning different symbols to different sub-channels will not change the final result, thus, for notational simplicity, we denote the eigenvalue corresponding to $x_k[n]$ as $\\lambda _{kn}$ .",
"Due to the channel decomposition, there is no interference and the achievable rate for each user is sum of the achievable rates at the corresponding sub-channels: $\\nonumber R_{r}=\\frac{1}{2N}\\sum _{i=1}^{N}{log_2\\left(1+\\frac{P_{ri}\\lambda _{ri}|h_r|^2}{\\sigma ^2_n}\\right)}$ The precoding at the BS also affects the power constraint.",
"The alignment of transmit covariance matrix in the direction of the eigen-vectors of matrix $R$ changes the product of $RQ$ to: $RQ=U_{R} \\left(\\begin{matrix}P_{11}\\lambda _{11}& 0& \\cdots & 0\\\\0& P_{12}\\lambda _{12}& \\cdots & 0\\\\\\vdots & \\ddots & \\ddots & \\vdots \\\\0& 0 & \\cdots &P_{KN}\\lambda _{KN}\\end{matrix} \\right)U_{R}^H$ As a result, the power constraint will be: $\\sum _{k=1}^{K}\\sum _{i=1}^{N}{P_{ki}\\lambda _{ki}}\\le NP$ Lemma 3 The achievable rate-region of T-NOMA method can be described as follows: $\\nonumber R_{r}&=\\frac{1}{2}log_2\\left(1+\\frac{P_r|h_r|^2}{\\sigma ^2_n}\\right)\\\\&s.t.",
"\\sum _{r=1}^{K}{P_r}\\le P$ The proof is presented in Appendix .",
"Note that the achievable rate-region of T-NOMA method is independent of the pulse shape and the timing offsets.",
"The only requirement to achieve this rate-region is the matrix $R$ to be full rank which is satisfied as long as the pulse shapes are time limited and the timed delays are distinct.",
"Also note that, using relatively band-limited signals like truncated sinc pulse shapes and truncated Raised Cosine pulse shapes will prevent the spectrum broadening caused by precoding [34].",
"Besides the individual performance of each user, the other significant criterion in a network is the amount of cumulative rate of all users which can be achieved at a same time termed as sum-rate.",
"In the next section, we compare the sum-rate provided by P-NOMA, AP-NOMA and T-NOMA methods.",
"We will show that for the P-NOMA method, achieving the maximum available sum-rate is equivalent to violating the fairness; however, AP-NOMA and T-NOMA methods can achieve the maximum sum-rate while maintaining fairness among users."
],
[
"Sum-Rate Analysis",
"Although analyzing the sum-rate in the general case of $K$ users is cumbersome, considering the case of $K=2$ will shed some light on behavior of sum-rate and its optimum points.",
"In the next lemma, we compare the sum-rate results for P-NOMA, AP-NOMA and T-NOMA and also their corresponding fairness.",
"Lemma 4 Denoting $R^*$ as the maximum of sum-rate, i.e., $R=R_1+R_2$ , we will have the following results: Sum-Rate Comparison: $R^*_{P-NOMA}<R^*_{AP-NOMA}<R^*_{T-NOMA}$ Fairness: In order to achieve the max sum-rate, one of the users is assigned zero power if: $\\left\\lbrace \\begin{matrix}|\\sigma _1-\\sigma _2|>0 \\ \\ & P-NOMA\\\\|\\sigma _1- \\sigma _2|>(1-g)P & AP-NOMA\\\\|\\sigma _1- \\sigma _2|> P & T-NOMA\\end{matrix}\\right.$ where $\\sigma _i=\\frac{\\sigma ^2_n}{|h_i|^2}$ and $g$ is equal to $G_{12}(=G_{21})$ defined in Eq.",
"(REF ).",
"The proof is presented in Appendix .",
"The results in the first part of Lemma REF indicates that the T-NOMA method provides the largest sum-rate.",
"The second part explains that in order to achieve the maximum sum-rate, P-NOMA almost always (i.e., for $\\sigma _1\\ne \\sigma _2$ ) assigns zero power to one of the users, thus, achieving the maximum sum-rate with fairness is not possible using the P-NOMA method.",
"However, AP-NOMA and T-NOMA methods assign zero power to one of the users only when the difference between $\\sigma _1$ and $\\sigma _2$ is greater than $(1-g)P$ and $P$ , respectively.",
"In other words, unlike the P-NOMA method, AP-NOMA and T-NOMA can achieve the maximum of the sum-rate while assigning non-zero power to both users.",
"Numerical results are provided in the next section."
],
[
"Numerical Results",
"In this section, we present numerical results to show the effectiveness of using asynchrony in providing a larger rate-region.",
"In particular, we show that AP-NOMA outperforms P-NOMA with a slight change of adding timing offset among transmitted symbols.",
"However, T-NOMA which uses the degrees of freedom provided by time asynchrony results in the best performance.",
"We assume that a transmit power of $P=10$ is available at the base station which is serving two users.",
"We consider the typical pulse shaping function in the literature, i.e., Rectangular pulse shape (Rect.",
"), and a more practical pulse shaping function, i.e., Root Raised Cosine (R.R.C.).",
"Theoretically, R.R.C.",
"pulse shaping is unlimited in time, however, it is truncated in practice and we have adopted the truncated version with 4 side lobes.",
"The symbol duration $T$ is normalized to be 1, $\\tau _1=0$ and $\\tau _2\\in [0, 0.5]$ due to the symmetry.",
"We first consider the Gaussian channel where the channel coefficients are determined by ${|h_1|^2} / {\\sigma ^2_n}= 10$ and ${|h_2|^2} / {\\sigma ^2_n}=1$ .",
"In Fig.",
"REF , we show the achievable rate-regions of AP-NOMA and T-NOMA with different symbol offsets.",
"Figure: Achievable rate-regions of AP-NOMA and T-NOMA for different symbol offsets and a Gaussian channel.In the AP-NOMA method, the choice of pulse shaping affects the amount of reduction in IUI and thus the overall performance.",
"Rect.",
"pulse shaping provides slightly better performance compared with R.R.C.",
"For both Rect.",
"and R.R.C.",
"pulse shapes, increasing the timing offset will improve the performance and $\\tau _2=0.5$ results in the best performance.",
"On the other hand, the performance of the T-NOMA method, which exploits the degrees of freedom available in the system, is independent of the pulse shape and time delays as long as matrix $R$ is full rank.",
"Fig.",
"REF shows that both T-NOMA and AP-NOMA outperform the conventional P-NOMA.",
"In Fig.",
"REF , there is sufficient discrepancy between channel coefficients of the two users to be exploited by P-NOMA and AP-NOMA methods.",
"However, in Fig.",
"REF , the quality of channel coefficients are assumed to be the same, i.e., ${|h_1|^2} / {\\sigma ^2_n}= 1$ and ${|h_2|^2} / {\\sigma ^2_n}=1$ , thus the P-NOMA performance coincides with that of the OMA systems like TDMA.",
"In such a case, the AP-NOMA method provides slightly better performance; however, T-NOMA significantly improves the performance showing the capability of this method even without power discrepancy.",
"Figure: The maximum achievable rate-regions of three schemes: P-NOMA, AP-NOMA and T-NOMA in Gaussian channels with |h 1 | 2 /σ n 2 =1{|h_1|^2} / {\\sigma ^2_n}= 1 and |h 2 | 2 /σ n 2 =1{|h_2|^2} / {\\sigma ^2_n}=1.The rate-region for 3 users with rectangular pulse shape and $\\tau =[0,0.3,0.7]$ is shown in Fig.",
"REF .",
"The rate-regions for P-NOMA and T-NOMA are calculated similar to those of the 2-user scenario.",
"However, there are two main differences in calculating the rate-region of AP-NOMA.",
"First, there are 3!=6 different assignments of time delays to users, and the total region is found by taking the convex hull of all possible assignments.",
"The other difference is that when one of the power assignments is equal to zero, the remaining time delays for the other two users need to be updated.",
"In more details: $R_{AP-NOMA}=\\left\\lbrace \\begin{matrix}R_{AP-NOMA}[0,0.3,0.7] \\ \\ \\ \\ P_1\\ne 0, P_2\\ne 0,P_3\\ne 0\\\\ R_{AP-NOMA}[0,0.4] \\ \\ \\ \\ \\ P_1= 0, P_2\\ne 0,P_3\\ne 0\\\\R_{AP-NOMA}[0,0.7] \\ \\ \\ \\ \\ P_1\\ne 0, P_2=0,P_3\\ne 0\\\\R_{AP-NOMA}[0,0.3] \\ \\ \\ \\ \\ P_1\\ne 0, P_2\\ne 0,P_3= 0\\\\\\end{matrix}\\right.$ Figure: The maximum achievable rate-regions of three schemes: P-NOMA, AP-NOMA and T-NOMA in Gaussian channels with |h 1 | 2 /σ n 2 =10{|h_1|^2} / {\\sigma ^2_n}= 10, |h 2 | 2 /σ n 2 =2{|h_2|^2} / {\\sigma ^2_n}=2 and |h 3 | 2 /σ n 2 =1{|h_3|^2} / {\\sigma ^2_n}= 1.To have a better understanding of the 3 dimensional rate-region provided in Fig.",
"(REF ), we show different two-dimensional cuts when $R_1=1$ , $R_2=1$ and $R_3=1$ in Figs.",
"(REF ), (REF ) and (REF ), respectively.",
"Figure: Different 2D cuts of the 3 dimensional rate-region in Fig.",
"Next, we consider the Rayleigh block-fading channel where the channel coefficients are independent Rayleigh distribution with unit variance, and noise variance is set to 0.1.",
"The ergodic rate is averaged over $10^5$ different realizations of the channel.",
"In Fig.",
"REF , we show the achievable rate-regions of P-NOMA, AP-NOMA and T-NOMA in Rayleigh block-fading channels.",
"As was the case in Gaussian channels, the achievable rate-region of P-NOMA is improved by adding asynchrony.",
"In addition, T-NOMA provides a large improvement compared with other schemes.",
"Figure: The maximum achievable rate-regions of three schemes: P-NOMA, AP-NOMA and T-NOMA in Rayleigh fading channels.In Fig.",
"REF , the maximum sum-rate with respect to the available transmit power is presented.",
"In the case where channels have the same qualities (i.e., $\\sigma _1=\\sigma _2$ ), both AP-NOMA and T-NOMA methods are strictly better than P-NOMA.",
"For the case of different channel qualities (i.e., $\\sigma _1\\ne \\sigma _2$ ), AP-NOMA outperforms the P-NOMA method when $P\\ge \\frac{\\sigma _2-\\sigma _1}{g}\\approx 13 dB$ and T-NOMA outperforms the P-NOMA method when $P\\ge \\frac{\\sigma _2-\\sigma _1}{g}\\approx 10 dB$ .",
"When $\\sigma _1=0.1$ and $\\sigma _2=1$ , P-NOMA assigns all the power to the first user, however, AP-NOMA and T-NOMA assign non-zero powers to both users when $P\\ge 13$ and $P\\ge 10$ , respectively.",
"For example when $P=45$ dB power is available, AP-NOMA assigns $P_1=20.3, P_2=24.7$ and T-NOMA assigns $P_1=22.8, P_2=22.2$ to the first and second users, respectively.",
"Therefore, not only do AP-NOMA and T-NOMA outperform P-NOMA but also they maintain the fairness among the users.",
"Figure: The maximum sum-rate with respect to available transmit power for two cases: different channel qualities and same channel qualities."
],
[
"Conclusion",
"In this work, we propose novel symbol-asynchronous downlink NOMA schemes.",
"In contrast to the conventional P-NOMA, we propose to intentionally add timing offsets among superimposed symbols.",
"The receiver architecture in AP-NOMA includes oversampling and a SIC scheme similar to the P-NOMA, however, asynchrony reduces IUI and improves the overall performance.",
"T-NOMA exploits the degrees of freedom introduced by time asynchrony, using novel precoding and simple post processing at users.",
"In other words, T-NOMA decomposes the channel into independent sub-channels and eliminates the interference.",
"Our analysis shows that both AP-NOMA and T-NOMA methods can improve the achievable rate-regions.",
"In addition, we showed that the proposed methods provides higher sum-rate while maintaining the fairness among the users."
],
[
"Proof of Lemma 1",
"Denoting g(t) as a pulse shape with real spectrum, we show that $IUI(\\tau )$ calculated as: $IUI(\\tau )=\\sum _{i=-\\infty }^{\\infty }{|g(\\tau +iT)|^2}$ is maximized at $\\tau =0$ .",
"In other words, $IUI(\\tau )<IUI(0), \\ \\ \\tau \\ne 0$ .",
"Assume that $G(f)$ is the Fourier transform of the pulse shape $g(t)$ .",
"Then, the Fourier transform of the shifted version of $g(t)$ , i.e., $g(\\tau +t)$ , will be $G(f)e^{j2\\pi f \\tau }$ .",
"The DTFT of the samples of $g(\\tau +t)$ , i.e., $g(\\tau +iT), i\\in Z$ can be expressed as: $G^{\\prime }_{\\tau }(f)=\\sum _{i=-\\infty }^{\\infty }{G(f+i/T)e^{j2\\pi (f+i/T) \\tau }}$ Note that $G^{\\prime }_{\\tau }(F)$ is periodic with period of $1/T$ .",
"Based on the Parseval's theorem, the IUI energy, i.e., $IUI(\\tau )=\\sum _{i=-\\infty }^{\\infty }{|g(\\tau +iT)|^2}$ , will be equal to: $IUI(\\tau )=\\int _{-1/2T}^{1/2T}{|G^{\\prime }_{\\tau }(f)|^2df}$ Then, with the assumption of having real spectrum, we will have: $\\nonumber IUI(\\tau )&=\\int _{-1/2T}^{1/2T}{\\left|\\sum _{i=-\\infty }^{\\infty }{G(f+i/T)e^{j2\\pi (f+i/T) \\tau }}\\right|^2df}\\\\&\\le \\int _{-1/2T}^{1/2T}{\\left|\\sum _{i=-\\infty }^{\\infty }{G(f+i/T)}\\right|^2df}=IUI(0)$ which concludes the proof."
],
[
"Proof of Lemma 2",
"The power of the asynchronous signal can be written as: $P_{asynch}&=E\\left[\\int _{-\\infty }^{\\infty }x_{asynch}(t)x^*_{asynch}(t)dt \\right]\\\\\\nonumber &=E\\left[\\int _{-\\infty }^{\\infty } \\left(\\sum _{k=1}^{K}x_k(t-\\tau _k)\\right)\\left(\\sum _{k=1}^{K}x_k(t-\\tau _k)\\right)^*dt \\right]$ Then, we have $\\nonumber P_{asynch}&=E\\left[\\int _{-\\infty }^{\\infty } \\left(\\sum _{k=1}^K\\sum _{n=1}^{N}{x_k[n]p(t-(n-1)T-\\tau _k)}\\right)\\left(\\sum _{k=1}^K\\sum _{n=1}^{N}{x_k[n]p(t-(n-1)T-\\tau _k)}\\right)^*dt \\right]\\\\\\nonumber &=\\sum _{k_1=1}^K\\sum _{k_2=1}^{K}\\sum _{n_1=1}^N\\sum _{n_2=1}^{N}E\\left[x_{k_1}[n_1]x_{k_2}[n_2]\\right] \\int _{-\\infty }^{\\infty } p(t-(n_1-1)T-\\tau _{k_1})p(t-(n_2-1)T-\\tau _{k_2})dt \\\\&=\\sum _{k_1=1}^K\\sum _{k_2=1}^{K}trace(R_{k_1k_2}COV[x_{k_1},x_{k_2}])=trace(R\\ COV[x])$ which concludes the proof."
],
[
"Proof of Lemma 3",
"The achievable rate for each user is the sum of the achievable rates for the corresponding sub-channels: $R_{r}=\\frac{1}{2N}\\sum _{i=1}^{N}{log_2\\left(1+\\frac{P_{ri}\\lambda _{ri}|h_r|^2}{\\sigma ^2_n}\\right)}$ with the power constraint of: $\\sum _{k=1}^{K}\\sum _{i=1}^{N}{P_{ki}\\lambda _{ki}}\\le NP$ Denoting $p_{ri}$ as $\\frac{P_{ri}\\lambda _{ri}}{N}$ and $P_r$ as $\\sum _{i=1}^{N}{p_{ri}}$ , the above optimization problem can be rewritten in $K$ simpler problems as: $\\nonumber R_{r}&=\\frac{1}{2N}\\sum _{i=1}^{N}{log_2\\left(1+\\frac{p_{ri}N|h_r|^2}{\\sigma ^2_n}\\right)}\\\\&s.t.\\ \\ \\sum _{i=1}^{N}{p_{ri}}=P_r$ The sum of the power constraints for the sub-problems should add up to $P$ , i.e., $\\sum _{r=1}^{K}{P_r}\\le P$ .",
"It can be easily shown that the power assignment that maximizes $R_r$ is such that: $p_{ri}=\\cdots =p_{rN}=P_r/N$ Therefore, by simple substitution, we can conclude that the achievable rate for each user is: $\\nonumber R_{r}=\\frac{1}{2}{log_2\\left(1+\\frac{P_{r}|h_r|^2}{\\sigma ^2_n}\\right)}$ such that $\\sum _{r=1}^{K}{P_r}\\le P$ .",
"Note that the different assignments of sub-channels to users only change the power assignment, otherwise, the final result remains the same."
],
[
"Proof of Lemma 4",
"Using (REF ) for two users, the achievable sum-rate by AP-NOMA can be calculated as: $R=\\frac{1}{2}\\log _2\\left( 1+\\frac{P_1}{\\sigma _1}\\right)+\\frac{1}{2}\\log _2\\left( 1+\\frac{P_2}{gP_1+\\sigma _2}\\right)$ where $\\sigma _i=\\frac{\\sigma ^2_n}{|h_i|^2}$ (assuming $\\sigma _2\\ge \\sigma _1$ without loss of generality).",
"By inserting $P_2=P-P_1$ , taking the derivative with respect to $P_1$ , making it equal to zero, and discarding the non-relevant terms, we will have: $g(g-1)(P^*_1)^2+2(g-1)\\sigma _2(P^*_1)+P(\\sigma _2-g\\sigma _1)+\\sigma _2(\\sigma _2-\\sigma _1)=0$ where $P_1^*$ is the optimal power allocated to the first user.",
"By inserting $g=1$ , Eq.",
"(REF ) results in $(P+\\sigma _2)(\\sigma _2-\\sigma _1)$ for P-NOMA.",
"Thus, for P-NOMA, if $\\sigma _2=\\sigma _1$ , then the derivative is always equal to zero, meaning that the sum-rate is a constant value.",
"In fact, if we put $\\sigma _1=\\sigma _2=\\sigma $ and $g=1$ , Eq.",
"(REF ) simplifies to: $R=\\frac{1}{2}\\log _2\\left( 1+\\frac{P}{\\sigma }\\right)$ In other words, when the channels have the same quality, the sum-rate of P-NOMA is always fixed, independent of the power assignment.",
"However, if $\\sigma _2>\\sigma _1$ , the derivative with respect to $P_1$ is always positive, implying that the sum-rate is strictly increasing in $P_1$ .",
"Thus, assigning $P_1=P$ results in the maximum sum-rate of P-NOMA which will be equal to $R^*=\\frac{1}{2}\\log _2\\left( 1+\\frac{P}{\\sigma _1}\\right)$ .",
"Note that for $\\sigma _1>\\sigma _2$ , all the power will be assigned to $P_2$ and the sum-rate will be $R^*=\\frac{1}{2}\\log _2\\left( 1+\\frac{P}{\\sigma _2}\\right)$ .",
"Unlike P-NOMA where the derivative is always positive (or zero when $\\sigma _1=\\sigma _2$ ), for AP-NOMA, Eq.",
"(REF ) has two roots which one of them is infeasible and the other one is: $P_1^*=-\\frac{\\sigma _2}{g}+\\frac{\\sqrt{\\sigma _2^2+\\frac{g}{1-g}A}}{g}$ where $A=P(\\sigma _2-g\\sigma _1)+\\sigma _2(\\sigma _2-\\sigma _1)$ .",
"If $P^*_1\\ge P$ , i.e., $P\\le \\frac{\\sigma _2-\\sigma _1}{1-g}$ , then all the power is assigned to User 1, i.e., $P^*_1=P$ .",
"It can be concluded that as long as $|\\sigma _1- \\sigma _2|<(1-g)P$ , then non-zero powers will be assigned to both users.",
"To prove that $R^*_{P-NOMA}<R^*_{AP-NOMA}$ , we first find the values of $P_1^*$ such that: $\\frac{1}{2}\\log _2\\left( 1+\\frac{P^*_1}{\\sigma _1}\\right)+\\frac{1}{2}\\log _2\\left( 1+\\frac{P-P_1^*}{gP^*_1+\\sigma _2}\\right)>\\frac{1}{2}\\log _2\\left( 1+\\frac{P}{\\sigma _1}\\right)$ After some calculations, it can be found that Inequality (REF ) is satisfied if $P_1^*>\\frac{\\sigma _2-\\sigma _1}{1-g}$ .",
"On the other hand, considering the assumption of $P> \\frac{\\sigma _2-\\sigma _1}{1-g}$ in Eq.",
"(REF ) results in $P_1^*>\\frac{\\sigma _2-\\sigma _1}{1-g}$ .",
"Hence, the we can conclude that $R^*_{P-NOMA}<R^*_{AP-NOMA}$ for $P> \\frac{\\sigma _2-\\sigma _1}{1-g}$ .",
"In summary, if $P\\le \\frac{|\\sigma _2-\\sigma _1|}{1-g}$ , then $R^*_{AP-NOMA}=R^*_{P-NOMA}=\\frac{1}{2}\\log _2\\left( 1+\\frac{P}{min\\lbrace \\sigma _1,\\sigma _2\\rbrace }\\right)$ which is achieved by assigning the total available power to the stronger user, otherwise, AP-NOMA assigns non-zero power to both users and $R^*_{P-NOMA}<R^*_{AP-NOMA}$ .",
"Note that, when $\\sigma _2=\\sigma _1$ , the sum-rate achieved by AP-NOMA is strictly greater than the one achieved by P-NOMA and both users will be assigned non-zero powers.",
"The sum-rate for T-NOMA can be calculated as: $R=\\frac{1}{2}log_2\\left(1+\\frac{P_1}{\\sigma _1}\\right)+\\frac{1}{2}log_2\\left(1+\\frac{P_2}{\\sigma _2}\\right)$ By inserting $P_2=P-P_1$ , taking the derivative with respect to $P_1$ , making it equal to zero we can find the optimal $P_1$ as: $P_1^*=\\frac{P+\\sigma _2-\\sigma _1}{2}$ Therefore, if $\\sigma _2-\\sigma _1>P$ , then all available power is assigned to User 1, i.e., $P^*_1=P$ .",
"Similarly, it can be shown that if $P\\le {|\\sigma _2-\\sigma _1|}$ , then $R^*_{T-NOMA}=R^*_{P-NOMA}=\\frac{1}{2}\\log _2\\left( 1+\\frac{P}{min\\lbrace \\sigma _1,\\sigma _2\\rbrace }\\right)$ which is achieved by assigning the total available power to one of the users, otherwise, T-NOMA assigns non-zero power to both users and $R^*_{P-NOMA}<R^*_{AP-NOMA}<R^*_{T-NOMA}$ .",
"The superiority of the maximum sum-rate achieved by T-NOMA can be easily verified by the fact that every optimal pair of $(P^*_1,P^*_2)$ for P-NOMA or AP-NOMA will result in a higher sum-rate for T-NOMA."
]
] | 1808.08665 |
[
[
"Fidelity based unitary operation-induced quantum correlation for\n continuous-variable systems"
],
[
"Abstract We propose a measure of nonclassical correlation $N_{\\mathcal F}^{\\mathcal G}$ in terms of local Gaussian unitary operations based on square of the fidelity $\\mathcal F$ for bipartite continuous-variable systems.",
"This quantity is easier to calculate or estimate and is a remedy for the local ancilla problem associated with the geometric measurement-induced nonlocality.",
"A simple computation formula of $N_{\\mathcal F}^{\\mathcal G}$ for any $(1+1)$-mode Gaussian states is presented and an estimation of $N_{\\mathcal F}^{\\mathcal G}$ for any $(n+m)$-mode Gaussian states is given.",
"For any $(1+1)$-mode Gaussian states, $N_{\\mathcal F}^{\\mathcal G}$ does not increase after performing a local Gaussian channel on the unmeasured subsystem.",
"Comparing $N_{\\mathcal F}^{\\mathcal G}(\\rho_{AB})$ in scale with other quantum correlations such as Gaussian geometric discord for two-mode symmetric squeezed thermal states reveals that $N_{\\mathcal F}^{\\mathcal G}$ is much better in detecting quantum correlations of Gaussian states."
],
[
"Introduction",
"The presence of correlations in bipartite quantum systems is one of the main features of quantum mechanics.",
"The most important among such correlations is surely entanglement [1].",
"However, much attention has been devoted to studying and characterizing the quantum correlations that go beyond the paradigm of entanglement recently.",
"Non-entangled quantum correlations are also physical resources which play important roles in various quantum communications and quantum computing tasks.",
"For the last two decades, various methods have been proposed to describe quantum correlations, such as quantum discord (QD) [2], geometric quantum discord [3], [4], [5], measurement-induced nonlocality (MIN) [6] and measurement-induced disturbance (MID) [7] for discrete-variable systems.",
"For continuous-variable systems, Giorda, Paris [8] and Adesso, Datta [9] independently gave the definition of Gaussian QD for two-mode Gaussian states and discussed its properties.",
"G. Adesso, D. Girolami in [10] proposed the concept of Gaussian geometric discord for Gaussian states.",
"Measurement-induced disturbance of Gaussian states was studied in [11].",
"In [12], the MIN for Gaussian states was discussed.",
"For other related results, see [13], [14], [15], [16], [17], [18], [19] and the references therein.",
"Also, many efforts have been made to find simpler methods to quantify these correlations.",
"However, it seems that this is a very difficult task, too.",
"By now, for example, almost all known quantifications of various correlations, including entanglement measures, for continuous-variable systems are difficult to evaluate and can only be calculated for $(1+1)$ -mode Gaussian states or some special states.",
"Even for finite-dimensional cases, the authors in [20] proved that computing quantum discord is NP-hard.",
"So it makes sense and is important to find more helpful quantifications of quantum correlations.",
"The purpose of this paper is to propose a correlation $N_{\\mathcal {F}}^{\\mathcal {G}}$ for bipartite Gaussian systems in terms of local Gaussian unitary operations based on square of the fidelity $\\mathcal {F}$ introduced by Wang, Yu and Yi in [21].",
"This correlation $N_{\\mathcal {F}}^{\\mathcal {G}}$ describes the same correlation as Gaussian geometric discord for Gaussian states but have some remarkable nice properties that the known quantifications are not possed: (1) $N_{\\mathcal {F}}^{\\mathcal {G}}$ is a quantum correlation without ancilla problem; (2) $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ can be easily estimated for any $(n+m)$ -mode Gaussian states and calculated for any $(1+1)$ -mode Gaussian states; (3) $N_{\\mathcal {F}}^{\\mathcal {G}}$ is non-increasing after performing local Gaussian operations on the unmeasured subsystem.",
"Comparison $N_{\\mathcal {F}}^{\\mathcal {G}}$ in scale with other quantum correlations for two-mode symmetric squeezed thermal states reveals that $N_{\\mathcal {F}}^{\\mathcal {G}}$ is better in detecting the nonclassicality in Gaussian states."
],
[
"Gaussian States and Gaussian unitary operations",
"We recall briefly some notions and notations concerning Gaussian states and Gaussian unitary operations.",
"For arbitrary state $\\rho $ in a $n$ -mode continuous-variable system with state space $H$ , its characteristic function $\\chi _{\\rho }$ is defined as $\\chi _{\\rho }(z)={\\rm tr}(\\rho W(z)),$ where $z=(x_{1}, y_{1}, \\cdots , x_{n}, y_{n})^{\\rm T}\\in {\\mathbb {R}}^{2n}$ , $W(z)=\\exp (i{R}^{T}z)$ is the Weyl displacement operator, ${R}=(R_1,R_2,\\cdots ,R_{2n})=(\\hat{Q}_1,\\hat{P}_1,\\cdots ,\\hat{Q}_n,\\hat{P}_n)$ .",
"As usual, $\\hat{Q_k}=(\\hat{a_k}+\\hat{a_k}^\\dag )/\\sqrt{2}$ and $\\hat{P_k}=-i(\\hat{a_k}-\\hat{a_k}^\\dag )/\\sqrt{2}$ ($k=1,2,\\cdots ,n$ ) stand for respectively the position and momentum operators, where $\\hat{a}_k^\\dag $ and $\\hat{a}_k$ are the creation and annihilation operators in the $k$ th mode satisfying the Canonical Commutation Relation (CCR) $[\\hat{a}_k,\\hat{a}_l^\\dag ]=\\delta _{kl}I\\ {\\rm and}\\ [\\hat{a}_k^\\dag ,\\hat{a}_l^\\dag ]=[\\hat{a}_k,\\hat{a}_l]=0,\\ \\ k,l=1,2,\\cdots ,n.$ $\\rho $ is called a Gaussian state if $\\chi _{\\rho }(z)$ is of the form $\\chi _{\\rho }(z)=\\exp [-\\frac{1}{4}z^{\\rm T}\\Gamma z+i{\\mathbf {d}}^{\\rm T}z],$ where ${\\mathbf {d}}=(\\langle \\hat{R}_1 \\rangle , \\langle \\hat{R}_2\\rangle , \\ldots ,\\langle \\hat{R}_{2n} \\rangle )^{\\rm T}=({\\rm tr}(\\rho R_1), {\\rm tr}(\\rho R_2), \\ldots , {\\rm tr}(\\rho R_{2n}))^{\\rm T}\\in {\\mathbb {R}}^{2n}$ is called the mean or the displacement vector of $\\rho $ and $\\Gamma =(\\gamma _{kl})\\in M_{2n}(\\mathbb {R})$ is the covariance matrix (CM) of $\\rho $ defined by $\\gamma _{kl}={\\rm tr}[\\rho (\\Delta \\hat{R}_k\\Delta \\hat{R}_l+\\Delta \\hat{R}_l\\Delta \\hat{R}_k)]$ with $\\Delta \\hat{R}_k=\\hat{R}_k-\\langle \\hat{R}_k\\rangle $ ([22]).",
"Note that $\\Gamma $ is real symmetric and satisfies the condition $\\Gamma +i\\Delta \\ge 0$ , where $\\Delta =\\oplus _{k=1}^n\\Delta _k$ with $\\Delta _k=\\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix}$ for each $k$ .",
"Here $M_d(\\mathbb {R})$ stands for the algebra of all $d\\times d$ matrices over the real field $\\mathbb {R}$ .",
"Now assume that $\\rho _{AB}$ is an $(n+m)$ -mode Gaussian state with state space $H=H_A\\otimes H_B$ .",
"Then the CM $\\Gamma $ of $\\rho _{AB}$ can be written as $\\Gamma =\\left(\\begin{array}{cc}A& C\\\\C^T & B\\end{array}\\right),$ where $A \\in M_{2n}({\\mathbb {R}})$ , $B\\in M_{2m}({\\mathbb {R}})$ and $C\\in M_{2n\\times 2m}({\\mathbb {R}})$ .",
"Particularly, if $n=m=1$ , by means of local Gaussian unitary (symplectic at the CM level) operations, $\\Gamma $ has a standard form: $\\Gamma _0=\\left(\\begin{array}{cc}A_0&C_0\\\\ C_0^T & B_0\\end{array}\\right),$ where $A_0=\\left(\\begin{array}{cc}a&0\\\\0 & a\\end{array}\\right)$ , $B_0=\\left(\\begin{array}{cc}b&0\\\\0 & b\\end{array}\\right)$ , $C_0=\\left(\\begin{array}{cc}c&0\\\\0 & d\\end{array}\\right)$ , $a,b\\ge 1$ and $ab-1\\ge c ^2(d^2)$ .",
"For any unitary operator $U$ acting on $H$ , the unitary operation $\\rho \\mapsto U\\rho U^\\dag $ is said to be Gaussian if it sends Gaussian states into Gaussian states, and such $U$ is called a Gaussian unitary operator.",
"It is well-known that a unitary operator $U$ is Gaussian if and only if $U^{\\dag }RU = \\mathbf {S}R +\\mathbf {m}$ for some vector $\\mathbf {m}$ in ${\\mathbb {R}}^{2n}$ and some $\\mathbf {S} \\in {\\rm Sp}(2n,\\mathbb {R})$ , the symplectic group of all $2n\\times 2n$ real matrices $\\bf S$ that satisfy $\\mathbf {S}\\in {\\rm Sp}(2n,\\mathbb {R})\\Leftrightarrow \\mathbf {S}\\Delta \\mathbf {S}^{\\rm T}=\\Delta .$ Thus, every Gaussian unitary operator $U$ is determined by some affine symplectic map $(\\mathbf {S}, \\mathbf {m})$ acting on the phase space, and can be denoted by $U=U_{\\mathbf {S}, \\mathbf {m}}$ ([23], [24]).",
"We list some simple facts for Gaussian states and Gaussian unitary operations, and some useful results for matrix theory, which will be used frequently in the present paper.",
"Lemma 1 ([23]) For any $(n+m)$ -mode Gaussian state $\\rho _{AB}$ , write its CM $\\Gamma $ as in Eq.(1).",
"Then the CMs of the reduced states $\\rho _{A}= {\\rm tr}_{B}\\rho _{AB}$ and $\\rho _{B}={\\rm tr}_{A}\\rho _{AB}$ are matrices $A$ and $B$ , respectively.",
"Denote by $S(H)$ the set of all quantum states of the system with state space $H$ .",
"Lemma 2 ([25]) Assume that $\\rho _{AB} \\in S(H_{A}\\otimes H_{B})$ is a $(n+m)$ -mode Gaussian state.",
"Then $\\rho _{AB}$ is a product state, that is, $\\rho _{AB}=\\sigma _{A}\\otimes \\sigma _{B}$ for some $\\sigma _A\\in {\\mathcal {S}}(H_A)$ and $\\sigma _B\\in {\\mathcal {S}}(H_B)$ , if and only if $\\Gamma =\\Gamma _{A }\\oplus \\Gamma _{B}$ , where $\\Gamma $ , $\\Gamma _{A}$ and $\\Gamma _{B}$ are the CMs of $\\rho _{AB}$ , $\\sigma _{A}$ and $\\sigma _{B}$ , respectively.",
"Lemma 3 ([23], [24]) Assume that $\\rho $ is any $n$ -mode Gaussian state with CM $\\Gamma $ and displacement vector $\\mathbf {d}$ , and assume that $U_{\\mathbf {S},\\mathbf {m}}$ is a Gaussian unitary operator.",
"Then the characteristic function of the Gaussian state $\\sigma =U\\rho U^{\\dag }$ is of the form $\\exp (-\\frac{1}{4}z^{\\rm T}\\Gamma _{\\sigma } z+i\\mathbf {d}_{\\sigma }^{\\rm T}z)$ , where $\\Gamma _{\\sigma } = \\mathbf {S}\\Gamma \\mathbf {S}^{\\rm T}$ and $\\mathbf {d}_{\\sigma } =\\mathbf {m} + \\mathbf {S}\\mathbf {d}$ .",
"Lemma 4 ([26]) For any quantum states $\\rho $ , $\\sigma $ and any numbers $a>1$ , we have $\\mathop {\\rm tr}\\nolimits (\\rho \\sigma )\\le (\\mathop {\\rm tr}\\nolimits \\rho ^{a})^{\\frac{1}{a}}(\\mathop {\\rm tr}\\nolimits \\sigma ^{b})^{\\frac{1}{b}},$ where $b=\\frac{a}{a-1}$ .",
"Lemma 5 ([27]) Let $M =\\left(\\begin{array}{cc}A & B \\\\C & D\\end{array}\\right)$ be a square matrix.",
"(1) If $A$ is invertible, then its determinant $\\det \\left(\\begin{array}{cc}A & B \\\\C & D\\end{array}\\right)=(\\det A)( \\det (D-CA^{-1}B))$ .",
"(2) If $D$ is invertible, then its determinant $\\det \\left(\\begin{array}{cc}A & B \\\\C & D\\end{array}\\right)=(\\det D )(\\det (A-BD^{-1}C))$ ."
],
[
"Fidelity based nonclassicality of Gaussian states by Gaussian unitary operations",
"Fidelity is a measure of closeness between two arbitrary states $\\rho $ and $\\sigma $ , defined as $F(\\rho ,\\sigma )=(\\rm tr\\sqrt{\\sqrt{\\rho }\\sigma \\sqrt{\\rho }})^{2}$[28].",
"This measure has been explored in various context of quantum information processing such as cloning [29], teleportation [30], quantum states tomography [31], quantum chaos [32] and spotlighting phase transition in physical systems [33].",
"Though fidelity itself is not a metric, one can define a metric $D(\\rho ,\\sigma )=g(F(\\rho ,\\sigma )),$ where $g$ is a monotonically decreasing function of distance measure.",
"A few such fidelity induced metrics we mentioned here are Bures angle $A(\\rho ,\\sigma )=\\arccos \\sqrt{F(\\rho ,\\sigma )}$ , Bures metric $B(\\rho ,\\sigma )=(2-2\\sqrt{F(\\rho ,\\sigma )})^{\\frac{1}{2}}$ and sine metric $C(\\rho ,\\sigma )=\\sqrt{1-F(\\rho ,\\sigma )}$ [34].",
"Since the computation of fidelity involves square root of density matrix, various forms of fidelity have been proposed to simplify the computation.",
"In [21], the authors proposed another form $\\mathcal {F}$ of fidelity as $\\mathcal {F}(\\rho ,\\sigma )=\\frac{|\\rm tr\\rho \\sigma |}{\\sqrt{\\rm tr\\rho ^{2} \\rm tr \\sigma ^{2}}},$ In [35], to capture global nonlocal effect of a quantum state of discrete system due to locally invariant projective measurements, the authors use the fidelity in Eq.",
"(3) to define a metric $C(\\rho ,\\sigma )=\\sqrt{1-\\mathcal {F}^2(\\rho ,\\sigma )}$ for any states $\\rho $ and $\\sigma $ .",
"Furthermore, for any finite-dimensional bipartite quantum state $\\rho _{AB}$ , a new kind of MIN in terms of this metric was defined as $N_{\\mathcal {F}}(\\rho _{AB})=\\max \\limits _{\\Pi ^{A}}C^{2}(\\rho _{AB},\\Pi ^{A}(\\rho _{AB})),$ where the maximum is taken over all von Neumann measurements performing on subsystem A that are invariant at $\\rho _A={\\rm tr}_B(\\rho _{AB})$ , the reduced state of $\\rho _{AB}$ .",
"They presented an analytic expression of this version of MIN for pure bipartite states and $2\\times n$ dimensional mixed states.",
"In the present paper, motivated by the work of [35], we propose a quantum nonclassicality $N_{\\mathcal {F}}^{\\mathcal {G}}$ for continuous-variable systems by local Gaussian unitary operations for $(n+m)$ -mode states using the same metric based on the fidelity Eq.(3).",
"Definition 1 For any $(n+m)$ -mode state $\\rho _{AB}\\in \\mathcal {S}(H_{A}\\otimes H_{B})$ , the quantity $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ is defined by $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}) = \\sup _{U} C^{2}(\\rho _{AB},(U\\otimes \\textit {I})\\rho _{AB}(U^{\\dag }\\otimes \\textit {I}))=\\sup _{U}\\lbrace 1-\\frac{({\\rm tr}\\rho _{AB}(U\\otimes \\textit {I})\\rho _{AB}(U^{\\dag }\\otimes \\textit {I}))^{2}}{{\\rm tr}(\\rho _{AB}^{2}){\\rm tr}((U\\otimes \\textit {I})\\rho _{AB}(U^{\\dag }\\otimes \\textit {I}))^{2}} \\rbrace ,$ where the supremum is taken over all Gaussian unitary operators $U$ on $H_A$ satisfying $U\\rho _{A}U^{\\dag }=\\rho _{A}$ .",
"Remark 1 For any Gaussian state $\\rho _{AB}$ , there are many nontrivial Gaussian unitary operators $U$ (other than the identity $I$ ) satisfying $U\\rho _{A}U^{\\dag }=\\rho _{A}$ [16], and hence Definition 1 makes sense.",
"Different from [16], in which a quantum nonclassicality $\\mathcal {N}$ is proposed by Gaussian unitary operations based on the Hilbert-Schmidt norm, the quantity $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ measures the global nonlocal effect of a quantum state due to locally invariant Gaussian unitary operations by the metric $C^2(\\rho ,\\sigma )={1-\\mathcal {F}^2(\\rho ,\\sigma )}$ with the fidelity $\\mathcal {F}$ as in Eq.(3).",
"Recall that, the MIN [6] is defined as the square of Hilbert-Schmidt norm $\\Vert \\cdot \\Vert _2$ ($\\Vert A\\Vert _2=\\sqrt{{\\rm tr}(A^\\dag A)}\\ $ ) of difference of pre- and post-measurement states.",
"i.e., $N(\\rho _{AB})=\\max _{\\Pi ^{A}}\\Vert \\rho _{AB}-(\\Pi ^{A}\\otimes I)\\rho _{AB}(\\Pi ^{A}\\otimes I)^{\\dagger }\\Vert _2^{2},$ where the maximum is taken over all von Neumann measurements which maintain the reduced state $\\rho _A$ invariant corresponding to part A.",
"In [16], a kind of quantum correlation $\\mathcal {N}$ for $(n+m)$ -mode continuous-variable systems is defined as the square of Hilbert-Schmidt norm of difference of pre- and post-transform states $\\mathcal {N}(\\rho _{AB})=\\frac{1}{2}\\sup _{U}\\Vert \\rho _{AB}-(U\\otimes I)\\rho _{AB}(U\\otimes I)^{\\dagger }\\Vert ^{2}_2,$ where the supremum is taken over all unitary operators which maintain $\\rho _A$ invariant corresponding to party A.",
"There are other quantum correlations defined by Hilbert-Schmidt norm, for example, the Gaussian geometric discord and the quantum correlation proposed respectively in [10], [13].",
"These kinds of quantity defined by Hilbert-Schmidt norm mentioned above may change rather wildly through some trivial and uncorrelated actions on the unoperated party B.",
"For example, if we append an uncorrelated ancilla C, and regarding the state $\\rho _{ABC}=\\rho _{AB}\\otimes \\rho _{C}$ as a bipartite state with the partition A:BC.",
"After some straight-forward calculations, one gets $ \\mathcal {N}(\\rho _{ABC})=\\mathcal {N}(\\rho _{AB}){\\rm tr}\\rho _{C}^{2},$ which means that the quantity $\\mathcal {N}$ differs arbitrarily due to local ancilla C as long as $\\rho _{C}$ is mixed.",
"While this problem can be avoided if one employs ${\\mathcal {N}}_{\\mathcal {F}}^{\\mathcal {G}}$ as in Definition 1 since $&\\mathcal {F}(\\rho _{ABC},(U\\otimes I\\otimes I)\\rho _{ABC}) =\\mathcal {F}(\\rho _{AB}\\otimes \\rho _{C},(U\\otimes I)\\rho _{AB}\\otimes I\\rho _{C})\\\\=& \\mathcal {F}(\\rho _{AB},(U\\otimes I)\\rho _{AB})\\cdot \\mathcal {F}(\\rho _{C},\\rho _{C})=\\mathcal {F}(\\rho _{AB},(U\\otimes I)\\rho _{AB}),$ according to the multiplicativity of the fidelity [21].",
"Thus, we reach the following conclusion.",
"Theorem 1 $N_{\\mathcal {F}}^{\\mathcal {G}}$ is a quantum nonclassicality without ancilla problem.",
"We explore further the properties of ${ N}_{\\mathcal {F}}^{\\mathcal {G}}$ below.",
"Denote by ${\\mathcal {B}}(H)$ the algebra of all bounded linear operators acting on $H$ .",
"Theorem 2 $ N_{\\mathcal {F}}^{\\mathcal {G}}$ is locally Gaussian unitary invariant, that is, for any $(n+m)$ -mode Gaussian state $\\rho _{AB}\\in \\mathcal {S}(H_{A}\\otimes H_{B})$ and any Gaussian unitary operators $W\\in {\\mathcal {B}}(H_A)$ and $V\\in {\\mathcal {B}}(H_{B})$ , we have $N_{\\mathcal {F}}^{\\mathcal {G}}((W\\otimes V)\\rho _{AB}(W^{\\dag }\\otimes V^{\\dag })) = N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}) $ .",
"Proof.",
"Assume that $\\rho _{AB}\\in \\mathcal {S}(H_{A}\\otimes H_{B})$ is an $(n+m)$ -mode Gaussian state.",
"For given Gaussian unitary operators $W\\in {\\mathcal {B}}(H_A)$ and $V\\in {\\mathcal {B}}(H_{B})$ , let $\\sigma _{AB} =(W\\otimes V)\\rho _{AB} (W^{\\dag }\\otimes V^{\\dag })$ .",
"Denote ${\\mathcal {U}}^{\\mathcal {G}}(H_A)$ the set of all Gaussian unitary operators acting on $H_A$ .",
"Since $ N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}) = &\\sup _{U\\in {\\mathcal {U}^{\\mathcal {G}}(H_A)},\\ U\\rho _AU^\\dag =\\rho _A}C^{2}(\\rho _{AB}, (U\\otimes \\textit {I})\\rho _{AB}(U^{\\dag }\\otimes \\textit {I}))\\\\=& \\sup _{U\\in {\\mathcal {U}^{\\mathcal {G}}(H_A)},\\ U\\rho _AU^\\dag =\\rho _A}\\lbrace 1-\\mathcal {F}^2(\\rho _{AB},(U\\otimes I)\\rho _{AB}(U^{\\dag }\\otimes I))\\rbrace \\\\ = &1-\\inf _{U\\in {\\mathcal {U}^{\\mathcal {G}}(H_A)},\\ U\\rho _AU^\\dag =\\rho _A}\\mathcal {F}^2(\\rho _{AB},(U\\otimes I)\\rho _{AB}(U^{\\dag }\\otimes I)) ,$ to demonstrate that $N_{\\mathcal {F}}^\\mathcal {G}$ is locally Gaussian unitary invariant, it is sufficient to prove $ & \\inf _{U\\in {\\mathcal {U}^{\\mathcal {G}}(H_A)},\\ U\\rho _AU^\\dag =\\rho _A}\\mathcal {F}(\\rho _{AB},(U\\otimes I)\\rho _{AB}(U^{\\dag }\\otimes I))\\\\\\nonumber = & \\inf _{ U^\\prime \\in {\\mathcal {U}^{\\mathcal {G}}(H_A)},\\ U^\\prime \\sigma _AU^{\\prime \\dag }=\\sigma _A }\\mathcal {F}(\\sigma _{AB},(U^{\\prime }\\otimes I)\\sigma _{AB}(U^{\\prime \\dag }\\otimes I)),$ where $\\sigma _{AB}=(W\\otimes V)\\rho _{AB}(W^{\\dag }\\otimes V^{\\dag })$ , $W$ and $V$ are given Gaussian unitary operators acting on Hilbert spaces $H_{A}$ and $H_{B}$ , respectively.",
"Note that $\\sigma _{A} = W\\rho _{A}W^{\\dag }$ .",
"For any Gaussian unitary operator $U\\in {\\mathcal {B}}(H_{A})$ satisfying $U\\rho _{A} U^{\\dag }=\\rho _{A}$ , let $U^{\\prime }= WUW^{\\dag }$ .",
"Then $U^\\prime $ is a Gaussian unitary operator satisfing $U^{\\prime }\\sigma _{A}U^{\\prime \\dag } =WUW^{\\dag }W\\rho _{A}W^{\\dag } WU^{\\dag }W^\\dag =\\sigma _{A}$ .",
"Conversely, if $U^\\prime \\sigma _{A}U^{\\prime \\dag }=\\sigma _{A}$ , $U=W^\\dag U^\\prime W$ will satisfy $U\\rho _AU^\\dag =\\rho _A$ .",
"By Eq.",
"(3), we have $&\\mathcal {F}^2(\\rho _{AB},(U\\otimes I)\\rho _{AB}(U^{\\dag }\\otimes I))=\\frac{(\\mathop {\\rm tr}\\nolimits \\rho _{AB}(U\\otimes I)\\rho _{AB}(U^{\\dag }\\otimes I))^{2}}{\\mathop {\\rm tr}\\nolimits \\rho _{AB}^{2}\\mathop {\\rm tr}\\nolimits ((U\\otimes I)\\rho _{AB}(U^{\\dag }\\otimes I))^{2}} \\\\=& \\frac{(\\mathop {\\rm tr}\\nolimits (W^\\dag \\otimes V^\\dag )\\sigma _{AB} (W \\otimes V)(U\\otimes I)(W^\\dag \\otimes V^\\dag )\\sigma _{AB} (W \\otimes V)(U^{\\dag }\\otimes I))^{2}}{\\mathop {\\rm tr}\\nolimits ((W^\\dag \\otimes V^\\dag )\\sigma _{AB} (W \\otimes V ))^{2}\\mathop {\\rm tr}\\nolimits ((U\\otimes I)(W^\\dag \\otimes V^\\dag )\\sigma _{AB} (W \\otimes V )(U^{\\dag }\\otimes I))^{2}} \\\\=&\\frac{(\\mathop {\\rm tr}\\nolimits \\sigma _{AB} (U^{\\prime }\\otimes I)\\sigma _{AB}(U^{\\prime \\dag }\\otimes I))^{2}}{\\mathop {\\rm tr}\\nolimits \\sigma _{AB}^{2}\\mathop {\\rm tr}\\nolimits ((U^{\\prime }\\otimes I)\\sigma _{AB}(U^{\\prime \\dag }\\otimes I))^{2}} =\\mathcal {F}^2(\\sigma _{AB},(U^{\\prime }\\otimes I)\\sigma _{AB}(U^{\\prime \\dag }\\otimes I)).$ Therefore, Eq.",
"(5) holds, as desired.",
"$\\Box $ Notice that, for any $(n+m)$ -mode product quantum state $\\rho _{AB}$ , one must have $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})=0$ by the definition.",
"But for Gaussian states, the converse is also true.",
"Hence, when restricted to Gaussian states, the correlation $N_{\\mathcal {F}}^{\\mathcal {G}}$ describes the same nonclassicality as that described by Gaussian QD (two-mode) [8], [9], Gaussian geometric discord [10], the correlations $Q$ , $Q_{\\mathcal {P}}$ discussed in [13] and the correlation ${\\mathcal {N}}$ discussed in [16].",
"Theorem 3 For any $(n+m)$ -mode Gaussian state $\\rho _{AB}\\in {\\mathcal {S}}(H_A\\otimes H_B)$ , $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}) = 0$ if and only if $\\rho _{AB}$ is a product state.",
"Proof.",
"By Definition 1, the “if” part is apparent.",
"Let us check the “only if” part.",
"Since the mean of any Gaussian state can be transformed to zero under some local Gaussian unitary operation, by Theorem 2, it is sufficient to consider the Gaussian states whose mean are zero.",
"Assume that $\\rho _{AB}$ is an $(n+m)$ -mode Gaussian state with CM $\\Gamma =\\left(\\begin{array}{cc}A & C \\\\C^{T} & B\\end{array}\\right)$ as in Eq.",
"(1) and zero mean such that $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})=0$ .",
"By Lemma 1, the CM of $\\rho _A$ is $A$ .",
"According to the Williamson Theorem, there exists a symplectic matrix $\\mathbf {S}_{0}$ such that $\\mathbf {S}_{0}A\\mathbf {S}_{0}^{\\rm T} = \\oplus _{i=1}^n v_{i}I$ and $U_{0}\\rho _{A}U_{0}^{\\dag }=\\otimes _{i=1}^n\\rho _{i}$ , where $U_0=U_{{\\bf S}_0,{\\bf 0}}$ and $\\rho _{i}$ s are some thermal states.",
"Write $\\sigma _{AB}=(U_{0}\\otimes I)\\rho _{AB}(U_{0}^{\\dag }\\otimes I)$ .",
"It follows from Theorem 2 that $N_{\\mathcal {F}}^{\\mathcal {G}}(\\sigma _{AB})=N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})=0$ .",
"Obviously, $\\sigma _{AB}$ has the CM $\\Gamma ^{\\prime } = \\left(\\begin{array}{cc}\\oplus _{i}^nv_{i}I & C^{\\prime } \\\\C^{\\prime \\rm T} & B^{\\prime }\\end{array}\\right)$ and the mean 0.",
"By Lemma 3 and [16], for any Gaussian unitary operator $U_{\\mathbf {S},\\mathbf {m}}\\in {\\mathcal {B}}(H_A)$ so that $\\mathbf {m}=0$ and $\\mathbf {S}=\\oplus _{i=1}^n \\mathbf {S}_{\\theta _{i}}$ with $\\mathbf {S}_{\\theta _{i}} = \\left(\\begin{array}{cc}\\cos \\theta _{i} & \\sin \\theta _{i} \\\\-\\sin \\theta _{i} & \\cos \\theta _{i} \\\\\\end{array}\\right)$ for some $\\theta _{i}\\in [0, \\frac{\\pi }{2}]$ , we have $U_{\\mathbf {S},\\mathbf {m}}\\sigma _{A}U_{\\mathbf {S},\\mathbf {m}}^{\\dag } =\\sigma _{A}={\\rm tr}_B(\\sigma _{AB})$ .",
"Then, by the definition Eq.",
"(4), $N_{\\mathcal {F}}^{\\mathcal {G}}(\\sigma _{AB})=0$ entails $(\\rm tr\\sigma _{AB}(U_{\\mathbf {S},\\mathbf {m}}\\otimes I)\\sigma _{AB}(U_{\\mathbf {S},\\mathbf {m}}^\\dag \\otimes I))^{2}=\\rm tr\\sigma _{AB}^{2}{\\rm tr}((U_{\\mathbf {S},\\mathbf {m}}\\otimes I)\\sigma _{AB}(U_{\\mathbf {S},\\mathbf {m}}^\\dag \\otimes I))^{2}.$ Since the Holder's inequality (Lemma 4) asserts that ${\\rm tr}(\\rho \\sigma )^2\\le {\\rm tr}\\rho ^2{\\rm tr}\\sigma ^2$ and clearly, the equality holds if and only if $\\sigma =\\rho $ , we must have $\\sigma _{AB}=(U_{\\mathbf {S},\\mathbf {m}}\\otimes I)\\sigma _{AB}(U_{\\mathbf {S},\\mathbf {m}}^\\dag \\otimes I).$ Hence $\\sigma _{AB}$ and $(U_{\\mathbf {S},\\mathbf {m}}\\otimes I)\\sigma _{AB}(U_{\\mathbf {S},\\mathbf {m}}^\\dag \\otimes I)$ have the same CMs, that is, $\\left(\\begin{array}{cc}\\oplus _{i=1}^nv_{i}I & C^{\\prime } \\\\C^{\\prime \\rm T} & B^{\\prime }\\end{array}\\right) = \\left(\\begin{array}{cc}\\oplus _{i=1}^nv_{i}I & {\\bf S}C^{\\prime } \\\\C^{\\prime \\rm T}{\\bf S}^{\\rm T} & B^{\\prime } \\\\\\end{array}\\right).$ If we take $\\theta _{i}\\in (0, \\frac{\\pi }{2})$ for each $i$ , then $I- {\\bf S}$ is an invertible matrix, which forces $C^{\\prime } = 0.$ So $\\sigma _{AB}$ is a product state by Lemma 2.",
"It follows that $\\rho _{AB} = (U_0^{\\dag } \\otimes I )\\sigma _{AB}(U_0 \\otimes I)$ is also a product state.",
"$\\Box $ In the rest of this paper, we mainly consider the case when the states $\\rho _{AB}$ are Gaussian.",
"A remarkable virtue of $N_{\\mathcal {F}}^{\\mathcal {G}}$ is that it can be evaluated easily.",
"For any two-mode Gaussian state $\\rho _{AB}$ , we can give an analytic computation formula.",
"Theorem 4 For any $(1+1)$ -mode Gaussian state $\\rho _{AB}$ whose CM has the standard form $\\Gamma _0=\\left(\\begin{array}{cc}A_{0} & C_{0} \\\\C_{0}^{\\rm T} & B_{0}\\end{array}\\right)=\\left(\\begin{array}{cccc}a & 0 & c & 0\\\\0 & a & 0 & d\\\\c & 0 & b & 0\\\\0 & d & 0 & b\\end{array}\\right)$ , we have ${ N}_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}) =1-\\frac{(ab-c^{2})(ab-d^{2})}{(ab-c^{2}/2)(ab-d^{2}/2)}.$ Particularly, the value of ${ N}_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ is independent of the mean of the state $\\rho _{AB}$ .",
"Proof.",
"For any $(1+1)$ -mode Gaussian state $\\rho _{AB}$ with CM $\\Gamma ^{\\prime }$ and mean $(\\mathbf {d}_{A}^{\\prime }, \\mathbf {d}_{B}^{\\prime })$ , we can always find two Gaussian operators $U$ and $V$ so that the CM $\\Gamma _0$ of $\\sigma _{AB}=(U\\otimes V)\\rho _{AB}(U^{\\dag }\\otimes V^{\\dag })$ is of the standard form $\\Gamma _0=\\left(\\begin{array}{cc}A_{0} & C_{0} \\\\C_{0}^{\\rm T} & B_{0}\\end{array}\\right)=\\left(\\begin{array}{cccc}a & 0 & c & 0\\\\0 & a & 0 & d\\\\c & 0 & b & 0\\\\0 & d & 0 & b\\end{array}\\right).$ Denote the mean of $\\sigma _{AB}$ by $(\\mathbf {d}_{A}, \\mathbf {d}_{B})$ .",
"Since $N_{\\mathcal {F}}^{\\mathcal {G}}$ is locally Gaussian unitary invariant, one has $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})=N_{\\mathcal {F}}^{\\mathcal {G}}(\\sigma _{AB})$ .",
"Hence, we may assume that the CM of $\\rho _{AB}$ is $\\Gamma _0$ and the mean of $\\rho _{AB}$ is $(\\mathbf {d}_{A}, \\mathbf {d}_{B})$ .",
"For any Gaussian unitary operator $U_{\\mathbf {S},\\mathbf {m}}$ such that $U_{\\mathbf {S},\\mathbf {m}}\\rho _{A} U_{\\mathbf {S},\\mathbf {m}}^{\\dag }=\\rho _{A}$ , we see that $\\mathbf {S}$ and $\\mathbf {m}$ meet the conditions $\\mathbf {S}A_0\\mathbf {S}^{\\rm T} =A_0$ and $ \\mathbf {S}\\mathbf {d}_{A} + \\mathbf {m}=\\mathbf {d}_{A}$ .",
"As $A_0=aI_2$ , we have $\\mathbf {S}\\mathbf {S}^{\\rm T}=I_2$ .",
"It follows from $\\mathbf {S}\\Delta \\mathbf {S}^{\\rm T} = \\Delta $ that there exists some $\\theta \\in [0,\\frac{\\pi }{2}]$ such that ${\\bf S} = {\\bf S}_{\\theta } = \\left(\\begin{array}{cc}\\cos \\theta & \\sin \\theta \\\\-\\sin \\theta & \\cos \\theta \\end{array}\\right)$ .",
"So the CM of Gaussian state $(U_{\\mathbf {S},\\mathbf {m}}\\otimes I)\\rho _{AB}(U_{\\mathbf {S},\\mathbf {m}}^{\\dag }\\otimes I)$ is $\\Gamma _{\\theta }= \\left(\\begin{array}{cccc}a & 0 & c\\cos \\theta & d\\sin \\theta \\\\0 & a & -c\\sin \\theta & d\\cos \\theta \\\\c\\cos \\theta & -c\\sin \\theta & b & 0 \\\\d\\sin \\theta & d\\cos \\theta & 0 & b\\end{array}\\right),$ and the mean of $(U_{\\mathbf {S},\\mathbf {m}}\\otimes I)\\rho _{AB}(U_{\\mathbf {S},\\mathbf {m}}^{\\dag }\\otimes I)$ is $({\\bf S}\\oplus I)(\\mathbf {d}_{A}\\oplus \\mathbf {d}_{B})+\\mathbf {m}\\oplus 0=({\\bf S} \\mathbf {d}_{A}+\\mathbf {m}) \\oplus \\mathbf {d}_{B}=\\mathbf {d}_{A}\\oplus \\mathbf {d}_{B}=(\\mathbf {d}_{A}, \\mathbf {d}_{B})$ as $\\mathbf {S}\\mathbf {d}_{A} + \\mathbf {m}=\\mathbf {d}_{A}$ .",
"Conversely, for any $\\mathbf {S}_\\theta $ , taking $\\mathbf {m}=\\mathbf {d}_A-\\mathbf {S}_\\theta \\mathbf {d}_A$ , we have $U_{\\mathbf {S}_\\theta ,\\mathbf {m}}$ satisfies the condition $U_{\\mathbf {S}_\\theta ,\\mathbf {m}}\\rho _{A}U_{\\mathbf {S}_\\theta ,\\mathbf {m}}^{\\dag }=\\rho _{A}$ .",
"Also, notice that, for any $n$ -mode Gaussian states $\\rho ,\\sigma $ with CMs $V_{\\rho }, V_{\\sigma }$ and means $\\mathbf {d}_{\\rho }, \\mathbf {d}_{\\sigma }$ , respectively, it is shown in [36] that $\\textrm {Tr}(\\rho \\sigma ) =\\frac{1}{\\sqrt{\\det [(V_{\\rho }+V_{\\sigma })/2]}}\\exp [-\\frac{1}{2}\\delta \\langle \\mathbf {d}\\rangle ^{T}\\det [(V_{\\rho }+V_{\\sigma })/2]^{-1}\\delta \\langle \\mathbf {d}\\rangle ],$ where $ \\delta \\langle \\mathbf {d}\\rangle =\\mathbf {d}_{\\rho } - \\mathbf {d}_{\\sigma }$ .",
"Hence, by Eq.",
"(4) and Eq.",
"(6) as well as the fact that $\\det \\Gamma _\\theta =\\det \\Gamma _0=(ab-c^{2})(ab-d^{2})$ , one obtains $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})=&\\sup _{U\\in {\\mathcal {U}^{\\mathcal {G}}(H_A)},\\ U\\rho _AU^\\dag =\\rho _A}C^{2}(\\rho _{AB}, (U\\otimes \\textit {I})\\rho _{AB}(U^{\\dag }\\otimes \\textit {I}))\\\\= & \\sup _{U\\in {\\mathcal {U}^{\\mathcal {G}}(H_A)},\\ U\\rho _AU^\\dag =\\rho _A} \\lbrace 1-\\frac{(\\textrm {tr}\\rho _{AB}(U\\otimes \\textit {I})\\rho _{AB}(U^{\\dag }\\otimes \\textit {I}))^{2}}{\\textrm {tr}(\\rho _{AB}^{2})\\textrm {tr}((U\\otimes \\textit {I})\\rho _{AB}(U^{\\dag }\\otimes \\textit {I}))^{2}} \\rbrace \\\\= &\\sup _{\\theta \\in [0, \\frac{\\pi }{2}]}\\lbrace 1-\\frac{\\sqrt{\\det \\Gamma _0\\det \\Gamma _\\theta }}{{\\det ((\\Gamma _0 +\\Gamma _\\theta )/2)}} \\rbrace \\\\= & \\max _{\\theta \\in [0, \\frac{\\pi }{2}]}\\lbrace 1-\\frac{(ab-c^{2})(ab-d^{2})}{{[ab-c^{2}(1+\\cos \\theta )/2][ab-d^{2}(1+\\cos \\theta )/2]}}\\rbrace \\\\= & 1-\\frac{(ab-c^{2})(ab-d^{2})}{{(ab-c^{2}/2)(ab-d^{2}/2)}},$ and, this quantity is independent of the mean of $\\rho _{AB}$ , completing the proof.",
"$\\Box $ Next, we are going to give an estimate of $N_{\\mathcal {F}}^{\\mathcal {G}}$ for any $(n+m)$ -mode Gaussian state $\\rho _{AB}$ .",
"Theorem 5 For any $(n+m)$ -mode Gaussian state $\\rho _{AB}$ with CM $\\Gamma =\\left(\\begin{array}{cc}A & C \\\\C^{\\rm T} & B\\end{array}\\right)$ , $ N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ is independent of the mean of $\\rho _{AB}$ and $0 \\le N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}) \\le 1- \\frac{\\det (B-C^{\\rm T}A^{-1}C)}{{\\det B}}< 1.$ Furthermore, the upper bound 1 is tight.",
"Proof.",
"Let $\\rho _{AB}$ be any $(n+m)$ -mode Gaussian state with CM $\\Gamma =\\left(\\begin{array}{cc}A & C \\\\C^{\\rm T} & B\\end{array}\\right)$ and mean $\\mathbf {d}=(\\mathbf {d}_{A},\\mathbf {d}_{B})$ .",
"Note that, by Lemma 1, the CM of $\\rho _A$ is $A$ .",
"Write $\\sigma _{AB}=(U_{\\mathbf {S},\\mathbf {m}}\\otimes I)\\rho _{AB}(U_{\\mathbf {S},\\mathbf {m}}^{\\dag }\\otimes I)$ , where $U_{\\mathbf {S},\\mathbf {m}}$ is any Gaussian unitary operator of the subsystem $A$ .",
"Clearly, $U_{\\mathbf {S},\\mathbf {m}}\\rho _{A} U_{\\mathbf {S},\\mathbf {m}}^{\\dag } = \\rho _{A}$ if and only if the symplectic matrix $\\mathbf {S}$ satisfies $\\mathbf {S} A \\mathbf {S}^{\\rm T} = A$ and the vector $\\mathbf {m}=\\mathbf {d}_{A}- \\mathbf {S}\\mathbf {d}_{A} $ .",
"In this case $\\sigma _{AB}$ has the CM $\\Gamma _{\\mathbf {S}}= \\left(\\begin{array}{cc}A & {\\bf S}C \\\\C^{\\rm T}{\\bf S}^{\\rm T} & B\\end{array}\\right)$ and the mean $\\mathbf {d}_{\\mathbf {S}}=({\\bf S}\\oplus I)(\\mathbf {d}_{A}\\oplus \\mathbf {d}_{B})+\\mathbf {m}\\oplus 0=({\\bf S}\\mathbf {d}_{A}+\\mathbf {m}) \\oplus \\mathbf {d}_{B}=\\mathbf {d}_{A}\\oplus \\mathbf {d}_{B}=(\\mathbf {d}_{A}, \\mathbf {d}_{B})=\\mathbf {d}$ .",
"Denote by $\\mathcal {S} (2n)={\\rm Sp}(2n, \\mathbb {R}),$ the set of all $2n\\times 2n$ symplectic matrices.",
"Then, by Eq.",
"(6), $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}) = &\\sup _{U\\in {\\mathcal {U}^{\\mathcal {G}}(H_A)},\\ U\\rho _AU^\\dag =\\rho _A}C^{2}(\\rho _{AB},(U\\otimes \\textit {I})\\rho _{AB}(U^{\\dag }\\otimes \\textit {I})) \\\\=&\\sup _{U\\in {\\mathcal {U}^{\\mathcal {G}}(H_A)},\\ U\\rho _AU^\\dag =\\rho _A}\\lbrace 1-\\frac{(\\textrm {tr}\\rho _{AB}(U\\otimes \\textit {I})\\rho _{AB}(U^{\\dag }\\otimes \\textit {I}))^{2}}{\\textrm {tr}(\\rho _{AB}^{2})\\textrm {tr}((U\\otimes \\textit {I})\\rho _{AB}(U^{\\dag }\\otimes \\textit {I}))^{2}} \\rbrace \\\\= & \\sup _{\\mathbf {S}\\in {\\mathcal {S}}(2n),\\ \\mathbf {S} A\\mathbf {S}^{\\rm T} = A}\\lbrace 1-\\frac{\\frac{1}{({\\det (\\Gamma +\\Gamma _{\\bf S})/2)}}}{\\frac{1}{\\sqrt{\\det \\Gamma }}\\frac{1}{\\sqrt{\\det \\Gamma _{\\bf S}}}} \\rbrace \\\\=& \\sup _{\\mathbf {S}\\in {\\mathcal {S}}(2n),\\ \\mathbf {S} A\\mathbf {S}^{\\rm T} = A}\\lbrace 1-\\frac{\\sqrt{\\det \\Gamma \\det \\Gamma _{\\bf S}}}{{\\det ((\\Gamma +\\Gamma _{\\bf S})/2)}} \\rbrace .$ That is, $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}) =\\sup _{\\mathbf {S}\\in {\\mathcal {S}}(2n),\\ \\mathbf {S} A \\mathbf {S}^{\\rm T} =A}\\lbrace 1-\\frac{\\sqrt{\\det \\Gamma \\det \\Gamma _{\\bf S}}}{{\\det ((\\Gamma +\\Gamma _{\\bf S})/2)}} \\rbrace .$ Obviously, $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ is independent of the mean $\\mathbf {d}$ .",
"It is easy to verify that $\\det \\Gamma =\\det \\Gamma _{\\bf S}$ .",
"Since $\\Gamma = \\left(\\begin{array}{cc}A & C \\\\C^{\\rm T} & B\\end{array} \\right)>0$ , by Lemma 4, we have $0<\\det \\Gamma =\\det A\\det (B-C^{\\rm T}A^{-1}C)=\\det \\Gamma _{\\bf S},$ which implies that $\\det (B-C^{\\rm T}A^{-1}C)>0$ .",
"In addition, as $\\frac{\\Gamma + \\Gamma _{\\bf S}}{2}=\\left(\\begin{array}{cc}A & \\frac{C +{\\bf S}C}{2} \\\\\\frac{C^{ T}+C^{ T}{\\bf S}^{T}}{2}& B\\end{array}\\right)$ and $A$ , $B$ are positive-definite, by Fischer's inequality ([27]), we have $\\det \\frac{\\Gamma +\\Gamma _{\\bf S}}{2}\\le \\det A\\det B$ .",
"Hence, by Eq.",
"(7), we get $0\\le N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}) \\le & 1- \\frac{{\\det A\\det (B-C^{\\rm T}A^{-1}C)}}{{\\det A\\det B }}=1- \\frac{\\det (B-C^{\\rm T}A^{-1}C)}{{\\det B}}< 1.$ We claim that the upper bound 1 is tight, that is, we have $ \\sup _{\\rho _{AB}} N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})=1.$ To see this, consider a two-mode squeezed vacuum state $\\rho (r) = S(r)|00\\rangle \\langle 00|S^{\\dag }(r)$ , where $S(r)=\\exp (-r\\hat{a}_{1}\\hat{a}_{2}+r\\hat{a}_{1}^{\\dag }\\hat{a}_{2}^{\\dag })$ is a two-mode squeezing operator with squeezed number $r\\ge 0$ and $|00\\rangle $ is the vacuum state ([37]).",
"The CM of $\\rho (r)$ is $\\frac{1}{2}\\left(\\begin{array}{cc}A_r& C_r \\\\C_r^{ T} & B_r\\end{array}\\right),$ where $A_r=B_r=\\left(\\begin{array}{cc}\\exp (-2r)+\\exp (2r) & 0\\\\0 & \\exp (-2r)+\\exp (2r)\\end{array}\\right)$ and $C_r=C_r^{ T}=\\left(\\begin{array}{cc}-\\exp (-2r)+\\exp (2r) & 0 \\\\0 & \\exp (-2r)-\\exp (2r)\\end{array}\\right).$ By Theorem 4, it is easily checked that $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho (r))=1-\\frac{16}{(\\frac{\\exp (-4r)+\\exp (4r)}{2}+3)^{2}}.$ Clearly, $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho (r)) \\rightarrow 1$ as $r\\rightarrow \\infty $ .",
"So $ \\sup _r N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho (r))=1$ and Eq.",
"(8) is true.",
"$\\Box $ Suppose that $\\rho _{AB}$ is an $(n+m)$ -mode Gaussian state with CM $\\Gamma =\\left(\\begin{array}{cc}A & C \\\\C^{\\rm T} & B\\end{array}\\right)$ as in Eq.(1).",
"One can always perform a local Gaussian unitary operation on the state $\\rho _{AB}$ , say $\\sigma _{AB}=(U_{S_A}\\otimes V_{S_B})\\rho _{AB}(U_{S_A}^{\\dag }\\otimes V_{S_B}^{\\dag })$ , such that the corresponding CM of $\\sigma _{AB}$ is of the form $\\Gamma ^{\\prime } = \\left(\\begin{array}{cc}\\oplus _{i}^nv_{i}I_2 & C^{\\prime } \\\\C^{\\prime \\rm T} & \\oplus _{i}^ms_{i}I_2\\end{array}\\right)$ , where $v_{i}$ s and $s_{i}$ s are the symplectic roots of $\\rho _A$ and $\\rho _B$ respectively, $C^\\prime =S_ACS_B^{\\rm T}$ .",
"By Theorem 2, $N_{\\mathcal {F}}^{\\mathcal {G}}(\\sigma _{AB})=N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ .",
"This gives an estimation of $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ for $(n+m)$ -mode Gaussian state $\\rho _{AB}$ in terms of symplectic roots of the CMs of the reduced states $\\rho _A$ and $\\rho _B$ : $0 \\le N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}) \\le 1- \\frac{\\det (\\oplus _{i}^m s_{i} I_2-S_BC^{\\rm T}S_A^{\\rm T}(\\oplus _{i}^n1/v_{i} I_2)S_ACS_B^{\\rm T})}{{\\prod _{i=1}^{m}s_{i}^{2}}}< 1.$"
],
[
"Nonlocality connected to Gaussian channels",
"In this section we intend to investigate the fidelity based nonlocality connected to a Gaussian quantum channel.",
"Here we mainly consider the $(1+1)$ -mode Gaussian states whose CM are of the standard form.",
"Since a Gaussian state $\\rho $ is described by its CM $\\Gamma $ and displacement vector $\\mathbf {d}$ , we can denote it as $\\rho =\\rho (\\Gamma ,\\mathbf {d})$ .",
"Recall that a Gaussian channel is a quantum channel that transforms Gaussian states into Gaussian states.",
"Assume that $\\Phi $ is a Gaussian channel of $n$ -mode Gaussian systems.",
"Then, there exist real matrices $M, K\\in M_{2n}(\\mathbb {R})$ satisfying $M=M^T\\ge 0$ and det$ M\\ge ({\\rm detK}-1)^2$ , and a vector $\\overline{\\mathbf {d}}\\in {\\mathbb {R}}^{2n}$ , such that, for any $n$ -mode Gaussian state $\\rho =\\rho (\\Gamma ,\\mathbf {d})$ , we have $\\Phi (\\rho (\\Gamma ,\\mathbf {d}))=\\rho (\\Gamma ^{\\prime },\\mathbf {d}^{\\prime })$ with $\\mathbf {d}^{\\prime }=K\\mathbf {d}+\\overline{\\mathbf {d}}\\ \\ {\\rm and}\\ \\ \\Gamma ^{\\prime }=K\\Gamma K^{T}+M.$ So we can parameterize the Gaussian channel $\\Phi $ as $\\Phi =\\Phi (K, M, \\overline{\\mathbf {d}})$ .",
"Theorem 6 Consider the $(1+1)$ -mode continuous-variable system AB.",
"Let $\\Phi =\\Phi (K, M, \\overline{\\mathbf {d}})$ be a Gaussian channel performed on the subsystem B with $K=\\left(\\begin{array}{cc}k_{11}&k_{12}\\\\k_{21} & k_{22}\\end{array}\\right)$ and $M=\\left(\\begin{array}{cc}m_{11} &m_{12}\\\\m_{12} & m_{22}\\end{array}\\right)$ .",
"Assume that $\\rho _{AB}\\in {\\mathcal {S}}(H_A\\otimes H_B)$ is any $(1+1)$ -mode Gaussian state with CM $\\Gamma _0=\\left(\\begin{array}{cccc}a & 0 & c & 0\\\\0 & a & 0 & d\\\\c & 0 & b & 0\\\\0 & d & 0 & b\\end{array}\\right)$ .",
"Then $N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )\\rho _{AB})=1-\\frac{(ab-c^{2})(ab-d^{2})n_{1}+a(ab-c^{2})n_{2}+a(ab-d^{2})n_{3}+a^{2}n_{4}}{(ab-c^{2}/2)(ab-d^{2}/2)n_{1}+a(ab-c^{2}/2)n_{2}+a(ab-d^{2}/2)n_{3}+a^{2}n_{4}},$ where $n_{0}=(1+\\cos \\theta )/2$ , $n_{1}=k^{2}_{11}k^{2}_{22}+k^{2}_{12}k^{2}_{21}-2k_{11}k_{12}k_{21}k_{22}$ , $n_{2}=m_{22}k^{2}_{11}+m_{11}k^{2}_{21}-2m_{12}k_{11}k_{21}$ , $n_{3}=m_{22}k^{2}_{12}+m_{11}k^{2}_{22}-2m_{12}k_{12}k_{22}$ and $n_{4}=m_{11}m_{22}-m_{12}^{2}$ .",
"Proof.",
"Suppose that the $(1+1)$ -mode Gaussian state $\\rho _{AB}$ has CM $\\Gamma _0=\\left(\\begin{array}{cccc}a & 0 & c & 0\\\\0 & a & 0 & d\\\\c & 0 & b & 0\\\\0 & d & 0 & b\\end{array}\\right)$ and the mean $(\\mathbf {d}_{A},\\mathbf {d}_{B})$ .",
"Then the CM $\\Gamma ^\\prime $ and the mean $\\mathbf {d}^\\prime $ of $\\sigma _{AB}=(I\\otimes \\Phi )\\rho _{AB}$ are respectively $&\\Gamma ^{\\prime } = \\left(\\begin{array}{cc}I & 0 \\\\0 & K\\end{array}\\right)\\left(\\begin{array}{cc}A_{0} & C_{0} \\\\C^{T}_{0} & B_{0}\\end{array}\\right)\\left(\\begin{array}{cc}I & 0 \\\\0 & K^{T}\\end{array}\\right)+\\left(\\begin{array}{cc}0 & 0 \\\\0 & M\\end{array}\\right)=\\left(\\begin{array}{cc}A_{0} & C_{0}K^{T} \\\\KC_{0}^{T} & KB_{0}K^{T}+M\\end{array}\\right)$ and $\\mathbf {d}^{\\prime }=(I\\oplus K)(\\mathbf {d}_{A}\\oplus \\mathbf {d}_{B})+0\\oplus \\overline{\\mathbf {d}}=\\mathbf {d}_{A}\\oplus (K\\mathbf {d}_{B}+\\overline{\\mathbf {d}}).$ After a local invariant Gaussian unitary operation on the subsystem A, one has $(U\\otimes I)\\sigma _{AB}(U^{\\dag }\\otimes I)=\\sigma _{AB}^{\\prime }$ .",
"Remind that $U\\rho _AU^\\dag =\\rho _A$ , which forces that, at the symplectic transformation level, $U=U_{\\mathbf {S},\\mathbf {m}}$ with $\\mathbf {m}=0$ and ${\\bf S} = {\\bf S}_{\\theta } =\\left(\\begin{array}{cc}\\cos \\theta & \\sin \\theta \\\\-\\sin \\theta & \\cos \\theta \\end{array}\\right)$ for some $\\theta \\in [0, \\frac{\\pi }{2}]$ .",
"Hence the CM and the mean of $\\sigma _{AB}^\\prime $ are respectively $&\\Gamma ^{\\prime } _{\\bf S}= \\left(\\begin{array}{cc}\\bf S & 0 \\\\0 & I\\end{array}\\right)\\left(\\begin{array}{cc}A_{0} & C_{0}K^{T} \\\\KC_{0}^{T} & KB_{0}K^{T}+M\\end{array}\\right)\\left(\\begin{array}{cc}\\bf S^{T} & 0 \\\\0 & \\textit {I}\\end{array}\\right)=\\left(\\begin{array}{cc}A_{0} & {\\bf S} C_{0}K^{T} \\\\KC_{0}^{T} \\bf S^{T} & KB_{0}K^{T}+M\\end{array}\\right)$ and $\\mathbf {d}^{\\prime }_{\\bf S}=({\\bf S}\\oplus I)(\\mathbf {d}_{A}\\oplus (K\\mathbf {d}_{B}+\\overline{\\mathbf {d}}))+\\mathbf {m} \\oplus 0=({\\bf S} \\mathbf {d}_{A}+\\mathbf {m} )\\oplus (K\\mathbf {d}_{B}+\\overline{\\mathbf {d}})=\\mathbf {d}_{A}\\oplus (K\\mathbf {d}_{B}+\\overline{\\mathbf {d}}).$ After some straight-forward calculations, one can immediately get $&N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )\\rho _{AB}) =N_{\\mathcal {F}}^{\\mathcal {G}}(\\sigma _{AB}) \\\\=& \\sup _{U\\in {\\mathcal {U}^{\\mathcal {G}}(H_A)},\\ U\\sigma _AU^\\dag =\\sigma _A} C^{2}(\\sigma _{AB},(U\\otimes \\textit {I})\\sigma _{AB}(U^{\\dag }\\otimes \\textit {I})) \\\\=& \\sup _{\\theta \\in [0,\\frac{\\pi }{2}]}\\lbrace 1-\\frac{\\sqrt{\\det \\Gamma ^{\\prime }\\det \\Gamma ^{\\prime }_{\\bf S_\\theta }}}{{\\det ((\\Gamma ^{\\prime } +\\Gamma ^{\\prime }_{\\bf S_\\theta })/2)}} \\rbrace .$ By the fact that $\\det \\Gamma ^{\\prime }=\\det \\Gamma ^{\\prime }_{\\bf S}=\\det A_{0} \\det (KB_{0}K^{T}+M-KC_{0}^{T}A_{0}^{-1}C_{0}K^{T})$ , the above formula can rewritten as the following $N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )\\rho _{AB})=& \\sup _{\\theta \\in [0,\\frac{\\pi }{2}]}\\lbrace 1-\\frac{\\det \\left(\\begin{array}{cc}A_{0} & C_{0}K^{T} \\\\KC_{0}^{T} & KB_{0}K^{T}+M\\end{array}\\right)}{\\det \\left(\\begin{array}{cc}A_{0} & \\frac{(I+{\\bf S_\\theta }) C_{0}K^{T}}{2} \\\\\\frac{KC_{0}^{T}(I+{\\bf S}_\\theta ^{T})}{2} &KB_{0}K^{T}+M\\end{array}\\right)}\\rbrace \\\\=&\\sup _{\\theta \\in [0,\\frac{\\pi }{2}]}\\lbrace 1-\\frac{\\det A_{0}\\det (KB_{0}K^{T}+M-KC_{0}^{T}A_{0}^{-1}C_{0}K^{T})}{\\det A_{0}\\det (KB_{0}K^{T}+M-\\frac{KC_{0}^{T}(I+{\\bf S}_\\theta ^{T})}{2}A_{0}^{-1}\\frac{(I+{\\bf S}_\\theta ) C_{0}K^{T}}{2})}\\rbrace \\\\=& \\sup _{\\theta \\in [0,\\frac{\\pi }{2}]}\\lbrace 1-\\frac{\\det (K(B_{0}-C_{0}^{T}A_{0}^{-1}C_{0})K^{T}+M)}{\\det (K(B_{0}-\\frac{C_{0}^{T}(I+{\\bf S}_\\theta ^{T})}{2}A_{0}^{-1}\\frac{(I+{\\bf S}_\\theta )C_{0}}{2})K^{T}+M)}\\rbrace .$ Clearly, the quantity $N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )\\rho _{AB})$ is independent of the parameter $\\overline{\\mathbf {d}}$ .",
"Notice that $K$ , $M$ can not be zero simultaneously, substituting ${\\bf S}_{\\theta } = \\left(\\begin{array}{cc}\\cos \\theta & \\sin \\theta \\\\-\\sin \\theta & \\cos \\theta \\end{array}\\right)$ into the above equation, after tedious calculations, one has $&&N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )\\rho _{AB}) \\\\&=&\\sup _{\\theta \\in [0,\\frac{\\pi }{2}]}\\lbrace 1-\\frac{\\det (K(B_{0}-C_{0}^{T}A_{0}^{-1}C_{0})K^{T}+M)}{\\det (K(B_{0}-\\frac{C_{0}^{T}(I+{\\bf S}_\\theta ^{T})}{2}A_{0}^{-1}\\frac{(I+{\\bf S}_\\theta ) C_{0}}{2})K^{T}+M)}\\rbrace \\\\&=&\\sup _{\\theta \\in [0,\\frac{\\pi }{2}]}\\lbrace 1-\\frac{(ab-c^{2})(ab-d^{2})n_{1}+a(ab-c^{2})n_{2}+a(ab-d^{2})n_{3}+a^{2}n_{4}}{(ab-c^{2}n_{0})(ab-d^{2}n_{0})n_{1}+a(ab-c^{2}n_{0})n_{2}+a(ab-d^{2}n_{0})n_{3}+a^{2}n_{4}}\\rbrace \\\\&=&1-\\frac{(ab-c^{2})(ab-d^{2})n_{1}+a(ab-c^{2})n_{2}+a(ab-d^{2})n_{3}+a^{2}n_{4}}{(ab-c^{2}/2)(ab-d^{2}/2)n_{1}+a(ab-c^{2}/2)n_{2}+a(ab-d^{2}/2)n_{3}+a^{2}n_{4}},$ where $ n_{0}=& (1+\\cos \\theta )/2,\\\\n_{1}=&k^{2}_{11}k^{2}_{22}+k^{2}_{12}k^{2}_{21}-2k_{11}k_{12}k_{21}k_{22},&n_{2}=& m_{22}k^{2}_{11}+m_{11}k^{2}_{21}-2m_{12}k_{11}k_{21},\\\\n_{3}=& m_{22}k^{2}_{12}+m_{11}k^{2}_{22}-2m_{12}k_{12}k_{22}, &n_{4}=& m_{11}m_{22}-m_{12}^{2}.",
"$ The proof is completed.",
"$\\Box $ Remark 2.",
"If $K=0$ , then $\\det M\\ge 1$ , and we have $N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi (0, M, \\overline{\\mathbf {d}}))\\rho _{AB}) =&\\lbrace 1-\\frac{ \\det M }{ \\det M}\\rbrace =0.$ In fact, in this case, the Gaussian channel $I\\otimes \\Phi (0, M,\\overline{\\mathbf {d}})$ maps any Gaussian state $\\rho _{AB}$ to a product state.",
"Thus, by Theorem 3, we always have $N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi (0, M, \\overline{\\mathbf {d}}))\\rho _{AB})=0$ .",
"Remark 3.",
"If $M=0$ , then $\\det K=1=\\det K^{T}$ , and $N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi (K, 0, \\overline{\\mathbf {d}}))\\rho _{AB})=& \\sup _{\\theta \\in [0,\\frac{\\pi }{2}]}\\lbrace 1-\\frac{\\det (K(B_{0}-C_{0}^{T}A_{0}^{-1}C_{0})K^{T})}{\\det (K(B_{0}-\\frac{C_{0}^{T}(I+{\\bf S}_\\theta ^{T})}{2}A_{0}^{-1}\\frac{(I+{\\bf S}_\\theta )C_{0}}{2})K^{T})}\\rbrace \\\\=& \\sup _{\\theta \\in [0,\\frac{\\pi }{2}]}\\lbrace 1-\\frac{ \\det (B_{0}-C_{0}^{T}A_{0}^{-1}C_{0})}{\\det (B_{0}-\\frac{C_{0}^{T}(I+{\\bf S}_\\theta ^{T})}{2}A_{0}^{-1}\\frac{(I+{\\bf S}_\\theta ) C_{0}}{2})}\\rbrace \\\\=& \\sup _{\\theta \\in [0,\\frac{\\pi }{2}]}\\lbrace 1-\\frac{\\det \\left(\\begin{array}{cc}A_{0} & C_{0} \\\\C_{0}^{T} & B_{0}\\end{array}\\right)}{\\det \\left(\\begin{array}{cc}A_{0} & \\frac{(I+{\\bf S}_\\theta ) C_{0}}{2} \\\\\\frac{C_{0}^{T}(I+{\\bf S}_\\theta ^{T})}{2} & B_{0}\\end{array}\\right)}\\rbrace = N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}).$ In this case, one can conclude that, after performing the Gaussian operation $I\\otimes \\Phi (K, 0, \\overline{\\mathbf {d}})$ , the quantity $N_{\\mathcal {F}}^{\\mathcal {G}}$ remains the same for those $(1+1)$ -mode Gaussian states whose CM are of the standard form.",
"The following result gives a kind of local Gaussian operation non-increasing property of $N_{\\mathcal {F}}^{\\mathcal {G}}$ , which was not discussed for other known similar correlations such as the Gaussian QD (two-mode) [8], [9], Gaussian geometric discord [10], the correlations $Q$ , $Q_{\\mathcal {P}}$ discussed in [13] and the correlation ${\\mathcal {N}}$ discussed in [16].",
"Theorem 7 Let $\\rho _{AB}$ be a $(1+1)$ -mode Gaussian state.",
"Then, for any Gaussian channel $\\Phi $ performed on the subsystem $B$ , we have $0\\le N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )\\rho _{AB})\\le N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}).$ Proof.",
"We first consider the case that the $(1+1)$ -mode Gaussian states $\\rho _{AB}$ whose CM $\\Gamma _0$ are of the standard form, that is, $\\Gamma _0=\\left(\\begin{array}{cccc}a & 0 & c & 0\\\\0 & a & 0 & d\\\\c & 0 & b & 0\\\\0 & d & 0 & b\\end{array}\\right).$ Let $\\Phi =\\Phi (K, M,\\overline{\\mathbf {d}})$ be any Gaussian channel performed on subsystem B with $K=\\left(\\begin{array}{cc}k_{11}&k_{12}\\\\k_{21} & k_{22}\\end{array}\\right)$ and $M=\\left(\\begin{array}{cc}m_{11} &m_{12}\\\\m_{12} & m_{22}\\end{array}\\right).$ We have to show that $N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )\\rho _{AB}) \\le N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ .",
"If $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})=0$ , then, by Theorem 3, $\\rho _{AB}$ is a product state.",
"So $(I\\otimes \\Phi )\\rho _{AB}$ is a product state, and hence $N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )\\rho _{AB})=0=N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ .",
"Assume that $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})\\ne 0$ .",
"Then $N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )\\rho _{AB}) \\le N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ holds if and only if $\\frac{N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )\\rho _{AB})}{N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})}\\le 1$ .",
"Let $\\alpha =(ab-c^{2})(ab-d^{2})$ , $\\beta =(ab-c^{2}/2)(ab-d^{2}/2)$ , $\\gamma =a(ab-c^{2})n_{2}+a(ab-d^{2})n_{3}+a^{2}n_{4}$ and $\\delta =a(ab-c^{2}/2)n_{2}+a(ab-d^{2}/2)n_{3}+a^{2}n_{4}$ with $n_2,n_3, n_4$ as in Theorem 6.",
"Then, according to Theorem 6, we have $\\frac{N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )\\rho _{AB})}{N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})}\\le 1 \\Leftrightarrow \\frac{1-\\frac{\\alpha n_{1}+\\gamma }{\\beta n_{1}+\\delta }}{1-\\frac{\\alpha }{\\beta }}\\le 1\\Leftrightarrow \\frac{\\alpha n_{1}+\\gamma }{\\beta n_{1}+\\delta }\\ge \\frac{\\alpha }{\\beta }\\Leftrightarrow \\gamma \\beta \\ge \\alpha \\delta .",
"$ Therefore, it suffices to prove that $\\gamma \\beta -\\alpha \\delta \\ge 0$ .",
"By some computations, one sees that $&&\\gamma \\beta =[a(ab-c^{2})n_{2}+a(ab-d^{2})n_{3}+a^{2}n_{4}](ab-\\frac{c^{2}}{2})(ab-\\frac{d^{2}}{2})\\\\&=&a(ab-c^{2})(ab-\\frac{c^{2}}{2})(ab-\\frac{d^{2}}{2})n_{2}+a(ab-d^{2})(ab-\\frac{c^{2}}{2})(ab-\\frac{d^{2}}{2})n_{3}+a^{2}(ab-\\frac{c^{2}}{2})(ab-\\frac{d^{2}}{2})n_{4}$ and $&&\\alpha \\delta =a(ab-c^{2})(ab-\\frac{c^{2}}{2})(ab-d^{2})n_{2}+a(ab-d^{2})(ab-c^{2})(ab-\\frac{d^{2}}{2})n_{3}+a^{2}(ab-c^{2})(ab-d^{2})n_{4}.$ Note that $n_{1}=k^{2}_{11}k^{2}_{22}+k^{2}_{12}k^{2}_{21}-2k_{11}k_{12}k_{21}k_{22}=(k_{11}k_{22}-k_{12}k_{21})^{2}\\ge 0$ and $n_{4}=m_{11}m_{22}-m_{12}^{2}=\\det M\\ge 0$ .",
"Since $m_{22}k^{2}_{11}+m_{11}k^{2}_{21}\\ge 2 \\sqrt{m}_{22} \\sqrt{m}_{11}k_{11}k_{21}\\ge 2 m_{12}k_{11}k_{21}$ , we have $n_{2}\\ge 0$ .",
"One can verify $n_{3}\\ge 0$ by the same way.",
"Also note that $a,b\\ge 1$ and $ab\\ge c ^2(d^2)$ by the constraint condition of the parameters in the definition of the Gaussian state.",
"Now it is clear that $&&\\gamma \\beta -\\alpha \\delta =a(ab-c^{2})(ab-\\frac{c^{2}}{2})\\frac{d^{2}}{2}n_{2}+a(ab-d^{2})(ab-\\frac{d^{2}}{2})\\frac{c^{2}}{2}n_{3}+a^{2}\\frac{c^{2}}{2}\\frac{d^{2}}{2}n_{4}\\ge 0,$ as desired.",
"To this end, we come to the conclusion that $N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )\\rho _{AB}) \\le N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ , and the equality holds if $M=0$ (See Remark 3 after the proof of Theorem 6).",
"Now let us consider the general case.",
"Let ${\\mathcal {U}}\\otimes {\\mathcal {V}}$ be a local Gaussian unitary operation, that is, for some Gaussian unitary operators $U$ and $V$ on the subsystem A and B respectively, so that $({\\mathcal {U}}\\otimes {\\mathcal {V}})(\\rho _{AB})=(U\\otimes V)\\rho _{AB}(U^\\dag \\otimes V^\\dag )$ for each state $\\rho _{AB}$ .",
"Then, $(I\\otimes \\Phi )\\circ ({\\mathcal {U}}\\otimes {\\mathcal {V}})={\\mathcal {U}}\\otimes (\\Phi \\circ {\\mathcal {V}})=({\\mathcal {U}}\\otimes I)\\circ (I\\otimes (\\Phi \\circ {\\mathcal {V}})).$ Note that, $\\Phi \\circ {\\mathcal {V}}$ is still a Gaussian channel which sends $\\rho _B$ to $\\Phi (V\\rho _BV^\\dag )$ .",
"Keep this in mind and let $\\rho _{AB}$ be any $(1+1)$ -mode Gaussian state.",
"Then there exists a local Gaussian unitary operation $U\\otimes V$ such that $\\sigma _{AB}=(U^\\dag \\otimes V^\\dag )\\rho _{AB}(U\\otimes V)$ has CM of the standard form.",
"By what we have proved above and Theorem 2, we see that $N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )\\rho _{AB})=& N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )((U\\otimes V)\\sigma _{AB}(U^\\dag \\otimes V^\\dag )))\\\\=& N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )\\circ ({\\mathcal {U}}\\otimes {\\mathcal {V}})\\sigma _{AB})=N_{\\mathcal {F}}^{\\mathcal {G}}(({\\mathcal {U}}\\otimes I)\\circ (I\\otimes (\\Phi \\circ {\\mathcal {V}}))\\sigma _{AB})\\\\=&N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes (\\Phi \\circ {\\mathcal {V}}))\\sigma _{AB})\\le N_{\\mathcal {F}}^{\\mathcal {G}}(\\sigma _{AB})=N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}),$ as desired, which completes the proof.",
"$\\Box $"
],
[
"Comparison between $N_{\\mathcal {F}}^{\\mathcal {G}}$ and other quantifications of the Gaussian quantum correlations",
"$N_{\\mathcal {F}}^{\\mathcal {G}}$ , $D_{G}$ and $Q$ describe the same quantum nonclassicality when they are restricted to Gaussian states because they take value 0 at a Gaussian state $\\rho _{AB}$ if and only if $\\rho _{AB}$ is a product state.",
"In this section, we calculate $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ for all two-mode symmetric squeezed thermal states $\\rho _{AB}$ and compare it with Gaussian geometric discord $D_{G}(\\rho _{AB})$ and $Q(\\rho _{AB})$ in scale.",
"Our result reveals that $N_{\\mathcal {F}}^{\\mathcal {G}}$ is bigger and thus is easier to detect the correlation in states.",
"Since the known computation formula of $D_{G}(\\rho _{AB})$ is only for symmetric squeezed thermal states $\\rho _{AB}$ , we compare them on such states.",
"Symmetric squeezed thermal states: Assume that $\\rho _{AB}$ is any two-mode Gaussian state; then its standard CM has the form as in Eq.(3).",
"Recall that the symmetric squeezed thermal states (SSTSs) are Gaussian states whose CMs are parameterized by $\\bar{n}$ and $\\mu $ such that $a=b=1+2\\bar{n}$ and $c=-d=2\\mu \\sqrt{\\bar{n}(1+\\bar{n})}$ , where $\\bar{n}$ is the mean photon number for each part and $\\mu $ is the mixing parameter with $0\\le \\mu \\le 1$ (ref.",
"[38]).",
"Thus every SSTS may be denoted by $\\rho _{AB}(\\bar{n},\\mu )$ .",
"Thus by Theorem 4, for any SSTS $\\rho _{AB}(\\bar{n},\\mu )$ , we have $ N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}(\\bar{n},\\mu ))=1-\\frac{((1+2\\bar{n})^{2}-4 \\mu ^{2} \\bar{n}(1+\\bar{n}))^{2}}{((1+2\\bar{n})^{2}-2 \\mu ^{2}\\bar{n}(1+\\bar{n}))^{2}}.$ For any two-mode Gaussian state $\\rho _{AB}$ , recall that the Gaussian geometric discord of $\\rho _{AB}$ ([10]) is defined as $D_{G}(\\rho _{AB})=\\inf _{\\Pi ^{A}}\\Vert \\rho _{AB}-\\Pi ^{A}(\\rho _{AB})\\Vert ^{2}_{2},$ where $\\Pi ^{A}={\\Pi ^{A}(\\alpha )}$ runs over all Gaussian positive operator valued measurements of subsystem A, $\\Pi ^{A}(\\rho _{AB})=\\int (\\Pi ^{A}(\\alpha )\\otimes I)^{\\frac{1}{2}}\\rho _{AB} (\\Pi ^{A}(\\alpha )\\otimes I)^{\\frac{1}{2}} {\\rm d}^{2}\\alpha $ .",
"According to the analytical formula of $D_{G}(\\rho _{AB})$ provided in [10], for any SSTS $\\rho _{AB}$ with parameters $\\bar{n}$ and $\\mu $ , one has $D_{G}(\\rho _{AB}(\\bar{n},\\mu ))=\\frac{1}{(1+2\\bar{n})^{2}-4\\mu ^{2}\\bar{n}(1+\\bar{n})}-\\frac{9}{[\\sqrt{4(1+2\\bar{n})^{2}-12\\mu ^{2}\\bar{n}(1+\\bar{n})}+(1+2\\bar{n})]^{2}}.$ By Eqs.",
"(9)-(10), it is clear that $\\lim _{\\bar{n}\\rightarrow \\infty }N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}(\\bar{n},\\mu ))=1-\\frac{(1-\\mu ^2)^2}{(1-\\frac{1}{2}\\mu ^2)^2}>0\\quad {\\rm for}\\ \\mu \\in (0,1),$ while $ \\lim _{\\bar{n}\\rightarrow \\infty }D_{G}(\\rho _{AB}(\\bar{n},\\mu ))=0 \\quad {\\rm for}\\ \\mu \\in (0,1).$ This shows that, for the case $\\mu \\ne 0, 1$ , $N_{\\mathcal {F}}^{\\mathcal {G}}$ is able to recognize well the quantum correlation in the states with large mean photon number but $D_{G}$ is not.",
"It is clear that $\\mu =0$ if and only if $\\rho _{AB} $ is a product SSTS, and in this case, $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}(\\bar{n},0))=D_{G}(\\rho _{AB}(\\bar{n},0))=0$ .",
"When $\\mu =1$ , we have $ N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}(\\bar{n},1))=1-\\frac{1}{(1+2\\bar{n} +2 \\bar{n}^{2})^{2}}$ and $D_{G}(\\rho _{AB}(\\bar{n},1))=1-\\frac{9}{[1+2\\bar{n}+2\\sqrt{1+\\bar{n}+ \\bar{n}^2}]^{2}},$ which reveals that we always have $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}(\\bar{n},1))>D_{G}(\\rho _{AB}(\\bar{n},1)).$ Moreover, we randomly chose 100000 pairs of $(\\bar{n},\\mu )$ with $\\bar{n}\\in (0,10000000000000)$ and $\\mu \\in (0,1)$ , numerical results show that $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}(\\bar{n},\\mu ) )> D_{G}(\\rho _{AB}(\\bar{n},\\mu ))$ .",
"On the other hand, the numerical method suggests that $N_{\\mathcal {F}}^{\\mathcal {G}}$ is better than $D_G$ in detecting the quantum correlation contained in any SSTS because we always have $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}(\\bar{n},\\mu ) )> D_{G}(\\rho _{AB}(\\bar{n},\\mu ))$ for all SSTSs $\\rho _{AB}(\\bar{n},\\mu )$ with $\\mu \\ne 0$ .",
"In Fig.1, we compare $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ with $D_{G}(\\rho _{AB})$ for SSTSs $\\rho _{AB}$ by considering $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})-D_{G}(\\rho _{AB})$ for $\\bar{n}\\le 50$ .",
"Fig.1 shows that $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})-D_{G}(\\rho _{AB})\\ge 0$ and $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})\\gg D_{G}(\\rho _{AB})$ for SSTSs $\\rho _{AB}$ with $\\mu $ near 1.",
"For example, considering the state $\\rho _{AB}$ with $\\bar{n}=49$ and $\\mu =0.9$ , we have $D_{G}(\\rho _{AB})\\approx 0.000356$ , which is very close to 0 and difficult to judge weather or not $\\rho _{AB}$ contains the correlation.",
"However, $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})\\approx 0.897995\\gg 0$ , which guarantees that $\\rho _{AB}$ does contain the quantum correlation.",
"For large mean photon number, for example, $\\bar{n}=10000$ , taking $\\mu =0.9$ , we have $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})\\approx 0.89803\\gg 0$ , but $D_{G}(\\rho _{AB})\\approx 8.72518 \\times 10^{-11}$ .",
"Furthermore, Fig.2 shows that $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})-D_{G}(\\rho _{AB})\\ge 0$ holds as well for $\\bar{n}\\in (100000,100500)$ and $\\mu \\in (0,1)$ .",
"$Q$ is a quantum correlation for $(m+n)$ -mode continuous-variable systems defined in terms of average distance between the reduced states under the local Gaussian positive operator valued measurements [13]: $Q(\\rho _{AB}):=\\sup \\limits _{\\Pi ^{A}}\\int p(\\alpha ){\\Vert \\rho _{B}-\\rho ^{(\\alpha )}_{B}\\Vert ^{2}_{2}}{\\rm d}^{2m}\\alpha ,$ where $\\Pi ^{A}={\\Pi ^{A}(\\alpha )}$ runs over all Gaussian positive operator valued measurements of subsystem A, $\\Pi ^{A}=\\lbrace \\Pi ^{A}(\\alpha )\\rbrace $ on the subsystem $H_A$ , $\\rho _{B}={\\rm Tr}_{A}(\\rho _{AB})$ , $p(\\alpha )={\\rm Tr}[(\\Pi ^{A}(\\alpha )\\otimes I_{B})\\rho _{AB}]$ and $\\rho ^{(\\alpha )}_{B}=\\frac{1}{p(\\alpha )}{\\rm Tr}_{A}[(\\Pi ^{A}(\\alpha )\\otimes I_{B})^{\\frac{1}{2}}\\rho _{AB}(\\Pi ^{A}(\\alpha )\\otimes I_{B})^{\\frac{1}{2}}]$ .",
"For any SSTS $\\rho _{AB}$ with parameters $\\bar{n}$ and $\\mu $ , by [13], $ Q(\\rho _{AB}(\\bar{n},\\mu ))=\\frac{1}{1+2\\bar{n}(1-\\mu ^{2})}-\\frac{1}{1+2\\bar{n}}.$ Obviously, $\\lim _{\\bar{n}\\rightarrow \\infty } Q(\\rho _{AB}(\\bar{n},\\mu ) )=0,\\quad {\\rm for}\\ \\mu \\in (0,1).$ which reveals that $Q$ is not valid for those states with $\\mu \\in (0,1)$ and large mean photon number.",
"For the case $\\mu =1$ , we have $Q(\\rho _{AB}(\\bar{n},1))=1-\\frac{1}{1+2\\bar{n}}<N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}(\\bar{n},1))$ for any $\\bar{n}$ .",
"Also, we always have $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})>Q(\\rho _{AB})$ for all SSTSs with $\\mu \\ne 0$ .",
"For random pairs $(\\bar{n},\\mu )$ with $\\bar{n}\\in (0,10000000000000)$ and $\\mu \\in (0,1)$ , 100000 numerical results illustrate that $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB}(\\bar{n},\\mu ))> Q(\\rho _{AB}(\\bar{n},\\mu ))$ .",
"Figure: z=N ℱ 𝒢 (ρ AB )-D G (ρ AB )N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})-D_{G}(\\rho _{AB})with SSTSs, and 0≤μ≤10\\le \\mu \\le 1, 0≤n ¯≤500\\le \\bar{n} \\le 50.Figure: z=N ℱ 𝒢 (ρ AB )-D G (ρ AB )N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})-D_{G}(\\rho _{AB})with SSTSs, and 0≤μ≤10\\le \\mu \\le 1, 100000≤n ¯≤100500100000\\le \\bar{n} \\le 100500.Figure: z=N ℱ 𝒢 (ρ AB )-Q(ρ AB )N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})-Q(\\rho _{AB})with SSTSs, and 0≤μ≤10\\le \\mu \\le 1, 0≤n ¯≤500\\le \\bar{n} \\le 50.Figure: z=N ℱ 𝒢 (ρ AB )-D G (ρ AB )N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})-D_{G}(\\rho _{AB})with SSTSs, and 0≤μ≤10\\le \\mu \\le 1, 100000≤n ¯≤100500100000\\le \\bar{n} \\le 100500.The deference of $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ and $Q(\\rho _{AB})$ for SSTSs is showed in Fig.3 for $\\bar{n}\\le 50$ .",
"It reveals that $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})\\gg Q(\\rho _{AB})$ for those SSTSs $\\rho _{AB}$ with large mean photon number $\\bar{n}$ and larger mixing parameter $\\mu $ .",
"Consider the states $\\rho _{AB}$ with respectively $(\\bar{n}, \\mu )=(49, 0.9)$ and $(\\bar{n}, \\mu )=(10000, 0.9)$ , the same examples as above.",
"We have respectively $Q(\\rho _{AB})\\approx 0.040867<N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})\\approx 0.897955$ and $Q(\\rho _{AB})\\approx 0.000021\\ll N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})\\approx 0.89803$ , which means that applying $N_{\\mathcal {F}}^{\\mathcal {G}}$ is much more easier than $Q$ to guarantee that $\\rho _{AB}$ contains the quantum correlation.",
"Fig.4 demonstrates that $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})-Q(\\rho _{AB})\\ge 0$ also holds for these $\\bar{n}\\in (100000,100500)$ and $\\mu \\in (0,1)$ ."
],
[
"Conclusion",
"In this paper, based on fidelity ${\\mathcal {F}}(\\rho ,\\sigma )=\\frac{|{\\rm tr}\\rho \\sigma |}{\\sqrt{{\\rm tr}\\rho ^2{\\rm tr}\\sigma ^2}}$ and the distance $C^2(\\rho ,\\sigma )=1-{\\mathcal {F}}^2(\\rho ,\\sigma )$ , we have proposed a new kind of quantum nonclassicality $N_{\\mathcal {F}}^{\\mathcal {G}}$ by local Guassian unitary operations for any states in $(n+m)$ -mode continuous-variable systems.",
"Though, when restricted to the Gaussian states, $N_{\\mathcal {F}}^{\\mathcal {G}}$ describes the same nonclassical correlation as several known correlations such as Gaussian QD, Gaussian geometric discord $D_G$ and the nonlocality $Q$ , it is comparatively much easier to be computed and estimated.",
"Furthermore, $N_{\\mathcal {F}}^{\\mathcal {G}}$ has several nice properties that other known quantifications of such correlation do not possess: $N_{\\mathcal {F}}^{\\mathcal {G}}$ is a quantum correlation without ancilla problem; $N_{\\mathcal {F}}^{\\mathcal {G}}((I\\otimes \\Phi )\\rho _{AB}) \\le N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ holds for any $(1+1)$ -mode Gaussian state $\\rho _{AB}$ and any Gaussian channel $\\Phi $ , that is, undergoing a local Gaussian channel performed on the unmeasured part, the quantity we proposed will not increase.",
"We guess that this nice property is still valid for $(n+m)$ -mode systems.",
"We give a computation formula of $N_{\\mathcal {F}}^{\\mathcal {G}}$ for any $(1+1)$ -mode Gaussian states and an upper bound for any $(n+m)$ -mode Gaussian states, which are simple and easily calculated.",
"Furthermore, by comparing $N_{\\mathcal {F}}^{\\mathcal {G}}(\\rho _{AB})$ with $D_G(\\rho _{AB})$ and $Q(\\rho _{AB})$ for two-mode symmetric squeezed thermal states, we find that $N_{\\mathcal {F}}^{\\mathcal {G}}$ is greater than $D_G$ and $Q$ , and so is better in detecting quantum correlation in Gaussian states.",
"Acknowledgement.",
"Authors thank Professor Kan He for discussion.",
"This work is supported by the National Natural Science Foundation of China (11671294,11671006) and Outstanding Youth Foundation of Shanxi Province (201701D211001)."
]
] | 1808.08546 |
[
[
"A Tutorial on Modular Ontology Modeling with Ontology Design Patterns:\n The Cooking Recipes Ontology"
],
[
"Abstract We provide a detailed example for modular ontology modeling based on ontology design patterns."
],
[
"Use Case",
"Step 1: Define use case or scope of use cases.",
"Every ontology is designed for a purpose; this purpose may be defined by a use case, or by a set of use cases, or possibly by a set of potential use cases, which may include the future extensions or refinements of the ontology, and future reuse of the ontology by others.",
"How specific should a use case be?",
"Conventional wisdom may suggest that it is always better to be more specific.",
"However, in the context of ontology modeling the case is not as clear-cut.",
"A very specific use case may give rise to an ontology which is very specialized, i.e.",
"modeling choices (so-called ontological commitments) may be made which fit only the very specific and detailed use case.",
"As a consequence, later modifications, e.g.",
"by widening the scope of the application (and therefore of the underlying ontology) become very cumbersome as they may conflict with ontological commitments made earlier.",
"Let us look at a very simple example of this.",
"Say, our application involves movies and actors from the casts of these movies.",
"It may first seem as if actor names could simply be attached to the movies using an OWL datatype property hasActor.",
"E.g., this could be written in RDF Turtle as :myMovie :hasActor \"JaneSmith\" .",
"This will be sufficient, e.g., if only the name of an actor is relevant for an application.",
"However, it is conceivable that the application (and thus the ontology) may later on be extended in order to be able to list all movies in which a given actor was a cast member.",
"Since it is likely that there may be different persons with the same name, such as Jane Smith, we would need to be able to identify which name strings identify the same, and which identify different actors, i.e., we have to disambiguate the name strings.",
"Furthermore, Jane Smith may also be listed as actor under a different name, say Jane W. Smith.",
"Using an OWL datatype property as above, however, was a modeling choice which prevents this.",
"What we would need is URIs for actors.",
"With this case our example would look like the following.",
":myMovie :hasActor :janeSmith1 .",
":janeSmith1 :hasName \"JaneSmith\" .",
"Further extensions of the application (or attempted reuses of the ontology), however, may pose yet additional problems.",
"E.g., it may be desired to also list the character played by an actor in a specific movie.",
"Given our current modeling choices, however, it seems unclear where to attach this information: If it is attached to the movie, then we would no longer be able to say which character in the movie was played by which actor.",
"If we attach the information to the person, then we would no longer know which movie the character appeared in.",
"If we attach it to both movie and person, then we would run into difficulties if characters appear in different movies, played by different actors.",
"The solution in this case is to create another node in the graph, which stands for the actor role.",
"Our example would then look as thus.",
":myMovie :hasActor :myMovieMissXRole .",
":myMovieMissXRole :assumedBy :janeSmith1 ; :asCharacter :MissX .",
":janeSmith1 :hasName \"JaneSmith\" .",
"We understand that we can make modeling choices which make future reuse easier, e.g., by making sure that we include enough nodes in the graph.",
"This, of course, begs the question where to stop?",
"If we follow this principle, then won't we end up with much too many nodes, blowing up the graphs?",
"This is a valid concern, of course, and there are not straightforward solutions for this issue which work in all circumstances.",
"Generally speaking, we should strive for a balance, i.e., finding a soft spot somewhere between the extremes.",
"Our approach using ontology design patterns addresses the issue as we will be able to reuse patterns which have been created and vetted by the community, and which provide a good trade-off between the extremes in many circumstances.",
"Returning to the movie example, there are two patterns which would be the standard choices in this situation, the AgentRole and the NameStub pattern.",
"Class diagrams for these two are shown in Figures REF and REF .",
"We can now take the AgentRole pattern and remove the TimeInstant class, and join it with the NameStub pattern by using Agent instead of owl:Thing, see Figure REF .",
"Finally, we use the resulting class diagram as a template and make renamings of class and property names to fit it to our more specific use case; the result is displayed in Figure REF .",
"This process exemplifies our intended use of ontology design patterns: We use them as templates and specialize and join them in order to obtain a draft of our desired model.",
"Figure: Generic AgentRole patternFigure: Generic NameStub patternFigure: Joining the AgentRole and NameStub patternsFigure: The movie snippet: using Figure as a templateAfter this discussion, let us now return to the actual task at hand, namely to define a use case scenario for our worked example.",
"The setting we have in mind concerns online cooking recepies, and in particular the task of integrating recipes from different websites in order to enable a fine-grained cross-website search for recipes: Design an ontology which can be used as part of a “recipe discovery” website.",
"The ontology shall be set up such that content from existing recipe websites can in principle be mapped to it (i.e., the ontology gets populated with data from the recipe websites).",
"On the discovery website, detailed graph-queries (using the ontology) shall produce links to recipes from different recipe websites as results.",
"The ontology should be extendable towards incorporation of additional external data, e.g., nutritional information about ingredients or detailed information about cooking equipment.",
"Let us make a few remarks about the scenario we have just defined.",
"First of all, we notice that data will come from multiple sources which are not exactly specified.",
"This means that our ontology needs to be general enough to accomodate different conceptual representations on the source side.",
"Second, the ontology shall be extendable towards additional related data, meaning that we have to accomodate such extension capabilities, to a reasonable extent, without knowing what these future extensions would exactly look like, and this again asks for a rather general model.",
"Third, fine-grained search for recipes shall be possible, meaning that our ontology needs to be specific enough to allow these.",
"The scenario thus calls for a reasonable trade-off between specifity and generality, i.e., it is a typical use-case for ontology design pattern based modular ontology modeling."
],
[
"Competency Questions and Data Sources",
"Step 2: Make competency questions while looking at possible data sources and scoping the problem, i.e., decide on what should be modeled now, and what should be left for a possible later extension.",
"Competency questions are queries, formulated in natural language, which could potentially used for retrieval of data from the knowledge base.",
"They help to further specify the use cases, i.e., main classes of potential queries should be represented.",
"For our scenario, possible competency questions are the following.",
"Gluten-free low-calorie desserts.",
"How do I make a low-carb pot roast?",
"How do I make a Chili without beans?",
"Sweet breakfast under 100 calories.",
"Breakfast dishes which can be prepared quickly with 2 potatoes, an egg, and some flour.",
"How do I prepare Chicken thighs in a slow cooker?",
"A simple recipe with pork shoulder and spring onions.",
"A side prepared using Brussels sprouts, bacon, and chestnuts.",
"The competency questions already indicate some parameters that will be important, e.g.",
": Retrieval of cooking instructions.",
"Search by ingredients.",
"Search by properties of the prepared food, e.g.",
"calorie or carb content.",
"Search by properties such as cooking time, simplicity.",
"At this stage, at the latest, it is also necessary to look at possible data sources.",
"A quick Web search provides a significant number of recepie websites, e.g., allrecipes.com, food.com, epicurious.com.",
"Pages commonly list ingredients, cooking instructions, and sometimes other information such as nutritional information.",
"Additional nutritional information is, e.g., available from Google Knowledge Graph nutrition data.https://search.googleblog.com/2013/05/time-to-back-away-from-cookie-jar.html Data is usually not available in structured form, i.e., for an application it will be necessary to extract content from text-based web pages, which can be a tricky task in itself; however herein we only concern ourselves with producing a suitable underlying ontology.",
"Looking at the data sources now prompts us to go back to the competency questions, to reevaluate them.",
"Some competency questions may have to be dropped or modified at first, e.g., recipe websites seem to rarely mention equipment, like slow cookers, separately, identify a breakfast as sweet, or use classification such as low-carb or gluten-free.",
"We should keep these in mind, though, and make sure that the ontology we produce is extendable towards future inclusion of such aspects.",
"At the same time, inspection of the data may yield further insights regarding data that could now or in the future be included, such as recipe authors, peer recommendations, cooking time, level of difficulty, or category tags such as dessert or side.",
"These can either be incorporated right away, or alternatively extensibility towards future inclusion can be kept in mind during modeling.",
"The decision process regarding what to model now versus later is called scoping.",
"At the end of this step, we should have arrived at a clear idea concerning the scope of the target ontology."
],
[
"Key Notions to Modules",
"Step 3: Identify key notions from the data and the use case and identify which pattern should be used for each.",
"Many can remain “stubs” if detailed modeling is not yet necessary.",
"Instantiate these key notions from the pattern templates, and adapt/change the result as needed.",
"Add axioms for each module, informed by the pattern axioms.",
"As a result of this step, we arrive at a set of modules for the final ontology.",
"Think of the key notions as the main classes of things appearing in the competency questions or which you identify from the data sources.",
"Obvious possible key notions which come to mind are recipe, food, time, equipment, classification of food prepared (e.g., as side), difficulty level, nutritional information.",
"Let's go through these one by one and refine the list while creating corresponding schema diagrams.",
"After that, we will talk about axiomatizing them."
],
[
"Recipe",
"Recipe is an obvious candidate for a class, it is central to what we intend to do, and in addition we may want to notice already that the name of the recipe, which is often identical with the food which is going to be prepared, should be recorded.",
"For the latter, we should probably use the NameStub.",
"We now also want to identify a pattern which will be the basis for the core of the recipe modeling.",
"Of course, we have not yet discussed other patterns than NameStub and AgentRole.",
"One way to approach this is to go through a list of known patternshtt://www.ontologydesignpatterns.org sports many patterns, however they and their documentations are of very differing quality.",
"and to contemplate which may fit best, and sometimes there seem to be more than one candidate.",
"Let us look at three more or less obvious candidates.",
"Let us first check whether it makes sense to think of recipes as documents.",
"There is certainly a perspective from this this seems valid: in the end, isn't it simply a document which we retrieve from the Web when we download a recipe?",
"However, document seems to be a rather generic notion which does not naturally cater for key aspects of a recipe such as having ingredients, or taking a particular time.",
"We also wouldn't say about a document whether it's low-carb or not.",
"This line of thinking may lead us to the conclusion that a document may contain a recipe description, but that a recipe as such is a different type of entity.",
"Since recipes usually contain step-by-step descriptions of the food preparation process, another alternative may be to think of a recipe as a sequence, which is another fundamental ontology design pattern.",
"But then it also seems clear that many of the aspects important for our competency questions are not naturally catered for by the notion of sequence, e.g., what are ingredients in relation to recipe as a sequence?",
"This line of thinking may lead us to the conclusion that some parts of the recipe – the cooking steps – may be representable as a sequence, but the whole of the recipe is much more than that.",
"On the other hand, our competency questions do not indicate that the preparation steps sequence as such is particularly relevant to our task, namely the discovery of recipes.",
"We could also think of recipes as processes which may help us to emphasize input and output aspects.",
"This may indeed be a valuable perspective.",
"However, the notion of process may usually allude to much more rigid and well-defined sequences of actions, so we would have to have a very detailed look at a process ontology design pattern to decide whether it provides the right perspective for our purpose.",
"The perspective we will actually take here is that a recipe is a type of description.",
"Indeed, the general description pattern has a specialization to plans, and indeed it seems a reasonable perspective to think of recipes as plans (to produce something).",
"Let us look at a part of the Plan pattern which is depicted in Figure REF .",
"A plan leads from an initial situation to a situation which is understood as the goal of the plan.",
"The initial situation would be one in which required ingredients (and equipment etc) would be available, while in the goal situation the prepared food would be available.",
"In fact, the required ingredients, equipment etc.",
"are necessary for these respective situations, i.e.",
"they are constituents for them.",
"Putting these thoughts together, we can arrive at a first piece of the Recipe module, as an instantiation of the Plan template.",
"Its schema diagram is depicted in Figure REF .",
"There is much more to be said about descriptions, plans, situations, etc., but we will not go into detail here.",
"See for a central reference.",
"Figure: Basic Plan ontology design pattern: schema diagram.",
"The dashed boxes indicate complex notions which would easily merit a pattern description in their own right.Figure: Recipe as planProceeding with our list of potential key notions, the next one is food.",
"This seems a little unspecific.",
"What exactly is meant by this?",
"Well, food can be things like cucumbers, potatoes, eggs, lasagna, and Chicken Kiev.",
"Perhaps we should distinguish between ingredients and results, i.e., full dishes?",
"But wait a second, what about, say, Pesto Genovese?",
"It's not a dish by itself, but an ingredient in some recipes; yet there are also recipes how to make Pesto Genovese.",
"Indeed, many cooking ingredients are already processed from even more basic ingredients.",
"So it probably will not make much sense to try to distinguish between ingredients and dishes when talking about recipes.The type of discussion exemplified in this paragraph is central to coming up with good key notions and modules.",
"It is extremely helpful to have this discussion in a group, as others are often so much better in finding flaws in our ideas than we are ourselves.",
"But, if we say Pesto Genovese, what exactly do we mean?",
"Do we mean Pesto Genovese in general, as such, or do we mean, say, two teaspoons of it, as required by some recipe?",
"Indeed it seems that for recipes the quantity of a required or produced food item is also important.",
"We're starting to narrow this down.",
"Quantity seems like an ontology design pattern which we should make use of for our purposes, and “a quantity of food” seems to be a central concept for modeling recipes as plans, to be used both on the input and on the output side of the recipe as plan.",
"So we understandt hat there should be a concept of QuantityOfFood (like, 2 tsp of Pesto) which is always of some quantity (like, 2 tsp) and at the same time is of some type of foodstuff (say, Pesto).",
"The foodstuff can thus be understood as a FoodType (like, Pesto, or potato), namely the type of stuff the quantity of food consists of.",
"See Figures REF and REF for schema diagrams.",
"Our pattern for Quantity is very much directly derived from QUDT.http://qudt.org/ Figure: The QuantityOfStuff pattern, the inner box indicates the Quantity pattern.",
"There is much more to be said about quantities, but we will not further dwell on this hereFigure: QuantityOfFood module, as an instance of the Quantity pattern from Figure .Next on our list of key notions is equipment, such as slow cooker, blender, etc.",
"However, while keeping track of the occasional special equipment may be helpful, our scenario does not call for a detailed modeling of kitchen equipments at this stage.",
"So let us decide to delay such detailed modeling for the moment, i.e., we consciously restrict the scope of our model.",
"Decisions such as this are very important during the modeling process, as they limit the scope of the ontology.",
"Indeed, it is impossible to always model all details, as we would end up with a model of almost everything in this case.",
"At the same time, however, we would like to keep in mind that our ontology may be reused later and possibly repurposed for a scenario in which detailed modeling of equipment may be more important.",
"This means, that we do not want to simply introduce a datatype property such as requiresEquipment with strings – the names of the equipment – as range.",
"We rather want to utilize a slightly more sophisticated approach where we at least have a node as placeholder for the equipment entity.",
"The corresponding ontology design pattern is called a Stub , and it is depicted in Figure REF together with the instantiation for equipment which we will use.",
"It is really essentially the same as the NameStub pattern introduced earlier, the only difference being that the identifying string is not necessarily a name of the thing identified.",
"Figure: Top, the Stub (meta)pattern.",
"Bottom, its instantiation for equipment.Note also that we attach the cooking equipment as constituent to a situation, which seems to be its natural place.",
"We opt for stubs also for other key notions we have identified, namely for DifficultyLevel and for RecipeClassification (such as low-carb, diabetic, etc.",
"), i.e., for now the ontology will be able to hold only strings for these, but the model remains extendable if so desired in the future.",
"The corresponding schema diagrams can be found in Figure REF .",
"Figure: Stubs for DifficultyLevel and RecipeClassification.We will use stubs also in other places, e.g., we have not further talked about FoodType as it appears in Figure REF .",
"As before, it is conceivable that there may be a sophisticated model of different food types, but we will use a stub at this stage.",
"Next we turn our attention to nutritional information, the next keyword on our list, and we opt to model this in somewhat more detail.",
"More precisely, we will model the contents of Nutritional Facts labels as mandated in the U.S.A. for most food products,see https://en.wikipedia.org/wiki/Nutrition_facts_label#United_States see Figure REF While this may seem overly specific, by virtue of our modular modeling approach it would be easy to replace the NutritionalInformation module with one tailored to, e.g., other countries, or other nutritional convictions.",
"In fact, we will highlight this by creating a class US-Nutrition-Label as a subclass of the generic NutritionalInformation class.",
"These Nutritional Facts labels have highly structured content.",
"We will of course not be concerned with layout issues, and it is also not necessary that we model all content.",
"E.g., we will not list “% Daily Value” amounts for fat or sodium.",
"We will list absolute amounts for Fat, Saturated Fat, Trans Fat, Cholesterol, Sodium, Carbs, Dietary Fiber, Sugars, and Protein, and “% Daily Value” amounts for Vitamins A and C, Calcium, and Iron, which we represent as instances of a class NutritionalContentType.Essentially, we are creating a small controlled vocabulary for substances of nutritional importance.",
"It seems obvious that we will reuse the QuantityOfStuff pattern again, however as a percentage value is not really a quantity, we add an alternative to giving the quantity, which consists simply of a datatype property isPercentageOfDailyValue with range xsd:positiveInteger.",
"See the right of Figure REF .",
"Figure: Nutritional Information module.",
"The box indicates a modified instance of the QuantityOfStuff pattern.Of course we also need to record the serving size to which the nutritional information refers, and this can again be done using the Quantity pattern.",
"We also list calorie content and calories-from-fat content as indicated in the figure.",
"We have worked through our list of keywords, but before we move on, let us briefly reflect whether there is anything else that needs modeling, which we can derive from our scenario description.",
"And indeed, our scenario states that queries shall produce links to recipes from different recipe websites – however, or modeling so far did not include anything which would make it possible to track where a recipe came from.",
"We thus need to do some provenance modeling.",
"The schema diagram of a generic provenance pattern, as derived from PROV-O and mentioned in , is provided in Figure REF .",
"The key idea of this is that everything (any owl:Thing) for which provenance is important, was generated by some activity (in our case, web retrieval) which used some other thing (in our case, a recipe website).",
"The item under consideration (the recipe) may also be directly related to its origin (the recipe website) using the property wasDerivedFrom.",
"In addition, agents may be involved in activities.",
"Figure: Provenance patternFor our purpose, it will suffice to reuse a small part of this pattern, namely the wasDerivedFrom property.",
"Derivation in this case is from a document which has a URL (i.e., a website), and we can use a Document stub for this.",
"The resulting module is depicted in Figure REF Figure: Recipe provenance module"
],
[
"Axiomatizations",
"We have now produced diagrams for all key notions we had identified, and have used schema diagrams of general ontology design patterns to produce them.",
"The list of key notions, together with the used patterns can be found in Figure REF .",
"We now turn to producing OWL axioms for all modules.",
"We use the earlier schema diagrams as guidance.",
"Usually, axioms would be derived from the axioms provided with the patterns, but we will recreate them from scratch, in order to gain a deeper understanding of them.",
"We will in fact produce a rather exhaustive list of axioms which seem appropriate for our model, while steering away from overly strong ontological commitments.",
"Figure: Generic node-edge-node schema diagram for explaining systematic axiomatizationThere is a systematic way to look at each node-edge-node triple in the schema diagram in order to decide which axioms should be added: Given a node-edge-node triple with nodes $A$ and $B$ and edge $R$ from $A$ to $B$ , as depicted in Figure REF , we check all of the following axioms whether they should be included.The OWLAx Protégé plug-in provides a convenient interface for adding these axioms.",
"We list them in natural language, see Figure REF for the formal versions in description logic notation, and Figure REF for the same in Manchester syntax, where we also list our names for these axioms.",
"$A$ and $B$ are disjoint.",
"The domain of $R$ is $A$ .",
"For every $B$ which has an inverse $R$ -filler, this inverse $R$ -filler is in $A$ .",
"In other words, the domain of $R$ scoped with $B$ is $A$ .",
"The range of $R$ is $B$ .",
"For every $A$ which has an $R$ -filler, this $R$ -filler is in $B$ .",
"In other words, the range of $R$ scoped with $A$ is $B$ .",
"For every $A$ there has to be an $R$ -filler in $B$ .",
"For every $B$ there has to be an inverse $R$ -filler in $A$ .",
"$R$ is functional.",
"$R$ has at most one filler in $B$ .",
"For every $A$ there is at most one $R$ -filler.",
"For every $A$ there is at most one $R$ -filler in $B$ .",
"$R$ is inverse functional.",
"$R$ has at most one inverse filler in $A$ .",
"For every $B$ there is at most one inverse $R$ -filler.",
"For every $B$ there is at most one inverse $R$ -filler in $A$ .",
"Domain and range axoims are items 2–5 in this list.",
"Items 6 and 7 are extistential axioms.",
"Items 8–15 are about variants of functionality and inverse functionality.",
"All axiom types except disjointness and those utilizing inverses also apply to datatype properties.",
"Figure: Most common axioms which could be produced from a single edge RR between nodes AA and BB in a schema diagram: description logic notation.Figure: Creating OWL Files"
]
] | 1808.08433 |
[
[
"3-folds CR-embedded in 5-dimensional real hyperquadrics"
],
[
"Abstract E. Cartan's method of moving frames is applied to 3-dimensional manifolds $M$ which are CR-embedded in 5-dimensional real hyperquadrics $Q$ in order to classify $M$ up to CR symmetries of $Q$ given by the action of one of the Lie groups $SU(3,1)$ or $SU(2,2)$.",
"In the latter case, the CR structure of $M$ derives from a shear-free null geodesic congruence on Minkowski spacetime, and the relationship to relativity is discussed.",
"In both cases, we compute which homogeneous CR 3-folds appear in $Q$."
],
[
"Introduction",
"For several physically significant solutions to Einstein's field equations in general relativity, spacetime is a 4-dimensional Lorentzian manifold that is foliated by a family of curves called a shear-free null geodesic congruence (SFNGC), which induces a CR structure on the 3-dimensional leaf space of the foliation.",
"Conversely, a 3-dimensional CR structure can be “lifted\" to a spacetime admitting a SFNGC.",
"The Robinson-Trautman metrics, for example, describe congruences which are hypersurface-orthogonal, and their corresponding CR structures are Levi-flat.",
"These include models of electromagnetic and gravitational fields radiating along the foliating curves, generating wave-fronts orthogonal to their direction of propagation.",
"Levi-nondegenerate CR structures, on the other hand, are associated with “twisting\" congruences, such as those appearing in Kerr's model of a rotating black hole.",
"The geometry of SFNGCs is explained in §REF , and a glimpse of their history in relativity is given in §REF .",
"CR maps between manifolds establish a notion of intrinsic CR equivalence, in terms of which Levi-flat CR structures are all locally the same in any fixed dimension.",
"By contrast, Levi-nondegenerate CR manifolds $M$ are locally classified by Cartan's method of equivalence, which constructs a principal Cartan bundle $\\mathcal {B}\\rightarrow M$ of (co)frames adapted to the CR structure of $M$ as well as a canonical Cartan connection $\\gamma $ on $\\mathcal {B}$ .",
"When the curvature tensor $\\text{d}\\gamma +\\tfrac{1}{2}[\\gamma ,\\gamma ]$ vanishes, $M$ is locally equivalent to a real hyperquadric $\\mathcal {Q}$ , and $\\mathcal {B}$ is a special unitary Lie group of CR symmetries of $\\mathcal {Q}$ , with $\\gamma $ playing the role of the Maurer-Cartan form on $\\mathcal {B}$ .",
"See §REF for basic definitions and references.",
"In dimension three, the real hyperquadric is the CR 3-sphere whose Lie group of CR symmetries is $SU(2,1)$ .",
"If $M$ is homogeneous under the action of its Lie group $\\mathcal {G}$ of intrinsic CR symmetries, then either $\\mathcal {G}=SU(2,1)$ and $M$ is necessarily flat, or else $\\mathcal {G}$ is 3-dimensional and the curvature of $M$ may be zero, nonzero, or undefined (if $M$ is Levi-flat).",
"In any case, $M$ is recognizable by the Lie algebra $\\mathfrak {g}$ of $\\mathcal {G}$ since 3-dimensional Lie algebras over $\\mathbb {R}$ were classified by Bianchi.",
"Homogeneous models of Levi-nondegenerate 3-folds are catalogued at the end of §REF .",
"The properties of a SFNGC are maintained under conformal rescaling of the spacetime metric; indeed, a given SFNGC belongs to a substantially larger family of metrics than a conformal class, and the ambiguity in the choice of a representative is related to that of a choice of adapted coframing on the underlying CR manifold.",
"It is therefore natural to ask which CR structures give rise to SFNGCs whose family of metrics contains one with special characteristics; e.g., a metric which is conformally flat.",
"The answer to the latter question – provided by a theorem attributed to Kerr – is those 3-folds that may be embedded by a CR map into the 5-dimensional real hyperquadric $\\mathcal {Q}\\subset \\mathbb {CP}^3$ whose CR symmetry group $SU(2,2)$ is infinitesimally isomorphic to the conformal symmetry group $SO(2,4)$ of compactified Minkowski spacetime.",
"The Kerr Theorem is presented in §REF in the framework of Penrose's twistor theory, emphasizing the role of these symmetry groups.",
"Unsurprisingly, homogeneous 3-folds are noteworthy in this context, both mathematically and physically.",
"Their symmetry simplifies their structure equations, making them ideal subjects to test any criteria for embeddability in $\\mathcal {Q}$ .",
"Those that are embedded with symmetries in $SU(2,2)$ correspond to SFNGC with conformal symmetries, while those that admit no embedding at all correspond to SFNGC in spacetimes with curvature.",
"The proof of the Kerr Theorem is sketched in §REF .",
"There is little added expense in generalizing beyond the context of the Kerr Theorem.",
"To wit, there are two 5-dimensional real hyperquadrics, differing in the signature of their (rank-2) Levi forms, which is either definite or split.",
"They are both called $\\mathcal {Q}$ ; when necessary we recognize them individually by their CR symmetry groups: $SU(3,1)$ for the CR 5-sphere and $SU(2,2)$ for the projectivized null twistors, respectively, or $SU_\\star $ to refer to both.",
"They are both fixed by a parabolic subgroup $\\mathcal {P}\\subset SU_\\star $ stabilizing a null line in $\\mathbb {C}^4$ .",
"We consider CR embeddings of 3-folds $M$ into both homogeneous spaces $\\mathcal {Q}=SU_\\star /\\mathcal {P}$ .",
"Of especial interest are embeddings that preserve intrinsic CR symmetries of $M$ : Definition 1.1 Let $\\mathcal {Q}$ be the 5-dimensional real hyperquadric with CR symmetry group $SU_\\star $ , where $SU_\\star $ is one of $SU(2,2)$ or $SU(3,1)$ , and let $M$ be a homogeneous CR 3-fold whose Lie group of CR symmetries is $\\mathcal {G}$ .",
"A CR embedding $f:M\\rightarrow \\mathcal {Q}$ is equivariant if for every $x\\in M$ and $\\varsigma \\in \\mathcal {G}$ , there exists $\\varsigma _\\star \\in SU_\\star $ such that $f(\\varsigma (x))=\\varsigma _\\star (f(x))$ .",
"If such an embedding exists, $M$ is equivariantly embeddable in $\\mathcal {Q}$ .",
"Curry and Gover ([5]) developed a general framework for analyzing CR embeddings using tractor calculus, which replaces the Cartan bundle $\\mathcal {B}$ and Cartan connection $\\gamma $ of $M$ with an associated vector bundle and differential operator, though the distinction is purely aesthetic since $\\mathcal {B}$ and $\\gamma $ are recoverable from the tractor bundle and connection.",
"As such, when the ambient CR manifold under consideration is a hyperquadric $\\mathcal {Q}$ , employing the Curry-Gover formalism to study embedded CR submanifolds $M\\subset \\mathcal {Q}$ should be equivalent to applying Cartan's method of moving frames ([9], [11]), which is the strategy of the present paper.",
"As an application of exterior differential systems, the method of moving frames is similar in technique to the method of equivalence, except that it involves adapting (co)frames of an ambient homogeneous space to the geometry of embedded submanifolds.",
"Accordingly, moving frames classify submanifolds up to extrinsic equivalence determined by the symmetry group of the homogeneous space.",
"This extrinsic action may discriminate CR submanifolds which are intrinsically equivalent.",
"In particular, Levi-flat manifolds are not necessarily locally identical, and the local classification of Levi-nondegenerate CR 3-folds is more refined.",
"In §REF , the Cartan bundle of $\\mathcal {Q}$ is labeled $\\mathcal {H}$ , in part to distinguish it from that of an abstract, Levi-nondegenerate 3-fold $M$ , but also because it is constructed as Hermitian frames or bases of $\\mathbb {C}^4$ .",
"This approach makes explicit the isomorphism $\\mathcal {H}\\cong SU_\\star $ .",
"Its Cartan connection – the Maurer-Cartan form of $SU_\\star $ – is denoted $\\mu $ so there is no confusion with $\\gamma $ , the Cartan connection of $M$ .",
"In §REF , the method of equivalence is implemented to build $\\mathcal {B}\\rightarrow M$ and $\\gamma $ by the standard procedure that incorporates all intrinsic symmetries of $M$ .",
"We begin adapting frames in $\\mathcal {H}$ over $M\\subset \\mathcal {Q}$ in §REF .",
"Informally speaking, this process divides $\\mu $ into two pieces: one that contains $\\gamma $ as though $\\mathcal {H}$ contains $\\mathcal {B}$ , and another that acts on the “normal bundle,\" encoded in a bilinear form of rank at most two over $M$ that we dub the second fundamental form and denote $\\mathbf {II}$ in §REF by analogy to the study of Riemannian hypersurfaces in Euclidean space.",
"The classification of Levi-nondegenerate $M\\subset \\mathcal {Q}$ in § parallels their intrinsic classification in §REF , including the distinction between flat 3-folds (§REF ) and those $M$ with nonzero curvature $\\text{d}\\gamma +\\tfrac{1}{2}[\\gamma ,\\gamma ]$ (§REF ).",
"An essential result of the paper is Theorem REF , which details how the second fundamental form of an embedded, Levi-nondegenerate 3-fold relates to its intrinsic CR invariants in the case that the curvature tensor is nonvanishing.",
"This characterizes up to symmetries in $SU_\\star $ the abstract, curved 3-folds which admit embeddings in $\\mathcal {Q}$ in the spirit of the Bonnet Theorem.",
"The praxis of Theorem REF is complicated in general, but it simplifies when $M$ is homogeneous, and even more so for equivariant embeddings as in Definition REF : Theorem 1.2 Let $\\mathcal {Q}$ be the 5-dimensional real hyperquadric with CR symmetry group $SU_\\star $ , where $SU_\\star $ is one of $SU(2,2)$ or $SU(3,1)$ , and let $M$ be a homogeneous, Levi-nondegenerate CR 3-fold with nonzero curvature whose Lie algebra of infinitesimal CR symmetries is $\\mathfrak {g}$ .",
"$M$ is locally equivariantly embeddable in $\\mathcal {Q}$ if and only if one of the following is true: $SU_\\star =SU(3,1)$ and $\\mathfrak {g}=\\mathfrak {su}(2)$ ; every member of the parameter family of these models is realized in $\\mathcal {Q}$ with rank$(\\mathbf {II})=2$ .",
"$SU_\\star =SU(2,2)$ and $\\mathfrak {g}=\\mathfrak {su}(1,1)$ ; every member of the parameter family of these models is realized in $\\mathcal {Q}$ with rank$(\\mathbf {II})=2$ .",
"$SU_\\star =SU(2,2)$ and $\\mathfrak {g}=\\mathfrak {su}(2)$ ; a single member of the parameter family of these models is realized in $\\mathcal {Q}$ with rank$(\\mathbf {II})=1$ .",
"Since Levi-nondegenerate 3-folds with vanishing curvature are all locally intrinsically equivalent to the 3-sphere, there is no question that they are locally embeddable in either hyperquadric.",
"Still, the method of moving frames serves to distinguish embeddings that are inequivalent under the action of $SU_\\star $ on $\\mathcal {Q}$ , and as always the rank of $\\mathbf {II}$ is a valuable invariant of this action.",
"Those $M\\subset \\mathcal {Q}$ with $\\mathbf {II}$ of submaximal rank are all equivariantly embeddable.",
"The maximally symmetric case $\\mathbf {II}=0$ realizes the full symmetry group of the 3-sphere, whereas rank$(\\mathbf {II})=1$ describes models with submaximal symmetry.",
"By contrast, embeddings with rank$(\\mathbf {II})=2$ are classified by structure equations (REF ) which only evince an equivariant embedding in the hyperquadric with symmetry group $SU(2,2)$ .",
"Theorem 1.3 Let $\\mathcal {Q}$ be the 5-dimensional real hyperquadric with CR symmetry group $SU_\\star $ , where $SU_\\star $ is one of $SU(2,2)$ or $SU(3,1)$ , and let $M$ be a homogeneous, Levi-nondegenerate CR 3-fold with zero curvature whose Lie group of CR symmetries is $\\mathcal {G}$ with Lie algebra $\\mathfrak {g}$ .",
"$M$ is locally equivariantly embeddable in $\\mathcal {Q}$ if and only if one of the following is true: $\\mathcal {G}=SU(2,1)$ and $M$ is the CR 3-sphere.",
"$M$ is embedded as an orbit of $U(2,1)\\subset SU_\\star $ with $\\mathbf {II}=0$ .",
"$SU_\\star =SU(3,1)$ and $M$ is the unique flat model with $\\mathfrak {g}=\\mathfrak {su}(2)$ .",
"$M$ is locally embedded as an orbit of $\\hat{\\mathcal {G}}\\subset SU(3,1)$ with rank$(\\mathbf {II})=1$ , where $\\hat{\\mathcal {G}}$ is an extension of $\\mathcal {G}$ by a central action of $U(1)\\subset SU(3,1)$ .",
"$SU_\\star =SU(2,2)$ and $M$ is the unique flat model with $\\mathfrak {g}=\\mathfrak {su}(1,1)$ .",
"$M$ is locally embedded as an orbit of $\\hat{\\mathcal {G}}\\subset SU(2,2)$ with rank$(\\mathbf {II})=1$ , where $\\hat{\\mathcal {G}}$ is an extension of $\\mathcal {G}$ by a central action of $U(1)\\subset SU(2,2)$ .",
"$SU_\\star =SU(2,2)$ and $M$ is the unique flat model with $\\mathfrak {g}$ as the extension of $\\mathbb {R}^2$ by the derivation $\\left[{\\begin{matrix}3&1\\\\1&3\\end{matrix}}\\right]$ .",
"$M$ is embedded with rank$(\\mathbf {II})=2$ .",
"For $\\mathcal {Q}$ whose Levi form has split signature, it is also possible that $M\\subset \\mathcal {Q}$ is Levi-flat, in which case there is no Cartan bundle or connection to speak of.",
"All such $M$ are locally the same, intrinsically.",
"Even so, the reduction of $\\mathcal {H}\\cong SU(2,2)\\rightarrow M$ in §REF and the definition of the second fundamental form $\\mathbf {II}$ in §REF are meaningful, and in § we classify Levi-flat $M\\subset \\mathcal {Q}$ up to the action of $SU(2,2)$ on $\\mathcal {Q}$ .",
"The maximally symmetric case $\\mathbf {II}=0$ has the structure equations of the 10-dimensional parabolic subgroup $\\mathcal {R}\\subset SU(2,2)$ that stabilizes a partial flag given by a null line in a 3-plane spanned by null lines.",
"When rank$(\\mathbf {II})=1$ , $\\mu $ is fully reduced to a coframing of $M$ with classifying structure equations (REF ), including one homogeneous model.",
"Finally, the maximal rank case branches based on first-order behavior of $\\mathbf {II}$ , where the generic subcase is fully reduced and classified by (REF ), and the alternative exhibits the Maurer-Cartan equations of $\\mathfrak {sl}_2\\mathbb {R}\\oplus \\mathfrak {su}(p,q)$ for either $(p,q)=(2,0)$ or $(p,q)=(1,1)$ .",
"Theorem 1.4 Let $\\mathcal {Q}$ be the 5-dimensional real hyperquadric with CR symmetry group $SU(2,2)$ , and let $M$ be a Levi-flat CR 3-fold embedded in $\\mathcal {Q}$ with second fundamental form $\\mathbf {II}$ .",
"$M$ is homogeneous for its Lie group of CR symmetries induced by the action of $SU(2,2)$ on $\\mathcal {Q}$ if and only if one of the following is true: $\\mathbf {II}=0$ and $M$ contains a complex line which is null for the Levi form of $\\mathcal {Q}$ .",
"In this case the group if CR symmetries of $M$ is the 10-dimensional parabolic subgroup $\\mathcal {R}\\subset SU(2,2)$ that stabilizes a partial flag given by a null line in a 3-plane spanned by null lines.",
"Rank$(\\mathbf {II})=1$ and the algebra of infinitesimal CR symmetries of $M$ is isomorphic to the extension of $\\mathbb {R}^2$ by the derivation $\\@root 3 \\of {3}\\left[{\\begin{matrix}1&0\\\\0&-1\\end{matrix}}\\right]$ .",
"Rank$(\\mathbf {II})=2$ and the algebra of infinitesimal CR symmetries of $M$ is $\\mathfrak {sl}_2\\mathbb {R}\\oplus \\mathfrak {su}(p,q)$ for either $(p,q)=(2,0)$ or $(p,q)=(1,1)$ .",
"Rank$(\\mathbf {II})=2$ and the algebra of infinitesimal CR symmetries of $M$ is isomorphic to the extension of $\\mathbb {R}^2$ by the derivation $\\left[{\\begin{matrix}1&0\\\\0&2\\end{matrix}}\\right]$ .",
"Acknowledgements: This project was initiated while the author participated in the Fall 2017 Simons Semester Symmetry and Geometric Structures hosted by the Mathematics Institute of the Polish Academy of Sciences.",
"The author also acknowledges the Czech Science Foundation (GAČR) for support via the program GAČR 19-14466Y."
],
[
"Basic Definitions, Adapted Coframings",
"For any fiber bundle $\\pi :E\\rightarrow M$ , $E_x=\\pi ^{-1}(x)$ denotes the fiber of $E$ over $x\\in M$ and $\\Gamma (E)$ denotes the sheaf of smooth (local) sections of $E$ .",
"If $E$ is a vector bundle, $\\mathbb {C}E$ is its complexification whose fibers are $\\mathbb {C}E_x=E_x\\otimes _\\mathbb {R}\\mathbb {C}$ .",
"We use bold text for the constants $\\mathbf {i}=\\sqrt{-1}$ and $\\mathbf {e}$ , the natural exponential.",
"Here, CR structure refers specifically to a hypersurface-type CR structure $(M,D,J)$ , which is a $(2n+1)$ -dimensional smooth manifold $M$ equipped with a corank-1 distribution $D\\subset TM$ carrying an almost-complex structure $&J:D\\rightarrow D,&J^2=-\\mathbb {1},$ where $J_x:D_x\\rightarrow D_x$ is linear for every $x\\in M$ , and $\\mathbb {1}$ is the identity map on the fibers of $D$ .",
"The induced action of $J$ on the complexified bundle splits $\\mathbb {C}D=H\\oplus \\overline{H},$ where the CR bundle $H$ is the $\\mathbf {i}$ -eigenspace and the anti-CR bundle $\\overline{H}$ the $(-\\mathbf {i})$ -eigenspace of $J$ .",
"The CR dimension of $M$ is $\\text{rank}_\\mathbb {C}H=n$ .",
"Given two CR structures $(M_1,D_1,J_1)$ and $(M_2,D_2,J_2)$ , a CR map between them is a smooth map $f:M_1\\rightarrow M_2$ whose pushforward $f_*:TM_1\\rightarrow TM_2$ satisfies $f_*D_1\\subset D_2$ and $f_*\\circ J_1=J_2\\circ f_*$ ; in other words, $f_*H_1\\subset H_2$ .",
"$M_1$ and $M_2$ are CR equivalent if there exists a CR map between them that is a diffeomorphism.",
"Often we are merely concerned with local equivalence maps, defined on some neighborhood of any given point: Definition 2.1 Let $M_1,M_2$ be CR manifolds of CR dimension $n_1,n_2$ , respectively, where $n_1\\le n_2$ .",
"$M_1$ is said to be (locally) CR-embeddable in $M_2$ if for each $x\\in M_1$ , there is a neighborhood $N_x\\subset M_1$ and a CR map $f:N_x\\rightarrow M_2$ which is a CR equivalence onto its image; such $f$ is a CR embedding.",
"If $n_2=n_1$ , a CR embedding is a (local) CR equivalence, and if $M_2=M_1$ a local CR equivalence is a (local) CR symmetry of $M_1$ .",
"All CR structures in this paper are CR-integrable; i.e., sections of the (anti-)CR bundle are closed under the Lie bracket of vector fields, $[\\Gamma (H),\\Gamma (H)]\\subset \\Gamma (H)\\Longleftrightarrow [\\Gamma (\\overline{H}),\\Gamma (\\overline{H})]\\subset \\Gamma (\\overline{H}).$ The failure of integrability of the underlying real distribution $D$ is measured by the Levi form, $&\\left.\\begin{array}{rl}\\ell :H_x\\times H_x&\\rightarrow \\mathbb {C}T_xM/\\mathbb {C}D_x\\\\(y_1,y_2)&\\mapsto \\mathbf {i}[Y_1,\\overline{Y}_2](x)\\mathbb {\\unknown.",
"}\\mod {\\mathbb {}{C}D_x}&&Y_i\\in \\Gamma (H),&&Y_i(x)=y_i\\quad (i=1,2).\\end{array}\\right.M is \\emph {Levi-flat} if \\ell vanishes identically.",
"The Newlander-Nirenberg Theorem implies that Levi-flat CR manifolds are locally CR equivalent to \\mathbb {R}\\times \\mathbb {C}^n.$ CR structures can locally be encoded into an adapted coframing.",
"Writing $D^\\bot \\subset T^*M,\\overline{H}^\\bot \\subset \\mathbb {C}T^*M$ for the annihilators of $D$ and $\\overline{H}$ , a 0-adapted coframing is given by a collection of 1-forms $&\\varphi ^0\\in \\Gamma (D^\\bot ),&\\varphi ^j\\in \\Gamma (\\overline{H}^\\bot )\\quad 1\\le j\\le n,&&\\text{such that}&&\\varphi ^0\\wedge \\Big (\\bigwedge _{j=1}^n\\varphi ^j\\Big )\\wedge \\Big (\\bigwedge _{j=1}^n\\overline{\\varphi }^j\\Big )\\ne 0.$ Equivalently, a 0-adapted coframing is a local section of the bundle $\\pi :\\mathcal {F}^0\\rightarrow M$ whose fiber over $x\\in M$ consists of 0-adapted coframes, which are linear isomorphisms $\\mathcal {F}^0_x=\\lbrace \\varphi _x:T_xM\\stackrel{\\simeq }{\\longrightarrow } \\mathbb {R}\\oplus \\mathbb {C}^n\\ |\\ \\varphi _x(D_x)=\\mathbb {C}^n,\\ \\varphi _x\\circ J_x=\\mathbf {i}\\varphi _x\\rbrace .$ We call $\\varphi ^0$ a characteristic form, while $\\varphi ^j$ and $\\overline{\\varphi }^j$ are CR and anti-CR forms, respectively.",
"The CR integrability condition (REF ) is expressed $&\\text{d}\\varphi ^i\\equiv 0\\mod {\\lbrace }\\varphi ^0,\\dots ,\\varphi ^n\\rbrace ,&0\\le i\\le n.$ In particular, a characteristic form is real-valued, so using the summation convention we can write $&\\text{d}\\varphi ^0\\equiv \\mathbf {i}\\ell _{jk}\\varphi ^j\\wedge \\overline{\\varphi }^k\\mod {\\lbrace }\\varphi ^0\\rbrace ,&\\ell _{kj}=\\overline{\\ell }_{jk}\\in C^\\infty (M,\\mathbb {C});&&1\\le j,k\\le n,$ where $\\ell _{jk}(x)$ is a local representation of the Levi form (REF ) as an $n\\times n$ Hermitian-symmetric matrix.",
"The signature $(p,q)$ of this matrix is an invariant of $M$ under CR equivalence (modulo $(p,q)\\sim (q,p)$ ).",
"Of course, the 0-adapted coframing $\\lbrace \\varphi ^0,\\varphi ^j\\rbrace $ is not uniquely determined by (REF ), but only up to a transformation of the form $&\\left[\\begin{array}{cc}u&0\\\\b&a\\end{array}\\right]\\left[\\begin{array}{c}\\varphi ^0\\\\\\varphi ^j\\end{array}\\right],&0\\ne u\\in C^\\infty (M),\\quad a\\in C^\\infty (M,GL_n\\mathbb {C}),\\quad b\\in C^\\infty (M,\\mathbb {C}^n).$ Equivalently, the bundle $\\pi :\\mathcal {F}^0\\rightarrow M$ carries a natural $G_0$ -principal action on its fibers (REF ), $G_0=\\left.\\left\\lbrace \\left[\\begin{array}{cc}u&0\\\\b&a\\end{array}\\right]\\in GL(\\mathbb {R}\\oplus \\mathbb {C}^n)\\ \\right|\\ 0\\ne u\\in \\mathbb {R},\\ a\\in GL_n\\mathbb {C},\\ b\\in \\mathbb {C}^n \\right\\rbrace .$ Thus we see that a CR structure is an example of a G-structure ([1]); there is a tautologically defined 1-form $\\Phi \\in \\Omega ^1(\\mathcal {F}^0,\\mathbb {R}\\oplus \\mathbb {C}^n)$ , $\\Phi |_{\\varphi _x}=\\varphi _x\\circ \\pi _*,$ and any local equivalence $f:M_1\\rightarrow M_2$ between CR manifolds lifts canonically to a diffeomorphism $\\hat{f}:\\mathcal {F}_1^0\\rightarrow \\mathcal {F}_2^0$ between their 0-adapted coframe bundles in a manner that identifies their tautological forms, $\\hat{f}^*\\Phi _2=\\Phi _1$ .",
"To find all local invariants of a CR structure via Cartan's method of equivalence ([8]), one attempts to complete the tautological form to a full coframing of a principal bundle over $M$ by choosing a complementary pseudoconnection form ([1]).",
"Such a choice depends on reducing the structure group of the coframe bundle as much as possible by successively adapting frames to higher order.",
"For example, if $M$ is not Levi-flat, we could define a 1-adapted coframing to be a 0-adapted coframing which has the additional property that the matrix entries (REF ) of the Levi form take constant, specified values (such as $\\ell $ being diagonalized with $p$ positive ones and $q$ negative ones on the diagonal).",
"This reduces $\\mathcal {F}^0$ to the subbundle of 1-adapted coframes whose structure group $G_1$ is matrices (REF ) with the additional constraint, $&G_1\\subset G_0: &\\overline{a}^t\\ell a=u\\ell .$ This reduction is not meaningful in the Levi-flat case; indeed, we have already noted that there are no local invariants for Levi-flat CR manifolds.",
"In general, the degree of (non)degeneracy of the Levi form has substantial bearing on the application of the method of equivalence.",
"The opposite extreme of Levi-flatness is Levi-nondegeneracy, when $\\ell $ has signature $(p,q)$ , $p+q=n$ .",
"In CR dimension $n=1$ , Levi-nondegeneracy is the same as pseudo-convexity, and the corresponding equivalence problem was solved by Cartan ([4], [12]).",
"The general case was treated by Tanaka ([25]) using his modified version of Cartan's method that would later provide a valuable framework for understanding all parabolic geometries with canonical Cartan connections ([26], [7]).",
"Chern-Moser ([6]) offered an alternative solution using the standard method and emphasizing the link between the intrinsic geometry of CR manifolds and the extrinsic analysis of normal forms.",
"The solution to the Levi-nondegenerate equivalence problem may be stated as follows.",
"Tanaka-Chern-Moser Classification Let $M$ be a hypersurface-type CR manifold of dimension $2n+1$ whose Levi form has signature $(p,q)$ , $p+q=n$ .",
"There exists a canonically defined principal bundle $\\mathcal {B}\\rightarrow M$ whose structure group is isomorphic to the parabolic subgroup $\\mathcal {P}\\subset SU(p+1,q+1)$ given by the stabilizer of a complex line in $\\mathbb {C}^{n+2}$ which is null for a Hermitian form $ of signature $ (p+1,q+1)$.$ There exists a canonical Cartan connection $\\gamma \\in \\Omega ^1(\\mathcal {B},\\mathfrak {su}(p+1,q+1))$ whose curvature tensor $\\text{d}\\gamma +\\tfrac{1}{2}[\\gamma ,\\gamma ]\\in \\Omega ^2(\\mathcal {B},\\mathfrak {su}(p+1,q+1))$ and its covariant derivatives determine a complete set of local invariants of $M$ .",
"The algebra of infinitesimal symmetries of $M$ has dimension $\\le n^2+4n+3$ , and the upper bound is only achieved where curvature locally vanishes.",
"In this case, $M$ is locally CR equivalent to the hyperquadric $\\mathcal {Q}\\subset \\mathbb {CP}^{n+1}$ given by the complex projectivization of the $-null cone in $ Cn+2$; i.e., $ Q=SU(p+1,q+1)/P$.$ The real hyperquadric $\\mathcal {Q}$ is the “flat model\" of Levi-nondegenerate CR geometry in the sense that it is locally characterized by a vanishing curvature tensor.",
"When $\\dim M=5$ , a nondegenerate Levi form either has definite signature $(2,0)$ or split signature $(1,1)$ , and the Cartan connection $\\gamma $ takes values in $\\mathfrak {su}(3,1)$ or $\\mathfrak {su}(2,2)$ , respectively.",
"Thus, for the flat models $M=\\mathcal {Q}$ , the principal bundle $\\mathcal {B}$ is isomorphic to one of the Lie groups $SU(3,1)$ or $SU(2,2)$ , and $\\gamma $ is exactly the Maurer-Cartan form of $\\mathcal {B}$ .",
"Remark 2.2 The proof of the Tanaka-Chern-Moser Classification is constructive, building the Cartan Bundle $\\mathcal {B}$ out of normalized jets of local CR coframings so that the Cartan connection $\\gamma $ prolongs the tautological form (REF ) to a full coframing of $\\mathcal {B}$ .",
"By design, a CR map $f:M_1\\rightarrow M_2$ lifts uniquely to a smooth map $\\hat{f}:\\mathcal {B}_1\\rightarrow \\mathcal {B}_2$ such that $\\hat{f}^*\\gamma _2=\\gamma _1$ , and conversely, a smooth map $\\hat{f}:\\mathcal {B}_1\\rightarrow \\mathcal {B}_2$ such that $\\hat{f}^*\\gamma _2=\\gamma _1$ descends uniquely to a CR map $f:M_1\\rightarrow M_2$ .",
"We are concerned with CR embeddings as in Definition REF where $M_1=M$ is 3-dimensional and $M_2=\\mathcal {Q}$ is a 5-dimensional hyperquadric.",
"To avoid confusion, the Cartan bundle and connection of $M$ – constructed in §REF – are called $\\mathcal {F}^3$ and $\\gamma $ , while those of $\\mathcal {Q}$ – exhibited in §REF – are called $\\mathcal {H}$ and $\\mu $ .",
"3-dimensional, Levi-nondegenerate CR Manifolds This section closely follows [3] with only minor changes to notation, and omitting several details.",
"Fix $n=1$ so that $\\dim M=3$ and a local 0-adapted coframing (REF ) consists of an $\\mathbb {R}$ -valued characteristic form $\\varphi ^0$ and a $\\mathbb {C}$ -valued CR form $\\varphi ^1$ such that $\\varphi ^0\\wedge \\varphi ^1\\wedge \\overline{\\varphi }^1\\ne 0$ .",
"This coframing is a local section of the bundle $\\pi :\\mathcal {F}^0\\rightarrow M$ of 0-adapted coframes, and as such it determines a local trivialization of $\\mathcal {F}^0$ over which the tautological form (REF ) is $&\\Phi =\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right]=\\left[\\begin{array}{cc}u&0\\\\b&a\\end{array}\\right]\\pi ^*\\left[\\begin{array}{c}\\varphi ^0\\\\\\varphi ^1\\end{array}\\right],&0\\ne u\\in C^\\infty (\\mathcal {F}^0),\\quad 0\\ne a\\in C^\\infty (\\mathcal {F}^0,\\mathbb {C}),\\quad b\\in C^\\infty (\\mathcal {F}^0,\\mathbb {C}),$ with the functions $u,a,b$ acting as $G_0$ -valued fiber coordinates for $\\mathcal {F}^0$ .",
"CR integrability (REF ) is automatic in dimension three, and in particular (REF ) reads $&\\text{d}\\varphi ^0\\equiv \\mathbf {i}\\ell \\varphi ^1\\wedge \\overline{\\varphi }^1\\mod {\\lbrace }\\varphi ^0\\rbrace ,&\\ell \\in C^\\infty (M).$ Levi-nondegeneracy says $\\ell $ is non-vanishing, so in this case we reduce to the bundle $\\mathcal {F}^1\\subset \\mathcal {F}^0$ of 1-adapted coframes with $\\ell =1$ , which reduces the structure group $G_0$ to $G_1=\\left.\\left\\lbrace \\left[\\begin{array}{cc}|a|^2&0\\\\b&a\\end{array}\\right]\\in GL(\\mathbb {R}\\oplus \\mathbb {C})\\ \\right|\\ a\\in \\mathbb {C}\\setminus \\lbrace 0\\rbrace ,\\ b\\in \\mathbb {C} \\right\\rbrace .$ After pulling back the tautological form (REF ) along the inclusion $\\mathcal {F}^1\\hookrightarrow \\mathcal {F}^0$ , its exterior derivative can be expressed in terms of a pseudoconnection form taking values in the Lie algebra $\\mathfrak {g}_1$ of (REF ), $&\\text{d}\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right]=-\\left[\\begin{array}{cc}\\alpha _0+\\overline{\\alpha }_0&0\\\\\\beta _0&\\alpha _0\\end{array}\\right]\\wedge \\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right]+\\left[\\begin{array}{c}\\mathbf {i}\\eta \\wedge \\overline{\\eta }\\\\0\\end{array}\\right];&\\alpha _0,\\beta _0\\in \\Omega ^1(\\mathcal {F}^1,\\mathbb {C}).$ However, the structure equations (REF ) do not uniquely determine $\\alpha _0$ and $\\beta _0$ as they remain the same after a replacement $&\\left[\\begin{array}{c}\\alpha ^{\\prime }_0\\\\\\beta ^{\\prime }_0\\end{array}\\right]=\\left[\\begin{array}{c}\\alpha _0\\\\\\beta _0\\end{array}\\right]+\\left[\\begin{array}{cc}s^1&0\\\\s^2&s^1\\end{array}\\right]\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right],&s^1,s^2\\in C^\\infty (\\mathcal {F}^1,\\mathbb {C}).$ For any $\\alpha _0^{\\prime },\\beta _0^{\\prime }$ of the form (REF ), the 1-forms $\\kappa ,\\eta ,\\overline{\\eta },\\alpha ^{\\prime }_0,\\overline{\\alpha }^{\\prime }_0,\\beta ^{\\prime }_0,\\overline{\\beta }^{\\prime }_0$ are called a 1-adapted coframing of $\\mathcal {F}^1$ .",
"The bundle $\\hat{\\pi }:\\hat{\\mathcal {F}}^1\\rightarrow \\mathcal {F}^1$ of 1-adapted coframes of $\\mathcal {F}^1$ features a tautological $\\mathbb {R}\\oplus \\mathbb {C}\\oplus \\mathfrak {g}_1$ -valued form whose $\\mathbb {R}\\oplus \\mathbb {C}$ -valued components are simply the $\\hat{\\pi }$ pullback of $\\Phi $ (we recycle the names of the individual 1-forms), $\\hat{\\pi }^*\\Phi =\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right]\\in \\Omega ^1(\\hat{\\mathcal {F}}^1,\\mathbb {R}\\oplus \\mathbb {C}),$ and whose $\\mathfrak {g}_1$ -valued components $\\left[{\\begin{matrix}\\alpha +\\overline{\\alpha }&0\\\\\\beta &\\alpha \\end{matrix}}\\right]\\in \\Omega ^1(\\hat{\\mathcal {F}}^1,\\mathfrak {g}_1)$ satisfy the “lifted\" structure equations, $&\\text{d}\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right]=-\\left[\\begin{array}{cc}\\alpha +\\overline{\\alpha }&0\\\\\\beta &\\alpha \\end{array}\\right]\\wedge \\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right]+\\left[\\begin{array}{c}\\mathbf {i}\\eta \\wedge \\overline{\\eta }\\\\0\\end{array}\\right];&\\alpha ,\\beta \\in \\Omega ^1(\\hat{\\mathcal {F}}^1,\\mathbb {C}).$ In particular, from (REF ) we see that $&\\alpha =\\hat{\\pi }^*\\alpha _0-s^1\\kappa ,&\\beta =\\hat{\\pi }^*\\beta _0-s^2\\kappa -s^1\\eta ,&&s^1,s^2\\in C^\\infty (\\hat{\\mathcal {F}}^1,\\mathbb {C}),$ where $s^1,s^2$ now serve as fiber coordinates for $\\hat{\\pi }:\\hat{\\mathcal {F}}^1\\rightarrow \\mathcal {F}^1$ .",
"Differentiating the structure equations (REF ) yields $\\text{d}\\left[\\begin{array}{c}\\alpha \\\\\\beta \\end{array}\\right]=-\\left[\\begin{array}{cc}\\sigma ^1_0&0\\\\\\sigma ^2_0&\\sigma ^1_0\\end{array}\\right]\\wedge \\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right]+\\left[\\begin{array}{c}-\\mathbf {i}\\beta \\wedge \\overline{\\eta }-2\\mathbf {i}\\overline{\\beta }\\wedge \\eta +R\\eta \\wedge \\overline{\\eta } \\\\-\\beta \\wedge \\overline{\\alpha }\\end{array}\\right],$ for some $R\\in C^\\infty (\\hat{\\mathcal {F}}^1)$ , with $\\sigma _0^1,\\sigma _0^2,\\kappa ,\\eta ,\\alpha ,\\beta $ and their conjugates furnishing a coframing of $\\hat{\\mathcal {F}}^1$ .",
"The identity $\\text{d}^2\\alpha =0$ then reveals that we can restrict to a subbundle $\\mathcal {F}^2\\subset \\hat{\\mathcal {F}}^1$ whose sections are 2-adapted coframings defined by $R=0$ , which reduces the real dimension of the fibers over $\\mathcal {F}^1$ by one and forces $s^1$ and $\\sigma =\\sigma ^1_0$ to be strictly $\\mathbb {R}$ -valued.",
"Next, the same identity shows that we can reduce further to 3-adapted coframings corresponding to a subbundle $\\mathcal {F}^3\\subset \\mathcal {F}^2$ where $s^2=0$ , hence the real fiber dimension of $\\mathcal {F}^3\\rightarrow \\mathcal {F}^1$ is one.",
"The coframing of $\\mathcal {F}^3$ given by the complex forms $\\eta ,\\alpha ,\\beta $ and their conjugates, along with the real forms $\\kappa ,\\sigma $ , is globally defined on $\\mathcal {F}^3$ and uniquely determined by the structure equations $\\begin{aligned}\\text{d}\\kappa &=\\mathbf {i}\\eta \\wedge \\overline{\\eta }-(\\alpha +\\overline{\\alpha })\\wedge \\kappa ,\\\\\\text{d}\\eta &=-\\beta \\wedge \\kappa -\\alpha \\wedge \\eta ,\\\\\\text{d}\\alpha &=-\\sigma \\wedge \\kappa -\\mathbf {i}\\beta \\wedge \\overline{\\eta }-2\\mathbf {i}\\overline{\\beta }\\wedge \\eta ,\\\\\\text{d}\\beta &=-\\sigma \\wedge \\eta +\\overline{\\alpha }\\wedge \\beta +S\\kappa \\wedge \\overline{\\eta },\\\\\\text{d}\\sigma &=(\\alpha +\\overline{\\alpha })\\wedge \\sigma +\\mathbf {i}\\beta \\wedge \\overline{\\beta }+\\kappa \\wedge (P\\overline{\\eta }+\\overline{P}\\eta ),\\end{aligned}$ where $S,P\\in C^\\infty (\\mathcal {F}^3,\\mathbb {C})$ have differential identities $\\begin{aligned}\\text{d}S&=S(3\\overline{\\alpha }+\\alpha )+U\\kappa +P\\eta +Q\\overline{\\eta },\\\\\\text{d}P&=P(3\\overline{\\alpha }+2\\alpha )-\\mathbf {i}S\\overline{\\beta }+W\\kappa +R\\eta +V\\overline{\\eta },\\end{aligned}$ for some $U,Q,W,V\\in C^\\infty (\\mathcal {F}^3,\\mathbb {C})$ and $R\\in C^\\infty (\\mathcal {F}^3)$ .",
"$\\mathcal {F}^3$ realizes the Cartan bundle $\\mathcal {B}$ of $M$ whose existence is guaranteed by the Tanaka-Chern-Moser Classification; the Cartan connection is $\\gamma =\\left[\\begin{array}{ccc}-\\tfrac{1}{3}(2\\alpha +\\overline{\\alpha })&-\\mathbf {i}\\overline{\\beta }&-\\mathbf {i}\\sigma \\\\\\eta &\\tfrac{1}{3}(\\alpha -\\overline{\\alpha })&\\mathbf {i}\\beta \\\\-\\mathbf {i}\\kappa &\\overline{\\eta }&\\tfrac{1}{3}(\\alpha +2\\overline{\\alpha })\\end{array}\\right],$ so that $&\\overline{\\gamma }^t=0,&\\left[\\begin{array}{crc}0&0&-1\\\\0&1&0\\\\-1&0&0\\end{array}\\right],$ and $\\gamma $ is indeed $\\mathfrak {su}(2,1)$ -valued.",
"The equations (REF ) engender the Bianchi identities of the curvature tensor $\\text{d}\\gamma +\\gamma \\wedge \\gamma $ .",
"Furthermore, when $S=0\\Rightarrow P=0$ so that curvature locally vanishes, (REF ) are exactly the Maurer-Cartan equations of $\\mathfrak {su}(2,1)$ , as previously discussed.",
"If the curvature tensor of $M$ never vanishes, we can adapt to even higher order (see [3]).",
"First, differentiate the identities (REF ) to obtain $\\begin{aligned}\\text{d}R&= R(3\\overline{\\alpha }+3\\alpha )+R_0^{\\prime }\\kappa +R_1^{\\prime }\\overline{\\eta }+\\overline{R}_1^{\\prime }\\eta ,\\\\\\text{d}Q&= Q(4\\overline{\\alpha }+\\alpha )-5\\mathbf {i}S\\beta +U_1^{\\prime }\\kappa +(V-\\mathbf {i}U)\\eta +Q^{\\prime }\\overline{\\eta },\\\\\\text{d}U&= U(4\\overline{\\alpha }+2\\alpha )+4S\\sigma +P\\beta +Q\\overline{\\beta }+W\\eta +U_1^{\\prime }\\overline{\\eta }+U_2^{\\prime }\\kappa ,\\\\\\text{d}V&\\equiv V(4\\overline{\\alpha }+2\\alpha )-\\mathbf {i}S\\sigma -4\\mathbf {i}P\\beta -\\mathbf {i}Q\\overline{\\beta }+(R_1^{\\prime }-\\mathbf {i}W)\\eta +V^{\\prime }\\kappa &\\mod {\\lbrace }\\overline{\\eta }\\rbrace ,\\\\\\text{d}W&=W(4\\overline{\\alpha }+3\\alpha )+5P\\sigma +R\\beta +(V-\\mathbf {i}U)\\overline{\\beta }+(R_0^{\\prime }-\\mathbf {i}|S|^2)\\eta +V^{\\prime }\\overline{\\eta }+W^{\\prime }\\kappa ,\\end{aligned}$ for some $R_0^{\\prime }\\in C^\\infty (\\mathcal {F}^3)$ and $R_1^{\\prime },Q^{\\prime },U_1^{\\prime },U_2^{\\prime },V^{\\prime },W^{\\prime }\\in C^\\infty (\\mathcal {F}^3,\\mathbb {C})$ .",
"For later use, we differentiate the first line of (REF ) in order to record $\\begin{aligned}\\text{d}R_1^{\\prime }&\\equiv R_1^{\\prime }(4\\overline{\\alpha }+3\\alpha )-3\\mathbf {i}R\\beta +(R_0^{\\prime \\prime }-\\tfrac{\\mathbf {i}}{2}R_0^{\\prime })\\eta +R_1^{\\prime \\prime }\\kappa &&\\mod {\\lbrace }\\overline{\\eta }\\rbrace ,\\end{aligned}$ with additional $R_0^{\\prime \\prime }\\in C^\\infty (\\mathcal {F}^3)$ and $R_1^{\\prime \\prime }\\in C^\\infty (\\mathcal {F}^3,\\mathbb {C})$ .",
"Now observe that the identity (REF ) for $\\text{d}S$ implies that if $S$ is nowhere zero, we can reduce to $\\mathcal {F}^4\\subset \\mathcal {F}^3$ where $S=1$ , and over $\\mathcal {F}^4$ we have $\\alpha =\\frac{1}{8}\\Big ( (U-3\\overline{U})\\kappa + (P-3\\overline{Q})\\eta +(Q-3\\overline{P})\\overline{\\eta }\\Big ).$ At this point, the equation (REF ) for $\\text{d}P$ shows that there is a subbundle $\\mathcal {F}^5\\subset \\mathcal {F}^4$ on which $P=0$ and $\\beta =\\mathbf {i}(\\overline{W}\\kappa +\\overline{V}\\eta +R\\overline{\\eta }).$ Finally, by the third line of (REF ) we can reduce to 6-adapted coframes $\\mathcal {F}^6\\subset \\mathcal {F}^5$ defined by $\\Re U=\\tfrac{1}{2}(U+\\overline{U})=0$ , with $&\\sigma =-\\frac{1}{4}\\Re (U(4\\overline{\\alpha }+2\\alpha )+Q\\overline{\\beta }+W\\eta +U_1^{\\prime }\\overline{\\eta }+U_2^{\\prime }\\kappa ),&\\text{subject to}&&(\\ref {alphareduction}),(\\ref {betareduction}).$ After these reductions to $\\mathcal {F}^6\\subset \\mathcal {F}^3$ , what remains of (REF ) is $\\begin{aligned}\\text{d}\\kappa &=\\mathbf {i}\\eta \\wedge \\overline{\\eta }-2\\kappa \\wedge (\\overline{A}\\eta +A\\overline{\\eta }),\\\\\\text{d}\\eta &=A\\eta \\wedge \\overline{\\eta }+\\mathbf {i}\\kappa \\wedge (B\\eta + C\\overline{\\eta }),\\end{aligned}$ where $&A=\\frac{Q}{8},&B=\\overline{V}+\\frac{\\mathbf {i}}{2}U,&&C=R.$ The structure equations (REF , REF ) for a 6-adapted coframing satisfy $\\text{d}^2\\kappa =\\text{d}^2\\eta =0$ by virtue of the identities (REF ).",
"When $M$ is homogeneous under the action of its CR symmetry group, $A,B,C$ are constant and the differential conditions $\\text{d}^2\\kappa =\\text{d}^2\\eta =0$ simplify to algebraic relations $&B=\\overline{B},&AB=\\overline{A}C.$ Conversely, suppose $M$ is a 3-dimensional CR manifold admitting a 1-adapted coframing $\\kappa ,\\eta $ satisfying (REF ), where $A,B,C$ are constant with $C\\in \\mathbb {R}$ .",
"Such $M$ is locally homogeneous, and the identities $\\text{d}^2\\kappa =\\text{d}^2\\eta =0$ once again imply (REF ).",
"Moreover, the 1-adapted coframing $\\kappa ,\\eta $ determines a section $M\\rightarrow \\mathcal {F}^3$ along which the Cartan connection forms pull back to $\\begin{aligned}\\alpha &=-\\mathbf {i}(|A|^2+\\tfrac{3}{4}B)\\kappa -3\\overline{A}\\eta +A\\overline{\\eta },\\\\\\beta &=\\tfrac{1}{3}(AB+4\\overline{A}C-4A^2\\overline{A})\\kappa +\\mathbf {i}(\\tfrac{1}{4}B-|A|^2)\\eta +\\mathbf {i}C\\overline{\\eta },\\\\\\sigma &=(\\tfrac{5}{3}C(A^2+\\overline{A}^2)-\\tfrac{1}{3}|A|^4-\\tfrac{13}{6}B|A|^2+\\tfrac{1}{16}B^2-C^2)\\kappa +\\tfrac{1}{3}\\Re \\Big (\\mathbf {i}(10CA-5B\\overline{A}-4A\\overline{A}^2)\\eta \\Big ),\\end{aligned}$ according to the structure equations (REF ), the penultimate of which necessitates the relation $40A^3\\overline{A}-10A^2B-28|A|^2C+9BC=6S.$ Remark 2.3 When $M$ is locally flat ($S=0\\Rightarrow P=0$ ), higher-order functions in (REF ) and (REF ) vanish in turn.",
"On the other hand, if (REF ) constitutes a 6-adapted coframing ($S=1, P=0, \\overline{U}=-U$ ) of a non-flat, homogeneous CR 3-fold, the section $M\\rightarrow \\mathcal {F}^3$ determined by $\\kappa ,\\eta $ pulls back the higher-order coefficients to $&\\begin{array}{l}R=C, \\\\Q=8A, \\\\U=-\\mathbf {i}(2|A|^2+\\tfrac{3}{2}B), \\\\V=\\tfrac{1}{4}B-|A|^2, \\\\W=\\tfrac{\\mathbf {i}}{3}(B\\overline{A}+4CA-4A\\overline{A}^2),\\\\R_1^{\\prime }=6CA,\\\\R_1^{\\prime \\prime }=\\mathbf {i}\\overline{A}C(4C-10A^2)-\\tfrac{7\\mathbf {i}}{2}ABC,\\end{array}\\begin{array}{l}R_0^{\\prime }=0,\\\\R_0^{\\prime \\prime }=33C|A|^2-\\tfrac{3}{4}BC,\\\\Q^{\\prime }=88A^2-5C,\\\\U_1^{\\prime }=\\tfrac{\\mathbf {i}}{3}(20C\\overline{A}-49BA-92A^2\\overline{A}),\\\\U_2^{\\prime }=8|A|^4-\\tfrac{5}{2}B^2+4C^2-\\tfrac{1}{3}C(52A^2+20\\overline{A}^2),\\\\V^{\\prime }=-\\mathbf {i}(9|A|^4-\\tfrac{3}{2}|A|^2B-\\tfrac{1}{3}C(37A^2+5\\overline{A}^2)+\\tfrac{5}{16}B^2+C^2),\\\\W^{\\prime }=\\tfrac{1}{3}(4|A|^2(5AC-B\\overline{A}-4\\overline{A}|A|^2)+7ABC+2\\overline{A}(B^2-2C^2)).\\end{array}$ To summarize, we see that (REF ) – with constants $A\\in \\mathbb {C}$ , $B,C\\in \\mathbb {R}$ constrained by (REF ) and (REF ) – locally characterizes all 3-dimensional, Levi-nondegenerate CR manifolds which are homogeneous under the action of their CR symmetry groups.",
"In the non-flat case $S\\ne 0$ , these are the structure equations for the maximal CR symmetry algebras tangent to these symmetry groups.",
"Homogeneous, Levi-nondegenerate CR 3-folds are classified (locally and globally) in [4], which exhibits local hypersurface realizations of each model labeled by capital Latin letters.",
"Nurowski and Tafel ([14]) offer alternative coordinate realizations and make explicit the reliance of Cartan's arguments on Bianchi's classification of 3-dimensional real Lie algebras, which are labeled with Roman numerals as in, e.g., [23].",
"Bianchi's type I algebra is abelian; it is simply the vector space $\\mathbb {R}^3$ , which cannot serve as the symmetry algebra of a Levi-nondegenerate CR 3-fold.",
"Indeed, for coordinates $(x,z)\\in \\mathbb {R}\\times \\mathbb {C}$ , the 0-adapted coframing $\\varphi ^0=\\text{d}x$ , $\\varphi ^1=\\text{d}z$ has trivial structure equations.",
"Type II is the Heisenberg Lie algebra – the negatively-graded part of $\\mathfrak {su}(2,1)$ with respect to the grading induced by the parabolic subgroup $\\mathcal {P}\\subset SU(2,1)$ appearing in the Tanaka-Chern-Moser Classification – whose structure equations are $(\\ref {FsixSE}) \\text{ with } A=B=C=0, \\text{ subject to (\\ref {hiddenrelation}) with } S=0.$ As the twin labels suggest, this is Cartan's A model, the CR 3-sphere, whose full group of CR symmetries is $SU(2,1)$ .",
"Skipping to the end of the list, Bianchi's VIII and IX are $\\mathfrak {su}(1,1)$ and $\\mathfrak {su}(2)$ , respectively, each of which is the symmetry algebra of a parameter-family of homogeneous models: $&(\\ref {FsixSE}, \\ref {hiddenrelation}) \\text{ with } A=0,\\ &B=-1,\\ &&&C=0,\\ &&S=0;\\\\&(\\ref {FsixSE}, \\ref {hiddenrelation}) \\text{ with } A=0,\\ &B<0,\\ &&&C=\\frac{2}{3B},\\ &&S=1;\\\\&(\\ref {FsixSE}, \\ref {hiddenrelation}) \\text{ with } A=0,\\ &B=1,\\ &&&C=0,\\ &&S=0;\\\\&(\\ref {FsixSE}, \\ref {hiddenrelation}) \\text{ with } A=0,\\ &B>0,\\ &&&C=\\frac{2}{3B},\\ &&S=1.$ The rest of the algebras on Bianchi's list may be represented as extensions of the abelian Lie algebra $\\mathbb {R}^2$ by a single, nontrivial derivation; i.e., a nonzero $2\\times 2$ matrix.",
"Type V extends $\\mathbb {R}^2$ by a diagonal matrix, and the resulting structure equations are Levi-flat (see §).",
"Type IV is represented as the extension by $\\left[{\\begin{matrix}1&1\\\\0&1\\end{matrix}}\\right]$ , and with a suitable choice of basis its structure equations are $&(\\ref {FsixSE}, \\ref {hiddenrelation}) \\text{ with } A=2\\@root 4 \\of {6},\\ &B=C=9\\sqrt{6},\\ &&S=1.$ The remaining Bianchi types are parameter-families of algebras, indexed by $0< t\\in \\mathbb {R}$ , each member of which has a unique homogeneous model.",
"Type $\\text{VII}_t$ is the extension of $\\mathbb {R}^2$ by $\\left[{\\begin{matrix}t&1\\\\-1&t\\end{matrix}}\\right]$ , which is rank-2 for every $t$ , and its corresponding homogeneous model is $&(\\ref {FsixSE}, \\ref {hiddenrelation}) \\text{ with } A=\\frac{2t\\sqrt{6}}{\\@root 4 \\of {6t^4+60t^2+54}},\\ &B=C=\\frac{54t^2+6}{\\sqrt{6t^4+60t^2+54}},\\ &&S=1.$ Type $\\text{VI}_t$ is the extension of $\\mathbb {R}^2$ by $\\left[{\\begin{matrix}t&m\\\\m&t\\end{matrix}}\\right]$ , where $m\\in \\mathbb {R}$ is determined by the curvature $S$ .",
"Homogeneous models are described in general by $&(\\ref {FsixSE}, \\ref {hiddenrelation}) \\text{ with } A=\\iota t\\quad (\\iota =1\\text{ or }\\mathbf {i}),\\ &B=\\iota ^2C=\\frac{9t^2-m^2}{4},\\ &&S=\\frac{\\iota ^2}{96}(t^2-m^2)(t^2-9m^2).$ Note some exceptional values of $t$ ($\\iota =m=1$ ) with zero curvature: $&(\\ref {FsixSE}, \\ref {hiddenrelation}) \\text{ with } A=1,\\ &B=C=2,\\ &&S=0;\\\\&(\\ref {FsixSE}, \\ref {hiddenrelation}) \\text{ with } A=3,\\ &B=C=20,\\ &&S=0.$ For the remaining values of $t$ , we have $&0<t\\ne 1,3;&(\\ref {sixEgeneral}) \\text{ with } m \\text{ such that } S=1.$ Physical Motivation Some History The subject matter of this article is closely related to relativistic theories of radiation, both electromagnetic and gravitational.",
"In general relativity, spacetime is an oriented 4-dimensional manifold $\\mathcal {S}$ .",
"The distribution of mass-energy in $\\mathcal {S}$ is encoded in a symmetric 2-tensor field, and Einstein's field equations say that this field prescribes the Ricci curvature of a Lorentzian metric; we suppose that $\\mathcal {S}$ admits a solution $g\\in \\bigodot ^2T^*\\mathcal {S}$ .",
"An electromagnetic (EM) field is a 2-form $F\\in \\Omega ^2(\\mathcal {S})$ that may be interpreted as ($-\\mathbf {i}$ times) the curvature of a principal connection on a $U(1)$ -bundle over $\\mathcal {S}$ .",
"$F$ is null if it is $g$ -orthogonal to itself and its Hodge dual.",
"Null EM fields are associated with electromagnetic radiation.",
"A gravitational field is the curvature of $g$ , called null if the Weyl tensor has Petrov type N (see [16] for a pleasant introduction to the Petrov classification) – the most degenerate type among non-conformally-flat metrics.",
"More generally, a metric is algebraically special if its Weyl tensor is at all degenerate; i.e., if it is of any Petrov type besides I.",
"It is helpful to think of the motivation for the present work in the context of the search for a theoretical framework for gravitational radiation analogous to that of electromagnetic radiation.",
"The study of gravitational waves was initiated by Einstein in the beginning of the twentieth century; a brief history with references is given in [28].",
"For our purposes, it suffices to join the story in medias res, when Trautman showed in [27] that gravitational fields satisfying a Sommerfeld radiation condition are asymptotically null.",
"The next year, Robinson reported to the Royaumont Conference that null EM and gravitational fields determine a foliation of spacetime by a family of curves known as a shear-free null geodesic congruence, or SFNGC.",
"These are discussed in detail in §REF .",
"Robinson also proved the converse for electromagnetic fields ([19]); i.e., a SFNGC gives rise to a null EM field.",
"Spacetimes admitting SFNGCs seemed to be natural candidates for a model of gravitational radiation, though the work [22] of Sachs established that these were not restricted to null gravitational fields.",
"Indeed, joint work [10] with Goldberg would show that, away from sources of mass-energy – that is, in a vacuum spacetime with a Ricci-flat metric – every non-flat, algebraically special metric admits a SFNGC tangent to principal null directions of algebraic multiplicity $>1$ .",
"Robinson and Trautman ([20]) produced a class of metrics corresponding to hypersurface-orthogonal SFNGC, including a model for radiation with spherical wavefronts.",
"Then Kerr sought metrics corresponding to SFNGC that were not hypersurface-orthogonal ([13]), and in the process generalized the Schwarzschild solution to incorporate angular momentum, generating a model for spinning black holes ([16]).",
"The Kerr metrics have Petrov type D. Kerr's name is also attached to a theorem relating SFNGC of flat (Minkowski) spacetime to the objects of study of this article.",
"In §REF , we offer a geometric overview of the correspondence between subsets of Minkowski spacetime and those of a 5-dimensional real hyperquadric in the spirit of Penrose's Twistor program ([17], [29]), emphasizing the various symmetry groups involved.",
"Then §REF delves into SFNGC for general spacetimes and explains their connection to CR geometry, following [21] and [15].",
"Finally, a sketch of the proof of the Kerr Theorem appears in §REF , using explicit coordinate calculations as in [24].",
"The Kerr Theorem $\\mathbb {R}^n$ equipped with a symmetric, bilinear form ${\\tt b}$ of signature $(p,q)$ , $p+q=n$ , will be denoted $\\mathbb {R}^{p,q}$ .",
"The complex-linear extension of ${\\tt b}$ to $\\mathbb {C}^n$ is also called ${\\tt b}$ , but in the complexification its signature is no longer significant.",
"Hence, we reserve the notation $\\mathbb {C}^{p,q}$ for when $\\mathbb {C}^n$ carries a Hermitian form $ of signature $ (p,q)$, $ p+q=n$, unrelated to any underlying real form.$ The setting of special relativity is Minkowski spacetime $\\mathbb {M}$ , an affine space with modeling vector space $\\mathbb {R}^{1,3}$ .",
"Therefore, the Lorentz group $O(1,3)$ – and its affine extension, the Poincaré group – plays a central role in relativistic theories.",
"However, when a relativistic theory (such as the electrodynamics expressed in Maxwell's equations) exhibits conformal invariance, the corresponding group of symmetries is larger.",
"Conformal compactification of $\\mathbb {R}^{p,q}$ is achieved by affixing a point “at infinity\" for each one in the ${\\tt b}$ -null cone in order that inversion may be globally defined.",
"The resulting quadric is the real-projectivization of the $\\hat{{\\tt b}}$ -null cone in $\\mathbb {R}^{p+1,q+1}$ , whose bilinear form wears a hat to distinguish it from that of $\\mathbb {R}^{p,q}$ .",
"The group of (oriented) conformal symmetries of compactified Minkowski spacetime $\\mathbb {M}^c$ is the symmetry group of the $\\hat{{\\tt b}}$ -null cone in $\\mathbb {R}^{2,4}$ ; i.e., $SO(2,4)$ .",
"The Plücker embedding sends the Grassmannian $Gr(2,\\mathbb {C}^4)$ of complex 2-planes in $\\mathbb {C}^4$ into the complex-projective space $\\mathbb {P}(\\Lambda ^2\\mathbb {C}^4)=\\mathbb {CP}^5$ , and its image is the quadric given by the projectivization of the $\\hat{{\\tt b}}$ -null cone in $\\mathbb {C}^6$ .",
"This may be considered a geometric analog of the Lie algebra isomorphism $\\mathfrak {sl}_4\\mathbb {C}\\cong \\mathfrak {so}_6\\mathbb {C}$ .",
"Moreover, the Grassmannian $Gr^0(2,\\mathbb {C}^{2,2})\\subset Gr(2,\\mathbb {C}^4)$ of totally $-null 2-planes embeds onto $ Mc$ by analogy to the isomorphism $ su(2,2)so(2,4)$.$ Both $\\mathbb {CP}^3$ and $Gr(2,\\mathbb {C}^4)$ are partial flag manifolds associated to $\\mathbb {C}^4$ ; to these we add $F_{1,2}\\mathbb {C}^4$ consisting of pairs $(l,\\Pi )$ of a complex line and plane (respectively) satisfying $l\\subset \\Pi \\subset \\mathbb {C}^4$ .",
"With the projection maps $\\lambda (l,\\Pi )=l$ and $\\pi (l,\\Pi )=\\Pi $ we obtain the double fibration $@=1em{&_{\\lambda }@{->}[dl] F_{1,2}\\mathbb {C}^4^{\\pi }@{->}[dr] &\\\\\\mathbb {CP}^3& &Gr(2,\\mathbb {C}^4)}$ lying at the heart of Penrose's Twistor theory, which concerns the correspondence between subsets of $\\mathbb {CP}^3$ and $Gr(2,\\mathbb {C}^4)$ via the images of $\\lambda \\circ \\pi ^{-1}$ and $\\pi \\circ \\lambda ^{-1}$ .",
"To clarify some of the physical motivation for this framework, we restrict to $-isotropic flags $ F1,20C2,2F1,2C4$, so that (\\ref {Cdoublefiber}) becomes{\\begin{@align}{1}{-1}@=1em{&_{\\lambda }@{->}[dl] F^0_{1,2}\\mathbb {C}^{2,2}^{\\pi }@{->}[dr] &\\\\\\mathcal {Q}& &\\mathbb {M}^c=Gr^0(2,\\mathbb {C}^{2,2}),}\\end{@align}}where $ QCP3$ is the 5-dimensional real hyperquadric given by the complex projectivization of the $ -null cone $\\mathcal {N}\\subset \\mathbb {C}^{2,2}$ .",
"The trajectory of a massless particle in $\\mathbb {M}$ is tangent to a ${\\tt b}$ -null (affine) line, and each such line corresponds to a point in $\\mathcal {Q}$ .",
"Physicists refer to a foliation of $\\mathbb {M}$ by null lines as a null congruence, the relevance of which to the present work is stated in the Kerr Theorem A null congruence of $\\mathbb {M}$ corresponds to a CR submanifold of $\\mathcal {Q}$ if and only if it is shear-free.",
"The Kerr Theorem first appeared in print in [17], where it is stated that a shear-free null congruence is representable in $\\mathbb {CP}^3$ as the intersection of $\\mathcal {Q}$ with a complex-analytic surface (or a limiting case of such intersections); see also [18].",
"The proof sketch in §REF makes this construction explicit.",
"The version we've stated is closer to [15].",
"Shear-Free Null Geodesic Congruences In this section we follow [21] and [15].",
"Let $\\mathcal {S}$ be a smooth, 4-dimensional manifold with a line bundle $K\\subset T\\mathcal {S}$ whose fibers are spanned by a nowhere-vanishing vector field $k\\in \\Gamma (K)$ , which determines a smooth flow $\\phi :I\\times \\mathcal {S}\\rightarrow \\mathcal {S},$ where $I\\subseteq \\mathbb {R}$ is some open interval containing zero.",
"For fixed $x\\in \\mathcal {S}$ and variable $t\\in I$ , $\\phi (t,x)$ is the integral curve of $k$ passing through $x$ when $t=0$ , and $\\mathcal {S}$ is foliated by these flow curves.",
"For fixed $t\\in I$ , $\\begin{aligned}\\phi _t:\\mathcal {S}&\\rightarrow \\mathcal {S}\\\\x&\\mapsto \\phi (t,x)\\end{aligned}$ is a diffeomorphism whose pushforward $\\phi _{t*}:T\\mathcal {S}\\rightarrow T\\mathcal {S}$ satisfies $\\phi _{t*}K_x=K_{\\phi (t,x)},$ and therefore descends to a well-defined map on the quotient bundle $T\\mathcal {S}/K\\rightarrow \\mathcal {S}$ .",
"Thus, the family $\\lbrace \\phi _t:t\\in I\\rbrace $ of diffeomorphisms provides linear isomorphisms between the spaces $T_{\\phi (t,x)}\\mathcal {S}/K_{\\phi (t,x)}$ for any fixed $x\\in \\mathcal {S}$ , and we see that the quotient bundle $T\\mathcal {S}/K$ has the same fibers as the tangent bundle of the leaf space $M $ ; i.e., the 3-dimensional quotient manifold of equivalence classes $[x]$ of points $x\\in \\mathcal {S}$ , where two points are equivalent if they lie in the same leaf of the foliation (the same flow curve), $&T_{[x]}M \\cong T_{\\phi (t,x)}\\mathcal {S}/K_{\\phi (t,x)}&\\forall t\\in I.$ Remark 3.1 In general, a 4-manifold need not admit a globally defined, non-vanishing tangent vector field, nor should the entire leaf space of a foliation necessarily inherit a global manifold structure.",
"However, our considerations are local in nature and we will continue to implicitly assume that $\\mathcal {S}$ is such that our constructions are well-defined.",
"In particular, we may also take $\\mathcal {S}$ to be orientable.",
"If $\\omega \\in \\Omega ^4(\\mathcal {S})$ is a volume form, then the contraction $k\\omega \\in \\Omega ^3(\\mathcal {S})$ vanishes on $K$ and descends to a 3-form on $T\\mathcal {S}/K$ .",
"Note that $k\\omega $ does not determine a well-defined volume form on $M $ unless $\\mathcal {L}_k\\omega =0$ , where $\\mathcal {L}_k$ denotes the Lie derivative along $k$ .",
"However, the sign of $k\\omega $ on any ordered basis of (REF ) is sufficient to determine whether a volume form on $M $ is positively or negatively oriented relative to $k\\omega $ , and so determines a choice of orientation on $M $ .",
"Suppose that $\\mathcal {S}$ is equipped with a non-degenerate metric $g\\in \\bigodot ^2T^*\\mathcal {S}$ .",
"For the moment, we make no assumptions about the signature of $g$ .",
"The one-form $\\kappa \\in \\Omega ^1(\\mathcal {S})$ dual to $k$ has as its kernel a rank-3 distribution $&\\kappa =kg&\\leadsto &&\\ker \\kappa =K^\\bot \\subset T\\mathcal {S}.$ Definition 3.2 The flow of $k$ is conformally geodesic if it preserves the distribution $K^\\bot $ , $&\\phi _{t*}K_x^\\bot =K_{\\phi (t,x)}^\\bot &\\forall t\\in I, x\\in \\mathcal {S}.$ Equivalently, the flow of $k$ is conformally geodesic when $&\\kappa \\wedge \\phi _t^*\\kappa =0&\\Rightarrow &&\\kappa \\wedge \\mathcal {L}_k\\kappa =0.$ Hence, a conformally geodesic flow not only identifies the fibers of $K$ along a flow curve as in (REF ), but also the fibers of $K^\\bot $ .",
"The implications of this for the leaf space $M $ depend on the metric properties of $k$ .",
"If $g(k_x,k_x)\\ne 0$ for every $x\\in \\mathcal {S}$ , then $T\\mathcal {S}=K\\oplus K^\\bot $ and $T_{[x]}M \\cong K^\\bot _{\\phi (t,x)}$ for every $t\\in I$ .",
"On the other hand, if $g$ has mixed signature and $g(k,k)=0$ , then $K\\subset K^\\bot $ and $M $ inherits a well-defined, rank-2 distribution $D\\subset TM \\quad \\text{with fibers}\\quad D_{[x]}\\cong K^\\bot _{\\phi (t,x)}/K_{\\phi (t,x)}\\quad \\forall t\\in I.$ We also have when $k$ is null that $\\kappa $ descends to the quotient bundle $T\\mathcal {S}/K$ , and the additional condition (REF ) that $k$ is conformally geodesic further implies that $\\kappa $ determines a well-defined, non-vanishing one-form (of the same name) on $M $ , which annihilates (REF ).",
"Remark 3.3 Condition (REF ) is always invariant under conformal scaling of $\\kappa $ , and when $k$ is null it is even invariant under scaling of $k$ by a non-vanishing function, which effects a reparameterization of the flow curves of $k$ .",
"Henceforth, we restrict to the case that $g$ has Lorentzian signature $(1,3)$ and $k$ is $g$ -null with a conformally geodesic flow.",
"The foliation of $\\mathcal {S}$ by flow curves is now called a null geodesic congruence, the fibers of the quotient bundle $K^\\bot /K$ are called screen spaces, and the geometry of the null congruence may be understood intuitively in terms of the following illustration regarding optical scalars ([16]).",
"In relativity, light propagates in null directions; imagine a beam of light casting the shadow of an opaque disk onto a 2-dimensional screen placed orthogonal to its (null) direction.",
"As the screen is moved along the flow curve, this circular image might be rotated, enlarged, or distorted into an ellipse of greater eccentricity.",
"If the latter, non-conformal distortion does not occur, the null congruence is shear-free.",
"The precise geometric definition applies to arbitrary conformally geodesic flows.",
"Definition 3.4 A conformally geodesic flow is shear-free if it preserves the conformal class of $g$ restricted to $K^\\bot $ ; i.e., for any $t\\in I$ and $x\\in \\mathcal {S}$ , there is some $s\\in \\mathbb {R}$ , $s>0$ such that $\\phi _{t}^*(g|_{K^\\bot _{\\phi (t,x)}})=sg|_{K^\\bot _x},$ so that in particular, $\\mathcal {L}_kg=ag+\\kappa \\odot \\alpha $ for some $a\\in C^\\infty (\\mathcal {S})$ and $\\alpha \\in \\Omega ^1(\\mathcal {S})$ .",
"Remark 3.5 Using general properties of the Lie derivative, it is straightforward to confirm that (REF ) is maintained under rescaling of $k$ by a non-vanishing function, albeit for different $a,\\alpha $ .",
"Along with Remark REF , this shows that a shear-free null geodesic congruence (SFNGC) is independent of the choice of $k$ spanning $K$ .",
"Note that if $k$ is $g$ -null, it is also $\\tilde{g}$ -null, where $&\\tilde{g}=fg+\\kappa \\odot \\xi ,&0<f\\in C^\\infty (\\mathcal {S}),\\ \\xi \\in \\Omega ^1(\\mathcal {S}),$ and $\\tilde{\\kappa }=k\\tilde{g}$ is a rescaling of $\\kappa $ .",
"Here again, the properties of the Lie derivative show that $\\tilde{g}$ satisfies (REF ) whenever $g$ does, so the class (REF ) of metrics associated to given SFNGC is manifestly larger than a conformal class of metrics.",
"For null $k$ , $g|_K=0$ and we see from (REF ) that a SFNGC determines a well-defined conformal structure on the subbundle (REF ) of the leaf space.",
"As such, we can define an almost-complex structure on $M $ , $&J:D\\rightarrow D,&J^2=-\\mathbb {1},$ by taking $J_{[x]}$ to be a rotation by $\\frac{\\pi }{2}$ in $D_{[x]}$ .",
"(There are two choices for the direction of the rotation – clockwise or counterclockwise – in each $D_{[x]}$ .",
"Take the one which is positively oriented for the orientation induced by the semi-Riemannian volume form on $\\mathcal {S}$ ; see Remark REF .)",
"Thus we see that a SFNGC induces a CR structure on the 3-dimensional leaf space $M $ , with $\\kappa $ serving as a characteristic form.",
"To this we may add a CR form $\\eta \\in \\Omega ^1(M ,\\mathbb {C})$ so that $\\kappa ,\\eta ,\\overline{\\eta }$ is a 0-adapted CR coframing.",
"CR integrability is automatic in dimension three, but pseudo-convexity is not.",
"In the Levi-flat case, $\\kappa \\wedge \\text{d}\\kappa =0$ and $M $ is foliated by complex curves; the original curves of our SFNGC are hypersurface-orthogonal, as one would expect from a spacetime featuring radiating wave fronts.",
"More interesting from a CR perspective is the Levi-nondegenerate case corresponding to “twisting\" congruences $\\kappa \\wedge \\text{d}\\kappa \\ne 0$ , including the Kerr spacetime which describes a rotating black hole.",
"Conversely, suppose that $M $ is a 3-dimensional CR manifold with a 0-adapted coframing $\\kappa ,\\eta ,\\overline{\\eta }$ , and set $\\mathcal {S}=\\mathbb {R}\\times M $ .",
"We use the same names $\\kappa ,\\eta ,\\overline{\\eta }$ to denote their pullbacks along the projection $\\mathcal {S}\\rightarrow M $ .",
"Take $k\\in \\Gamma (T\\mathcal {S})$ to be $k=\\frac{\\partial }{\\partial r}$ where $r$ is the Cartesian coordinate of $\\mathbb {R}$ , and choose any $\\rho \\in \\Omega ^1(\\mathcal {S})$ with $\\rho (k)=1$ ; i.e., $\\rho \\equiv \\text{d}r\\mod {\\lbrace }\\kappa ,\\eta ,\\overline{\\eta }\\rbrace $ .",
"The metric $g=\\kappa \\odot \\rho -\\eta \\odot \\overline{\\eta }$ has signature $(1,3)$ and satisfies $g(k,k)=0$ as well as $\\kappa =kg$ .",
"The flow curves of $k$ are the $r$ -coordinate curves of $\\mathcal {S}$ , and Lie derivatives along $k$ can be computed via H. Cartan's formula, yielding $&\\mathcal {L}_k\\kappa =\\mathcal {L}_k\\eta =\\mathcal {L}_k\\overline{\\eta }=0,&\\mathcal {L}_k\\rho \\equiv 0\\mod {\\lbrace }\\kappa ,\\eta ,\\overline{\\eta }\\rbrace ,$ whence both conditions (REF ) and (REF ) are verified.",
"This establishes a correspondence $\\lbrace \\text{SFNGC on 4-manifolds}\\rbrace \\stackrel{\\text{(local)}}{\\longleftrightarrow }\\lbrace \\text{CR structure on 3-manifolds}\\rbrace $ Now suppose we submit our coframing on $M $ to a 0-adapted transformation, $&\\left[\\begin{array}{c}\\kappa ^{\\prime }\\\\\\eta ^{\\prime }\\end{array}\\right]=\\left[\\begin{array}{cc}u&0\\\\b&a\\end{array}\\right]\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right];&u\\in C^\\infty (M ),\\ a,b\\in C^\\infty (M ,\\mathbb {C}),\\ u,a\\ne 0,$ and write the metric $\\tilde{g}$ as in (REF ).",
"In terms of our original coframing, we obtain $\\tilde{g}&=\\kappa ^{\\prime }\\odot \\rho -\\eta ^{\\prime }\\odot \\overline{\\eta }^{\\prime }\\\\&=|b|^2g+\\kappa \\odot ((u-|b|^2)\\rho -|b|^2\\kappa -a\\overline{b}\\eta -\\overline{a}b\\overline{\\eta })\\\\&=fg+\\kappa \\odot \\xi $ as in (REF ).",
"Following our initial selection of $\\rho $ , the ambiguity of the metric $g$ due to our choice (REF ) of coframing on $M $ is measured by 5 real functions of 3 variables, rather than the 5 functions of 4 variables apparent in the full class of metrics discussed in Remark REF .",
"However, if we allow the fiber coordinates $u,a,b$ of our G-structure (REF ) to vary with $r$ , then the structure group of our bundle of 0-adapted CR frames exactly parameterizes the class of metrics associated to this SFNGC (note that the Lie derivatives along $k$ of our CR coframing will no longer vanish identically as in (REF ) if $u,a,b$ depend on $r$ ; (REF ) and (REF ) will hold nonetheless).",
"The correspondence (REF ) raises several questions, the first of which is presented as Problem 1 in [15], and the second of which was communicated to the author by Paweł Nurowski: Which CR structures lift to a SFNGC whose class (REF ) of metrics contains a solution to Einstein's vacuum field equations?",
"The Goldberg-Sachs theorem says that there are two SFNGC associated to a metric of Petrov type D; are the two corresponding CR structures always equivalent?",
"Which CR structures lift to a SFNGC whose class (REF ) of metrics contains one that is (conformally) flat?",
"The Kerr Theorem offers the answer to the final question: those that are embedded within the real hyperquadric $\\mathcal {Q}$.",
"The present article attempts to answer the inevitable follow-up question: which are those?",
"Kerr Theorem Proof Sketch We argue as in [24].",
"Remark REF says that we can scale the vector field $k\\in \\Gamma (K)$ tangent to our SFNGC at will, so we are less occupied with null vectors tangent to Minkowski spacetime $\\mathbb {M}$ than we are with null directions.",
"The projectivized ${\\tt b}$ -null cone in $\\mathbb {R}^{1,3}$ is the (Riemann) 2-sphere $\\mathbb {CP}^1$ , hence a single stereographic coordinate $\\zeta \\in \\mathbb {C}$ suffices to parameterize all null tangent directions in each $T_x\\mathbb {M}$ , with the exception of one direction “at infinity.\"",
"In standard coordinates $(x_0,x_1,x_2,x_3)\\in \\mathbb {M}$ , the metric is diagonal, $g=\\text{d}x_0\\odot \\text{d}x_0-\\text{d}x_1\\odot \\text{d}x_1-\\text{d}x_2\\odot \\text{d}x_2-\\text{d}x_3\\odot \\text{d}x_3.$ Introducing null and complexified coordinates $&u=x_0-x_3,&v=x_0+x_3,&&w=x_1+\\mathbf {i}x_2,&&\\overline{w}=x_1-\\mathbf {i}x_2,$ brings $g$ into the form $g=\\text{d}u\\odot \\text{d}v-\\text{d}w\\odot \\text{d}\\overline{w}.$ A general null vector field and its dual form are, up to real scale, $&k=\\frac{\\partial }{\\partial v}-\\zeta \\frac{\\partial }{\\partial w}-\\overline{\\zeta }\\frac{\\partial }{\\partial \\overline{w}}+|\\zeta |^2\\frac{\\partial }{\\partial u},&\\kappa =\\text{d}u+\\zeta \\text{d}\\overline{w}+\\overline{\\zeta }(\\text{d}w+\\zeta \\text{d}v),&&\\zeta \\in C^\\infty (\\mathbb {M},\\mathbb {C}),$ while the SFNGC “at infinity\" is given by the $u$ -coordinate lines.",
"In the latter case, the remaining coordinates $v,w,\\overline{w}$ descend to the leaf-space of $u$ -coordinate lines, which is the Levi-flat $\\mathbb {R}\\times \\mathbb {C}$ , and this corresponds to a CR structure in $\\mathcal {Q}$ that is tangent to a complex curve.",
"In the general case we can write $&g=\\kappa \\odot \\text{d}v-\\eta \\odot \\overline{\\eta },&\\eta =\\text{d}w+\\zeta \\text{d}v,$ and after computing Lie derivatives, $&\\mathcal {L}_k\\kappa =\\text{d}\\zeta (k)\\text{d}\\overline{w}+\\text{d}\\overline{\\zeta }(k)\\text{d}w+(\\overline{\\zeta }\\text{d}\\zeta (k)+\\zeta \\text{d}\\overline{\\zeta }(k))\\text{d}v,&\\mathcal {L}_k\\eta =\\overline{\\mathcal {L}_k\\overline{\\eta }}=-\\text{d}\\zeta +\\text{d}\\zeta (k)\\text{d}v,$ we see that conditions (REF ) and (REF ) hold when $0&=\\text{d}\\zeta (k)=\\text{d}\\overline{\\zeta }(k),\\\\0&=\\kappa \\wedge \\eta \\wedge \\mathcal {L}_k\\eta ,$ where the second becomes equivalent to $\\text{d}(u+\\zeta \\overline{w})\\wedge \\text{d}(w+\\zeta v)\\wedge \\text{d}\\zeta =0.$ Name the three $\\mathbb {C}$ -valued functions $&z_1=u+\\zeta \\overline{w},&z_2=w+\\zeta v,&&z_3=\\zeta ,$ and observe that $\\mathbf {i}(z_1-\\overline{z}_1+z_2\\overline{z}_3-z_3\\overline{z}_2)=0.$ If $Z_0,Z_1,Z_2,Z_3$ are coordinates for $\\mathbb {C}^{2,2}$ with the Hermitian form $Z,W)=\\mathbf {i}(Z_1\\overline{W}_0-Z_0\\overline{W}_1+Z_2\\overline{W}_3-Z_3\\overline{W}_2),$ then (REF ) describes the projectivization in $\\mathbb {CP}^3$ of the $-null cone $ Z,Z)=0$ in the affine coordinate neighborhood $ Z00$, via projective coordinates $ [Z0:Z1:Z3:Z4]=[1:z1:z2:z3]$.",
"Moreover, if $ =z3$ is implicitly defined by $ H(z1,z2,z3)=0$, where $ H$ is holomorphic (and not constant) in $ z1,z2,z3$, then the 3-form $ dz1dz2dz3$ vanishes on the subbundle $ dH=0$ of the complexified tangent bundle of $ CP3$, and over the hyperquadric $ Q$ locally defined by (\\ref {Qlocalhypersurfaceeqn}), this is exactly the shear-free condition (\\ref {minksf}).",
"The level set $ H=0$ is a complex-analytic surface in $ CP3$ whose intersection with the real hyperquadric $ Q$ defines a 3-dimensional CR submanifold of $ Q$.$ For the remaining details, consult [17], [24], or [18].",
"Moving Frames Over Embedded 3-folds Hermitian Frames of $\\mathbb {C}^4$ Let $\\underline{e}=(e_0,e_1,e_2,e_3)$ denote the standard basis of column vectors for $\\mathbb {C}^4$ and recall that $\\mathbf {e}$ is the natural exponential and $\\mathbf {i}=\\sqrt{-1}$ .",
"Fix index ranges and constants $&0\\le i,j\\le 3,&\\epsilon =\\pm 1,&&\\delta _\\epsilon =\\left\\lbrace {\\begin{matrix}0,&\\epsilon =1\\\\1,&\\epsilon =-1\\end{matrix}}\\right.&&\\Rightarrow \\epsilon =(-1)^{\\delta _\\epsilon }.$ The Hermitian form $ of signature $ (3-,1+)$ acts on vectors $z=ziei$ and $w=wjej$ via{\\begin{@align*}{1}{-1}{\\tt w},{\\tt z})&=\\mathbf {i}(\\overline{w}^0z^3-\\overline{w}^3z^0)+\\overline{w}^1z^1+\\epsilon \\overline{w}^2z^2.\\end{@align*}}A \\emph {Hermitian frame} is an ordered, complex basis $v=(v0,v1,v2,v3)$ of $ C4$ such that{\\begin{@align}{1}{-1}{\\tt v}_i,{\\tt v}_j)=(\\pm 1)^{\\delta _\\epsilon }\\left[\\begin{array}{cccc}0&0&0&\\mathbf {i}\\\\0&1&0&0\\\\0&0&\\epsilon &0\\\\-\\mathbf {i}&0&0&0\\end{array}\\right].\\end{@align}}Denote by $ H$ the collection of all Hermitian frames, and note that $ H$ is identified with the unitary group $ U(3-,1+)$ by fixing $e$ as the identity and taking $v$ to be the matrix whose column vectors are the basis vectors of $v$.$ Remark 4.1 The notation $(\\pm 1)^{\\delta _\\epsilon }$ in () means the sign is allowed to change when $\\epsilon =-1$ but not when $\\epsilon =1$ .",
"This is to avoid privileging either of ${\\tt v}_1,{\\tt v}_2$ as necessarily positive-definite when $\\epsilon =-1$ .",
"Both vectors are called “orthonormal\" regardless of the value of $\\epsilon $ .",
"The symbol ${\\tt v}_i$ will also refer to the $\\mathbb {C}^4$ -valued function mapping a Hermitian frame to its $i^{\\text{th}}$ basis vector: ${\\tt v}_i\\in C^\\infty (\\hat{\\mathcal {H}},\\mathbb {C}^4)$ .",
"These functions are differentiated via the Maurer-Cartan (MC) forms of $U(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ , $&\\text{d}{\\tt v}_i=\\mu ^j_i{\\tt v}_j,&\\mu \\in \\Omega ^1(\\hat{\\mathcal {H}},\\mathfrak {u}(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )).$ In our representation of this Lie algebra, we can write $\\mu =\\left[\\begin{array}{cccc}\\lambda &-\\mathbf {i}\\overline{\\xi }&-\\mathbf {i}\\overline{\\phi }_2&\\psi \\\\\\eta &\\mathbf {i}\\rho &-\\overline{\\phi }_1&\\xi \\\\\\zeta &\\epsilon \\phi _1&\\mathbf {i}\\tau &\\epsilon \\phi _2\\\\\\kappa &\\mathbf {i}\\overline{\\eta }&\\epsilon \\mathbf {i}\\overline{\\zeta }&-\\overline{\\lambda }\\end{array}\\right],$ where $\\kappa ,\\psi \\in \\Omega ^1(\\hat{\\mathcal {H}})$ and the rest are $\\mathbb {C}$ -valued, so that $\\overline{\\mu }^t=0$ .",
"The MC equations $\\text{d}\\mu =-\\mu \\wedge \\mu $ are $\\begin{aligned}\\text{d}\\kappa &=\\mathbf {i}\\eta \\wedge \\overline{\\eta }+(\\lambda +\\overline{\\lambda })\\wedge \\kappa +\\epsilon \\mathbf {i}\\zeta \\wedge \\overline{\\zeta },\\\\\\text{d}\\eta &=(\\lambda -\\mathbf {i}\\rho )\\wedge \\eta -\\xi \\wedge \\kappa +\\overline{\\phi }_1\\wedge \\zeta ,\\\\\\text{d}\\psi &=\\psi \\wedge (\\lambda +\\overline{\\lambda })-\\mathbf {i}\\xi \\wedge \\overline{\\xi }-\\epsilon \\mathbf {i}\\phi _2\\wedge \\overline{\\phi }_2,\\\\\\text{d}\\xi &=\\psi \\wedge \\eta +\\xi \\wedge (\\overline{\\lambda }+\\mathbf {i}\\rho )+\\epsilon \\overline{\\phi }_1\\wedge \\phi _2,\\\\\\text{d}\\lambda &=\\mathbf {i}\\overline{\\xi }\\wedge \\eta +\\mathbf {i}\\overline{\\phi }_2\\wedge \\zeta -\\psi \\wedge \\kappa ,\\\\\\text{d}\\rho &=\\epsilon \\mathbf {i}\\phi _1\\wedge \\overline{\\phi }_1-\\overline{\\xi }\\wedge \\eta -\\xi \\wedge \\overline{\\eta },\\\\\\text{d}\\phi _1&=\\epsilon \\mathbf {i}\\zeta \\wedge \\overline{\\xi }+\\mathbf {i}(\\rho -\\tau )\\wedge \\phi _1-\\mathbf {i}\\phi _2\\wedge \\overline{\\eta },\\\\\\text{d}\\phi _2&=\\epsilon \\psi \\wedge \\zeta +\\phi _2\\wedge (\\overline{\\lambda }+\\mathbf {i}\\tau )+\\xi \\wedge \\phi _1,\\\\\\text{d}\\zeta &=(\\lambda -\\mathbf {i}\\tau )\\wedge \\zeta -\\epsilon \\phi _1\\wedge \\eta -\\epsilon \\phi _2\\wedge \\kappa ,\\\\\\text{d}\\tau &=\\zeta \\wedge \\overline{\\phi }_2+\\overline{\\zeta }\\wedge \\phi _2-\\epsilon \\mathbf {i}\\phi _1\\wedge \\overline{\\phi }_1.\\end{aligned}$ Define $\\det :\\hat{\\mathcal {H}}\\rightarrow U(1)\\subset \\mathbb {C}$ as usual by ${\\tt v}_0\\wedge {\\tt v}_1\\wedge {\\tt v}_2\\wedge {\\tt v}_3=\\det (\\underline{{\\tt v}})e_0\\wedge e_1\\wedge e_2\\wedge e_3,$ and let $\\mathcal {H}\\subset \\hat{\\mathcal {H}}$ denote the collection of oriented Hermitian frames satisfying $\\det (\\underline{{\\tt v}})=1$ .",
"From any Hermitian frame $\\underline{{\\tt v}}$ one obtains an oriented Hermitian frame in a variety of ways; e.g., $\\underline{{\\tt v}}\\mapsto ({\\tt v}_0,{\\tt v}_1,\\overline{\\det (\\underline{{\\tt v}})}{\\tt v}_2,{\\tt v}_3),$ in this case preserving the vectors ${\\tt v}_0,{\\tt v}_1,{\\tt v}_3$ .",
"$\\mathcal {H}$ is identified with the special unitary group $SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ .",
"Keeping the same names after pulling back the MC forms (REF ) along the inclusion $\\mathcal {H}\\hookrightarrow \\hat{\\mathcal {H}}$ exhibits the trace-free condition of the special unitary Lie algebra $\\mathfrak {su}(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ , $\\mathbf {i}\\rho +\\mathbf {i}\\tau +\\lambda -\\overline{\\lambda }=0.$ If two Hermitian frames $\\underline{{\\tt v}}$ and $\\tilde{\\underline{{\\tt v}}}$ share the same ${\\tt v}_0=\\tilde{{\\tt v}}_0$ , then they differ by a transformation $\\begin{aligned}&\\left[\\begin{array}{cccc}{\\tt v}_0&\\tilde{{\\tt v}}_1&\\tilde{{\\tt v}}_2&\\tilde{{\\tt v}}_3\\end{array}\\right]=\\left[\\begin{array}{cccc}{\\tt v}_0&{\\tt v}_1&{\\tt v}_2&{\\tt v}_3\\end{array}\\right]p_0,\\\\&p_0=\\left[\\begin{array}{cccr}1&-\\mathbf {i}(a_1\\overline{c}_1+\\epsilon a_2\\overline{c}_2)&\\epsilon \\mathbf {i}\\mathbf {e}^{\\mathbf {i}r}(\\overline{a}_2\\overline{c}_1-\\overline{a}_1\\overline{c}_2)&c_0(\\pm 1)^{\\delta _\\epsilon } \\\\0&a_1&-\\epsilon \\mathbf {e}^{\\mathbf {i}r}\\overline{a}_2&c_1(\\pm 1)^{\\delta _\\epsilon }\\\\0&a_2&\\mathbf {e}^{\\mathbf {i}r}\\overline{a}_1&c_2(\\pm 1)^{\\delta _\\epsilon }\\\\0&0&0&(\\pm 1)^{\\delta _\\epsilon }\\end{array}\\right]\\begin{array}{c}a_1,a_2,c_1,c_2\\in \\mathbb {C},\\\\|a_1|^2+\\epsilon |a_2|^2=(\\pm 1)^{\\delta _\\epsilon },\\\\c_0=t-\\tfrac{\\mathbf {i}}{2}(|c_1|^2+\\epsilon |c_2|^2),\\\\r,t\\in \\mathbb {R}.\\end{array}\\end{aligned}$ (See Remark REF regarding the sign ambiguity $(\\pm 1)^{\\delta _\\epsilon }$ when $\\epsilon =-1$ .)",
"All such $p_0\\in U(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ form a subgroup that we call $\\mathcal {P}_0$ .",
"Apparently, $\\det (p_0)=\\mathbf {e}^{\\mathbf {i}r}$ .",
"More generally, if the first basis vectors of two Hermitian frames $\\underline{{\\tt v}}$ and $\\tilde{\\underline{{\\tt v}}}$ are $\\mathbb {C}$ -collinear – i.e., $\\tilde{{\\tt v}}_0=l{\\tt v}_0$ for some $l\\in \\mathbb {C}\\setminus \\lbrace 0\\rbrace $ – then $&\\tilde{\\underline{{\\tt v}}}=\\underline{{\\tt v}} p_lp_0;&p_l=\\left[\\begin{array}{cccc}l&0&0&0\\\\0&1&0&0\\\\0&0&1&0\\\\0&0&0&\\overline{l}^{-1}\\end{array}\\right],\\quad l\\in \\mathbb {C}\\setminus \\lbrace 0\\rbrace ,&&p_0 \\text{ as in (\\ref {P0}).",
"}$ All such $p_l\\in U(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ form an abelian subgroup that we call $\\mathcal {P}_l$ .",
"We also name $&\\hat{\\mathcal {P}}=\\mathcal {P}_l\\mathcal {P}_0,&\\mathcal {P}=\\hat{\\mathcal {P}}\\cap SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon );$ the latter consists of multiples $p_lp_0$ with the $r$ -coordinate of (REF ) constrained by $\\mathbf {e}^{\\mathbf {i}r}=\\overline{l}l^{-1}$ , and it is exactly the parabolic subgroup $\\mathcal {P}\\subset SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ mentioned in the Tanaka-Chern-Moser Classification for CR-dimension $n=2$ .",
"We conclude this section by recording the dual version of the $\\hat{\\mathcal {P}}$ -transformation of Hermitian frames; namely, if we use superscripts to denote the dual coframe of $\\underline{{\\tt v}}$ (i.e., ${\\tt v}^i({\\tt v}_j)=\\delta ^i_j$ ) then the dual coframes of two Hermitian frames $\\tilde{\\underline{{\\tt v}}}$ and $\\underline{{\\tt v}}$ with $\\tilde{{\\tt v}}_0=l{\\tt v}_0$ are related by $&\\left[\\begin{array}{c}\\tilde{{\\tt v}}^0\\\\\\tilde{{\\tt v}}^1\\\\\\tilde{{\\tt v}}^2\\\\\\tilde{{\\tt v}}^3\\end{array}\\right]={p_0}^{-1}{p_l}^{-1}\\left[\\begin{array}{c}{\\tt v}^0\\\\{\\tt v}^1\\\\{\\tt v}^2\\\\{\\tt v}^3\\end{array}\\right];&p_lp_0\\in \\hat{\\mathcal {P}}.$ First Adaptations Let $\\mathcal {N}\\subset \\mathbb {C}^4$ be the null-cone of $,{\\begin{@align*}{1}{-1}&\\mathcal {N}=\\lbrace {\\tt v}\\in \\mathbb {C}^4\\setminus \\lbrace \\left[{\\begin{matrix}0&0&0&0\\end{matrix}}\\right]^t\\rbrace :{\\tt v},{\\tt v})=0\\rbrace &\\Longrightarrow &&T_{{\\tt v}}\\mathcal {N}=\\lbrace {\\tt w}\\in \\mathbb {C}^4:\\Re ({\\tt v},{\\tt w}))=0\\rbrace .\\end{@align*}}We recognize the following distinguished subbundles of $ TN$ by their fibers,{\\begin{@align*}{1}{-1}&L_{{\\tt v}}=\\lbrace l{\\tt v}:l\\in \\mathbb {C}\\rbrace ,&\\subset &&L_{{\\tt v}}^\\bot =\\lbrace {\\tt w}\\in \\mathbb {C}^4:{\\tt v},{\\tt w})=0\\rbrace .\\end{@align*}}Define the projection{\\begin{@align*}{1}{-1}h_\\mathcal {N}:\\hat{\\mathcal {H}}&\\rightarrow \\mathcal {N}\\\\\\underline{{\\tt v}}&\\mapsto {\\tt v}_0,\\end{@align*}}which identifies $ N$ with the homogeneous space $ U(3-,1+)/P0$.",
"For each $v=(v0,v1,v2,v3)hN-1(v0)$, we have spanning sets{\\begin{@align}{1}{-1}&\\langle {\\tt v}_0\\rangle _\\mathbb {C}=L_{{\\tt v}_0},&\\langle {\\tt v}_0,{\\tt v}_1,{\\tt v}_2\\rangle _\\mathbb {C}=L_{{\\tt v}_0}^\\bot ,&&\\langle {\\tt v}_0,{\\tt v}_1,{\\tt v}_2,\\mathbf {i}{\\tt v}_0,\\mathbf {i}{\\tt v}_1,\\mathbf {i}{\\tt v}_2,{\\tt v}_3\\rangle _\\mathbb {R}=T_{{\\tt v}_0}\\mathcal {N}.\\end{@align}}Conversely, one can assign an adapted basis of $ Tv0N$ to each $v0N$ in order to define a section $ s:NH$ as follows: take $v0$ itself to span $ Lv0$, choose two orthonormal vectors $v1,v2Lv0$ (varying smoothly in a neighborhood of $v0$), and then $v3$ is uniquely determined to complete the Hermitian frame $ s(v0)=(v0,v1,v2,v3)$.$ Remark 4.2 Remark REF explains why the $(\\pm 1)^{\\delta _\\epsilon }$ ambiguity in () and (REF ) allows us to choose any non-null ${\\tt v}_1\\in L_{{\\tt v}_0}^\\bot $ after ${\\tt v}_0$ is fixed.",
"The order of the basis elements in a frame reflects the ascending filtration $L\\subset L^\\bot \\subset T\\mathcal {N}$ , but of course it is possible to first choose ${\\tt v}_3$ with ${\\tt v}_0,{\\tt v}_3)=\\mathbf {i}(\\pm 1)^{\\delta _\\epsilon }$ and then take orthonormal ${\\tt v}_1,{\\tt v}_2$ in the orthogonal complement of $L_{{\\tt v}_0}\\oplus \\langle {\\tt v}_3\\rangle _\\mathbb {R}$ .",
"With a section $s:\\mathcal {N}\\rightarrow \\hat{\\mathcal {H}}$ , we pull back $\\text{d}{\\tt v}_0\\in \\Omega ^1(\\hat{\\mathcal {H}},\\mathbb {C}^4)$ from (REF ) to get $s^*\\text{d}{\\tt v}_0=s^*\\lambda {\\tt v}_0+s^*\\eta {\\tt v}_1+s^*\\zeta {\\tt v}_2+s^*\\kappa {\\tt v}_3,$ but $s^*{\\tt v}_0$ is just the identity map on $\\mathcal {N}$ , so its differential is the identity on $T\\mathcal {N}$ .",
"Thus, $&s^*\\lambda ={\\tt v}^0,&s^*\\eta ={\\tt v}^1,&&s^*\\zeta ={\\tt v}^2,&&s^*\\kappa ={\\tt v}^3.$ Citing (REF ) with $l=1$ , we can say in other words that $\\kappa \\in \\Omega ^1(\\hat{\\mathcal {H}})$ and $\\eta ,\\zeta ,\\lambda \\in \\Omega ^1(\\hat{\\mathcal {H}},\\mathbb {C})$ are semi-basic, tautological 1-forms for the (co)frame bundle fibration $\\mathcal {P}_0\\hookrightarrow U(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )\\stackrel{h_\\mathcal {N}}{\\longrightarrow }\\mathcal {N}.$ Write $\\mathbb {P}:\\mathbb {C}^4\\setminus \\lbrace \\left[{\\begin{matrix}0&0&0&0\\end{matrix}}\\right]^t\\rbrace \\rightarrow \\mathbb {CP}^3$ for the canonical complex projection, so the real hyperquadric is the image $\\mathcal {Q}=\\mathbb {P}(\\mathcal {N})$ , and $T_{\\mathbb {P}({\\tt v}_0)}\\mathcal {Q}=\\mathbb {P}_*(T_{{\\tt v}_0}\\mathcal {N})\\cong L_{{\\tt v}_0}^*\\otimes T_{{\\tt v}_0}\\mathcal {N}/L_{{\\tt v}_0}$ for ${\\tt v}_0\\in \\mathcal {N}$ .",
"The latter can be made explicit for a frame $\\underline{{\\tt v}}\\in h_\\mathcal {N}^{-1}({\\tt v}_0)$ , namely $T_{\\mathbb {P}({\\tt v}_0)}\\mathcal {Q}=\\langle \\mathbb {P}_*{\\tt v}_1,\\mathbb {P}_*{\\tt v}_2\\rangle _\\mathbb {C}\\oplus \\langle \\mathbb {P}_*{\\tt v}_3\\rangle _\\mathbb {R}\\cong \\langle {\\tt v}^0\\otimes {\\tt v}_1,{\\tt v}^0\\otimes {\\tt v}_2\\rangle _\\mathbb {C}\\oplus \\langle {\\tt v}^0\\otimes {\\tt v}_3\\rangle _\\mathbb {R}.$ Remark 4.3 By (REF ) and (REF ) we see that for $\\tilde{\\underline{{\\tt v}}}=\\underline{{\\tt v}}p_l\\in h_\\mathcal {N}^{-1}(\\mathbb {P}^{-1}({\\tt v}_0))$ , $\\tilde{{\\tt v}}^0\\otimes \\tilde{{\\tt v}}_3=|l|^{-2}{\\tt v}^0\\otimes {\\tt v}_3$ .",
"When $\\epsilon =-1$ , a $p_0$ -transformation (REF ) with $(\\pm 1)^{\\delta _\\epsilon }=-1$ changes the sign of ${\\tt v}^0\\otimes {\\tt v}_3$ (see Remark REF ).",
"It's clear from () that the fibers of $T\\mathcal {N}\\rightarrow \\mathcal {N}$ vary along those of $\\mathcal {N}\\rightarrow \\mathcal {Q}$ .",
"Nonetheless, Remark REF reassures us that (REF ) is well-defined.",
"Moreover, it always holds that $L_{{\\tt v}_0}=L_{\\tilde{{\\tt v}}_0}$ and $L_{{\\tt v}_0}^\\bot =L_{\\tilde{{\\tt v}}_0}^\\bot $ when $\\mathbb {P}({\\tt v}_0)=\\mathbb {P}(\\tilde{{\\tt v}}_0)$ .",
"In particular, for any ${\\tt v}_0\\in \\mathcal {N}$ , $L_{{\\tt v}_0}=\\ker \\mathbb {P}_*|_{T_{{\\tt v}_0}\\mathcal {N}}$ , and there is a well-defined, $\\mathbb {R}$ -corank-1 subbundle $D_{\\mathbb {P}({\\tt v}_0)}=\\mathbb {P}_*L_{{\\tt v}_0}^\\bot \\subset T_{\\mathbb {P}({\\tt v}_0)}\\mathcal {Q}.$ Scalar multiplication by $\\mathbf {i}$ in $\\mathbb {C}^4$ defines an endomorphism $J:L_{{\\tt v}_0}^\\bot \\rightarrow L_{{\\tt v}_0}^\\bot $ satisfying $J^2=-\\mathbb {1}$ , of which $L_{{\\tt v}_0}$ is an invariant subspace, hence $J$ descends to a well-defined almost-complex structure $J:D\\rightarrow D$ .",
"The Tanaka-Chern-Moser Classification for $n=2$ ensures that $\\mathcal {H}$ realizes the Cartan bundle of the CR structure $(\\mathcal {Q},D,J)$ , as encoded in the fibration $&\\mathcal {P}\\hookrightarrow SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )\\stackrel{h_\\mathcal {Q}}{\\longrightarrow }\\mathcal {Q};&h_\\mathcal {Q}=\\mathbb {P}\\circ h_\\mathcal {N}.$ It will be convenient to work instead with $\\hat{\\mathcal {P}}\\rightarrow \\hat{\\mathcal {H}}\\stackrel{h_\\mathcal {Q}}{\\longrightarrow }\\mathcal {Q}$ in order to factor through (REF ): $\\begin{tikzcd}\\mathcal {P}_0 [hook]{r}{\\underline{{\\tt v}}\\mapsto \\underline{{\\tt v}}p_0}& U(3-\\delta _\\epsilon ,1+\\delta _\\epsilon ){d}{h_\\mathcal {N}}\\\\\\mathcal {P}_l\\cong \\mathbb {C}\\setminus \\lbrace 0\\rbrace {r}{{\\tt v}_0\\mapsto l{\\tt v}_0}&\\mathcal {N}{d}{\\mathbb {P}}\\\\&\\mathcal {Q}\\end{tikzcd}$ To this end, let $\\varsigma :\\mathcal {Q}\\rightarrow \\hat{\\mathcal {H}}$ be a section that factors through $s:\\mathcal {N}\\rightarrow \\hat{\\mathcal {H}}$ as in (REF ).",
"Then $\\varsigma ^*\\kappa \\in \\Omega ^1(\\mathcal {Q})$ is a characteristic form annihilating (REF ), which along with $\\varsigma ^*\\eta ,\\varsigma ^*\\zeta \\in \\Omega ^1(\\mathcal {Q},\\mathbb {C})$ and their conjugates furnishes a 1-adapted coframing whose Levi form (REF ) is represented $\\ell =\\left[\\begin{array}{cc}1&0\\\\0&\\epsilon \\end{array}\\right]$ in the basis (REF ) given by $\\underline{{\\tt v}}=\\varsigma (\\mathbb {P}({\\tt v}_0))$ .",
"Remark REF explains why the representation (REF ) of $\\ell $ does not depend on the choice of $(\\pm 1)^{\\delta _\\epsilon }$ in (), (REF ) when $\\epsilon =-1$ .",
"If $M$ is a 3-dimensional manifold CR-embedded in $\\mathcal {Q}$ , then there is a rank-2, $J$ -invariant subbundle $D_M\\subset D$ tangent to $M$ .",
"Let $\\hat{M}=\\mathbb {P}^{-1}(M)\\subset \\mathcal {N}$ be the cone over $M$ and $L_M\\subset T\\hat{M}$ be $\\mathbb {P}_*^{-1}(D_M)$ .",
"Name the restriction and projections $&\\mathcal {H}^0=h_{\\mathcal {Q}}^{-1}(M)\\subset \\hat{\\mathcal {H}};&h_{\\hat{M}}=h_\\mathcal {N}|_{\\mathcal {H}^0},&&h_M=h_\\mathcal {Q}|_{\\mathcal {H}^0}.$ The Hermitian frames $\\mathcal {H}^0$ are “zero-adapted to $M$ \" and the fibers of $h_M$ have (real) dimension $\\dim \\hat{\\mathcal {P}}=11$ – see (REF ).",
"We will exploit the degrees of freedom (REF ) and (REF ) in the structure group $\\hat{\\mathcal {P}}$ of $\\mathcal {H}^0$ to reduce to subbundles of frames which are increasingly adapted to the CR geometry of $M$ .",
"Since $\\mathcal {Q}$ is Levi-nondegenerate, $D\\subset T\\mathcal {Q}$ is a rank-4 contact distribution, so there cannot be a 3-dimensional submanifold of $\\mathcal {Q}$ which is tangent to $D$ .",
"That is to say $\\varsigma ^*\\kappa $ cannot vanish identically on $TM$ or, equivalently, $s^*\\kappa |_{T\\hat{M}}\\ne 0$ .",
"Therefore, we restrict to those Hermitian frames $\\underline{{\\tt v}}\\in h_{\\hat{M}}^{-1}({\\tt v}_0) \\quad \\text{ with } {\\tt v}_3\\in T_{{\\tt v}_0}\\hat{M}.$ Further adaptation branches based on the (non)vanishing of (REF ) on $D_M$ .",
"If $M$ is Levi-nondegenerate, $\\ell |_{D_M}\\ne 0$ and we consider $\\underline{{\\tt v}}\\in \\mathcal {H}^0$ with $L_M=\\langle {\\tt v}_0,{\\tt v}_1\\rangle _\\mathbb {C}$ , which along with (REF ) reduces $\\mathcal {H}^0$ and its structure group over $\\hat{M}$ : $&\\mathcal {H}^1=\\left\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^0: \\begin{array}{c}L_M=\\langle {\\tt v}_0,{\\tt v}_1\\rangle _\\mathbb {C}\\\\ T\\hat{M}=L_M\\oplus \\langle {\\tt v}_3\\rangle _\\mathbb {R}\\end{array}\\right\\rbrace ,&\\mathcal {P}_1=\\left\\lbrace p_0\\in \\mathcal {P}_0 \\text{ as in } (\\ref {P0}) : \\begin{array}{r}a_2=c_2=0\\\\ (\\pm 1)^{\\delta _\\epsilon }=1\\end{array}\\right\\rbrace .$ When $\\epsilon =-1$ it is also possible that $\\ell _{D_M}=0$ .",
"Such $M$ is Levi-flat and contains a complex curve tangent to the $\\mathbb {P}_*$ image of an $-null distribution of complex rank two in $ TM$.",
"In this case we arrange for $ LM=v0,v1+v2C$ combined with (\\ref {TMv3}) to obtain{\\begin{@align}{1}{-1}&\\mathcal {H}^1=\\left\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^0: \\begin{array}{c}L_M=\\langle {\\tt v}_0,{\\tt v}_1+{\\tt v}_2\\rangle _\\mathbb {C}\\\\ T\\hat{M}=L_M\\oplus \\langle {\\tt v}_3\\rangle _\\mathbb {R}\\end{array}\\right\\rbrace &\\mathcal {P}_1=\\left\\lbrace p_0\\in \\mathcal {P}_0 \\text{ as in } (\\ref {P0}) :\\begin{array}{c}\\epsilon =-1, (\\pm 1)^{\\delta _{-1}}=1\\\\a_1+a_2=\\mathbf {e}^{\\mathbf {i}r}(\\overline{a}_2+\\overline{a}_1)\\\\ c_2=c_1\\end{array}\\right\\rbrace .\\end{@align}}In every case, we can say $v,vhM-1(v0)H1 p1P1 such that v=vp1$, so that the fibers of $ hM|H1$ have real dimension $ PlP1=7$.",
"Furthermore, our adaptations in both cases are undisturbed by the projection (\\ref {orient}), so we may descend as in (\\ref {CBQ}) to oriented frames over $ M${\\begin{@align}{1}{-1}&\\mathcal {P}_2\\rightarrow \\mathcal {H}^2\\stackrel{h_M}{\\longrightarrow }M,&\\mathcal {P}_2=\\mathcal {P}_l\\mathcal {P}_1\\cap \\mathcal {P},&&\\mathcal {H}^2=\\mathcal {H}^1\\cap \\mathcal {H}.\\end{@align}}$ Second Fundamental Form $(\\mathbf {II})$ All higher-order adaptations of moving frames over $M\\subset \\mathcal {Q}$ will be controlled by the MC equations (REF ).",
"Specifically, we pull back the MC form $\\mu $ as in (REF ) along the inclusion $\\mathcal {H}^1\\hookrightarrow \\mathcal {H}^0$ and explore the differential consequences of the algebraic relations imposed on the individual 1-forms (REF ) over $\\mathcal {H}^1$ .",
"Obviously we must separately consider 3-folds $M$ which are Levi-nondegenerate (REF ) or Levi-flat (), but in both cases the reduction () will force the relation (REF ) (our notation suppresses the pullback along the inclusion $\\mathcal {H}^2\\hookrightarrow \\hat{\\mathcal {H}}$ and keeps the same names for the MC forms with every such reduction).",
"Moreover, in both cases the method of moving frames exploits the remaining degrees of freedom in $\\mathcal {P}_2$ to normalize a symmetric, $\\mathbb {C}$ -bilinear tensor $\\mathbf {II}:{\\bigodot }^2(T\\mathcal {H}^2/T\\mathcal {P}_2)\\rightarrow \\mathbb {C},$ which we dub the Second Fundamental Form by analogy to the study of hypersurfaces in Euclidean space.",
"(One might consider the the Levi form $\\ell $ of $\\mathcal {Q}$ restricted to $D_M$ to be a first fundamental form of sorts, but for $\\dim M=3$ this is simply a real function whose only import is its non-vanishing.)",
"Levi-Nondegenerate 3-folds Our adaptation (REF ) together with (REF ) implies that $\\zeta =0 \\text{ on }\\mathcal {H}^2.$ Applying Cartan's Lemma to the equation (REF ) for $\\text{d}\\zeta $ yields $&\\left[\\begin{array}{c}\\phi _1\\\\\\phi _2\\end{array}\\right]=\\left[\\begin{array}{cc}a&b\\\\b&c\\end{array}\\right]\\left[\\begin{array}{c}\\eta \\\\\\kappa \\end{array}\\right]&\\text{ for some } a,b,c\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C}).$ Remark 4.4 It behooves us to remember that $\\mathcal {P}_2$ () is in part defined by $a_2=0$ (which forces $|a_1|=1$ ) in (REF ), as well as $\\mathbf {e}^{\\mathbf {i}r}=\\overline{l}l^{-1}$ in (REF ), (REF ).",
"Consequently, there is no crime in recycling notation by setting $a_1=\\mathbf {e}^{\\mathbf {i}r}$ in (REF ) so that $\\mathbf {i}\\rho $ in (REF ) measures the infinitesimal generator of this $U(1)$ -action on $\\mathcal {H}^2$ .",
"Differentiating (REF ) using $\\text{d}\\phi _1,\\text{d}\\phi _2$ from (REF ) reveals $\\begin{aligned}\\text{d}\\left[\\begin{array}{c}a\\\\b\\\\c\\end{array}\\right]=\\left[\\begin{array}{ccc}3\\mathbf {i}\\rho -\\overline{\\lambda }&0&0\\\\\\xi &2\\mathbf {i}\\rho -2\\overline{\\lambda }&0\\\\0&2\\xi &\\mathbf {i}\\rho -3\\overline{\\lambda }\\end{array}\\right]\\left[\\begin{array}{c}a\\\\b\\\\c\\end{array}\\right]+\\left[\\begin{array}{ccc}u_1&u_2&2\\mathbf {i}b\\\\u_2&u_3&\\mathbf {i}c\\\\u_3&u_4&0\\end{array}\\right]\\left[\\begin{array}{c}\\eta \\\\\\kappa \\\\\\overline{\\eta }\\end{array}\\right]\\\\\\text{ for some } u_1,u_2,u_3,u_4\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C}).\\end{aligned}$ The identities (REF ) imply that if one of $a,b,c$ vanishes identically on $\\mathcal {H}^2$ , they all vanish identically.",
"Furthermore, $\\text{d}(ac-b^2)\\equiv 4(ac-b^2)(\\mathbf {i}\\rho -\\overline{\\lambda })\\mod {\\lbrace }\\kappa ,\\eta \\rbrace ,$ so the determinant of the matrix (REF ) is either identically zero or non-vanishing on each fiber of $\\mathcal {H}^2\\rightarrow M$ .",
"Definition 4.5 The second fundamental form (REF ) of a Levi-nondegenerate 3-fold $M$ CR embedded in $\\mathcal {Q}$ is given by $\\mathbf {II}=a\\eta \\odot \\eta +2b\\eta \\odot \\kappa +c\\kappa \\odot \\kappa ,$ where the coefficients are derived from (REF ) via (REF ).",
"The condition that $\\mathbf {II}$ is of (sub)maximal rank on a fiber of $\\mathcal {H}^2\\rightarrow M$ is invariant under the action of CR symmetry group $SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ on $\\mathcal {Q}$ .",
"We update the remaining MC equations over $\\mathcal {H}^2$ , $\\begin{aligned}\\text{d}\\kappa &=\\mathbf {i}\\eta \\wedge \\overline{\\eta }+(\\lambda +\\overline{\\lambda })\\wedge \\kappa ,\\\\\\text{d}\\eta &=(\\lambda -\\mathbf {i}\\rho )\\wedge \\eta -\\xi \\wedge \\kappa ,\\\\\\text{d}\\psi &=\\psi \\wedge (\\lambda +\\overline{\\lambda })-\\mathbf {i}\\xi \\wedge \\overline{\\xi }+\\epsilon \\mathbf {i}\\kappa \\wedge (b\\overline{c}\\eta -\\overline{b}c\\overline{\\eta })-\\epsilon \\mathbf {i}|b|^2\\eta \\wedge \\overline{\\eta },\\\\\\text{d}\\xi &=\\psi \\wedge \\eta +\\xi \\wedge (\\overline{\\lambda }+\\mathbf {i}\\rho )+\\epsilon \\kappa \\wedge (|b|^2\\eta -ac\\overline{\\eta })-\\epsilon \\overline{a}b\\eta \\wedge \\overline{\\eta },\\\\\\text{d}\\lambda &=\\mathbf {i}\\overline{\\xi }\\wedge \\eta -\\psi \\wedge \\kappa ,\\\\\\text{d}\\rho &=-\\overline{\\xi }\\wedge \\eta -\\xi \\wedge \\overline{\\eta }-\\epsilon \\mathbf {i}\\kappa \\wedge (a\\overline{b}\\eta +\\overline{a}b\\overline{\\eta })+\\epsilon \\mathbf {i}|a|^2\\eta \\wedge \\overline{\\eta }.\\end{aligned}$ Recall from Remark REF that a CR embedding $M\\hookrightarrow \\mathcal {Q}$ lifts to a mapping $\\mathcal {F}^3\\hookrightarrow \\mathcal {H}$ between the Cartan bundles of $M$ and $\\mathcal {Q}$ in a manner that pulls back $\\mu $ (REF ) to $\\gamma $ (REF ).",
"To realize the image of $\\mathcal {F}^3$ over $M\\subset \\mathcal {Q}$ , we locate $\\gamma $ within $\\mu |_{\\mathcal {H}^2}$ using the fact that the semi-basic forms $\\kappa ,\\eta ,\\overline{\\eta }$ over both $\\mathcal {F}^3$ and $\\mathcal {H}^2$ encode adapted (co)framings of $M$ .",
"The equations (REF ) and (REF ) for $\\text{d}\\kappa $ , $\\text{d}\\eta $ alone demonstrate $&\\lambda =-\\alpha +\\mathbf {i}\\rho +a_0\\kappa ,&\\xi =\\beta -b_0\\kappa -a_0\\eta ,&&\\text{for some }a_0,b_0\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C}),$ and comparing $\\text{d}\\alpha ,\\text{d}\\beta $ in (REF ) to $\\text{d}\\lambda ,\\text{d}\\xi $ in (REF ) validates the substitution $&\\psi =-\\sigma +s_0\\kappa +s_1\\eta +\\overline{s}_1\\overline{\\eta },&&\\text{for some }s_0\\in C^\\infty (\\mathcal {H}^2), s_1\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C}),$ after which we solve for $a_0,b_0,s_0,s_1$ to bring the first five equations of (REF ) into the form (REF ): $a_0&=-\\epsilon \\tfrac{\\mathbf {i}}{4}|a|^2,&&b_0=-\\epsilon (\\mathbf {i}\\overline{a}b+\\tfrac{1}{6}a\\overline{u}_1),\\\\s_0&=\\tfrac{9}{16}|a|^4+\\epsilon (\\tfrac{3\\mathbf {i}}{4}(a\\overline{u}_2-\\overline{a}u_2)-3|b|^2-\\tfrac{1}{6}|u_1|^2)&&s_1=-\\epsilon \\tfrac{1}{12}(6a\\overline{b}+\\mathbf {i}\\overline{a}u_1).$ In conclusion, $\\begin{aligned}\\alpha &=\\mathbf {i}\\rho -\\lambda -\\epsilon \\tfrac{\\mathbf {i}}{4}|a|^2\\kappa ,\\quad \\quad \\beta =\\xi -\\epsilon (\\mathbf {i}\\overline{a}b+\\tfrac{1}{6}a\\overline{u}_1)\\kappa -\\epsilon \\tfrac{\\mathbf {i}}{4}|a|^2\\eta ,\\\\\\sigma &=-\\psi +(\\tfrac{9}{16}|a|^4+\\epsilon (\\tfrac{3\\mathbf {i}}{4}(a\\overline{u}_2-\\overline{a}u_2)-3|b|^2-\\tfrac{1}{6}|u_1|^2))\\kappa -\\epsilon \\tfrac{1}{12}(6a\\overline{b}+\\mathbf {i}\\overline{a}u_1)\\eta -\\epsilon \\tfrac{1}{12}(6\\overline{a}b-\\mathbf {i}a\\overline{u}_1)\\overline{\\eta }.\\end{aligned}$ Next we seek expressions for the coefficient functions $S,P$ of the curvature tensor $\\text{d}\\gamma +\\gamma \\wedge \\gamma $ as they appear in the equations for $\\text{d}\\beta ,\\text{d}\\sigma $ , as well as the higher-order coefficients in (REF ), (REF ), and (REF ).",
"To this end, we differentiate (REF ) and use the identity $\\text{d}^2=0$ to compute $\\begin{aligned}\\text{d}\\left[\\begin{array}{c}u_1\\\\u_2\\\\u_3\\\\u_4\\end{array}\\right]&=\\left[\\begin{array}{cccc}4\\mathbf {i}\\rho -\\overline{\\lambda }-\\lambda &0&0&0\\\\\\xi &3\\mathbf {i}\\rho -2\\overline{\\lambda }-\\lambda &0&0\\\\0&2\\xi &2\\mathbf {i}\\rho -3\\overline{\\lambda }-\\lambda &0\\\\0&0&3\\xi &\\mathbf {i}\\rho -4\\overline{\\lambda }-\\lambda \\end{array}\\right]\\left[\\begin{array}{c}u_1\\\\u_2\\\\u_3\\\\u_4\\end{array}\\right]\\\\&+\\left[\\begin{array}{cc}0&3\\mathbf {i}a\\\\-a&2\\mathbf {i}b\\\\-2b&\\mathbf {i}c\\\\-3c&0\\end{array}\\right]\\left[\\begin{array}{c}\\psi \\\\\\overline{\\xi }\\end{array}\\right]+\\left(\\left[\\begin{array}{cccc}v_1&v_2&3\\mathbf {i}u_2\\\\v_2&v_3&2\\mathbf {i}u_3\\\\v_3&v_4&\\mathbf {i}u_4\\\\v_4&v_5&0\\end{array}\\right]-\\epsilon \\left[\\begin{array}{cc}0&3a\\\\a&2b\\\\2b&c\\\\3c&0\\end{array}\\right]\\left[\\begin{array}{ccc}0&|b|^2&\\overline{a}b\\\\0&a\\overline{b}&|a|^2\\end{array}\\right]\\right)\\left[\\begin{array}{c}\\eta \\\\\\kappa \\\\\\overline{\\eta }\\end{array}\\right],\\end{aligned}$ for some $v_1,\\dots ,v_5\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ .",
"This is sufficient to express $\\begin{aligned}\\left[\\begin{array}{c}S\\\\P\\\\R\\end{array}\\right]&=\\frac{\\epsilon }{6}\\left[\\begin{array}{ccc}a&b&c\\\\u_1&u_2&u_3\\\\v_1&v_2&v_3\\end{array}\\right]\\left[\\begin{array}{c}\\overline{v}_1\\\\8\\mathbf {i}\\overline{u}_1\\\\-12\\overline{a}\\end{array}\\right]+\\frac{\\epsilon }{6}\\left[\\begin{array}{ccc}0&0&0\\\\a& b& c\\\\2u_1&2u_2&2u_3\\end{array}\\right]\\left[\\begin{array}{c}-4\\mathbf {i}\\overline{v}_2\\\\24\\overline{u}_2\\\\24\\mathbf {i}\\overline{b}\\end{array}\\right]\\\\&+\\frac{\\epsilon }{6}\\left[\\begin{array}{ccc}0&0&0\\\\0&0&0\\\\a&b&c\\end{array}\\right]\\left[\\begin{array}{c}-12\\overline{v}_3\\\\-48\\mathbf {i}\\overline{u}_3\\\\24\\overline{c}\\end{array}\\right]-\\frac{1}{6}\\left[\\begin{array}{ccc}0&0&0\\\\0&30b&0\\\\30|a|^2&54u_2&36u_1\\end{array}\\right]\\left[\\begin{array}{c}-\\epsilon |a|^4\\\\\\mathbf {i}\\overline{a}^2a\\\\\\mathbf {i}\\overline{a}^2b\\end{array}\\right]\\\\&+\\frac{1}{12}\\left[\\begin{array}{cc}0&0\\\\-20\\overline{a}&0\\\\72\\mathbf {i}\\overline{b}&108\\mathbf {i}\\overline{a}\\end{array}\\right]\\left[\\begin{array}{c}a^2\\overline{u}_1\\\\a^2\\overline{u}_2\\end{array}\\right]-\\left[\\begin{array}{cc}0&0\\\\0&0\\\\36b&5u_1\\end{array}\\right]\\left[\\begin{array}{l}|a|^2\\overline{b}\\\\|a|^2\\overline{u}_1\\end{array}\\right].\\end{aligned}$ Remark 4.6 The prototypical example of a 3-dimensional CR manifold is a hypersurface given by a regular level set of a smooth, non-constant function $\\varrho :\\mathbb {C}^2\\rightarrow \\mathbb {R}$ .",
"The curvature coefficient $S$ for the level set depends on derivatives of $\\varrho $ up to order six, which is to say that $S$ is a function of the 6-jet of $\\varrho $ .",
"In the present setting, $S$ for $M\\subset \\mathcal {Q}$ is a function of the 2-jet of $\\mathbf {II}$ .",
"The remaining functions in (REF ) require another derivative of $\\mathbf {II}$ , so we apply $\\text{d}^2=0$ to (REF ) to get $\\text{d}\\left[\\begin{array}{c}v_1\\\\v_2\\\\v_3\\\\v_4\\\\v_5\\end{array}\\right]&=\\left[\\begin{array}{ccccc}5\\mathbf {i}\\rho -\\overline{\\lambda }-2\\lambda &0&0&0&0\\\\\\xi &4\\mathbf {i}\\rho -2\\overline{\\lambda }-2\\lambda &0&0&0\\\\0&2\\xi &3\\mathbf {i}\\rho -3\\overline{\\lambda }-2\\lambda &0&0\\\\0&0&3\\xi &2\\mathbf {i}\\rho -4\\overline{\\lambda }-2\\lambda &0\\\\0&0&0&4\\xi &\\mathbf {i}\\rho -5\\overline{\\lambda }-2\\lambda \\end{array}\\right]\\left[\\begin{array}{c}v_1\\\\v_2\\\\v_3\\\\v_4\\\\v_5\\end{array}\\right]\\\\&+\\left[\\begin{array}{cc}0&8\\mathbf {i}u_1\\\\-2u_1&6\\mathbf {i}u_2\\\\-4u_2&4\\mathbf {i}u_3\\\\-6u_3&2\\mathbf {i}u_4\\\\-8u_4&0\\end{array}\\right]\\left[\\begin{array}{c}\\psi \\\\\\overline{\\xi }\\end{array}\\right]+\\left[\\begin{array}{ccc}w_1&w_2&4\\mathbf {i}v_2\\\\w_2&w_3&3\\mathbf {i}v_3\\\\w_3&w_4&2\\mathbf {i}v_4\\\\w_4&w_5&\\mathbf {i}v_5\\\\w_5&w_6&0\\end{array}\\right]\\left[\\begin{array}{c}\\eta \\\\\\kappa \\\\\\overline{\\eta }\\end{array}\\right]\\nonumber \\\\&-\\epsilon \\left(\\left[\\begin{array}{ccc}0&0&10u_1\\\\0&4u_1&6u_2\\\\u_1&6u_2&3u_3\\\\3u_2&6u_3&u_4\\\\6u_3&4u_4&0\\end{array}\\right]\\left[\\begin{array}{ccc}0&\\overline{b}c&\\overline{a}c\\\\0&|b|^2&\\overline{a}b\\\\0&a\\overline{b}&|a|^2\\end{array}\\right]-\\mathbf {i}\\left[\\begin{array}{ccc}0&0&6a\\\\0&3a&3b\\\\a&4b&c\\\\3b&3c&0\\\\6c&0&0\\end{array}\\right]\\left[\\begin{array}{ccc}0&|c|^2&\\overline{b}c\\\\0&b\\overline{c}&|b|^2\\\\0&a\\overline{c}&a\\overline{b}\\end{array}\\right]\\right)\\left[\\begin{array}{c}\\eta \\\\\\kappa \\\\\\overline{\\eta }\\end{array}\\right],\\nonumber $ for some $w_1,\\dots ,w_6\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ , and with this we can write $\\left[\\begin{array}{c}Q\\\\U\\\\V\\\\W\\\\R_1^{\\prime }\\end{array}\\right]&=\\frac{\\epsilon }{6}\\left[{\\begin{matrix}a&b&c\\\\0&0&0\\\\u_1&u_2&u_3\\\\0&0&0\\\\v_1&v_2&v_3\\end{matrix}}\\right]\\left[\\begin{array}{c}\\overline{w}_1\\\\10\\mathbf {i}\\overline{v}_1\\\\-20\\overline{u}_1\\end{array}\\right]-\\frac{\\epsilon }{6}\\left[{\\begin{matrix}0&0&0\\\\-a&-b&-c\\\\4\\mathbf {i}a&4\\mathbf {i}b&4\\mathbf {i}c\\\\-u_1&-u_2&-u_3\\\\8\\mathbf {i}u_1&8\\mathbf {i}u_2&8\\mathbf {i}u_3\\end{matrix}}\\right]\\left[\\begin{array}{c}\\overline{w}_2\\\\8\\mathbf {i}\\overline{v}_2\\\\-12\\overline{u}_2\\end{array}\\right]-\\frac{\\epsilon }{6}\\left[{\\begin{matrix}0&0&0\\\\0&0&0\\\\0&0&0\\\\-a& -b& -c\\\\3\\mathbf {i}a&3\\mathbf {i}b&3\\mathbf {i}c\\end{matrix}}\\right]\\left[\\begin{array}{c}-4\\mathbf {i}\\overline{w}_3\\\\24\\overline{v}_3\\\\24\\mathbf {i}\\overline{u}_3\\end{array}\\right]\\\\&+\\frac{\\epsilon }{6}\\left[{\\begin{matrix}0&0&0&0&0&0\\\\u_2&u_3&u_4&0&0&0\\\\\\mathbf {i}u_2&\\mathbf {i}u_3&\\mathbf {i}u_4&0&0&0\\\\v_2&v_3&v_4&u_2&u_3&u_4\\\\2\\mathbf {i}v_2&2\\mathbf {i}v_3&2\\mathbf {i}v_4&2\\mathbf {i}u_2&2\\mathbf {i}u_3&2\\mathbf {i}u_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{v}_1\\\\8\\mathbf {i}\\overline{u}_1\\\\-12\\overline{a}\\\\-4\\mathbf {i}\\overline{v}_2\\\\24\\overline{u}_2\\\\24\\mathbf {i}\\overline{b}\\\\\\end{matrix}}\\right]-\\frac{1}{6}\\left[{\\begin{matrix}0&0&0&0\\\\0&0&24b&0\\\\0&\\mathbf {i}45 c&\\mathbf {i}84 b&0\\\\\\tfrac{1}{2}(45\\overline{a}b+5\\mathbf {i}a\\overline{u}_1)&27u_3&54u_2&4u_1\\\\450\\mathbf {i}\\overline{a}b+180a\\overline{u}_1&144\\mathbf {i}u_3&288\\mathbf {i}u_2&48\\mathbf {i}u_1\\end{matrix}}\\right]\\left[\\begin{array}{c}-\\epsilon |a|^4\\\\\\mathbf {i}\\overline{a}^2a\\\\\\mathbf {i}\\overline{a}^2b\\\\\\mathbf {i}\\overline{a}^2c\\end{array}\\right]\\nonumber \\\\&-\\frac{1}{12}\\left[{\\begin{matrix}0&0&0\\\\0&\\mathbf {i}\\overline{a}&0\\\\20\\overline{u}_1&\\tfrac{53}{2}\\overline{a}&0\\\\20\\overline{u}_2&8\\overline{b}&22\\overline{a}\\\\-180\\mathbf {i}\\overline{u}_2&-84\\mathbf {i}\\overline{b}&-156\\mathbf {i}\\overline{a}\\end{matrix}}\\right]\\left[{\\begin{matrix}a^2\\overline{u}_1\\\\a^2\\overline{v}_1\\\\a^2\\overline{v}_2\\end{matrix}}\\right]-\\frac{1}{3}\\left[{\\begin{matrix}0&0&0&0&0&0&0&0\\\\0&3b&0&0&0&0&0&0\\\\0&63\\mathbf {i}b&0&0&0&0&0&0\\\\-3c&9u_2&21\\mathbf {i}b&\\tfrac{5}{24}\\mathbf {i}u_1&24\\overline{a}&8\\mathbf {i}a&0&\\tfrac{13}{3}\\overline{a}\\\\144\\mathbf {i}c&123\\mathbf {i}u_2&288b&20u_1&288\\mathbf {i}\\overline{a}&204a&15\\overline{u}_1&82\\mathbf {i}\\overline{a}\\end{matrix}}\\right]\\left[{\\begin{matrix}|a|^2\\overline{b}\\\\|a|^2\\overline{u}_1\\\\|a|^2\\overline{u}_2\\\\|a|^2\\overline{v}_1\\\\|b|^2b\\\\|b|^2\\overline{u}_1\\\\|u_1|^2a\\\\|u_1|^2b\\end{matrix}}\\right],\\nonumber $ as well as the totally real coefficients $\\left[\\begin{array}{c}R_0^{\\prime }\\\\R_{0}^{\\prime \\prime }\\end{array}\\right]&=\\epsilon \\left[{\\begin{matrix}-2a&-2b&-2c\\\\5\\mathbf {i}a&5\\mathbf {i}b&5\\mathbf {i}c\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_4\\\\ 4\\mathbf {i}\\overline{v}_4\\\\ -2\\overline{u}_4\\end{matrix}}\\right]-\\frac{\\epsilon }{3}\\left[{\\begin{matrix}-u_1&-u_2&-u_3\\\\5\\mathbf {i}u_1&5\\mathbf {i}u_2&5\\mathbf {i}u_3\\end{matrix}}\\right]\\left[{\\begin{matrix}-4\\mathbf {i}\\overline{w}_3\\\\ 24\\overline{v}_3\\\\ 24\\mathbf {i}\\overline{u}_3\\end{matrix}}\\right]-\\frac{\\epsilon }{12}\\left[{\\begin{matrix}-2v_1&-2v_2&-2v_3\\\\25\\mathbf {i}v_1&25\\mathbf {i}v_2&25\\mathbf {i}v_3\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_2\\\\ 8\\mathbf {i}\\overline{v}_2\\\\ -12\\overline{u}_2\\end{matrix}}\\right]\\\\&-\\epsilon \\left[{\\begin{matrix}2u_2&2u_3&2u_4\\\\5\\mathbf {i}u_2&5\\mathbf {i}u_3&5\\mathbf {i}u_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{v}_3\\\\ 4\\mathbf {i}\\overline{u}_3\\\\ -2\\overline{c}\\end{matrix}}\\right]+\\frac{\\epsilon }{6}\\left[{\\begin{matrix}2v_2&2v_3&2v_4\\\\5\\mathbf {i}v_2&5\\mathbf {i}v_3&5\\mathbf {i}v_4\\end{matrix}}\\right]\\left[{\\begin{matrix}-4\\mathbf {i}\\overline{v}_2\\\\ 24\\overline{u}_2\\\\ 24\\mathbf {i}\\overline{b}\\end{matrix}}\\right]+\\frac{\\epsilon }{12}\\left[{\\begin{matrix}2w_2&2w_3&2w_4\\\\5\\mathbf {i}w_2&5\\mathbf {i}w_3&5\\mathbf {i}w_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{v}_1\\\\ 8\\mathbf {i}\\overline{u}_1\\\\ -12\\overline{a}\\end{matrix}}\\right]\\nonumber \\\\&+\\frac{\\epsilon }{6}\\left(\\left[{\\begin{matrix}0&0&0\\\\w_1&w_2&w_3\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_1\\\\ 10\\mathbf {i}\\overline{v}_1\\\\ -20\\overline{u}_1\\end{matrix}}\\right]+|a|^4\\left[{\\begin{matrix}0\\\\6210|b|^2+\\tfrac{2195}{2}|u_1|^2\\end{matrix}}\\right]\\right)-\\frac{|a|^2}{6}\\left[{\\begin{matrix}0\\\\\\tfrac{1035}{2}|a|^6 +342|c|^2+1710|u_2|^2+\\tfrac{217}{4}|v_1|^2\\end{matrix}}\\right]\\nonumber \\\\&+\\Re \\left(\\frac{\\epsilon \\overline{a}^3a}{3}\\left[{\\begin{matrix}90&90\\\\\\tfrac{2385}{2}\\mathbf {i}&1650\\mathbf {i}\\end{matrix}}\\right]\\left[{\\begin{matrix}au_2\\\\bu_1\\end{matrix}}\\right]+\\frac{\\overline{a}^2}{3}\\left[{\\begin{matrix}-54\\mathbf {i}& -60\\mathbf {i}& -6\\mathbf {i}& -36\\mathbf {i}& -54\\mathbf {i}\\\\ 270& 390& 51& 210& 315\\end{matrix}}\\right]\\left[{\\begin{matrix}av_3\\\\ bv_2\\\\ cv_1\\\\ u_1u_3\\\\ {u_2}^2\\end{matrix}}\\right]+\\frac{|a|^2}{3}\\left[{\\begin{matrix}-180&-30\\\\1350\\mathbf {i}&375\\mathbf {i}\\end{matrix}}\\right]\\left[{\\begin{matrix}b\\overline{u}_3\\\\u_1\\overline{v}_2\\end{matrix}}\\right]\\right)\\nonumber \\\\&-\\Re \\left(\\frac{1}{3}\\left[{\\begin{matrix}(288|b|^2+30|u_1|^2)a\\overline{u}_2+a(\\overline{b}(10u_1\\overline{v}_1-24\\mathbf {i}\\overline{u}_1u_2)+36b\\overline{c}\\overline{u}_1)\\\\(720b\\overline{c}+199\\mathbf {i}\\overline{u}_1v_1)\\overline{a}b-\\mathbf {i}(2880|b|^2+425|u_1|^2)a\\overline{u}_2+40(30b\\overline{u}_2+u_1\\overline{v}_1)\\overline{a}u_1-498\\mathbf {i}ab\\overline{c}\\overline{u}_1\\end{matrix}}\\right]\\right)\\nonumber \\\\&-\\frac{1}{6}\\left[{\\begin{matrix}0\\\\30|u_1|^4+1440|b|^4+832|bu_1|^2\\end{matrix}}\\right].\\nonumber $ One last derivative of (REF ) yields $\\text{d}\\left[\\begin{array}{c}w_1\\\\w_2\\\\w_3\\\\w_4\\\\w_5\\\\w_6\\end{array}\\right]&=\\left[{\\begin{matrix}6\\mathbf {i}\\rho -\\overline{\\lambda }-3\\lambda &0&0&0&0&0\\\\\\xi &5\\mathbf {i}\\rho -2\\overline{\\lambda }-3\\lambda &0&0&0&0\\\\0&2\\xi &4\\mathbf {i}\\rho -3\\overline{\\lambda }-3\\lambda &0&0&0\\\\0&0&3\\xi &3\\mathbf {i}\\rho -4\\overline{\\lambda }-3\\lambda &0&0\\\\0&0&0&4\\xi &2\\mathbf {i}\\rho -5\\overline{\\lambda }-3\\lambda &0\\\\0&0&0&0&5\\xi &\\mathbf {i}\\rho -6\\overline{\\lambda }-3\\lambda \\end{matrix}}\\right]\\left[\\begin{array}{c}w_1\\\\w_2\\\\w_3\\\\w_4\\\\w_5\\\\w_6\\end{array}\\right]\\\\&+\\left[{\\begin{matrix}0&15\\mathbf {i}v_1\\\\-3v_1&12\\mathbf {i}v_2\\\\-6v_2&9\\mathbf {i}v_3\\\\-9v_3&6\\mathbf {i}v_4\\\\-12v_4&3\\mathbf {i}v_5\\\\-15v_5&0\\end{matrix}}\\right]\\left[\\begin{array}{c}\\psi \\\\\\overline{\\xi }\\end{array}\\right]+\\left(\\left[{\\begin{matrix}z_1&z_2&5\\mathbf {i}w_2\\\\z_2&z_3&4\\mathbf {i}w_3\\\\z_3&z_4&3\\mathbf {i}w_4\\\\z_4&z_5&2\\mathbf {i}w_5\\\\z_5&z_6&\\mathbf {i}w_6\\\\z_6&z_7&0\\end{matrix}}\\right]-\\epsilon \\left[{\\begin{matrix}10{u_1}^2\\\\10u_1u_2\\\\4u_1u_3+6{u_2}^2\\\\u_1u_4+9u_2u_3\\\\4u_2u_4+6{u_3}^2\\\\10u_3u_4\\end{matrix}}\\right]\\left[\\begin{array}{ccc}0&\\overline{b}&\\overline{a}\\end{array}\\right]\\right)\\left[\\begin{array}{c}\\eta \\\\\\kappa \\\\\\overline{\\eta }\\end{array}\\right]\\nonumber \\\\-&\\epsilon \\left(\\left[{\\begin{matrix}0&0&15v_1\\\\0&5v_1&10v_2\\\\v_1&8v_2&6v_3\\\\3v_2&9v_3&3v_4\\\\6v_3&8v_4&v_5\\\\10v_4&5v_5&0\\end{matrix}}\\right]\\left[\\begin{array}{ccc}0&\\overline{b}c&\\overline{a}c\\\\0&|b|^2&\\overline{a}b\\\\0&a\\overline{b}&|a|^2\\end{array}\\right]-\\mathbf {i}\\left[{\\begin{matrix}0&0&30u_1\\\\0&12u_1&18u_2\\\\3u_1&18u_2&9u_3\\\\9u_2&18u_3&3u_4\\\\18u_3&12u_4&0\\\\30u_4&0&0\\end{matrix}}\\right]\\left[\\begin{array}{ccc}0&|c|^2&\\overline{b}c\\\\0&b\\overline{c}&|b|^2\\\\0&a\\overline{c}&a\\overline{b}\\end{array}\\right]\\right)\\left[\\begin{array}{c}\\eta \\\\\\kappa \\\\\\overline{\\eta }\\end{array}\\right],\\nonumber $ for some $z_1,\\dots ,z_7\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ .",
"In the language of Remark REF , we have functions of the 4-jet of $\\mathbf {II}$ , $\\begin{aligned}\\left[\\begin{array}{c}Q^{\\prime }\\\\U_1^{\\prime }\\\\U_2^{\\prime }\\\\V^{\\prime }\\end{array}\\right]&=\\frac{\\epsilon }{6}\\left(\\left[{\\begin{matrix}a&b&c\\\\0&0&0\\\\0&0&0\\\\0&0&0\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{z}_1\\\\ 12\\mathbf {i}\\overline{w}_1\\\\ -30\\overline{v}_1\\end{matrix}}\\right]+\\left[{\\begin{matrix}0&0&0\\\\a&b&c\\\\0&0&0\\\\u_1&u_2&u_3\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{z}_2\\\\ 10\\mathbf {i}\\overline{w}_2\\\\ -20\\overline{v}_2\\end{matrix}}\\right]+\\left[{\\begin{matrix}0&0&0\\\\0&0&0\\\\a&b&c\\\\-4\\mathbf {i}a&-4\\mathbf {i}b&-4\\mathbf {i}c\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{z}_3\\\\ 8\\mathbf {i}\\overline{w}_3\\\\-12\\overline{v}_3\\end{matrix}}\\right]\\right)\\\\&+\\frac{\\epsilon }{6}\\left(\\left[{\\begin{matrix}0&0&0\\\\u_2&u_3&u_4\\\\0&0&0\\\\v_2&v_3&v_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_1\\\\ 10\\mathbf {i}\\overline{v}_1\\\\ -20\\overline{u}_1\\end{matrix}}\\right]+\\left[{\\begin{matrix}0&0&0\\\\0&0&0\\\\2u_2&2u_3&2u_4\\\\-3\\mathbf {i}u_2&-3\\mathbf {i}u_3&-3\\mathbf {i}u_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_2\\\\ 8\\mathbf {i}\\overline{v}_2\\\\ -12\\overline{u}_2\\end{matrix}}\\right]+\\left[{\\begin{matrix}0&0&0\\\\0&0&0\\\\v_3&v_4&v_5\\\\\\mathbf {i}v_3&\\mathbf {i}v_4&\\mathbf {i}v_5\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{v}_1\\\\ 8\\mathbf {i}\\overline{u}_1\\\\ -12\\overline{a}\\end{matrix}}\\right]\\right)\\\\&+\\frac{|a|^2}{36}\\left(\\epsilon \\left[{\\begin{matrix}0&0&0&0&0\\\\0&0&0&0&0\\\\216&144&-150\\mathbf {i}&-15&-10\\\\1296\\mathbf {i}&\\tfrac{603}{2}\\mathbf {i}&390&\\mathbf {i}\\tfrac{345}{8}&70\\mathbf {i}\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{a}^2b^2\\\\|a|^2\\overline{a}c\\\\|a|^2b\\overline{u}_1\\\\|a|^2a\\overline{v}_1\\\\a^2{\\overline{u}_1}^2\\end{matrix}}\\right]-\\left[{\\begin{matrix}0&0&0&0&0&0\\\\30\\mathbf {i}c&0&15b&0&\\tfrac{9\\mathbf {i}}{2}a&0\\\\24u_3&-144\\mathbf {i}c&-9\\mathbf {i}u_2&48b&-u_1&6\\mathbf {i}a\\\\624\\mathbf {i}u_3&-306c&\\frac{243}{2}u_2&708\\mathbf {i}b&4\\mathbf {i}u_1&\\tfrac{183}{2}a\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{u}_1\\\\\\overline{u}_2\\\\\\overline{v}_1\\\\\\overline{v}_2\\\\\\overline{w}_1\\\\\\overline{w}_2\\end{matrix}}\\right]\\right)\\\\&-\\frac{1}{36}\\left(\\left[{\\begin{matrix}0&0&0\\\\0&-360c&0\\\\-36\\mathbf {i}u_4&504\\mathbf {i}u_3&252\\mathbf {i}u_2\\\\-234u_4&-1224u_3&-612u_2\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{a}^2a\\\\\\overline{a}^2b\\\\\\overline{a}^2c\\end{matrix}}\\right]+\\left[{\\begin{matrix}0&0&0\\\\5\\mathbf {i}\\overline{v}_1&0&0\\\\4\\mathbf {i}\\overline{v}_2&21\\mathbf {i}\\overline{u}_2&6\\mathbf {i}\\overline{b}\\\\136\\overline{v}_2&84\\overline{u}_2&24\\overline{b}\\end{matrix}}\\right]\\left[{\\begin{matrix}a^2\\overline{u}_1\\\\a^2\\overline{v}_1\\\\a^2\\overline{w}_1\\end{matrix}}\\right]\\right)\\\\&-\\frac{1}{36}\\left(\\left[{\\begin{matrix}0&0&0&0\\\\20b&0&360\\mathbf {i}\\overline{a}&0\\\\-8\\mathbf {i}u_2&-32\\mathbf {i}u_1&-288\\mathbf {i}\\overline{b}&432\\mathbf {i}\\overline{a}\\\\88u_2&52u_1&288\\overline{b}&-432\\overline{a}\\end{matrix}}\\right]\\left[{\\begin{matrix}a{\\overline{u}_1}^2\\\\b{\\overline{u}_1}^2\\\\b^2\\overline{u}_1\\\\b^2\\overline{u}_2\\end{matrix}}\\right]+\\left[{\\begin{matrix}0&0&0&0\\\\0&0&0&0\\\\-84\\overline{v}_1&288c&-5\\overline{v}_1&68c\\\\156\\mathbf {i}\\overline{v}_1&1008\\mathbf {i}c&5\\mathbf {i}\\overline{v}_1&148\\mathbf {i}c\\end{matrix}}\\right]\\left[{\\begin{matrix}a|b|^2\\\\\\overline{a}|b|^2\\\\a|u_1|^2\\\\\\overline{a}|u_1|^2\\end{matrix}}\\right]\\right)\\\\&+\\frac{1}{9}\\left[{\\begin{matrix}0\\\\0\\\\6\\mathbf {i}a\\overline{b}c\\overline{u}_1+4\\mathbf {i}\\overline{a}bu_1\\overline{v}_1-72\\overline{a}b\\overline{u}_1u_2+18 ab\\overline{u}_1\\overline{u}_2\\\\24a\\overline{b}c\\overline{u}_1-14\\overline{a}bu_1\\overline{v}_1-297\\mathbf {i}\\overline{a}b\\overline{u}_1u_2-162\\mathbf {i}ab\\overline{u}_1\\overline{u}_2\\end{matrix}}\\right],\\end{aligned}$ and our collection is completed by $\\left[\\begin{array}{c}W^{\\prime }\\\\R_1^{\\prime \\prime }\\end{array}\\right]&=-\\frac{\\epsilon }{6}\\left(4\\left[{\\begin{matrix}\\mathbf {i}a&\\mathbf {i}b&\\mathbf {i}c\\\\3a&3b&3c\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{z}_4\\\\ 6\\mathbf {i}\\overline{w}_4\\\\ -6\\overline{v}_4\\end{matrix}}\\right]+\\left[{\\begin{matrix}-u_1&-u_2&-u_3\\\\8\\mathbf {i}u_1&8\\mathbf {i}u_2&8\\mathbf {i}u_3\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{z}_3\\\\ 8\\mathbf {i}\\overline{w}_3\\\\-12\\overline{v}_3\\end{matrix}}\\right]-\\left[{\\begin{matrix}0&0&0\\\\v_1&v_2&v_3\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{z}_2\\\\ 10\\mathbf {i}\\overline{w}_2\\\\ -20\\overline{v}_2\\end{matrix}}\\right]\\right)\\\\&-\\frac{\\epsilon }{6}\\left(4\\left[{\\begin{matrix}2\\mathbf {i}u_2&2\\mathbf {i}u_3&2\\mathbf {i}u_4\\\\u_2&u_3&u_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_3\\\\ 6\\mathbf {i}\\overline{v}_3\\\\ -6\\overline{u}_3\\end{matrix}}\\right]+2\\left[{\\begin{matrix}-v_2&-v_3&-v_4\\\\3\\mathbf {i}v_2&3\\mathbf {i}v_3&3\\mathbf {i}v_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_2\\\\ 8\\mathbf {i}\\overline{v}_2\\\\ -12\\overline{u}_2\\end{matrix}}\\right]-\\left[{\\begin{matrix}0&0&0\\\\w_2&w_3&w_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_1\\\\ 10\\mathbf {i}\\overline{v}_1\\\\ -20\\overline{u}_1\\end{matrix}}\\right]\\right)\\nonumber \\\\&+\\frac{\\epsilon }{6}\\left(4\\left[{\\begin{matrix}-\\mathbf {i}v_3&-\\mathbf {i}v_4&-\\mathbf {i}v_5\\\\2v_3&2v_4&2v_5\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{v}_2\\\\ 6\\mathbf {i}\\overline{u}_2\\\\ -6\\overline{b}\\end{matrix}}\\right]+\\left[{\\begin{matrix}w_3&w_4&w_5\\\\2\\mathbf {i}w_3&2\\mathbf {i}w_4&2\\mathbf {i}w_5\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{v}_1\\\\ 8\\mathbf {i}\\overline{u}_1\\\\ -12\\overline{a}\\end{matrix}}\\right]+\\overline{a}^3a\\left[{\\begin{matrix}\\tfrac{99}{2}u_3 &252u_2 &4u_1\\\\414\\mathbf {i}u_3& 1404\\mathbf {i}u_2& 228\\mathbf {i}u_1\\end{matrix}}\\right]\\left[{\\begin{matrix}a\\\\b\\\\c\\end{matrix}}\\right]\\right)\\nonumber \\\\&+\\frac{\\epsilon }{6} \\left(a^3\\left[{\\begin{matrix}16\\mathbf {i}\\overline{b}\\overline{u}_1& 54\\mathbf {i}\\overline{a}\\overline{u}_1&16\\mathbf {i}\\overline{a}\\overline{b}& \\tfrac{33}{2}\\mathbf {i}\\overline{a}^2\\\\108\\overline{b}\\overline{u}_1 &\\tfrac{819}{2}\\overline{a}\\overline{u}_1&\\tfrac{261}{2}\\overline{a}\\overline{b}&\\tfrac{399}{2}\\overline{a}^2\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{u}_1\\\\ \\overline{u}_2\\\\ \\overline{v}_1\\\\ \\overline{v}_2\\end{matrix}}\\right]+|a|^4\\left[{\\begin{matrix}15\\mathbf {i}c_1&-57\\mathbf {i}u_2&-81b&-\\tfrac{125}{24}u_1\\\\360c& \\tfrac{903}{2}u_2& 918\\mathbf {i}b &\\ 10\\mathbf {i}u_1\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{b}\\\\ \\overline{u}_1\\\\ \\overline{u}_2\\\\ \\overline{v}_1\\end{matrix}}\\right]\\right)\\nonumber \\\\&+\\frac{\\epsilon }{6}\\left[{\\begin{matrix}-(152a\\overline{u}_1+324\\mathbf {i}\\overline{a}b)&-(15a\\overline{u}_1+\\tfrac{224}{3}\\mathbf {i}\\overline{a}b)& 72\\\\1008\\overline{a}b+546\\mathbf {i}a\\overline{u}_1&431\\overline{a}b+\\tfrac{45}{2}\\mathbf {i}a\\overline{u}_1& 384\\mathbf {i}\\end{matrix}}\\right]\\left[{\\begin{matrix}|ab|^2\\\\ |au_1|^2\\\\ \\overline{a}^3b^2u_1\\end{matrix}}\\right]+\\frac{\\overline{a}^2}{6}\\left[{\\begin{matrix}-24\\mathbf {i}&-90\\mathbf {i}& -10\\mathbf {i}& -4\\mathbf {i}& -126\\mathbf {i}\\\\138& 360& 120& 48& 432\\end{matrix}}\\right]\\left[{\\begin{matrix}av_4\\\\ bv_3\\\\ cv_2\\\\ u_1u_4\\\\ u_2u_3\\end{matrix}}\\right]\\nonumber \\\\&+\\frac{a^2}{6}\\left[{\\begin{matrix}-12& -12& 0& -12& -36& -4\\\\81\\mathbf {i}& 66\\mathbf {i}& 6\\mathbf {i}& 96\\mathbf {i}& 168\\mathbf {i}& 42\\mathbf {i}\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{a}\\overline{w}_3\\\\ \\overline{b}\\overline{w}_2\\\\ \\overline{c}\\overline{w}_1\\\\ \\overline{u}_1\\overline{v}_3\\\\ \\overline{u}_2\\overline{v}_2\\\\ \\overline{u}_3\\overline{v}_1\\end{matrix}}\\right]-\\frac{|a|^2}{6}\\left[{\\begin{matrix}36\\mathbf {i}&24& \\tfrac{3}{2}\\mathbf {i}& 24& 54\\mathbf {i}& -24&0& \\tfrac{2}{3}\\mathbf {i}& 18\\\\558&198\\mathbf {i}& 42& 108\\mathbf {i}& 252& 252\\mathbf {i}&\\tfrac{\\mathbf {i}}{4}& 37& 231\\mathbf {i}\\end{matrix}}\\right]\\left[{\\begin{matrix}b\\overline{v}_3\\\\ c\\overline{u}_3\\\\ u_1\\overline{w}_2\\\\ u_2\\overline{v}_2\\\\ u_3\\overline{u}_2\\\\ u_4\\overline{b}\\\\ v_1\\overline{w}_1\\\\ v_2\\overline{v}_1\\\\ v_3\\overline{u}_1\\end{matrix}}\\right]\\nonumber \\\\&-\\frac{1}{6}\\left(|b|^2\\left[{\\begin{matrix}96\\mathbf {i}& -144& -144\\mathbf {i}& -6& -40\\mathbf {i}& 288\\\\528& 432\\mathbf {i}& 288& 8\\mathbf {i}&\\ 480& 1296\\mathbf {i}\\end{matrix}}\\right]+|u_1|^2\\left[{\\begin{matrix}\\tfrac{14}{3}\\mathbf {i}& -\\tfrac{4}{3}&0&-\\tfrac{5}{6}&-\\tfrac{20}{3}\\mathbf {i}& 22\\\\64& 84\\mathbf {i}& 60& 0& 30& 164\\mathbf {i}\\end{matrix}}\\right]\\right)\\left[{\\begin{matrix}a\\overline{v}_2\\\\ b\\overline{u}_2\\\\ c\\overline{b}\\\\ u_1\\overline{v}_1\\\\ u_2\\overline{u}_1\\\\ u_3\\overline{a}\\end{matrix}}\\right]\\nonumber \\\\&-\\frac{1}{6}\\left(a\\overline{u}_1\\left[{\\begin{matrix}-\\tfrac{135}{4}&24&0&-\\tfrac{1}{6}&8\\mathbf {i}&-\\tfrac{4}{3}\\mathbf {i}\\\\60\\mathbf {i}&48\\mathbf {i}&90\\mathbf {i}&\\tfrac{\\mathbf {i}}{2}&384&26\\end{matrix}}\\right]+b\\overline{a}\\left[{\\begin{matrix}-\\frac{405}{4}\\mathbf {i}& 120\\mathbf {i}& 36\\mathbf {i}& -\\mathbf {i}& 72&\\tfrac{80}{3}\\\\\\tfrac{675}{2}&0&288&12&504\\mathbf {i}&260\\mathbf {i}\\end{matrix}}\\right]\\right)\\left[{\\begin{matrix}|a|^6\\\\|c|^2\\\\|u_2|^2\\\\|v_1|^2\\\\b\\overline{u}_3\\\\v_2\\overline{u}_1\\end{matrix}}\\right]\\nonumber \\\\&-\\frac{1}{6}\\left(\\overline{a}u_1\\left[{\\begin{matrix}4& 16\\mathbf {i}& -\\tfrac{1}{6}& -5\\mathbf {i}&0\\\\108\\mathbf {i}& 48& 0& 0&0\\end{matrix}}\\right]+\\overline{b}a\\left[{\\begin{matrix}0&-72& \\mathbf {i}& 4& 12\\mathbf {i}\\\\0&216\\mathbf {i}& 10& -12\\mathbf {i}& 276\\end{matrix}}\\right]\\right)\\left[{\\begin{matrix}b\\overline{v}_2\\\\ c\\overline{u}_2\\\\ u_1\\overline{w}_1\\\\ u_2\\overline{v}_1\\\\ u_3\\overline{u}_1\\end{matrix}}\\right]\\nonumber \\\\&-\\frac{1}{6}\\left[{\\begin{matrix}\\mathbf {i}(108b\\overline{u}_2+\\tfrac{25}{6}u_1\\overline{v}_1)a\\overline{u}_2+24(\\overline{b}c+2u_2\\overline{u}_1)\\overline{a}u_2-(10\\mathbf {i}a\\overline{v}_1-56b\\overline{u}_1)b\\overline{c}\\\\(60\\overline{c}\\overline{v}_1+504{\\overline{u}_2}^2)ab+(648\\mathbf {i}\\overline{b}c+40u_1\\overline{v}_1+246\\mathbf {i}\\overline{u}_1u_2)\\overline{a}u_2+10bv_1{\\overline{u}_1}^2+72\\mathbf {i}b^2\\overline{c}\\overline{u}_1+30\\mathbf {i}\\overline{a}cv_1\\overline{u}_1+40au_1\\overline{u}_2\\overline{v}_1\\end{matrix}}\\right].\\nonumber $ Levi-Flat 3-folds Our adaptation (REF ) together with (REF ) implies that $\\zeta -\\eta =0 \\text{ on }\\mathcal {H}^2.$ Using Cartan's Lemma with (REF ) while recalling that $\\rho $ is $\\mathbb {R}$ -valued, $&\\left[\\begin{array}{c}\\phi _2\\\\2\\mathbf {i}\\rho \\end{array}\\right]=-\\left[\\begin{array}{c}\\xi \\\\\\phi _1-\\overline{\\phi }_1+\\lambda -\\overline{\\lambda }\\end{array}\\right]+\\left[\\begin{array}{cc}a&\\mathbf {i}b\\\\\\mathbf {i}b&0\\end{array}\\right]\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right]&\\text{ for some } a\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})\\quad b\\in C^\\infty (\\mathcal {H}^2).$ To see how these functions vary on $\\mathcal {H}^2$ , we invoke the MC equations (REF ) again to differentiate $\\text{d}\\left[\\begin{array}{c}b\\\\a\\end{array}\\right]=-\\left[\\begin{array}{cc}\\phi _1+\\overline{\\phi }_1+\\lambda +\\overline{\\lambda }&0\\\\-2\\mathbf {i}\\xi &\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1+\\lambda +5\\overline{\\lambda })\\end{array}\\right]\\left[\\begin{array}{c}b\\\\a\\end{array}\\right]+\\left[\\begin{array}{cc}u_0&0\\\\u_1&\\mathbf {i}u_0+b^2\\end{array}\\right]\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right],\\\\\\text{for some } u_0\\in C^\\infty (\\mathcal {H}^2),\\quad u_1\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C}),\\nonumber $ and we find: if $a$ vanishes identically on each fiber $\\mathcal {H}^2\\rightarrow M$ , then so does $b$ ; $b$ is either zero or non-vanishing on each fiber.",
"Definition 4.7 The second fundamental form (REF ) of a Levi-flat 3-fold $M$ CR embedded in $\\mathcal {Q}=SU(2,2)/\\mathcal {P}$ is given by $\\mathbf {II}=a\\kappa \\odot \\kappa +2\\mathbf {i}b\\kappa \\odot \\eta ,$ where the coefficients are derived from (REF ) via (REF ).",
"The condition that $\\mathbf {II}$ is of (sub)maximal rank on a fiber of $\\mathcal {H}^2\\rightarrow M$ is invariant under the action of CR symmetry group $SU(2,2)$ on $\\mathcal {Q}$ .",
"We update the remaining MC equations on $\\mathcal {H}^2$ , $\\begin{aligned}\\text{d}\\kappa &=(\\lambda +\\overline{\\lambda })\\wedge \\kappa ,\\\\\\text{d}\\eta &=-\\xi \\wedge \\kappa +\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1+3\\lambda -\\overline{\\lambda })\\wedge \\eta -\\tfrac{\\mathbf {i}}{2}b\\kappa \\wedge \\eta ,\\\\\\text{d}\\phi _1&=-\\phi _1\\wedge \\overline{\\phi }_1+\\mathbf {i}\\xi \\wedge \\overline{\\eta }+\\mathbf {i}\\overline{\\xi }\\wedge \\eta -\\mathbf {i}b\\phi _1\\wedge \\kappa -\\mathbf {i}a\\kappa \\wedge \\overline{\\eta }+b\\eta \\wedge \\overline{\\eta },\\\\\\text{d}\\xi &=\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1+\\lambda -3\\overline{\\lambda })\\wedge \\xi +\\psi \\wedge \\eta +(\\tfrac{\\mathbf {i}}{2}b\\xi -a\\overline{\\phi }_1)\\wedge \\kappa -\\mathbf {i}b\\overline{\\phi }_1\\wedge \\eta ,\\\\\\text{d}\\lambda &=-\\psi \\wedge \\kappa +\\mathbf {i}\\overline{a}\\kappa \\wedge \\eta -b\\eta \\wedge \\overline{\\eta },\\\\\\text{d}\\psi &=-(\\lambda +\\overline{\\lambda })\\wedge \\psi +\\mathbf {i}(a\\overline{\\xi }-\\overline{a}\\xi )\\wedge \\kappa -b(\\xi \\wedge \\overline{\\eta }+\\overline{\\xi }\\wedge \\eta )+b\\kappa \\wedge (\\overline{a}\\eta +a\\overline{\\eta })+\\mathbf {i}b^2\\eta \\wedge \\overline{\\eta },\\end{aligned}$ and gather more differential identities by applying $\\text{d}^2=0$ to the exterior derivative of (REF ), $\\text{d}\\left[\\begin{array}{c}u_0\\\\u_1\\end{array}\\right]&=-\\left[\\begin{array}{cc}\\phi _1+\\overline{\\phi }_1+2(\\lambda +\\overline{\\lambda })&0\\\\-3\\mathbf {i}\\xi &\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1+3\\lambda +7\\overline{\\lambda })\\end{array}\\right]\\left[\\begin{array}{c}u_0\\\\u_1\\end{array}\\right]+\\left[\\begin{array}{cc}-2b&0\\\\-3a&2b^2\\end{array}\\right]\\left[\\begin{array}{l}\\psi \\\\\\xi \\end{array}\\right]\\\\&+\\mathbf {i}b\\left[\\begin{array}{cc}- b& b\\\\-\\tfrac{1}{2}a&\\tfrac{5}{2}a\\end{array}\\right]\\left[\\begin{array}{l}\\phi _1\\\\\\overline{\\phi }_1\\end{array}\\right]+\\mathbf {i}\\left[\\begin{array}{cc}-2\\overline{a}b&2 ab\\\\-|a|^2&3 a^2\\end{array}\\right]\\left[\\begin{array}{c}\\eta \\\\\\overline{\\eta }\\end{array}\\right]+\\left[\\begin{array}{cc}v_0&0\\\\v_1&\\mathbf {i}v_0+\\tfrac{5}{2}bu_0-\\tfrac{\\mathbf {i}}{2}b^3\\end{array}\\right]\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right],\\nonumber $ for some $v_0\\in C^\\infty (\\mathcal {H}^2)$ , $v_1\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ .",
"Differentiate (REF ) to get $\\text{d}\\left[\\begin{array}{c}v_0\\\\v_1\\end{array}\\right]&=-\\left[\\begin{array}{cc}\\phi _1+\\overline{\\phi }_1+3(\\lambda +\\overline{\\lambda })&0\\\\-4\\mathbf {i}\\xi &\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1+5\\lambda +9\\overline{\\lambda })\\end{array}\\right]\\left[\\begin{array}{c}v_0\\\\v_1\\end{array}\\right]\\nonumber \\\\&+\\left[\\begin{array}{ccc}-6u_0&-4\\mathbf {i}\\overline{a}b&4\\mathbf {i}ab\\\\-8u_1&-\\tfrac{3\\mathbf {i}}{2}b^3-4\\mathbf {i}|a|^2+8bu_0&6\\mathbf {i}a^2\\end{array}\\right]\\left[{\\begin{matrix}\\psi \\\\\\xi \\\\\\overline{\\xi }\\end{matrix}}\\right]\\nonumber \\\\&+\\left[\\begin{array}{cc}b(b^2-3\\mathbf {i}u_0)& b(b^2+3\\mathbf {i}u_0)\\\\\\tfrac{1}{2}a(b^2-\\mathbf {i}u_0)-\\mathbf {i}bu_1&\\tfrac{1}{2}a(9b^2+11\\mathbf {i}u_0)+3\\mathbf {i}bu_1\\end{array}\\right]\\left[\\begin{array}{l}\\phi _1\\\\\\overline{\\phi }_1\\end{array}\\right]\\\\&-\\left[\\begin{array}{cc}4\\overline{a}b^2+5\\mathbf {i}\\overline{a}u_0+2\\mathbf {i}b\\overline{u}_1&4ab^2-5\\mathbf {i}au_0-2\\mathbf {i}bu_1\\\\6|a|^2b+3\\mathbf {i}\\overline{a}u_1+\\mathbf {i}a\\overline{u}_1&5a^2b-10\\mathbf {i}au_1\\end{array}\\right]\\left[\\begin{array}{c}\\eta \\\\\\overline{\\eta }\\end{array}\\right]\\nonumber \\\\&+\\left[\\begin{array}{cc}w_0&0\\\\w_1&\\mathbf {i}w_0+\\tfrac{5}{2}{u_0}^2+3bv_0-\\tfrac{11\\mathbf {i}}{4}b^2u_0-\\tfrac{1}{4}b^4\\end{array}\\right]\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right];\\nonumber $ $w_0\\in C^\\infty (\\mathcal {H}^2)$ , $w_1\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ , and a final derivative provides $\\text{d}\\left[\\begin{array}{c}w_0\\\\w_1\\end{array}\\right]&=-\\left[\\begin{array}{cc}\\phi _1+\\overline{\\phi }_1+4(\\lambda +\\overline{\\lambda })&0\\\\-5\\mathbf {i}\\xi &\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1+7\\lambda +11\\overline{\\lambda })\\end{array}\\right]\\left[\\begin{array}{c}w_0\\\\w_1\\end{array}\\right]\\nonumber \\\\&-\\left[{\\begin{matrix}12v_0&6\\overline{a}b^2+15\\mathbf {i}\\overline{a}u_0+6\\mathbf {i}b\\overline{u}_1&6ab^2-15\\mathbf {i}au_0-6\\mathbf {i}bu_1\\\\15v_1&b(b^3+8|a|^2-13v_0)+5\\mathbf {i}(3\\overline{a}u_1+a\\overline{u}_1+\\tfrac{9}{4}b^2u_0)-\\tfrac{21}{2}{u_0}^2&8a^2b-30\\mathbf {i}au_1\\end{matrix}}\\right]\\left[{\\begin{matrix}\\psi \\\\\\xi \\\\\\overline{\\xi }\\end{matrix}}\\right]\\nonumber \\\\&+\\left[\\begin{array}{cc}\\mathbf {i}b(4|a|^2+b^3-4v_0)& -\\mathbf {i}b(4|a|^2+b^3-4v_0)\\\\\\mathbf {i}a(6|a|^2+\\tfrac{1}{2}(b^3-v_0))&-\\mathbf {i}a(4|a|^2+6b^3-\\tfrac{19}{2}v_0)\\end{array}\\right]\\left[\\begin{array}{l}\\phi _1\\\\\\overline{\\phi }_1\\end{array}\\right]\\nonumber \\\\&+\\left[\\begin{array}{cc}3u_0(2b^2-\\mathbf {i}u_0)& 3u_0(2b^2+\\mathbf {i}u_0)\\\\\\tfrac{3}{2}(b(au_0+bu_1-\\mathbf {i}v_1)-\\mathbf {i}u_0u_1)&\\tfrac{1}{2}(b(45au_0+15bu_1+7\\mathbf {i}v_1)+17\\mathbf {i}u_0u_1)\\end{array}\\right]\\left[\\begin{array}{l}\\phi _1\\\\\\overline{\\phi }_1\\end{array}\\right]\\\\&-\\left[{\\begin{matrix}3\\overline{a}b(\\tfrac{13}{2}u_0-\\mathbf {i}b^2)+\\overline{u}_1(5b^2+7\\mathbf {i}u_0)+9\\mathbf {i}\\overline{a}v_0+2\\mathbf {i}b\\overline{v}_1&3ab(\\tfrac{13}{2}u_0+\\mathbf {i}b^2)+u_1(5b^2-7\\mathbf {i}u_0)-9\\mathbf {i}av_0-2\\mathbf {i}bv_1\\\\\\tfrac{1}{2}(23|a|^2u_0+b(37\\overline{a}u_1+13a\\overline{u}_1))+\\mathbf {i}(6\\overline{a}v_1+a\\overline{v}_1+4|u_1|^2)&3ab(8u_1+\\mathbf {i}ab)+\\tfrac{11}{2}a^2u_0-15\\mathbf {i}av_1-10\\mathbf {i}{u_1}^2\\end{matrix}}\\right]\\left[\\begin{array}{c}\\eta \\\\\\overline{\\eta }\\end{array}\\right]\\nonumber \\\\&+\\left[\\begin{array}{cc}z_0&0\\\\z_1&\\mathbf {i}z_0-\\tfrac{27\\mathbf {i}}{4}b{u_0}^2+8u_0v_0+\\tfrac{7}{2}bw_0+\\tfrac{\\mathbf {i}}{2}b^2(15|a|^2-\\tfrac{17}{2}v_0)-\\tfrac{19}{8}b^3u_0+\\tfrac{\\mathbf {i}}{8}b^5\\end{array}\\right]\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right],\\nonumber $ for $z_0\\in C^\\infty (\\mathcal {H}^2)$ , $z_1\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ .",
"Classification: Levi-Nondegenerate Case We have seen that if any coefficient of the second fundamental form (REF ) vanishes identically on $\\mathcal {H}^2$ , then $\\mathbf {II}=0$ everywhere over $M$ .",
"The leading coefficient $a\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ is particularly indicative, since it is either identically zero or non-vanishing on each fiber of $\\mathcal {H}^2\\rightarrow M$ , so the process of adapting frames “branches\" based on the (non)degeneracy of $a$ .",
"If $a=0\\Rightarrow \\mathbf {II}=0$ everywhere, (REF ) subsides to the Maurer-Cartan equations of $U(2,1)$ , which translate via (REF ) to the structure equations (REF ) of the flat CR 3-sphere over a quotient of $\\mathcal {H}^2$ by the central action of $U(1)$ mentioned in Remark REF .",
"In the terminology of Definition REF , $M$ is equivariantly embedded in $\\mathcal {Q}$ as an orbit of $U(2,1)\\subset SU_\\star $ , and in this case no further reduction of $\\mathcal {H}^2$ is admissible.",
"Otherwise, we make the generic assumption that $a\\ne 0$ and arrive at another branching point based on whether $M$ is flat.",
"Locally, all flat 3-folds are equivalent to the 3-sphere, hence the question embeddability is trivial.",
"Rather, we focus on equivariant embeddings by normalizing $\\mathbf {II}$ and tracking which algebras emerge as the symmetries of constant-coefficient structure equations.",
"The results are summarized in Theorem REF .",
"For non-flat 3-folds, we reduce $\\mathcal {H}^2\\rightarrow M$ to the Cartan bundle $\\mathcal {F}^3\\rightarrow M$ and carry out the same normalization procedure as in §REF .",
"The resulting structure equations classify embedded, curved 3-folds as detailed in Theorem REF .",
"Equivariant embeddings are gathered into Theorem REF .",
"Flat 3-folds If $M$ is flat, the coefficients $S,P$ of the curvature tensor appearing in (REF ) are zero, as are the coefficients (REF ) of the covariant derivative of curvature, the second covariant derivative (REF ), and those we've named (REF ) within the third covariant derivative.",
"Since $a\\ne 0$ , we can use the expressions (REF ), (REF ), (REF ), (REF ), and (REF ) of these coefficients to solve for the jet coordinates of $\\mathbf {II}$ .",
"When more than one variable is eligible for solution – e.g., we can solve $S=\\tfrac{\\epsilon }{6}(a\\overline{v}_1+8\\mathbf {i}b\\overline{u}_1-12c\\overline{a})=0$ for either of $c,\\overline{v}_1$ – we default to the higher-order jet: $&\\text{solve }S=0\\text{ for }\\overline{v}_1,&\\text{solve }P=0\\text{ for }\\overline{v}_2,&&\\text{etc.",
"},$ unless the higher-order variable is obtained from another equation.",
"It is also understood that we solve the complex-conjugated equation for the conjugate coordinate.",
"Onward, $&\\begin{array}{l}\\text{solve }R=0\\text{ for }\\Re v_3, \\\\\\text{solve }Q=0\\text{ for }\\overline{w}_1, \\\\\\text{solve }U=0\\text{ for }\\overline{w}_2, \\\\\\text{solve }V=0\\text{ for }u_4, \\\\\\text{solve }W=0\\text{ for }\\overline{w}_3,\\\\\\text{solve }R_1^{\\prime }=0\\text{ for }v_4,\\\\\\text{solve }R_1^{\\prime \\prime }=0\\text{ for }w_5,\\end{array}\\begin{array}{l}\\text{solve }R_0^{\\prime }=0\\text{ and }R_0^{\\prime \\prime }=0\\text{ for }\\overline{w}_4,w_4,\\\\\\text{solve }Q^{\\prime }=0\\text{ for }\\overline{z}_1,\\\\\\text{solve }U_1^{\\prime }=0\\text{ for }\\overline{z}_2,\\\\\\text{solve }U_2^{\\prime }=0\\text{ for }v_5,\\\\\\text{solve }V^{\\prime }=0\\text{ for }\\overline{z}_3,\\\\\\text{solve }W^{\\prime }=0\\text{ for }\\overline{z}_4.\\end{array}$ Now we resume the process of adapting frames.",
"The identities (REF ) for $\\text{d}a$ and $\\text{d}b$ tell us that we can restrict to frames whose second fundamental form is diagonalized with leading coefficient 1, $\\mathcal {H}^3&=\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^2: a(\\underline{{\\tt v}})=1, b(\\underline{{\\tt v}})=0\\rbrace ,$ over which we have $\\begin{aligned}\\lambda &=-3\\mathbf {i}\\rho +\\overline{u}_1\\overline{\\eta }+\\overline{u}_2\\kappa ,\\\\\\xi &=-\\mathbf {i}c\\overline{\\eta }-u_2\\eta -u_3\\kappa .\\end{aligned}$ With this normalization, we have constrained the coordinates $c_1$ (REF ) and $l$ (REF ) in the fibers of $\\mathcal {H}^2\\rightarrow M$ .",
"In particular, fixing $l$ implies that sections $M\\rightarrow \\mathcal {H}^3$ factor through a unique lift $M\\rightarrow \\hat{M}$ .",
"The real part of $c_0$ (REF ) is similarly determined by the reducing to $\\mathcal {H}^4&=\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^3: \\Re u_2(\\underline{{\\tt v}})=0\\rbrace ,$ as suggested by the equation (REF ) for $\\text{d}u_2$ , which reveals that on $T\\mathcal {H}^4$ , $\\begin{aligned}\\psi &=\\left(\\tfrac{5}{4}+7|c|^2+\\tfrac{5\\epsilon }{12}|u_1|^2-3{u_2}^2-\\tfrac{\\mathbf {i}}{6}u_2(8|u_1|^2+27\\epsilon )-c{u_1}^2-\\overline{c}{\\overline{u}_1}^2+\\tfrac{1}{2}u_1u_3+\\tfrac{1}{2}\\overline{u}_1\\overline{u}_3\\right)\\kappa \\\\&+\\tfrac{\\mathbf {i}}{4}(8\\overline{c}\\overline{u}_1-5\\epsilon u_1+8\\mathbf {i}u_1u_2-10\\overline{u}_3)\\eta -\\tfrac{\\mathbf {i}}{4}(8cu_1-5\\epsilon \\overline{u}_1+8\\mathbf {i}\\overline{u}_1u_2-10u_3)\\overline{\\eta }.\\end{aligned}$ The rank of $\\mathbf {II}$ will index our final instance of branching in the flat setting.",
"If $c=0$ so that $\\text{rank}(\\mathbf {II})=1$ , $&\\text{d}c=0\\Rightarrow u_1=u_3=0,&\\text{d}u_1=0\\Rightarrow u_2=-\\mathbf {i}\\frac{\\epsilon }{2},&&\\text{d}u_2=0\\Rightarrow \\Im v_3=0,$ and we are left with structure equations $\\text{d}\\kappa &=\\mathbf {i}\\eta \\wedge \\overline{\\eta },\\\\\\text{d}\\eta &=\\mathbf {i}(\\epsilon \\kappa -4\\rho )\\wedge \\eta ,\\\\\\text{d}\\rho &=0,$ which describe a central extension of the symmetry algebra of (VIII,C) when $\\epsilon =-1$ or (IX,D) when $\\epsilon =1$ (see the end of §REF ).",
"On the other hand, if $\\mathbf {II}$ has maximal rank 2 it must be that $c$ is nonvanishing, so the identity (REF ) for $\\text{d}c$ shows that we can reduce to those frames where $c$ takes values in $\\mathbb {R}\\setminus \\lbrace 0\\rbrace $ , $\\mathcal {H}^5&=\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^4: \\Im c(\\underline{{\\tt v}})=0\\rbrace ,$ over which $\\rho =\\frac{1}{96c}(\\epsilon (18c+{u_1}^2+{\\overline{u}_1}^2)+6(u_1\\overline{u}_3+\\overline{u}_1u_3)-4c|u_1|^2)\\kappa +\\frac{\\mathbf {i}}{16c}(3u_1c-u_3)\\eta -\\frac{\\mathbf {i}}{16c}(3\\overline{u}_1c-\\overline{u}_3)\\overline{\\eta }.$ Differentiating this and comparing to (REF ) provides $\\epsilon (9cu_1-{u_1}^3)-18c^2\\overline{u}_1+2c\\overline{u}_1{u_1}^2+54c\\overline{u}_3-6\\overline{u}_3{u_1}^2=0.$ Therefore, the most generic CR embedding in $\\mathcal {Q}$ of a flat 3-fold is encoded in the structure equations $\\begin{aligned}\\text{d}\\kappa &=\\mathbf {i}\\eta \\wedge \\overline{\\eta }-\\kappa \\wedge (u_1\\eta +\\overline{u}_1\\overline{\\eta }),\\\\\\text{d}\\eta &=-\\frac{c\\overline{u}_1+\\overline{u}_3}{4c}\\eta \\wedge \\overline{\\eta }-\\mathbf {i}\\frac{\\epsilon (18c+{u_1}^2+{\\overline{u}_1}^2)+6(u_1\\overline{u}_3+\\overline{u}_1u_3)-4c|u_1|^2-48\\mathbf {i}cu_2}{24c}\\kappa \\wedge \\eta -\\mathbf {i}c\\kappa \\wedge \\overline{\\eta },\\end{aligned}$ where $c,u_1,u_3\\in C^\\infty (\\mathcal {H}^5,\\mathbb {C})$ and $-\\mathbf {i}u_2,\\Im v_3\\in C^\\infty (\\mathcal {H}^5,\\mathbb {R})$ satisfy (REF ), (REF ), and (REF ) subject to $a=1,b=0$ , (REF ), (REF ), (REF ), (REF ), (REF ), and (REF ).",
"The equations (REF ) remain invariant under the action of the CR symmetry group $SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ on $\\mathcal {Q}$ , so they classify embedded flat 3-folds whose second fundamental form has rank 2.",
"Moreover, when (REF ) has constant coefficients, the equations classify equivariant embeddings as in Definition REF .",
"Taking $c,u_1,u_3$ to be constant implies $&c=\\tfrac{1}{9}{u_1}^2,&{u_1}^2={\\overline{u}_1}^2,&&u_2=-\\tfrac{\\mathbf {i}}{6}(3\\epsilon +|u_1|^2),&&u_3=\\tfrac{1}{3}{u_1}^3,&&\\Im v_3=0,$ so $u_1\\ne 0$ is either real or imaginary, and by $\\text{d}u_2=0$ from (REF ) we see $16\\epsilon |u_1|^2+9=0.$ The latter only has solutions when $\\epsilon =-1$ , given by $u_1=\\pm \\tfrac{3}{4}$ or $u_1=\\pm \\tfrac{3}{4}\\mathbf {i}$ .",
"Submitting the CR coframing $\\kappa ,\\eta $ to the 1-adapted transformation $\\frac{2}{9}\\left[\\begin{array}{rc}2|u_1|^2&0\\\\-9\\mathbf {i}|u_1|^2&3u_1\\end{array}\\right]\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right]$ brings (REF ) into the form (REF ) for the homogeneous 3-fold ($\\text{VI}_3$ , E).",
"Curved 3-folds Suppose the coefficients $S\\in C^\\infty (\\mathcal {F}^3,\\mathbb {C})$ of the curvature tensor and $a\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ of the second fundamental form of $M$ are non-vanishing and invoke the identity (REF ) for $\\text{d}a$ to reduce to Hermitian frames where $a$ is $\\mathbb {R}$ -valued: $\\mathcal {H}^3=\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^2 : a(\\underline{{\\tt v}})-\\overline{a}(\\underline{{\\tt v}})=0\\rbrace .$ On $T\\mathcal {H}^3$ we have $\\rho =\\frac{\\mathbf {i}}{6a}(a(\\lambda -\\overline{\\lambda })+(u_2-\\overline{u}_2)\\kappa +(u_1+2\\mathbf {i}\\overline{b})\\eta -(\\overline{u}_1-2\\mathbf {i}b)\\overline{\\eta }),$ and the transformation (REF ) is the infinitesimal version of the identification $\\mathcal {H}^3=\\mathcal {F}^3$ as bundles of coframes over $M$ .",
"Now we pursue the same process of reduction as in §REF .",
"There, $\\mathcal {F}^4$ is defined by normalizing $S$ , which is achieved in the present setting by solving the equation (REF ) $S=1$ for $\\overline{v}_1$ , thereby exhausting one complex degree of freedom in the fibers of $\\mathcal {H}^3\\supset \\mathcal {H}^4=\\mathcal {F}^4$ over $M$ .",
"Accordingly, $\\lambda $ is no longer an independent 1-form, but is determined by (REF ) via (REF ) and (REF ), (REF ).",
"Next we constrain $\\overline{v}_2$ in (REF ) by $P=0$ so that $\\mathcal {H}^5\\subset \\mathcal {H}^4$ coincides with $\\mathcal {F}^5$ and $\\xi $ satisfies (REF ) by virtue of (REF ), (REF ), and (REF ).",
"Finally, the condition $\\Re U=0$ fixes $\\Re w_2$ on $\\mathcal {F}^6=\\mathcal {H}^6\\subset \\mathcal {H}^5$ , where $\\psi $ is subject to (REF ) with coefficients (REF ), (REF ), and (REF ).",
"Our construction proves the following Theorem 5.1 Let $M$ be a 3-dimensional, Levi-nondegenerate CR manifold whose curvature tensor is non-vanishing.",
"$M$ is CR embeddable in the 5-dimensional real hyperquadric $\\mathcal {Q}=SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )/\\mathcal {P}$ if and only if $M$ admits a 1-adapted CR coframing $\\kappa ,\\eta \\in \\Omega ^1(M,\\mathbb {C})$ and functions $a,b,c,u_1,\\dots ,u_4,v_1,\\dots ,v_5,w_1,\\dots ,w_5,z_1,\\dots ,z_4\\in C^\\infty (M,\\mathbb {C})$ satisfying the structure equations (REF ) for (REF ) given by (REF ), (REF ), where $S=1,P=\\Re U=0$ , and the differential identities (REF ), (REF ), (REF ), and (REF ) hold for (REF ) determined by (REF ), (REF ), and (REF ) with (REF ).",
"For such $M$ , the structure equations (REF ) remain invariant under the action of $SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ on $\\mathcal {Q}$ .",
"Let us implement Theorem REF to treat the examples of non-flat homogeneous 3-folds.",
"In addition to $v_1,v_2$ decided by $S=1$ and $P=0$ , respectively, we find most of the functions (REF ) by following the prescription (REF ), except that the coefficients of the covariant derivatives of curvature take the values listed in Remark REF instead of zero.",
"A general embedding now depends on the existence of functions $a,b,c,u_1,u_2,u_3,\\Im v_3$ which satisfy (REF ) and the identities $\\text{d}^2=0$ , and these conditions become algebraic for equivariant embeddings.",
"Namely, if $\\mathbf {II}$ is constant, $c&=\\tfrac{2}{a}b^2-3\\mathbf {i}Ab-Ca,&&u_1=2\\mathbf {i}\\overline{b}+2\\overline{A}a,\\\\u_2&=\\tfrac{2\\mathbf {i}}{a}|b|^2+\\overline{A}b+\\mathbf {i}(|A|^2-\\tfrac{B}{4})a-\\epsilon \\tfrac{\\mathbf {i}}{4}a^3,&&u_3=\\tfrac{4\\mathbf {i}}{a^2}\\overline{b}b^2+\\tfrac{6A}{a}|b|^2-\\tfrac{\\mathbf {i}}{2}(Bb+4C\\overline{b}-4|A|^2b)-\\epsilon \\tfrac{\\mathbf {i}}{2}ba^2,$ which in turn implies $&0=A\\overline{b}+\\overline{A}b,&B=\\tfrac{4}{3a^2}|b|^2+\\tfrac{2\\mathbf {i}}{a}\\overline{A}b+\\tfrac{8}{3}|A|^2+\\epsilon a^2,&&C=-\\tfrac{10}{9}\\overline{A}^2+\\tfrac{38\\mathbf {i}}{9a}\\overline{Ab}+\\tfrac{2}{3a^2}(2\\overline{b}^2+\\epsilon ),$ along with $\\Im v_3=0$ .",
"Polynomial relationships between the remaining quantities $a,b,A$ are clarified by addressing separately the possible values of $\\overline{A}=\\pm A$ .",
"First consider $A=0$ , yielding $&\\overline{b}=\\frac{2b}{a^4},&b(3a^8+6\\epsilon b^2)=0.$ If $b=0$ , $a\\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace $ is free and the resulting structure equations describe the homogeneous models (VIII, K) when $\\epsilon =-1$ or (IX, L) when $\\epsilon =1$ .",
"Note that $\\mathbf {II}$ is diagonalized with determinant $-\\epsilon \\tfrac{2}{3}$ .",
"The alternative ($b\\ne 0$ ) is only possible when $\\epsilon =-1$ ; namely, $a=\\pm \\@root 4 \\of {2}$ and $b=\\pm \\sqrt{2}$ , which is (IX, L) for $B=\\tfrac{\\sqrt{2}}{3}$ , and in this case $\\text{rank}\\mathbf {II}=1$ .",
"For $A\\ne 0$ , (REF ) shows $\\overline{b}=\\tfrac{\\overline{A}}{A}b$ while $a,b,A$ are governed by three polynomials of degree six or seven, $&D_{\\overline{\\eta }}a^2\\text{d}c,&D_\\kappa a^2A\\text{d}\\overline{b},&&D_\\kappa a^2A\\text{d}u_1,$ where the hook $$ indicates contraction with the vector field dual to the subscripted 1-form.",
"Solutions exist only when $\\epsilon =-1$ , and up to signs they are given by $&A=\\mathbf {i}\\frac{4\\@root 4 \\of {10}}{\\sqrt{5}},&a=\\frac{\\@root 4 \\of {10}}{\\sqrt{5}},&&b=-\\sqrt{10}.$ The homogeneous model is therefore (VI$_{t}$ , E) as in (REF ) with $\\iota =\\mathbf {i}$ , $t=\\tfrac{4\\@root 4 \\of {10}}{\\sqrt{5}}$ , and $m=\\tfrac{t}{2}$ so that $S=1$ .",
"Classification: Levi-Flat Case All Levi-flat 3-folds $M$ are locally CR equivalent, so embeddability $M\\subset \\mathcal {Q}$ is a question of the signature of $\\mathcal {Q}$ 's Levi form.",
"In this respect we recall that our discussion here applies only to the real hyperquadric whose CR symmetry group is $SU(2,2)$ ($\\epsilon =-1$ in the notation of §), hence the embeddings of interest will be equivariant for some action of this Lie group in the sense of Definition REF .",
"Our list of homogeneous models in §REF omitted the Bianchi algebras that serve as infinitesimal CR symmetries of Levi-flat $M$ ; let us record here two models with symmetry of Bianchi type V. For an appropriate choice of bases, the extension of $\\mathbb {R}^2$ by $\\left[{\\begin{matrix}\\@root 3 \\of {3}&0\\\\0&-\\@root 3 \\of {3}\\end{matrix}}\\right]$ has structure equations $\\begin{aligned}\\text{d}\\kappa &=\\@root 3 \\of {3}(\\eta +\\overline{\\eta })\\wedge \\kappa ,\\\\\\text{d}\\eta &=\\@root 3 \\of {3}\\eta \\wedge \\overline{\\eta }+\\tfrac{\\mathbf {i}}{15\\@root 3 \\of {3}}(4\\eta -\\overline{\\eta })\\wedge \\kappa ,\\end{aligned}$ and the extension by $\\left[{\\begin{matrix}1&0\\\\0&2\\end{matrix}}\\right]$ is described by $\\begin{aligned}\\text{d}\\kappa &=\\tfrac{\\mathbf {i}}{2}(\\eta -\\overline{\\eta })\\wedge \\kappa ,\\\\\\text{d}\\eta &=\\mathbf {i}\\eta \\wedge \\overline{\\eta }.\\end{aligned}$ As in the Levi-nondegenerate case, the process of reducing $\\mathcal {H}^2\\rightarrow M$ to these and other homogeneous models branches based on the rank of the second fundamental form $\\mathbf {II}$ .",
"This section constitutes the proof of Theorem REF .",
"$\\mathbf {II}=0$ : Maximal Symmetry Let $\\underline{{\\tt v}}=({\\tt v}_0,{\\tt v}_1,{\\tt v}_2,{\\tt v}_3)$ be a Hermitian frame () whose vectors we rearrange, $&\\left[\\begin{array}{cccc}{\\tt n}_1&{\\tt n}_2&{\\tt n}_3&{\\tt n}_4\\end{array}\\right]=\\left[\\begin{array}{cccc}{\\tt v}_0&{\\tt v}_1&{\\tt v}_2&{\\tt v}_3\\end{array}\\right]U,&U=\\frac{1}{\\sqrt{2}}\\left[\\begin{array}{rrrr}0&\\sqrt{2}&0&0\\\\1&0&0&-1\\\\1&0&0&1\\\\0&0&\\sqrt{2}&0\\end{array}\\right].$ The transformation $U$ is “unitary\" in the sense that $U^{-1}=\\overline{U}^t$ , and even though $\\overline{U}^tỦ\\ne , we refer to the symmetry groups of both forms as $ SU(2,2)$.",
"Let $ RCSL2C$ be the 10-dimensional parabolic subgroup that stabilizes the partial flag{\\begin{@align}{1}{-1}\\langle {\\tt n}_1\\rangle _\\mathbb {C}\\subset \\langle {\\tt n}_1,{\\tt n}_2,{\\tt n}_3\\rangle _\\mathbb {C}\\subset \\mathbb {C}^{4},\\end{@align}}and name $ R=RCSU(2,2)$ with Lie algebra $ rsu(2,2)$.$ The new basis (REF ) consists entirely of $-null vectors, and for $vH2$ they are adapted to $ M$ -- cf.",
"(\\ref {LFfirstadapt}) -- in that $n1,n2,n3$ span $ Tv0M$ with $n1$ descending to the CR bundle of $ M$.",
"If the complex curve tangent to the CR bundle of $ M$ is a complex line in $ Q$ (see \\cite [Example 1.5]{BryanthololorentzCR}), then the osculating flag (\\ref {oscflag}) is constant along $ M$ and $ M$ itself is contained in the fixed subspace $n1,n2,n3C$.",
"Hence, one expects the extrinsic CR symmetries of such $ M$ given by the action of $ SU(2,2)$ on $ Q$ to lie in $ R$.",
"In general, $ |H2$ is (\\ref {UMCform}) subject to (\\ref {trace-free}), (\\ref {zetaiseta}), and (\\ref {LFIIcoeff}), so transforming according to (\\ref {LFUtrans}) we get{\\begin{@align}{1}{-1}U^{-1}\\mu |_{\\mathcal {H}^2}U=\\left[\\begin{array}{cccc}-\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1+\\lambda -\\overline{\\lambda })&\\sqrt{2}\\eta &\\sqrt{2}\\xi -\\tfrac{1}{\\sqrt{2}}(a\\kappa +\\mathbf {i}b\\eta )&\\phi _1-\\overline{\\phi }_1-\\tfrac{\\mathbf {i}}{2}b\\kappa \\\\-\\tfrac{1}{\\sqrt{2}}(b\\overline{\\eta }+\\mathbf {i}\\overline{a}\\kappa )&\\lambda &\\psi &\\sqrt{2}\\mathbf {i}\\overline{\\xi }-\\tfrac{1}{\\sqrt{2}}(b\\overline{\\eta }+\\mathbf {i}\\overline{a}\\kappa )\\\\0&\\kappa &-\\overline{\\lambda }&-\\mathbf {i}\\sqrt{2}\\overline{\\eta }\\\\-\\tfrac{\\mathbf {i}}{2}b\\kappa &0&-\\tfrac{1}{\\sqrt{2}}(a\\kappa +\\mathbf {i}b\\eta )&\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1-\\lambda +\\overline{\\lambda })\\end{array}\\right].\\end{@align}}In particular, if $ a=b=0$, (\\ref {LFMCtrans}) takes values $ r$, and (\\ref {LFHtwoMC}) are the Maurer-Cartan equations of $ R$.",
"Thus, if $ II=0$, $ H2$ is locally $ RSU(2,2)$.$ Rank$(\\mathbf {II})=1$ The second fundamental form (REF ) will have rank one only if $b=0$ in (REF ), which in turn requires $u_0=0$ (REF ), $v_0=0$ (REF ), $w_0=0$ (REF ), and $z_0=0$ (REF ), but it must be that $a$ is nonvanishing.",
"The identity (REF ) for $\\text{d}a$ therefore implies that we can reduce to $\\mathcal {H}^3=\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^2: a(\\underline{{\\tt v}})=1\\rbrace ,$ and on $\\mathcal {H}^3$ we have $\\lambda =-\\tfrac{1}{6}(\\phi _1+\\overline{\\phi }_1)-\\tfrac{1}{12}(u_1-5\\overline{u}_1)\\kappa .$ Looking to $\\text{d}u_1$ (REF ) and $\\text{d}v_1$ (REF ), we see the opportunity to reduce further, $\\mathcal {H}^4=\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^3: \\Re u_1(\\underline{{\\tt v}})=0=v_1(\\underline{{\\tt v}})\\rbrace .$ After renaming $u=\\Im u_1\\in C^\\infty (\\mathcal {H}^4),$ we can write the newly imposed constraints $\\psi &=\\tfrac{1}{3}(u^2\\kappa -2\\mathbf {i}\\eta +2\\mathbf {i}\\overline{\\eta }),\\\\\\xi &=\\tfrac{1}{30}(40u^3+6\\mathbf {i}w_1-9\\mathbf {i}\\overline{w}_1)\\kappa +\\tfrac{\\mathbf {i}}{15}(11u\\eta +u\\overline{\\eta }),$ as well as the updated identity $\\text{d}w_1=(6\\mathbf {i}+w_1)\\phi _1+(w_1-4\\mathbf {i})\\overline{\\phi }_1+(z_1-\\tfrac{80}{3}u^4-8\\mathbf {i}uw_1+3\\mathbf {i}u\\overline{w}_1)\\kappa +\\tfrac{\\mathbf {i}}{3}u^2(16\\eta +38\\overline{\\eta }).$ The latter suggests a final reduction $\\mathcal {H}^5=\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^4: w_1(\\underline{{\\tt v}})=0\\rbrace ,$ over which $\\phi _1&=-\\tfrac{\\mathbf {i}}{30}(80u^4-9z_1+6\\overline{z}_1)\\kappa -\\tfrac{1}{15}u^2(62\\eta +73\\overline{\\eta }),\\\\\\text{d}u&=-\\tfrac{1}{3}uz\\kappa +(1-3u^3)\\eta +(1-3u^3)\\overline{\\eta };&&z=\\Im z_1.$ What's left of (REF ) reads $\\begin{aligned}\\text{d}\\kappa &=3u^2(\\eta +\\overline{\\eta })\\wedge \\kappa ,\\\\\\text{d}\\eta &=3u^2\\eta \\wedge \\overline{\\eta }-\\tfrac{1}{15}(4\\mathbf {i}u+5z)\\kappa \\wedge \\eta +\\tfrac{\\mathbf {i}}{15}u\\kappa \\wedge \\overline{\\eta },\\end{aligned}$ for $u,z\\in C^\\infty (\\mathcal {H}^5)$ .",
"The equations (REF ), along with differential identities necessitated by $\\text{d}^2=0$ , classify embedded Levi-flat 3-folds $M\\subset \\mathcal {Q}$ with rank$(\\mathbf {II})=1$ up to the action of $SU(2,2)$ on $\\mathcal {Q}$ , including the unique equivariant embedding when $\\text{d}u=0\\Rightarrow u=3^{-\\tfrac{1}{3}}$ , $z=0$ , which is exactly the model (REF ).",
"Rank$(\\mathbf {II})=2$ The second fundamental form (REF ) has full rank if and only if $b$ is nonvanishing; in this case the identities (REF ), (REF ) for $\\text{d}b, \\text{d}a, \\text{d}u_0$ show that we can reduce to $\\mathcal {H}^3=\\left\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^2:\\begin{array}{c}b(\\underline{{\\tt v}})=\\pm 1\\\\a(\\underline{{\\tt v}})=0\\\\u_0(\\underline{{\\tt v}})=0\\end{array}\\right\\rbrace ,$ over which we have relations $\\phi _1&=-\\frac{1}{2}(\\lambda +\\overline{\\lambda })+\\frac{\\mathbf {i}}{b}\\psi -\\frac{v_0}{2b^2}\\kappa ,\\\\\\xi &=\\frac{\\mathbf {i}}{2}b\\eta +\\frac{\\mathbf {i}u_1}{2b}\\kappa ,$ as well as the identity $\\text{d}u_1=-u_1(\\lambda +3\\overline{\\lambda })+(v_1+\\mathbf {i}bu_1)\\kappa +\\mathbf {i}(v_0+\\tfrac{1}{2}b^3)\\eta .$ Thus we are obliged to consider branching based on the possible values of $u_1$ .",
"If $u_1=0$ identically, then $v_0=-\\tfrac{1}{2}b^3$ , $v_1=0$ , and the remaining coefficients in (REF ), (REF ) are zero, leaving $\\begin{aligned}\\text{d}\\kappa &=(\\lambda +\\overline{\\lambda })\\wedge \\kappa ,\\\\\\text{d}\\eta &=(\\lambda -\\overline{\\lambda })\\wedge \\eta ,\\\\\\text{d}\\lambda &=-\\psi \\wedge \\kappa -b\\eta \\wedge \\overline{\\eta },\\\\\\text{d}\\psi &=\\psi \\wedge (\\lambda +\\overline{\\lambda }).\\end{aligned}$ Note that (REF ) are the Maurer-Cartan equations of the MC forms $&\\left[\\begin{array}{cc}\\tfrac{1}{2}(\\lambda +\\overline{\\lambda })&\\psi \\\\\\kappa &-\\tfrac{1}{2}(\\lambda +\\overline{\\lambda })\\end{array}\\right]\\in \\Omega ^1(\\mathcal {H}^3,\\mathfrak {sl}_2\\mathbb {R}),&\\left[\\begin{array}{cc}\\tfrac{1}{2}(\\lambda -\\overline{\\lambda })&-\\sqrt{b}\\overline{\\eta }\\\\\\sqrt{b}\\eta &-\\tfrac{1}{2}(\\lambda -\\overline{\\lambda })\\end{array}\\right]\\in \\Omega ^1(\\mathcal {H}^3,\\mathfrak {su}(p,q)),$ where $(p,q)=(2,0)$ if $b=1$ and $(p,q)=(1,1)$ if $b=-1$ .",
"Otherwise, in any neighborhood where $u_1\\ne 0$ we can normalize it to define $\\mathcal {H}^4=\\left\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^3:\\begin{array}{c}u_1(\\underline{{\\tt v}})=1\\\\\\Re v_1(\\underline{{\\tt v}})=0\\end{array}\\right\\rbrace ,$ so that on $\\mathcal {H}^4$ , $\\lambda &=-\\tfrac{\\mathbf {i}}{2}(b+v)\\kappa -\\tfrac{\\mathbf {i}}{16}(b^3+2v_0)\\eta -\\tfrac{3\\mathbf {i}}{16}(b^3+2v_0)\\overline{\\eta } &(v=\\Im v_1),\\\\\\psi &=\\tfrac{1}{16}(vb^3 + b^4+ 2bv_0 + 2v_0v + 2\\mathbf {i}w_0 )\\eta +\\tfrac{1}{16}(vb^3 + b^4+ 2bv_0 + 2v_0v - 2\\mathbf {i}w_0 )\\overline{\\eta }\\\\&+\\frac{1}{16b}(4v^2b + 4vb^2 + 3b^3 + 2b\\overline{w}_1 + 2bw_1 - 16v_0)\\kappa ,$ and the structure equations are $\\begin{aligned}\\text{d}\\kappa &=\\tfrac{\\mathbf {i}}{8}(b^3+2v_0)(\\eta -\\overline{\\eta })\\wedge \\kappa ,\\\\\\text{d}\\eta &=\\tfrac{\\mathbf {i}}{4}(b^3+2v_0)\\eta \\wedge \\overline{\\eta }-\\mathbf {i}(b+v)\\kappa \\wedge \\eta ,\\end{aligned}$ with identities $\\begin{aligned}\\text{d}v_0&=w_0\\kappa -\\tfrac{\\mathbf {i}}{8}(b^6+4b^3v_0+4{v_0}^2+16b)\\eta +\\tfrac{\\mathbf {i}}{8}(b^6+4b^3v_0+4{v_0}^2+16b)\\overline{\\eta },\\\\\\text{d}v&=\\tfrac{\\mathbf {i}}{2}(\\overline{w}_1-w_1)\\kappa -\\tfrac{\\mathbf {i}}{8}(3vb^3+3b^4+6v_0v+4\\mathbf {i}w_0+6bv_0)\\eta +\\tfrac{\\mathbf {i}}{8}(3vb^3+3b^4+6v_0v-4\\mathbf {i}w_0+6bv_0)\\overline{\\eta }.\\end{aligned}$ We conclude that embedded Levi-flat 3-folds $M\\subset \\mathcal {Q}$ with rank$(\\mathbf {II})=2$ are classified up to the action of $SU(2,2)$ on $\\mathcal {Q}$ by the Maurer-Cartan equations (REF ) of $SL_2\\mathbb {R}\\times SU(p,q)$ – $(p,q)=(2,0)$ or $(1,1)$ – when $u_1=0$ , or (REF ) with (REF ) when $u_1\\ne 0$ .",
"The latter evince a homogeneous action in $SU(2,2)$ when $&w_0=0,&v=1,&&v_0=-\\tfrac{3}{2}\\text{ or }\\tfrac{5}{2},&&b=-1,$ in which case they coincide (up to sign) with (REF )."
],
[
"Some History",
"The subject matter of this article is closely related to relativistic theories of radiation, both electromagnetic and gravitational.",
"In general relativity, spacetime is an oriented 4-dimensional manifold $\\mathcal {S}$ .",
"The distribution of mass-energy in $\\mathcal {S}$ is encoded in a symmetric 2-tensor field, and Einstein's field equations say that this field prescribes the Ricci curvature of a Lorentzian metric; we suppose that $\\mathcal {S}$ admits a solution $g\\in \\bigodot ^2T^*\\mathcal {S}$ .",
"An electromagnetic (EM) field is a 2-form $F\\in \\Omega ^2(\\mathcal {S})$ that may be interpreted as ($-\\mathbf {i}$ times) the curvature of a principal connection on a $U(1)$ -bundle over $\\mathcal {S}$ .",
"$F$ is null if it is $g$ -orthogonal to itself and its Hodge dual.",
"Null EM fields are associated with electromagnetic radiation.",
"A gravitational field is the curvature of $g$ , called null if the Weyl tensor has Petrov type N (see [16] for a pleasant introduction to the Petrov classification) – the most degenerate type among non-conformally-flat metrics.",
"More generally, a metric is algebraically special if its Weyl tensor is at all degenerate; i.e., if it is of any Petrov type besides I.",
"It is helpful to think of the motivation for the present work in the context of the search for a theoretical framework for gravitational radiation analogous to that of electromagnetic radiation.",
"The study of gravitational waves was initiated by Einstein in the beginning of the twentieth century; a brief history with references is given in [28].",
"For our purposes, it suffices to join the story in medias res, when Trautman showed in [27] that gravitational fields satisfying a Sommerfeld radiation condition are asymptotically null.",
"The next year, Robinson reported to the Royaumont Conference that null EM and gravitational fields determine a foliation of spacetime by a family of curves known as a shear-free null geodesic congruence, or SFNGC.",
"These are discussed in detail in §REF .",
"Robinson also proved the converse for electromagnetic fields ([19]); i.e., a SFNGC gives rise to a null EM field.",
"Spacetimes admitting SFNGCs seemed to be natural candidates for a model of gravitational radiation, though the work [22] of Sachs established that these were not restricted to null gravitational fields.",
"Indeed, joint work [10] with Goldberg would show that, away from sources of mass-energy – that is, in a vacuum spacetime with a Ricci-flat metric – every non-flat, algebraically special metric admits a SFNGC tangent to principal null directions of algebraic multiplicity $>1$ .",
"Robinson and Trautman ([20]) produced a class of metrics corresponding to hypersurface-orthogonal SFNGC, including a model for radiation with spherical wavefronts.",
"Then Kerr sought metrics corresponding to SFNGC that were not hypersurface-orthogonal ([13]), and in the process generalized the Schwarzschild solution to incorporate angular momentum, generating a model for spinning black holes ([16]).",
"The Kerr metrics have Petrov type D. Kerr's name is also attached to a theorem relating SFNGC of flat (Minkowski) spacetime to the objects of study of this article.",
"In §REF , we offer a geometric overview of the correspondence between subsets of Minkowski spacetime and those of a 5-dimensional real hyperquadric in the spirit of Penrose's Twistor program ([17], [29]), emphasizing the various symmetry groups involved.",
"Then §REF delves into SFNGC for general spacetimes and explains their connection to CR geometry, following [21] and [15].",
"Finally, a sketch of the proof of the Kerr Theorem appears in §REF , using explicit coordinate calculations as in [24]."
],
[
"The Kerr Theorem",
"$\\mathbb {R}^n$ equipped with a symmetric, bilinear form ${\\tt b}$ of signature $(p,q)$ , $p+q=n$ , will be denoted $\\mathbb {R}^{p,q}$ .",
"The complex-linear extension of ${\\tt b}$ to $\\mathbb {C}^n$ is also called ${\\tt b}$ , but in the complexification its signature is no longer significant.",
"Hence, we reserve the notation $\\mathbb {C}^{p,q}$ for when $\\mathbb {C}^n$ carries a Hermitian form $ of signature $ (p,q)$, $ p+q=n$, unrelated to any underlying real form.$ The setting of special relativity is Minkowski spacetime $\\mathbb {M}$ , an affine space with modeling vector space $\\mathbb {R}^{1,3}$ .",
"Therefore, the Lorentz group $O(1,3)$ – and its affine extension, the Poincaré group – plays a central role in relativistic theories.",
"However, when a relativistic theory (such as the electrodynamics expressed in Maxwell's equations) exhibits conformal invariance, the corresponding group of symmetries is larger.",
"Conformal compactification of $\\mathbb {R}^{p,q}$ is achieved by affixing a point “at infinity\" for each one in the ${\\tt b}$ -null cone in order that inversion may be globally defined.",
"The resulting quadric is the real-projectivization of the $\\hat{{\\tt b}}$ -null cone in $\\mathbb {R}^{p+1,q+1}$ , whose bilinear form wears a hat to distinguish it from that of $\\mathbb {R}^{p,q}$ .",
"The group of (oriented) conformal symmetries of compactified Minkowski spacetime $\\mathbb {M}^c$ is the symmetry group of the $\\hat{{\\tt b}}$ -null cone in $\\mathbb {R}^{2,4}$ ; i.e., $SO(2,4)$ .",
"The Plücker embedding sends the Grassmannian $Gr(2,\\mathbb {C}^4)$ of complex 2-planes in $\\mathbb {C}^4$ into the complex-projective space $\\mathbb {P}(\\Lambda ^2\\mathbb {C}^4)=\\mathbb {CP}^5$ , and its image is the quadric given by the projectivization of the $\\hat{{\\tt b}}$ -null cone in $\\mathbb {C}^6$ .",
"This may be considered a geometric analog of the Lie algebra isomorphism $\\mathfrak {sl}_4\\mathbb {C}\\cong \\mathfrak {so}_6\\mathbb {C}$ .",
"Moreover, the Grassmannian $Gr^0(2,\\mathbb {C}^{2,2})\\subset Gr(2,\\mathbb {C}^4)$ of totally $-null 2-planes embeds onto $ Mc$ by analogy to the isomorphism $ su(2,2)so(2,4)$.$ Both $\\mathbb {CP}^3$ and $Gr(2,\\mathbb {C}^4)$ are partial flag manifolds associated to $\\mathbb {C}^4$ ; to these we add $F_{1,2}\\mathbb {C}^4$ consisting of pairs $(l,\\Pi )$ of a complex line and plane (respectively) satisfying $l\\subset \\Pi \\subset \\mathbb {C}^4$ .",
"With the projection maps $\\lambda (l,\\Pi )=l$ and $\\pi (l,\\Pi )=\\Pi $ we obtain the double fibration $@=1em{&_{\\lambda }@{->}[dl] F_{1,2}\\mathbb {C}^4^{\\pi }@{->}[dr] &\\\\\\mathbb {CP}^3& &Gr(2,\\mathbb {C}^4)}$ lying at the heart of Penrose's Twistor theory, which concerns the correspondence between subsets of $\\mathbb {CP}^3$ and $Gr(2,\\mathbb {C}^4)$ via the images of $\\lambda \\circ \\pi ^{-1}$ and $\\pi \\circ \\lambda ^{-1}$ .",
"To clarify some of the physical motivation for this framework, we restrict to $-isotropic flags $ F1,20C2,2F1,2C4$, so that (\\ref {Cdoublefiber}) becomes{\\begin{@align}{1}{-1}@=1em{&_{\\lambda }@{->}[dl] F^0_{1,2}\\mathbb {C}^{2,2}^{\\pi }@{->}[dr] &\\\\\\mathcal {Q}& &\\mathbb {M}^c=Gr^0(2,\\mathbb {C}^{2,2}),}\\end{@align}}where $ QCP3$ is the 5-dimensional real hyperquadric given by the complex projectivization of the $ -null cone $\\mathcal {N}\\subset \\mathbb {C}^{2,2}$ .",
"The trajectory of a massless particle in $\\mathbb {M}$ is tangent to a ${\\tt b}$ -null (affine) line, and each such line corresponds to a point in $\\mathcal {Q}$ .",
"Physicists refer to a foliation of $\\mathbb {M}$ by null lines as a null congruence, the relevance of which to the present work is stated in the Kerr Theorem A null congruence of $\\mathbb {M}$ corresponds to a CR submanifold of $\\mathcal {Q}$ if and only if it is shear-free.",
"The Kerr Theorem first appeared in print in [17], where it is stated that a shear-free null congruence is representable in $\\mathbb {CP}^3$ as the intersection of $\\mathcal {Q}$ with a complex-analytic surface (or a limiting case of such intersections); see also [18].",
"The proof sketch in §REF makes this construction explicit.",
"The version we've stated is closer to [15]."
],
[
"Shear-Free Null Geodesic Congruences",
"In this section we follow [21] and [15].",
"Let $\\mathcal {S}$ be a smooth, 4-dimensional manifold with a line bundle $K\\subset T\\mathcal {S}$ whose fibers are spanned by a nowhere-vanishing vector field $k\\in \\Gamma (K)$ , which determines a smooth flow $\\phi :I\\times \\mathcal {S}\\rightarrow \\mathcal {S},$ where $I\\subseteq \\mathbb {R}$ is some open interval containing zero.",
"For fixed $x\\in \\mathcal {S}$ and variable $t\\in I$ , $\\phi (t,x)$ is the integral curve of $k$ passing through $x$ when $t=0$ , and $\\mathcal {S}$ is foliated by these flow curves.",
"For fixed $t\\in I$ , $\\begin{aligned}\\phi _t:\\mathcal {S}&\\rightarrow \\mathcal {S}\\\\x&\\mapsto \\phi (t,x)\\end{aligned}$ is a diffeomorphism whose pushforward $\\phi _{t*}:T\\mathcal {S}\\rightarrow T\\mathcal {S}$ satisfies $\\phi _{t*}K_x=K_{\\phi (t,x)},$ and therefore descends to a well-defined map on the quotient bundle $T\\mathcal {S}/K\\rightarrow \\mathcal {S}$ .",
"Thus, the family $\\lbrace \\phi _t:t\\in I\\rbrace $ of diffeomorphisms provides linear isomorphisms between the spaces $T_{\\phi (t,x)}\\mathcal {S}/K_{\\phi (t,x)}$ for any fixed $x\\in \\mathcal {S}$ , and we see that the quotient bundle $T\\mathcal {S}/K$ has the same fibers as the tangent bundle of the leaf space $M $ ; i.e., the 3-dimensional quotient manifold of equivalence classes $[x]$ of points $x\\in \\mathcal {S}$ , where two points are equivalent if they lie in the same leaf of the foliation (the same flow curve), $&T_{[x]}M \\cong T_{\\phi (t,x)}\\mathcal {S}/K_{\\phi (t,x)}&\\forall t\\in I.$ Remark 3.1 In general, a 4-manifold need not admit a globally defined, non-vanishing tangent vector field, nor should the entire leaf space of a foliation necessarily inherit a global manifold structure.",
"However, our considerations are local in nature and we will continue to implicitly assume that $\\mathcal {S}$ is such that our constructions are well-defined.",
"In particular, we may also take $\\mathcal {S}$ to be orientable.",
"If $\\omega \\in \\Omega ^4(\\mathcal {S})$ is a volume form, then the contraction $k\\omega \\in \\Omega ^3(\\mathcal {S})$ vanishes on $K$ and descends to a 3-form on $T\\mathcal {S}/K$ .",
"Note that $k\\omega $ does not determine a well-defined volume form on $M $ unless $\\mathcal {L}_k\\omega =0$ , where $\\mathcal {L}_k$ denotes the Lie derivative along $k$ .",
"However, the sign of $k\\omega $ on any ordered basis of (REF ) is sufficient to determine whether a volume form on $M $ is positively or negatively oriented relative to $k\\omega $ , and so determines a choice of orientation on $M $ .",
"Suppose that $\\mathcal {S}$ is equipped with a non-degenerate metric $g\\in \\bigodot ^2T^*\\mathcal {S}$ .",
"For the moment, we make no assumptions about the signature of $g$ .",
"The one-form $\\kappa \\in \\Omega ^1(\\mathcal {S})$ dual to $k$ has as its kernel a rank-3 distribution $&\\kappa =kg&\\leadsto &&\\ker \\kappa =K^\\bot \\subset T\\mathcal {S}.$ Definition 3.2 The flow of $k$ is conformally geodesic if it preserves the distribution $K^\\bot $ , $&\\phi _{t*}K_x^\\bot =K_{\\phi (t,x)}^\\bot &\\forall t\\in I, x\\in \\mathcal {S}.$ Equivalently, the flow of $k$ is conformally geodesic when $&\\kappa \\wedge \\phi _t^*\\kappa =0&\\Rightarrow &&\\kappa \\wedge \\mathcal {L}_k\\kappa =0.$ Hence, a conformally geodesic flow not only identifies the fibers of $K$ along a flow curve as in (REF ), but also the fibers of $K^\\bot $ .",
"The implications of this for the leaf space $M $ depend on the metric properties of $k$ .",
"If $g(k_x,k_x)\\ne 0$ for every $x\\in \\mathcal {S}$ , then $T\\mathcal {S}=K\\oplus K^\\bot $ and $T_{[x]}M \\cong K^\\bot _{\\phi (t,x)}$ for every $t\\in I$ .",
"On the other hand, if $g$ has mixed signature and $g(k,k)=0$ , then $K\\subset K^\\bot $ and $M $ inherits a well-defined, rank-2 distribution $D\\subset TM \\quad \\text{with fibers}\\quad D_{[x]}\\cong K^\\bot _{\\phi (t,x)}/K_{\\phi (t,x)}\\quad \\forall t\\in I.$ We also have when $k$ is null that $\\kappa $ descends to the quotient bundle $T\\mathcal {S}/K$ , and the additional condition (REF ) that $k$ is conformally geodesic further implies that $\\kappa $ determines a well-defined, non-vanishing one-form (of the same name) on $M $ , which annihilates (REF ).",
"Remark 3.3 Condition (REF ) is always invariant under conformal scaling of $\\kappa $ , and when $k$ is null it is even invariant under scaling of $k$ by a non-vanishing function, which effects a reparameterization of the flow curves of $k$ .",
"Henceforth, we restrict to the case that $g$ has Lorentzian signature $(1,3)$ and $k$ is $g$ -null with a conformally geodesic flow.",
"The foliation of $\\mathcal {S}$ by flow curves is now called a null geodesic congruence, the fibers of the quotient bundle $K^\\bot /K$ are called screen spaces, and the geometry of the null congruence may be understood intuitively in terms of the following illustration regarding optical scalars ([16]).",
"In relativity, light propagates in null directions; imagine a beam of light casting the shadow of an opaque disk onto a 2-dimensional screen placed orthogonal to its (null) direction.",
"As the screen is moved along the flow curve, this circular image might be rotated, enlarged, or distorted into an ellipse of greater eccentricity.",
"If the latter, non-conformal distortion does not occur, the null congruence is shear-free.",
"The precise geometric definition applies to arbitrary conformally geodesic flows.",
"Definition 3.4 A conformally geodesic flow is shear-free if it preserves the conformal class of $g$ restricted to $K^\\bot $ ; i.e., for any $t\\in I$ and $x\\in \\mathcal {S}$ , there is some $s\\in \\mathbb {R}$ , $s>0$ such that $\\phi _{t}^*(g|_{K^\\bot _{\\phi (t,x)}})=sg|_{K^\\bot _x},$ so that in particular, $\\mathcal {L}_kg=ag+\\kappa \\odot \\alpha $ for some $a\\in C^\\infty (\\mathcal {S})$ and $\\alpha \\in \\Omega ^1(\\mathcal {S})$ .",
"Remark 3.5 Using general properties of the Lie derivative, it is straightforward to confirm that (REF ) is maintained under rescaling of $k$ by a non-vanishing function, albeit for different $a,\\alpha $ .",
"Along with Remark REF , this shows that a shear-free null geodesic congruence (SFNGC) is independent of the choice of $k$ spanning $K$ .",
"Note that if $k$ is $g$ -null, it is also $\\tilde{g}$ -null, where $&\\tilde{g}=fg+\\kappa \\odot \\xi ,&0<f\\in C^\\infty (\\mathcal {S}),\\ \\xi \\in \\Omega ^1(\\mathcal {S}),$ and $\\tilde{\\kappa }=k\\tilde{g}$ is a rescaling of $\\kappa $ .",
"Here again, the properties of the Lie derivative show that $\\tilde{g}$ satisfies (REF ) whenever $g$ does, so the class (REF ) of metrics associated to given SFNGC is manifestly larger than a conformal class of metrics.",
"For null $k$ , $g|_K=0$ and we see from (REF ) that a SFNGC determines a well-defined conformal structure on the subbundle (REF ) of the leaf space.",
"As such, we can define an almost-complex structure on $M $ , $&J:D\\rightarrow D,&J^2=-\\mathbb {1},$ by taking $J_{[x]}$ to be a rotation by $\\frac{\\pi }{2}$ in $D_{[x]}$ .",
"(There are two choices for the direction of the rotation – clockwise or counterclockwise – in each $D_{[x]}$ .",
"Take the one which is positively oriented for the orientation induced by the semi-Riemannian volume form on $\\mathcal {S}$ ; see Remark REF .)",
"Thus we see that a SFNGC induces a CR structure on the 3-dimensional leaf space $M $ , with $\\kappa $ serving as a characteristic form.",
"To this we may add a CR form $\\eta \\in \\Omega ^1(M ,\\mathbb {C})$ so that $\\kappa ,\\eta ,\\overline{\\eta }$ is a 0-adapted CR coframing.",
"CR integrability is automatic in dimension three, but pseudo-convexity is not.",
"In the Levi-flat case, $\\kappa \\wedge \\text{d}\\kappa =0$ and $M $ is foliated by complex curves; the original curves of our SFNGC are hypersurface-orthogonal, as one would expect from a spacetime featuring radiating wave fronts.",
"More interesting from a CR perspective is the Levi-nondegenerate case corresponding to “twisting\" congruences $\\kappa \\wedge \\text{d}\\kappa \\ne 0$ , including the Kerr spacetime which describes a rotating black hole.",
"Conversely, suppose that $M $ is a 3-dimensional CR manifold with a 0-adapted coframing $\\kappa ,\\eta ,\\overline{\\eta }$ , and set $\\mathcal {S}=\\mathbb {R}\\times M $ .",
"We use the same names $\\kappa ,\\eta ,\\overline{\\eta }$ to denote their pullbacks along the projection $\\mathcal {S}\\rightarrow M $ .",
"Take $k\\in \\Gamma (T\\mathcal {S})$ to be $k=\\frac{\\partial }{\\partial r}$ where $r$ is the Cartesian coordinate of $\\mathbb {R}$ , and choose any $\\rho \\in \\Omega ^1(\\mathcal {S})$ with $\\rho (k)=1$ ; i.e., $\\rho \\equiv \\text{d}r\\mod {\\lbrace }\\kappa ,\\eta ,\\overline{\\eta }\\rbrace $ .",
"The metric $g=\\kappa \\odot \\rho -\\eta \\odot \\overline{\\eta }$ has signature $(1,3)$ and satisfies $g(k,k)=0$ as well as $\\kappa =kg$ .",
"The flow curves of $k$ are the $r$ -coordinate curves of $\\mathcal {S}$ , and Lie derivatives along $k$ can be computed via H. Cartan's formula, yielding $&\\mathcal {L}_k\\kappa =\\mathcal {L}_k\\eta =\\mathcal {L}_k\\overline{\\eta }=0,&\\mathcal {L}_k\\rho \\equiv 0\\mod {\\lbrace }\\kappa ,\\eta ,\\overline{\\eta }\\rbrace ,$ whence both conditions (REF ) and (REF ) are verified.",
"This establishes a correspondence $\\lbrace \\text{SFNGC on 4-manifolds}\\rbrace \\stackrel{\\text{(local)}}{\\longleftrightarrow }\\lbrace \\text{CR structure on 3-manifolds}\\rbrace $ Now suppose we submit our coframing on $M $ to a 0-adapted transformation, $&\\left[\\begin{array}{c}\\kappa ^{\\prime }\\\\\\eta ^{\\prime }\\end{array}\\right]=\\left[\\begin{array}{cc}u&0\\\\b&a\\end{array}\\right]\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right];&u\\in C^\\infty (M ),\\ a,b\\in C^\\infty (M ,\\mathbb {C}),\\ u,a\\ne 0,$ and write the metric $\\tilde{g}$ as in (REF ).",
"In terms of our original coframing, we obtain $\\tilde{g}&=\\kappa ^{\\prime }\\odot \\rho -\\eta ^{\\prime }\\odot \\overline{\\eta }^{\\prime }\\\\&=|b|^2g+\\kappa \\odot ((u-|b|^2)\\rho -|b|^2\\kappa -a\\overline{b}\\eta -\\overline{a}b\\overline{\\eta })\\\\&=fg+\\kappa \\odot \\xi $ as in (REF ).",
"Following our initial selection of $\\rho $ , the ambiguity of the metric $g$ due to our choice (REF ) of coframing on $M $ is measured by 5 real functions of 3 variables, rather than the 5 functions of 4 variables apparent in the full class of metrics discussed in Remark REF .",
"However, if we allow the fiber coordinates $u,a,b$ of our G-structure (REF ) to vary with $r$ , then the structure group of our bundle of 0-adapted CR frames exactly parameterizes the class of metrics associated to this SFNGC (note that the Lie derivatives along $k$ of our CR coframing will no longer vanish identically as in (REF ) if $u,a,b$ depend on $r$ ; (REF ) and (REF ) will hold nonetheless).",
"The correspondence (REF ) raises several questions, the first of which is presented as Problem 1 in [15], and the second of which was communicated to the author by Paweł Nurowski: Which CR structures lift to a SFNGC whose class (REF ) of metrics contains a solution to Einstein's vacuum field equations?",
"The Goldberg-Sachs theorem says that there are two SFNGC associated to a metric of Petrov type D; are the two corresponding CR structures always equivalent?",
"Which CR structures lift to a SFNGC whose class (REF ) of metrics contains one that is (conformally) flat?",
"The Kerr Theorem offers the answer to the final question: those that are embedded within the real hyperquadric $\\mathcal {Q}$.",
"The present article attempts to answer the inevitable follow-up question: which are those?"
],
[
"Kerr Theorem Proof Sketch",
"We argue as in [24].",
"Remark REF says that we can scale the vector field $k\\in \\Gamma (K)$ tangent to our SFNGC at will, so we are less occupied with null vectors tangent to Minkowski spacetime $\\mathbb {M}$ than we are with null directions.",
"The projectivized ${\\tt b}$ -null cone in $\\mathbb {R}^{1,3}$ is the (Riemann) 2-sphere $\\mathbb {CP}^1$ , hence a single stereographic coordinate $\\zeta \\in \\mathbb {C}$ suffices to parameterize all null tangent directions in each $T_x\\mathbb {M}$ , with the exception of one direction “at infinity.\"",
"In standard coordinates $(x_0,x_1,x_2,x_3)\\in \\mathbb {M}$ , the metric is diagonal, $g=\\text{d}x_0\\odot \\text{d}x_0-\\text{d}x_1\\odot \\text{d}x_1-\\text{d}x_2\\odot \\text{d}x_2-\\text{d}x_3\\odot \\text{d}x_3.$ Introducing null and complexified coordinates $&u=x_0-x_3,&v=x_0+x_3,&&w=x_1+\\mathbf {i}x_2,&&\\overline{w}=x_1-\\mathbf {i}x_2,$ brings $g$ into the form $g=\\text{d}u\\odot \\text{d}v-\\text{d}w\\odot \\text{d}\\overline{w}.$ A general null vector field and its dual form are, up to real scale, $&k=\\frac{\\partial }{\\partial v}-\\zeta \\frac{\\partial }{\\partial w}-\\overline{\\zeta }\\frac{\\partial }{\\partial \\overline{w}}+|\\zeta |^2\\frac{\\partial }{\\partial u},&\\kappa =\\text{d}u+\\zeta \\text{d}\\overline{w}+\\overline{\\zeta }(\\text{d}w+\\zeta \\text{d}v),&&\\zeta \\in C^\\infty (\\mathbb {M},\\mathbb {C}),$ while the SFNGC “at infinity\" is given by the $u$ -coordinate lines.",
"In the latter case, the remaining coordinates $v,w,\\overline{w}$ descend to the leaf-space of $u$ -coordinate lines, which is the Levi-flat $\\mathbb {R}\\times \\mathbb {C}$ , and this corresponds to a CR structure in $\\mathcal {Q}$ that is tangent to a complex curve.",
"In the general case we can write $&g=\\kappa \\odot \\text{d}v-\\eta \\odot \\overline{\\eta },&\\eta =\\text{d}w+\\zeta \\text{d}v,$ and after computing Lie derivatives, $&\\mathcal {L}_k\\kappa =\\text{d}\\zeta (k)\\text{d}\\overline{w}+\\text{d}\\overline{\\zeta }(k)\\text{d}w+(\\overline{\\zeta }\\text{d}\\zeta (k)+\\zeta \\text{d}\\overline{\\zeta }(k))\\text{d}v,&\\mathcal {L}_k\\eta =\\overline{\\mathcal {L}_k\\overline{\\eta }}=-\\text{d}\\zeta +\\text{d}\\zeta (k)\\text{d}v,$ we see that conditions (REF ) and (REF ) hold when $0&=\\text{d}\\zeta (k)=\\text{d}\\overline{\\zeta }(k),\\\\0&=\\kappa \\wedge \\eta \\wedge \\mathcal {L}_k\\eta ,$ where the second becomes equivalent to $\\text{d}(u+\\zeta \\overline{w})\\wedge \\text{d}(w+\\zeta v)\\wedge \\text{d}\\zeta =0.$ Name the three $\\mathbb {C}$ -valued functions $&z_1=u+\\zeta \\overline{w},&z_2=w+\\zeta v,&&z_3=\\zeta ,$ and observe that $\\mathbf {i}(z_1-\\overline{z}_1+z_2\\overline{z}_3-z_3\\overline{z}_2)=0.$ If $Z_0,Z_1,Z_2,Z_3$ are coordinates for $\\mathbb {C}^{2,2}$ with the Hermitian form $Z,W)=\\mathbf {i}(Z_1\\overline{W}_0-Z_0\\overline{W}_1+Z_2\\overline{W}_3-Z_3\\overline{W}_2),$ then (REF ) describes the projectivization in $\\mathbb {CP}^3$ of the $-null cone $ Z,Z)=0$ in the affine coordinate neighborhood $ Z00$, via projective coordinates $ [Z0:Z1:Z3:Z4]=[1:z1:z2:z3]$.",
"Moreover, if $ =z3$ is implicitly defined by $ H(z1,z2,z3)=0$, where $ H$ is holomorphic (and not constant) in $ z1,z2,z3$, then the 3-form $ dz1dz2dz3$ vanishes on the subbundle $ dH=0$ of the complexified tangent bundle of $ CP3$, and over the hyperquadric $ Q$ locally defined by (\\ref {Qlocalhypersurfaceeqn}), this is exactly the shear-free condition (\\ref {minksf}).",
"The level set $ H=0$ is a complex-analytic surface in $ CP3$ whose intersection with the real hyperquadric $ Q$ defines a 3-dimensional CR submanifold of $ Q$.$ For the remaining details, consult [17], [24], or [18]."
],
[
"Hermitian Frames of $\\mathbb {C}^4$",
"Let $\\underline{e}=(e_0,e_1,e_2,e_3)$ denote the standard basis of column vectors for $\\mathbb {C}^4$ and recall that $\\mathbf {e}$ is the natural exponential and $\\mathbf {i}=\\sqrt{-1}$ .",
"Fix index ranges and constants $&0\\le i,j\\le 3,&\\epsilon =\\pm 1,&&\\delta _\\epsilon =\\left\\lbrace {\\begin{matrix}0,&\\epsilon =1\\\\1,&\\epsilon =-1\\end{matrix}}\\right.&&\\Rightarrow \\epsilon =(-1)^{\\delta _\\epsilon }.$ The Hermitian form $ of signature $ (3-,1+)$ acts on vectors $z=ziei$ and $w=wjej$ via{\\begin{@align*}{1}{-1}{\\tt w},{\\tt z})&=\\mathbf {i}(\\overline{w}^0z^3-\\overline{w}^3z^0)+\\overline{w}^1z^1+\\epsilon \\overline{w}^2z^2.\\end{@align*}}A \\emph {Hermitian frame} is an ordered, complex basis $v=(v0,v1,v2,v3)$ of $ C4$ such that{\\begin{@align}{1}{-1}{\\tt v}_i,{\\tt v}_j)=(\\pm 1)^{\\delta _\\epsilon }\\left[\\begin{array}{cccc}0&0&0&\\mathbf {i}\\\\0&1&0&0\\\\0&0&\\epsilon &0\\\\-\\mathbf {i}&0&0&0\\end{array}\\right].\\end{@align}}Denote by $ H$ the collection of all Hermitian frames, and note that $ H$ is identified with the unitary group $ U(3-,1+)$ by fixing $e$ as the identity and taking $v$ to be the matrix whose column vectors are the basis vectors of $v$.$ Remark 4.1 The notation $(\\pm 1)^{\\delta _\\epsilon }$ in () means the sign is allowed to change when $\\epsilon =-1$ but not when $\\epsilon =1$ .",
"This is to avoid privileging either of ${\\tt v}_1,{\\tt v}_2$ as necessarily positive-definite when $\\epsilon =-1$ .",
"Both vectors are called “orthonormal\" regardless of the value of $\\epsilon $ .",
"The symbol ${\\tt v}_i$ will also refer to the $\\mathbb {C}^4$ -valued function mapping a Hermitian frame to its $i^{\\text{th}}$ basis vector: ${\\tt v}_i\\in C^\\infty (\\hat{\\mathcal {H}},\\mathbb {C}^4)$ .",
"These functions are differentiated via the Maurer-Cartan (MC) forms of $U(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ , $&\\text{d}{\\tt v}_i=\\mu ^j_i{\\tt v}_j,&\\mu \\in \\Omega ^1(\\hat{\\mathcal {H}},\\mathfrak {u}(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )).$ In our representation of this Lie algebra, we can write $\\mu =\\left[\\begin{array}{cccc}\\lambda &-\\mathbf {i}\\overline{\\xi }&-\\mathbf {i}\\overline{\\phi }_2&\\psi \\\\\\eta &\\mathbf {i}\\rho &-\\overline{\\phi }_1&\\xi \\\\\\zeta &\\epsilon \\phi _1&\\mathbf {i}\\tau &\\epsilon \\phi _2\\\\\\kappa &\\mathbf {i}\\overline{\\eta }&\\epsilon \\mathbf {i}\\overline{\\zeta }&-\\overline{\\lambda }\\end{array}\\right],$ where $\\kappa ,\\psi \\in \\Omega ^1(\\hat{\\mathcal {H}})$ and the rest are $\\mathbb {C}$ -valued, so that $\\overline{\\mu }^t=0$ .",
"The MC equations $\\text{d}\\mu =-\\mu \\wedge \\mu $ are $\\begin{aligned}\\text{d}\\kappa &=\\mathbf {i}\\eta \\wedge \\overline{\\eta }+(\\lambda +\\overline{\\lambda })\\wedge \\kappa +\\epsilon \\mathbf {i}\\zeta \\wedge \\overline{\\zeta },\\\\\\text{d}\\eta &=(\\lambda -\\mathbf {i}\\rho )\\wedge \\eta -\\xi \\wedge \\kappa +\\overline{\\phi }_1\\wedge \\zeta ,\\\\\\text{d}\\psi &=\\psi \\wedge (\\lambda +\\overline{\\lambda })-\\mathbf {i}\\xi \\wedge \\overline{\\xi }-\\epsilon \\mathbf {i}\\phi _2\\wedge \\overline{\\phi }_2,\\\\\\text{d}\\xi &=\\psi \\wedge \\eta +\\xi \\wedge (\\overline{\\lambda }+\\mathbf {i}\\rho )+\\epsilon \\overline{\\phi }_1\\wedge \\phi _2,\\\\\\text{d}\\lambda &=\\mathbf {i}\\overline{\\xi }\\wedge \\eta +\\mathbf {i}\\overline{\\phi }_2\\wedge \\zeta -\\psi \\wedge \\kappa ,\\\\\\text{d}\\rho &=\\epsilon \\mathbf {i}\\phi _1\\wedge \\overline{\\phi }_1-\\overline{\\xi }\\wedge \\eta -\\xi \\wedge \\overline{\\eta },\\\\\\text{d}\\phi _1&=\\epsilon \\mathbf {i}\\zeta \\wedge \\overline{\\xi }+\\mathbf {i}(\\rho -\\tau )\\wedge \\phi _1-\\mathbf {i}\\phi _2\\wedge \\overline{\\eta },\\\\\\text{d}\\phi _2&=\\epsilon \\psi \\wedge \\zeta +\\phi _2\\wedge (\\overline{\\lambda }+\\mathbf {i}\\tau )+\\xi \\wedge \\phi _1,\\\\\\text{d}\\zeta &=(\\lambda -\\mathbf {i}\\tau )\\wedge \\zeta -\\epsilon \\phi _1\\wedge \\eta -\\epsilon \\phi _2\\wedge \\kappa ,\\\\\\text{d}\\tau &=\\zeta \\wedge \\overline{\\phi }_2+\\overline{\\zeta }\\wedge \\phi _2-\\epsilon \\mathbf {i}\\phi _1\\wedge \\overline{\\phi }_1.\\end{aligned}$ Define $\\det :\\hat{\\mathcal {H}}\\rightarrow U(1)\\subset \\mathbb {C}$ as usual by ${\\tt v}_0\\wedge {\\tt v}_1\\wedge {\\tt v}_2\\wedge {\\tt v}_3=\\det (\\underline{{\\tt v}})e_0\\wedge e_1\\wedge e_2\\wedge e_3,$ and let $\\mathcal {H}\\subset \\hat{\\mathcal {H}}$ denote the collection of oriented Hermitian frames satisfying $\\det (\\underline{{\\tt v}})=1$ .",
"From any Hermitian frame $\\underline{{\\tt v}}$ one obtains an oriented Hermitian frame in a variety of ways; e.g., $\\underline{{\\tt v}}\\mapsto ({\\tt v}_0,{\\tt v}_1,\\overline{\\det (\\underline{{\\tt v}})}{\\tt v}_2,{\\tt v}_3),$ in this case preserving the vectors ${\\tt v}_0,{\\tt v}_1,{\\tt v}_3$ .",
"$\\mathcal {H}$ is identified with the special unitary group $SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ .",
"Keeping the same names after pulling back the MC forms (REF ) along the inclusion $\\mathcal {H}\\hookrightarrow \\hat{\\mathcal {H}}$ exhibits the trace-free condition of the special unitary Lie algebra $\\mathfrak {su}(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ , $\\mathbf {i}\\rho +\\mathbf {i}\\tau +\\lambda -\\overline{\\lambda }=0.$ If two Hermitian frames $\\underline{{\\tt v}}$ and $\\tilde{\\underline{{\\tt v}}}$ share the same ${\\tt v}_0=\\tilde{{\\tt v}}_0$ , then they differ by a transformation $\\begin{aligned}&\\left[\\begin{array}{cccc}{\\tt v}_0&\\tilde{{\\tt v}}_1&\\tilde{{\\tt v}}_2&\\tilde{{\\tt v}}_3\\end{array}\\right]=\\left[\\begin{array}{cccc}{\\tt v}_0&{\\tt v}_1&{\\tt v}_2&{\\tt v}_3\\end{array}\\right]p_0,\\\\&p_0=\\left[\\begin{array}{cccr}1&-\\mathbf {i}(a_1\\overline{c}_1+\\epsilon a_2\\overline{c}_2)&\\epsilon \\mathbf {i}\\mathbf {e}^{\\mathbf {i}r}(\\overline{a}_2\\overline{c}_1-\\overline{a}_1\\overline{c}_2)&c_0(\\pm 1)^{\\delta _\\epsilon } \\\\0&a_1&-\\epsilon \\mathbf {e}^{\\mathbf {i}r}\\overline{a}_2&c_1(\\pm 1)^{\\delta _\\epsilon }\\\\0&a_2&\\mathbf {e}^{\\mathbf {i}r}\\overline{a}_1&c_2(\\pm 1)^{\\delta _\\epsilon }\\\\0&0&0&(\\pm 1)^{\\delta _\\epsilon }\\end{array}\\right]\\begin{array}{c}a_1,a_2,c_1,c_2\\in \\mathbb {C},\\\\|a_1|^2+\\epsilon |a_2|^2=(\\pm 1)^{\\delta _\\epsilon },\\\\c_0=t-\\tfrac{\\mathbf {i}}{2}(|c_1|^2+\\epsilon |c_2|^2),\\\\r,t\\in \\mathbb {R}.\\end{array}\\end{aligned}$ (See Remark REF regarding the sign ambiguity $(\\pm 1)^{\\delta _\\epsilon }$ when $\\epsilon =-1$ .)",
"All such $p_0\\in U(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ form a subgroup that we call $\\mathcal {P}_0$ .",
"Apparently, $\\det (p_0)=\\mathbf {e}^{\\mathbf {i}r}$ .",
"More generally, if the first basis vectors of two Hermitian frames $\\underline{{\\tt v}}$ and $\\tilde{\\underline{{\\tt v}}}$ are $\\mathbb {C}$ -collinear – i.e., $\\tilde{{\\tt v}}_0=l{\\tt v}_0$ for some $l\\in \\mathbb {C}\\setminus \\lbrace 0\\rbrace $ – then $&\\tilde{\\underline{{\\tt v}}}=\\underline{{\\tt v}} p_lp_0;&p_l=\\left[\\begin{array}{cccc}l&0&0&0\\\\0&1&0&0\\\\0&0&1&0\\\\0&0&0&\\overline{l}^{-1}\\end{array}\\right],\\quad l\\in \\mathbb {C}\\setminus \\lbrace 0\\rbrace ,&&p_0 \\text{ as in (\\ref {P0}).",
"}$ All such $p_l\\in U(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ form an abelian subgroup that we call $\\mathcal {P}_l$ .",
"We also name $&\\hat{\\mathcal {P}}=\\mathcal {P}_l\\mathcal {P}_0,&\\mathcal {P}=\\hat{\\mathcal {P}}\\cap SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon );$ the latter consists of multiples $p_lp_0$ with the $r$ -coordinate of (REF ) constrained by $\\mathbf {e}^{\\mathbf {i}r}=\\overline{l}l^{-1}$ , and it is exactly the parabolic subgroup $\\mathcal {P}\\subset SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ mentioned in the Tanaka-Chern-Moser Classification for CR-dimension $n=2$ .",
"We conclude this section by recording the dual version of the $\\hat{\\mathcal {P}}$ -transformation of Hermitian frames; namely, if we use superscripts to denote the dual coframe of $\\underline{{\\tt v}}$ (i.e., ${\\tt v}^i({\\tt v}_j)=\\delta ^i_j$ ) then the dual coframes of two Hermitian frames $\\tilde{\\underline{{\\tt v}}}$ and $\\underline{{\\tt v}}$ with $\\tilde{{\\tt v}}_0=l{\\tt v}_0$ are related by $&\\left[\\begin{array}{c}\\tilde{{\\tt v}}^0\\\\\\tilde{{\\tt v}}^1\\\\\\tilde{{\\tt v}}^2\\\\\\tilde{{\\tt v}}^3\\end{array}\\right]={p_0}^{-1}{p_l}^{-1}\\left[\\begin{array}{c}{\\tt v}^0\\\\{\\tt v}^1\\\\{\\tt v}^2\\\\{\\tt v}^3\\end{array}\\right];&p_lp_0\\in \\hat{\\mathcal {P}}.$"
],
[
"First Adaptations",
"Let $\\mathcal {N}\\subset \\mathbb {C}^4$ be the null-cone of $,{\\begin{@align*}{1}{-1}&\\mathcal {N}=\\lbrace {\\tt v}\\in \\mathbb {C}^4\\setminus \\lbrace \\left[{\\begin{matrix}0&0&0&0\\end{matrix}}\\right]^t\\rbrace :{\\tt v},{\\tt v})=0\\rbrace &\\Longrightarrow &&T_{{\\tt v}}\\mathcal {N}=\\lbrace {\\tt w}\\in \\mathbb {C}^4:\\Re ({\\tt v},{\\tt w}))=0\\rbrace .\\end{@align*}}We recognize the following distinguished subbundles of $ TN$ by their fibers,{\\begin{@align*}{1}{-1}&L_{{\\tt v}}=\\lbrace l{\\tt v}:l\\in \\mathbb {C}\\rbrace ,&\\subset &&L_{{\\tt v}}^\\bot =\\lbrace {\\tt w}\\in \\mathbb {C}^4:{\\tt v},{\\tt w})=0\\rbrace .\\end{@align*}}Define the projection{\\begin{@align*}{1}{-1}h_\\mathcal {N}:\\hat{\\mathcal {H}}&\\rightarrow \\mathcal {N}\\\\\\underline{{\\tt v}}&\\mapsto {\\tt v}_0,\\end{@align*}}which identifies $ N$ with the homogeneous space $ U(3-,1+)/P0$.",
"For each $v=(v0,v1,v2,v3)hN-1(v0)$, we have spanning sets{\\begin{@align}{1}{-1}&\\langle {\\tt v}_0\\rangle _\\mathbb {C}=L_{{\\tt v}_0},&\\langle {\\tt v}_0,{\\tt v}_1,{\\tt v}_2\\rangle _\\mathbb {C}=L_{{\\tt v}_0}^\\bot ,&&\\langle {\\tt v}_0,{\\tt v}_1,{\\tt v}_2,\\mathbf {i}{\\tt v}_0,\\mathbf {i}{\\tt v}_1,\\mathbf {i}{\\tt v}_2,{\\tt v}_3\\rangle _\\mathbb {R}=T_{{\\tt v}_0}\\mathcal {N}.\\end{@align}}Conversely, one can assign an adapted basis of $ Tv0N$ to each $v0N$ in order to define a section $ s:NH$ as follows: take $v0$ itself to span $ Lv0$, choose two orthonormal vectors $v1,v2Lv0$ (varying smoothly in a neighborhood of $v0$), and then $v3$ is uniquely determined to complete the Hermitian frame $ s(v0)=(v0,v1,v2,v3)$.$ Remark 4.2 Remark REF explains why the $(\\pm 1)^{\\delta _\\epsilon }$ ambiguity in () and (REF ) allows us to choose any non-null ${\\tt v}_1\\in L_{{\\tt v}_0}^\\bot $ after ${\\tt v}_0$ is fixed.",
"The order of the basis elements in a frame reflects the ascending filtration $L\\subset L^\\bot \\subset T\\mathcal {N}$ , but of course it is possible to first choose ${\\tt v}_3$ with ${\\tt v}_0,{\\tt v}_3)=\\mathbf {i}(\\pm 1)^{\\delta _\\epsilon }$ and then take orthonormal ${\\tt v}_1,{\\tt v}_2$ in the orthogonal complement of $L_{{\\tt v}_0}\\oplus \\langle {\\tt v}_3\\rangle _\\mathbb {R}$ .",
"With a section $s:\\mathcal {N}\\rightarrow \\hat{\\mathcal {H}}$ , we pull back $\\text{d}{\\tt v}_0\\in \\Omega ^1(\\hat{\\mathcal {H}},\\mathbb {C}^4)$ from (REF ) to get $s^*\\text{d}{\\tt v}_0=s^*\\lambda {\\tt v}_0+s^*\\eta {\\tt v}_1+s^*\\zeta {\\tt v}_2+s^*\\kappa {\\tt v}_3,$ but $s^*{\\tt v}_0$ is just the identity map on $\\mathcal {N}$ , so its differential is the identity on $T\\mathcal {N}$ .",
"Thus, $&s^*\\lambda ={\\tt v}^0,&s^*\\eta ={\\tt v}^1,&&s^*\\zeta ={\\tt v}^2,&&s^*\\kappa ={\\tt v}^3.$ Citing (REF ) with $l=1$ , we can say in other words that $\\kappa \\in \\Omega ^1(\\hat{\\mathcal {H}})$ and $\\eta ,\\zeta ,\\lambda \\in \\Omega ^1(\\hat{\\mathcal {H}},\\mathbb {C})$ are semi-basic, tautological 1-forms for the (co)frame bundle fibration $\\mathcal {P}_0\\hookrightarrow U(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )\\stackrel{h_\\mathcal {N}}{\\longrightarrow }\\mathcal {N}.$ Write $\\mathbb {P}:\\mathbb {C}^4\\setminus \\lbrace \\left[{\\begin{matrix}0&0&0&0\\end{matrix}}\\right]^t\\rbrace \\rightarrow \\mathbb {CP}^3$ for the canonical complex projection, so the real hyperquadric is the image $\\mathcal {Q}=\\mathbb {P}(\\mathcal {N})$ , and $T_{\\mathbb {P}({\\tt v}_0)}\\mathcal {Q}=\\mathbb {P}_*(T_{{\\tt v}_0}\\mathcal {N})\\cong L_{{\\tt v}_0}^*\\otimes T_{{\\tt v}_0}\\mathcal {N}/L_{{\\tt v}_0}$ for ${\\tt v}_0\\in \\mathcal {N}$ .",
"The latter can be made explicit for a frame $\\underline{{\\tt v}}\\in h_\\mathcal {N}^{-1}({\\tt v}_0)$ , namely $T_{\\mathbb {P}({\\tt v}_0)}\\mathcal {Q}=\\langle \\mathbb {P}_*{\\tt v}_1,\\mathbb {P}_*{\\tt v}_2\\rangle _\\mathbb {C}\\oplus \\langle \\mathbb {P}_*{\\tt v}_3\\rangle _\\mathbb {R}\\cong \\langle {\\tt v}^0\\otimes {\\tt v}_1,{\\tt v}^0\\otimes {\\tt v}_2\\rangle _\\mathbb {C}\\oplus \\langle {\\tt v}^0\\otimes {\\tt v}_3\\rangle _\\mathbb {R}.$ Remark 4.3 By (REF ) and (REF ) we see that for $\\tilde{\\underline{{\\tt v}}}=\\underline{{\\tt v}}p_l\\in h_\\mathcal {N}^{-1}(\\mathbb {P}^{-1}({\\tt v}_0))$ , $\\tilde{{\\tt v}}^0\\otimes \\tilde{{\\tt v}}_3=|l|^{-2}{\\tt v}^0\\otimes {\\tt v}_3$ .",
"When $\\epsilon =-1$ , a $p_0$ -transformation (REF ) with $(\\pm 1)^{\\delta _\\epsilon }=-1$ changes the sign of ${\\tt v}^0\\otimes {\\tt v}_3$ (see Remark REF ).",
"It's clear from () that the fibers of $T\\mathcal {N}\\rightarrow \\mathcal {N}$ vary along those of $\\mathcal {N}\\rightarrow \\mathcal {Q}$ .",
"Nonetheless, Remark REF reassures us that (REF ) is well-defined.",
"Moreover, it always holds that $L_{{\\tt v}_0}=L_{\\tilde{{\\tt v}}_0}$ and $L_{{\\tt v}_0}^\\bot =L_{\\tilde{{\\tt v}}_0}^\\bot $ when $\\mathbb {P}({\\tt v}_0)=\\mathbb {P}(\\tilde{{\\tt v}}_0)$ .",
"In particular, for any ${\\tt v}_0\\in \\mathcal {N}$ , $L_{{\\tt v}_0}=\\ker \\mathbb {P}_*|_{T_{{\\tt v}_0}\\mathcal {N}}$ , and there is a well-defined, $\\mathbb {R}$ -corank-1 subbundle $D_{\\mathbb {P}({\\tt v}_0)}=\\mathbb {P}_*L_{{\\tt v}_0}^\\bot \\subset T_{\\mathbb {P}({\\tt v}_0)}\\mathcal {Q}.$ Scalar multiplication by $\\mathbf {i}$ in $\\mathbb {C}^4$ defines an endomorphism $J:L_{{\\tt v}_0}^\\bot \\rightarrow L_{{\\tt v}_0}^\\bot $ satisfying $J^2=-\\mathbb {1}$ , of which $L_{{\\tt v}_0}$ is an invariant subspace, hence $J$ descends to a well-defined almost-complex structure $J:D\\rightarrow D$ .",
"The Tanaka-Chern-Moser Classification for $n=2$ ensures that $\\mathcal {H}$ realizes the Cartan bundle of the CR structure $(\\mathcal {Q},D,J)$ , as encoded in the fibration $&\\mathcal {P}\\hookrightarrow SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )\\stackrel{h_\\mathcal {Q}}{\\longrightarrow }\\mathcal {Q};&h_\\mathcal {Q}=\\mathbb {P}\\circ h_\\mathcal {N}.$ It will be convenient to work instead with $\\hat{\\mathcal {P}}\\rightarrow \\hat{\\mathcal {H}}\\stackrel{h_\\mathcal {Q}}{\\longrightarrow }\\mathcal {Q}$ in order to factor through (REF ): $\\begin{tikzcd}\\mathcal {P}_0 [hook]{r}{\\underline{{\\tt v}}\\mapsto \\underline{{\\tt v}}p_0}& U(3-\\delta _\\epsilon ,1+\\delta _\\epsilon ){d}{h_\\mathcal {N}}\\\\\\mathcal {P}_l\\cong \\mathbb {C}\\setminus \\lbrace 0\\rbrace {r}{{\\tt v}_0\\mapsto l{\\tt v}_0}&\\mathcal {N}{d}{\\mathbb {P}}\\\\&\\mathcal {Q}\\end{tikzcd}$ To this end, let $\\varsigma :\\mathcal {Q}\\rightarrow \\hat{\\mathcal {H}}$ be a section that factors through $s:\\mathcal {N}\\rightarrow \\hat{\\mathcal {H}}$ as in (REF ).",
"Then $\\varsigma ^*\\kappa \\in \\Omega ^1(\\mathcal {Q})$ is a characteristic form annihilating (REF ), which along with $\\varsigma ^*\\eta ,\\varsigma ^*\\zeta \\in \\Omega ^1(\\mathcal {Q},\\mathbb {C})$ and their conjugates furnishes a 1-adapted coframing whose Levi form (REF ) is represented $\\ell =\\left[\\begin{array}{cc}1&0\\\\0&\\epsilon \\end{array}\\right]$ in the basis (REF ) given by $\\underline{{\\tt v}}=\\varsigma (\\mathbb {P}({\\tt v}_0))$ .",
"Remark REF explains why the representation (REF ) of $\\ell $ does not depend on the choice of $(\\pm 1)^{\\delta _\\epsilon }$ in (), (REF ) when $\\epsilon =-1$ .",
"If $M$ is a 3-dimensional manifold CR-embedded in $\\mathcal {Q}$ , then there is a rank-2, $J$ -invariant subbundle $D_M\\subset D$ tangent to $M$ .",
"Let $\\hat{M}=\\mathbb {P}^{-1}(M)\\subset \\mathcal {N}$ be the cone over $M$ and $L_M\\subset T\\hat{M}$ be $\\mathbb {P}_*^{-1}(D_M)$ .",
"Name the restriction and projections $&\\mathcal {H}^0=h_{\\mathcal {Q}}^{-1}(M)\\subset \\hat{\\mathcal {H}};&h_{\\hat{M}}=h_\\mathcal {N}|_{\\mathcal {H}^0},&&h_M=h_\\mathcal {Q}|_{\\mathcal {H}^0}.$ The Hermitian frames $\\mathcal {H}^0$ are “zero-adapted to $M$ \" and the fibers of $h_M$ have (real) dimension $\\dim \\hat{\\mathcal {P}}=11$ – see (REF ).",
"We will exploit the degrees of freedom (REF ) and (REF ) in the structure group $\\hat{\\mathcal {P}}$ of $\\mathcal {H}^0$ to reduce to subbundles of frames which are increasingly adapted to the CR geometry of $M$ .",
"Since $\\mathcal {Q}$ is Levi-nondegenerate, $D\\subset T\\mathcal {Q}$ is a rank-4 contact distribution, so there cannot be a 3-dimensional submanifold of $\\mathcal {Q}$ which is tangent to $D$ .",
"That is to say $\\varsigma ^*\\kappa $ cannot vanish identically on $TM$ or, equivalently, $s^*\\kappa |_{T\\hat{M}}\\ne 0$ .",
"Therefore, we restrict to those Hermitian frames $\\underline{{\\tt v}}\\in h_{\\hat{M}}^{-1}({\\tt v}_0) \\quad \\text{ with } {\\tt v}_3\\in T_{{\\tt v}_0}\\hat{M}.$ Further adaptation branches based on the (non)vanishing of (REF ) on $D_M$ .",
"If $M$ is Levi-nondegenerate, $\\ell |_{D_M}\\ne 0$ and we consider $\\underline{{\\tt v}}\\in \\mathcal {H}^0$ with $L_M=\\langle {\\tt v}_0,{\\tt v}_1\\rangle _\\mathbb {C}$ , which along with (REF ) reduces $\\mathcal {H}^0$ and its structure group over $\\hat{M}$ : $&\\mathcal {H}^1=\\left\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^0: \\begin{array}{c}L_M=\\langle {\\tt v}_0,{\\tt v}_1\\rangle _\\mathbb {C}\\\\ T\\hat{M}=L_M\\oplus \\langle {\\tt v}_3\\rangle _\\mathbb {R}\\end{array}\\right\\rbrace ,&\\mathcal {P}_1=\\left\\lbrace p_0\\in \\mathcal {P}_0 \\text{ as in } (\\ref {P0}) : \\begin{array}{r}a_2=c_2=0\\\\ (\\pm 1)^{\\delta _\\epsilon }=1\\end{array}\\right\\rbrace .$ When $\\epsilon =-1$ it is also possible that $\\ell _{D_M}=0$ .",
"Such $M$ is Levi-flat and contains a complex curve tangent to the $\\mathbb {P}_*$ image of an $-null distribution of complex rank two in $ TM$.",
"In this case we arrange for $ LM=v0,v1+v2C$ combined with (\\ref {TMv3}) to obtain{\\begin{@align}{1}{-1}&\\mathcal {H}^1=\\left\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^0: \\begin{array}{c}L_M=\\langle {\\tt v}_0,{\\tt v}_1+{\\tt v}_2\\rangle _\\mathbb {C}\\\\ T\\hat{M}=L_M\\oplus \\langle {\\tt v}_3\\rangle _\\mathbb {R}\\end{array}\\right\\rbrace &\\mathcal {P}_1=\\left\\lbrace p_0\\in \\mathcal {P}_0 \\text{ as in } (\\ref {P0}) :\\begin{array}{c}\\epsilon =-1, (\\pm 1)^{\\delta _{-1}}=1\\\\a_1+a_2=\\mathbf {e}^{\\mathbf {i}r}(\\overline{a}_2+\\overline{a}_1)\\\\ c_2=c_1\\end{array}\\right\\rbrace .\\end{@align}}In every case, we can say $v,vhM-1(v0)H1 p1P1 such that v=vp1$, so that the fibers of $ hM|H1$ have real dimension $ PlP1=7$.",
"Furthermore, our adaptations in both cases are undisturbed by the projection (\\ref {orient}), so we may descend as in (\\ref {CBQ}) to oriented frames over $ M${\\begin{@align}{1}{-1}&\\mathcal {P}_2\\rightarrow \\mathcal {H}^2\\stackrel{h_M}{\\longrightarrow }M,&\\mathcal {P}_2=\\mathcal {P}_l\\mathcal {P}_1\\cap \\mathcal {P},&&\\mathcal {H}^2=\\mathcal {H}^1\\cap \\mathcal {H}.\\end{@align}}$"
],
[
"Second Fundamental Form $(\\mathbf {II})$",
"All higher-order adaptations of moving frames over $M\\subset \\mathcal {Q}$ will be controlled by the MC equations (REF ).",
"Specifically, we pull back the MC form $\\mu $ as in (REF ) along the inclusion $\\mathcal {H}^1\\hookrightarrow \\mathcal {H}^0$ and explore the differential consequences of the algebraic relations imposed on the individual 1-forms (REF ) over $\\mathcal {H}^1$ .",
"Obviously we must separately consider 3-folds $M$ which are Levi-nondegenerate (REF ) or Levi-flat (), but in both cases the reduction () will force the relation (REF ) (our notation suppresses the pullback along the inclusion $\\mathcal {H}^2\\hookrightarrow \\hat{\\mathcal {H}}$ and keeps the same names for the MC forms with every such reduction).",
"Moreover, in both cases the method of moving frames exploits the remaining degrees of freedom in $\\mathcal {P}_2$ to normalize a symmetric, $\\mathbb {C}$ -bilinear tensor $\\mathbf {II}:{\\bigodot }^2(T\\mathcal {H}^2/T\\mathcal {P}_2)\\rightarrow \\mathbb {C},$ which we dub the Second Fundamental Form by analogy to the study of hypersurfaces in Euclidean space.",
"(One might consider the the Levi form $\\ell $ of $\\mathcal {Q}$ restricted to $D_M$ to be a first fundamental form of sorts, but for $\\dim M=3$ this is simply a real function whose only import is its non-vanishing.)"
],
[
"Levi-Nondegenerate 3-folds",
"Our adaptation (REF ) together with (REF ) implies that $\\zeta =0 \\text{ on }\\mathcal {H}^2.$ Applying Cartan's Lemma to the equation (REF ) for $\\text{d}\\zeta $ yields $&\\left[\\begin{array}{c}\\phi _1\\\\\\phi _2\\end{array}\\right]=\\left[\\begin{array}{cc}a&b\\\\b&c\\end{array}\\right]\\left[\\begin{array}{c}\\eta \\\\\\kappa \\end{array}\\right]&\\text{ for some } a,b,c\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C}).$ Remark 4.4 It behooves us to remember that $\\mathcal {P}_2$ () is in part defined by $a_2=0$ (which forces $|a_1|=1$ ) in (REF ), as well as $\\mathbf {e}^{\\mathbf {i}r}=\\overline{l}l^{-1}$ in (REF ), (REF ).",
"Consequently, there is no crime in recycling notation by setting $a_1=\\mathbf {e}^{\\mathbf {i}r}$ in (REF ) so that $\\mathbf {i}\\rho $ in (REF ) measures the infinitesimal generator of this $U(1)$ -action on $\\mathcal {H}^2$ .",
"Differentiating (REF ) using $\\text{d}\\phi _1,\\text{d}\\phi _2$ from (REF ) reveals $\\begin{aligned}\\text{d}\\left[\\begin{array}{c}a\\\\b\\\\c\\end{array}\\right]=\\left[\\begin{array}{ccc}3\\mathbf {i}\\rho -\\overline{\\lambda }&0&0\\\\\\xi &2\\mathbf {i}\\rho -2\\overline{\\lambda }&0\\\\0&2\\xi &\\mathbf {i}\\rho -3\\overline{\\lambda }\\end{array}\\right]\\left[\\begin{array}{c}a\\\\b\\\\c\\end{array}\\right]+\\left[\\begin{array}{ccc}u_1&u_2&2\\mathbf {i}b\\\\u_2&u_3&\\mathbf {i}c\\\\u_3&u_4&0\\end{array}\\right]\\left[\\begin{array}{c}\\eta \\\\\\kappa \\\\\\overline{\\eta }\\end{array}\\right]\\\\\\text{ for some } u_1,u_2,u_3,u_4\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C}).\\end{aligned}$ The identities (REF ) imply that if one of $a,b,c$ vanishes identically on $\\mathcal {H}^2$ , they all vanish identically.",
"Furthermore, $\\text{d}(ac-b^2)\\equiv 4(ac-b^2)(\\mathbf {i}\\rho -\\overline{\\lambda })\\mod {\\lbrace }\\kappa ,\\eta \\rbrace ,$ so the determinant of the matrix (REF ) is either identically zero or non-vanishing on each fiber of $\\mathcal {H}^2\\rightarrow M$ .",
"Definition 4.5 The second fundamental form (REF ) of a Levi-nondegenerate 3-fold $M$ CR embedded in $\\mathcal {Q}$ is given by $\\mathbf {II}=a\\eta \\odot \\eta +2b\\eta \\odot \\kappa +c\\kappa \\odot \\kappa ,$ where the coefficients are derived from (REF ) via (REF ).",
"The condition that $\\mathbf {II}$ is of (sub)maximal rank on a fiber of $\\mathcal {H}^2\\rightarrow M$ is invariant under the action of CR symmetry group $SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ on $\\mathcal {Q}$ .",
"We update the remaining MC equations over $\\mathcal {H}^2$ , $\\begin{aligned}\\text{d}\\kappa &=\\mathbf {i}\\eta \\wedge \\overline{\\eta }+(\\lambda +\\overline{\\lambda })\\wedge \\kappa ,\\\\\\text{d}\\eta &=(\\lambda -\\mathbf {i}\\rho )\\wedge \\eta -\\xi \\wedge \\kappa ,\\\\\\text{d}\\psi &=\\psi \\wedge (\\lambda +\\overline{\\lambda })-\\mathbf {i}\\xi \\wedge \\overline{\\xi }+\\epsilon \\mathbf {i}\\kappa \\wedge (b\\overline{c}\\eta -\\overline{b}c\\overline{\\eta })-\\epsilon \\mathbf {i}|b|^2\\eta \\wedge \\overline{\\eta },\\\\\\text{d}\\xi &=\\psi \\wedge \\eta +\\xi \\wedge (\\overline{\\lambda }+\\mathbf {i}\\rho )+\\epsilon \\kappa \\wedge (|b|^2\\eta -ac\\overline{\\eta })-\\epsilon \\overline{a}b\\eta \\wedge \\overline{\\eta },\\\\\\text{d}\\lambda &=\\mathbf {i}\\overline{\\xi }\\wedge \\eta -\\psi \\wedge \\kappa ,\\\\\\text{d}\\rho &=-\\overline{\\xi }\\wedge \\eta -\\xi \\wedge \\overline{\\eta }-\\epsilon \\mathbf {i}\\kappa \\wedge (a\\overline{b}\\eta +\\overline{a}b\\overline{\\eta })+\\epsilon \\mathbf {i}|a|^2\\eta \\wedge \\overline{\\eta }.\\end{aligned}$ Recall from Remark REF that a CR embedding $M\\hookrightarrow \\mathcal {Q}$ lifts to a mapping $\\mathcal {F}^3\\hookrightarrow \\mathcal {H}$ between the Cartan bundles of $M$ and $\\mathcal {Q}$ in a manner that pulls back $\\mu $ (REF ) to $\\gamma $ (REF ).",
"To realize the image of $\\mathcal {F}^3$ over $M\\subset \\mathcal {Q}$ , we locate $\\gamma $ within $\\mu |_{\\mathcal {H}^2}$ using the fact that the semi-basic forms $\\kappa ,\\eta ,\\overline{\\eta }$ over both $\\mathcal {F}^3$ and $\\mathcal {H}^2$ encode adapted (co)framings of $M$ .",
"The equations (REF ) and (REF ) for $\\text{d}\\kappa $ , $\\text{d}\\eta $ alone demonstrate $&\\lambda =-\\alpha +\\mathbf {i}\\rho +a_0\\kappa ,&\\xi =\\beta -b_0\\kappa -a_0\\eta ,&&\\text{for some }a_0,b_0\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C}),$ and comparing $\\text{d}\\alpha ,\\text{d}\\beta $ in (REF ) to $\\text{d}\\lambda ,\\text{d}\\xi $ in (REF ) validates the substitution $&\\psi =-\\sigma +s_0\\kappa +s_1\\eta +\\overline{s}_1\\overline{\\eta },&&\\text{for some }s_0\\in C^\\infty (\\mathcal {H}^2), s_1\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C}),$ after which we solve for $a_0,b_0,s_0,s_1$ to bring the first five equations of (REF ) into the form (REF ): $a_0&=-\\epsilon \\tfrac{\\mathbf {i}}{4}|a|^2,&&b_0=-\\epsilon (\\mathbf {i}\\overline{a}b+\\tfrac{1}{6}a\\overline{u}_1),\\\\s_0&=\\tfrac{9}{16}|a|^4+\\epsilon (\\tfrac{3\\mathbf {i}}{4}(a\\overline{u}_2-\\overline{a}u_2)-3|b|^2-\\tfrac{1}{6}|u_1|^2)&&s_1=-\\epsilon \\tfrac{1}{12}(6a\\overline{b}+\\mathbf {i}\\overline{a}u_1).$ In conclusion, $\\begin{aligned}\\alpha &=\\mathbf {i}\\rho -\\lambda -\\epsilon \\tfrac{\\mathbf {i}}{4}|a|^2\\kappa ,\\quad \\quad \\beta =\\xi -\\epsilon (\\mathbf {i}\\overline{a}b+\\tfrac{1}{6}a\\overline{u}_1)\\kappa -\\epsilon \\tfrac{\\mathbf {i}}{4}|a|^2\\eta ,\\\\\\sigma &=-\\psi +(\\tfrac{9}{16}|a|^4+\\epsilon (\\tfrac{3\\mathbf {i}}{4}(a\\overline{u}_2-\\overline{a}u_2)-3|b|^2-\\tfrac{1}{6}|u_1|^2))\\kappa -\\epsilon \\tfrac{1}{12}(6a\\overline{b}+\\mathbf {i}\\overline{a}u_1)\\eta -\\epsilon \\tfrac{1}{12}(6\\overline{a}b-\\mathbf {i}a\\overline{u}_1)\\overline{\\eta }.\\end{aligned}$ Next we seek expressions for the coefficient functions $S,P$ of the curvature tensor $\\text{d}\\gamma +\\gamma \\wedge \\gamma $ as they appear in the equations for $\\text{d}\\beta ,\\text{d}\\sigma $ , as well as the higher-order coefficients in (REF ), (REF ), and (REF ).",
"To this end, we differentiate (REF ) and use the identity $\\text{d}^2=0$ to compute $\\begin{aligned}\\text{d}\\left[\\begin{array}{c}u_1\\\\u_2\\\\u_3\\\\u_4\\end{array}\\right]&=\\left[\\begin{array}{cccc}4\\mathbf {i}\\rho -\\overline{\\lambda }-\\lambda &0&0&0\\\\\\xi &3\\mathbf {i}\\rho -2\\overline{\\lambda }-\\lambda &0&0\\\\0&2\\xi &2\\mathbf {i}\\rho -3\\overline{\\lambda }-\\lambda &0\\\\0&0&3\\xi &\\mathbf {i}\\rho -4\\overline{\\lambda }-\\lambda \\end{array}\\right]\\left[\\begin{array}{c}u_1\\\\u_2\\\\u_3\\\\u_4\\end{array}\\right]\\\\&+\\left[\\begin{array}{cc}0&3\\mathbf {i}a\\\\-a&2\\mathbf {i}b\\\\-2b&\\mathbf {i}c\\\\-3c&0\\end{array}\\right]\\left[\\begin{array}{c}\\psi \\\\\\overline{\\xi }\\end{array}\\right]+\\left(\\left[\\begin{array}{cccc}v_1&v_2&3\\mathbf {i}u_2\\\\v_2&v_3&2\\mathbf {i}u_3\\\\v_3&v_4&\\mathbf {i}u_4\\\\v_4&v_5&0\\end{array}\\right]-\\epsilon \\left[\\begin{array}{cc}0&3a\\\\a&2b\\\\2b&c\\\\3c&0\\end{array}\\right]\\left[\\begin{array}{ccc}0&|b|^2&\\overline{a}b\\\\0&a\\overline{b}&|a|^2\\end{array}\\right]\\right)\\left[\\begin{array}{c}\\eta \\\\\\kappa \\\\\\overline{\\eta }\\end{array}\\right],\\end{aligned}$ for some $v_1,\\dots ,v_5\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ .",
"This is sufficient to express $\\begin{aligned}\\left[\\begin{array}{c}S\\\\P\\\\R\\end{array}\\right]&=\\frac{\\epsilon }{6}\\left[\\begin{array}{ccc}a&b&c\\\\u_1&u_2&u_3\\\\v_1&v_2&v_3\\end{array}\\right]\\left[\\begin{array}{c}\\overline{v}_1\\\\8\\mathbf {i}\\overline{u}_1\\\\-12\\overline{a}\\end{array}\\right]+\\frac{\\epsilon }{6}\\left[\\begin{array}{ccc}0&0&0\\\\a& b& c\\\\2u_1&2u_2&2u_3\\end{array}\\right]\\left[\\begin{array}{c}-4\\mathbf {i}\\overline{v}_2\\\\24\\overline{u}_2\\\\24\\mathbf {i}\\overline{b}\\end{array}\\right]\\\\&+\\frac{\\epsilon }{6}\\left[\\begin{array}{ccc}0&0&0\\\\0&0&0\\\\a&b&c\\end{array}\\right]\\left[\\begin{array}{c}-12\\overline{v}_3\\\\-48\\mathbf {i}\\overline{u}_3\\\\24\\overline{c}\\end{array}\\right]-\\frac{1}{6}\\left[\\begin{array}{ccc}0&0&0\\\\0&30b&0\\\\30|a|^2&54u_2&36u_1\\end{array}\\right]\\left[\\begin{array}{c}-\\epsilon |a|^4\\\\\\mathbf {i}\\overline{a}^2a\\\\\\mathbf {i}\\overline{a}^2b\\end{array}\\right]\\\\&+\\frac{1}{12}\\left[\\begin{array}{cc}0&0\\\\-20\\overline{a}&0\\\\72\\mathbf {i}\\overline{b}&108\\mathbf {i}\\overline{a}\\end{array}\\right]\\left[\\begin{array}{c}a^2\\overline{u}_1\\\\a^2\\overline{u}_2\\end{array}\\right]-\\left[\\begin{array}{cc}0&0\\\\0&0\\\\36b&5u_1\\end{array}\\right]\\left[\\begin{array}{l}|a|^2\\overline{b}\\\\|a|^2\\overline{u}_1\\end{array}\\right].\\end{aligned}$ Remark 4.6 The prototypical example of a 3-dimensional CR manifold is a hypersurface given by a regular level set of a smooth, non-constant function $\\varrho :\\mathbb {C}^2\\rightarrow \\mathbb {R}$ .",
"The curvature coefficient $S$ for the level set depends on derivatives of $\\varrho $ up to order six, which is to say that $S$ is a function of the 6-jet of $\\varrho $ .",
"In the present setting, $S$ for $M\\subset \\mathcal {Q}$ is a function of the 2-jet of $\\mathbf {II}$ .",
"The remaining functions in (REF ) require another derivative of $\\mathbf {II}$ , so we apply $\\text{d}^2=0$ to (REF ) to get $\\text{d}\\left[\\begin{array}{c}v_1\\\\v_2\\\\v_3\\\\v_4\\\\v_5\\end{array}\\right]&=\\left[\\begin{array}{ccccc}5\\mathbf {i}\\rho -\\overline{\\lambda }-2\\lambda &0&0&0&0\\\\\\xi &4\\mathbf {i}\\rho -2\\overline{\\lambda }-2\\lambda &0&0&0\\\\0&2\\xi &3\\mathbf {i}\\rho -3\\overline{\\lambda }-2\\lambda &0&0\\\\0&0&3\\xi &2\\mathbf {i}\\rho -4\\overline{\\lambda }-2\\lambda &0\\\\0&0&0&4\\xi &\\mathbf {i}\\rho -5\\overline{\\lambda }-2\\lambda \\end{array}\\right]\\left[\\begin{array}{c}v_1\\\\v_2\\\\v_3\\\\v_4\\\\v_5\\end{array}\\right]\\\\&+\\left[\\begin{array}{cc}0&8\\mathbf {i}u_1\\\\-2u_1&6\\mathbf {i}u_2\\\\-4u_2&4\\mathbf {i}u_3\\\\-6u_3&2\\mathbf {i}u_4\\\\-8u_4&0\\end{array}\\right]\\left[\\begin{array}{c}\\psi \\\\\\overline{\\xi }\\end{array}\\right]+\\left[\\begin{array}{ccc}w_1&w_2&4\\mathbf {i}v_2\\\\w_2&w_3&3\\mathbf {i}v_3\\\\w_3&w_4&2\\mathbf {i}v_4\\\\w_4&w_5&\\mathbf {i}v_5\\\\w_5&w_6&0\\end{array}\\right]\\left[\\begin{array}{c}\\eta \\\\\\kappa \\\\\\overline{\\eta }\\end{array}\\right]\\nonumber \\\\&-\\epsilon \\left(\\left[\\begin{array}{ccc}0&0&10u_1\\\\0&4u_1&6u_2\\\\u_1&6u_2&3u_3\\\\3u_2&6u_3&u_4\\\\6u_3&4u_4&0\\end{array}\\right]\\left[\\begin{array}{ccc}0&\\overline{b}c&\\overline{a}c\\\\0&|b|^2&\\overline{a}b\\\\0&a\\overline{b}&|a|^2\\end{array}\\right]-\\mathbf {i}\\left[\\begin{array}{ccc}0&0&6a\\\\0&3a&3b\\\\a&4b&c\\\\3b&3c&0\\\\6c&0&0\\end{array}\\right]\\left[\\begin{array}{ccc}0&|c|^2&\\overline{b}c\\\\0&b\\overline{c}&|b|^2\\\\0&a\\overline{c}&a\\overline{b}\\end{array}\\right]\\right)\\left[\\begin{array}{c}\\eta \\\\\\kappa \\\\\\overline{\\eta }\\end{array}\\right],\\nonumber $ for some $w_1,\\dots ,w_6\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ , and with this we can write $\\left[\\begin{array}{c}Q\\\\U\\\\V\\\\W\\\\R_1^{\\prime }\\end{array}\\right]&=\\frac{\\epsilon }{6}\\left[{\\begin{matrix}a&b&c\\\\0&0&0\\\\u_1&u_2&u_3\\\\0&0&0\\\\v_1&v_2&v_3\\end{matrix}}\\right]\\left[\\begin{array}{c}\\overline{w}_1\\\\10\\mathbf {i}\\overline{v}_1\\\\-20\\overline{u}_1\\end{array}\\right]-\\frac{\\epsilon }{6}\\left[{\\begin{matrix}0&0&0\\\\-a&-b&-c\\\\4\\mathbf {i}a&4\\mathbf {i}b&4\\mathbf {i}c\\\\-u_1&-u_2&-u_3\\\\8\\mathbf {i}u_1&8\\mathbf {i}u_2&8\\mathbf {i}u_3\\end{matrix}}\\right]\\left[\\begin{array}{c}\\overline{w}_2\\\\8\\mathbf {i}\\overline{v}_2\\\\-12\\overline{u}_2\\end{array}\\right]-\\frac{\\epsilon }{6}\\left[{\\begin{matrix}0&0&0\\\\0&0&0\\\\0&0&0\\\\-a& -b& -c\\\\3\\mathbf {i}a&3\\mathbf {i}b&3\\mathbf {i}c\\end{matrix}}\\right]\\left[\\begin{array}{c}-4\\mathbf {i}\\overline{w}_3\\\\24\\overline{v}_3\\\\24\\mathbf {i}\\overline{u}_3\\end{array}\\right]\\\\&+\\frac{\\epsilon }{6}\\left[{\\begin{matrix}0&0&0&0&0&0\\\\u_2&u_3&u_4&0&0&0\\\\\\mathbf {i}u_2&\\mathbf {i}u_3&\\mathbf {i}u_4&0&0&0\\\\v_2&v_3&v_4&u_2&u_3&u_4\\\\2\\mathbf {i}v_2&2\\mathbf {i}v_3&2\\mathbf {i}v_4&2\\mathbf {i}u_2&2\\mathbf {i}u_3&2\\mathbf {i}u_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{v}_1\\\\8\\mathbf {i}\\overline{u}_1\\\\-12\\overline{a}\\\\-4\\mathbf {i}\\overline{v}_2\\\\24\\overline{u}_2\\\\24\\mathbf {i}\\overline{b}\\\\\\end{matrix}}\\right]-\\frac{1}{6}\\left[{\\begin{matrix}0&0&0&0\\\\0&0&24b&0\\\\0&\\mathbf {i}45 c&\\mathbf {i}84 b&0\\\\\\tfrac{1}{2}(45\\overline{a}b+5\\mathbf {i}a\\overline{u}_1)&27u_3&54u_2&4u_1\\\\450\\mathbf {i}\\overline{a}b+180a\\overline{u}_1&144\\mathbf {i}u_3&288\\mathbf {i}u_2&48\\mathbf {i}u_1\\end{matrix}}\\right]\\left[\\begin{array}{c}-\\epsilon |a|^4\\\\\\mathbf {i}\\overline{a}^2a\\\\\\mathbf {i}\\overline{a}^2b\\\\\\mathbf {i}\\overline{a}^2c\\end{array}\\right]\\nonumber \\\\&-\\frac{1}{12}\\left[{\\begin{matrix}0&0&0\\\\0&\\mathbf {i}\\overline{a}&0\\\\20\\overline{u}_1&\\tfrac{53}{2}\\overline{a}&0\\\\20\\overline{u}_2&8\\overline{b}&22\\overline{a}\\\\-180\\mathbf {i}\\overline{u}_2&-84\\mathbf {i}\\overline{b}&-156\\mathbf {i}\\overline{a}\\end{matrix}}\\right]\\left[{\\begin{matrix}a^2\\overline{u}_1\\\\a^2\\overline{v}_1\\\\a^2\\overline{v}_2\\end{matrix}}\\right]-\\frac{1}{3}\\left[{\\begin{matrix}0&0&0&0&0&0&0&0\\\\0&3b&0&0&0&0&0&0\\\\0&63\\mathbf {i}b&0&0&0&0&0&0\\\\-3c&9u_2&21\\mathbf {i}b&\\tfrac{5}{24}\\mathbf {i}u_1&24\\overline{a}&8\\mathbf {i}a&0&\\tfrac{13}{3}\\overline{a}\\\\144\\mathbf {i}c&123\\mathbf {i}u_2&288b&20u_1&288\\mathbf {i}\\overline{a}&204a&15\\overline{u}_1&82\\mathbf {i}\\overline{a}\\end{matrix}}\\right]\\left[{\\begin{matrix}|a|^2\\overline{b}\\\\|a|^2\\overline{u}_1\\\\|a|^2\\overline{u}_2\\\\|a|^2\\overline{v}_1\\\\|b|^2b\\\\|b|^2\\overline{u}_1\\\\|u_1|^2a\\\\|u_1|^2b\\end{matrix}}\\right],\\nonumber $ as well as the totally real coefficients $\\left[\\begin{array}{c}R_0^{\\prime }\\\\R_{0}^{\\prime \\prime }\\end{array}\\right]&=\\epsilon \\left[{\\begin{matrix}-2a&-2b&-2c\\\\5\\mathbf {i}a&5\\mathbf {i}b&5\\mathbf {i}c\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_4\\\\ 4\\mathbf {i}\\overline{v}_4\\\\ -2\\overline{u}_4\\end{matrix}}\\right]-\\frac{\\epsilon }{3}\\left[{\\begin{matrix}-u_1&-u_2&-u_3\\\\5\\mathbf {i}u_1&5\\mathbf {i}u_2&5\\mathbf {i}u_3\\end{matrix}}\\right]\\left[{\\begin{matrix}-4\\mathbf {i}\\overline{w}_3\\\\ 24\\overline{v}_3\\\\ 24\\mathbf {i}\\overline{u}_3\\end{matrix}}\\right]-\\frac{\\epsilon }{12}\\left[{\\begin{matrix}-2v_1&-2v_2&-2v_3\\\\25\\mathbf {i}v_1&25\\mathbf {i}v_2&25\\mathbf {i}v_3\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_2\\\\ 8\\mathbf {i}\\overline{v}_2\\\\ -12\\overline{u}_2\\end{matrix}}\\right]\\\\&-\\epsilon \\left[{\\begin{matrix}2u_2&2u_3&2u_4\\\\5\\mathbf {i}u_2&5\\mathbf {i}u_3&5\\mathbf {i}u_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{v}_3\\\\ 4\\mathbf {i}\\overline{u}_3\\\\ -2\\overline{c}\\end{matrix}}\\right]+\\frac{\\epsilon }{6}\\left[{\\begin{matrix}2v_2&2v_3&2v_4\\\\5\\mathbf {i}v_2&5\\mathbf {i}v_3&5\\mathbf {i}v_4\\end{matrix}}\\right]\\left[{\\begin{matrix}-4\\mathbf {i}\\overline{v}_2\\\\ 24\\overline{u}_2\\\\ 24\\mathbf {i}\\overline{b}\\end{matrix}}\\right]+\\frac{\\epsilon }{12}\\left[{\\begin{matrix}2w_2&2w_3&2w_4\\\\5\\mathbf {i}w_2&5\\mathbf {i}w_3&5\\mathbf {i}w_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{v}_1\\\\ 8\\mathbf {i}\\overline{u}_1\\\\ -12\\overline{a}\\end{matrix}}\\right]\\nonumber \\\\&+\\frac{\\epsilon }{6}\\left(\\left[{\\begin{matrix}0&0&0\\\\w_1&w_2&w_3\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_1\\\\ 10\\mathbf {i}\\overline{v}_1\\\\ -20\\overline{u}_1\\end{matrix}}\\right]+|a|^4\\left[{\\begin{matrix}0\\\\6210|b|^2+\\tfrac{2195}{2}|u_1|^2\\end{matrix}}\\right]\\right)-\\frac{|a|^2}{6}\\left[{\\begin{matrix}0\\\\\\tfrac{1035}{2}|a|^6 +342|c|^2+1710|u_2|^2+\\tfrac{217}{4}|v_1|^2\\end{matrix}}\\right]\\nonumber \\\\&+\\Re \\left(\\frac{\\epsilon \\overline{a}^3a}{3}\\left[{\\begin{matrix}90&90\\\\\\tfrac{2385}{2}\\mathbf {i}&1650\\mathbf {i}\\end{matrix}}\\right]\\left[{\\begin{matrix}au_2\\\\bu_1\\end{matrix}}\\right]+\\frac{\\overline{a}^2}{3}\\left[{\\begin{matrix}-54\\mathbf {i}& -60\\mathbf {i}& -6\\mathbf {i}& -36\\mathbf {i}& -54\\mathbf {i}\\\\ 270& 390& 51& 210& 315\\end{matrix}}\\right]\\left[{\\begin{matrix}av_3\\\\ bv_2\\\\ cv_1\\\\ u_1u_3\\\\ {u_2}^2\\end{matrix}}\\right]+\\frac{|a|^2}{3}\\left[{\\begin{matrix}-180&-30\\\\1350\\mathbf {i}&375\\mathbf {i}\\end{matrix}}\\right]\\left[{\\begin{matrix}b\\overline{u}_3\\\\u_1\\overline{v}_2\\end{matrix}}\\right]\\right)\\nonumber \\\\&-\\Re \\left(\\frac{1}{3}\\left[{\\begin{matrix}(288|b|^2+30|u_1|^2)a\\overline{u}_2+a(\\overline{b}(10u_1\\overline{v}_1-24\\mathbf {i}\\overline{u}_1u_2)+36b\\overline{c}\\overline{u}_1)\\\\(720b\\overline{c}+199\\mathbf {i}\\overline{u}_1v_1)\\overline{a}b-\\mathbf {i}(2880|b|^2+425|u_1|^2)a\\overline{u}_2+40(30b\\overline{u}_2+u_1\\overline{v}_1)\\overline{a}u_1-498\\mathbf {i}ab\\overline{c}\\overline{u}_1\\end{matrix}}\\right]\\right)\\nonumber \\\\&-\\frac{1}{6}\\left[{\\begin{matrix}0\\\\30|u_1|^4+1440|b|^4+832|bu_1|^2\\end{matrix}}\\right].\\nonumber $ One last derivative of (REF ) yields $\\text{d}\\left[\\begin{array}{c}w_1\\\\w_2\\\\w_3\\\\w_4\\\\w_5\\\\w_6\\end{array}\\right]&=\\left[{\\begin{matrix}6\\mathbf {i}\\rho -\\overline{\\lambda }-3\\lambda &0&0&0&0&0\\\\\\xi &5\\mathbf {i}\\rho -2\\overline{\\lambda }-3\\lambda &0&0&0&0\\\\0&2\\xi &4\\mathbf {i}\\rho -3\\overline{\\lambda }-3\\lambda &0&0&0\\\\0&0&3\\xi &3\\mathbf {i}\\rho -4\\overline{\\lambda }-3\\lambda &0&0\\\\0&0&0&4\\xi &2\\mathbf {i}\\rho -5\\overline{\\lambda }-3\\lambda &0\\\\0&0&0&0&5\\xi &\\mathbf {i}\\rho -6\\overline{\\lambda }-3\\lambda \\end{matrix}}\\right]\\left[\\begin{array}{c}w_1\\\\w_2\\\\w_3\\\\w_4\\\\w_5\\\\w_6\\end{array}\\right]\\\\&+\\left[{\\begin{matrix}0&15\\mathbf {i}v_1\\\\-3v_1&12\\mathbf {i}v_2\\\\-6v_2&9\\mathbf {i}v_3\\\\-9v_3&6\\mathbf {i}v_4\\\\-12v_4&3\\mathbf {i}v_5\\\\-15v_5&0\\end{matrix}}\\right]\\left[\\begin{array}{c}\\psi \\\\\\overline{\\xi }\\end{array}\\right]+\\left(\\left[{\\begin{matrix}z_1&z_2&5\\mathbf {i}w_2\\\\z_2&z_3&4\\mathbf {i}w_3\\\\z_3&z_4&3\\mathbf {i}w_4\\\\z_4&z_5&2\\mathbf {i}w_5\\\\z_5&z_6&\\mathbf {i}w_6\\\\z_6&z_7&0\\end{matrix}}\\right]-\\epsilon \\left[{\\begin{matrix}10{u_1}^2\\\\10u_1u_2\\\\4u_1u_3+6{u_2}^2\\\\u_1u_4+9u_2u_3\\\\4u_2u_4+6{u_3}^2\\\\10u_3u_4\\end{matrix}}\\right]\\left[\\begin{array}{ccc}0&\\overline{b}&\\overline{a}\\end{array}\\right]\\right)\\left[\\begin{array}{c}\\eta \\\\\\kappa \\\\\\overline{\\eta }\\end{array}\\right]\\nonumber \\\\-&\\epsilon \\left(\\left[{\\begin{matrix}0&0&15v_1\\\\0&5v_1&10v_2\\\\v_1&8v_2&6v_3\\\\3v_2&9v_3&3v_4\\\\6v_3&8v_4&v_5\\\\10v_4&5v_5&0\\end{matrix}}\\right]\\left[\\begin{array}{ccc}0&\\overline{b}c&\\overline{a}c\\\\0&|b|^2&\\overline{a}b\\\\0&a\\overline{b}&|a|^2\\end{array}\\right]-\\mathbf {i}\\left[{\\begin{matrix}0&0&30u_1\\\\0&12u_1&18u_2\\\\3u_1&18u_2&9u_3\\\\9u_2&18u_3&3u_4\\\\18u_3&12u_4&0\\\\30u_4&0&0\\end{matrix}}\\right]\\left[\\begin{array}{ccc}0&|c|^2&\\overline{b}c\\\\0&b\\overline{c}&|b|^2\\\\0&a\\overline{c}&a\\overline{b}\\end{array}\\right]\\right)\\left[\\begin{array}{c}\\eta \\\\\\kappa \\\\\\overline{\\eta }\\end{array}\\right],\\nonumber $ for some $z_1,\\dots ,z_7\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ .",
"In the language of Remark REF , we have functions of the 4-jet of $\\mathbf {II}$ , $\\begin{aligned}\\left[\\begin{array}{c}Q^{\\prime }\\\\U_1^{\\prime }\\\\U_2^{\\prime }\\\\V^{\\prime }\\end{array}\\right]&=\\frac{\\epsilon }{6}\\left(\\left[{\\begin{matrix}a&b&c\\\\0&0&0\\\\0&0&0\\\\0&0&0\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{z}_1\\\\ 12\\mathbf {i}\\overline{w}_1\\\\ -30\\overline{v}_1\\end{matrix}}\\right]+\\left[{\\begin{matrix}0&0&0\\\\a&b&c\\\\0&0&0\\\\u_1&u_2&u_3\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{z}_2\\\\ 10\\mathbf {i}\\overline{w}_2\\\\ -20\\overline{v}_2\\end{matrix}}\\right]+\\left[{\\begin{matrix}0&0&0\\\\0&0&0\\\\a&b&c\\\\-4\\mathbf {i}a&-4\\mathbf {i}b&-4\\mathbf {i}c\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{z}_3\\\\ 8\\mathbf {i}\\overline{w}_3\\\\-12\\overline{v}_3\\end{matrix}}\\right]\\right)\\\\&+\\frac{\\epsilon }{6}\\left(\\left[{\\begin{matrix}0&0&0\\\\u_2&u_3&u_4\\\\0&0&0\\\\v_2&v_3&v_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_1\\\\ 10\\mathbf {i}\\overline{v}_1\\\\ -20\\overline{u}_1\\end{matrix}}\\right]+\\left[{\\begin{matrix}0&0&0\\\\0&0&0\\\\2u_2&2u_3&2u_4\\\\-3\\mathbf {i}u_2&-3\\mathbf {i}u_3&-3\\mathbf {i}u_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_2\\\\ 8\\mathbf {i}\\overline{v}_2\\\\ -12\\overline{u}_2\\end{matrix}}\\right]+\\left[{\\begin{matrix}0&0&0\\\\0&0&0\\\\v_3&v_4&v_5\\\\\\mathbf {i}v_3&\\mathbf {i}v_4&\\mathbf {i}v_5\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{v}_1\\\\ 8\\mathbf {i}\\overline{u}_1\\\\ -12\\overline{a}\\end{matrix}}\\right]\\right)\\\\&+\\frac{|a|^2}{36}\\left(\\epsilon \\left[{\\begin{matrix}0&0&0&0&0\\\\0&0&0&0&0\\\\216&144&-150\\mathbf {i}&-15&-10\\\\1296\\mathbf {i}&\\tfrac{603}{2}\\mathbf {i}&390&\\mathbf {i}\\tfrac{345}{8}&70\\mathbf {i}\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{a}^2b^2\\\\|a|^2\\overline{a}c\\\\|a|^2b\\overline{u}_1\\\\|a|^2a\\overline{v}_1\\\\a^2{\\overline{u}_1}^2\\end{matrix}}\\right]-\\left[{\\begin{matrix}0&0&0&0&0&0\\\\30\\mathbf {i}c&0&15b&0&\\tfrac{9\\mathbf {i}}{2}a&0\\\\24u_3&-144\\mathbf {i}c&-9\\mathbf {i}u_2&48b&-u_1&6\\mathbf {i}a\\\\624\\mathbf {i}u_3&-306c&\\frac{243}{2}u_2&708\\mathbf {i}b&4\\mathbf {i}u_1&\\tfrac{183}{2}a\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{u}_1\\\\\\overline{u}_2\\\\\\overline{v}_1\\\\\\overline{v}_2\\\\\\overline{w}_1\\\\\\overline{w}_2\\end{matrix}}\\right]\\right)\\\\&-\\frac{1}{36}\\left(\\left[{\\begin{matrix}0&0&0\\\\0&-360c&0\\\\-36\\mathbf {i}u_4&504\\mathbf {i}u_3&252\\mathbf {i}u_2\\\\-234u_4&-1224u_3&-612u_2\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{a}^2a\\\\\\overline{a}^2b\\\\\\overline{a}^2c\\end{matrix}}\\right]+\\left[{\\begin{matrix}0&0&0\\\\5\\mathbf {i}\\overline{v}_1&0&0\\\\4\\mathbf {i}\\overline{v}_2&21\\mathbf {i}\\overline{u}_2&6\\mathbf {i}\\overline{b}\\\\136\\overline{v}_2&84\\overline{u}_2&24\\overline{b}\\end{matrix}}\\right]\\left[{\\begin{matrix}a^2\\overline{u}_1\\\\a^2\\overline{v}_1\\\\a^2\\overline{w}_1\\end{matrix}}\\right]\\right)\\\\&-\\frac{1}{36}\\left(\\left[{\\begin{matrix}0&0&0&0\\\\20b&0&360\\mathbf {i}\\overline{a}&0\\\\-8\\mathbf {i}u_2&-32\\mathbf {i}u_1&-288\\mathbf {i}\\overline{b}&432\\mathbf {i}\\overline{a}\\\\88u_2&52u_1&288\\overline{b}&-432\\overline{a}\\end{matrix}}\\right]\\left[{\\begin{matrix}a{\\overline{u}_1}^2\\\\b{\\overline{u}_1}^2\\\\b^2\\overline{u}_1\\\\b^2\\overline{u}_2\\end{matrix}}\\right]+\\left[{\\begin{matrix}0&0&0&0\\\\0&0&0&0\\\\-84\\overline{v}_1&288c&-5\\overline{v}_1&68c\\\\156\\mathbf {i}\\overline{v}_1&1008\\mathbf {i}c&5\\mathbf {i}\\overline{v}_1&148\\mathbf {i}c\\end{matrix}}\\right]\\left[{\\begin{matrix}a|b|^2\\\\\\overline{a}|b|^2\\\\a|u_1|^2\\\\\\overline{a}|u_1|^2\\end{matrix}}\\right]\\right)\\\\&+\\frac{1}{9}\\left[{\\begin{matrix}0\\\\0\\\\6\\mathbf {i}a\\overline{b}c\\overline{u}_1+4\\mathbf {i}\\overline{a}bu_1\\overline{v}_1-72\\overline{a}b\\overline{u}_1u_2+18 ab\\overline{u}_1\\overline{u}_2\\\\24a\\overline{b}c\\overline{u}_1-14\\overline{a}bu_1\\overline{v}_1-297\\mathbf {i}\\overline{a}b\\overline{u}_1u_2-162\\mathbf {i}ab\\overline{u}_1\\overline{u}_2\\end{matrix}}\\right],\\end{aligned}$ and our collection is completed by $\\left[\\begin{array}{c}W^{\\prime }\\\\R_1^{\\prime \\prime }\\end{array}\\right]&=-\\frac{\\epsilon }{6}\\left(4\\left[{\\begin{matrix}\\mathbf {i}a&\\mathbf {i}b&\\mathbf {i}c\\\\3a&3b&3c\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{z}_4\\\\ 6\\mathbf {i}\\overline{w}_4\\\\ -6\\overline{v}_4\\end{matrix}}\\right]+\\left[{\\begin{matrix}-u_1&-u_2&-u_3\\\\8\\mathbf {i}u_1&8\\mathbf {i}u_2&8\\mathbf {i}u_3\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{z}_3\\\\ 8\\mathbf {i}\\overline{w}_3\\\\-12\\overline{v}_3\\end{matrix}}\\right]-\\left[{\\begin{matrix}0&0&0\\\\v_1&v_2&v_3\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{z}_2\\\\ 10\\mathbf {i}\\overline{w}_2\\\\ -20\\overline{v}_2\\end{matrix}}\\right]\\right)\\\\&-\\frac{\\epsilon }{6}\\left(4\\left[{\\begin{matrix}2\\mathbf {i}u_2&2\\mathbf {i}u_3&2\\mathbf {i}u_4\\\\u_2&u_3&u_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_3\\\\ 6\\mathbf {i}\\overline{v}_3\\\\ -6\\overline{u}_3\\end{matrix}}\\right]+2\\left[{\\begin{matrix}-v_2&-v_3&-v_4\\\\3\\mathbf {i}v_2&3\\mathbf {i}v_3&3\\mathbf {i}v_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_2\\\\ 8\\mathbf {i}\\overline{v}_2\\\\ -12\\overline{u}_2\\end{matrix}}\\right]-\\left[{\\begin{matrix}0&0&0\\\\w_2&w_3&w_4\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{w}_1\\\\ 10\\mathbf {i}\\overline{v}_1\\\\ -20\\overline{u}_1\\end{matrix}}\\right]\\right)\\nonumber \\\\&+\\frac{\\epsilon }{6}\\left(4\\left[{\\begin{matrix}-\\mathbf {i}v_3&-\\mathbf {i}v_4&-\\mathbf {i}v_5\\\\2v_3&2v_4&2v_5\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{v}_2\\\\ 6\\mathbf {i}\\overline{u}_2\\\\ -6\\overline{b}\\end{matrix}}\\right]+\\left[{\\begin{matrix}w_3&w_4&w_5\\\\2\\mathbf {i}w_3&2\\mathbf {i}w_4&2\\mathbf {i}w_5\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{v}_1\\\\ 8\\mathbf {i}\\overline{u}_1\\\\ -12\\overline{a}\\end{matrix}}\\right]+\\overline{a}^3a\\left[{\\begin{matrix}\\tfrac{99}{2}u_3 &252u_2 &4u_1\\\\414\\mathbf {i}u_3& 1404\\mathbf {i}u_2& 228\\mathbf {i}u_1\\end{matrix}}\\right]\\left[{\\begin{matrix}a\\\\b\\\\c\\end{matrix}}\\right]\\right)\\nonumber \\\\&+\\frac{\\epsilon }{6} \\left(a^3\\left[{\\begin{matrix}16\\mathbf {i}\\overline{b}\\overline{u}_1& 54\\mathbf {i}\\overline{a}\\overline{u}_1&16\\mathbf {i}\\overline{a}\\overline{b}& \\tfrac{33}{2}\\mathbf {i}\\overline{a}^2\\\\108\\overline{b}\\overline{u}_1 &\\tfrac{819}{2}\\overline{a}\\overline{u}_1&\\tfrac{261}{2}\\overline{a}\\overline{b}&\\tfrac{399}{2}\\overline{a}^2\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{u}_1\\\\ \\overline{u}_2\\\\ \\overline{v}_1\\\\ \\overline{v}_2\\end{matrix}}\\right]+|a|^4\\left[{\\begin{matrix}15\\mathbf {i}c_1&-57\\mathbf {i}u_2&-81b&-\\tfrac{125}{24}u_1\\\\360c& \\tfrac{903}{2}u_2& 918\\mathbf {i}b &\\ 10\\mathbf {i}u_1\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{b}\\\\ \\overline{u}_1\\\\ \\overline{u}_2\\\\ \\overline{v}_1\\end{matrix}}\\right]\\right)\\nonumber \\\\&+\\frac{\\epsilon }{6}\\left[{\\begin{matrix}-(152a\\overline{u}_1+324\\mathbf {i}\\overline{a}b)&-(15a\\overline{u}_1+\\tfrac{224}{3}\\mathbf {i}\\overline{a}b)& 72\\\\1008\\overline{a}b+546\\mathbf {i}a\\overline{u}_1&431\\overline{a}b+\\tfrac{45}{2}\\mathbf {i}a\\overline{u}_1& 384\\mathbf {i}\\end{matrix}}\\right]\\left[{\\begin{matrix}|ab|^2\\\\ |au_1|^2\\\\ \\overline{a}^3b^2u_1\\end{matrix}}\\right]+\\frac{\\overline{a}^2}{6}\\left[{\\begin{matrix}-24\\mathbf {i}&-90\\mathbf {i}& -10\\mathbf {i}& -4\\mathbf {i}& -126\\mathbf {i}\\\\138& 360& 120& 48& 432\\end{matrix}}\\right]\\left[{\\begin{matrix}av_4\\\\ bv_3\\\\ cv_2\\\\ u_1u_4\\\\ u_2u_3\\end{matrix}}\\right]\\nonumber \\\\&+\\frac{a^2}{6}\\left[{\\begin{matrix}-12& -12& 0& -12& -36& -4\\\\81\\mathbf {i}& 66\\mathbf {i}& 6\\mathbf {i}& 96\\mathbf {i}& 168\\mathbf {i}& 42\\mathbf {i}\\end{matrix}}\\right]\\left[{\\begin{matrix}\\overline{a}\\overline{w}_3\\\\ \\overline{b}\\overline{w}_2\\\\ \\overline{c}\\overline{w}_1\\\\ \\overline{u}_1\\overline{v}_3\\\\ \\overline{u}_2\\overline{v}_2\\\\ \\overline{u}_3\\overline{v}_1\\end{matrix}}\\right]-\\frac{|a|^2}{6}\\left[{\\begin{matrix}36\\mathbf {i}&24& \\tfrac{3}{2}\\mathbf {i}& 24& 54\\mathbf {i}& -24&0& \\tfrac{2}{3}\\mathbf {i}& 18\\\\558&198\\mathbf {i}& 42& 108\\mathbf {i}& 252& 252\\mathbf {i}&\\tfrac{\\mathbf {i}}{4}& 37& 231\\mathbf {i}\\end{matrix}}\\right]\\left[{\\begin{matrix}b\\overline{v}_3\\\\ c\\overline{u}_3\\\\ u_1\\overline{w}_2\\\\ u_2\\overline{v}_2\\\\ u_3\\overline{u}_2\\\\ u_4\\overline{b}\\\\ v_1\\overline{w}_1\\\\ v_2\\overline{v}_1\\\\ v_3\\overline{u}_1\\end{matrix}}\\right]\\nonumber \\\\&-\\frac{1}{6}\\left(|b|^2\\left[{\\begin{matrix}96\\mathbf {i}& -144& -144\\mathbf {i}& -6& -40\\mathbf {i}& 288\\\\528& 432\\mathbf {i}& 288& 8\\mathbf {i}&\\ 480& 1296\\mathbf {i}\\end{matrix}}\\right]+|u_1|^2\\left[{\\begin{matrix}\\tfrac{14}{3}\\mathbf {i}& -\\tfrac{4}{3}&0&-\\tfrac{5}{6}&-\\tfrac{20}{3}\\mathbf {i}& 22\\\\64& 84\\mathbf {i}& 60& 0& 30& 164\\mathbf {i}\\end{matrix}}\\right]\\right)\\left[{\\begin{matrix}a\\overline{v}_2\\\\ b\\overline{u}_2\\\\ c\\overline{b}\\\\ u_1\\overline{v}_1\\\\ u_2\\overline{u}_1\\\\ u_3\\overline{a}\\end{matrix}}\\right]\\nonumber \\\\&-\\frac{1}{6}\\left(a\\overline{u}_1\\left[{\\begin{matrix}-\\tfrac{135}{4}&24&0&-\\tfrac{1}{6}&8\\mathbf {i}&-\\tfrac{4}{3}\\mathbf {i}\\\\60\\mathbf {i}&48\\mathbf {i}&90\\mathbf {i}&\\tfrac{\\mathbf {i}}{2}&384&26\\end{matrix}}\\right]+b\\overline{a}\\left[{\\begin{matrix}-\\frac{405}{4}\\mathbf {i}& 120\\mathbf {i}& 36\\mathbf {i}& -\\mathbf {i}& 72&\\tfrac{80}{3}\\\\\\tfrac{675}{2}&0&288&12&504\\mathbf {i}&260\\mathbf {i}\\end{matrix}}\\right]\\right)\\left[{\\begin{matrix}|a|^6\\\\|c|^2\\\\|u_2|^2\\\\|v_1|^2\\\\b\\overline{u}_3\\\\v_2\\overline{u}_1\\end{matrix}}\\right]\\nonumber \\\\&-\\frac{1}{6}\\left(\\overline{a}u_1\\left[{\\begin{matrix}4& 16\\mathbf {i}& -\\tfrac{1}{6}& -5\\mathbf {i}&0\\\\108\\mathbf {i}& 48& 0& 0&0\\end{matrix}}\\right]+\\overline{b}a\\left[{\\begin{matrix}0&-72& \\mathbf {i}& 4& 12\\mathbf {i}\\\\0&216\\mathbf {i}& 10& -12\\mathbf {i}& 276\\end{matrix}}\\right]\\right)\\left[{\\begin{matrix}b\\overline{v}_2\\\\ c\\overline{u}_2\\\\ u_1\\overline{w}_1\\\\ u_2\\overline{v}_1\\\\ u_3\\overline{u}_1\\end{matrix}}\\right]\\nonumber \\\\&-\\frac{1}{6}\\left[{\\begin{matrix}\\mathbf {i}(108b\\overline{u}_2+\\tfrac{25}{6}u_1\\overline{v}_1)a\\overline{u}_2+24(\\overline{b}c+2u_2\\overline{u}_1)\\overline{a}u_2-(10\\mathbf {i}a\\overline{v}_1-56b\\overline{u}_1)b\\overline{c}\\\\(60\\overline{c}\\overline{v}_1+504{\\overline{u}_2}^2)ab+(648\\mathbf {i}\\overline{b}c+40u_1\\overline{v}_1+246\\mathbf {i}\\overline{u}_1u_2)\\overline{a}u_2+10bv_1{\\overline{u}_1}^2+72\\mathbf {i}b^2\\overline{c}\\overline{u}_1+30\\mathbf {i}\\overline{a}cv_1\\overline{u}_1+40au_1\\overline{u}_2\\overline{v}_1\\end{matrix}}\\right].\\nonumber $"
],
[
"Levi-Flat 3-folds",
"Our adaptation (REF ) together with (REF ) implies that $\\zeta -\\eta =0 \\text{ on }\\mathcal {H}^2.$ Using Cartan's Lemma with (REF ) while recalling that $\\rho $ is $\\mathbb {R}$ -valued, $&\\left[\\begin{array}{c}\\phi _2\\\\2\\mathbf {i}\\rho \\end{array}\\right]=-\\left[\\begin{array}{c}\\xi \\\\\\phi _1-\\overline{\\phi }_1+\\lambda -\\overline{\\lambda }\\end{array}\\right]+\\left[\\begin{array}{cc}a&\\mathbf {i}b\\\\\\mathbf {i}b&0\\end{array}\\right]\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right]&\\text{ for some } a\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})\\quad b\\in C^\\infty (\\mathcal {H}^2).$ To see how these functions vary on $\\mathcal {H}^2$ , we invoke the MC equations (REF ) again to differentiate $\\text{d}\\left[\\begin{array}{c}b\\\\a\\end{array}\\right]=-\\left[\\begin{array}{cc}\\phi _1+\\overline{\\phi }_1+\\lambda +\\overline{\\lambda }&0\\\\-2\\mathbf {i}\\xi &\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1+\\lambda +5\\overline{\\lambda })\\end{array}\\right]\\left[\\begin{array}{c}b\\\\a\\end{array}\\right]+\\left[\\begin{array}{cc}u_0&0\\\\u_1&\\mathbf {i}u_0+b^2\\end{array}\\right]\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right],\\\\\\text{for some } u_0\\in C^\\infty (\\mathcal {H}^2),\\quad u_1\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C}),\\nonumber $ and we find: if $a$ vanishes identically on each fiber $\\mathcal {H}^2\\rightarrow M$ , then so does $b$ ; $b$ is either zero or non-vanishing on each fiber.",
"Definition 4.7 The second fundamental form (REF ) of a Levi-flat 3-fold $M$ CR embedded in $\\mathcal {Q}=SU(2,2)/\\mathcal {P}$ is given by $\\mathbf {II}=a\\kappa \\odot \\kappa +2\\mathbf {i}b\\kappa \\odot \\eta ,$ where the coefficients are derived from (REF ) via (REF ).",
"The condition that $\\mathbf {II}$ is of (sub)maximal rank on a fiber of $\\mathcal {H}^2\\rightarrow M$ is invariant under the action of CR symmetry group $SU(2,2)$ on $\\mathcal {Q}$ .",
"We update the remaining MC equations on $\\mathcal {H}^2$ , $\\begin{aligned}\\text{d}\\kappa &=(\\lambda +\\overline{\\lambda })\\wedge \\kappa ,\\\\\\text{d}\\eta &=-\\xi \\wedge \\kappa +\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1+3\\lambda -\\overline{\\lambda })\\wedge \\eta -\\tfrac{\\mathbf {i}}{2}b\\kappa \\wedge \\eta ,\\\\\\text{d}\\phi _1&=-\\phi _1\\wedge \\overline{\\phi }_1+\\mathbf {i}\\xi \\wedge \\overline{\\eta }+\\mathbf {i}\\overline{\\xi }\\wedge \\eta -\\mathbf {i}b\\phi _1\\wedge \\kappa -\\mathbf {i}a\\kappa \\wedge \\overline{\\eta }+b\\eta \\wedge \\overline{\\eta },\\\\\\text{d}\\xi &=\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1+\\lambda -3\\overline{\\lambda })\\wedge \\xi +\\psi \\wedge \\eta +(\\tfrac{\\mathbf {i}}{2}b\\xi -a\\overline{\\phi }_1)\\wedge \\kappa -\\mathbf {i}b\\overline{\\phi }_1\\wedge \\eta ,\\\\\\text{d}\\lambda &=-\\psi \\wedge \\kappa +\\mathbf {i}\\overline{a}\\kappa \\wedge \\eta -b\\eta \\wedge \\overline{\\eta },\\\\\\text{d}\\psi &=-(\\lambda +\\overline{\\lambda })\\wedge \\psi +\\mathbf {i}(a\\overline{\\xi }-\\overline{a}\\xi )\\wedge \\kappa -b(\\xi \\wedge \\overline{\\eta }+\\overline{\\xi }\\wedge \\eta )+b\\kappa \\wedge (\\overline{a}\\eta +a\\overline{\\eta })+\\mathbf {i}b^2\\eta \\wedge \\overline{\\eta },\\end{aligned}$ and gather more differential identities by applying $\\text{d}^2=0$ to the exterior derivative of (REF ), $\\text{d}\\left[\\begin{array}{c}u_0\\\\u_1\\end{array}\\right]&=-\\left[\\begin{array}{cc}\\phi _1+\\overline{\\phi }_1+2(\\lambda +\\overline{\\lambda })&0\\\\-3\\mathbf {i}\\xi &\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1+3\\lambda +7\\overline{\\lambda })\\end{array}\\right]\\left[\\begin{array}{c}u_0\\\\u_1\\end{array}\\right]+\\left[\\begin{array}{cc}-2b&0\\\\-3a&2b^2\\end{array}\\right]\\left[\\begin{array}{l}\\psi \\\\\\xi \\end{array}\\right]\\\\&+\\mathbf {i}b\\left[\\begin{array}{cc}- b& b\\\\-\\tfrac{1}{2}a&\\tfrac{5}{2}a\\end{array}\\right]\\left[\\begin{array}{l}\\phi _1\\\\\\overline{\\phi }_1\\end{array}\\right]+\\mathbf {i}\\left[\\begin{array}{cc}-2\\overline{a}b&2 ab\\\\-|a|^2&3 a^2\\end{array}\\right]\\left[\\begin{array}{c}\\eta \\\\\\overline{\\eta }\\end{array}\\right]+\\left[\\begin{array}{cc}v_0&0\\\\v_1&\\mathbf {i}v_0+\\tfrac{5}{2}bu_0-\\tfrac{\\mathbf {i}}{2}b^3\\end{array}\\right]\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right],\\nonumber $ for some $v_0\\in C^\\infty (\\mathcal {H}^2)$ , $v_1\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ .",
"Differentiate (REF ) to get $\\text{d}\\left[\\begin{array}{c}v_0\\\\v_1\\end{array}\\right]&=-\\left[\\begin{array}{cc}\\phi _1+\\overline{\\phi }_1+3(\\lambda +\\overline{\\lambda })&0\\\\-4\\mathbf {i}\\xi &\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1+5\\lambda +9\\overline{\\lambda })\\end{array}\\right]\\left[\\begin{array}{c}v_0\\\\v_1\\end{array}\\right]\\nonumber \\\\&+\\left[\\begin{array}{ccc}-6u_0&-4\\mathbf {i}\\overline{a}b&4\\mathbf {i}ab\\\\-8u_1&-\\tfrac{3\\mathbf {i}}{2}b^3-4\\mathbf {i}|a|^2+8bu_0&6\\mathbf {i}a^2\\end{array}\\right]\\left[{\\begin{matrix}\\psi \\\\\\xi \\\\\\overline{\\xi }\\end{matrix}}\\right]\\nonumber \\\\&+\\left[\\begin{array}{cc}b(b^2-3\\mathbf {i}u_0)& b(b^2+3\\mathbf {i}u_0)\\\\\\tfrac{1}{2}a(b^2-\\mathbf {i}u_0)-\\mathbf {i}bu_1&\\tfrac{1}{2}a(9b^2+11\\mathbf {i}u_0)+3\\mathbf {i}bu_1\\end{array}\\right]\\left[\\begin{array}{l}\\phi _1\\\\\\overline{\\phi }_1\\end{array}\\right]\\\\&-\\left[\\begin{array}{cc}4\\overline{a}b^2+5\\mathbf {i}\\overline{a}u_0+2\\mathbf {i}b\\overline{u}_1&4ab^2-5\\mathbf {i}au_0-2\\mathbf {i}bu_1\\\\6|a|^2b+3\\mathbf {i}\\overline{a}u_1+\\mathbf {i}a\\overline{u}_1&5a^2b-10\\mathbf {i}au_1\\end{array}\\right]\\left[\\begin{array}{c}\\eta \\\\\\overline{\\eta }\\end{array}\\right]\\nonumber \\\\&+\\left[\\begin{array}{cc}w_0&0\\\\w_1&\\mathbf {i}w_0+\\tfrac{5}{2}{u_0}^2+3bv_0-\\tfrac{11\\mathbf {i}}{4}b^2u_0-\\tfrac{1}{4}b^4\\end{array}\\right]\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right];\\nonumber $ $w_0\\in C^\\infty (\\mathcal {H}^2)$ , $w_1\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ , and a final derivative provides $\\text{d}\\left[\\begin{array}{c}w_0\\\\w_1\\end{array}\\right]&=-\\left[\\begin{array}{cc}\\phi _1+\\overline{\\phi }_1+4(\\lambda +\\overline{\\lambda })&0\\\\-5\\mathbf {i}\\xi &\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1+7\\lambda +11\\overline{\\lambda })\\end{array}\\right]\\left[\\begin{array}{c}w_0\\\\w_1\\end{array}\\right]\\nonumber \\\\&-\\left[{\\begin{matrix}12v_0&6\\overline{a}b^2+15\\mathbf {i}\\overline{a}u_0+6\\mathbf {i}b\\overline{u}_1&6ab^2-15\\mathbf {i}au_0-6\\mathbf {i}bu_1\\\\15v_1&b(b^3+8|a|^2-13v_0)+5\\mathbf {i}(3\\overline{a}u_1+a\\overline{u}_1+\\tfrac{9}{4}b^2u_0)-\\tfrac{21}{2}{u_0}^2&8a^2b-30\\mathbf {i}au_1\\end{matrix}}\\right]\\left[{\\begin{matrix}\\psi \\\\\\xi \\\\\\overline{\\xi }\\end{matrix}}\\right]\\nonumber \\\\&+\\left[\\begin{array}{cc}\\mathbf {i}b(4|a|^2+b^3-4v_0)& -\\mathbf {i}b(4|a|^2+b^3-4v_0)\\\\\\mathbf {i}a(6|a|^2+\\tfrac{1}{2}(b^3-v_0))&-\\mathbf {i}a(4|a|^2+6b^3-\\tfrac{19}{2}v_0)\\end{array}\\right]\\left[\\begin{array}{l}\\phi _1\\\\\\overline{\\phi }_1\\end{array}\\right]\\nonumber \\\\&+\\left[\\begin{array}{cc}3u_0(2b^2-\\mathbf {i}u_0)& 3u_0(2b^2+\\mathbf {i}u_0)\\\\\\tfrac{3}{2}(b(au_0+bu_1-\\mathbf {i}v_1)-\\mathbf {i}u_0u_1)&\\tfrac{1}{2}(b(45au_0+15bu_1+7\\mathbf {i}v_1)+17\\mathbf {i}u_0u_1)\\end{array}\\right]\\left[\\begin{array}{l}\\phi _1\\\\\\overline{\\phi }_1\\end{array}\\right]\\\\&-\\left[{\\begin{matrix}3\\overline{a}b(\\tfrac{13}{2}u_0-\\mathbf {i}b^2)+\\overline{u}_1(5b^2+7\\mathbf {i}u_0)+9\\mathbf {i}\\overline{a}v_0+2\\mathbf {i}b\\overline{v}_1&3ab(\\tfrac{13}{2}u_0+\\mathbf {i}b^2)+u_1(5b^2-7\\mathbf {i}u_0)-9\\mathbf {i}av_0-2\\mathbf {i}bv_1\\\\\\tfrac{1}{2}(23|a|^2u_0+b(37\\overline{a}u_1+13a\\overline{u}_1))+\\mathbf {i}(6\\overline{a}v_1+a\\overline{v}_1+4|u_1|^2)&3ab(8u_1+\\mathbf {i}ab)+\\tfrac{11}{2}a^2u_0-15\\mathbf {i}av_1-10\\mathbf {i}{u_1}^2\\end{matrix}}\\right]\\left[\\begin{array}{c}\\eta \\\\\\overline{\\eta }\\end{array}\\right]\\nonumber \\\\&+\\left[\\begin{array}{cc}z_0&0\\\\z_1&\\mathbf {i}z_0-\\tfrac{27\\mathbf {i}}{4}b{u_0}^2+8u_0v_0+\\tfrac{7}{2}bw_0+\\tfrac{\\mathbf {i}}{2}b^2(15|a|^2-\\tfrac{17}{2}v_0)-\\tfrac{19}{8}b^3u_0+\\tfrac{\\mathbf {i}}{8}b^5\\end{array}\\right]\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right],\\nonumber $ for $z_0\\in C^\\infty (\\mathcal {H}^2)$ , $z_1\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ ."
],
[
"Classification: Levi-Nondegenerate Case",
"We have seen that if any coefficient of the second fundamental form (REF ) vanishes identically on $\\mathcal {H}^2$ , then $\\mathbf {II}=0$ everywhere over $M$ .",
"The leading coefficient $a\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ is particularly indicative, since it is either identically zero or non-vanishing on each fiber of $\\mathcal {H}^2\\rightarrow M$ , so the process of adapting frames “branches\" based on the (non)degeneracy of $a$ .",
"If $a=0\\Rightarrow \\mathbf {II}=0$ everywhere, (REF ) subsides to the Maurer-Cartan equations of $U(2,1)$ , which translate via (REF ) to the structure equations (REF ) of the flat CR 3-sphere over a quotient of $\\mathcal {H}^2$ by the central action of $U(1)$ mentioned in Remark REF .",
"In the terminology of Definition REF , $M$ is equivariantly embedded in $\\mathcal {Q}$ as an orbit of $U(2,1)\\subset SU_\\star $ , and in this case no further reduction of $\\mathcal {H}^2$ is admissible.",
"Otherwise, we make the generic assumption that $a\\ne 0$ and arrive at another branching point based on whether $M$ is flat.",
"Locally, all flat 3-folds are equivalent to the 3-sphere, hence the question embeddability is trivial.",
"Rather, we focus on equivariant embeddings by normalizing $\\mathbf {II}$ and tracking which algebras emerge as the symmetries of constant-coefficient structure equations.",
"The results are summarized in Theorem REF .",
"For non-flat 3-folds, we reduce $\\mathcal {H}^2\\rightarrow M$ to the Cartan bundle $\\mathcal {F}^3\\rightarrow M$ and carry out the same normalization procedure as in §REF .",
"The resulting structure equations classify embedded, curved 3-folds as detailed in Theorem REF .",
"Equivariant embeddings are gathered into Theorem REF ."
],
[
"Flat 3-folds",
"If $M$ is flat, the coefficients $S,P$ of the curvature tensor appearing in (REF ) are zero, as are the coefficients (REF ) of the covariant derivative of curvature, the second covariant derivative (REF ), and those we've named (REF ) within the third covariant derivative.",
"Since $a\\ne 0$ , we can use the expressions (REF ), (REF ), (REF ), (REF ), and (REF ) of these coefficients to solve for the jet coordinates of $\\mathbf {II}$ .",
"When more than one variable is eligible for solution – e.g., we can solve $S=\\tfrac{\\epsilon }{6}(a\\overline{v}_1+8\\mathbf {i}b\\overline{u}_1-12c\\overline{a})=0$ for either of $c,\\overline{v}_1$ – we default to the higher-order jet: $&\\text{solve }S=0\\text{ for }\\overline{v}_1,&\\text{solve }P=0\\text{ for }\\overline{v}_2,&&\\text{etc.",
"},$ unless the higher-order variable is obtained from another equation.",
"It is also understood that we solve the complex-conjugated equation for the conjugate coordinate.",
"Onward, $&\\begin{array}{l}\\text{solve }R=0\\text{ for }\\Re v_3, \\\\\\text{solve }Q=0\\text{ for }\\overline{w}_1, \\\\\\text{solve }U=0\\text{ for }\\overline{w}_2, \\\\\\text{solve }V=0\\text{ for }u_4, \\\\\\text{solve }W=0\\text{ for }\\overline{w}_3,\\\\\\text{solve }R_1^{\\prime }=0\\text{ for }v_4,\\\\\\text{solve }R_1^{\\prime \\prime }=0\\text{ for }w_5,\\end{array}\\begin{array}{l}\\text{solve }R_0^{\\prime }=0\\text{ and }R_0^{\\prime \\prime }=0\\text{ for }\\overline{w}_4,w_4,\\\\\\text{solve }Q^{\\prime }=0\\text{ for }\\overline{z}_1,\\\\\\text{solve }U_1^{\\prime }=0\\text{ for }\\overline{z}_2,\\\\\\text{solve }U_2^{\\prime }=0\\text{ for }v_5,\\\\\\text{solve }V^{\\prime }=0\\text{ for }\\overline{z}_3,\\\\\\text{solve }W^{\\prime }=0\\text{ for }\\overline{z}_4.\\end{array}$ Now we resume the process of adapting frames.",
"The identities (REF ) for $\\text{d}a$ and $\\text{d}b$ tell us that we can restrict to frames whose second fundamental form is diagonalized with leading coefficient 1, $\\mathcal {H}^3&=\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^2: a(\\underline{{\\tt v}})=1, b(\\underline{{\\tt v}})=0\\rbrace ,$ over which we have $\\begin{aligned}\\lambda &=-3\\mathbf {i}\\rho +\\overline{u}_1\\overline{\\eta }+\\overline{u}_2\\kappa ,\\\\\\xi &=-\\mathbf {i}c\\overline{\\eta }-u_2\\eta -u_3\\kappa .\\end{aligned}$ With this normalization, we have constrained the coordinates $c_1$ (REF ) and $l$ (REF ) in the fibers of $\\mathcal {H}^2\\rightarrow M$ .",
"In particular, fixing $l$ implies that sections $M\\rightarrow \\mathcal {H}^3$ factor through a unique lift $M\\rightarrow \\hat{M}$ .",
"The real part of $c_0$ (REF ) is similarly determined by the reducing to $\\mathcal {H}^4&=\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^3: \\Re u_2(\\underline{{\\tt v}})=0\\rbrace ,$ as suggested by the equation (REF ) for $\\text{d}u_2$ , which reveals that on $T\\mathcal {H}^4$ , $\\begin{aligned}\\psi &=\\left(\\tfrac{5}{4}+7|c|^2+\\tfrac{5\\epsilon }{12}|u_1|^2-3{u_2}^2-\\tfrac{\\mathbf {i}}{6}u_2(8|u_1|^2+27\\epsilon )-c{u_1}^2-\\overline{c}{\\overline{u}_1}^2+\\tfrac{1}{2}u_1u_3+\\tfrac{1}{2}\\overline{u}_1\\overline{u}_3\\right)\\kappa \\\\&+\\tfrac{\\mathbf {i}}{4}(8\\overline{c}\\overline{u}_1-5\\epsilon u_1+8\\mathbf {i}u_1u_2-10\\overline{u}_3)\\eta -\\tfrac{\\mathbf {i}}{4}(8cu_1-5\\epsilon \\overline{u}_1+8\\mathbf {i}\\overline{u}_1u_2-10u_3)\\overline{\\eta }.\\end{aligned}$ The rank of $\\mathbf {II}$ will index our final instance of branching in the flat setting.",
"If $c=0$ so that $\\text{rank}(\\mathbf {II})=1$ , $&\\text{d}c=0\\Rightarrow u_1=u_3=0,&\\text{d}u_1=0\\Rightarrow u_2=-\\mathbf {i}\\frac{\\epsilon }{2},&&\\text{d}u_2=0\\Rightarrow \\Im v_3=0,$ and we are left with structure equations $\\text{d}\\kappa &=\\mathbf {i}\\eta \\wedge \\overline{\\eta },\\\\\\text{d}\\eta &=\\mathbf {i}(\\epsilon \\kappa -4\\rho )\\wedge \\eta ,\\\\\\text{d}\\rho &=0,$ which describe a central extension of the symmetry algebra of (VIII,C) when $\\epsilon =-1$ or (IX,D) when $\\epsilon =1$ (see the end of §REF ).",
"On the other hand, if $\\mathbf {II}$ has maximal rank 2 it must be that $c$ is nonvanishing, so the identity (REF ) for $\\text{d}c$ shows that we can reduce to those frames where $c$ takes values in $\\mathbb {R}\\setminus \\lbrace 0\\rbrace $ , $\\mathcal {H}^5&=\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^4: \\Im c(\\underline{{\\tt v}})=0\\rbrace ,$ over which $\\rho =\\frac{1}{96c}(\\epsilon (18c+{u_1}^2+{\\overline{u}_1}^2)+6(u_1\\overline{u}_3+\\overline{u}_1u_3)-4c|u_1|^2)\\kappa +\\frac{\\mathbf {i}}{16c}(3u_1c-u_3)\\eta -\\frac{\\mathbf {i}}{16c}(3\\overline{u}_1c-\\overline{u}_3)\\overline{\\eta }.$ Differentiating this and comparing to (REF ) provides $\\epsilon (9cu_1-{u_1}^3)-18c^2\\overline{u}_1+2c\\overline{u}_1{u_1}^2+54c\\overline{u}_3-6\\overline{u}_3{u_1}^2=0.$ Therefore, the most generic CR embedding in $\\mathcal {Q}$ of a flat 3-fold is encoded in the structure equations $\\begin{aligned}\\text{d}\\kappa &=\\mathbf {i}\\eta \\wedge \\overline{\\eta }-\\kappa \\wedge (u_1\\eta +\\overline{u}_1\\overline{\\eta }),\\\\\\text{d}\\eta &=-\\frac{c\\overline{u}_1+\\overline{u}_3}{4c}\\eta \\wedge \\overline{\\eta }-\\mathbf {i}\\frac{\\epsilon (18c+{u_1}^2+{\\overline{u}_1}^2)+6(u_1\\overline{u}_3+\\overline{u}_1u_3)-4c|u_1|^2-48\\mathbf {i}cu_2}{24c}\\kappa \\wedge \\eta -\\mathbf {i}c\\kappa \\wedge \\overline{\\eta },\\end{aligned}$ where $c,u_1,u_3\\in C^\\infty (\\mathcal {H}^5,\\mathbb {C})$ and $-\\mathbf {i}u_2,\\Im v_3\\in C^\\infty (\\mathcal {H}^5,\\mathbb {R})$ satisfy (REF ), (REF ), and (REF ) subject to $a=1,b=0$ , (REF ), (REF ), (REF ), (REF ), (REF ), and (REF ).",
"The equations (REF ) remain invariant under the action of the CR symmetry group $SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ on $\\mathcal {Q}$ , so they classify embedded flat 3-folds whose second fundamental form has rank 2.",
"Moreover, when (REF ) has constant coefficients, the equations classify equivariant embeddings as in Definition REF .",
"Taking $c,u_1,u_3$ to be constant implies $&c=\\tfrac{1}{9}{u_1}^2,&{u_1}^2={\\overline{u}_1}^2,&&u_2=-\\tfrac{\\mathbf {i}}{6}(3\\epsilon +|u_1|^2),&&u_3=\\tfrac{1}{3}{u_1}^3,&&\\Im v_3=0,$ so $u_1\\ne 0$ is either real or imaginary, and by $\\text{d}u_2=0$ from (REF ) we see $16\\epsilon |u_1|^2+9=0.$ The latter only has solutions when $\\epsilon =-1$ , given by $u_1=\\pm \\tfrac{3}{4}$ or $u_1=\\pm \\tfrac{3}{4}\\mathbf {i}$ .",
"Submitting the CR coframing $\\kappa ,\\eta $ to the 1-adapted transformation $\\frac{2}{9}\\left[\\begin{array}{rc}2|u_1|^2&0\\\\-9\\mathbf {i}|u_1|^2&3u_1\\end{array}\\right]\\left[\\begin{array}{c}\\kappa \\\\\\eta \\end{array}\\right]$ brings (REF ) into the form (REF ) for the homogeneous 3-fold ($\\text{VI}_3$ , E)."
],
[
"Curved 3-folds",
"Suppose the coefficients $S\\in C^\\infty (\\mathcal {F}^3,\\mathbb {C})$ of the curvature tensor and $a\\in C^\\infty (\\mathcal {H}^2,\\mathbb {C})$ of the second fundamental form of $M$ are non-vanishing and invoke the identity (REF ) for $\\text{d}a$ to reduce to Hermitian frames where $a$ is $\\mathbb {R}$ -valued: $\\mathcal {H}^3=\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^2 : a(\\underline{{\\tt v}})-\\overline{a}(\\underline{{\\tt v}})=0\\rbrace .$ On $T\\mathcal {H}^3$ we have $\\rho =\\frac{\\mathbf {i}}{6a}(a(\\lambda -\\overline{\\lambda })+(u_2-\\overline{u}_2)\\kappa +(u_1+2\\mathbf {i}\\overline{b})\\eta -(\\overline{u}_1-2\\mathbf {i}b)\\overline{\\eta }),$ and the transformation (REF ) is the infinitesimal version of the identification $\\mathcal {H}^3=\\mathcal {F}^3$ as bundles of coframes over $M$ .",
"Now we pursue the same process of reduction as in §REF .",
"There, $\\mathcal {F}^4$ is defined by normalizing $S$ , which is achieved in the present setting by solving the equation (REF ) $S=1$ for $\\overline{v}_1$ , thereby exhausting one complex degree of freedom in the fibers of $\\mathcal {H}^3\\supset \\mathcal {H}^4=\\mathcal {F}^4$ over $M$ .",
"Accordingly, $\\lambda $ is no longer an independent 1-form, but is determined by (REF ) via (REF ) and (REF ), (REF ).",
"Next we constrain $\\overline{v}_2$ in (REF ) by $P=0$ so that $\\mathcal {H}^5\\subset \\mathcal {H}^4$ coincides with $\\mathcal {F}^5$ and $\\xi $ satisfies (REF ) by virtue of (REF ), (REF ), and (REF ).",
"Finally, the condition $\\Re U=0$ fixes $\\Re w_2$ on $\\mathcal {F}^6=\\mathcal {H}^6\\subset \\mathcal {H}^5$ , where $\\psi $ is subject to (REF ) with coefficients (REF ), (REF ), and (REF ).",
"Our construction proves the following Theorem 5.1 Let $M$ be a 3-dimensional, Levi-nondegenerate CR manifold whose curvature tensor is non-vanishing.",
"$M$ is CR embeddable in the 5-dimensional real hyperquadric $\\mathcal {Q}=SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )/\\mathcal {P}$ if and only if $M$ admits a 1-adapted CR coframing $\\kappa ,\\eta \\in \\Omega ^1(M,\\mathbb {C})$ and functions $a,b,c,u_1,\\dots ,u_4,v_1,\\dots ,v_5,w_1,\\dots ,w_5,z_1,\\dots ,z_4\\in C^\\infty (M,\\mathbb {C})$ satisfying the structure equations (REF ) for (REF ) given by (REF ), (REF ), where $S=1,P=\\Re U=0$ , and the differential identities (REF ), (REF ), (REF ), and (REF ) hold for (REF ) determined by (REF ), (REF ), and (REF ) with (REF ).",
"For such $M$ , the structure equations (REF ) remain invariant under the action of $SU(3-\\delta _\\epsilon ,1+\\delta _\\epsilon )$ on $\\mathcal {Q}$ .",
"Let us implement Theorem REF to treat the examples of non-flat homogeneous 3-folds.",
"In addition to $v_1,v_2$ decided by $S=1$ and $P=0$ , respectively, we find most of the functions (REF ) by following the prescription (REF ), except that the coefficients of the covariant derivatives of curvature take the values listed in Remark REF instead of zero.",
"A general embedding now depends on the existence of functions $a,b,c,u_1,u_2,u_3,\\Im v_3$ which satisfy (REF ) and the identities $\\text{d}^2=0$ , and these conditions become algebraic for equivariant embeddings.",
"Namely, if $\\mathbf {II}$ is constant, $c&=\\tfrac{2}{a}b^2-3\\mathbf {i}Ab-Ca,&&u_1=2\\mathbf {i}\\overline{b}+2\\overline{A}a,\\\\u_2&=\\tfrac{2\\mathbf {i}}{a}|b|^2+\\overline{A}b+\\mathbf {i}(|A|^2-\\tfrac{B}{4})a-\\epsilon \\tfrac{\\mathbf {i}}{4}a^3,&&u_3=\\tfrac{4\\mathbf {i}}{a^2}\\overline{b}b^2+\\tfrac{6A}{a}|b|^2-\\tfrac{\\mathbf {i}}{2}(Bb+4C\\overline{b}-4|A|^2b)-\\epsilon \\tfrac{\\mathbf {i}}{2}ba^2,$ which in turn implies $&0=A\\overline{b}+\\overline{A}b,&B=\\tfrac{4}{3a^2}|b|^2+\\tfrac{2\\mathbf {i}}{a}\\overline{A}b+\\tfrac{8}{3}|A|^2+\\epsilon a^2,&&C=-\\tfrac{10}{9}\\overline{A}^2+\\tfrac{38\\mathbf {i}}{9a}\\overline{Ab}+\\tfrac{2}{3a^2}(2\\overline{b}^2+\\epsilon ),$ along with $\\Im v_3=0$ .",
"Polynomial relationships between the remaining quantities $a,b,A$ are clarified by addressing separately the possible values of $\\overline{A}=\\pm A$ .",
"First consider $A=0$ , yielding $&\\overline{b}=\\frac{2b}{a^4},&b(3a^8+6\\epsilon b^2)=0.$ If $b=0$ , $a\\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace $ is free and the resulting structure equations describe the homogeneous models (VIII, K) when $\\epsilon =-1$ or (IX, L) when $\\epsilon =1$ .",
"Note that $\\mathbf {II}$ is diagonalized with determinant $-\\epsilon \\tfrac{2}{3}$ .",
"The alternative ($b\\ne 0$ ) is only possible when $\\epsilon =-1$ ; namely, $a=\\pm \\@root 4 \\of {2}$ and $b=\\pm \\sqrt{2}$ , which is (IX, L) for $B=\\tfrac{\\sqrt{2}}{3}$ , and in this case $\\text{rank}\\mathbf {II}=1$ .",
"For $A\\ne 0$ , (REF ) shows $\\overline{b}=\\tfrac{\\overline{A}}{A}b$ while $a,b,A$ are governed by three polynomials of degree six or seven, $&D_{\\overline{\\eta }}a^2\\text{d}c,&D_\\kappa a^2A\\text{d}\\overline{b},&&D_\\kappa a^2A\\text{d}u_1,$ where the hook $$ indicates contraction with the vector field dual to the subscripted 1-form.",
"Solutions exist only when $\\epsilon =-1$ , and up to signs they are given by $&A=\\mathbf {i}\\frac{4\\@root 4 \\of {10}}{\\sqrt{5}},&a=\\frac{\\@root 4 \\of {10}}{\\sqrt{5}},&&b=-\\sqrt{10}.$ The homogeneous model is therefore (VI$_{t}$ , E) as in (REF ) with $\\iota =\\mathbf {i}$ , $t=\\tfrac{4\\@root 4 \\of {10}}{\\sqrt{5}}$ , and $m=\\tfrac{t}{2}$ so that $S=1$ ."
],
[
"Classification: Levi-Flat Case",
"All Levi-flat 3-folds $M$ are locally CR equivalent, so embeddability $M\\subset \\mathcal {Q}$ is a question of the signature of $\\mathcal {Q}$ 's Levi form.",
"In this respect we recall that our discussion here applies only to the real hyperquadric whose CR symmetry group is $SU(2,2)$ ($\\epsilon =-1$ in the notation of §), hence the embeddings of interest will be equivariant for some action of this Lie group in the sense of Definition REF .",
"Our list of homogeneous models in §REF omitted the Bianchi algebras that serve as infinitesimal CR symmetries of Levi-flat $M$ ; let us record here two models with symmetry of Bianchi type V. For an appropriate choice of bases, the extension of $\\mathbb {R}^2$ by $\\left[{\\begin{matrix}\\@root 3 \\of {3}&0\\\\0&-\\@root 3 \\of {3}\\end{matrix}}\\right]$ has structure equations $\\begin{aligned}\\text{d}\\kappa &=\\@root 3 \\of {3}(\\eta +\\overline{\\eta })\\wedge \\kappa ,\\\\\\text{d}\\eta &=\\@root 3 \\of {3}\\eta \\wedge \\overline{\\eta }+\\tfrac{\\mathbf {i}}{15\\@root 3 \\of {3}}(4\\eta -\\overline{\\eta })\\wedge \\kappa ,\\end{aligned}$ and the extension by $\\left[{\\begin{matrix}1&0\\\\0&2\\end{matrix}}\\right]$ is described by $\\begin{aligned}\\text{d}\\kappa &=\\tfrac{\\mathbf {i}}{2}(\\eta -\\overline{\\eta })\\wedge \\kappa ,\\\\\\text{d}\\eta &=\\mathbf {i}\\eta \\wedge \\overline{\\eta }.\\end{aligned}$ As in the Levi-nondegenerate case, the process of reducing $\\mathcal {H}^2\\rightarrow M$ to these and other homogeneous models branches based on the rank of the second fundamental form $\\mathbf {II}$ .",
"This section constitutes the proof of Theorem REF ."
],
[
"$\\mathbf {II}=0$ : Maximal Symmetry",
"Let $\\underline{{\\tt v}}=({\\tt v}_0,{\\tt v}_1,{\\tt v}_2,{\\tt v}_3)$ be a Hermitian frame () whose vectors we rearrange, $&\\left[\\begin{array}{cccc}{\\tt n}_1&{\\tt n}_2&{\\tt n}_3&{\\tt n}_4\\end{array}\\right]=\\left[\\begin{array}{cccc}{\\tt v}_0&{\\tt v}_1&{\\tt v}_2&{\\tt v}_3\\end{array}\\right]U,&U=\\frac{1}{\\sqrt{2}}\\left[\\begin{array}{rrrr}0&\\sqrt{2}&0&0\\\\1&0&0&-1\\\\1&0&0&1\\\\0&0&\\sqrt{2}&0\\end{array}\\right].$ The transformation $U$ is “unitary\" in the sense that $U^{-1}=\\overline{U}^t$ , and even though $\\overline{U}^tỦ\\ne , we refer to the symmetry groups of both forms as $ SU(2,2)$.",
"Let $ RCSL2C$ be the 10-dimensional parabolic subgroup that stabilizes the partial flag{\\begin{@align}{1}{-1}\\langle {\\tt n}_1\\rangle _\\mathbb {C}\\subset \\langle {\\tt n}_1,{\\tt n}_2,{\\tt n}_3\\rangle _\\mathbb {C}\\subset \\mathbb {C}^{4},\\end{@align}}and name $ R=RCSU(2,2)$ with Lie algebra $ rsu(2,2)$.$ The new basis (REF ) consists entirely of $-null vectors, and for $vH2$ they are adapted to $ M$ -- cf.",
"(\\ref {LFfirstadapt}) -- in that $n1,n2,n3$ span $ Tv0M$ with $n1$ descending to the CR bundle of $ M$.",
"If the complex curve tangent to the CR bundle of $ M$ is a complex line in $ Q$ (see \\cite [Example 1.5]{BryanthololorentzCR}), then the osculating flag (\\ref {oscflag}) is constant along $ M$ and $ M$ itself is contained in the fixed subspace $n1,n2,n3C$.",
"Hence, one expects the extrinsic CR symmetries of such $ M$ given by the action of $ SU(2,2)$ on $ Q$ to lie in $ R$.",
"In general, $ |H2$ is (\\ref {UMCform}) subject to (\\ref {trace-free}), (\\ref {zetaiseta}), and (\\ref {LFIIcoeff}), so transforming according to (\\ref {LFUtrans}) we get{\\begin{@align}{1}{-1}U^{-1}\\mu |_{\\mathcal {H}^2}U=\\left[\\begin{array}{cccc}-\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1+\\lambda -\\overline{\\lambda })&\\sqrt{2}\\eta &\\sqrt{2}\\xi -\\tfrac{1}{\\sqrt{2}}(a\\kappa +\\mathbf {i}b\\eta )&\\phi _1-\\overline{\\phi }_1-\\tfrac{\\mathbf {i}}{2}b\\kappa \\\\-\\tfrac{1}{\\sqrt{2}}(b\\overline{\\eta }+\\mathbf {i}\\overline{a}\\kappa )&\\lambda &\\psi &\\sqrt{2}\\mathbf {i}\\overline{\\xi }-\\tfrac{1}{\\sqrt{2}}(b\\overline{\\eta }+\\mathbf {i}\\overline{a}\\kappa )\\\\0&\\kappa &-\\overline{\\lambda }&-\\mathbf {i}\\sqrt{2}\\overline{\\eta }\\\\-\\tfrac{\\mathbf {i}}{2}b\\kappa &0&-\\tfrac{1}{\\sqrt{2}}(a\\kappa +\\mathbf {i}b\\eta )&\\tfrac{1}{2}(\\phi _1+\\overline{\\phi }_1-\\lambda +\\overline{\\lambda })\\end{array}\\right].\\end{@align}}In particular, if $ a=b=0$, (\\ref {LFMCtrans}) takes values $ r$, and (\\ref {LFHtwoMC}) are the Maurer-Cartan equations of $ R$.",
"Thus, if $ II=0$, $ H2$ is locally $ RSU(2,2)$.$"
],
[
"Rank$(\\mathbf {II})=1$",
"The second fundamental form (REF ) will have rank one only if $b=0$ in (REF ), which in turn requires $u_0=0$ (REF ), $v_0=0$ (REF ), $w_0=0$ (REF ), and $z_0=0$ (REF ), but it must be that $a$ is nonvanishing.",
"The identity (REF ) for $\\text{d}a$ therefore implies that we can reduce to $\\mathcal {H}^3=\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^2: a(\\underline{{\\tt v}})=1\\rbrace ,$ and on $\\mathcal {H}^3$ we have $\\lambda =-\\tfrac{1}{6}(\\phi _1+\\overline{\\phi }_1)-\\tfrac{1}{12}(u_1-5\\overline{u}_1)\\kappa .$ Looking to $\\text{d}u_1$ (REF ) and $\\text{d}v_1$ (REF ), we see the opportunity to reduce further, $\\mathcal {H}^4=\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^3: \\Re u_1(\\underline{{\\tt v}})=0=v_1(\\underline{{\\tt v}})\\rbrace .$ After renaming $u=\\Im u_1\\in C^\\infty (\\mathcal {H}^4),$ we can write the newly imposed constraints $\\psi &=\\tfrac{1}{3}(u^2\\kappa -2\\mathbf {i}\\eta +2\\mathbf {i}\\overline{\\eta }),\\\\\\xi &=\\tfrac{1}{30}(40u^3+6\\mathbf {i}w_1-9\\mathbf {i}\\overline{w}_1)\\kappa +\\tfrac{\\mathbf {i}}{15}(11u\\eta +u\\overline{\\eta }),$ as well as the updated identity $\\text{d}w_1=(6\\mathbf {i}+w_1)\\phi _1+(w_1-4\\mathbf {i})\\overline{\\phi }_1+(z_1-\\tfrac{80}{3}u^4-8\\mathbf {i}uw_1+3\\mathbf {i}u\\overline{w}_1)\\kappa +\\tfrac{\\mathbf {i}}{3}u^2(16\\eta +38\\overline{\\eta }).$ The latter suggests a final reduction $\\mathcal {H}^5=\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^4: w_1(\\underline{{\\tt v}})=0\\rbrace ,$ over which $\\phi _1&=-\\tfrac{\\mathbf {i}}{30}(80u^4-9z_1+6\\overline{z}_1)\\kappa -\\tfrac{1}{15}u^2(62\\eta +73\\overline{\\eta }),\\\\\\text{d}u&=-\\tfrac{1}{3}uz\\kappa +(1-3u^3)\\eta +(1-3u^3)\\overline{\\eta };&&z=\\Im z_1.$ What's left of (REF ) reads $\\begin{aligned}\\text{d}\\kappa &=3u^2(\\eta +\\overline{\\eta })\\wedge \\kappa ,\\\\\\text{d}\\eta &=3u^2\\eta \\wedge \\overline{\\eta }-\\tfrac{1}{15}(4\\mathbf {i}u+5z)\\kappa \\wedge \\eta +\\tfrac{\\mathbf {i}}{15}u\\kappa \\wedge \\overline{\\eta },\\end{aligned}$ for $u,z\\in C^\\infty (\\mathcal {H}^5)$ .",
"The equations (REF ), along with differential identities necessitated by $\\text{d}^2=0$ , classify embedded Levi-flat 3-folds $M\\subset \\mathcal {Q}$ with rank$(\\mathbf {II})=1$ up to the action of $SU(2,2)$ on $\\mathcal {Q}$ , including the unique equivariant embedding when $\\text{d}u=0\\Rightarrow u=3^{-\\tfrac{1}{3}}$ , $z=0$ , which is exactly the model (REF )."
],
[
"Rank$(\\mathbf {II})=2$",
"The second fundamental form (REF ) has full rank if and only if $b$ is nonvanishing; in this case the identities (REF ), (REF ) for $\\text{d}b, \\text{d}a, \\text{d}u_0$ show that we can reduce to $\\mathcal {H}^3=\\left\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^2:\\begin{array}{c}b(\\underline{{\\tt v}})=\\pm 1\\\\a(\\underline{{\\tt v}})=0\\\\u_0(\\underline{{\\tt v}})=0\\end{array}\\right\\rbrace ,$ over which we have relations $\\phi _1&=-\\frac{1}{2}(\\lambda +\\overline{\\lambda })+\\frac{\\mathbf {i}}{b}\\psi -\\frac{v_0}{2b^2}\\kappa ,\\\\\\xi &=\\frac{\\mathbf {i}}{2}b\\eta +\\frac{\\mathbf {i}u_1}{2b}\\kappa ,$ as well as the identity $\\text{d}u_1=-u_1(\\lambda +3\\overline{\\lambda })+(v_1+\\mathbf {i}bu_1)\\kappa +\\mathbf {i}(v_0+\\tfrac{1}{2}b^3)\\eta .$ Thus we are obliged to consider branching based on the possible values of $u_1$ .",
"If $u_1=0$ identically, then $v_0=-\\tfrac{1}{2}b^3$ , $v_1=0$ , and the remaining coefficients in (REF ), (REF ) are zero, leaving $\\begin{aligned}\\text{d}\\kappa &=(\\lambda +\\overline{\\lambda })\\wedge \\kappa ,\\\\\\text{d}\\eta &=(\\lambda -\\overline{\\lambda })\\wedge \\eta ,\\\\\\text{d}\\lambda &=-\\psi \\wedge \\kappa -b\\eta \\wedge \\overline{\\eta },\\\\\\text{d}\\psi &=\\psi \\wedge (\\lambda +\\overline{\\lambda }).\\end{aligned}$ Note that (REF ) are the Maurer-Cartan equations of the MC forms $&\\left[\\begin{array}{cc}\\tfrac{1}{2}(\\lambda +\\overline{\\lambda })&\\psi \\\\\\kappa &-\\tfrac{1}{2}(\\lambda +\\overline{\\lambda })\\end{array}\\right]\\in \\Omega ^1(\\mathcal {H}^3,\\mathfrak {sl}_2\\mathbb {R}),&\\left[\\begin{array}{cc}\\tfrac{1}{2}(\\lambda -\\overline{\\lambda })&-\\sqrt{b}\\overline{\\eta }\\\\\\sqrt{b}\\eta &-\\tfrac{1}{2}(\\lambda -\\overline{\\lambda })\\end{array}\\right]\\in \\Omega ^1(\\mathcal {H}^3,\\mathfrak {su}(p,q)),$ where $(p,q)=(2,0)$ if $b=1$ and $(p,q)=(1,1)$ if $b=-1$ .",
"Otherwise, in any neighborhood where $u_1\\ne 0$ we can normalize it to define $\\mathcal {H}^4=\\left\\lbrace \\underline{{\\tt v}}\\in \\mathcal {H}^3:\\begin{array}{c}u_1(\\underline{{\\tt v}})=1\\\\\\Re v_1(\\underline{{\\tt v}})=0\\end{array}\\right\\rbrace ,$ so that on $\\mathcal {H}^4$ , $\\lambda &=-\\tfrac{\\mathbf {i}}{2}(b+v)\\kappa -\\tfrac{\\mathbf {i}}{16}(b^3+2v_0)\\eta -\\tfrac{3\\mathbf {i}}{16}(b^3+2v_0)\\overline{\\eta } &(v=\\Im v_1),\\\\\\psi &=\\tfrac{1}{16}(vb^3 + b^4+ 2bv_0 + 2v_0v + 2\\mathbf {i}w_0 )\\eta +\\tfrac{1}{16}(vb^3 + b^4+ 2bv_0 + 2v_0v - 2\\mathbf {i}w_0 )\\overline{\\eta }\\\\&+\\frac{1}{16b}(4v^2b + 4vb^2 + 3b^3 + 2b\\overline{w}_1 + 2bw_1 - 16v_0)\\kappa ,$ and the structure equations are $\\begin{aligned}\\text{d}\\kappa &=\\tfrac{\\mathbf {i}}{8}(b^3+2v_0)(\\eta -\\overline{\\eta })\\wedge \\kappa ,\\\\\\text{d}\\eta &=\\tfrac{\\mathbf {i}}{4}(b^3+2v_0)\\eta \\wedge \\overline{\\eta }-\\mathbf {i}(b+v)\\kappa \\wedge \\eta ,\\end{aligned}$ with identities $\\begin{aligned}\\text{d}v_0&=w_0\\kappa -\\tfrac{\\mathbf {i}}{8}(b^6+4b^3v_0+4{v_0}^2+16b)\\eta +\\tfrac{\\mathbf {i}}{8}(b^6+4b^3v_0+4{v_0}^2+16b)\\overline{\\eta },\\\\\\text{d}v&=\\tfrac{\\mathbf {i}}{2}(\\overline{w}_1-w_1)\\kappa -\\tfrac{\\mathbf {i}}{8}(3vb^3+3b^4+6v_0v+4\\mathbf {i}w_0+6bv_0)\\eta +\\tfrac{\\mathbf {i}}{8}(3vb^3+3b^4+6v_0v-4\\mathbf {i}w_0+6bv_0)\\overline{\\eta }.\\end{aligned}$ We conclude that embedded Levi-flat 3-folds $M\\subset \\mathcal {Q}$ with rank$(\\mathbf {II})=2$ are classified up to the action of $SU(2,2)$ on $\\mathcal {Q}$ by the Maurer-Cartan equations (REF ) of $SL_2\\mathbb {R}\\times SU(p,q)$ – $(p,q)=(2,0)$ or $(1,1)$ – when $u_1=0$ , or (REF ) with (REF ) when $u_1\\ne 0$ .",
"The latter evince a homogeneous action in $SU(2,2)$ when $&w_0=0,&v=1,&&v_0=-\\tfrac{3}{2}\\text{ or }\\tfrac{5}{2},&&b=-1,$ in which case they coincide (up to sign) with (REF )."
]
] | 1808.08625 |
[
[
"On Strong Stability and Robust Strong Stability of Linear Difference\n Equations with Two Delays"
],
[
"Abstract This paper provides a necessary and sufficient condition for guaranteeing exponential stability of the linear difference equation $x(t)=Ax(t-a)+Bx(t-b)$ where $a>0,b>0$ are constants and $A,B$ are $n\\times n$ square matrices, in terms of a linear matrix inequality (LMI) of size $\\left( k+1\\right) n\\times \\left( k+1\\right) n$ where $k\\geq1$ is some integer.",
"Different from an existing condition where the coefficients $\\left( A,B\\right) $ appear as highly nonlinear functions, the proposed LMI condition involves matrices that are linear functions of $\\left( A,B\\right) .$ Such a property is further used to deal with the robust stability problem in case of norm bounded uncertainty and polytopic uncertainty, and the state feedback stabilization problem.",
"Solutions to these two problems are expressed by LMIs.",
"A time domain interpretation of the proposed LMI condition in terms of Lyapunov-Krasovskii functional is given, which helps to reveal the relationships among the existing methods.",
"Numerical example demonstrates the effectiveness of the proposed method."
],
[
"Introduction and Literature Review",
"Throughout this paper, we use $A\\otimes B$ to denote the Kronecker product of matrices $A$ and $B.$ For a matrix $A,$ the symbols $\\left|A\\right|,\\left\\Vert A\\right\\Vert ,A^{\\mathrm {T}},A^{\\mathrm {H}}$ , and $\\rho \\left(A\\right) $ denote respectively its determinant, norm, transpose, conjugate transpose, and spectral radius.",
"For a square matrix $P$ , $P>0$ denotes that it is positive definite.",
"The linear (continuous-time) difference equation $x(t)=\\sum \\limits _{i=1}^{N}A_{i}x(t-r_{i}), $ where $r_{i}>0$ are constants and $A_{i}$ are square matrices, is frequently encountered in neutral-type time-delay systems [11], [18] and coupled differential-functional equations [10], [15].",
"The stability of system (REF ) is usually the necessary condition for ensuring the asymptotic stability of the above two types of time-delay systems, and thus has attracted considerable attentions in the literature [3], [4], [7], [10], [12], [21], [23].",
"It is known that (REF ) is stable if and only if its spectral abscissa is less than zero [12].",
"However, the spectral abscissa of (REF ) is not continuous in delays and the stability might be destroyed by arbitrarily small changes in the delay [1], [12].",
"Therefore, the concept of strong stability was introduced by [12] to handle this hypersensitivity of the stability with respect to delays, which has been generalized in [18].",
"To go further, we introduce the following result from Theorem 6.1 (Chapter 9, p. 286) in [12].",
"Lemma 1 System (REF ) is strongly stable if and only if $\\max _{\\theta _{i}\\in \\left[ 0,2\\pi \\right] ,i=1,2,\\ldots ,N}\\rho \\left(\\sum \\limits _{i=1}^{N}A_{i}\\mathrm {e}^{\\mathrm {j}\\theta _{i}}\\right)<1.$ The strong stability concept is important since in practical applications the delays are generally subject to small errors [10].",
"The test of strong stability is however rather complex [13].",
"Indeed, condition (REF ) is not tractable in general since the spectral radius should be tested for all $\\theta _{i}\\in \\left[ 0,2\\pi \\right] ,i=1,2,\\ldots ,N.$ Strong stability of (REF ) was tested via deciding positive definiteness of a multivariate trigonometric polynomial matrix, which is then solved as a converging hierarchy of LMIs [13].",
"The condition in [13] needs to compute the characteristic equation of (REF ), which is not explicitly expressed as functions of the coefficients, and thus seems difficult to be used for robust stability analysis.",
"For a single delay, strong stability can be checked by computing the generalized eigenvalues of a pair of matrices [16], [17] as well as the matrix pencil based approach [19].",
"The method of cluster treatment of characteristic roots was used in [20] to derive the stability maps of (REF ) with three delays.",
"For more related work, see [10], [12], [13], [20] and the references therein.",
"In this note, we restrict ourself to a special case of (REF ) where $N=2$ , for which we rewrite (REF ) as $x(t)=Ax(t-a)+Bx(t-b), $ where $a,b$ are positive constants, and $A,B$ are $n\\times n$ square matrices.",
"Regarding the existence of a solution, the continuity/discontinuity of the solution, and definitions for stability of the solution, readers are suggested to refer [3] and [12] for details.",
"Notice that, by Lemma REF , system (REF ) is strongly stable if and only if $\\rho \\left( \\Delta _{\\theta }\\right) <1,\\; \\theta \\in \\left[ 0,2\\pi \\right],\\text{ }\\Delta _{\\theta }=A+B\\mathrm {e}^{-\\mathrm {j}\\theta }.",
"$ It came to our attention that condition (REF ) happens to be equivalent to the stability of the 2-D linear system described by the Fornasini-Marchesini second model $x\\left( i+1,j+1\\right) =Ax\\left( i,j+1\\right) +Bx\\left( i+1,j\\right) ,$ which has been well studied in the literature [6], [9].",
"For stability analysis of (REF ), a necessary and sufficient condition expressed by an LMI of size $3n^{2}\\times 3n^{2}$ was established in [6].",
"Lemma 2 The system (REF )/(REF ) is strongly/exponentially stable if and only if $\\rho \\left( A+B\\right) <1, $ and there exist two symmetric matrices $P_{1}\\in \\mathbf {R}^{n^{2}\\times n^{2}},P_{2}\\in \\mathbf {R}^{n^{2}\\times n^{2}}$ and a matrix $P_{3}\\in \\mathbf {R}^{n^{2}\\times n^{2}}$ such that $\\left[\\begin{array}[c]{ccc}-P_{1} & 0 & -P_{3}\\\\0 & -P_{2} & P_{3}^{\\mathrm {T}}\\\\-P_{3}^{\\mathrm {T}} & P_{3} & P_{1}+P_{2}\\end{array}\\right] <E^{\\mathrm {T}}E, $ where $E=[B^{\\mathrm {T}}\\otimes A,A^{\\mathrm {T}}\\otimes B,A^{\\mathrm {T}}\\otimes A+B^{\\mathrm {T}}\\otimes B-I_{n}\\otimes I_{n}].$ This result is almost the same as Theorem 1 in [6], where $E$ is replaced by $E_{\\ast }=[B\\otimes A,A\\otimes B,A\\otimes A+B\\otimes B-I_{n}\\otimes I_{n}].$ The proof given in [6] is based on the Guardian map and the positive real lemma.",
"Motivated by [5], we provide in Appendix a simple proof based on the well-known Yakubovich-Kalman-Popov (YKP) lemma.",
"Another necessary and sufficient conditions, which involve the generalized eigenvalues of two matrices with size $2n^{2}\\times 2n^{2},$ were obtained in [9], which were also established initially for testing stability of the 2-D linear system (REF ).",
"Bliman established in [2] another LMI based necessary and sufficient conditions for testing stability of (REF ).",
"To introduce this result, for any $k\\in \\mathbf {N}^{+},$ we define $\\overline{A}_{k} & =\\left[\\begin{array}[c]{ccccc}0 & B & AB & \\cdots & A^{k-2}B\\\\& 0 & B & \\cdots & A^{k-3}B\\\\& & \\ddots & \\ddots & \\vdots \\\\& & & 0 & B\\\\& & & & 0\\end{array}\\right] \\in \\mathbf {R}^{kn\\times kn},\\overline{B}_{k}=\\left[\\begin{array}[c]{c}A^{k-1}\\\\A^{k-2}\\\\\\vdots \\\\A\\\\I_{n}\\end{array}\\right] \\in \\mathbf {R}^{kn\\times n},\\\\\\overline{{A}}_{k} & =\\left[\\begin{array}[c]{ccccc}B & AB & A^{2}B & \\cdots & A^{k-1}B\\\\& B & AB & \\cdots & A^{k-2}B\\\\& & \\ddots & \\ddots & \\vdots \\\\& & & B & AB\\\\& & & & B\\end{array}\\right] \\in \\mathbf {R}^{kn\\times kn},\\overline{{B}}_{k}=\\left[\\begin{array}[c]{c}A^{k}\\\\A^{k-1}\\\\\\vdots \\\\A^{2}\\\\A\\end{array}\\right] \\in \\mathbf {R}^{kn\\times n}.",
"$ For two symmetric matrices $\\overline{P},\\overline{Q}\\in \\mathbf {R}^{kn\\times kn},$ we define a linear function $\\overline{\\mathit {\\Omega }}_{k}\\left(\\overline{P},\\overline{Q}\\right) \\in \\mathbf {R}^{\\left( k+1\\right)n\\times \\left( k+1\\right) n}$ as $\\overline{\\mathit {\\Omega }}_{k}\\left( \\overline{P},\\overline{Q}\\right)=\\left[\\begin{array}[c]{cc}\\overline{A}_{k}^{\\mathrm {T}}\\overline{P}\\overline{A}_{k}-\\overline{P}+\\overline{{A}}_{k}^{\\mathrm {T}}\\overline{Q}\\overline{{A}}_{k}-\\overline{A}_{k}^{\\mathrm {T}}\\overline{Q}\\overline{A}_{k} &\\overline{A}_{k}^{\\mathrm {T}}\\overline{P}\\overline{B}_{k}+\\overline{{A}}_{k}^{\\mathrm {T}}\\overline{Q}\\overline{{B}}_{k}-\\overline{A}_{k}^{\\mathrm {T}}\\overline{Q}\\overline{B}_{k}\\\\\\overline{B}_{k}^{\\mathrm {T}}\\overline{P}\\overline{A}_{k}+\\overline{{B}}_{k}^{\\mathrm {T}}\\overline{Q}\\overline{{A}}_{k}-\\overline{B}_{k}^{\\mathrm {T}}\\overline{Q}\\overline{A}_{k} & \\overline{B}_{k}^{\\mathrm {T}}\\overline{P}\\overline{B}_{k}+\\overline{{B}}_{k}^{\\mathrm {T}}\\overline{Q}\\overline{{B}}_{k}-\\overline{B}_{k}^{\\mathrm {T}}\\overline{Q}\\overline{B}_{k}\\end{array}\\right] .",
"$ Lemma 3 [2] If there exist positive definite matrices $\\overline{P}_{k},\\overline{Q}_{k}\\in \\mathbf {R}^{kn\\times kn}$ such that $\\overline{\\mathit {\\Omega }}_{k}\\left( \\overline{P}_{k},\\overline{Q}_{k}\\right) <0, $ then system (REF )/(REF ) is stable.",
"Moreover, if (REF )/(REF ) is stable, there exists an integer $k^{\\ast }\\ge 1$ , such that (REF ) is solvable with $\\overline{P}_{k}>0,\\overline{Q}_{k}>0,\\forall k\\ge k^{\\ast }.$ Notice that Lemma REF is slightly different from the original one in [2] where the result is built for a general 2-D linear system, and is expressed in a recursive form.",
"Even for $k=2,$ the LMI in Lemma REF is nonlinear in $A$ and $B$ , and thus can not be used for robust stability analysis.",
"In this note, motivated by [2], we will establish a new necessary and sufficient condition for testing strong stability of system (REF ).",
"Different from Lemmas REF and REF , the proposed LMI condition involves matrices that are linear functions of $\\left( A,B\\right).$ With the help of this property, the robust stability problem in case of norm bounded uncertainty is investigated, and the results are also expressed by LMIs (see Section ).",
"We also give time-domain interpretations of the proposed LMI condition and the Bliman condition, which help to reveal the relationships among them and the other existing methods such as those in [3] and [4] (see Section )."
],
[
"The Necessary and Sufficient Conditions",
"For any $k\\in \\mathbf {N}^{+},$ we denote $A_{k}=\\left[\\begin{array}[c]{cc}0 & I_{\\left( k-1\\right) n}\\\\0 & 0\\end{array}\\right] \\in \\mathbf {R}^{kn\\times kn},\\;B_{k}=\\left[\\begin{array}[c]{c}0\\\\I_{n}\\end{array}\\right] \\in \\mathbf {R}^{kn\\times n}, $ and $L_{k}=\\left[\\begin{array}[c]{cc}I_{kn} & 0_{kn\\times n}\\end{array}\\right] \\in \\mathbf {R}^{kn\\times \\left( k+1\\right) n}, $ which are independent of $\\left( A,B\\right) ,$ and ${A}_{k}=\\left[\\begin{array}[c]{ccccc}B & A & 0 & \\cdots & 0\\\\& B & A & \\ddots & \\vdots \\\\& & \\ddots & \\ddots & 0\\\\& & & B & A\\\\& & & & B\\end{array}\\right] \\in \\mathbf {R}^{kn\\times kn},\\; {B}_{k}=\\left[\\begin{array}[c]{c}0\\\\0\\\\\\vdots \\\\0\\\\A\\end{array}\\right] \\in \\mathbf {R}^{kn\\times n}, $ which are linear matrix functions of $\\left( A,B\\right) .$ For two symmetric matrices $P,Q\\in \\mathbf {R}^{kn\\times kn},$ we define $\\mathit {\\Omega }_{k1}\\left( P,Q\\right) = & \\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] ^{\\mathrm {T}}\\left( P-Q\\right) \\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] -L_{k}^{\\mathrm {T}}PL_{k},\\nonumber \\\\\\mathit {\\Omega }_{k}\\left( P,Q\\right) = & \\mathit {\\Omega }_{k1}\\left(P,Q\\right) +\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] ^{\\mathrm {T}}Q\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] , $ which are linear functions of $P,Q$ and, moreover, $\\mathit {\\Omega }_{k1}\\left( P,Q\\right) $ is independent of $\\left( A,B\\right) .$ Theorem 1 If there exist positive definite matrices $P_{k},Q_{k}\\in \\mathbf {R}^{kn\\times kn}$ such that $\\mathit {\\Omega }_{k}\\left( P_{k},Q_{k}\\right) <0, $ then system (REF ) is strongly stable.",
"Moreover, if system (REF ) is strongly stable, there exists an integer $k^{\\ast }\\ge 1$ , such that (REF ) is solvable with $P_{k}>0,Q_{k}>0,\\forall k\\ge k^{\\ast }.$ Proof.",
"Let $z_{k}=\\left[\\begin{array}[c]{c}z_{k,k}\\\\\\vdots \\\\z_{k,2}\\\\z_{k,1}\\end{array}\\right] =\\left( \\mathrm {e}^{\\mathrm {j}\\theta }I_{kn}-A_{k}\\right) ^{-1}B_{k}, $ which is equivalent to $\\left[\\begin{array}[c]{ccccc}\\mathrm {e}^{\\mathrm {j}\\theta }I_{n} & -I_{n} & 0 & \\cdots & 0\\\\& \\mathrm {e}^{\\mathrm {j}\\theta }I_{n} & \\ddots & \\ddots & \\vdots \\\\& & \\ddots & -I_{n} & 0\\\\& & & \\mathrm {e}^{\\mathrm {j}\\theta }I_{n} & -I_{n}\\\\& & & & \\mathrm {e}^{\\mathrm {j}\\theta }I_{n}\\end{array}\\right] \\left[\\begin{array}[c]{c}z_{k,k}\\\\z_{k,k-1}\\\\\\vdots \\\\z_{k,2}\\\\z_{k,1}\\end{array}\\right] =\\left[\\begin{array}[c]{c}0\\\\0\\\\\\vdots \\\\0\\\\I_{n}\\end{array}\\right] .$ Solving this equation recursively from the bottom to the up gives $z_{k}=\\left[\\begin{array}[c]{c}\\mathrm {e}^{-k\\mathrm {j}\\theta }I_{n}\\\\\\vdots \\\\\\mathrm {e}^{-2\\mathrm {j}\\theta }I_{n}\\\\\\mathrm {e}^{-\\mathrm {j}\\theta }I_{n}\\end{array}\\right] .",
"$ With this we get from (REF ), (REF ) and (REF ) that $\\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] \\left[\\begin{array}[c]{c}\\left( \\mathrm {e}^{\\mathrm {j}\\theta }I_{kn}-A_{k}\\right) ^{-1}B_{k}\\\\I_{n}\\end{array}\\right] & =\\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] \\left[\\begin{array}[c]{c}z_{k}\\\\I_{n}\\end{array}\\right] =\\left[\\begin{array}[c]{c}\\mathrm {e}^{-\\mathrm {j}\\left( k-1\\right) \\theta }I_{n}\\\\\\vdots \\\\\\mathrm {e}^{-\\mathrm {j}\\theta }I_{n}\\\\I_{n}\\end{array}\\right] =\\mathrm {e}^{\\mathrm {j}\\theta }z_{k},\\\\\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] \\left[\\begin{array}[c]{c}\\left( \\mathrm {e}^{\\mathrm {j}\\theta }I_{kn}-A_{k}\\right) ^{-1}B_{k}\\\\I_{n}\\end{array}\\right] & =\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] \\left[\\begin{array}[c]{c}z_{k}\\\\I_{n}\\end{array}\\right] =\\left[\\begin{array}[c]{c}\\mathrm {e}^{-\\mathrm {j}\\left( k-1\\right) \\theta }\\Delta _{\\theta }\\\\\\vdots \\\\\\mathrm {e}^{-\\mathrm {j}\\theta }\\Delta _{\\theta }\\\\\\Delta _{\\theta }\\end{array}\\right] =\\mathrm {e}^{\\mathrm {j}\\theta }z_{k}\\Delta _{\\theta }, $ and $L_{k}\\left[\\begin{array}[c]{c}\\left( \\mathrm {e}^{\\mathrm {j}\\theta }I_{kn}-A_{k}\\right) ^{-1}B_{k}\\\\I_{n}\\end{array}\\right] =\\left( \\mathrm {e}^{\\mathrm {j}\\theta }I_{kn}-A_{k}\\right) ^{-1}B_{k}=z_{k}.",
"$ Therefore, we can obtain $& \\left[\\begin{array}[c]{c}z_{k}\\\\I_{n}\\end{array}\\right] ^{\\mathrm {H}}\\mathit {\\Omega }_{k}\\left( P_{k},Q_{k}\\right) \\left[\\begin{array}[c]{c}z_{k}\\\\I_{n}\\end{array}\\right] \\nonumber \\\\= & \\left[\\begin{array}[c]{c}z_{k}\\\\I_{n}\\end{array}\\right] ^{\\mathrm {H}}\\left( \\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] ^{\\mathrm {T}}P_{k}\\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] -L_{k}^{\\mathrm {T}}P_{k}L_{k}\\right) \\left[\\begin{array}[c]{c}z_{k}\\\\I_{n}\\end{array}\\right] \\nonumber \\\\& +\\left[\\begin{array}[c]{c}z_{k}\\\\I_{n}\\end{array}\\right] ^{\\mathrm {H}}\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] ^{\\mathrm {T}}Q_{k}\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] \\left[\\begin{array}[c]{c}z_{k}\\\\I_{n}\\end{array}\\right] \\nonumber \\\\& -\\left[\\begin{array}[c]{c}z_{k}\\\\I_{n}\\end{array}\\right] ^{\\mathrm {H}}\\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] ^{\\mathrm {T}}Q_{k}\\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] \\left[\\begin{array}[c]{c}z_{k}\\\\I_{n}\\end{array}\\right] \\nonumber \\\\= & \\left( \\mathrm {e}^{\\mathrm {j}\\theta }z_{k}\\Delta _{\\theta }\\right)^{\\mathrm {H}}Q_{k}\\mathrm {e}^{\\mathrm {j}\\theta }z_{k}\\Delta _{\\theta }-\\left(\\mathrm {e}^{\\mathrm {j}\\theta }z_{k}\\right) ^{\\mathrm {H}}Q_{k}\\mathrm {e}^{\\mathrm {j}\\theta }z_{k}+\\left( \\mathrm {e}^{\\mathrm {j}\\theta }z_{k}\\right)^{\\mathrm {H}}P_{k}\\mathrm {e}^{\\mathrm {j}\\theta }z_{k}-z_{k}^{\\mathrm {H}}P_{k}z_{k}\\nonumber \\\\= & \\Delta _{\\theta }^{\\mathrm {H}}z_{k}^{\\mathrm {H}}Q_{k}z_{k}\\Delta _{\\theta }-z_{k}^{\\mathrm {H}}Q_{k}z_{k}\\nonumber \\\\< & 0, $ which implies (REF ) since $z_{k}^{\\mathrm {H}}Q_{k}z_{k}>0.$ We next prove the converse.",
"By Lemma REF in Appendix A2, we know that there exists a $k^{\\ast }\\ge 1$ such that $\\left( \\Delta _{\\theta }^{k}\\right) ^{\\mathrm {H}}\\Delta _{\\theta }^{k}<I_{n},\\;\\forall \\theta \\in \\left[ 0,2\\pi \\right] ,\\;k\\ge k^{\\ast }.",
"$ Denote $Q_{k}^{\\ast }=W_{k}^{\\mathrm {T}}W_{k}$ where (see the notation in Appendix A2) $W_{k}=\\left[\\begin{array}[c]{ccccc}B^{\\left[ k-1\\right] } & B^{\\left[ k-2\\right] }A^{\\left[ 1\\right] } &\\cdots & B^{\\left[ 1\\right] }A^{\\left[ k-2\\right] } & A^{\\left[k-1\\right] }\\\\& B^{\\left[ k-2\\right] } & B^{\\left[ k-3\\right] }A^{\\left[ 1\\right] } &\\ddots & A^{\\left[ k-2\\right] }\\\\& & \\ddots & \\ddots & \\vdots \\\\& & & B^{\\left[ 1\\right] } & A^{\\left[ 1\\right] }\\\\& & & & I_{n}\\end{array}\\right] .",
"$ It follows that $Q_{k}^{\\ast }\\ge 0$ and, moreover, $Q_{k}^{\\ast }>0$ if $B$ is nonsingular.",
"For any integer $i\\ge 1,$ by the binomial expansion theorem, we have $\\left( A+B\\mathrm {e}^{-\\mathrm {j}\\theta }\\right) ^{i} & =A^{\\left[i\\right] }+B^{\\left[ 1\\right] }A^{\\left[ i-1\\right] }\\mathrm {e}^{-\\mathrm {j}\\theta }+B^{\\left[ 2\\right] }A^{\\left[ i-2\\right] }\\mathrm {e}^{-2\\mathrm {j}\\theta }+\\cdots +B^{\\left[ i-1\\right] }A^{\\left[1\\right] }\\mathrm {e}^{-\\left( i-1\\right) \\mathrm {j}\\theta }+B^{\\left[i\\right] }\\mathrm {e}^{-i\\mathrm {j}\\theta }\\\\& =\\left[\\begin{array}[c]{ccccc}B^{\\left[ i\\right] } & B^{\\left[ i-1\\right] }A^{\\left[ 1\\right] } &\\cdots & B^{\\left[ 1\\right] }A^{\\left[ i-1\\right] } & A^{\\left[ i\\right]}\\end{array}\\right] \\left[\\begin{array}[c]{c}\\mathrm {e}^{-i\\mathrm {j}\\theta }I_{n}\\\\\\mathrm {e}^{-\\left( i-1\\right) \\mathrm {j}\\theta }I_{n}\\\\\\vdots \\\\\\mathrm {e}^{-\\mathrm {j}\\theta }I_{n}\\\\I_{n}\\end{array}\\right] .$ It follows that $W_{k}\\mathrm {e}^{\\mathrm {j}\\theta }z_{k}=W_{k}\\left[\\begin{array}[c]{c}\\mathrm {e}^{-\\left( k-1\\right) \\mathrm {j}\\theta }I_{n}\\\\\\mathrm {e}^{-\\left( k-2\\right) \\mathrm {j}\\theta }I_{n}\\\\\\vdots \\\\\\mathrm {e}^{-\\mathrm {j}\\theta }I_{n}\\\\I_{n}\\end{array}\\right] =\\left[\\begin{array}[c]{c}\\Delta _{\\theta }^{k-1}\\\\\\Delta _{\\theta }^{k-2}\\\\\\vdots \\\\\\Delta _{\\theta }\\\\I_{n}\\end{array}\\right] .$ Let $\\mathit {\\Theta }_{k}\\left( Q\\right) =\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] ^{\\mathrm {T}}Q\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] -\\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] ^{\\mathrm {T}}Q\\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] .$ We then have from (REF ) and equations (REF ) and () that $& \\left[\\begin{array}[c]{c}z_{k}\\\\I_{n}\\end{array}\\right] ^{\\mathrm {H}}\\mathit {\\Theta }_{k}\\left( Q_{k}^{\\ast }\\right) \\left[\\begin{array}[c]{c}z_{k}\\\\I_{n}\\end{array}\\right] \\\\= & \\left( \\mathrm {e}^{\\mathrm {j}\\theta }z_{k}\\Delta _{\\theta }\\right)^{\\mathrm {H}}Q_{k}^{\\ast }\\mathrm {e}^{\\mathrm {j}\\theta }z_{k}\\Delta _{\\theta }-\\left( \\mathrm {e}^{\\mathrm {j}\\theta }z_{k}\\right) ^{\\mathrm {H}}Q_{k}^{\\ast }\\mathrm {e}^{\\mathrm {j}\\theta }z_{k}\\\\= & \\Delta _{\\theta }^{\\mathrm {H}}\\left( \\left( W_{k}\\mathrm {e}^{\\mathrm {j}\\theta }z_{k}\\right) ^{\\mathrm {H}}W_{k}\\mathrm {e}^{\\mathrm {j}\\theta }z_{k}\\right) \\Delta _{\\theta }-\\left( W_{k}\\mathrm {e}^{\\mathrm {j}\\theta }z_{k}\\right) ^{\\mathrm {H}}W_{k}\\mathrm {e}^{\\mathrm {j}\\theta }z_{k}\\\\= & \\Delta _{\\theta }^{\\mathrm {H}}\\left[\\begin{array}[c]{c}\\Delta _{\\theta }^{k-1}\\\\\\vdots \\\\\\Delta _{\\theta }\\\\I_{n}\\end{array}\\right] ^{\\mathrm {H}}\\left[\\begin{array}[c]{c}\\Delta _{\\theta }^{k-1}\\\\\\vdots \\\\\\Delta _{\\theta }\\\\I_{n}\\end{array}\\right] \\Delta _{\\theta }-\\left[\\begin{array}[c]{c}\\Delta _{\\theta }^{k-1}\\\\\\vdots \\\\\\Delta _{\\theta }\\\\I_{n}\\end{array}\\right] ^{\\mathrm {H}}\\left[\\begin{array}[c]{c}\\Delta _{\\theta }^{k-1}\\\\\\vdots \\\\\\Delta _{\\theta }\\\\I_{n}\\end{array}\\right] \\\\= & \\left( \\Delta _{\\theta }^{k}\\right) ^{\\mathrm {H}}\\Delta _{\\theta }^{k}-I_{n}\\\\< & 0.$ As $A_{k}$ is Schur stable, by the YKP lemma in Appendix A2, the above inequality holds true if and only if there exists a symmetric matrix $P_{k}^{\\ast }\\in \\mathbf {R}^{kn\\times kn}$ such that $0> & \\left[\\begin{array}[c]{cc}A_{k}^{\\mathrm {T}}P_{k}^{\\ast }A_{k}-P_{k}^{\\ast } & A_{k}^{\\mathrm {T}}P_{k}^{\\ast }B_{k}\\\\B_{k}^{\\mathrm {T}}P_{k}^{\\ast }A_{k} & B_{k}^{\\mathrm {T}}P_{k}^{\\ast }B_{k}\\end{array}\\right] +\\mathit {\\Theta }_{k}\\left( Q_{k}^{\\ast }\\right) \\nonumber \\\\= & \\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] ^{\\mathrm {T}}P_{k}^{\\ast }\\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] -L_{k}^{\\mathrm {T}}P_{k}^{\\ast }L_{k}\\nonumber \\\\& +\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] ^{\\mathrm {T}}Q_{k}^{\\ast }\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] -\\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] ^{\\mathrm {T}}Q_{k}^{\\ast }\\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] \\nonumber \\\\= & \\mathit {\\Omega }_{k}\\left( P_{k}^{\\ast },Q_{k}^{\\ast }\\right) .$ By comparing (REF ) with (REF ), we know that the LMI in (REF ) is feasible with $\\left( P_{k},Q_{k}\\right) =\\left(P_{k}^{\\ast },Q_{k}^{\\ast }\\right) .$ In the following, we will show that $P_{k}^{\\ast }>0$ .",
"Straightforward computation gives that $W_{k}\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] =\\left[\\begin{array}[c]{ccccc}B^{\\left[ k\\right] } & B^{\\left[ k-1\\right] }A^{\\left[ 1\\right] } &\\cdots & B^{\\left[ 1\\right] }A^{\\left[ k-1\\right] } & A^{\\left[ k\\right]}\\\\& B^{\\left[ k-1\\right] } & B^{\\left[ k-2\\right] }A^{\\left[ 1\\right] } &\\ddots & A^{\\left[ k-1\\right] }\\\\& & \\ddots & \\ddots & \\vdots \\\\& & B^{\\left[ 2\\right] } & B^{\\left[ 1\\right] }A^{\\left[ 1\\right] } &A^{\\left[ 2\\right] }\\\\& & & B^{\\left[ 1\\right] } & A^{\\left[ 1\\right] }\\end{array}\\right] ,$ and $W_{k}\\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] =\\left[\\begin{array}[c]{cccccc}0 & B^{\\left[ k-1\\right] } & B^{\\left[ k-2\\right] }A^{\\left[ 1\\right] }& \\cdots & B^{\\left[ 1\\right] }A^{\\left[ k-2\\right] } & A^{\\left[k-1\\right] }\\\\& 0 & B^{\\left[ k-2\\right] } & B^{\\left[ k-3\\right] }A^{\\left[ 1\\right]} & \\ddots & A^{\\left[ k-2\\right] }\\\\& & 0 & \\ddots & \\ddots & \\vdots \\\\& & & \\ddots & B^{\\left[ 1\\right] } & A^{\\left[ 1\\right] }\\\\& & & & 0 & I_{n}\\end{array}\\right] .$ It follows that we can write $W_{k}A_{k} & =\\left[\\begin{array}[c]{cc}0_{\\left( k-1\\right) n\\times n} & U_{k}\\\\0_{n\\times n} & 0_{n\\times \\left( k-1\\right) n}\\end{array}\\right] ,W_{k}B_{k}=\\left[\\begin{array}[c]{c}A^{\\left[ k-1\\right] }\\\\\\vdots \\\\A^{\\left[ 1\\right] }\\\\I_{n}\\end{array}\\right] ,\\\\W_{k}{A}_{k} & =\\left[\\begin{array}[c]{cc}B^{\\left[ k\\right] } & V_{k}\\\\0_{\\left( k-1\\right) n\\times n} & U_{k}\\end{array}\\right] ,W_{k}{B}_{k}=\\left[\\begin{array}[c]{c}A^{\\left[ k\\right] }\\\\\\vdots \\\\A^{\\left[ 2\\right] }\\\\A^{\\left[ 1\\right] }\\end{array}\\right] ,$ where $U_{k} & =\\left[\\begin{array}[c]{cccc}B^{\\left[ k-1\\right] } & B^{\\left[ k-2\\right] }A^{\\left[ 1\\right] } &\\cdots & B^{\\left[ 1\\right] }A^{\\left[ k-2\\right] }\\\\& \\ddots & \\ddots & \\vdots \\\\& & B^{\\left[ 2\\right] } & B^{\\left[ 1\\right] }A^{\\left[ 1\\right] }\\\\& & & B^{\\left[ 1\\right] }\\end{array}\\right] ,\\\\V_{k} & =\\left[\\begin{array}[c]{cccc}B^{\\left[ k-1\\right] }A^{\\left[ 1\\right] } & B^{\\left[ k-2\\right]}A^{\\left[ 2\\right] } & \\cdots & B^{\\left[ 1\\right] }A^{\\left[k-1\\right] }\\end{array}\\right] .$ We also denote $C_{k} & =\\left[\\begin{array}[c]{cc}B^{\\left[ k\\right] } & V_{k}\\end{array}\\right] \\\\& =\\left[\\begin{array}[c]{cccc}B^{\\left[ k\\right] } & B^{\\left[ k-1\\right] }A^{\\left[ 1\\right] } &\\cdots & B^{\\left[ 1\\right] }A^{\\left[ k-1\\right] }\\end{array}\\right] \\in \\mathbf {R}^{n\\times kn},\\; \\\\D_{k} & =A^{\\left[ k\\right] }\\in \\mathbf {R}^{n\\times n}.$ Then, by straightforward computations, we obtain ${A}_{k}^{\\mathrm {T}}W_{k}^{\\mathrm {T}}W_{k}{A}_{k}-A_{k}^{\\mathrm {T}}W_{k}^{\\mathrm {T}}W_{k}A_{k} & =\\left[\\begin{array}[c]{cc}\\left( B^{\\left[ k\\right] }\\right) ^{\\mathrm {T}}B^{\\left[ k\\right] } &\\left( B^{\\left[ k\\right] }\\right) ^{\\mathrm {T}}V_{k}\\\\V_{k}^{\\mathrm {T}}B^{\\left[ k\\right] } & V_{k}^{\\mathrm {T}}V_{k}+U_{k}^{\\mathrm {T}}U_{k}\\end{array}\\right] -\\left[\\begin{array}[c]{cc}0_{n\\times n} & 0_{n\\times \\left( k-1\\right) n}\\\\0_{\\left( k-1\\right) n\\times n} & U_{k}^{\\mathrm {T}}U_{k}\\end{array}\\right] \\\\& =\\left[\\begin{array}[c]{cc}\\left( B^{\\left[ k\\right] }\\right) ^{\\mathrm {T}}B^{\\left[ k\\right] } &\\left( B^{\\left[ k\\right] }\\right) ^{\\mathrm {T}}V_{k}\\\\V_{k}^{\\mathrm {T}}B^{\\left[ k\\right] } & V_{k}^{\\mathrm {T}}V_{k}\\end{array}\\right] \\\\& =C_{k}^{\\mathrm {T}}C_{k}.$ Similarly, we have ${A}_{k}^{\\mathrm {T}}W_{k}^{\\mathrm {T}}W_{k}{B}_{k}-A_{k}^{\\mathrm {T}}W_{k}^{\\mathrm {T}}W_{k}B_{k} & =\\left[\\begin{array}[c]{c}\\left( B^{\\left[ k\\right] }\\right) ^{\\mathrm {T}}A^{\\left[ k\\right] }\\\\V_{k}^{\\mathrm {T}}A^{\\left[ k\\right] }+U_{k}^{\\mathrm {T}}\\left[\\begin{array}[c]{c}A^{\\left[ k-1\\right] }\\\\\\vdots \\\\A^{\\left[ 1\\right] }\\end{array}\\right]\\end{array}\\right] -\\left[\\begin{array}[c]{c}0_{n\\times n}\\\\U_{k}^{\\mathrm {T}}\\left[\\begin{array}[c]{c}A^{\\left[ k-1\\right] }\\\\\\vdots \\\\A^{\\left[ 1\\right] }\\end{array}\\right]\\end{array}\\right] \\\\& =\\left[\\begin{array}[c]{c}\\left( B^{\\left[ k\\right] }\\right) ^{\\mathrm {T}}A^{\\left[ k\\right] }\\\\V_{k}^{\\mathrm {T}}A^{\\left[ k\\right] }\\end{array}\\right] \\\\& =C_{k}^{\\mathrm {T}}D_{k},$ and ${B}_{k}^{\\mathrm {T}}W_{k}^{\\mathrm {T}}W_{k}{B}_{k}-B_{k}^{\\mathrm {T}}W_{k}^{\\mathrm {T}}W_{k}B_{k}=\\left( A^{\\left[ k\\right]}\\right) ^{\\mathrm {T}}A^{\\left[ k\\right] }-I_{n}=D_{k}^{\\mathrm {T}}D_{k}-I_{n}.$ Therefore, we can get $\\mathit {\\Omega }_{k}\\left( P_{k}^{\\ast },Q_{k}^{\\ast }\\right) =\\left[\\begin{array}[c]{cc}A_{k}^{\\mathrm {T}}P_{k}^{\\ast }A_{k}-P_{k}^{\\ast }+C_{k}^{\\mathrm {T}}C_{k} &A_{k}^{\\mathrm {T}}P_{k}B_{k}+C_{k}^{\\mathrm {T}}D_{k}\\\\B_{k}^{\\mathrm {T}}P_{k}^{\\ast }A_{k}+D_{k}^{\\mathrm {T}}C_{k} & B_{k}^{\\mathrm {T}}P_{k}^{\\ast }B_{k}+D_{k}^{\\mathrm {T}}D_{k}-I_{n}\\end{array}\\right] , $ which, together with (REF ), implies that $A_{k}^{\\mathrm {T}}P_{k}^{\\ast }A_{k}-P_{k}^{\\ast }+C_{k}^{\\mathrm {T}}C_{k}<0.$ As $A_{k}$ is Schur stable, the above equation implies $P_{k}^{\\ast }>0.$ By now we have shown that, if $B$ is nonsingular, the LMI in (REF ) is solvable with positive definite matrices $P_{k}^{\\ast }$ and $Q_{k}^{\\ast }=W_{k}^{\\mathrm {T}}W_{k}.$ However, if $B$ is singular, the matrix $Q_{k}^{\\ast }=W_{k}^{\\mathrm {T}}W_{k}$ is only semi-positive definite.",
"In the following, we will show that the LMI in (REF ) is also feasible with $\\left( P_{k},Q_{k}\\right) =\\left( P_{k}^{\\ast },Q_{k}^{\\ast }+\\varepsilon I_{kn}\\right) $ where $\\varepsilon >0$ is sufficiently small,$\\ $ namely, $\\mathit {\\Omega }_{k}\\left( P_{k}^{\\ast },Q_{k}^{\\ast }+\\varepsilon I_{kn}\\right) <0.",
"$ In fact, it follows from (REF ) that $\\mathit {\\Omega }_{k}\\left( P_{k}^{\\ast },Q_{k}^{\\ast }+\\varepsilon I_{kn}\\right) & =\\mathit {\\Omega }_{k}\\left( P_{k}^{\\ast },Q_{k}^{\\ast }\\right) +\\mathit {\\Omega }_{k}\\left( 0_{kn\\times kn},\\varepsilon I_{kn}\\right) \\\\& =\\mathit {\\Omega }_{k}\\left( P_{k}^{\\ast },Q_{k}^{\\ast }\\right)+\\varepsilon \\left( \\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] ^{\\mathrm {T}}\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] -\\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] ^{\\mathrm {T}}\\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] \\right) \\\\& \\le \\mathit {\\Omega }_{k}\\left( P_{k}^{\\ast },Q_{k}^{\\ast }\\right)+\\varepsilon \\left( \\left\\Vert \\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] \\right\\Vert ^{2}+\\left\\Vert \\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] \\right\\Vert ^{2}\\right) .$ Since $\\mathit {\\Omega }_{k}\\left( P_{k}^{\\ast },Q_{k}^{\\ast }\\right) $ is independent of $\\varepsilon $ and satisfies (REF ), there exists a sufficiently small $\\varepsilon >0$ such that (REF ) is satisfied.",
"The proof is finished.",
"By a Schur complement, the LMI (REF ) can be written as $\\left[\\begin{array}[c]{cc}\\mathit {\\Omega }_{k1}\\left( P_{k},Q_{k}\\right) & [{A}_{k},{B}_{k}]^{\\mathrm {T}}Q_{k}\\\\Q_{k}[{A}_{k},{B}_{k}] & -Q_{k}\\end{array}\\right] <0,$ whose left hand side is a linear function of $\\left( A,B\\right) .$ Thus, the most important feature of Theorem REF , when compared with the results in [2] (see Lemma REF ), the result in [6] (see Lemma REF ) and the method in [13], is that the coefficient $\\left( A,B\\right) $ appears as a linear function.",
"Such a property is helpful for solving the robust stability analysis problem, as made clear below.",
"Consider the perturbed system of (REF ) $x\\left( t\\right) =\\left( A+\\Delta A\\right) x\\left( t-a\\right) +\\left(B+\\Delta B\\right) x\\left( t-b\\right) , $ where $A\\in \\mathbf {R}^{n\\times n}$ and $B\\in \\mathbf {R}^{n\\times n}$ are the same as that in (REF ) and $\\left[\\begin{array}[c]{cc}\\Delta B & \\Delta A\\end{array}\\right] =E_{0}F\\left[\\begin{array}[c]{cc}B_{0} & A_{0}\\end{array}\\right] , $ where $E_{0}\\in \\mathbf {R}^{n\\times p},B_{0}\\in \\mathbf {R}^{q\\times n},A_{0}\\in \\mathbf {R}^{q\\times n}$ are known matrices, and $F\\in \\mathbf {R}^{p\\times q}$ denotes the norm bounded uncertainty (which can be time-varying) that satisfies $F^{\\mathrm {T}}F\\le I_{q}.",
"$ For further using, we denote $\\left[\\begin{array}[c]{cc}{A}_{k0} & {B}_{k0}\\end{array}\\right] =\\left[\\begin{array}[c]{ccccc}B_{0} & A_{0} & 0 & \\cdots & 0\\\\& \\ddots & \\ddots & \\ddots & \\vdots \\\\& & B_{0} & A_{0} & 0\\\\& & & B_{0} & A_{0}\\end{array}\\right] \\in \\mathbf {R}^{kq\\times \\left( k+1\\right) n}.$ Theorem 2 The uncertain linear difference equation (REF ) is exponentially stable for any $F\\in \\mathbf {R}^{p\\times q}$ satisfying (REF ) if there exists an integer $k\\ge 1,$ positive definite matrices $P_{k},Q_{k}\\in \\mathbf {R}^{kn\\times kn}$ and a positive definite matrix $S_{k}\\in \\mathbf {R}^{k\\times k}$ such that the following LMI is satisfied: $\\left[\\begin{array}[c]{cc}\\mathit {\\Omega }_{k}\\left( P_{k},Q_{k}\\right) +\\left[\\begin{array}[c]{cc}{A}_{k0} & {B}_{k0}\\end{array}\\right] ^{\\mathrm {T}}\\left( S_{k}\\otimes I_{q}\\right) \\left[\\begin{array}[c]{cc}{A}_{k0} & {B}_{k0}\\end{array}\\right] & \\left[\\begin{array}[c]{cc}{A}_{k0} & {B}_{k0}\\end{array}\\right] ^{\\mathrm {T}}Q_{k}\\left( I_{k}\\otimes E_{0}\\right) \\\\\\left( I_{k}\\otimes E_{0}^{\\mathrm {T}}\\right) Q_{k}\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] & \\left( I_{k}\\otimes E_{0}^{\\mathrm {T}}\\right) Q_{k}\\left(I_{k}\\otimes E_{0}\\right) -S_{k}\\otimes I_{p}\\end{array}\\right] <0.",
"$ Proof.",
"For notation simplicity, we denote ${C}_{k}=\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] ,\\; {C}_{k0}=\\left[\\begin{array}[c]{cc}{A}_{k0} & {B}_{k0}\\end{array}\\right] ,\\nonumber $ $\\mathit {\\Omega }_{k}=\\mathit {\\Omega }_{k}\\left( P_{k},Q_{k}\\right) ,$ and $\\mathit {\\Omega }_{k1}=\\mathit {\\Omega }_{k1}\\left( P_{k},Q_{k}\\right) .$ Notice that we can write $0 & >\\left[\\begin{array}[c]{cc}\\mathit {\\Omega }_{k}+{C}_{k0}^{\\mathrm {T}}\\left( S_{k}\\otimes I_{q}\\right) {C}_{k0} & {C}_{k}^{\\mathrm {T}}Q_{k}\\left(I_{k}\\otimes E_{0}\\right) \\\\\\left( I_{k}\\otimes E_{0}^{\\mathrm {T}}\\right) Q_{k}{C}_{k} & \\left(I_{k}\\otimes E_{0}^{\\mathrm {T}}\\right) Q_{k}\\left( I_{k}\\otimes E_{0}\\right) -S_{k}\\otimes I_{p}\\end{array}\\right] \\\\& =\\left[\\begin{array}[c]{cc}\\mathit {\\Omega }_{k1}+{C}_{k}^{\\mathrm {T}}Q_{k}{C}_{k}+{C}_{k0}^{\\mathrm {T}}\\left( S_{k}\\otimes I_{q}\\right){C}_{k0} & {C}_{k}^{\\mathrm {T}}Q_{k}\\left( I_{k}\\otimes E_{0}\\right) \\\\\\left( I_{k}\\otimes E_{0}^{\\mathrm {T}}\\right) Q_{k}{C}_{k} & \\left(I_{k}\\otimes E_{0}^{\\mathrm {T}}\\right) Q_{k}\\left( I_{k}\\otimes E_{0}\\right) -S_{k}\\otimes I_{p}\\end{array}\\right] \\\\& =\\left[\\begin{array}[c]{cc}\\mathit {\\Omega }_{k1}+{C}_{k0}^{\\mathrm {T}}\\left( S_{k}\\otimes I_{q}\\right) {C}_{k0} & 0_{\\left( k+1\\right) n\\times kp}\\\\0_{kp\\times \\left( k+1\\right) n} & -S_{k}\\otimes I_{p}\\end{array}\\right] +\\left[\\begin{array}[c]{c}{C}_{k}^{\\mathrm {T}}Q_{k}\\\\\\left( I_{k}\\otimes E_{0}^{\\mathrm {T}}\\right) Q_{k}\\end{array}\\right] Q_{k}^{-1}\\left[\\begin{array}[c]{c}{C}_{k}^{\\mathrm {T}}Q_{k}\\\\\\left( I_{k}\\otimes E_{0}^{\\mathrm {T}}\\right) Q_{k}\\end{array}\\right] ^{\\mathrm {T}},$ which, by a Schur complement, is equivalent to $\\left[\\begin{array}[c]{ccc}\\mathit {\\Omega }_{k1}+{C}_{k0}^{\\mathrm {T}}\\left( S_{k}\\otimes I_{q}\\right) {C}_{k0} & 0_{\\left( k+1\\right) n\\times kp} &{C}_{k}^{\\mathrm {T}}Q_{k}\\\\0_{kp\\times \\left( k+1\\right) n} & -S_{k}\\otimes I_{p} & \\left( I_{k}\\otimes E_{0}^{\\mathrm {T}}\\right) Q_{k}\\\\Q_{k}{C}_{k} & Q_{k}\\left( I_{k}\\otimes E_{0}\\right) & -Q_{k}\\end{array}\\right] <0.$ By a congruence transformation, this is equivalent to $\\left[\\begin{array}[c]{ccc}\\mathit {\\Omega }_{k1}+{C}_{k0}^{\\mathrm {T}}\\left( S_{k}\\otimes I_{q}\\right) {C}_{k0} & {C}_{k}^{\\mathrm {T}}Q_{k} &0_{\\left( k+1\\right) n\\times kp}\\\\Q_{k}{C}_{k} & -Q_{k} & Q_{k}\\left( I_{k}\\otimes E_{0}\\right) \\\\0_{kp\\times \\left( k+1\\right) n} & \\left( I_{k}\\otimes E_{0}^{\\mathrm {T}}\\right) Q_{k} & -S_{k}\\otimes I_{p}\\end{array}\\right] <0.$ By a Schur complement, the above inequality holds true if and only if $0> & \\left[\\begin{array}[c]{cc}\\mathit {\\Omega }_{k1} & {C}_{k}^{\\mathrm {T}}Q_{k}\\\\Q_{k}{C}_{k} & -Q_{k}\\end{array}\\right] +\\left[\\begin{array}[c]{c}0_{\\left( k+1\\right) n\\times kp}\\\\Q_{k}\\left( I_{k}\\otimes E_{0}\\right)\\end{array}\\right] \\left( S_{k}^{-1}\\otimes I_{p}\\right) \\left[\\begin{array}[c]{c}0_{\\left( k+1\\right) n\\times kp}\\\\Q_{k}\\left( I_{k}\\otimes E_{0}\\right)\\end{array}\\right] ^{\\mathrm {T}}\\nonumber \\\\& +\\left[\\begin{array}[c]{cc}{C}_{k0} & 0_{kq\\times kn}\\end{array}\\right] ^{\\mathrm {T}}\\left( S_{k}\\otimes I_{q}\\right) \\left[\\begin{array}[c]{cc}{C}_{k0} & 0_{kq\\times kn}\\end{array}\\right] .",
"$ By (REF ) we have $\\Delta {C}_{k} & \\triangleq \\left[\\begin{array}[c]{cccccc}\\Delta B & \\Delta A & 0 & \\cdots & 0 & 0\\\\& \\Delta B & \\Delta A & \\ddots & \\vdots & \\vdots \\\\& & \\ddots & \\ddots & 0 & 0\\\\& & & \\Delta B & \\Delta A & 0\\\\& & & & \\Delta B & \\Delta A\\end{array}\\right] \\\\& =\\left[\\begin{array}[c]{cccccc}E_{0}FB_{0} & E_{0}FA_{0} & 0 & \\cdots & 0 & 0\\\\& E_{0}FB_{0} & E_{0}FA_{0} & \\ddots & \\vdots & 0\\\\& & \\ddots & \\ddots & 0 & \\vdots \\\\& & & E_{0}FB_{0} & E_{0}FA_{0} & 0\\\\& & & & E_{0}FB_{0} & E_{0}FA_{0}\\end{array}\\right] \\\\& =\\left( I_{k}\\otimes E_{0}\\right) \\left( I_{k}\\otimes F\\right){C}_{k0}.$ By using (REF ) we can compute $\\left( I_{k}\\otimes F^{\\mathrm {T}}\\right) \\left( S_{k}\\otimes I_{p}\\right)\\left( I_{k}\\otimes F\\right) =S_{k}\\otimes F^{\\mathrm {T}}F\\le S_{k}\\otimes I_{q}.$ Therefore, by using Lemma REF , we have from (REF ) that $& \\left[\\begin{array}[c]{cc}\\mathit {\\Omega }_{k1} & \\left( {C}_{k}+\\Delta {C}_{k}\\right)^{\\mathrm {T}}Q_{k}\\\\Q_{k}\\left( {C}_{k}+\\Delta {C}_{k}\\right) & -Q_{k}\\end{array}\\right] \\nonumber \\\\= & \\left[\\begin{array}[c]{cc}\\mathit {\\Omega }_{k1} & {C}_{k}^{\\mathrm {T}}Q_{k}\\\\Q_{k}{C}_{k} & -Q_{k}\\end{array}\\right] +\\left[\\begin{array}[c]{cc}0_{\\left( k+1\\right) n\\times \\left( k+1\\right) n} & \\Delta {C}_{k}^{\\mathrm {T}}Q_{k}\\\\Q_{k}\\Delta {C}_{k} & 0_{kn\\times kn}\\end{array}\\right] \\nonumber \\\\= & \\left[\\begin{array}[c]{cc}\\mathit {\\Omega }_{k1} & {C}_{k}^{\\mathrm {T}}Q_{k}\\\\Q_{k}{C}_{k} & -Q_{k}\\end{array}\\right] +\\left[\\begin{array}[c]{c}0_{\\left( k+1\\right) n\\times kp}\\\\Q_{k}\\left( I_{k}\\otimes E_{0}\\right)\\end{array}\\right] \\left( I_{k}\\otimes F\\right) \\left[\\begin{array}[c]{cc}{C}_{k0} & 0_{kq\\times kn}\\end{array}\\right] \\nonumber \\\\& +\\left[\\begin{array}[c]{cc}{C}_{k0} & 0_{kq\\times kn}\\end{array}\\right] ^{\\mathrm {T}}\\left( I_{k}\\otimes F^{\\mathrm {T}}\\right) \\left[\\begin{array}[c]{c}0_{\\left( k+1\\right) n\\times kp}\\\\Q_{k}\\left( I_{k}\\otimes E_{0}\\right)\\end{array}\\right] ^{\\mathrm {T}}\\nonumber \\\\\\le & \\left[\\begin{array}[c]{cc}\\mathit {\\Omega }_{k1} & {C}_{k}^{\\mathrm {T}}Q_{k}\\\\Q_{k}{C}_{k} & -Q_{k}\\end{array}\\right] +\\left[\\begin{array}[c]{c}0_{\\left( k+1\\right) n\\times kp}\\\\Q_{k}\\left( I_{k}\\otimes E_{0}\\right)\\end{array}\\right] \\left( S_{k}^{-1}\\otimes I_{p}\\right) \\left[\\begin{array}[c]{c}0_{\\left( k+1\\right) n\\times kp}\\\\Q_{k}\\left( I_{k}\\otimes E_{0}\\right)\\end{array}\\right] ^{\\mathrm {T}}\\nonumber \\\\& +\\left[\\begin{array}[c]{cc}{C}_{k0} & 0_{kq\\times kn}\\end{array}\\right] ^{\\mathrm {T}}\\left( I_{k}\\otimes F^{\\mathrm {T}}\\right) \\left(S_{k}\\otimes I_{p}\\right) \\left( I_{k}\\otimes F\\right) \\left[\\begin{array}[c]{cc}{C}_{k0} & 0_{kq\\times kn}\\end{array}\\right] \\nonumber \\\\\\le & \\left[\\begin{array}[c]{cc}\\mathit {\\Omega }_{k1} & {C}_{k}^{\\mathrm {T}}Q_{k}\\\\Q_{k}{C}_{k} & -Q_{k}\\end{array}\\right] +\\left[\\begin{array}[c]{c}0_{\\left( k+1\\right) n\\times kp}\\\\Q_{k}\\left( I_{k}\\otimes E_{0}\\right)\\end{array}\\right] \\left( S_{k}^{-1}\\otimes I_{p}\\right) \\left[\\begin{array}[c]{c}0_{\\left( k+1\\right) n\\times kp}\\\\Q_{k}\\left( I_{k}\\otimes E_{0}\\right)\\end{array}\\right] ^{\\mathrm {T}}\\nonumber \\\\& +\\left[\\begin{array}[c]{cc}{C}_{k0} & 0_{kq\\times kn}\\end{array}\\right] ^{\\mathrm {T}}\\left( S_{k}\\otimes I_{q}\\right) \\left[\\begin{array}[c]{cc}{C}_{k0} & 0_{kq\\times kn}\\end{array}\\right] \\nonumber \\\\< & 0.",
"$ By a Schur complement, the above inequality is equivalent to $0> & \\mathit {\\Omega }_{k1}+\\left( {C}_{k}+\\Delta {C}_{k}\\right) ^{\\mathrm {T}}Q_{k}\\left( {C}_{k}+\\Delta {C}_{k}\\right) \\\\= & \\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] ^{\\mathrm {T}}\\left( P_{k}-Q_{k}\\right) \\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] -L_{k}^{\\mathrm {T}}P_{k}L_{k}\\\\& +\\left[\\begin{array}[c]{cc}{A}_{k}+\\Delta {A}_{k} & {B}_{k}+\\Delta {B}_{k}\\end{array}\\right] ^{\\mathrm {T}}Q_{k}\\left[\\begin{array}[c]{cc}{A}_{k}+\\Delta {A}_{k} & {B}_{k}+\\Delta {B}_{k}\\end{array}\\right] .$ By Theorem REF , we know that system (REF ) is exponentially stable.",
"The proof is finished.",
"The merit of the proof of Theorem REF is that we have utilized the fact that $\\left( A,B\\right) $ appears as a linear function in the LMIs, which helps to eliminate the uncertain matrix $F$ in the LMI (REF ).",
"This can not be achieved for the LMI in Lemmas REF and REF .",
"Moreover, from the proof we can see that the only conservatism comes from the usage of the inequality in Lemma REF .",
"Thus the condition in Theorem REF is considered to be quite tight.",
"By using again the property that $\\left( A,B\\right) $ appears in the matrix $\\mathit {\\Omega }_{k}\\left( P_{k},Q_{k}\\right) $ as a quadratic function, we can extend easily the results in Theorem REF to the case of polytopic type uncertainty, say, $\\left[\\begin{array}[c]{cc}\\Delta A & \\Delta B\\end{array}\\right] \\in \\mathrm {co}\\left\\lbrace \\left[\\begin{array}[c]{cc}A^{\\left( i\\right) } & B^{\\left( i\\right) }\\end{array}\\right] ,i=1,2,\\ldots ,N\\right\\rbrace , $ where $A^{\\left( i\\right) },B^{\\left( i\\right) },i=1,2,\\ldots ,N$ are given matrices.",
"Denote $\\left[\\begin{array}[c]{cc}{A}_{k}^{\\left( i\\right) } & {B}_{k}^{\\left( i\\right) }\\end{array}\\right] =\\left[\\begin{array}[c]{ccccc}B+B^{\\left( i\\right) } & A+A^{\\left( i\\right) } & \\cdots & 0 & 0\\\\& & \\ddots & 0 & \\vdots \\\\& & B+B^{\\left( i\\right) } & A+A^{\\left( i\\right) } & 0\\\\& & & B+B^{\\left( i\\right) } & A+A^{\\left( i\\right) }\\end{array}\\right] \\in \\mathbf {R}^{kn\\times \\left( k+1\\right) n}.$ Then we obtain immediately the following theorem.",
"Theorem 3 The uncertain linear difference equation (REF ), where $\\Delta A$ and $\\Delta B$ satisfy (REF ), is exponentially stable if there exists positive definite matrices $P_{k},Q_{k}\\in \\mathbf {R}^{kn\\times kn}$ such that $\\mathit {\\Omega }_{k}^{\\left( i\\right) }\\left( P_{k},Q_{k}\\right) = &\\mathit {\\Omega }_{k1}\\left( P_{k},Q_{k}\\right) +\\left[\\begin{array}[c]{cc}{A}_{k}^{\\left( i\\right) } & {B}_{k}^{\\left( i\\right) }\\end{array}\\right] ^{\\mathrm {T}}Q_{k}\\left[\\begin{array}[c]{cc}{A}_{k}^{\\left( i\\right) } & {B}_{k}^{\\left( i\\right) }\\end{array}\\right] \\nonumber \\\\< & 0, $ are satisfied for $i=1,2,\\ldots ,N.$ Proof.",
"Notice that (REF ) implies $\\left[\\begin{array}[c]{cc}\\mathit {\\Omega }_{k1}\\left( P_{k},Q_{k}\\right) & \\left[\\begin{array}[c]{cc}{A}_{k}^{\\left( i\\right) } & {B}_{k}^{\\left( i\\right) }\\end{array}\\right] ^{\\mathrm {T}}Q_{k}\\\\Q_{k}\\left[\\begin{array}[c]{cc}{A}_{k}^{\\left( i\\right) } & {B}_{k}^{\\left( i\\right) }\\end{array}\\right] & -Q_{k}\\end{array}\\right] <0,$ where $i=1,2,\\ldots ,N$ .",
"It follows that, for any $\\alpha _{i}\\ge 0,i=1,2,\\ldots ,N$ with $\\alpha _{1}+\\alpha _{2}+\\cdots +\\alpha _{N}=1,$ and $\\left[\\begin{array}[c]{cc}\\Delta A & \\Delta B\\end{array}\\right] =\\sum \\limits _{i=1}^{N}\\alpha _{i}\\left[\\begin{array}[c]{cc}A^{\\left( i\\right) } & B^{\\left( i\\right) }\\end{array}\\right] ,$ we have $0 & >\\left[\\begin{array}[c]{cc}\\sum \\limits _{i=1}^{N}\\alpha _{i}\\mathit {\\Omega }_{k1}\\left( P_{k},Q_{k}\\right)& \\sum \\limits _{i=1}^{N}\\alpha _{i}\\left[\\begin{array}[c]{cc}{A}_{k}^{\\left( i\\right) } & {B}_{k}^{\\left( i\\right) }\\end{array}\\right] ^{\\mathrm {T}}Q_{k}\\\\Q_{k}\\sum \\limits _{i=1}^{N}\\alpha _{i}\\left[\\begin{array}[c]{cc}{A}_{k}^{\\left( i\\right) } & {B}_{k}^{\\left( i\\right) }\\end{array}\\right] & -\\sum \\limits _{i=1}^{N}\\alpha _{i}Q_{k}\\end{array}\\right] \\\\& =\\left[\\begin{array}[c]{cc}\\mathit {\\Omega }_{k1}\\left( P_{k},Q_{k}\\right) & \\left( {C}_{k}+\\Delta {C}_{k}\\right) ^{\\mathrm {T}}Q_{k}\\\\Q_{k}\\left( {C}_{k}+\\Delta {C}_{k}\\right) & -Q_{k}\\end{array}\\right] ,$ which is exactly in the form of (REF ).",
"The remaining of the proof is similar to that of Theorem REF and is omitted."
],
[
"Interpretations and Relationships",
"We first provide time-domain interpretations of Theorem REF and Lemma REF by establishing LKFs.",
"Lemma 4 For any integer $k\\ge 1,$ there holds $x(t)=\\sum \\limits _{i=0}^{k}A^{\\left[ i\\right] }B^{\\left[ k-i\\right]}x\\left( t-ia-\\left( k-i\\right) b\\right) .",
"$ Proof.",
"Clearly, it follows from (REF ) that (REF ) holds true with $k=1.$ Assume that (REF ) is true with $k=m,$ namely, $x(t)=\\sum \\limits _{i=0}^{m}A^{\\left[ i\\right] }B^{\\left[ m-i\\right]}x\\left( t-ia-\\left( m-i\\right) b\\right) .",
"$ Then, by inserting (REF ) into (REF ), we have $x(t)= & \\sum \\limits _{i=0}^{m}A^{\\left[ i\\right] }B^{\\left[ m-i\\right]}\\left( Ax\\left( t-\\left( i+1\\right) a-\\left( m-i\\right) b\\right)+Bx\\left( t-ia-\\left( m+1-i\\right) b\\right) \\right) \\nonumber \\\\= & \\sum \\limits _{i=0}^{m}A^{\\left[ i\\right] }B^{\\left[ m-i\\right]}Bx\\left( t-ia-\\left( m+1-i\\right) b\\right) +\\sum \\limits _{i=0}^{m}A^{\\left[ i\\right] }B^{\\left[ m-i\\right] }Ax\\left( t-\\left(i+1\\right) a-\\left( m-i\\right) b\\right) \\nonumber \\\\= & \\sum \\limits _{i=0}^{m}A^{\\left[ i\\right] }B^{\\left[ m-i\\right]}Bx\\left( t-ia-\\left( m+1-i\\right) b\\right) +\\sum \\limits _{j=1}^{m+1}A^{\\left[ j-1\\right] }B^{\\left[ m+1-j\\right] }Ax\\left( t-ja-\\left(m+1-j\\right) b\\right) \\nonumber \\\\= & A^{\\left[ 0\\right] }B^{\\left[ m\\right] }Bx\\left( t-\\left(m+1\\right) b\\right) +\\sum \\limits _{i=1}^{m}A^{\\left[ i\\right] }B^{\\left[m-i\\right] }Bx\\left( t-ia-\\left( m+1-i\\right) b\\right) \\nonumber \\\\& +\\sum \\limits _{j=1}^{m}A^{\\left[ j-1\\right] }B^{\\left[ m+1-j\\right]}Ax\\left( t-ja-\\left( m+1-j\\right) b\\right) +A^{\\left[ m\\right]}B^{\\left[ 0\\right] }Ax\\left( t-\\left( m+1\\right) a\\right) \\nonumber \\\\= & B^{\\left[ m+1\\right] }x\\left( t-\\left( m+1\\right) b\\right)+A^{\\left[ m+1\\right] }x\\left( t-\\left( m+1\\right) a\\right) \\nonumber \\\\& +\\sum \\limits _{i=1}^{m}\\left( A^{\\left[ i\\right] }B^{\\left[ m-i\\right]}B+A^{\\left[ i-1\\right] }B^{\\left[ m+1-i\\right] }A\\right) x\\left(t-ia-\\left( m+1-i\\right) b\\right) .",
"$ Notice that (see (REF ) in Appendix A2) $A^{\\left[ i\\right] }B^{\\left[ m-i\\right] }B+A^{\\left[ i-1\\right]}B^{\\left[ m+1-i\\right] }A=A^{\\left[ i\\right] }B^{\\left[ m+1-i\\right]},i=1,2,\\ldots ,m,$ substitution of which into (REF ) gives $x(t) & =B^{\\left[ m+1\\right] }x\\left( t-\\left( m+1\\right) b\\right)+\\sum \\limits _{i=1}^{m}A^{\\left[ i\\right] }B^{\\left[ m+1-i\\right] }x\\left(t-ia-\\left( m+1-i\\right) b\\right) +A^{\\left[ m+1\\right] }x\\left(t-\\left( m+1\\right) a\\right) \\\\& =\\sum \\limits _{i=0}^{m+1}A^{\\left[ i\\right] }B^{\\left[ m+1-i\\right]}x\\left( t-ia-\\left( m+1-i\\right) b\\right) .$ Therefore, (REF ) holds with $k=m+1.$ The proof is finished by mathematical induction.",
"In the following, we assume, without loss of generality, that $b>a$ since otherwise we can change the roles of $a$ and $b.$ Lemma 5 For any integer $k\\ge 1,$ let $\\left\\lbrace \\begin{array}[c]{rl}X_{k}\\left( t\\right) & =\\left[\\begin{array}[c]{c}x\\left( t-kb\\right) \\\\x\\left( t-\\left( k-1\\right) b-a\\right) \\\\\\vdots \\\\x\\left( t-2b-\\left( k-2\\right) a\\right) \\\\x\\left( t-b-\\left( k-1\\right) a\\right)\\end{array}\\right] \\in \\mathbf {R}^{kn},\\\\U_{k}\\left( t\\right) & =x\\left( t-ka\\right) \\in \\mathbf {R}^{n},\\\\Y_{k}\\left( t\\right) & =x\\left( t\\right) \\in \\mathbf {R}^{n}.\\end{array}\\right.",
"$ Then $\\left( U_{k}\\left( t\\right) ,X_{k}\\left( t\\right) ,Y_{k}\\left(t\\right) \\right) $ satisfies $\\left.\\begin{array}[c]{rl}X_{k}\\left( t+b-a\\right) & =A_{k}X_{k}\\left( t\\right) +B_{k}U_{k}\\left(t\\right) ,\\\\Y_{k}\\left( t\\right) & =C_{k}X_{k}\\left( t\\right) +D_{k}U_{k}\\left(t\\right) .\\end{array}\\right\\rbrace $ Proof.",
"This can be verified by direct computation.",
"In fact, by definition, we have $X_{k}\\left( t+b-a\\right) & =\\left[\\begin{array}[c]{c}x\\left( t-\\left( k-1\\right) b-a\\right) \\\\x\\left( t-\\left( k-2\\right) b-2a\\right) \\\\\\vdots \\\\x\\left( t-b-\\left( k-1\\right) a\\right) \\\\x\\left( t-ka\\right)\\end{array}\\right] \\\\& =A_{k}X_{k}\\left( t\\right) +B_{k}U\\left( t\\right) ,$ and it follows from Lemma REF that $Y_{k}\\left( t\\right) & =\\sum \\limits _{i=0}^{k}A^{\\left[ i\\right]}B^{\\left[ k-i\\right] }x\\left( t-ia-\\left( k-i\\right) b\\right) \\\\& =\\left[\\begin{array}[c]{cccc}B^{\\left[ k\\right] } & A^{\\left[ 1\\right] }B^{\\left[ k-1\\right] } &\\cdots & A^{\\left[ k-1\\right] }B^{\\left[ 1\\right] }\\end{array}\\right] X_{k}\\left( t\\right) +A^{\\left[ k\\right] }x\\left( t-ka\\right) \\\\& =C_{k}X_{k}\\left( t\\right) +D_{k}U_{k}\\left( t\\right) .$ The proof is finished.",
"We next provide a time-domain interpretation of Theorem REF by establishing an LKF for the system.",
"Proposition 1 For any integer $k\\ge 1,$ let $\\mathit {\\Omega }_{k}\\left(P,Q\\right) $ be defined by (REF ) where $\\left( A_{k},B_{k},{A}_{k},{B}_{k}\\right) $ is defined by (REF )-(REF ).",
"Consider the following LKF $V_{k}\\left( x_{t}\\right) =\\int _{t-b}^{t-a}X_{k}^{\\mathrm {T}}\\left(s\\right) P_{k}X_{k}\\left( s\\right) \\mathrm {d}s+\\int _{t-a}^{t}X_{k}^{\\mathrm {T}}\\left( s\\right) Q_{k}X_{k}\\left( s\\right) \\mathrm {d}s,$ where $P_{k}=P_{k}^{\\mathrm {T}}\\in \\mathbf {R}^{kn\\times kn}$ and $Q_{k}=Q_{k}^{\\mathrm {T}}\\in \\mathbf {R}^{kn\\times kn}.$ Then $\\dot{V}_{k}\\left( x_{t}\\right) =\\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] ^{\\mathrm {T}}\\mathit {\\Omega }_{k}\\left( P_{k},Q_{k}\\right) \\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] .",
"$ Proof.",
"From (REF ) and (REF ) we know $U_{k}\\left( t+a-b\\right)=x\\left( t-\\left( k-1\\right) a-b\\right) $ and $X_{k}\\left( t-a\\right) =A_{k}X_{k}\\left( t-b\\right) +B_{k}x\\left(t-ka-b\\right) .$ By using (REF ) and noting the structures of ${A}_{k}$ and ${B}_{k},$ we have $X_{k}\\left( t\\right) & =\\left[\\begin{array}[c]{c}x\\left( t-kb\\right) \\\\x\\left( t-\\left( k-1\\right) b-a\\right) \\\\\\vdots \\\\x\\left( t-2b-\\left( k-2\\right) a\\right) \\\\x\\left( t-b-\\left( k-1\\right) a\\right)\\end{array}\\right] \\\\& =\\left[\\begin{array}[c]{cccccc}B & A & 0 & \\cdots & 0 & 0\\\\& B & A & \\ddots & \\vdots & 0\\\\& & \\ddots & \\ddots & 0 & \\vdots \\\\& & & B & A & 0\\\\& & & & B & A\\end{array}\\right] \\left[\\begin{array}[c]{c}x\\left( t-\\left( k+1\\right) b\\right) \\\\x\\left( t-kb-a\\right) \\\\\\vdots \\\\x\\left( t-2b-\\left( k-1\\right) a\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] \\\\& =\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] \\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] .$ Therefore, it follows from (REF ) that $\\dot{V}_{k}\\left( x_{t}\\right) = & X_{k}^{\\mathrm {T}}\\left( t-a\\right)P_{k}X_{k}\\left( t-a\\right) -X_{k}^{\\mathrm {T}}\\left( t-b\\right)P_{k}X_{k}\\left( t-b\\right) \\\\& +X_{k}^{\\mathrm {T}}\\left( t\\right) Q_{k}X_{k}\\left( t\\right)-X_{k}^{\\mathrm {T}}\\left( t-a\\right) Q_{k}X_{k}\\left( t-a\\right) \\\\= & \\left( A_{k}X_{k}\\left( t-b\\right) +B_{k}x\\left( t-b-ka\\right)\\right) ^{\\mathrm {T}}\\left( P_{k}-Q_{k}\\right) \\left( A_{k}X_{k}\\left(t-b\\right) +B_{k}x\\left( t-b-ka\\right) \\right) \\\\& +X_{k}^{\\mathrm {T}}\\left( t\\right) Q_{k}X_{k}\\left( t\\right)-X_{k}^{\\mathrm {T}}\\left( t-b\\right) P_{k}X_{k}\\left( t-b\\right) \\\\= & \\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] ^{\\mathrm {T}}\\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] ^{\\mathrm {T}}\\left( P_{k}-Q_{k}\\right) \\left[\\begin{array}[c]{cc}A_{k} & B_{k}\\end{array}\\right] \\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] \\\\& -\\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] ^{\\mathrm {T}}\\left[\\begin{array}[c]{cc}I_{kn} & 0_{kn\\times n}\\end{array}\\right] ^{\\mathrm {T}}P_{k}\\left[\\begin{array}[c]{cc}I_{kn} & 0_{kn\\times n}\\end{array}\\right] \\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] \\\\& +\\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] ^{\\mathrm {T}}\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] ^{\\mathrm {T}}Q_{k}\\left[\\begin{array}[c]{cc}{A}_{k} & {B}_{k}\\end{array}\\right] \\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] \\\\= & \\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] ^{\\mathrm {T}}\\mathit {\\Omega }_{k}\\left( P_{k},Q_{k}\\right) \\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] .$ The proof is finished.",
"Similar to Lemma REF , we can present the following lemma.",
"Lemma 6 For any integer $k\\ge 1,$ let $\\overline{A}_{k},\\overline{B}_{k}$ be defined in (REF ) and $\\overline{C}_{k}=\\left[\\begin{array}[c]{cccc}B & AB & \\cdots & A^{k-1}B\\end{array}\\right] \\in \\mathbf {R}^{n\\times kn},\\; \\overline{D}_{k}=A^{k}\\in \\mathbf {R}^{n\\times n}.",
"$ Let $\\left\\lbrace \\begin{array}[c]{rl}\\overline{X}_{k}\\left( t\\right) & =\\left[\\begin{array}[c]{c}x\\left( t\\right) \\\\x\\left( t-a\\right) \\\\\\vdots \\\\x\\left( t-\\left( k-1\\right) a\\right)\\end{array}\\right] \\in \\mathbf {R}^{kn},\\\\\\overline{U}_{k}\\left( t\\right) & =x\\left( t+b-ka\\right) \\in \\mathbf {R}^{n},\\\\\\overline{Y}_{k}\\left( t\\right) & =x\\left( t+b\\right) \\in \\mathbf {R}^{n}.\\end{array}\\right.",
"$ Then $\\left( \\overline{U}_{k}\\left( t\\right) ,\\overline{X}_{k}\\left(t\\right) ,\\overline{Y}_{k}\\left( t\\right) \\right) $ satisfies $\\left.\\begin{array}[c]{rl}\\overline{X}_{k}\\left( t+b-a\\right) & =\\overline{A}_{k}\\overline{X}_{k}\\left( t\\right) +\\overline{B}_{k}\\overline{U}_{k}\\left( t\\right) ,\\\\\\overline{Y}_{k}\\left( t\\right) & =\\overline{C}_{k}\\overline{X}_{k}\\left(t\\right) +\\overline{D}_{k}\\overline{U}_{k}\\left( t\\right) .\\end{array}\\right\\rbrace $ Proof.",
"It is straightforward to see that, for any $i=0,1,\\ldots ,k-1,$ $x\\left( t-ia\\right) = & Bx\\left( t-ia-b\\right) +Ax\\left( t-\\left(i+1\\right) a\\right) \\nonumber \\\\= & Bx\\left( t-ia-b\\right) +A\\left( Bx\\left( t-\\left( i+1\\right)a-b\\right) +Ax\\left( t-\\left( i+2\\right) a\\right) \\right) \\nonumber \\\\= & Bx\\left( t-ia-b\\right) +ABx\\left( t-\\left( i+1\\right) a-b\\right)+A^{2}x\\left( t-\\left( i+2\\right) a\\right) \\nonumber \\\\= & \\cdots \\nonumber \\\\= & Bx\\left( t-ia-b\\right) +ABx\\left( t-\\left( i+1\\right) a-b\\right)+\\cdots \\nonumber \\\\& +A^{k-i-1}Bx\\left( t-\\left( k-1\\right) a-b\\right) +A^{k-i}x\\left(t-ka\\right) .",
"$ For $i=1,2,\\ldots ,k-1,$ we write the above $k-1$ equations in the dense form $\\left[\\begin{array}[c]{c}x\\left( t-a\\right) \\\\x\\left( t-2a\\right) \\\\\\vdots \\\\x\\left( t-\\left( k-1\\right) a\\right) \\\\x\\left( t-ka\\right)\\end{array}\\right] =\\left[\\begin{array}[c]{ccccc}0 & B & AB & \\cdots & A^{k-2}B\\\\& 0 & B & \\cdots & A^{k-3}B\\\\& & \\ddots & \\ddots & \\vdots \\\\& & & 0 & B\\\\& & & & 0\\end{array}\\right] \\left[\\begin{array}[c]{c}x\\left( t-b\\right) \\\\x\\left( t-a-b\\right) \\\\\\vdots \\\\x\\left( t-\\left( k-2\\right) a-b\\right) \\\\x\\left( t-\\left( k-1\\right) a-b\\right)\\end{array}\\right] +\\left[\\begin{array}[c]{c}A^{k-1}\\\\A^{k-2}\\\\\\vdots \\\\A\\\\I_{n}\\end{array}\\right] x\\left( t-ka\\right) ,$ which can be written as $\\overline{X}_{k}\\left( t-a\\right) =\\overline{A}_{k}\\overline{X}_{k}\\left(t-b\\right) +\\overline{B}_{k}x\\left( t-ka\\right) , $ which is just the first equation (REF ).",
"On the other hand, with $i=0$ in (REF ), we have $x\\left( t\\right) & =Bx\\left( t-b\\right) +ABx\\left( t-a-b\\right)+\\cdots +A^{k-1}Bx\\left( t-\\left( k-1\\right) a-b\\right) +A^{k}x\\left(t-ka\\right) \\\\& =\\left[\\begin{array}[c]{cccc}B & AB & \\cdots & A^{k-1}B\\end{array}\\right] X_{k}\\left( t-b\\right) +A^{k}x\\left( t-ka\\right) \\\\& =\\overline{C}_{k}X_{k}\\left( t-b\\right) +\\overline{D}_{k}x\\left(t-ka\\right) ,$ which is just the second equation in (REF ).",
"The proof is finished.",
"We then can present for Lemma REF a time-domain interpretation, which parallels Proposition REF .",
"Proposition 2 For any integer $k\\ge 1,$ let $\\overline{\\mathit {\\Omega }}_{k}$ be defined in (REF ).",
"Consider the following LKF $\\overline{V}_{k}\\left( x_{t}\\right) =\\int _{t-a}^{t}\\overline{X}_{k}^{\\mathrm {T}}\\left( s\\right) \\overline{Q}_{k}\\overline{X}_{k}\\left(s\\right) \\mathrm {d}s+\\int _{t-b}^{t-a}\\overline{X}_{k}^{\\mathrm {T}}\\left(s\\right) \\overline{P}_{k}\\overline{X}_{k}\\left( s\\right) \\mathrm {d}s,$ where $\\overline{P}_{k}=\\overline{P}_{k}^{\\mathrm {T}}\\in \\mathbf {R}^{kn\\times kn}$ and $\\overline{Q}_{k}=\\overline{Q}_{k}^{\\mathrm {T}}\\in \\mathbf {R}^{kn\\times kn}.$ Then $\\dot{\\overline{V}}_{k}\\left( x_{t}\\right) =\\left[\\begin{array}[c]{c}\\overline{X}_{k}\\left( t-b\\right) \\\\x\\left( t-ka\\right)\\end{array}\\right] ^{\\mathrm {T}}\\overline{\\mathit {\\Omega }}_{k}\\left( \\overline{P}_{k},\\overline{Q}_{k}\\right) \\left[\\begin{array}[c]{c}\\overline{X}_{k}\\left( t-b\\right) \\\\x\\left( t-ka\\right)\\end{array}\\right] .",
"$ Proof.",
"By using (REF ) and noting the structures of $\\overline{{A}}_{k}$ and $\\overline{{B}}_{k}$ in (), we have $\\overline{X}_{k}\\left( t\\right) & =\\left[\\begin{array}[c]{cccccc}B & AB & A^{2}B & \\cdots & A^{k-1}B & A^{k}\\\\& B & AB & \\ddots & A^{k-2}B & A^{k-1}\\\\& & \\ddots & \\ddots & \\vdots & \\vdots \\\\& & & B & AB & A^{2}\\\\& & & & B & A\\end{array}\\right] \\left[\\begin{array}[c]{c}x\\left( t-b\\right) \\\\x\\left( t-a-b\\right) \\\\\\vdots \\\\x\\left( t-\\left( k-1\\right) a-b\\right) \\\\x\\left( t-ka\\right)\\end{array}\\right] \\\\& =\\left[\\begin{array}[c]{cc}\\overline{{A}}_{k} & \\overline{{B}}_{k}\\end{array}\\right] \\left[\\begin{array}[c]{c}\\overline{X}_{k}\\left( t-b\\right) \\\\x\\left( t-ka\\right)\\end{array}\\right] .$ Therefore, it follows from (REF ) that $\\dot{\\overline{V}}_{k}\\left( x_{t}\\right) = & \\overline{X}_{k}^{\\mathrm {T}}\\left( t-a\\right) \\overline{P}_{k}\\overline{X}_{k}\\left(t-a\\right) -\\overline{X}_{k}^{\\mathrm {T}}\\left( t-b\\right) \\overline{P}_{k}\\overline{X}_{k}\\left( t-b\\right) \\\\& +\\overline{X}_{k}^{\\mathrm {T}}\\left( t\\right) \\overline{Q}_{k}\\overline{X}_{k}\\left( t\\right) -\\overline{X}_{k}^{\\mathrm {T}}\\left(t-a\\right) \\overline{Q}_{k}\\overline{X}_{k}\\left( t-a\\right) \\\\= & \\left( \\overline{A}_{k}\\overline{X}_{k}\\left( t-b\\right)+\\overline{B}_{k}x\\left( t-ka\\right) \\right) ^{\\mathrm {T}}\\left(\\overline{P}_{k}-\\overline{Q}_{k}\\right) \\left( \\overline{A}_{k}\\overline{X}_{k}\\left( t-b\\right) +\\overline{B}_{k}x\\left( t-ka\\right) \\right) \\\\& +\\overline{X}_{k}^{\\mathrm {T}}\\left( t\\right) \\overline{Q}_{k}\\overline{X}_{k}\\left( t\\right) -\\overline{X}_{k}^{\\mathrm {T}}\\left(t-b\\right) \\overline{P}_{k}\\overline{X}_{k}\\left( t-b\\right) \\\\= & \\left[\\begin{array}[c]{c}\\overline{X}_{k}\\left( t-b\\right) \\\\x\\left( t-ka\\right)\\end{array}\\right] ^{\\mathrm {T}}\\left[\\begin{array}[c]{cc}\\overline{A}_{k} & \\overline{B}_{k}\\end{array}\\right] ^{\\mathrm {T}}\\left( \\overline{P}_{k}-\\overline{Q}_{k}\\right)\\left[\\begin{array}[c]{cc}\\overline{A}_{k} & \\overline{B}_{k}\\end{array}\\right] \\left[\\begin{array}[c]{c}\\overline{X}_{k}\\left( t-b\\right) \\\\x\\left( t-ka\\right)\\end{array}\\right] \\\\& -\\left[\\begin{array}[c]{c}\\overline{X}_{k}\\left( t-b\\right) \\\\x\\left( t-ka\\right)\\end{array}\\right] ^{\\mathrm {T}}\\left[\\begin{array}[c]{cc}I_{kn} & 0_{kn\\times n}\\end{array}\\right] ^{\\mathrm {T}}\\overline{P}_{k}\\left[\\begin{array}[c]{cc}I_{kn} & 0_{kn\\times n}\\end{array}\\right] \\left[\\begin{array}[c]{c}\\overline{X}_{k}\\left( t-b\\right) \\\\x\\left( t-ka\\right)\\end{array}\\right] \\\\& +\\left[\\begin{array}[c]{c}\\overline{X}_{k}\\left( t-b\\right) \\\\x\\left( t-ka\\right)\\end{array}\\right] ^{\\mathrm {T}}\\left[\\begin{array}[c]{cc}\\overline{{A}}_{k} & \\overline{{B}}_{k}\\end{array}\\right] ^{\\mathrm {T}}\\overline{Q}_{k}\\left[\\begin{array}[c]{cc}\\overline{{A}}_{k} & \\overline{{B}}_{k}\\end{array}\\right] \\left[\\begin{array}[c]{c}\\overline{X}_{k}\\left( t-b\\right) \\\\x\\left( t-ka\\right)\\end{array}\\right] \\\\= & \\left[\\begin{array}[c]{c}\\overline{X}_{k}\\left( t-b\\right) \\\\x\\left( t-ka\\right)\\end{array}\\right] ^{\\mathrm {T}}\\overline{\\mathit {\\Omega }}_{k}\\left( \\overline{P}_{k},\\overline{Q}_{k}\\right) \\left[\\begin{array}[c]{c}\\overline{X}_{k}\\left( t-b\\right) \\\\x\\left( t-ka\\right)\\end{array}\\right] ,$ which completes the proof.",
"One may wonder the relationship between Theorem REF and Lemma REF .",
"Such a relationship should be revealed from the time-domain interpretations of these two LMIs.",
"To investigate this problem, we need to find the relationship between $\\overline{\\mathit {\\Omega }}_{k}$ and $\\mathit {\\Omega }_{k}.$ Such a relationship should be revealed from the time-domain interpretations of these two LMIs, say, the relationship between $X_{k}\\left( t\\right) $ and $\\overline{X}_{k}\\left( t\\right) ,$ and the relationship between $\\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] \\text{ and }\\left[\\begin{array}[c]{c}\\overline{X}_{k}\\left( t-b\\right) \\\\x\\left( t-ka\\right)\\end{array}\\right] .$ To this end, we denote, for any integer $k\\ge 1,$ $W_{k}=\\left[\\begin{array}[c]{cccc}B^{\\left[ k-1\\right] } & B^{\\left[ k-2\\right] }A^{\\left[ 1\\right] } &\\cdots & A^{\\left[ k-1\\right] }\\\\& \\ddots & \\ddots & \\vdots \\\\& & B^{\\left[ 1\\right] } & A^{\\left[ 1\\right] }\\\\& & & I_{n}\\end{array}\\right] ,T_{k}=\\left[\\begin{array}[c]{cc}W_{k} & \\\\& I_{n}\\end{array}\\right] .$ Then we have the following result.",
"Proposition 3 Let $\\mathit {\\Omega }_{k}\\left( P_{k},Q_{k}\\right) $ and $\\overline{\\mathit {\\Omega }}_{k}\\left( \\overline{P}_{k},\\overline{Q}_{k}\\right) $ be defined respectively in (REF ) and (REF ).",
"Let $P_{k}=W_{k}^{\\mathrm {T}}\\overline{P}_{k}W_{k},\\;Q_{k}=W_{k}^{\\mathrm {T}}\\overline{Q}_{k}W_{k}.",
"$ Then there holds $\\mathit {\\Omega }_{k}\\left( P_{k},Q_{k}\\right) =T_{k}^{\\mathrm {T}}\\overline{\\mathit {\\Omega }}_{k}\\left( \\overline{P}_{k},\\overline{Q}_{k}\\right) T_{k}.",
"$ Therefore, the LMI in (REF ) is feasible if and only if the LMI in (REF ) is feasible.",
"Proof.",
"By using Lemma REF we have $\\overline{X}_{k}\\left( t-b\\right) & =\\left[\\begin{array}[c]{c}x\\left( t-b\\right) \\\\x\\left( t-a-b\\right) \\\\\\vdots \\\\x\\left( t-\\left( k-1\\right) a-b\\right)\\end{array}\\right] \\\\& =\\left[\\begin{array}[c]{cccc}B^{\\left[ k-1\\right] } & B^{\\left[ k-2\\right] }A^{\\left[ 1\\right] } &\\cdots & A^{\\left[ k-1\\right] }\\\\& \\ddots & \\ddots & \\vdots \\\\& & B^{\\left[ 1\\right] } & A^{\\left[ 1\\right] }\\\\& & & I_{n}\\end{array}\\right] \\left[\\begin{array}[c]{c}x\\left( t-kb\\right) \\\\x\\left( t-\\left( k-1\\right) b-a\\right) \\\\\\vdots \\\\x\\left( t-2b-\\left( k-2\\right) a\\right) \\\\x\\left( t-b-\\left( k-1\\right) a\\right)\\end{array}\\right] \\\\& =W_{k}X_{k}\\left( t\\right) ,$ from which we get $\\left[\\begin{array}[c]{c}\\overline{X}_{k}\\left( t-2b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] & =\\left[\\begin{array}[c]{c}W_{k}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] \\\\& =\\left[\\begin{array}[c]{cc}W_{k} & 0\\\\0 & I_{n}\\end{array}\\right] \\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] \\\\& =T_{k}\\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] .$ Therefore, we have from (REF ) that $\\overline{V}_{k}\\left( x_{t-b}\\right) & =\\int _{t-a}^{t}\\overline{X}_{k}^{\\mathrm {T}}\\left( s-b\\right) \\overline{Q}_{k}\\overline{X}_{k}\\left(s-b\\right) \\mathrm {d}s+\\int _{t-b}^{t-a}\\overline{X}_{k}^{\\mathrm {T}}\\left(s-b\\right) \\overline{P}_{k}\\overline{X}_{k}\\left( s-b\\right) \\mathrm {d}s,\\nonumber \\\\& =\\int _{t-a}^{t}X_{k}^{\\mathrm {T}}\\left( s\\right) W_{k}^{\\mathrm {T}}\\overline{Q}_{k}W_{k}X_{k}\\left( s\\right) \\mathrm {d}s+\\int _{t-b}^{t-a}X_{k}^{\\mathrm {T}}\\left( s\\right) W_{k}^{\\mathrm {T}}\\overline{P}_{k}W_{k}X_{k}\\left( s\\right) \\mathrm {d}s,$ and from (REF ) that $\\dot{\\overline{V}}_{k}\\left( x_{t-b}\\right) & =\\left[\\begin{array}[c]{c}\\overline{X}_{k}\\left( t-2b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] ^{\\mathrm {T}}\\overline{\\mathit {\\Omega }}_{k}\\left( \\overline{P}_{k},\\overline{Q}_{k}\\right) \\left[\\begin{array}[c]{c}\\overline{X}_{k}\\left( t-2b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] \\nonumber \\\\& =\\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] ^{\\mathrm {T}}T_{k}^{\\mathrm {T}}\\overline{\\mathit {\\Omega }}_{k}\\left(\\overline{P}_{k},\\overline{Q}_{k}\\right) T_{k}\\left[\\begin{array}[c]{c}X_{k}\\left( t-b\\right) \\\\x\\left( t-b-ka\\right)\\end{array}\\right] .$ By comparing (REF ) and (REF ) with (REF ) and (REF ) we know that, if $\\left( P_{k},Q_{k}\\right) $ satisfies (REF ), then $\\mathit {\\Omega }_{k}$ and $\\overline{\\mathit {\\Omega }}_{k}$ satisfies (REF ).",
"The proof is finished.",
"It follows that Theorem REF is equivalent to Lemma REF .",
"Even so, Theorem REF possesses great advantage over Lemma REF since the system parameters appear linearly (quadratically) in the LMIs (REF ), which has been very important in the robust stability analysis.",
"We next show the connection to the Carvalho Condition.",
"Lemma 7 [3] The linear difference equation (REF ) is exponentially stable if there exist two positive definite matrices $X_{1}\\in \\mathbf {R}^{n\\times n}$ and $Y_{1}\\in \\mathbf {R}^{n\\times n}$ such that the following LMI is satisfied $\\mathit {\\Phi }_{1}\\left( X_{1},Y_{1}\\right) =\\left[\\begin{array}[c]{cc}A & B\\\\I_{n} & 0\\end{array}\\right] ^{\\mathrm {T}}\\left[\\begin{array}[c]{cc}X_{1} & 0\\\\0 & Y_{1}\\end{array}\\right] \\left[\\begin{array}[c]{cc}A & B\\\\I_{n} & 0\\end{array}\\right] -\\left[\\begin{array}[c]{cc}X_{1} & 0\\\\0 & Y_{1}\\end{array}\\right] <0.",
"$ Proof.",
"For future use, we give a simple proof here.",
"Choose the following LK functional $W_{1}\\left( x_{t}\\right) =\\int _{t-a}^{t}x^{\\mathrm {T}}\\left( s\\right)X_{1}x\\left( s\\right) \\mathrm {d}s+\\int _{t-b}^{t-a}x^{\\mathrm {T}}\\left(s\\right) Y_{1}x\\left( s\\right) \\mathrm {d}s, $ which is such that $\\dot{W}_{1}\\left( x_{t}\\right) & =x^{\\mathrm {T}}\\left( t\\right)X_{1}x\\left( t\\right) -x^{\\mathrm {T}}\\left( t-a\\right) X_{1}x\\left(t-a\\right) +x^{\\mathrm {T}}\\left( t-a\\right) Y_{1}x\\left( t-a\\right)-x^{\\mathrm {T}}\\left( t-b\\right) Y_{1}x\\left( t-b\\right) \\nonumber \\\\& =\\left[\\begin{array}[c]{c}x\\left( t-a\\right) \\\\x\\left( t-b\\right)\\end{array}\\right] ^{\\mathrm {T}}\\mathit {\\Phi }_{1}\\left( X_{1},Y_{1}\\right) \\left[\\begin{array}[c]{c}x\\left( t-a\\right) \\\\x\\left( t-b\\right)\\end{array}\\right] .",
"$ Since $\\mathit {\\Phi }_{1}\\left( X_{1},Y_{1}\\right) <0,$ the stability follows from the Lyapunov stability theorem [3].",
"If we set $k=1$ in Theorem REF and denote $E_{2}=\\left[\\begin{array}[c]{cc}0 & I_{n}\\\\I_{n} & 0\\end{array}\\right] ,E_{3}=\\left[\\begin{array}[c]{ccc}0 & 0 & I_{n}\\\\0 & I_{n} & 0\\\\I_{n} & 0 & 0\\end{array}\\right] ,$ we obtain the following result.",
"Lemma 8 Let $\\mathit {\\Omega }_{k}$ be defined in (REF ), $\\overline{\\mathit {\\Omega }}_{k}$ be defined in (REF ) and $\\mathit {\\Phi }_{1}$ be defined in (REF ).",
"Then, for $k=1$ , there holds $\\mathit {\\Omega }_{1}\\left( P_{1},Q_{1}\\right) & =E_{2}^{\\mathrm {T}}\\mathit {\\Phi }_{1}\\left( Q_{1},P_{1}\\right) E_{2},\\\\\\overline{\\mathit {\\Omega }}_{1}\\left( \\overline{P}_{1},\\overline{Q}_{1}\\right) & =E_{2}^{\\mathrm {T}}\\mathit {\\Phi }_{1}\\left( \\overline{Q}_{1},\\overline{P}_{1}\\right) E_{2}.",
"$ Thus the result in Lemma REF [3] is a special case of Lemma REF and Theorem REF .",
"Proof.",
"Let $k=1.$ Then it follows from (REF ) that $V_{1}\\left( x_{t+b}\\right) & =\\int _{t-b}^{t-a}X_{1}^{\\mathrm {T}}\\left(s+b\\right) P_{1}X_{1}\\left( s+b\\right) \\mathrm {d}s+\\int _{t-a}^{t}X_{1}^{\\mathrm {T}}\\left( s+b\\right) Q_{1}X_{1}\\left( s+b\\right)\\mathrm {d}s\\nonumber \\\\& =\\int _{t-b}^{t-a}x^{\\mathrm {T}}\\left( s\\right) P_{1}x\\left( s\\right)\\mathrm {d}s+\\int _{t-a}^{t}x^{\\mathrm {T}}\\left( s\\right) Q_{1}x\\left(s\\right) \\mathrm {d}s, $ and from (REF ) that $\\dot{V}_{1}\\left( x_{t+b}\\right) & =\\left[\\begin{array}[c]{c}x\\left( t-b\\right) \\\\x\\left( t-a\\right)\\end{array}\\right] ^{\\mathrm {T}}\\mathit {\\Omega }_{1}\\left( P_{1},Q_{1}\\right) \\left[\\begin{array}[c]{c}x\\left( t-b\\right) \\\\x\\left( t-a\\right)\\end{array}\\right] \\nonumber \\\\& =\\left[\\begin{array}[c]{c}x\\left( t-a\\right) \\\\x\\left( t-b\\right)\\end{array}\\right] ^{\\mathrm {T}}E_{2}^{\\mathrm {T}}\\mathit {\\Omega }_{1}\\left( P_{1},Q_{1}\\right) E_{2}\\left[\\begin{array}[c]{c}x\\left( t-a\\right) \\\\x\\left( t-b\\right)\\end{array}\\right] .",
"$ By comparing (REF ) and (REF ) with (REF ) and (REF ), respectively, we get (REF ).",
"Similarly, we have from (REF ) that $\\overline{V}_{1}\\left( x_{t}\\right) & =\\int _{t-a}^{t}\\overline{X}_{1}^{\\mathrm {T}}\\left( s\\right) \\overline{Q}_{1}\\overline{X}_{1}\\left(s\\right) \\mathrm {d}s+\\int _{t-b}^{t-a}\\overline{X}_{1}^{\\mathrm {T}}\\left(s\\right) \\overline{P}_{1}\\overline{X}_{1}\\left( s\\right) \\mathrm {d}s\\nonumber \\\\& =\\int _{t-a}^{t}x^{\\mathrm {T}}\\left( s\\right) \\overline{Q}_{1}x\\left(s\\right) \\mathrm {d}s+\\int _{t-b}^{t-a}x^{\\mathrm {T}}\\left( s\\right)\\overline{P}_{1}x\\left( s\\right) \\mathrm {d}s, $ and from (REF ) that $\\dot{\\overline{V}}_{1}\\left( x_{t}\\right) & =\\left[\\begin{array}[c]{c}x\\left( t-b\\right) \\\\x\\left( t-a\\right)\\end{array}\\right] ^{\\mathrm {T}}\\overline{\\mathit {\\Omega }}_{1}\\left( \\overline{P}_{1},\\overline{Q}_{1}\\right) \\left[\\begin{array}[c]{c}x\\left( t-b\\right) \\\\x\\left( t-a\\right)\\end{array}\\right] \\nonumber \\\\& =\\left[\\begin{array}[c]{c}x\\left( t-a\\right) \\\\x\\left( t-b\\right)\\end{array}\\right] ^{\\mathrm {T}}E_{2}^{\\mathrm {T}}\\overline{\\mathit {\\Omega }}_{1}\\left(\\overline{P}_{1},\\overline{Q}_{1}\\right) E_{2}\\left[\\begin{array}[c]{c}x\\left( t-a\\right) \\\\x\\left( t-b\\right)\\end{array}\\right] .",
"$ By comparing (REF ) and (REF ) with (REF ) and (REF ), respectively, we get (REF ).",
"The proof is finished.",
"We next investigate the relationship between Theorem REF and a result in [4].",
"To this end, we denote $N_{21} & =\\left[\\begin{array}[c]{ccc}A & B & 0\\\\I_{n} & 0 & 0\\end{array}\\right] ,\\;N_{22}=\\left[\\begin{array}[c]{ccc}0 & 0 & I_{n}\\\\0 & I_{n} & 0\\end{array}\\right] ,\\\\M_{21} & =\\left[\\begin{array}[c]{ccc}A & B & 0\\\\0 & 0 & I_{n}\\end{array}\\right] ,\\;M_{22}=\\left[\\begin{array}[c]{ccc}I_{n} & 0 & 0\\\\0 & I_{n} & 0\\end{array}\\right] .$ Lemma 9 [4] The linear difference equation (REF ) is exponentially stable if there exist four positive definite matrices $X_{2},Y_{2}\\in \\mathbf {R}^{2n\\times 2n},U_{2},V_{2}\\in \\mathbf {R}^{n\\times n},$ such that the following LMI is satisfied $\\mathit {\\Phi }_{2}\\left( X_{2}^{\\ast },Y_{2}^{\\ast }\\right) =N_{21}^{\\mathrm {T}}X_{2}^{\\ast }N_{21}-N_{22}^{\\mathrm {T}}X_{2}^{\\ast }N_{22}+M_{21}^{\\mathrm {T}}Y_{2}^{\\ast }M_{21}-M_{22}^{\\mathrm {T}}Y_{2}^{\\ast }M_{22}<0, $ where $Y_{2}^{\\ast } & =Y_{2}+\\left[\\begin{array}[c]{cc}U_{2}+V_{2} & 0\\\\0 & 0_{n\\times n}\\end{array}\\right] >0,\\\\X_{2}^{\\ast } & =X_{2}+\\left[\\begin{array}[c]{cc}0_{n\\times n} & 0\\\\0 & V_{2}\\end{array}\\right] >0.",
"$ Proof.",
"This lemma is a little different from the original result in [4] and thus a simple proof will be provided for completeness (also for the purpose of further using).",
"Choose a more general LKF candidate as [4] (where we have assumed without loss of generality that $\\mu =0$ ) $W_{2}\\left( x_{t}\\right) = & \\int _{t-a}^{t}x^{\\mathrm {T}}\\left( s\\right)U_{2}x\\left( s\\right) \\mathrm {d}s+\\int _{t-b}^{t}x^{\\mathrm {T}}\\left(s\\right) V_{2}x\\left( s\\right) \\mathrm {d}s,\\nonumber \\\\& +\\int _{t-c}^{t}\\left[\\begin{array}[c]{c}x\\left( s\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] ^{\\mathrm {T}}X_{2}\\left[\\begin{array}[c]{c}x\\left( s\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] \\mathrm {d}s,\\nonumber \\\\& +\\int _{t-b}^{t-c}\\left[\\begin{array}[c]{c}x\\left( s+c\\right) \\\\x\\left( s\\right)\\end{array}\\right] ^{\\mathrm {T}}Y_{2}\\left[\\begin{array}[c]{c}x\\left( s+c\\right) \\\\x\\left( s\\right)\\end{array}\\right] \\mathrm {d}s,\\nonumber $ where $c=b-a$ .",
"It can be verified that $& \\int _{t-a}^{t}x^{\\mathrm {T}}\\left( s\\right) U_{2}x\\left( s\\right)\\mathrm {d}s+\\int _{t-b}^{t}x^{\\mathrm {T}}\\left( s\\right) V_{2}x\\left(s\\right) \\mathrm {d}s\\\\& =\\int _{t-a}^{t}x^{\\mathrm {T}}\\left( s\\right) \\left( U_{2}+V_{2}\\right)x\\left( s\\right) \\mathrm {d}s+\\int _{t-b}^{t-a}x^{\\mathrm {T}}\\left( s\\right)V_{2}x\\left( s\\right) \\mathrm {d}s\\\\& =\\int _{t-b}^{t-c}x^{\\mathrm {T}}\\left( s+c\\right) \\left( U_{2}+V_{2}\\right) x\\left( s+c\\right) \\mathrm {d}s+\\int _{t-c}^{t}x^{\\mathrm {T}}\\left( s-a\\right) V_{2}x\\left( s-a\\right) \\mathrm {d}s,$ from which it follows that $W_{2}\\left( x_{t}\\right) = & \\int _{t-c}^{t}\\left[\\begin{array}[c]{c}x\\left( s\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] ^{\\mathrm {T}}X_{2}^{\\ast }\\left[\\begin{array}[c]{c}x\\left( s\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] \\mathrm {d}s\\nonumber \\\\& +\\int _{t-b}^{t-c}\\left[\\begin{array}[c]{c}x\\left( s+c\\right) \\\\x\\left( s\\right)\\end{array}\\right] ^{\\mathrm {T}}Y_{2}^{\\ast }\\left[\\begin{array}[c]{c}x\\left( s+c\\right) \\\\x\\left( s\\right)\\end{array}\\right] \\mathrm {d}s, $ whose time-derivative can be evaluated as $\\dot{W}_{2}\\left( x_{t}\\right) = & \\left[\\begin{array}[c]{c}x\\left( t\\right) \\\\x\\left( t-a\\right)\\end{array}\\right] ^{\\mathrm {T}}X_{2}^{\\ast }\\left[\\begin{array}[c]{c}x\\left( t\\right) \\\\x\\left( t-a\\right)\\end{array}\\right] -\\left[\\begin{array}[c]{c}x\\left( t-c\\right) \\\\x\\left( t-b\\right)\\end{array}\\right] ^{\\mathrm {T}}X_{2}^{\\ast }\\left[\\begin{array}[c]{c}x\\left( t-c\\right) \\\\x\\left( t-b\\right)\\end{array}\\right] \\nonumber \\\\& +\\left[\\begin{array}[c]{c}x\\left( t\\right) \\\\x\\left( t-c\\right)\\end{array}\\right] ^{\\mathrm {T}}Y_{2}^{\\ast }\\left[\\begin{array}[c]{c}x\\left( t\\right) \\\\x\\left( t-c\\right)\\end{array}\\right] -\\left[\\begin{array}[c]{c}x\\left( t-a\\right) \\\\x\\left( t-b\\right)\\end{array}\\right] ^{\\mathrm {T}}Y_{2}^{\\ast }\\left[\\begin{array}[c]{c}x\\left( t-a\\right) \\\\x\\left( t-b\\right)\\end{array}\\right] \\nonumber \\\\= & \\xi _{2}^{\\mathrm {T}}\\left( t\\right) \\mathit {\\Phi }_{2}\\xi _{2}\\left(t\\right) , $ where $\\xi _{2}\\left( t\\right) =[x^{\\mathrm {T}}\\left( t-a\\right),x^{\\mathrm {T}}\\left( t-b\\right) ,x^{\\mathrm {T}}\\left( t-c\\right)]^{\\mathrm {T}}$ .",
"The result then follows again from the Lyapunov stability theorem [3].",
"The decision matrices $U_{2}$ and $V_{2}$ in (REF ) are in fact redundant, as shown in the following corollary.",
"Corollary 1 There exist four positive definite matrices $X_{2},Y_{2}\\in \\mathbf {R}^{2n\\times 2n},U_{2},V_{2}\\in \\mathbf {R}^{n\\times n}$ such that (REF ) is satisfied if and only if there exist two positive definite matrices $X_{2}^{\\ast },Y_{2}^{\\ast }\\in \\mathbf {R}^{2n\\times 2n}$ such that (REF ) is satisfied.",
"Proof.",
"If $X_{2}>0,Y_{2}>0,U_{2}>0,V_{2}>0,$ then it follows from (REF )-() that $X_{2}^{\\ast }>0,Y_{2}^{\\ast }>0.$ On the other hand, if $X_{2}^{\\ast }>0,Y_{2}^{\\ast }>0$ , we can always find $X_{2}>0,Y_{2}>0,U_{2}>0,V_{2}>0,$ satisfying (REF )-(), for example, $U_{2}=V_{2}=\\varepsilon I_{n},$ where $\\varepsilon >0$ is sufficiently small.",
"The proof is finished.",
"We then can state the following result which connects the result in this paper and the one in [4].",
"Proposition 4 Let $\\left( P_{2},Q_{2}\\right) ,\\left( \\overline{P}_{2},\\overline{Q}_{2}\\right) $ and $\\left( X_{2}^{\\ast },Y_{2}^{\\ast }\\right) $ be related with $P_{2} & =W_{2}^{\\mathrm {T}}X_{2}^{\\ast }W_{2},\\;Q_{2}=E_{2}^{\\mathrm {T}}Y_{2}^{\\ast }E_{2},\\\\X_{2}^{\\ast } & =\\overline{P}_{2},\\;Y_{2}^{\\ast }=E_{2}^{\\mathrm {T}}W_{2}^{\\mathrm {T}}\\overline{Q}_{2}W_{2}E_{2}.",
"$ Then $\\mathit {\\Omega }_{2}\\left( P_{2},Q_{2}\\right) $ and $\\mathit {\\Phi }_{2}\\left( X_{2}^{\\ast },Y_{2}^{\\ast }\\right) $ satisfy $\\mathit {\\Omega }_{2}\\left( P_{2},Q_{2}\\right) & =T_{3}^{\\mathrm {T}}E_{3}^{\\mathrm {T}}\\mathit {\\Phi }_{2}\\left( X_{2}^{\\ast },Y_{2}^{\\ast }\\right)E_{3}T_{3},\\\\\\mathit {\\Phi }_{2}\\left( X_{2}^{\\ast },Y_{2}^{\\ast }\\right) & =E_{3}^{\\mathrm {T}}\\overline{\\mathit {\\Omega }}_{2}\\left( \\overline{P}_{2},\\overline{Q}_{2}\\right) E_{3}.",
"$ Thus the result in Lemma REF [4] is a special case of Lemma REF and Theorem REF .",
"Proof.",
"Notice from (REF ) that $V_{2}\\left( x_{t+b}\\right) & =\\int _{t-b}^{t-a}X_{2}^{\\mathrm {T}}\\left(s+b\\right) P_{2}X_{2}\\left( s+b\\right) \\mathrm {d}s+\\int _{t-a}^{t}X_{2}^{\\mathrm {T}}\\left( s+b\\right) Q_{2}X_{2}\\left( s+b\\right)\\mathrm {d}s\\nonumber \\\\& =\\int _{t-b}^{t-a}\\left[\\begin{array}[c]{c}x\\left( s-b\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] ^{\\mathrm {T}}P_{2}\\left[\\begin{array}[c]{c}x\\left( s-b\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] \\mathrm {d}s+\\int _{t-a}^{t}\\left[\\begin{array}[c]{c}x\\left( s-b\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] ^{\\mathrm {T}}Q_{2}\\left[\\begin{array}[c]{c}x\\left( s-b\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] \\mathrm {d}s, $ and from (REF ) that $\\dot{V}_{2}\\left( x_{t+b}\\right) & =\\left[\\begin{array}[c]{c}X_{2}\\left( t\\right) \\\\x\\left( t-2a\\right)\\end{array}\\right] ^{\\mathrm {T}}\\mathit {\\Omega }_{2}\\left( P_{2},Q_{2}\\right) \\left[\\begin{array}[c]{c}X_{2}\\left( t\\right) \\\\x\\left( t-2a\\right)\\end{array}\\right] \\nonumber \\\\& =\\left[\\begin{array}[c]{c}x\\left( t-2b\\right) \\\\x\\left( t-a-b\\right) \\\\x\\left( t-2a\\right)\\end{array}\\right] ^{\\mathrm {T}}\\mathit {\\Omega }_{2}\\left( P_{2},Q_{2}\\right) \\left[\\begin{array}[c]{c}x\\left( t-2b\\right) \\\\x\\left( t-a-b\\right) \\\\x\\left( t-2a\\right)\\end{array}\\right] .",
"$ On the other hand, we get from (REF ) that $W_{2}\\left( x_{t}\\right) & =\\int _{t-\\left( b-a\\right) }^{t}\\left[\\begin{array}[c]{c}x\\left( s\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] ^{\\mathrm {T}}X_{2}^{\\ast }\\left[\\begin{array}[c]{c}x\\left( s\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] \\mathrm {d}s+\\int _{t-b}^{t-\\left( b-a\\right) }\\left[\\begin{array}[c]{c}x\\left( s+b-a\\right) \\\\x\\left( s\\right)\\end{array}\\right] ^{\\mathrm {T}}Y_{2}^{\\ast }\\left[\\begin{array}[c]{c}x\\left( s+b-a\\right) \\\\x\\left( s\\right)\\end{array}\\right] \\mathrm {d}s\\\\& =\\int _{t-b}^{t-a}\\left[\\begin{array}[c]{c}x\\left( s+a\\right) \\\\x\\left( s\\right)\\end{array}\\right] ^{\\mathrm {T}}X_{2}^{\\ast }\\left[\\begin{array}[c]{c}x\\left( s+a\\right) \\\\x\\left( s\\right)\\end{array}\\right] \\mathrm {d}s+\\int _{t-a}^{t}\\left[\\begin{array}[c]{c}x\\left( s\\right) \\\\x\\left( s-b+a\\right)\\end{array}\\right] ^{\\mathrm {T}}Y_{2}^{\\ast }\\left[\\begin{array}[c]{c}x\\left( s\\right) \\\\x\\left( s-b+a\\right)\\end{array}\\right] \\mathrm {d}s,$ from which we have $W_{2}\\left( x_{t-a}\\right) & =\\int _{t-b}^{t-a}\\left[\\begin{array}[c]{c}x\\left( s\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] ^{\\mathrm {T}}X_{2}^{\\ast }\\left[\\begin{array}[c]{c}x\\left( s\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] \\mathrm {d}s+\\int _{t-a}^{t}\\left[\\begin{array}[c]{c}x\\left( s-a\\right) \\\\x\\left( s-b\\right)\\end{array}\\right] ^{\\mathrm {T}}Y_{2}^{\\ast }\\left[\\begin{array}[c]{c}x\\left( s-a\\right) \\\\x\\left( s-b\\right)\\end{array}\\right] \\mathrm {d}s\\nonumber \\\\& =\\int _{t-b}^{t-a}\\left[\\begin{array}[c]{c}x\\left( s-b\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] ^{\\mathrm {T}}W_{2}^{\\mathrm {T}}X_{2}^{\\ast }W_{2}\\left[\\begin{array}[c]{c}x\\left( s-b\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] \\mathrm {d}s+\\int _{t-a}^{t}\\left[\\begin{array}[c]{c}x\\left( s-b\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] ^{\\mathrm {T}}E_{2}^{\\mathrm {T}}Y_{2}^{\\ast }E_{2}\\left[\\begin{array}[c]{c}x\\left( s-b\\right) \\\\x\\left( s-a\\right)\\end{array}\\right] \\mathrm {d}s. $ Moreover, from (REF ) we obtain $\\dot{W}_{2}\\left( x_{t-a}\\right) & =\\left[\\begin{array}[c]{c}x\\left( t-2a\\right) \\\\x\\left( t-a-b\\right) \\\\x\\left( t-b\\right)\\end{array}\\right] ^{\\mathrm {T}}\\mathit {\\Phi }_{2}\\left( X_{2}^{\\ast },Y_{2}^{\\ast }\\right) \\left[\\begin{array}[c]{c}x\\left( t-2a\\right) \\\\x\\left( t-a-b\\right) \\\\x\\left( t-b\\right)\\end{array}\\right] \\nonumber \\\\& =\\left[\\begin{array}[c]{c}x\\left( t-2b\\right) \\\\x\\left( t-a-b\\right) \\\\x\\left( t-2a\\right)\\end{array}\\right] ^{\\mathrm {T}}T_{3}^{\\mathrm {T}}E_{3}^{\\mathrm {T}}\\mathit {\\Phi }_{2}\\left( X_{2}^{\\ast },Y_{2}^{\\ast }\\right) E_{3}T_{3}\\left[\\begin{array}[c]{c}x\\left( t-2b\\right) \\\\x\\left( t-a-b\\right) \\\\x\\left( t-2a\\right)\\end{array}\\right] .",
"$ Thus, by comparing (REF ) and (REF ) with (REF ) and (REF ) respectively, if (REF ) is satisfied, we obtain (REF ).",
"The relation () and () can be proven in a similar way."
],
[
"Numerical Examples",
"We consider the linear difference equation (REF ) with $A\\left( \\alpha \\right) =\\left[\\begin{array}[c]{cc}-0.4 & -0.3\\\\0.1+\\alpha & 0.15\\end{array}\\right] ,\\;B\\left( \\beta \\right) =\\left[\\begin{array}[c]{cc}0.1 & 0.25\\\\-0.9 & -0.1+\\beta \\end{array}\\right] ,$ where $\\alpha ,\\beta \\in \\mathbf {R}$ are free parameters [23].",
"We look for the pair $(\\alpha ,\\beta )$ such that system (REF ) is strongly stable.",
"By a linear search technique, the regions of $(\\alpha ,\\beta )$ obtained by different methods are plotted in Fig.",
"REF .",
"One can verify that the obtained region $(\\alpha ,\\beta )$ by Theorem REF with $k=2$ coincides with the exact region of stability obtained in [23].",
"This indicates that $k=2$ is already very efficient.",
"Actually, thousands of numerical examples show that $k=2$ in Theorem REF can lead to necessary and sufficient stability conditions.",
"Thus, the advantage of Theorem REF over Lemma REF is that the size of the LMI has been reduced significantly, especially, for large $n$ .",
"We now treat $\\alpha $ and $\\beta $ as uncertainties (which might be time-varying) and solve the robust stability problem, particularly, we want to find the maximal value of $r>0$ (denoted by $r^{\\ast }$ ) such that the system (REF ) is strongly stable for all $\\alpha \\in \\left[ -r,r\\right] $ and $\\beta \\in \\left[ -r,r\\right] .$ To this end, we rewrite $A\\left(\\alpha \\right) =A+\\Delta A$ and $B\\left( \\beta \\right) =B+\\Delta B,$ where $A & =\\left[\\begin{array}[c]{cc}-0.4 & -0.3\\\\0.1 & 0.15\\end{array}\\right] ,\\; \\Delta A=\\left[\\begin{array}[c]{cc}0 & 0\\\\\\alpha & 0\\end{array}\\right] ,\\\\B & =\\left[\\begin{array}[c]{cc}0.1 & 0.25\\\\-0.9 & -0.1\\end{array}\\right] ,\\; \\Delta B=\\left[\\begin{array}[c]{cc}0 & 0\\\\0 & \\beta \\end{array}\\right] .$ It can be verified that $\\left( \\Delta B,\\Delta A\\right) $ satisfies (REF ) where $F=[\\frac{\\beta }{r},\\frac{\\alpha }{r}]$ and $E_{0}=\\left[\\begin{array}[c]{c}0\\\\1\\end{array}\\right] ,\\;B_{0}=\\left[\\begin{array}[c]{cc}0 & r\\\\0 & 0\\end{array}\\right] ,\\;A_{0}=\\left[\\begin{array}[c]{cc}0 & 0\\\\r & 0\\end{array}\\right] .$ We clearly have $F^{\\mathrm {T}}F\\le I_{2}.$ Then, by applying Theorem REF for different $k$ and applying a linear search technique on $r$ , we can get $r_{\\ast }\\left( k\\right) .$ It is found that $r_{\\ast }\\left(1\\right) =0.4979$ and $r_{\\ast }\\left( 2\\right) =r_{\\ast }\\left( 3\\right)=0.5001.$ Denote the square $\\square _{k}=\\lbrace \\left( \\alpha ,\\beta \\right):\\alpha \\in \\left[ -r_{\\ast }\\left( k\\right) ,r_{\\ast }\\left( k\\right)\\right] ,\\beta \\in \\left[ -r_{\\ast }\\left( k\\right) ,r_{\\ast }\\left(k\\right) \\right] \\rbrace .$ It follows that $\\square _{1}$ is very close to $\\square _{2}$ which is recorded in Fig.",
"REF .",
"We can see that the square $\\square _{2}$ turns to be the maximal square that can be included in the region where the system is strongly stable for fixed $\\left( \\alpha ,\\beta \\right) .$ This indicates that Theorem REF can even provide necessary and sufficient conditions for robust strong stability for this example."
],
[
"Conclusion",
"This note established a necessary and sufficient condition for guaranteeing strong stability of linear difference equations with two delays.",
"The most important advantage of the proposed method is that the coefficients of the linear difference equation appear as linear functions in the proposed conditions, which helps to deal the robust stability analysis problem.",
"The relationships among the proposed condition and the existing ones were revealed by establishing a time-domain interpretation of the proposed LMI condition."
],
[
"A1: A Proof of Lemma ",
"Notice that $\\rho \\left( \\Delta _{\\theta }\\right) <1,\\forall \\theta \\in \\mathbf {R}$ , is equivalent to that $\\Delta _{0}$ is Schur stable and $0 & \\ne \\left|\\Delta _{\\theta }^{\\mathrm {H}}\\otimes \\Delta _{\\theta }-I_{n}\\otimes I_{n}\\right|\\nonumber \\\\& =\\left|A^{\\mathrm {T}}\\otimes B\\mathrm {e}^{-\\mathrm {j}\\theta }+B^{\\mathrm {T}}\\otimes A\\mathrm {e}^{\\mathrm {j}\\theta }+\\left( A^{\\mathrm {T}}\\otimes A+B^{\\mathrm {T}}\\otimes B-I_{n}\\otimes I_{n}\\right) \\right|\\nonumber \\\\& =\\mathrm {e}^{-n^{2}\\mathrm {j}\\theta }\\left|A^{\\mathrm {T}}\\otimes B+B^{\\mathrm {T}}\\otimes A\\mathrm {e}^{-2\\mathrm {j}\\theta }+\\left(A^{\\mathrm {T}}\\otimes A+B^{\\mathrm {T}}\\otimes B-I_{n}\\otimes I_{n}\\right)\\mathrm {e}^{-\\mathrm {j}\\theta }\\right|\\nonumber \\\\& =\\mathrm {e}^{-n^{2}\\mathrm {j}\\theta }\\left|\\mathcal {C}_{0}\\left(\\mathrm {e}^{\\mathrm {j}\\theta }I_{2n^{2}}-\\mathcal {A}_{0}\\right) ^{-1}\\mathcal {B}_{0}+\\mathcal {D}_{0}\\right|\\nonumber \\\\& =\\mathrm {e}^{-n^{2}\\mathrm {j}\\theta }\\left|G_{0}\\left( \\mathrm {e}^{\\mathrm {j}\\theta }\\right) \\right|,\\; \\forall \\theta \\in \\mathbf {R},$ where $G_{0}\\left( s\\right) =\\mathcal {C}_{0}\\left( sI_{2n^{2}}-\\mathcal {A}_{0}\\right) ^{-1}\\mathcal {B}_{0}+\\mathcal {D}_{0}$ with $\\mathcal {A}_{0} & =\\left[\\begin{array}[c]{cc}0_{n^{2}\\times n^{2}} & I_{n^{2}}\\\\0_{n^{2}\\times n^{2}} & 0_{n^{2}\\times n^{2}}\\end{array}\\right] ,\\ \\mathcal {B}_{0}=\\left[\\begin{array}[c]{c}0_{n^{2}\\times n^{2}}\\\\I_{n^{2}}\\end{array}\\right] ,\\\\\\mathcal {C}_{0} & =\\left[\\begin{array}[c]{cc}B^{\\mathrm {T}}\\otimes A & A^{\\mathrm {T}}\\otimes A+B^{\\mathrm {T}}\\otimes B-I_{n}\\otimes I_{n}\\end{array}\\right] ,\\\\\\mathcal {D}_{0} & =A^{\\mathrm {T}}\\otimes B.$ The condition (REF ) is also equivalent to $0 & >-G_{0}^{\\mathrm {H}}\\left( \\mathrm {e}^{\\mathrm {j}\\theta }\\right)G_{0}\\left( \\mathrm {e}^{\\mathrm {j}\\theta }\\right) \\\\& =\\left[\\begin{array}[c]{c}\\left( \\mathrm {e}^{\\mathrm {j}\\theta }I_{2n^{2}}-\\mathcal {A}_{0}\\right)^{-1}\\mathcal {B}_{0}\\\\I_{n^{2}}\\end{array}\\right] ^{\\mathrm {H}}M_{0}\\left[\\begin{array}[c]{c}\\left( \\mathrm {e}^{\\mathrm {j}\\theta }I_{2n^{2}}-\\mathcal {A}_{0}\\right)^{-1}\\mathcal {B}_{0}\\\\I_{n^{2}}\\end{array}\\right]$ where $\\theta \\in \\mathbf {R}$ and $M_{0}=-\\left[\\begin{array}[c]{cc}\\mathcal {C}_{0} & \\mathcal {D}_{0}\\end{array}\\right] ^{\\mathrm {T}}\\left[\\begin{array}[c]{cc}\\mathcal {C}_{0} & \\mathcal {D}_{0}\\end{array}\\right] .$ Thus, by the YKP lemma (Lemma REF ), this is equivalent to the existence of a symmetric matrix $P\\in \\mathbf {R}^{2n^{2}\\times 2n^{2}}$ such that $\\left[\\begin{array}[c]{cc}\\mathcal {A}_{0}^{\\mathrm {T}}P\\mathcal {A}_{0}-P & \\mathcal {A}_{0}^{\\mathrm {T}}P\\mathcal {B}_{0}\\\\\\mathcal {B}_{0}^{\\mathrm {T}}P\\mathcal {A}_{0} & \\mathcal {B}_{0}^{\\mathrm {T}}P\\mathcal {B}_{0}\\end{array}\\right] +M_{0}<0.$ Let $P$ be partitioned as $P=\\left[\\begin{array}[c]{cc}P_{1} & P_{3}\\\\P_{3}^{\\mathrm {T}} & -P_{2}\\end{array}\\right] ,$ where $P_{i},i=1,2,3,$ are $n^{2}\\times n^{2}$ matrices with $P_{i},i=1,2$ being symmetric.",
"Then, in view of the special structures of $\\left(\\mathcal {A}_{0},\\mathcal {B}_{0}\\right) ,$ (REF ) is equivalent to the LMI $\\left[\\begin{array}[c]{ccc}-P_{1} & -P_{3} & 0\\\\-P_{3}^{\\mathrm {T}} & P_{1}+P_{2} & P_{3}\\\\0 & P_{3}^{\\mathrm {T}} & -P_{2}\\end{array}\\right] +M_{0}<0.$ Applying the congruent transformation $T=\\left[\\begin{array}[c]{ccc}I_{n^{2}} & 0 & 0\\\\0 & 0 & I_{n^{2}}\\\\0 & I_{n^{2}} & 0\\end{array}\\right] ,$ on the LMI (REF ) gives (REF ).",
"The proof is finished by noting that $\\Delta _{0}$ is Schur stable if and only if (REF ) is satisfied.",
"For two matrices $A\\in \\mathbf {R}^{n\\times n},B\\in \\mathbf {R}^{n\\times n},$ the shuffle product (power) is defined as [8] $A^{\\left[ i\\right] }B^{\\left[ j\\right] }=\\sum \\limits _{\\begin{array}{c}i_{1}+i_{2}+\\cdots +i_{s}=i\\\\j_{1}+j_{2}+\\cdots +j_{s}=j\\end{array}}A^{i_{1}}B^{j_{1}}A^{i_{2}}B^{i_{2}}\\cdots A^{i_{s}}B^{j_{s}},$ where $\\left( i,j\\right) $ is a pair of nonnegative integers, and $i_{k},j_{k}\\ge 0,k=1,2,\\ldots ,s.$ For example, $A^{\\left[ 1\\right] }B^{\\left[ 2\\right] }=AB^{2}+BAB+B^{2}A.$ There are several simple properties of the shuffle product.",
"For example, [8] $A^{\\left[ i\\right] }B^{\\left[ j\\right] } & =B^{\\left[ j\\right]}A^{\\left[ i\\right] },\\nonumber \\\\A^{\\left[ i\\right] }B^{\\left[ 0\\right] } & =A^{i},\\;A^{\\left[ 0\\right]}B^{\\left[ j\\right] }=B^{j},\\nonumber \\\\A^{\\left[ i\\right] }B^{\\left[ j\\right] } & =A\\left( A^{\\left[i-1\\right] }B^{\\left[ j\\right] }\\right) +B\\left( A^{\\left[ i\\right]}B^{\\left[ j-1\\right] }\\right) \\nonumber \\\\& =\\left( A^{\\left[ i-1\\right] }B^{\\left[ j\\right] }\\right) A+\\left(A^{\\left[ i\\right] }B^{\\left[ j-1\\right] }\\right) B.",
"$ We next recall the so-called Yakubovich-Kalman-Popov (YKP) Lemma.",
"This lemma in the discrete-time setting is also known as the Szego-Kalman-Popov (SKP) Lemma [14], [22], [25].",
"Lemma 10 (YKP Lemma) Given $A\\in \\mathbf {R}^{n\\times n},B\\in \\mathbf {R}^{n\\times m}$ and $M\\in \\mathbf {R}^{\\left( n+m\\right) \\times \\left(n+m\\right) }$ with $\\left|\\mathrm {e}^{\\mathrm {j}\\theta }I_{n}-A\\right|\\ne 0,\\forall \\theta \\in \\mathbf {R}.$ Then $\\left[\\begin{array}[c]{c}\\left( \\mathrm {e}^{\\mathrm {j}\\theta }I_{n}-A\\right) ^{-1}B\\\\I_{m}\\end{array}\\right] ^{\\mathrm {H}}M\\left[\\begin{array}[c]{c}\\left( \\mathrm {e}^{\\mathrm {j}\\theta }I_{n}-A\\right) ^{-1}B\\\\I_{m}\\end{array}\\right] <0,$ holds for all $\\theta \\in \\mathbf {R}$ if and only if there exists a symmetric matrix $P\\in \\mathbf {R}^{n\\times n}$ such that $\\left[\\begin{array}[c]{cc}A^{\\mathrm {T}}PA-P & A^{\\mathrm {T}}PB\\\\B^{\\mathrm {T}}PA & B^{\\mathrm {T}}PB\\end{array}\\right] +M<0.$ The next lemma is adopted from [2].",
"Lemma 11 [2] If the inequality (REF ) is satisfied, namely, $\\sup _{\\theta \\in \\left[ 0,2\\pi \\right] }\\left\\lbrace \\rho \\left( \\Delta _{\\theta }\\right) \\right\\rbrace <1, $ then there exists a $k_{\\ast }\\in \\mathbf {N}^{+}$ such that $\\sup _{\\theta \\in \\left[ 0,2\\pi \\right] }\\left\\lbrace \\left\\Vert \\Delta _{\\theta }^{k}\\right\\Vert \\right\\rbrace <1,\\; \\forall k\\ge k_{\\ast }.",
"$ We finally recall a well-known result that was frequently used in robust control literature.",
"Lemma 12 [7] Let $X$ and $Y$ be real matrices of appropriate dimensions.",
"For $Q>0$ the following inequality is satisfied $XY+Y^{\\mathrm {T}}X^{\\mathrm {T}}\\le XQX^{\\mathrm {T}}+Y^{\\mathrm {T}}Q^{-1}Y.$"
]
] | 1808.08540 |
[
[
"Inductive Learning of Answer Set Programs from Noisy Examples"
],
[
"Abstract In recent years, non-monotonic Inductive Logic Programming has received growing interest.",
"Specifically, several new learning frameworks and algorithms have been introduced for learning under the answer set semantics, allowing the learning of common-sense knowledge involving defaults and exceptions, which are essential aspects of human reasoning.",
"In this paper, we present a noise-tolerant generalisation of the learning from answer sets framework.",
"We evaluate our ILASP3 system, both on synthetic and on real datasets, represented in the new framework.",
"In particular, we show that on many of the datasets ILASP3 achieves a higher accuracy than other ILP systems that have previously been applied to the datasets, including a recently proposed differentiable learning framework."
],
[
"Introduction",
"The ultimate aim of cognitive systems is to achieve human-like intelligence.",
"As humans, we are capable of performing many cognitive activities such as learning from past experience, predicting outcomes of our actions based on what we have learned, and reasoning using our learned knowledge.",
"Each of these cognitive processes uses existing knowledge and generates new knowledge.",
"They are underpinned by our ability to perform inductive reasoning, one of our most important high-level cognitive functions.",
"Inductive reasoning is a complex process by which new knowledge is inferred from a series of observations in a way that can be transferred from past experiences to new situations.",
"When performing inductive reasoning, observations perceived through the environment are often noisy and the existing knowledge that we use during the reasoning process is also limited and incomplete.",
"The human inductive reasoning process is therefore capable of handling noise in the observations, reasoning with incomplete and defeasible knowledge, applying knowledge learned in one scenario to many other scenarios, and learning complex knowledge expressed in terms of rules, constraints and preferences that can be communicated to others.",
"To realise cognitive systems able to perform human-like inductive reasoning, Machine Learning (ML) solutions have to meet the above properties.",
"Research in ML has yielded approaches and systems that, although capable of identifying patterns in datasets consisting of millions of (noisy) data points, cannot express the learned knowledge in a form that could be understood by a human.",
"Moreover, their learned knowledge can only be used in exactly the scenario in which it was learned: for example, a system trained to play Go on a standard 19x19 board may not perform very well at Go played on a 20x20 board.",
"Lack of interpretability and transferability of the learned knowledge make these approaches far from human cognition.",
"On the other hand, Inductive Logic Programming (ILP [22]) has been shown to be suited for learning knowledge that can be understood by humans and applied to new scenarios.",
"Although approaches for performing ILP in the context of noisy examples have been presented in the literature (e.g.",
"[28], [20], [24]), many existing ILP systems can only learn knowledge expressed as definite logic programs, so they are not capable of learning common-sense knowledge involving defaults and exceptions, which are essential aspects of human reasoning.",
"This type of knowledge can be modelled using negation as failure.",
"Recently, ILP has been extended to enable learning programs containing negation as failure (e.g.",
"[26], [27]), and interpreted under the answer set semantics [8].",
"In particular, our recent results in inductive learning of answer set programs (ILASP, [14], [18]) have demonstrated the ability to support automated acquisition of complex knowledge structures in the language of Answer Set Programming (ASP).",
"The theoretical framework underpinning ILASP, called Learning from Answer Sets (LAS), enables the learning of constraints, preferences and non-deterministic concepts.",
"For instance, LAS can learn the concept that a coin may non-deterministically land on either heads or tails, but never both.",
"When learning, humans are also capable of disregarding information that does not fit the general pattern.",
"Any cognitive system that aims to mimic human-level learning should therefore be capable of learning in the presence of noisy data.",
"A realistic approach to cognitive knowledge acquisition is therefore the learning of knowledge that covers the majority of the examples, but which at the same time weights coverage against its complexity.",
"In this paper, we present a noise tolerant extension of our LAS framework, Learning from noisy answer sets ($ILP_{LOAS}^{noise}$ ) and show that our ILASP3 system is capable of learning complex knowledge from noisy data in an effective and scalable way.",
"A collection of datasets, ranging from synthetically generated to real datasets, is used to evaluate the performance of the system with respect to the percentage of noise in the examples and to compare it to existing ILP systems.",
"Specifically, we consider two classes of synthetically generated datasets, called Hamiltonian and Journey preferences, and show that ILASP3 is able in both cases to achieve a high accuracy (of well over 90%), even with 20% of the examples labelled incorrectly.",
"We also evaluate ILASP3 on datasets concerning learning event theories [10], sentence chunking [2], preference learning [9], [1] and the synthetic datasets of [7].",
"Our results show that in most cases the ability of ILASP3 to compute optimal solutions for a given learning task allows it to reach higher accuracy than the other systems, which do not guarantee the computation of an optimal solution.",
"Next, in Section , we review relevant background material.",
"Section introduces our new framework for learning ASP from noisy examples; Section discusses the ILASP algorithms; Sections and present an extensive evaluation of our ILASP3 system; and finally, we conclude with a discussion of related and future work."
],
[
"Background",
"We briefly introduce basic notions and terminologies used throughout the paper.",
"Given any atoms $\\mbox{$\\mathtt {h, h_1,\\ldots ,h_k, b_1,\\ldots ,b_n,c_1,\\ldots ,c_m}$}$ , a normal rule is of the form $\\mbox{$\\mathtt {h \\operatorname{\\mathtt {:-} }b_1,\\ldots , b_n, \\operatorname{\\;\\mathtt {not}\\;}c_{1},\\ldots ,}$}$ $\\mbox{$\\mathtt {\\operatorname{\\;\\mathtt {not}\\;}c_{m}}$}$ , where “$\\mathtt {not}$ ” is negation as failure, $h$ is the head of the rule and $\\mbox{$\\mathtt {b_1,\\ldots , b_n, \\operatorname{\\;\\mathtt {not}\\;}c_{1},\\ldots , \\operatorname{\\;\\mathtt {not}\\;}c_{m}}$}$ is the body of the rule.",
"For example, $\\mbox{$\\mathtt {fly(X) \\operatorname{\\mathtt {:-} }bird(X),\\operatorname{\\;\\mathtt {not}\\;}ab(X)}$}$ is a normal rule stating that any bird can fly, unless it is abnormal.",
"The negated condition $\\mbox{$\\mathtt {\\operatorname{\\;\\mathtt {not}\\;}ab(X)}$}$ is assumed to hold unless there is a way of proving $\\mbox{$\\mathtt {ab(X)}$}$ for some value of $X$ .",
"So, the normal rule essentially models that by default, birds can fly, unless there is a proof that the bird is abnormal.",
"ASP programs include three other types of rule: choice rules, hard and weak constraints.",
"A choice rule is of the form $\\mbox{$\\mathtt {l\\lbrace h_{1},\\ldots , h_{k}\\rbrace u\\operatorname{\\mathtt {:-} }b_1,\\ldots , b_n,}$}$ $\\mbox{$\\mathtt {\\operatorname{\\;\\mathtt {not}\\;}c_{1},\\ldots , \\operatorname{\\;\\mathtt {not}\\;}c_{m}}$}$ , where $\\mbox{$\\mathtt {l}$}$ and $\\mbox{$\\mathtt {u}$}$ are integers and $\\mbox{$\\mathtt {l\\lbrace h_{1}, \\ldots ,h_{k}\\rbrace u}$}$ is called an aggregate.",
"A hard constraint is of the form $\\mbox{$\\mathtt {\\operatorname{\\mathtt {:-} }b_1,\\ldots , b_n,\\operatorname{\\;\\mathtt {not}\\;}c_{1},\\ldots ,\\operatorname{\\;\\mathtt {not}\\;}c_{m}}$}$ and a weak constraint is of the form $\\mbox{$\\mathtt {:\\sim b_1,\\ldots ,b_n, \\operatorname{\\;\\mathtt {not}\\;}c_1,\\ldots ,}$}$ $\\mbox{$\\mathtt {\\operatorname{\\;\\mathtt {not}\\;}c_m.",
"[w@l,t_1,\\ldots , t_k]}$}$ where $\\mbox{$\\mathtt {w}$}$ and $\\mbox{$\\mathtt {l}$}$ are terms specifying weight and priority level, and $\\mbox{$\\mathtt {t_1,\\ldots ,t_k}$}$ are terms.",
"The Herbrand Base of an ASP program $P$ , denoted $HB_P$ , is the set of ground (variable free) atoms that can be formed from predicates and constants in $P$ .",
"Subsets of $HB_P$ are called (Herbrand) interpretations of $P$ .",
"The semantics of ASP programs $P$ are defined in terms of answer sets – a specialFor a formal definition of answer sets of the programs in this paper see [17].",
"subset of interpretations of $P$ , denoted as $AS(P)$ , that satisfy every rule in $P$ .",
"Given an answer set $A$ , a ground normal or choice rule is satisfied if the head is satisfied by $A$ whenever all positive atoms and none of the negated atoms of the body are in $A$ , that is when the body is satisfied.",
"A ground aggregate $\\mbox{$\\mathtt {l\\lbrace h_{1}, \\ldots ,h_{k}\\rbrace u}$}$ is satisfied by an interpretation $I$ iff $\\mbox{$\\mathtt {l}$}\\le |I\\cap \\lbrace \\mbox{$\\mathtt {h_{1}, \\ldots ,h_{k}}$}\\rbrace |\\le \\mbox{$\\mathtt {u}$}$ .",
"So, informally, a ground choice rule is satisfied by an answer set $A$ if whenever its body is satisfied by an answer set $A$ , a number between $\\mbox{$\\mathtt {l}$}$ and $\\mbox{$\\mathtt {u}$}$ (inclusive) of the atoms in the aggregate are also in $A$ .",
"A ground constraint is satisfied when its body is not satisfied.",
"A constraint therefore has the effect of eliminating all answer sets that satisfy its body.",
"Weak constraints do not affect what is, or is not, an answer set of a program $P$ .",
"Instead, they create an ordering $\\succ _{P}$ over $AS(P)$ specifying which answer sets are “preferred” to others.",
"Informally, at each priority level $\\mbox{$\\mathtt {l}$}$ , satisfying weak constraints with level $l$ means discarding any answer set that does not minimise the sum of the weights of the ground weak constraints (with level $\\mbox{$\\mathtt {l}$}$ ) whose bodies are satisfied.",
"Higher levels are minimised first.",
"For example, the two weak constraints $\\mbox{$\\mathtt {:\\sim mode(L, walk), distance(L,D).",
"[D@2, L]}$}$ and $\\mbox{$\\mathtt {:\\sim cost(L, C).",
"[C@1, L]}$}$ express a preference ordering over alternative journeys.",
"The first constraint (at priority 2) expresses that the total walking distance (the sum of the distances of journey legs whose mode of transport is $\\mbox{$\\mathtt {walk}$}$ ) should be minimised, and the second constraint expresses that the total cost of the journey should be minimised.",
"As the first weak constraint has a higher priority level than the second, it is minimised first – so given a journey $j_1$ with a higher cost than another journey $j_2$ , $j_1$ is still preferred to $j_2$ so long as the walking distance of $j_1$ is lower than that of $j_2$ .",
"The set $ord(P)$ captures the ordering of interpretations induced by $P$ and generalises the $\\succ _{P}$ relation, so it not only includes $\\langle A_1, A_2, <\\rangle $ if $A_1 \\succ _{P} A_2$ , but includes tuples for each binary comparison operator ($<$ , $>$ , $=$ , $\\le $ , $\\ge $ and $\\ne $ ).",
"A partial interpretation, $e_{pi}$ , is a pair of sets of ground atoms $\\langle e^{inc}, e_{pi}^{exc}\\rangle $ .",
"An interpretation $I$ extends $e_{pi}$ iff $e_{pi}^{inc} \\subseteq I$ and $e_{pi}^{exc} \\cap I = \\emptyset $ .",
"Examples for learning come in two forms: context-dependent partial interpretations (CDPIs) and context-dependent ordering examples (CDOEs).",
"A CDPI example $e$ is a pair $\\langle e_{pi}, e_{ctx}\\rangle $ , where $e_{pi}$ is a partial interpretation and $e_{ctx}$ is a program with no weak constraints called the context of $e$ .",
"A program $P$ is said to bravely accept $e$ if there is at least one answer set $A$ of $P \\cup e_{ctx}$ that extends $e_{pi}$ – such an $A$ is called an accepting answer set of $P$ wrt $e$ .",
"Essentially, a CDPI says that the learned program, together with the context of $e$ , should bravelyA program $P$ bravely entails an atom $\\mbox{$\\mathtt {a}$}$ if there is at least one answer set of $P$ that contains $a$ .",
"entail all inclusion atoms and none of the exclusion atoms of $e$ .",
"CDPIs can be used for classification tasks, as they specify that given contexts should entail given conjunctions of atoms.",
"But as learned programs may have multiple answer sets, accepting a CDPI may require additional assumptions to be made.",
"A CDOE $o$ is a tuple $\\langle e_1, e_2, op\\rangle $ , where the first two elements are CDPIs and $op$ is a binary comparison operator.",
"A program $P$ is said to bravely respect $o$ if there is a pair of accepting answer sets, $A_1$ and $A_2$ , of $P$ wrt $e_1$ and $e_2$ , respectively, such that $\\langle A_1, A_2, op\\rangle \\in ord(P)$ .",
"$P$ is said to cautiously respect $o$ if for every pair, $A_1$ and $A_2$ , of accepting answer sets of $P$ (wrt $e_1$ and $e_2$ , respectively), $\\langle A_1, A_2, op\\rangle \\in ord(P)$ .",
"CDOEs enable preference learning as they specify which answer sets should be prefered to other answer sets.",
"An $ILP_{LOAS}^{context}$ task $T$ consists of an ASP background knowledge $B$ , a hypothesis space $S_M$ , labelled CDPIs, $E^{+}$ (positive examples) and $E^{-}$ (negative examples), and labelled CDOEs, $O^{b}$ (brave orderings) and $O^{c}$ (cautious orderings).",
"$S_M$ is the set of rules allowed in hypotheses.",
"A hypothesis $H\\subseteq S_M$ covers a positive (resp.",
"negative) example $e$ if $B\\cup H$ accepts (resp.",
"does not accept) $e$ .",
"$H$ covers a brave (resp.",
"cautious) ordering $o$ if $B\\cup H$ bravely (resp.",
"cautiously) respects $o$ .",
"$H$ is an inductive solution of $T$ iff $H$ covers every example in $T$ ."
],
[
"Learning Framework",
"This section presents the $ILP_{LOAS}^{noise}$ framework, which extends our previous (non-noisy) learning framework $ILP_{LOAS}^{context}$ ([18]), by allowing examples to be weighted context-dependent partial interpretations and weighted context-dependent ordering examples.",
"These are essentially the same as CDPIs and CDOEs, but weighted with a notion of penalty.",
"If a hypothesis does not cover an example, we say that it pays the penalty of that example.",
"Informally, penalties are used to calculate the cost associated with a hypothesis for not covering examples.",
"The cost function of a hypothesis $H$ is the sum over the penalties of all of the examples that are not covered by $H$ , augmented with the length of the hypothesis.",
"The goal of $ILP_{LOAS}^{noise}$ is to find a hypothesis that minimises the cost function over a given hypothesis space with respect to a given set of examples.",
"Definition 3.1 A weighted context-dependent partial interpretation $e$ is a tuple $\\langle e_{id}, e_{pen}, e_{cdpi}\\rangle $ , where $e_{id}$ is a constant, called the identifier of $e$ (unique to each example), $e_{pen}$ is the penalty of $e$ and $e_{cdpi}$ is a context-dependent partial interpretation.",
"The penalty $e_{pen}$ is either a positive integer, or $\\infty $ .",
"A program $P$ accepts $e$ iff it accepts $e_{cdpi}$ .",
"A weighted context-dependent ordering example $o$ is a tuple $\\langle o_{id}, o_{pen}, o_{ord}\\rangle $ , where $o_{id}$ is a constant, called the identifier of $o$ , $o_{pen}$ is the penalty of $o$ and $o_{ord}$ is a CDOE.",
"The penalty $o_{pen}$ is either a positive integer, or $\\infty $ .",
"A program $P$ bravely (resp.",
"cautiously) respects $o$ iff it bravely (resp.",
"cautiously) respects $o_{ord}$ .",
"In learning tasks without noise, each example must be covered by any inductive solution.",
"However, when examples are noisy (i.e.",
"they have a weight), inductive solutions need not cover every example, but they incur penalties for each uncovered example.",
"Multiple occurrences of the same CDPI example have different identifiers.",
"So hypotheses that do not cover that example will pay the penalty multiple times (for instance, if a CDPI occurs twice then a hypothesis will have to pay twice the penalty for not covering it).",
"In most of the learning tasks presented in this paper, all examples have the same penalty.",
"In some cases, however, penalties are used to simulate oversampling; for example, in tasks with far more positive examples than negative examples, we may choose to give the negative examples a higher weight – otherwise it is likely that the learned hypothesis will treat all negative examples as noisy.",
"Our learning task with noisy examples consists of an ASP background knowledge, weighted CDPI and CDOE examples and a hypothesis space,For details of hypothesis spaces in this paper, see https://www.doc.ic.ac.uk/~ml1909/ILASP/.",
"which defines the set of rules allowed to be used in constructing solutions of the task.",
"These tasks are supervised learning tasks, as all examples are labelled, as positive/negative, or with an operator in the case of the ordering examples.",
"Definition 3.2 An $ILP_{LOAS}^{noise}$ task $T$ is a tuple of the form $\\langle B, S_M, \\langle E^+,E^-, O^b, O^c\\rangle \\rangle $ , where $B$ is an ASP program, $S_M$ is a hypothesis space, $E^+$ and $E^-$ are sets of weighted CDPIs and $O^b$ and $O^c$ are sets of weighted CDOEs.",
"Given a hypothesis $H \\subseteq S_M$ , $uncov(H, T)$ is the set consisting of all examples $e \\in E^+$ (resp.",
"$E^-$ ) such that $B \\cup H$ does not accept (resp.",
"accepts) $e$ and all ordering examples $o \\in O^b$ (resp.",
"$O^c$ ) such that $B \\cup H$ does not bravely (resp.",
"cautiously) respect $o$ .",
"the penalty of $H$ , denoted as $pen(H, T)$ , is the sum $\\sum _{e \\in uncov(H, T)} e_{pen}$ .",
"the score of $H$ , denoted as $\\mathcal {S}(H, T)$ , is the sum $|H| + pen(H, T)$ .",
"$H$ is an inductive solution of $T$ (written $H \\in ILP_{LOAS}^{noise}(T)$ ) if and only if $\\mathcal {S}(H, T)$ is finite.",
"$H$ is an optimal inductive solution of $T$ (written $H \\in $ $^*ILP_{LOAS}^{noise}(T)$ ) if and only if $\\mathcal {S}(H, T)$ is finite and $\\nexists H^{\\prime } \\subseteq S_M$ such that $\\mathcal {S}(H, T) >\\mathcal {S}(H^{\\prime }, T)$ .",
"Examples with infinite penalty must be covered by any inductive solution, as any hypothesis that does not cover such an example will have an infinite score.",
"An $ILP_{LOAS}^{noise}$ task $T$ is said to be satisfiable if $ILP_{LOAS}^{noise}(T)$ is non-empty.",
"If $ILP_{LOAS}^{noise}(T)$ is empty, then $T$ is said to be unsatisfiable.",
"Theorem REF shows that for propositional tasks (where all hypothesis spaces, contexts and background knowledge are propositional) the complexity of $ILP_{LOAS}^{noise}$ is the same as $ILP_{LOAS}^{context}$ for the decision problems of verification – deciding if a given hypothesis is a solution of a given task – and satisfiability – deciding if a given task has any solutions – investigated in [19].",
"Theorem 3.1 $ $ Deciding verification for an arbitrary propositional $ILP_{LOAS}^{noise}$ task is $DP$ -complete Deciding satisfiability for an arbitrary propositional $ILP_{LOAS}^{noise}$ task is $\\Sigma ^P_2$ -complete Like its predecessor $ILP_{LOAS}^{context}$ , our new learning framework $ILP_{LOAS}^{noise}$ for noisy examples is capable of learning complex human-interpretable knowledge, containing defaults, non-determinism, exceptions and preferences.",
"The generalisation to allow penalties on the examples means that the new framework can be deployed in realistic settings where examples are not guaranteed to be correctly labelled.",
"Theorem REF shows that this generalisation does not come at any additional cost in terms of the computational complexity of important decision problems of the framework."
],
[
"The ILASP system",
"ILASP (Inductive Learning of Answer Set Programs, [14], [15], [16], [18]) is a collection of algorithms for solving LAS tasks.",
"The general idea behind the ILASP approach is to transform a learning task into a meta-level ASP program, which can be iteratively solved (extending the program in each iteration) until the optimal answer sets of the program correspond to solutions of the learning task.",
"Unlike many other ILP systems, such as [23], [26], [11], the ILASP algorithms are guaranteed to return an optimal solution of the input learning task (with respect to the cost function).",
"This can of course mean that ILASP may take longer to compute a solution than approximate systems (which are not guaranteed to return an optimal solution); however, as we demonstrate in Section , the hypotheses found by ILASP are often more accurate than those found by approximate systems.",
"Each version of ILASP has aimed to address scalability issues of the previous versions.ILASP1 [14] was a prototype implementation, with a major efficiency issue with respect to negative examples.",
"ILASP2 [16] addressed this issue by introducing a notion of an violating reason.",
"In each iteration, each answer set of the ILASP2 meta-level program $T_{meta}$ contains a representation of a hypothesis which covers every positive example and every brave ordering example.",
"An answer set representing a hypothesis that is not an inductive solution, contains a “reason” why at least one negative example or cautious ordering is not covered, which can be translated into an ASP representation that, when added to $T_{meta}$ , rules out any hypothesis that is not a solution for this reason.",
"This process is performed iteratively until no more violating reasons are detected.",
"For full details of violating reasons, see [16].",
"Both ILASP1 and ILASP2 scale poorly with respect to the number of examples, as the number of rules in the ground instantiation of their meta-level representation is proportional to the number of examples in the learning task.",
"As many examples may be similar, and thus covered by the same hypotheses, in non-noisy tasks (where all examples must be covered), it is often sufficient to consider a small subset of the examples called a relevant subset of the examples.",
"ILASP2i [18] uses this property to further improve the scalability of ILASP2.",
"It starts with an empty set of relevant examples $RE$ , and, at each iteration, it calls ILASP2 on a learning task using only the examples in $RE$ .",
"The hypothesis returned by ILASP2 is guaranteed to cover the current relevant examples, but is not necessarily an inductive solution of the original task.",
"So, if ILASP2 returns a hypothesis that does not cover at least one example, then an arbitrary uncovered example is added to $RE$ and the next iteration is started.",
"If no such example exists, then the hypothesis is returned as an optimal inductive solution of the original task.",
"[18] showed that ILASP2i can be up to two orders of magnitude faster than ILASP2 on tasks with 500 (noise-free) examples.",
"Both ILASP2 and ILASP2i can be extended to solve $ILP_{LOAS}^{noise}$ tasks; however, neither algorithm is well suited to solving tasks with a large number of noise examples with finite penalties.",
"ILASP2 does not scale with respect to the number of examples (regardless of whether examples have finite penalties), and the relevant example feature of ILASP2i is not equally effective when examples have penalties.",
"One reason for this is that many noisy examples may have to be added to the relevant example set before the cost of not covering a particular class of relevant examples is enough to outweigh the cost of learning an extra rule in the hypothesis.",
"The most recent ILASP algorithm, ILASP3, iteratively translates examples into hypothesis constraints – constraints on the structure of a hypothesis that are satisfied if and only if the hypothesis covers the example.",
"This leads to a much more compact meta-level program, defined in terms of these hypothesis constraints.",
"Once hypothesis constraints have been computed for one example $e$ , it is possible to compute the set of other examples (which have not yet been translated into hypothesis constraints) that are definitely not covered if $e$ is not covered.",
"This means that one relevant example can effectively have a much higher penalty than just the penalty for that example, meaning that the number of relevant examples that are needed in ILASP3 is often lower than those needed by ILASP2i."
],
[
"Evaluation of ILASP3 on synthetic datasets",
"In this section ILASP3 is evaluated on two synthetic datasets, the first of which is aimed at learning normal rules, choice rules and hard constraints, while the second is aimed at learning weak constraints.",
"The value of using synthetic datasets is that we can control the amount of noise and investigate how the accuracy and running time of ILASP3 varies with the amount of noise."
],
[
"Hamilton Graphs",
"In this experiment the task is to learn the definition of what it means for a graph to be Hamiltonian.",
"This concept was chosen as it requires learning a hypothesis that contains choice rules, recursive rules and hard constraints, and also contains negation as failure.",
"In these experiments, we show that ILASP3 could learn this hypothesis in the presence of noise, and we test how the running time of ILASP3 is affected by the number of examples and the number of incorrectly labeled examples.",
"For $n = 20, 40, \\ldots , 200$ , $n$ random graphs of size one to four were generated, half of which were Hamiltonian.",
"The graphs were labelled as either positive or negative, where positive indicates that the graph is Hamiltonian.",
"The correct ASP representation of Hamiltonian and a discussion of the representation of examples in this task is given in Appendix .",
"We ran three sets of experiments to evaluate ILASP3 on the Hamilton learning problem, with 5%, 10% and 20% of the examples being labelled incorrectly.",
"In each experiment, an equal number of Hamiltonian graphs and non-Hamiltonian graphs were randomly generated and 5%, 10% or 20% of the examples were chosen at random to be labelled incorrectly.",
"This set of examples were labelled as positive (resp.",
"negative) if the graph was not (resp.",
"was) Hamiltonian.",
"The remaining examples were labelled correctly (positive if the graph was Hamiltonian; negative if the graph was not Hamiltonian).",
"Figure REF shows the average accuracy and running time of ILASP3 with up to 200 example graphs.",
"Each experiment was repeated 50 times (with different randomly generated examples).",
"In each case, the accuracy was tested by generating a further 1,000 graphs and using the learned hypothesis to classify the graphs as either Hamiltonian or non-Hamiltonian (based on whether the hypothesis was satisfiable when combined with the representation of the graph).",
"The experiments show that on average ILASP3 is able to achieve a high accuracy (of well over 90%), even with 20% of the examples labelled incorrectly.",
"A larger percentage of noise means that ILASP3 requires a larger number of examples to achieve a high accuracy.",
"This is to be expected, as with few examples, the hypothesis is more likely to “overfit” to the noise, or pay the penalty of some non-noisy examples.",
"With large numbers of examples, it is more likely that ignoring some non-noisy examples would mean not covering others, and thus paying a larger penalty.",
"The computation time rises in all three graphs as the number of examples increases.",
"This is because larger numbers of examples are likely to require larger numbers of iterations of the ILASP3 algorithm.",
"Similarly, more noise is also likely to mean a larger number of iterations."
],
[
"Noisy Journey Preferences",
"The experiment in this section is a noisy extension of the journey preference learning setting used in [18], where the goal is to learn a user's preferences from a set of ordered pairs of journeys.",
"These experiments aim to show that ILASP3 is capable of preference learning in the presence of noise, and to test how the accuracy and running time of ILASP3 are affected by the numbers of examples and the proportion of examples which are incorrectly labelled.",
"In each experiment, we selected a “target hypothesis” consisting of between one and three weak constraints from a hypothesis space of weak constraints (discussed in Appendix ).",
"For each set of weak constraints, we then ran learning tasks with 0, 20, $\\ldots $ , 200 examples and with $5\\%$ , $10\\%$ and $20\\%$ noise.",
"The ordering examples for these learning tasks were generated from the weak constraints such that half of the (brave) ordering examples represented pairs of journeys $J_1$ and $J_2$ where $J_1$ was strictly preferred to $J_2$ , given the weak constraints, and the other half represented journeys such that $J_1$ was equally preferred to $J_2$ .",
"Depending on the level of noise, either $5\\%$ , $10\\%$ or $20\\%$ of the examples were given with the wrong operator ($>$ instead of $<$ and $\\ne $ instead of $=$ ).",
"Each ordering example was given a penalty of one.",
"Figure: (a) and (c) the average accuracy and (b) averagecomputation time of ILASP3 for the journey preference learning task, withvarying numbers of examples, and varying noise.",
"Each point in the graphs isan average over 50 different tasks.The results (Figure REF (a)) show that even with $20\\%$ noise, ILASP3 was able to learn hypotheses with an average accuracy of over 90%.",
"There was not much difference between ILASP3's accuracy with 5%, 10% and 20% noise, although the noisier tasks had a higher computation time (this is shown in Figure REF (b)), as in general ILASP3 requires more iterations on noisier tasks.",
"Even with 20% noise and 200 ordering examples, ILASP3 terminated in just over 60 seconds on average.",
"As the results for 20% noise were so close to the results for 5% noise, we ran a further set of examples to check that there was some limit to the level of noise where ILASP3 would no longer learn such an accurate hypothesis.If ILASP could achieve such a high accuracy, even with very high levels of noise, then this would indicate that the hypothesis space was too restrictive, and it was impossible to learn anything other than an accurate hypothesis.",
"In this second set of experiments, we tested ILASP3 with up to 40% noise, and investigated with 0, 10, $\\ldots $ , 100 examples.",
"With 40% noise, the accuracy was lower, but ILASP still achieved an average accuracy of just under 80%.",
"These experiments show that ILASP3 is able to accurately learn a set of weak constraints from examples of the orderings of answer sets given by these weak constraints, even when 20% of the orderings are incorrect.",
"Although the running time of ILASP3 is affected by the number of examples and the proportion of incorrectly labelled examples, ILASP3 is able to find an optimal solution in an average of 60 seconds, even with 200 ordering examples, 20% of which are incorrectly labelled.",
"Learning weak constraints is significant, as they can be used to represent user preferences.",
"In Sections REF and REF , we apply ILASP3 on two real preference learning datasets."
],
[
"Comparison with Other Systems",
"The experiments in this section use datasets that have previously been used to evaluate other ILP systems in the presence of noise.",
"Unlike ILASP3, none of the systems we compare with aim to find optimal solutions.",
"The aim of this set of experiments is therefore to test whether finding optimal solutions leads to any gain in accuracy over systems which may return sub-optimal solutions."
],
[
"CAVIAR Dataset",
"In this experiment ILASP3 was tested on the recent CAVIAR dataset that has been used to evaluate the OLED [10] system, which is an extension of the XHAIL [26] algorithm, for learning Event Calculus [12] theories.",
"The dataset contains data gathered from a video stream.",
"Information such as the positions of people has been extracted from the stream, and humans have annotated the data to specify when any two people are interacting.",
"Specifically, we consider a task from [10], in which the aim is to learn rules to define initiating and terminating conditions for two people meeting.",
"In the evaluation of the OLED system, examples were generated for every pair of consecutive timepoints $t$ and $t+1$ .",
"Each example is a pair $\\langle N\\cup A_t, A_{t+1}\\rangle $ , where $N$ is the “narrative” at time $t$ (a collection of information about the people in the video stream, such as their location and direction), and $A_i$ is the “annotation” at time $i$ (exactly which pairs of people in the video have been labelled as meeting).",
"This is very simple to express using context-dependent examples.",
"The context of an example is simply the narrative and annotation of time $t$ together with a set of constraints that enforce that the meetings at time $t$ are exactly those in the annotation.",
"The aim of this experiment is to compare ILASP3 to OLED, which was specifically designed to solve this kind of task efficiently.",
"We aimed to discover whether ILASP3 is able to find better quality hypotheses than OLED (in terms of the $F_1$ measure used to evaluate the hypotheses found by OLED), and whether ILASP3's guarantee of finding an optimal solution comes at a cost in terms of running time.",
"In total there are 24,530 consecutive pairs in the dataset.We used the data from users.iit.demokritos.gr/~nkatz/OLED-data/caviar.json.tar.gz We performed ten-fold cross validation by randomly partitioning the dataset.",
"As there were only twenty-two timepoints where the group of people meeting was different to the timepoint before, these examples were given a high penalty (of 100).",
"Effectively this is the same as oversampling this class of examples.",
"If all examples had been given a penalty of one, then ILASP3 would have likely learned the empty hypothesis, as the twenty-two examples in a task of many thousands of examples would likely be treated as noise.",
"We compare ILASP3 to OLED on the measures of precision, recall and the $F_1$ score.Let $tp$ , $tn$ , $fp$ , $fn$ represent the number of true positives, true negatives, false positives and false negatives achieved by a classifier on some test data.",
"The precision of the classifier (on this test data) is equal to $\\frac{tp}{tp + fp}$ and the recall is equal to $\\frac{tp}{tp + fn}$ .",
"The $F_1$ score is equal to $\\frac{2 \\times precision\\times recall}{precision + recall}$ .",
"ILASP3 achieved a precision of 0.832 and a recall of 0.853, giving an $F_1$ score of 0.842, compared with OLED's precision of 0.678 and recall of 0.953, with an average $F_1$ score of 0.792.",
"ILASP3's average running time was significantly higher at 576.3s compared with OLED's 107s.",
"This is explained by the fact that the OLED system computes hypotheses through theory revision, iteratively processing examples in sequence to continuously revise its hypothesis.",
"This means that, unlike ILASP3, OLED is not guaranteed to find an optimal solution of a learning task.",
"We note several key differences between our experiments and those reported in [10].",
"Firstly, to reduce the number of irrelevant answer sets (which lead to slow computation), we constrained the hypothesis space stating that rules for $\\mbox{$\\mathtt {terminatedAt(meeting(V1, V2), T)}$}$ had to contain $\\mbox{$\\mathtt {holdsAt(meeting(V1, V2), T)}$}$ in the body, which ensures that a fluent can only be terminated if it is currently happening.",
"Similarly, any rule for $\\mbox{$\\mathtt {initiatedAt(meeting(V1, V2), T)}$}$ had to contain $\\mbox{$\\mathtt {\\operatorname{\\;\\mathtt {not}\\;}holdsAt(meeting(V1, V2), T)}$}$ in the body.",
"OLED does not employ this constraint, but when processing an example pair of time points, only considers learning a new rule for $\\mbox{$\\mathtt {initiatedAt}$}$ , for example, if two people are meeting at time $t+1$ , but not at time $t$ .",
"The second difference in our experiment is that ILASP3 enumerates the hypothesis space in full.",
"As the hypothesis space in this task is potentially very large, several “common sense” constraints were enforced on the hypothesis space; for instance, two people cannot be both close to and far away from each other at the same time (rules with both conditions in the body were not generated).",
"In total, the hypothesis space contained 3,370 rules.",
"OLED does not enumerate the hypothesis space in full, but uses an approach similar to XHAIL, and derives a “bottom clause” from the background knowledge and the example.",
"In most cases (unless there is noise in the narrative, suggesting that two people are both close to and far away from each other) OLED will therefore only consider rules that respect the “common sense” constraints, as other rules would not be derivable.",
"This experiment has shown that, at least on this dataset, ILASP3's guarantee of finding an optimal solution can lead to better quality hypotheses than those found by OLED; however, this quality comes at a cost, as ILASP3's running time is significantly higher than OLED's."
],
[
"Sentence Chunking",
"In [11], the Inspire system was evaluated on a sentence chunking [32] dataset [2].",
"The task in this setting is to learn to split a sentence into short phrases called chunks.",
"For instance, according to the dataset [2], the sentence “Thai opposition party to boycott general election.” should be split into the three chunks “Thai opposition party”, “to boycott” and “general election”.",
"[11] describe how to transform each sentence into a set of facts consisting of part of speach (POS) tags.",
"We use each of these sets of facts as the context of a context dependent example.",
"In Inspire (which is a brave induction system), the facts are all put into the background knowledge.",
"The task is to learn a predicate $\\mbox{$\\mathtt {split/1}$}$ , which expresses where sentences should be split.",
"Inspire does not guarantee finding an optimal solution.",
"The hypothesis can be suboptimal for three reasons: firstly, the abductive phase may find an abductive solution which leads to a suboptimal inductive solution; secondly, Inspire's pruning may remove some hypotheses from the hypothesis space; and finally, Inspire was set to interrupt the inductive phase after 1,800 seconds, returning the most optimal hypothesis found so far.",
"In these experiments, we aimed to show that ILASP3's guarantee of finding an optimal solution leads to a better quality hypotheses than Inspire's approximations, and if so, whether ILASP3's running time was higher Inspire's timeout of 1,800s.",
"Note that the Inspire tasks in [11] group the multiple $\\mbox{$\\mathtt {split}$}$ examples for a chunk into a single example (using a $\\mbox{$\\mathtt {goodchunk}$}$ predicate); for example, the background knowledge may contain a rule $\\mbox{$\\mathtt {goodchunk(1) \\operatorname{\\mathtt {:-} }split(1), \\operatorname{\\;\\mathtt {not}\\;}split(2), \\operatorname{\\;\\mathtt {not}\\;}split(3), split(4)}$}$ expressing that there is a chunk between words one and four of a sentence.",
"It is noted in [11] that this increased performance.",
"This is because there is no benefit in covering some of the $\\mbox{$\\mathtt {split}$}$ atoms that make up a chunk, as hypotheses are tested over full chunks rather than splits.",
"In our framework, we represent this directly with no need for the $\\mbox{$\\mathtt {goodchunk}$}$ rules, with the individual split atoms being inclusions and exclusions in the partial interpretation of the example and the penalty being on the full example.",
"In our learning task, the example corresponding to the rule for $\\mbox{$\\mathtt {goodchunk(1)}$}$ would have the partial interpretation $\\langle \\lbrace $ ;,$ $ $\\mathtt {}$split(1);split(4)$\\rbrace ,\\lbrace $ ;,$ $ $\\mathtt {}$split(2);split(3)$\\rbrace \\rangle $ .",
"In [11], eleven-fold cross validation was performed on five different datasets, with 100 and 500 examples.",
"As Inspire has a parameter which determines how aggressive the pruning should be, [11] present several $F_1$ scores, for different values of this parameter.",
"Each entry for Inspire in Table REF is Inspire's best $F_1$ score over all pruning parameters.",
"Table: F 1 F_1 scores for Inspire andILASP3 and ILASP3's average running time on the sentence chunking tasks.Inspire approximates the optimal inductive solution of the task and has a timeout of 1,800s on the inductive phase – in contrast, ILASP3 terminated in less than 1,800 seconds on every task.",
"ILASP3 achieved a higher average $F_1$ score than Inspire on every one of the ten tasks.",
"This shows that computing the optimal inductive solution of a task can lead to a better quality hypothesis than approximating the optimal solution.",
"Note that for four out of the five datasets, Inspire performs better with 100 examples than with 500 examples.",
"A possible explanation for this is that with more examples, Inspire does not get as close to the optimal solution as it does with fewer examples, thus leading to a lower $F_1$ score on the test data.",
"With 500 examples, ILASP3 does take longer to terminate than it does for 100 examples, but in four out of the five cases, ILASP's average $F_1$ score is higher, confirming the expected result that more data should tend to lead to a better hypothesis."
],
[
"Car Preference Learning",
"We tested ILASP3's ability to learn real user preferences with the car preference dataset from [1].",
"This dataset consists of responses from 60 different users, who were each asked to give their preferences about ten cars.",
"They were asked to order each (distinct) pair of cars, leading to 45 orderings.",
"The cars had four attributes, shown in Table REF (a).",
"Through this experiment, we aim to show that ILASP3 is capable of learning real user preferences, encoded as weak constraints.",
"There is not much work on applying ILP systems to preference learning, but one such work [25] applied the Aleph [31] system to the car preference dataset.",
"Aleph is not guaranteed to find an optimal solution,Aleph processes the examples sequentially, and searches for the best clause to add in each iteration (in terms of coverage).",
"Although each iteration adds the best clause, this may still lead to a sub-optimal hypothesis overall.",
"and is only capable of learning rules (and not of learning weak constraints).",
"[25] used Aleph to learn rules defining the predicate $\\mbox{$\\mathtt {bt/2}$}$ , where $\\mbox{$\\mathtt {bt(c_1, c_2)}$}$ represents that $\\mbox{$\\mathtt {c_1}$}$ is preferred to $\\mbox{$\\mathtt {c_2}$}$ .",
"For comparison, we present the results of [25] on this dataset.",
"Table: (a) The attributes of the car preference dataset, along with thepossible range of values for each attribute.",
"The integer next to each valueis how that value is represented in the data.",
"(b) The accuracy results ofILASP3 compared with the three methods in on thecar preference dataset.Our initial experiment was based on an experiment in [25], where the Aleph [31] system was used to learn the preferences of each user in the dataset and compared with support vector machines (SVM) and decision trees (DT).",
"Ten-fold cross validation was performed for each of the 60 users on the 45 orderings.",
"In each fold, 10% of the orderings were omitted from the training data and used to test the learned hypothesis.",
"The flaw in this approach is that the omitted examples will often be implied by the rest of the examples (i.e.",
"if $a\\prec b$ and $b \\prec c$ are given as examples it does not make sense to omit $a \\prec c$ ).",
"For this reason, we also experimented with leaving out all the examples for a single car in each fold (i.e.",
"every pair that contains that car), and using these examples to test (again leading to ten folds).",
"This new task corresponds to learning preferences from a complete ordering of nine cars, and testing the preferences on an unseen car.",
"Table REF (b) shows the accuracy of the approach in [25] and ILASP3 accuracy on the two versions of the experiment.",
"The easier task (with 10% of the orderings omitted) is denoted as experiment A in the table, and the harder task is denoted as experiment B.",
"In fact, even on the harder version of the task, ILASP3 performs better than the approaches in [25] perform on the easier version of the task.",
"In one fold for the first user (in experiment A), ILASP3 learns the following weak constraints: $\\mbox{$\\mathtt {:\\sim fuel(2).",
"[1@4]}$}$ ; $\\mbox{$\\mathtt {:\\sim body(1), transmission(2).",
"[-1@3]}$}$ ; $\\mbox{$\\mathtt {:\\sim engine\\_cap(V0).",
"[V0@2, V0]}$}$ ; $\\mbox{$\\mathtt {:\\sim body(1).",
"[-1@1]}$}$ .",
"This hypothesis corresponds to the following set of prioritised preferences (ordered from most to least important): the user (1) prefers hybrid cars to non-hybrid cars; (2) likes automatic sedans; (3) would like to minimise the engine capacity of the car; and (4) prefers sedans to SUVs.",
"The noise in this experiment comes from the fact that some of the answers given by participants in the survey may contradict other answers.",
"Some participants gave inconsistent orderings (breaking transitivity) meaning that there is no set of weak constraints that covers every ordering example.",
"The results of these experiments have shown that ILASP3 is able to learn hypotheses that accurately represent real user preferences, even in the presence of noise.",
"On average, ILASP3 learns a hypothesis with a higher accuracy than the hypothesis learned by [25].",
"This could be for two reasons: (1) the fact that Aleph might return a sub-optimal inductive solution; or (2), the representation of hypotheses as weak constraints allows for preferences to be expressed that cannot be expressed using the definite search space in [25]."
],
[
"SUSHI Preference Learning",
"Another dataset for preference learning is the SUSHI dataset [9].",
"The dataset is comprised of peoples' preference orderings over different types of sushi.",
"The purpose of these experiments is to show that ILASP3 is capable of learning weak constraints that accurately capture real user preferences.",
"[25] also tested their approach on these datasets, and we compare ILASP3's accuracy with their results in order to test whether the optimal solution found by ILASP3 is more accurate than their solutions.",
"Each type of sushi has several attributes, described in Table REF (a).",
"There is a mix of categorical and continuous attributes.",
"In the language bias for these experiments, the categorical attributes are used as constants, whereas the continuous attributes are variables that can be used as the weight of the weak constraint.",
"This allows weak constraints to express that the continuous attributes should be minimised or maximised.",
"The dataset was constructed from a survey in which people were asked to order ten different types of sushi.",
"This ordering leads to 45 ordering examples per person.",
"This experiment is based on a similar experiment in [25].",
"For each of the first 60 people in the dataset ten-fold cross validation was performed, omitting 10% of the orderings in each fold.",
"This experiment suffers from the same flaw as Experiment A on the car dataset in that some of the omitted examples may be implied by the training examples, but we give the results for a comparison to [25].",
"As shown in Table REF (b), ILASP3 achieved an average accuracy of 0.84, comparing favourably to each result from [25].",
"Although in this experiment each participant gave a consistent total ordering of the ten types of sushi, it might be the case that there is no hypothesis in the hypothesis space that covers all of the examples.",
"This could be the case when we are not modelling a feature of the sushi that the participant considers to be important.",
"For this reason, we treated this as a noisy learning setting, and used ILASP3 to maximise the coverage of the examples.",
"This experiment has shown that ILASP3 is capable of learning weak constraints that accurately capture users' preferences, and that ILASP3's approach of finding an optimal hypothesis comprising of weak constraints is (on average) more accurate than the approach of [25], which finds a (potentially sub-optimal) set of definite clauses."
],
[
"Comparison to $\\delta $ ILP",
"Although the work in this paper concerns learning ASP programs from noisy examples, work has been done in the area of extending definite clause learning to handle noisy examples (for example, [28], [31], [24]).",
"In [7], it was claimed that ILP approaches are unable “to handle noisy, erroneous, or ambiguous data” and that “If the positive or negative examples contain any mislabelled data, [ILP approaches] will not be able to learn the intended rule”.",
"The experiments in this section aim to refute this claim.",
"To learn from noisy data, [7] introduced the $\\delta $ ILP algorithm, based on artificial neural networks.",
"They demonstrated that $\\delta $ ILP is able to achieve a high accuracy even with a large proportion of noise in the examples.",
"[7] evaluated $\\delta $ ILP on six synthetic datasets, with noise ranging from 0% to 90%.",
"In these experiments, we investigated the accuracy of ILASP3 on five of these six datasets.The authors of [7] provided us with the training and test data for these five problems.",
"In the original experiments, examples were atoms, and noise corresponded to swapping positive and negative examples.",
"In each of the $ILP_{LOAS}^{noise}$ tasks, we ensured that the hypothesis space was such that for each $H\\subseteq S_M$ , $B\\cup H\\cup e_{ctx}$ was stratified for each example $e$ .",
"This allowed atomic examples to be represented as (positive) partial interpretations – a positive example $\\mbox{$\\mathtt {e}$}$ was represented as a partial interpretation $\\langle \\lbrace $ ;,$ $ $\\mathtt {}$e$\\rbrace ,\\emptyset \\rangle $ , and a negative example $\\mbox{$\\mathtt {e}$}$ was represented as a partial interpretation $\\langle \\emptyset ,\\lbrace $ ;,$ $ $\\mathtt {}$e$\\rbrace \\rangle $ .",
"Due to the differences in language biases used by ILASP and $\\delta $ ILP, the hypothesis spaces of the two systems are not equivalent.",
"Due to the imbalance of positive and negative examples in many of the tasks, we weight the positive examples at $w \\times |E^-|/(|E^+|+|E^-|)$ and the negative examples at $w \\times |E^+|/(|E^+|+|E^-|)$ , where in this experiment $w$ is 100.",
"The weight for each example class (positive or negative) is equal to $w$ multiplied by the proportion of the whole set of examples which are in the other class.",
"This “corrects” any imbalance between positive and negative examples (i.e.",
"the penalty for not covering a proportion of the positive examples is the same as the penalty for not covering the same proportion of negative examples).",
"The constant $w$ can be thought of as the difference in importance between the hypothesis length and the number of examples covered.",
"In these experiments we chose 100, as it is high enough to ensure that coverage is considered far more important than hypothesis length.",
"Figure: A comparison of δ\\delta ILPand ILASP3 on five datasets from .",
"Specifically thegraphs correspond to the (a) predecessor, (b) less than, (c) member, (d)connected and (e) undirected edge experiments in .In each graph, the X and Y axes represent the noise level and mean squarederror, respectively.Figure REF shows the mean squared error of the two systems, where the results for $\\delta $ ILP are taken from [7].",
"In most tasks ILASP3 achieves similar results to $\\delta $ ILP when the noise is in the range of 0% to 40%.",
"However, at the other end of the scale (with more than 50% noise), there are some tasks where ILASP3 finds hypotheses with close to 100% error, where $\\delta $ ILP's error is much lower (less than 20% in the “member” problem).",
"We argue that when the noisy examples outnumber the correctly labelled examples, the learner should start learning the negation of the target hypothesis; for instance, in the case of “less than”, ILASP3 correctly learned the “greater than or equal to” relation.",
"The ideal outcome of these kinds of experiments, where the proportion of noise is varied, is that the learner achieves close to 0% error until around 50% noise and close to 100% error thereafter.",
"This is roughly what seems to happen for ILASP3 in the “predecessor”, “less than”, “member” and “undirected edge” experiments.",
"In “predecessor”, the graph is less symmetric, with the “crossover” from low to high error occurring later.",
"This is likely because the hypothesis for “not predecessor” is longer than the hypothesis for “predecessor”.",
"The failure of $\\delta $ ILP to get close to 100% error in many of the tasks (for example in “member”, $\\delta $ ILP has an error of less than 20% with the noise level at 90%) may indicate that the negation of the target concept is not representable given the language bias used by $\\delta $ ILP in these experiments, instead of $\\delta $ ILP being particularly robust to noise.",
"In some cases (such as “member”), this is likely because the negation of the concept requires negation as failure (which is not supported by $\\delta $ ILP), but in others such as “less than”, the negation of the concept is expressible without negation as failure.",
"These results show that, on the ILP problems investigated by [7], ILASP3 is certainly robust to noise, thus refuting their claim that ILP systems cannot handle noise."
],
[
"Related Work",
"Several other ILP systems use ASP solvers in the search for hypotheses.",
"For example, [4] presented an early system for learning action descriptions, where the search for inductive solutions is encoded in ASP.",
"Many of these systems, such as [4], [3], [5], [11] operate under a brave semantics – the learned program should have at least one answer set that satisfies some given properties (such as covering examples).",
"But our results on the generality of learning frameworks in [19] prove that there are ASP programs that can be learned by our framework and that cannot be learned by any of these systems.",
"For example, brave induction systems cannot learn hard or weak constraints, no matter what examples are given.",
"In a different line of research [30], [29], ASP solvers have also been used together with relational reinforcement learning (RRL).",
"[30] present an architecture that combines RRL with ASP-based inference.",
"RRL and decision tree induction were used to identify a set of candidate axioms.",
"The candidates deemed to have the highest likelihood are then represented in an ASP program, which is used for planning.",
"Early approaches to relational learning (e.g.",
"[13], [21], [6]) were able to learn definite rules from noisy data.",
"[21] presented an ILP system based on theory revision, where hypotheses are only modified if the modification leads to the additional coverage of more than one example.",
"In practice, however, it is possible that given a large enough set of examples, two noisy examples may be covered by exactly the same class of hypotheses.",
"Under the $ILP_{LOAS}^{noise}$ approach, the penalty for not covering a set of examples which forms a small proportion of examples is low, even if there are multiple examples in this set.",
"[6] introduces algorithms which learn from noisy examples, learning one clause at a time.",
"ILP systems which iteratively learn single clauses, removing covered positive examples after each iteration, are common when the target hypotheses are definite logic programs (with no negation), as the programs being learned are monotonic.",
"Learning non-monotonic ASP programs with negation (allowing for the learning of exceptions) requires a different approach ([26]).",
"This is because, due to the non-monotonicity of the learned programs, examples which are covered in one iteration may become uncovered when further rules are learned.",
"In order to search for good hypotheses, ILP systems often use a cost function, defined in terms of the coverage of the examples and the length of the hypothesis (e.g.",
"[31], [23], [5]).",
"When examples are noisy, this cost function is sometimes combined with a notion of maximum threshold, and the search is not for a hypothesis that minimises the cost function, but for a hypothesis that does not fail to cover more than a defined maximum threshold number of examples (e.g.",
"[31], [24], [3]).",
"In this way, once an acceptable hypothesis (i.e.",
"a hypothesis that covers a sufficient number of examples) is computed the system does not search for a better one.",
"As such, the computational task is simpler, and therefore the time needed to compute a hypothesis is shorter, but there may be other hypotheses which have a lower cost.",
"Furthermore, to guess the “correct” maximum threshold requires some idea of how much noise there is in the given set of examples.",
"For instance, one of the inputs to the HYPER/N [24] system is the proportion of noise in the examples.",
"When the proportion of noise is unknown, too small a threshold could result in the learning task being unsatisfiable, or in learning a hypothesis that overfits the data.",
"On the other hand, too high a threshold could result in poor accuracy, as the hypothesis may not cover many of the examples.",
"Our $ILP_{LOAS}^{noise}$ framework addresses the problem of computing optimal solutions (with respect to the cost function) and in doing so does not require knowledge a priori of the level of noise in the data.",
"Note that optimal hypotheses are not guaranteed to outperform other hypotheses on unseen data, but based on the evidence (i.e.",
"the training examples) they minimise the cost function, and so if the cost function is reasonable, they should be more likely to be correct.",
"This can be seen in the sentence chunking experiments, where we used ILASP with the same cost function as Inspire (which does not guarantee minimising the cost function).",
"In future work, we intend to explore alternative cost functions, and formalise what makes a cost function “reasonable” in a given learning setting."
],
[
"Conclusion",
"Learning interpretable knowledge is a key requirement for cognitive systems that are required to communicate with each other, or with humans.",
"Our research addresses the problem of learning ASP programs, which are capable of representing complex knowledge, such as defaults, exceptions and preferences.",
"In practice, cognitive systems are required to learn knowledge from noisy data sources, where there is no guarantee that all examples are perfectly labelled.",
"This paper has presented the $ILP_{LOAS}^{noise}$ framework for learning ASP from noisy examples and evaluated the ILASP3 system, designed to solve the learning tasks of this framework.",
"We used several synthetic datasets to show that ILASP3 can learn even in the presence of high proportions noisy examples.",
"We have also tested ILASP3's performance on several datasets used by other ILP systems The results of these experiments show that in most cases ILASP3 is able to learn with a higher accuracy than the other systems, which, unlike ILASP3, are not guaranteed to find optimal solutions of the tasks.",
"Although ILASP3 is a significant improvement on previous ILASP systems with respect to the running time on noisy tasks, some scalability issues remain, especially with the size of the hypothesis space.",
"Every ILASP system begins by computing the hypothesis space in full, which limits the feasible size of the hypothesis space.",
"In future work, we plan to design ILASP systems which do not begin by computing the hypothesis space in full.",
"We would like to thank the reviewers for their useful comments and suggestions.",
"Details of the Representations used in the Hamilton Graph and Journey Preference Experiments Hamiltonian Graphs The Hamilton graph learning tasks in Section REF were aimed at learning how to decide whether a graph is Hamiltonian of not.",
"The four node Hamiltonian graph $G$ in figure REF can be represented by the set of facts $F_{G}$ .",
"Figure: An example of a Hamiltonian Graph GG and its corresponding representation in ASP, F G F_{G}.To decide whether a graph is Hamiltonian or not, we can use the program $H$ below: reach(V0) :- in(1,V0).",
"reach(V1) :- in(V0,V1), reach(V0).",
"0 {in(V0,V1) } 1 :- edge(V0, V1).",
":- node(V0), not reach(V0).",
":- in(V0,V1), in(V0,V2), V1 != V2.",
"If for a graph $G$ and its corresponding set of facts $F_{G}$ , $G$ is Hamiltonian if and only if $F_{G}\\cup H$ is satisfiable.",
"In the $ILP_{LOAS}^{noise}$ tasks, we made use of (weighted) CDPI examples to represent graphs.",
"We did not use any background knowledge (i.e.",
"the background knowledge in each task was empty), and instead encoded the graphs in the contexts of examples such as in the positive CDPI example: $\\langle \\langle \\emptyset ,\\emptyset \\rangle ,\\lbrace \\mbox{$\\mathtt {node(1..4).",
"}$}\\;\\; \\mbox{$\\mathtt {edge(1,2).",
"}$}\\;\\;\\mbox{$\\mathtt {edge(2,3).",
"}$}\\;\\; \\mbox{$\\mathtt {edge(3,4).",
"}$}\\;\\; \\mbox{$\\mathtt {edge(4,1).",
"}$}\\rbrace \\rangle $ , which represents the graph $G$ .",
"Journey Preferences We now describe the structure of journeys in the experiments in Section REF .",
"A journey consists of a set of legs.",
"The attributes of journey legs in these experiments were: $\\mbox{$\\mathtt {mode}$}$ , which took one of the values $\\mbox{$\\mathtt {bus}$}$ , $\\mbox{$\\mathtt {car}$}$ , $\\mbox{$\\mathtt {walk}$}$ or $\\mbox{$\\mathtt {bicycle}$}$ ; $\\mbox{$\\mathtt {distance}$}$ , which took an integer value between $\\mbox{$\\mathtt {1}$}$ and $\\mbox{$\\mathtt {20,000}$}$ ; and $\\mbox{$\\mathtt {crime\\_rating}$}$ .",
"As the crime ratings were not readily available from the simulator, we used a randomly generated value between $\\mbox{$\\mathtt {1}$}$ and $\\mbox{$\\mathtt {5}$}$ for each journey leg.",
"In the experiments, we assumed that a user's preferences could be represented by a set of weak constraints based on the attributes of a leg.",
"$S_J$ denotes the set of possible weak constraints that we used in the experiments, each of which includes at most three literals (characterised by a mode bias which can be found at https://www.doc.ic.ac.uk/~ml1909/ILASP/).",
"Most of these literals capture the leg's attributes, e.g., $\\mbox{$\\mathtt {mode(L, bus)}$}$ or $\\mbox{$\\mathtt {crime\\_rating(L, R)}$}$ (if the attribute's values range over integers this is represented by a variable, otherwise each possible value is used as a constant).",
"For the crime rating ($\\mbox{$\\mathtt {crime\\_rating(L,R)}$}$ ), we also allow comparisons of the form $R > \\mbox{$\\mathtt {c}$}$ where $\\mbox{$\\mathtt {c}$}$ is an integer from 1 to 4.",
"The weight of each weak constraint is a variable representing the distance of the leg in the body of the weak constraint, or 1 and the priority is 1, 2 or 3.",
"One possible set of preferences is represented by the weak constraints $W^*$ .",
"$W^* = \\left\\lbrace \\begin{array}{l}\\mbox{$\\mathtt {:\\sim leg\\_mode(L, walk), leg\\_crime\\_rating(L, C), C > 3.",
"[1@3, L, C]}$}\\\\\\mbox{$\\mathtt {:\\sim leg\\_mode(L, car).",
"[1@2, L]}$}\\\\\\mbox{$\\mathtt {:\\sim leg\\_mode(L, walk), leg\\_distance(L, D).",
"[D@1, L, D]}$}\\\\\\end{array}\\right\\rbrace $ These preferences represent that the user's top priority is to avoid walking through areas with a high crime rating.",
"Second, the user would like to avoid driving, and finally, the user would like to minimise the total walking distance of the journey.",
"We now describe how to represent the journey preferences scenario in $ILP_{LOAS}^{noise}$ .",
"We assume that each journey is encoded as a set of attributes of the legs of the journey; for example the journey $\\lbrace $ ;,$ $ $\\mathtt {}$ distance(leg(1), 2000); distance(leg(2), 100); mode(leg(1), bus); mode(leg(2), walk)$\\rbrace $ has two legs; in the first leg, the person must take a bus for 2,000m and in the second, he/she must walk 100m.",
"Each of our learning tasks had an empty background knowledge.",
"Each positive example in our tasks was a weighted CDPI $\\langle e_{id}, \\infty ,\\langle \\langle \\emptyset ,\\emptyset \\rangle , J\\rangle \\rangle $ , where $J$ is the set of facts representing a journey.",
"The brave ordering examples were defined over pairs of the positive examples with appropriate ordering operators, and each with a penalty of 1.",
"Note that the positive examples are automatically satisfied as the (empty) background knowledge (combined with the context) already covers them.",
"Also, as the background knowledge together with each context has exactly one answer set, the notions of brave and cautious orderings coincide; hence, we do not need cautious ordering examples for this task.",
"Furthermore, since only weak constraints are being learned, the task also has no negative examples (a negative example would correspond to an invalid journey).",
"ILASP Flags Used in the Experiments ILASP3 has various optional features, used to improve the speed of the algorithm on different kinds of learning task.",
"Table REF shows the option flags that were used in the calls to ILASP in each experiment.",
"In addition to these “core” ILASP options, in all but the Hamilton and CAVIAR experiments, a flag was passed to run Clingo 5 with the option --opt-strat=usc,stratify.",
"Table: The flags that were passed to ILASP when runningthe experiments in this paper.",
"Proofs In this section, we prove Theorem REF , showing that $ILP_{LOAS}^{noise}$ shares the same computational complexity as $ILP_{LOAS}^{context}$ on the two decision problems of verification and satisfiability.",
"The proof relies on two propositions.",
"Proposition C.1 Deciding verification and satisfiability for a propositional $ILP_{LOAS}^{context}$ task both reduce polynomially to the same problem for a propositional $ILP_{LOAS}^{noise}$ task.",
"Let $T$ be the $ILP_{LOAS}^{context}$ task $\\langle B, S_M, \\langle E^+,E^-, O^b, O^c\\rangle \\rangle $ .",
"Consider the $ILP_{LOAS}^{noise}$ task $T^{\\prime }= \\langle B, S_M, \\langle E^+_2, E^-_2, O^b_2, O^c_2\\rangle \\rangle $ , where the examples are defined as follows: $E^+_2 = \\left\\lbrace \\langle e_{id}, \\infty , e\\rangle | e \\in E^+\\right\\rbrace $ $E^-_2 = \\left\\lbrace \\langle e_{id}, \\infty , e\\rangle | e \\in E^-\\right\\rbrace $ $O^b_2 = \\left\\lbrace \\langle o_{id}, \\infty , o\\rangle | o \\in O^b\\right\\rbrace $ $O^c_2 = \\left\\lbrace \\langle o_{id}, \\infty , o\\rangle | o \\in O^c\\right\\rbrace $ First note that $H \\in ILP_{LOAS}^{context}(T) \\Leftrightarrow \\mathcal {S}(H,T^{\\prime })$ is finite.",
"Hence $H \\in ILP_{LOAS}^{context}(T) \\Leftrightarrow H \\in ILP_{LOAS}^{noise}(T^{\\prime })$ .",
"So verification for $ILP_{LOAS}^{context}$ reduces to verification for $ILP_{LOAS}^{noise}$ .",
"As this also means that $ILP_{LOAS}^{context}(T) = \\emptyset \\Leftrightarrow ILP_{LOAS}^{noise}(T^{\\prime }) =\\emptyset $ , this also shows that satisfiability for $ILP_{LOAS}^{context}$ reduces to satisfiability for $ILP_{LOAS}^{noise}$ .",
"Proposition C.2 Deciding verification and satisfiability for a propositional $ILP_{LOAS}^{noise}$ task both reduce polynomially to the same problem for a propositional $ILP_{LOAS}^{context}$ task.",
"Let $T$ be the $ILP_{LOAS}^{noise}$ task $\\langle B, S_M, \\langle E^+, E^-, O^b, O^c\\rangle \\rangle $ .",
"Consider the $ILP_{LOAS}^{context}$ task $T^{\\prime } = \\langle B, S_M, \\langle E^+_2,E^-_2, O^b_2, O^c_2\\rangle \\rangle $ , where the examples are defined as follows: $E^+_2 = \\left\\lbrace \\langle e_{pi}, e_{ctx}\\rangle | \\langle e_{id}, e_{pen}, \\langle e_{pi}, e_{ctx}\\rangle \\rangle \\in E^+, e_{pen} =\\infty \\right\\rbrace $ $E^-_2 = \\left\\lbrace \\langle e_{pi}, e_{ctx}\\rangle | \\langle e_{id}, e_{pen}, \\langle e_{pi}, e_{ctx}\\rangle \\rangle \\in E^-, e_{pen} =\\infty \\right\\rbrace $ $O^b_2 = \\left\\lbrace o_{ord} | \\langle o_{id}, o_{pen},o_{ord}\\rangle \\in O^b, o_{pen} = \\infty \\right\\rbrace $ $O^c_2 = \\left\\lbrace o_{ord} | \\langle o_{id}, o_{pen},o_{ord}\\rangle \\in O^c, o_{pen} = \\infty \\right\\rbrace $ $\\forall H \\subseteq S_M$ , $H \\in ILP_{LOAS}^{context}(T^{\\prime })$ if and only if $H$ covers all examples in $T$ that have a finite penalty.",
"Hence, $ILP_{LOAS}^{context}(T^{\\prime }) = ILP_{LOAS}^{noise}(T)$ .",
"This means that both verification and satisfiability for $ILP_{LOAS}^{noise}$ reduce to verification and satisfiability for $ILP_{LOAS}^{context}$ (as $H \\in ILP_{LOAS}^{noise}(T)\\Leftrightarrow H \\in ILP_{LOAS}^{context}(T^{\\prime })$ and $ILP_{LOAS}^{noise}(T)=\\emptyset \\Leftrightarrow ILP_{LOAS}^{context}(T^{\\prime })=\\emptyset $ ).",
"We can now prove Theorem REF .",
"Theorem REF $ $ Deciding verification for an arbitrary propositional $ILP_{LOAS}^{noise}$ task is $DP$ -complete Deciding satisfiability for an arbitrary propositional $ILP_{LOAS}^{noise}$ task is $\\Sigma ^P_2$ -complete $ $ As Propositions REF and REF show polynomial reductions in both directions from this problem to deciding verification for an arbitrary propositional $ILP_{LOAS}^{context}$ task, it remains to show that the corresponding decision problem for $ILP_{LOAS}^{context}$ is $DP$ -complete.",
"This was shown in [19].",
"Similarly, as Propositions REF and REF show polynomial reductions in both directions from this problem to deciding satisfiability for an arbitrary propositional $ILP_{LOAS}^{context}$ task, it remains to show that the corresponding decision problem for $ILP_{LOAS}^{context}$ is $DP$ -complete.",
"This was shown in [19]."
],
[
"Hamiltonian Graphs",
"The Hamilton graph learning tasks in Section REF were aimed at learning how to decide whether a graph is Hamiltonian of not.",
"The four node Hamiltonian graph $G$ in figure REF can be represented by the set of facts $F_{G}$ .",
"Figure: An example of a Hamiltonian Graph GG and its corresponding representation in ASP, F G F_{G}.To decide whether a graph is Hamiltonian or not, we can use the program $H$ below: reach(V0) :- in(1,V0).",
"reach(V1) :- in(V0,V1), reach(V0).",
"0 {in(V0,V1) } 1 :- edge(V0, V1).",
":- node(V0), not reach(V0).",
":- in(V0,V1), in(V0,V2), V1 != V2.",
"If for a graph $G$ and its corresponding set of facts $F_{G}$ , $G$ is Hamiltonian if and only if $F_{G}\\cup H$ is satisfiable.",
"In the $ILP_{LOAS}^{noise}$ tasks, we made use of (weighted) CDPI examples to represent graphs.",
"We did not use any background knowledge (i.e.",
"the background knowledge in each task was empty), and instead encoded the graphs in the contexts of examples such as in the positive CDPI example: $\\langle \\langle \\emptyset ,\\emptyset \\rangle ,\\lbrace \\mbox{$\\mathtt {node(1..4).",
"}$}\\;\\; \\mbox{$\\mathtt {edge(1,2).",
"}$}\\;\\;\\mbox{$\\mathtt {edge(2,3).",
"}$}\\;\\; \\mbox{$\\mathtt {edge(3,4).",
"}$}\\;\\; \\mbox{$\\mathtt {edge(4,1).",
"}$}\\rbrace \\rangle $ , which represents the graph $G$ ."
],
[
"Journey Preferences",
"We now describe the structure of journeys in the experiments in Section REF .",
"A journey consists of a set of legs.",
"The attributes of journey legs in these experiments were: $\\mbox{$\\mathtt {mode}$}$ , which took one of the values $\\mbox{$\\mathtt {bus}$}$ , $\\mbox{$\\mathtt {car}$}$ , $\\mbox{$\\mathtt {walk}$}$ or $\\mbox{$\\mathtt {bicycle}$}$ ; $\\mbox{$\\mathtt {distance}$}$ , which took an integer value between $\\mbox{$\\mathtt {1}$}$ and $\\mbox{$\\mathtt {20,000}$}$ ; and $\\mbox{$\\mathtt {crime\\_rating}$}$ .",
"As the crime ratings were not readily available from the simulator, we used a randomly generated value between $\\mbox{$\\mathtt {1}$}$ and $\\mbox{$\\mathtt {5}$}$ for each journey leg.",
"In the experiments, we assumed that a user's preferences could be represented by a set of weak constraints based on the attributes of a leg.",
"$S_J$ denotes the set of possible weak constraints that we used in the experiments, each of which includes at most three literals (characterised by a mode bias which can be found at https://www.doc.ic.ac.uk/~ml1909/ILASP/).",
"Most of these literals capture the leg's attributes, e.g., $\\mbox{$\\mathtt {mode(L, bus)}$}$ or $\\mbox{$\\mathtt {crime\\_rating(L, R)}$}$ (if the attribute's values range over integers this is represented by a variable, otherwise each possible value is used as a constant).",
"For the crime rating ($\\mbox{$\\mathtt {crime\\_rating(L,R)}$}$ ), we also allow comparisons of the form $R > \\mbox{$\\mathtt {c}$}$ where $\\mbox{$\\mathtt {c}$}$ is an integer from 1 to 4.",
"The weight of each weak constraint is a variable representing the distance of the leg in the body of the weak constraint, or 1 and the priority is 1, 2 or 3.",
"One possible set of preferences is represented by the weak constraints $W^*$ .",
"$W^* = \\left\\lbrace \\begin{array}{l}\\mbox{$\\mathtt {:\\sim leg\\_mode(L, walk), leg\\_crime\\_rating(L, C), C > 3.",
"[1@3, L, C]}$}\\\\\\mbox{$\\mathtt {:\\sim leg\\_mode(L, car).",
"[1@2, L]}$}\\\\\\mbox{$\\mathtt {:\\sim leg\\_mode(L, walk), leg\\_distance(L, D).",
"[D@1, L, D]}$}\\\\\\end{array}\\right\\rbrace $ These preferences represent that the user's top priority is to avoid walking through areas with a high crime rating.",
"Second, the user would like to avoid driving, and finally, the user would like to minimise the total walking distance of the journey.",
"We now describe how to represent the journey preferences scenario in $ILP_{LOAS}^{noise}$ .",
"We assume that each journey is encoded as a set of attributes of the legs of the journey; for example the journey $\\lbrace $ ;,$ $ $\\mathtt {}$ distance(leg(1), 2000); distance(leg(2), 100); mode(leg(1), bus); mode(leg(2), walk)$\\rbrace $ has two legs; in the first leg, the person must take a bus for 2,000m and in the second, he/she must walk 100m.",
"Each of our learning tasks had an empty background knowledge.",
"Each positive example in our tasks was a weighted CDPI $\\langle e_{id}, \\infty ,\\langle \\langle \\emptyset ,\\emptyset \\rangle , J\\rangle \\rangle $ , where $J$ is the set of facts representing a journey.",
"The brave ordering examples were defined over pairs of the positive examples with appropriate ordering operators, and each with a penalty of 1.",
"Note that the positive examples are automatically satisfied as the (empty) background knowledge (combined with the context) already covers them.",
"Also, as the background knowledge together with each context has exactly one answer set, the notions of brave and cautious orderings coincide; hence, we do not need cautious ordering examples for this task.",
"Furthermore, since only weak constraints are being learned, the task also has no negative examples (a negative example would correspond to an invalid journey)."
],
[
"ILASP Flags Used in the Experiments",
"ILASP3 has various optional features, used to improve the speed of the algorithm on different kinds of learning task.",
"Table REF shows the option flags that were used in the calls to ILASP in each experiment.",
"In addition to these “core” ILASP options, in all but the Hamilton and CAVIAR experiments, a flag was passed to run Clingo 5 with the option --opt-strat=usc,stratify.",
"Table: The flags that were passed to ILASP when runningthe experiments in this paper."
],
[
"Proofs",
"In this section, we prove Theorem REF , showing that $ILP_{LOAS}^{noise}$ shares the same computational complexity as $ILP_{LOAS}^{context}$ on the two decision problems of verification and satisfiability.",
"The proof relies on two propositions.",
"Proposition C.1 Deciding verification and satisfiability for a propositional $ILP_{LOAS}^{context}$ task both reduce polynomially to the same problem for a propositional $ILP_{LOAS}^{noise}$ task.",
"Let $T$ be the $ILP_{LOAS}^{context}$ task $\\langle B, S_M, \\langle E^+,E^-, O^b, O^c\\rangle \\rangle $ .",
"Consider the $ILP_{LOAS}^{noise}$ task $T^{\\prime }= \\langle B, S_M, \\langle E^+_2, E^-_2, O^b_2, O^c_2\\rangle \\rangle $ , where the examples are defined as follows: $E^+_2 = \\left\\lbrace \\langle e_{id}, \\infty , e\\rangle | e \\in E^+\\right\\rbrace $ $E^-_2 = \\left\\lbrace \\langle e_{id}, \\infty , e\\rangle | e \\in E^-\\right\\rbrace $ $O^b_2 = \\left\\lbrace \\langle o_{id}, \\infty , o\\rangle | o \\in O^b\\right\\rbrace $ $O^c_2 = \\left\\lbrace \\langle o_{id}, \\infty , o\\rangle | o \\in O^c\\right\\rbrace $ First note that $H \\in ILP_{LOAS}^{context}(T) \\Leftrightarrow \\mathcal {S}(H,T^{\\prime })$ is finite.",
"Hence $H \\in ILP_{LOAS}^{context}(T) \\Leftrightarrow H \\in ILP_{LOAS}^{noise}(T^{\\prime })$ .",
"So verification for $ILP_{LOAS}^{context}$ reduces to verification for $ILP_{LOAS}^{noise}$ .",
"As this also means that $ILP_{LOAS}^{context}(T) = \\emptyset \\Leftrightarrow ILP_{LOAS}^{noise}(T^{\\prime }) =\\emptyset $ , this also shows that satisfiability for $ILP_{LOAS}^{context}$ reduces to satisfiability for $ILP_{LOAS}^{noise}$ .",
"Proposition C.2 Deciding verification and satisfiability for a propositional $ILP_{LOAS}^{noise}$ task both reduce polynomially to the same problem for a propositional $ILP_{LOAS}^{context}$ task.",
"Let $T$ be the $ILP_{LOAS}^{noise}$ task $\\langle B, S_M, \\langle E^+, E^-, O^b, O^c\\rangle \\rangle $ .",
"Consider the $ILP_{LOAS}^{context}$ task $T^{\\prime } = \\langle B, S_M, \\langle E^+_2,E^-_2, O^b_2, O^c_2\\rangle \\rangle $ , where the examples are defined as follows: $E^+_2 = \\left\\lbrace \\langle e_{pi}, e_{ctx}\\rangle | \\langle e_{id}, e_{pen}, \\langle e_{pi}, e_{ctx}\\rangle \\rangle \\in E^+, e_{pen} =\\infty \\right\\rbrace $ $E^-_2 = \\left\\lbrace \\langle e_{pi}, e_{ctx}\\rangle | \\langle e_{id}, e_{pen}, \\langle e_{pi}, e_{ctx}\\rangle \\rangle \\in E^-, e_{pen} =\\infty \\right\\rbrace $ $O^b_2 = \\left\\lbrace o_{ord} | \\langle o_{id}, o_{pen},o_{ord}\\rangle \\in O^b, o_{pen} = \\infty \\right\\rbrace $ $O^c_2 = \\left\\lbrace o_{ord} | \\langle o_{id}, o_{pen},o_{ord}\\rangle \\in O^c, o_{pen} = \\infty \\right\\rbrace $ $\\forall H \\subseteq S_M$ , $H \\in ILP_{LOAS}^{context}(T^{\\prime })$ if and only if $H$ covers all examples in $T$ that have a finite penalty.",
"Hence, $ILP_{LOAS}^{context}(T^{\\prime }) = ILP_{LOAS}^{noise}(T)$ .",
"This means that both verification and satisfiability for $ILP_{LOAS}^{noise}$ reduce to verification and satisfiability for $ILP_{LOAS}^{context}$ (as $H \\in ILP_{LOAS}^{noise}(T)\\Leftrightarrow H \\in ILP_{LOAS}^{context}(T^{\\prime })$ and $ILP_{LOAS}^{noise}(T)=\\emptyset \\Leftrightarrow ILP_{LOAS}^{context}(T^{\\prime })=\\emptyset $ ).",
"We can now prove Theorem REF .",
"Theorem REF $ $ Deciding verification for an arbitrary propositional $ILP_{LOAS}^{noise}$ task is $DP$ -complete Deciding satisfiability for an arbitrary propositional $ILP_{LOAS}^{noise}$ task is $\\Sigma ^P_2$ -complete $ $ As Propositions REF and REF show polynomial reductions in both directions from this problem to deciding verification for an arbitrary propositional $ILP_{LOAS}^{context}$ task, it remains to show that the corresponding decision problem for $ILP_{LOAS}^{context}$ is $DP$ -complete.",
"This was shown in [19].",
"Similarly, as Propositions REF and REF show polynomial reductions in both directions from this problem to deciding satisfiability for an arbitrary propositional $ILP_{LOAS}^{context}$ task, it remains to show that the corresponding decision problem for $ILP_{LOAS}^{context}$ is $DP$ -complete.",
"This was shown in [19]."
]
] | 1808.08441 |
[
[
"Dimitrov's question for the polynomials of degree 1,2,3,4,5,6"
],
[
"Abstract In 2002 Dimitar Dimitrov posted the problem of finding the optimal polynomials that provide the sharpness of Koebe Quarter Theorem for polynomials and asked whether Suffridge polynomials are optimal ones.",
"We disproved Dimitrov's conjecture for polynomials of degree 3,4,5 and 6.",
"For polynomials of degree 1 and 2 the conjecture is valid."
],
[
"Introduction",
"One of the fundamental results in the geometric complex analysis is the famous Koebe Quarter Theorem.",
"It states that for any function $f\\in \\mathcal {U}_n$ the image $f(\\mathbb {D})$ contains a disc of radius 1/4, whether $\\mathbb {D}=\\lbrace |z|<1\\rbrace $ and ${\\mathcal {U}_n}=\\lbrace f(z): f(0)=0, f^\\prime (0)=1,\\; f(z) \\;\\mbox{is univalent in}\\;\\mathbb {D}\\rbrace .$ The 1/4 bound is sharp as it is indicated by the Koebe function $K(z)=z/(1-z)^2.$ A natural question is whether the constant 1/4 can be improved for polynomial of specific degree.",
"Say, for polynomials of the first degree it is trivially 1; a simple computation demonstrates that for polynomials of degree 2 it is 1/2.",
"The task was formalized by Dimitrov [4] who posted the following problem For any $n\\in \\mathbb {Z}_+$ find the polynomial $p_n(z)\\in \\mathcal {U}_n$ , for which the infimum $\\inf \\lbrace |p_n(z)| : z = e^{it}, 0\\le t\\le 2\\pi \\rbrace $ is attained.",
"By the Koebe Quarter Theorem the above infimums are bounded from below by 1/4.",
"Córdova and Ruscheweyh [3] considered the Suffridge polynomials [7] $S_{n,j}(z) = \\sum ^n_{k=1}\\left(1 -\\frac{k-1}{n}\\right) \\frac{\\sin (\\pi jk /(n + 1))}{\\sin (\\pi j /(n + 1))}z^k.$ Note that $S_{n,j}(z)\\in \\mathcal {U}_n$ and $|S_{n,1}(-1)|=\\frac{1}{4}\\frac{n+1}{n}\\sec ^2\\frac{\\pi }{2(n+1)}\\rightarrow −1/4.$ Hence these polynomials solve the latter problem at least asymptotically.",
"Note that the value $\\frac{1}{4}\\frac{n+1}{n}\\sec ^2\\frac{\\pi }{2(n+1)}$ is the Koebe radius only for polynomials $S_{n,1}(z)$ of even degree.",
"For the polynomials of odd degree the quantity $\\inf \\lbrace |S_{n,1}(z)|:|z|=1\\rbrace $ is not achived at the point $z=-1,$ rather a different point $\\xi ,$ such that $S^\\prime _{n,1}(\\xi )=0.$ (see Fig 1).",
"Figure: NO_CAPTIONFigure: NO_CAPTION Fig 1: The image and fragment for $S_{3,1}(\\mathbb {D}).$ Note that for $n=3,$ $|S_{3,1}(-1)|\\approx 0.3905$ while the Koebe radius is $r_3\\approx 0.3849.$ For $n=5$ $|S_{5,1}(-1)|\\approx 0.3215$ while the Koebe radius is $r_5\\approx 0.3196.$ Note that $r_2=0.5,$ $r_4\\approx 0.3455,$ $r_6\\approx 0.3069.$ Dimitrov [4] asked a specific question about the Suffridge polynomial: Is it the extremal one for every fixed $N$?",
"Note that they are indeed extremal for $N=1,2.$ Below we prove that the answer is negative for $N=3,4,5,6.$"
],
[
"New extremal polynomials",
"Univalent polynomials are classical objects of complex analysis.",
"Perhaps, the first systematic approach was suggested by Alexander [1] who proved that the truncated sums of the Taylor series of the function $f(z)=\\log (1/(1-z))$ are univalent in $\\mathbb {D}$ polynomials.",
"Note that Alexander's paper contains many ideas that were not properly estimated at that time , c.f.",
"[5].",
"The subtlety of the situation well illustrates the fact that a necessary condition of univalency - the derivative does not vanish in $\\mathbb {D}$ - implies that the n-th coefficient of the polynomial of degree $n$ cannot exceed $1/n$ in absolute value.",
"This is perfectly fine with the logarithm function and awfully wrong with the Koebe function.",
"Thus, Suffridge polynomials can be treated as reasonable substitutions for the function $K(z).$ These polynomials are extremal in a way that they have the $n$ -th coefficient exactly $1/n.$ Thus, so far we have two families of extremal univalent polynomials in play - Alexander polynomials and Suffridge polynomials.",
"The main discovery of the current paper is a new extremal family of polynomials that seem to be univalent in $\\mathbb {D}$ and might be as important as the two mentioned above series.",
"Namely, the following polynomials were introduced in [8].",
"$P_N(z)=\\frac{1}{ { { U^{\\prime }_N}\\left({\\cos \\frac{\\pi }{ N+2 } } \\right) } }\\sum _{k=1}^N{ U^{\\prime }_{ N - k + 1 } }\\left(\\cos \\frac{\\pi }{ N+2 } \\right){ U_ { k - 1 } }\\left(\\cos \\frac{\\pi }{ N+2 } \\right)z^k,$ where ${ U_k}\\left(x \\right) $ is a family of Chebyshev polynomials of the second kind and ${ U^{\\prime }_k}\\left(x \\right) $ is a derivative.",
"One given below some examples: $P_1(z)=z,\\quad P_2(z)=z+\\frac{1}{2}z^2,\\quad $ $P_3(z)=z+\\frac{2}{\\sqrt{5}}z^2+\\frac{1}{2}\\left(1- \\frac{1}{\\sqrt{5}}\\right)z^3,\\quad P_4(z)=z+\\frac{7}{6}z^2+\\frac{2}{3}z^3+\\frac{1}{6}z^4,$ $P_5(z)=z+&{\\frac{8-40\\, \\left( \\cos \\left( \\pi /7 \\right) \\right) ^{2}+32\\,\\left( \\cos \\left( \\pi /7 \\right) \\right) ^{3}-24\\,\\cos \\left( \\pi /7\\right) }{40\\, \\left( \\cos \\left( \\pi /7 \\right) \\right) ^{3}-30\\,\\cos \\left( \\pi /7 \\right) -32\\, \\left( \\cos \\left( \\pi /7 \\right)\\right) ^{2}+7}}{z}^{2}+ \\\\&{\\frac{24\\, \\left( \\cos \\left( \\pi /7\\right) \\right) ^{3}-28\\, \\left( \\cos \\left( \\pi /7 \\right) \\right)^{2}-18\\,\\cos \\left( \\pi /7 \\right) +4}{40\\, \\left( \\cos \\left( \\pi /7\\right) \\right) ^{3}-30\\,\\cos \\left( \\pi /7 \\right) -32\\, \\left( \\cos \\left( \\pi /7 \\right) \\right) ^{2}+7}}{z}^{3}+\\\\&{\\frac{16\\, \\left(\\cos \\left( \\pi /7 \\right) \\right) ^{3}-16\\, \\left( \\cos \\left( \\pi /7\\right) \\right) ^{2}-12\\,\\cos \\left( \\pi /7 \\right) +4}{40\\, \\left(\\cos \\left( \\pi /7 \\right) \\right) ^{3}-30\\,\\cos \\left( \\pi /7 \\right)-32\\, \\left( \\cos \\left( \\pi /7 \\right) \\right) ^{2}+7}}{z}^{4}+\\\\&{\\frac{8\\, \\left( \\cos \\left( \\pi /7 \\right) \\right) ^{3}-4\\, \\left(\\cos \\left( \\pi /7 \\right) \\right) ^{2}-6\\,\\cos \\left( \\pi /7 \\right) +1}{40\\, \\left( \\cos \\left( \\pi /7 \\right) \\right) ^{3}-30\\,\\cos \\left( \\pi /7 \\right) -32\\, \\left( \\cos \\left( \\pi /7 \\right) \\right)^{2}+7}}{z}^{5}$ $P_6(z)=z+{\\frac{9+8\\,\\sqrt{2}}{4\\,\\sqrt{2}+8}}{z}^{2}+{\\frac{6\\,\\sqrt{2}+10}{4\\,\\sqrt{2}+8}}{z}^{3}+{\\frac{4\\,\\sqrt{2}+6}{4\\,\\sqrt{2}+8}}{z}^{4}+{\\frac{2\\,\\sqrt{2}+2}{4\\,\\sqrt{2}+8}}{z}^{5}+ \\frac{1}{4\\,\\sqrt{2}+8}{z}^{6}$ Theorem 1 The following presentation is valid for $t\\in (0,\\pi ),\\; t\\ne \\frac{2\\pi }{N+2}$ $P_N(e^{it})=\\frac{1}{2\\left( \\cos t- \\cos \\frac{2\\pi }{N+2}\\right)}+\\frac{1-\\cos \\frac{2\\pi }{N+2}}{(N+2)(1-\\cos t)}\\frac{\\sin t\\sin \\frac{N+2}{2}t}{\\left(\\cos t-\\cos \\frac{2\\pi }{N+2}\\right)^2} e^{\\frac{N+2}{2}it}.$ First, let us write $P_N(z)$ in terms of trigonometric expressions [8] $P_N(z)=\\frac{1}{ \\left(N+2\\right)\\sin \\frac{2\\pi }{N+2}}\\sum _{k=1}^N\\bigg [&\\left(N-k+3\\right)\\sin \\frac{\\left(k+1\\right)\\pi }{N+2}-\\\\&\\left(N-k+1\\right)\\sin \\frac{\\left(k-1\\right)\\pi }{N+2}\\bigg ]\\sin \\frac{k\\pi }{N+2}z^k$ Having in mind that $\\bigg [2\\sin (\\pi )-0\\cdot \\sin \\frac{N\\pi }{N+2}\\bigg ]\\sin \\frac{\\left(N+1\\right)\\pi }{N+2}z^{N+1} \\equiv 0$ we can change the upper bound for the range in the sum from $N$ to $N+1$ .",
"Further modification produces $P_N(z)=\\frac{1}{ \\left(N+2\\right)\\sin \\frac{2\\pi }{N+2}}\\sum _{k=1}^{N+1}\\bigg [\\left(N-k+2\\right)\\sin \\frac{2k\\pi }{N+2}+2\\cdot \\frac{\\cos \\frac{\\pi }{N+2}}{\\sin \\frac{\\pi }{N+2}}\\sin ^2\\frac{k\\pi }{N+2}\\bigg ]z^k.$ An important observation is that $\\frac{N+1}{N+2}\\cdot S_{N+1,2}(z)=\\frac{1}{ \\left(N+2\\right)\\sin \\frac{2\\pi }{N+2}}\\sum _{k=1}^{N+1}\\left(N-k+2\\right)\\sin \\frac{2k\\pi }{N+2}\\cdot {z^k},$ where $S_{N+1,2}(z)$ is the second Suffridge polynom of order $N+1$ .",
"By formula (5) in [7], for $n=N+1$ and $j=2$ we get $\\frac{N+2}{N+1}\\cdot S_{N+1,2}\\left(e^{it}\\right)=\\frac{1}{2\\left(\\cos t - \\cos \\frac{2\\pi }{N+2}\\right)}+\\frac{1}{N+2}\\cdot \\frac{\\sin t \\cdot \\sin \\frac{N+2}{2}t}{\\left(\\cos t-\\cos \\frac{2\\pi }{N+2}\\right)^2}\\cdot e^{\\frac{N+2}{2}it}$ Meanwhile $\\sum _{k=1}^{N+1}\\sin ^2\\frac{k\\pi }{N+2}e^{ikt}=\\sin ^2\\frac{\\pi }{N+2}\\cdot \\frac{\\sin \\frac{N+2}{2}t}{\\cos {t}-\\cos \\frac{2\\pi }{N+2}}\\cdot \\frac{\\sin t}{1-\\cos t}\\cdot e^{i\\frac{N+2}{2}t}$ By combining both formulas, we get the formula in the theorem.",
"Note that the right hand side has removable singularities, thus in fact it is a trigonometric polynomial.",
"Let us fix a positive integer $N$ and let $R_N(e^{it})=|P_N(e^{it})|^2.$ The following theorem can be directly verified by tedious standard computations.",
"Theorem 2 The following presentation is valid for $t\\in (0,\\pi ),\\; t\\ne \\frac{2\\pi }{N+2}$ : $4|P_N(e^{it})|^2=&\\left(\\frac{\\cos \\frac{N+2}{2}t}{\\cos t- \\cos \\frac{2\\pi }{N+2}}+\\frac{2}{N+2}\\frac{1-\\cos \\frac{2\\pi }{N+2}}{1-\\cos t}\\frac{\\sin t}{(\\cos t- \\cos \\frac{2\\pi }{N+2})^2}\\sin \\frac{N+2}{2}t\\right)^2+\\\\&\\left(\\frac{\\sin \\frac{N+2}{2}t}{\\cos t- \\cos \\frac{2\\pi }{N+2}} \\right)^2.$ Because the real coefficients symmetry of $P_N(e^{it})$ (the real part is an even function and the imaginary is an odd function of $t$ ), we denote $|P_N(e^{it})|^2=R_N(x)$ as a polynomial of $x=\\cos (t).$ Let $b=\\cos \\frac{2\\pi }{N+2}$ and $T_N$ be the Chebyshev polynomial of the first kind.",
"From Theorem REF one can get the following formulas by straightforward computations: $4R_N(x)=\\frac{1}{\\left( x-b \\right) ^{2}}+2\\,{\\frac{ \\left( 1-b\\right) \\left( 1+x \\right) U_{N+1} \\left(x\\right) }{ \\left( N+2 \\right)\\left( x-b \\right) ^{3}}}+2\\,{\\frac{ \\left( 1-b \\right) ^{2} \\left( 1+x\\right) \\left( 1- T_{N+2} \\left( x \\right) \\right) }{ \\left( N+2 \\right) ^{2} \\left( x-b \\right) ^{4} \\left( 1-x\\right) }},$ $4(R_N(x))^\\prime =\\frac{2}{\\left( b-x \\right) ^{3}}\\left(1-\\frac{1-b}{1-x}\\left(1-T_{N+2}(x)\\right)\\left(1-\\frac{4(1-b)(1+x)}{(N+2)^2(b-x)^2}-\\frac{2(1-b)}{(N+2)^2(1-x)(b-x)}\\right)+\\right.\\\\\\frac{1-b}{1-x}-\\frac{1-b}{1-x}\\frac{1}{N+2}U_{N+1}(x)\\frac{1-bx+3(1-x^2)}{b-x}\\bigg ).$ Theorem 3 If $(R_N(x))^\\prime >0$ for $x\\in (-1,1)$ then the polynomial $P_N(z)$ is univalent in $\\mathbb {D}$ and the Koebe radius of this polynomial is $\\sqrt{R_N(-1)}.$ It is proved in [8] that the polynomial $P_N(z)$ is typically real and thus the image of the unit circle has no self intersections, the theorem is proved.",
"Note, that [8] $\\sqrt{R_N(-1)}=\\frac{1}{4}\\sec ^2\\frac{\\pi }{N+2}.$"
],
[
"The case N=1.",
"In this case $R_1(x)=1,$ thus the Koebe radius is 1."
],
[
"The case N=2.",
"In this case $R_2(x)=5/4+x,$ thus the Koebe radius is $\\sqrt{R_2(-1)}=1/2.$"
],
[
"The case N=3.",
"In this case the polynomial $P_3(z)$ is univalent that can be verified using Brennan's criteria [2].",
"Also $4R_3(x)=-&\\frac{2}{25}\\,{\\frac{37\\,\\cos \\left( 1/5\\,\\pi \\right) -69+56\\,\\left( \\cos \\left( 1/5\\,\\pi \\right) \\right) ^{2}}{ \\left( \\cos \\left( 1/5\\,\\pi \\right) \\right) ^{2}+1-2\\,\\cos \\left( 1/5\\,\\pi \\right) }}-{\\frac{32}{25}}\\,{\\frac{ \\left( 23\\,\\cos \\left( 1/5\\,\\pi \\right) -51+49\\, \\left( \\cos \\left( 1/5\\,\\pi \\right) \\right) ^{2} \\right) x}{-9-5\\,\\cos \\left( 1/5\\,\\pi \\right) +20\\, \\left( \\cos \\left( 1/5\\,\\pi \\right) \\right) ^{2}}}-\\\\&{\\frac{32}{5}}\\,{\\frac{\\left( 10\\,\\cos \\left( 1/5\\,\\pi \\right) -14+9\\, \\left( \\cos \\left( 1/5\\,\\pi \\right) \\right) ^{2} \\right) {x}^{2}}{14\\, \\left( \\cos \\left( 1/5\\,\\pi \\right) \\right) ^{2}+3-15\\,\\cos \\left( 1/5\\,\\pi \\right) }}$ $4R_3^\\prime (x)=-{\\frac{32}{25}}\\,{\\frac{23\\,\\cos \\left( 1/5\\,\\pi \\right) -51+49\\,\\left( \\cos \\left( 1/5\\,\\pi \\right) \\right) ^{2}}{-9-5\\,\\cos \\left( 1/5\\,\\pi \\right) +20\\, \\left( \\cos \\left( 1/5\\,\\pi \\right)\\right) ^{2}}}-{\\frac{64}{5}}\\,{\\frac{ \\left( 10\\,\\cos \\left( 1/5\\,\\pi \\right) -14+9\\, \\left( \\cos \\left( 1/5\\,\\pi \\right) \\right) ^{2} \\right) x}{14\\, \\left( \\cos \\left( 1/5\\,\\pi \\right) \\right) ^{2}+3-15\\,\\cos \\left( 1/5\\,\\pi \\right) }}$ One can check that $R_3^\\prime (x)$ is positive on [-1,1], which implies the estimate from above on Koebe radius $|P_3(-1)|=\\frac{3-\\sqrt{5}}{2}\\approx 0.382.$"
],
[
"The case N=4.",
"In this case the polynomial $P_4(z)$ is univalent, c.f.",
"[6].",
"$4R_4(x)=40/9+(112/9)x+(124/9)x^2+(16/3)x^3$ and $4R_4^\\prime (x)=112/9+(248/9)x+(48/3)x^2$ The discriminant is $-37.13...$ therefore the smallest value for $R_4(x)$ is at -1, which implies the estimate from above on Koebe radius $|P_4(-1)|=1/3.$"
],
[
"The case N=5.",
"In the particular case $N=5$ we get $4R_5^\\prime (x)=&{\\frac{16}{49}}\\,{\\frac{42\\, \\left( \\cos \\left( 1/7\\,\\pi \\right)\\right) ^{3}-31\\,\\cos \\left( 1/7\\,\\pi \\right) +9-47\\, \\left( \\cos \\left( 1/7\\,\\pi \\right) \\right) ^{2}}{2\\, \\left( \\cos \\left( 1/7\\,\\pi \\right) \\right) ^{3}+\\cos \\left( 1/7\\,\\pi \\right) -10\\, \\left(\\cos \\left( 1/7\\,\\pi \\right) \\right) ^{2}+5}}+\\\\&{\\frac{64}{49}}\\,{\\frac{ \\left( 762\\, \\left( \\cos \\left( 1/7\\,\\pi \\right) \\right) ^{3}-618\\,\\cos \\left( 1/7\\,\\pi \\right) -3-323\\, \\left( \\sin \\left( 1/7\\,\\pi \\right) \\right) ^{2} \\right) x}{-11+2\\, \\left( \\sin \\left( 1/7\\,\\pi \\right) \\right) ^{2}-39\\,\\cos \\left( 1/7\\,\\pi \\right) +60\\,\\left( \\cos \\left( 1/7\\,\\pi \\right) \\right) ^{3}}}+\\\\&{\\frac{192}{49}}\\,{\\frac{ \\left( 940\\, \\left( \\cos \\left( 1/7\\,\\pi \\right)\\right) ^{3}-761\\,\\cos \\left( 1/7\\,\\pi \\right) +81-536\\, \\left( \\sin \\left( 1/7\\,\\pi \\right) \\right) ^{2} \\right) {x}^{2}}{-18+14\\,\\left( \\cos \\left( 1/7\\,\\pi \\right) \\right) ^{3}+35\\, \\left( \\sin \\left( 1/7\\,\\pi \\right) \\right) ^{2}}}+\\\\&{\\frac{128}{7}}\\,{\\frac{\\left( 380\\, \\left( \\cos \\left( 1/7\\,\\pi \\right) \\right) ^{3}-293\\,\\cos \\left( 1/7\\,\\pi \\right) +17-176\\, \\left( \\sin \\left( 1/7\\,\\pi \\right) \\right) ^{2} \\right) {x}^{3}}{-22+9\\, \\left( \\sin \\left( 1/7\\,\\pi \\right) \\right) ^{2}-70\\,\\cos \\left( 1/7\\,\\pi \\right) +112\\,\\left( \\cos \\left( 1/7\\,\\pi \\right) \\right) ^{3}}}.$ Figure: NO_CAPTIONFigure: NO_CAPTION Fig 2: The graphs $R_5(x)$ and $R^\\prime _5(x)$ By decomposing into Taylor polynomial centered at -1 we get $4R_5^\\prime (x)=-&{\\frac{8}{49}}\\,{\\frac{778868087\\, \\left( \\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) ^{2}-791395834+2270258054\\, \\left( \\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) ^{3}-1666223113\\,\\cos \\left(\\frac{1}{7}\\,\\pi \\right) }{10381281\\, \\left( \\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) ^{3}-5460108\\,\\cos \\left(\\frac{1}{7}\\,\\pi \\right) -4219605\\, \\left( \\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) ^{2}+752170}}\\\\-&{\\frac{32}{49}}\\,{\\frac{ \\left(-58704325+88578183\\, \\left( \\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) ^{2}+57393568\\, \\left( \\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) ^{3}-61237982\\,\\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) \\left( 1+x \\right) }{-360031\\, \\left( \\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) ^{2}+121875+515550\\, \\left( \\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) ^{3}-229418\\,\\cos \\left(\\frac{1}{7}\\,\\pi \\right) }}+\\\\&{\\frac{192}{49}}\\,{\\frac{ \\left( -212691+312494\\, \\left( \\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) ^{2}-238981\\,\\cos \\left(\\frac{1}{7}\\,\\pi \\right) +238432\\, \\left( \\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) ^{3} \\right) \\left( 1+x \\right) ^{2}}{2543-6451\\, \\left( \\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) ^{2}-2632\\,\\cos \\left(\\frac{1}{7}\\,\\pi \\right) +6916\\, \\left( \\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) ^{3}}}+\\\\&{\\frac{128}{7}}\\,{\\frac{ \\left( 17-176\\,\\left( \\sin \\left( \\frac{1}{7}\\,\\pi \\right) \\right) ^{2}+380\\, \\left( \\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) ^{3}-293\\,\\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) \\left( 1+x \\right) ^{3}}{-22+9\\, \\left( \\sin \\left(\\frac{1}{7}\\,\\pi \\right) \\right) ^{2}-70\\,\\cos \\left(\\frac{1}{7}\\,\\pi \\right) +112\\, \\left( \\cos \\left(\\frac{1}{7}\\,\\pi \\right) \\right) ^{3}}}.$ Thus, $R_5^\\prime (x)=A_0+(x+1)(A_1+A_2(x+1)+A_3(x+1)^2)$ with the obviuos choice of $A_j.$ Since for $|x|\\le 1$ the value $x+1$ is positive and $A_i\\ge 0$ for $i=0,1$ then the inequality $A_2^2-4A_1A_3<0$ implies that $R^\\prime (x)> 0;\\quad x\\in [-1,1].$ The verification of (REF ) is an elementary issue based on approximations of $\\cos {\\pi /7}$ and $\\sin {\\pi /7}$ from above and below with sufficiently large number of digits.",
"This proves that the derivative does not intersect the interval and that $R^\\prime _5(z)\\ge {0}$ .",
"Thus, $R_5(z)$ is not decreasing on $\\left[-1,1\\right]$ therefore $P_5(z)$ is univalent by Theorem REF .",
"This gives us an estimate on the Koebe radius $|P_5(-1)|\\approx 0.3080.$"
],
[
"The case $N=6.$",
"In this case $4R_6(x) &=&2+\\left( 8\\,\\sqrt{2}-4 \\right) x+ \\left( 38-12\\,\\sqrt{2} \\right) {x}^{2}+ \\left( 28\\,\\sqrt{2}-4 \\right) {x}^{3}+\\\\& & \\left( 28\\,\\sqrt{2}-10\\right) {x}^{4}+\\left( -16\\,\\sqrt{2}+32 \\right) {x}^{5}.$ $4R^\\prime _6(x)&=&8\\,\\sqrt{2}-4+2\\, \\left( 38-12\\,\\sqrt{2} \\right) x+3\\, \\left( 28\\,\\sqrt{2}-4 \\right) {x}^{2}+\\\\& & 4\\, \\left( 28\\,\\sqrt{2}-10 \\right) {x}^{3}+5\\, \\left( -16\\,\\sqrt{2}+32 \\right) {x}^{4}\\\\&= & -76\\,\\sqrt{2}+108+ \\left( 464\\,\\sqrt{2}-660 \\right) \\left( 1+x\\right) + \\left( -732\\,\\sqrt{2}+1068 \\right) \\left( 1+x \\right) ^{2}+\\\\& & \\left( 432\\,\\sqrt{2}-680 \\right) \\left( 1+x \\right) ^{3}+ \\left( -80\\,\\sqrt{2}+160 \\right) \\left( 1+x \\right) ^{4}\\\\&=&\\left( 108-76\\, \\sqrt{2}+\\frac{ -660+464\\, \\sqrt{2}}{2\\, \\sqrt{108-76\\, \\sqrt{2}}} \\left( x+1 \\right) \\right) ^{2}+\\left[{\\frac{ \\left( -660+464\\, \\sqrt{2} \\right) ^{2} }{4 \\left( 108-76\\, \\sqrt{2} \\right) }}-732\\, \\sqrt{2}+1068 \\right.+\\\\& & \\left.",
"\\left( 432\\, \\sqrt{2}-680\\right) \\left( x+1 \\right)+\\left( -80\\, \\sqrt{2}+160 \\right)\\left( x+1 \\right) ^{2}\\right] \\left( x+1 \\right) ^{2}$ Applying on argument similar to the formula (REF ) we get formula (REF ) which implies the estimate for the Koebe radius $|P_6(-1)|\\approx 0.2929.$ We conjecture that the obtained estimates in fact are true values."
],
[
"Conclusion",
"In [8] a new class of polynomials was introduced $P_N(z)=\\frac{1}{ { { U^{\\prime }_N}\\left({\\cos \\frac{\\pi }{ N+2 } } \\right) } }\\sum _{k=1}^N{ U^{\\prime }_{ N - k + 1 } }\\left(\\cos \\frac{\\pi }{ N+2 } \\right){ U_ { k - 1 } }\\left(\\cos \\frac{\\pi }{ N+2 } \\right)z^k,$ and the extremal property of these polynomials was mentioned $\\sup _{p_N(z)=z+\\sum _{k=2}^Na_kz^k}\\min _t\\lbrace \\Re (p_N(e^{it})): \\Im (p_N(e^{it})=0\\rbrace =P_N(-1).$ It was conjectured that these polynomials are univalent and solves Dimitrov problem.",
"In the present article the first conjecture is proved for $N=1,...,6$ thus for those $N$ the estimates from below on the radius Koebe of polynomials from $\\mathcal {U}_N$ are obtained.",
"It is shown that those values are smaller then the corresponding ones for Suffridge polynomials $S_{N,1}(z).$ To prove the case $N>6$ one needs to verify the criteria given by Theorem REF , which is a notrivial task.",
"Currently we are working on this subject.",
"Also, let us mention that the polynomials $P_N(z), S_{N,1}(z)$ and their generalizations turnes out to be very helpful in the problem of stabilization of of cycles in nonlinear discrete systems [10], [9]."
]
] | 1808.08636 |
[
[
"Mean density inversions for red giants and red clump stars"
],
[
"Abstract Since the CoRoT and Kepler missions, the availability of high quality seismic spectra for red giants has made them the standard clocks and rulers for Galactic Archeology.",
"With the expected excellent data from the TESS and PLATO missions, red giants will again play a key role in Galactic studies and stellar physics, thanks to the precise masses and radii determined by asteroseismology.",
"The determination of these quantities is often based on so-called scaling laws, which have been used extensively for main-sequence stars.",
"We show how the SOLA inversion technique can provide robust determinations of the mean density of red giants within 1 per cent of the real value, using only radial oscillations.",
"Combined with radii determinations from Gaia of around 2 per cent precision, this approach provides robust, less model-dependent masses with an error lower than 10 per cent.",
"It will improve age determinations, helping to accurately dissect the Galactic structure and history.",
"We present results on artificial data of standard models, models including an extended atmosphere from averaged 3D simulations and non-adiabatic frequency calculations to test surface effects, and on eclipsing binaries.",
"We show that the inversions provide very robust mean density estimates, using at best seismic information.",
"However, we also show that a distinction between red-giant branch and red-clump stars is required to determine a reliable estimate of the mean density.",
"The stability of the inversion enables an implementation in automated pipelines, making it suitable for large samples of stars."
],
[
"Introduction",
"Red giants play a key role in stellar physics.",
"Since the detection of mixed modes in their oscillation spectra thanks to the CoRoT and Kepler missions, they are at the origin of multiple questions on the reliability of our depiction of stellar structure and evolution , .",
"The availability of thousands of high quality seismic spectra for these stars led to their use as the standard clocks and rulers for Galactic Archeology , .",
"Today, they are used as tracers of the structure and chemical evolution of our Galaxy .",
"New accurate data for these stars are also expected to be delivered by the TESS and PLATO missions, which will play a key role in Galactic studies .",
"These successes originate from the ability of asteroseismology to provide precise masses and radii for a large number of stars.",
"The seismic determination of these quantities is often based on so-called scaling laws, which have also been used extensively for main-sequence stars.",
"However, while the precision of these determinations is excellent, due to the high precision of the space-based photometry data, their accuracy is far from perfect.",
"Multiple studies , , , , have shown that they could lead to inaccurate results.",
"From a physical point of view, their limited accuracy is not surprising as they do not fully exploit the information contained in the seismic spectra.",
"Therefore, providing a more robust way of determining the mean density of the observed targets using seismology is required so that, using constraints from the second Gaia data release, more accurate masses can be determined for thousands of stars.",
"These accurate masses will help with dissecting the structure of the Galaxy, thus providing new insights on its evolution and formation history.",
"In addition to this potential, the determination of accurate fundamental parameters of red giants in stellar clusters is also crucial to constrain the mass loss rate on the red giant branch, a still uncertain key phenomenon of stellar evolution , .",
"In this study, we will show how the adaptation of the SOLA inversion technique for the mean density developed by used on the radial oscillations of red giant stars can provide more robust determinations of the mean density than values obtained from the fitting of the average large frequency separation or the usual scaling laws.",
"In section , we briefly recall the principles of the inversion techniques.",
"In section , we test the inversion in various numerical exercises, using artificial targets on the red giant branch (hereafter denoted RGB), in the red clump, and an RGB target including an averaged 3D atmosphere model for which the frequencies are computed using adiabatic and non-adiabatic oscillation codes to test various surface effects correction.",
"In section , we apply our method to a subsample of eclipsing binaries studied by and .",
"This is then followed by a conclusion."
],
[
"The inversion technique",
"The inversion procedure used to obtain the mean density is that of .",
"We only briefly recall a few specific aspects of the method for the sake of clarity.",
"The goal of the approach is to determine through the SOLA inversion technique an estimate of the mean density of a given observed star using the linear integral structural relations between individual relative frequency differences and corrections of thermodynamic quantities such as density, $\\rho $ , the squared adiabatic sound speed, $c^{2}$ or the adiabatic exponent, $\\Gamma _{1}=\\frac{\\partial \\ln P}{\\partial \\ln \\rho }\\vert _{S}$ , with $P$ the pressure and $S$ the entropy , .",
"This can be done by using the linear perturbation of the mean density with the integral formula of the stellar mass $\\frac{\\delta \\bar{\\rho }}{\\bar{\\rho }}=\\frac{3}{4\\pi R^{3}\\bar{\\rho }}\\int _{0}^{R}4\\pi r^{2}\\delta \\rho dr, $ to define the target function of the SOLA inversion.",
"Using a little algebra in Eq.",
"REF and by non-dimensionalising the integral, one has as $\\mathcal {T}_{\\bar{\\rho }}=4\\pi x^{2} \\frac{\\rho }{\\rho _{R}},$ with the radial position of a layer of stellar material normalized by the photospheric stellar radius $x=r/R$ , $\\rho $ the density of stellar material and $\\rho _{R}=M/R^{3}$ , with $M$ the stellar mass and $R$ the photospheric stellar radius.",
"Using this target, the cost function of the SOLA method becomes $\\mathcal {J}_{\\bar{\\rho }}(c_{i})=&\\int _{0}^{1}\\left[K_{\\mathrm {Avg}} - \\mathcal {T}_{\\bar{\\rho }}\\right]^{2}dx + \\beta \\int _{0}^{1}\\left( K_{\\mathrm {Cross}}\\right)^{2}dx \\nonumber \\\\ &+ \\lambda \\left[ 2 -\\sum _{i}c_{i} \\right] + \\tan \\theta \\frac{\\sum _{i}\\left(c_{i}\\sigma _{i}\\right)^{2}}{<\\sigma ^{2}>}, $ where we have introduced the averaging and cross-term kernels, defined as follows $K_{\\mathrm {Avg}}=\\sum _{i}c_{i}K^{i}_{\\rho ,\\Gamma _{1}}, \\\\K_{\\mathrm {Cross}}=\\sum _{i}c_{i}K^{i}_{\\Gamma _{1},\\rho },$ and the parameters $\\beta $ and $\\theta $ which define the trade-off problem between the fit of the target, the contribution of the cross term and the amplification of observational error bars of the individual frequencies, denoted $\\sigma _{i}$ .",
"The $K^{i}_{\\rho ,\\Gamma _{1}}$ and $K^{i}_{\\Gamma _{1},\\rho }$ are the so-called structural kernel functions, derived from the variational analysis of the pulsation equations and $<\\sigma ^{2}>=\\frac{1}{N}\\sum _{i=1}^{N}\\sigma ^{2}_{i}$ with $N$ the number of observed frequencies.",
"In Eq.",
"REF , we have also introduced the inversion coefficients $c_{i}$ and $\\lambda $ , a Lagrange multiplier.",
"The third term is based on homologous reasoning described in which also leads to a non-linear generalization of the method where the inverted mean density, $\\bar{\\rho }_{\\mathrm {Inv}}$ , is determined using the formula $\\bar{\\rho }_{\\mathrm {Inv}}=\\left(1+\\frac{1}{2}\\sum _{i}c_{i}\\frac{\\delta \\nu _{i}}{\\nu _{i}}\\right)^{2}\\bar{\\rho }_{\\mathrm {Ref}},$ with $\\bar{\\rho }_{\\mathrm {Ref}}$ the mean density of the reference model of the inversion and $\\frac{\\delta \\nu _{i}}{\\nu _{i}}$ the relative differences between the observed and theoretical frequencies defined as $\\frac{\\nu _{\\mathrm {Obs}}-\\nu _{\\mathrm {Ref}}}{\\nu _{\\mathrm {Ref}}}$ .",
"In the following sections, we will always use this non-linear generalization.",
"Using this approach, the errors on the inverted mean density are given by $\\sigma _{\\bar{\\rho }_{\\mathrm {Inv}}}=\\bar{\\rho }_{\\mathrm {Ref}}\\left(1+\\frac{1}{2}\\sum _{i}c_{i}\\frac{\\delta \\nu _{i}}{\\nu _{i}} \\right)\\sqrt{\\sum _{i}c^{2}_{i}\\sigma ^{2}_{i}}.$ A few additional comments can be made on Eq.",
"REF .",
"We have intentionally dropped the classical surface term commonly used in helioseismology .",
"In Section REF , we will comment on this choice and discuss in more details the optimal approach to implement surface corrections and provide examples using the formulation of and for the behaviour of the surface term for patched models and frequencies including non-adiabatic effects.",
"We also note that we make the choice of using the $\\left( \\rho ,\\Gamma _{1} \\right)$ structural pair to carry out the inversions.",
"Other pairs, such as the $\\left(\\rho ,c^{2}\\right)$ or the $\\left( \\rho ,Y \\right)$ pair, with $Y$ the helium mass fraction, could be used, but the former shows strong contributions from the cross-term kernel and is thus inadequate, whereas the latter requires an implementation of the equation of state which leads to accurate derivations of state derivatives of $\\Gamma _{1}$ and leads to the same accuracy as the $\\left( \\rho , \\Gamma _{1} \\right)$ pair.",
"The reliability of the inversion procedure is usually assessed in terms of the norm of the averaging and cross-term kernels, defined as follows $\\vert \\vert K_{\\mathrm {Avg}} \\vert \\vert ^{2}&=\\int _{0}^{1}\\left[ K_{\\mathrm {Avg}}-\\mathcal {T}_{\\bar{\\rho }}\\right]^{2}dx, \\\\\\vert \\vert K_{\\mathrm {Cross}} \\vert \\vert ^{2}&=\\int _{0}^{1} K_{\\mathrm {Cross}}^{2}dx.$ In addition to this analysis, it can be more thoroughly assessed when the method is applied to artificial data with the help of three other quantities.",
"The definition of these error contributions is $\\epsilon _{\\mathrm {Avg}}=&-\\int _{0}^{1}\\left( \\mathcal {T}_{\\bar{\\rho }}-K_{Avg}\\right)\\frac{\\delta \\rho }{\\rho }dx, \\\\\\epsilon _{\\mathrm {Cross}}=&-\\int _{0}^{1} K_{\\mathrm {Cross}} \\frac{\\delta \\Gamma _{1}}{\\Gamma _{1}}dx, \\\\\\epsilon _{\\mathrm {Res}}=&\\frac{\\bar{\\rho }_{\\mathrm {Inv}}-\\bar{\\rho }_{\\mathrm {Obs}}}{\\bar{\\rho }_{\\mathrm {Ref}}}-\\epsilon _{\\mathrm {Avg}}-\\epsilon _{\\mathrm {Cross}}, $ where we have defined $\\epsilon _{\\mathrm {Avg}}$ , the contribution from the inaccurate fit of the target by the averaging kernel, $\\epsilon _{\\mathrm {Cross}}$ , the error contribution from the non-zero value of the cross-term kernel and $\\epsilon _{\\mathrm {Res}}$ , the residual error, which contains all other sources of uncertainties such as non-linear contributions, surface effects, linearization of the equation of state or systematic errors in the values of the observed frequencies.",
"We have also used the notations $\\bar{\\rho }_{\\mathrm {Ref}}$ and $\\bar{\\rho }_{\\mathrm {Obs}}$ for the mean density of the reference model and the “observed” artificial target respectively."
],
[
"Numerical exercises",
"To test the reliability and robustness of the inversion technique, we defined a set of artificial targets that would be fitted using various constraints to define reference models for the inversion.",
"A first set of targets on the RGB and their properties are summarized in Table REF .",
"Table: Properties of the target models used for the inversion.These targets have been computed using the Liège stellar evolution code (CLES ) and their frequencies have been computed using the Liège oscillation code (LOSC ).",
"The formalism used for convection is that of the classical mixing-length theory and overshooting, when applied, is implemented in the form of an instantaneous mixing.",
"The temperature gradient in the overshooting region is forced to be adiabatic.",
"Target models from Table REF and reference models from Table REF also included opacities at low temperature from and the effects of conductivity from and .",
"The nuclear reaction rates we used are from [1].",
"We used only low order radial modes in our study, with $n=1$ up to $n=15$ .",
"Tests were also carried out by reducing the number of observed frequencies down to 10 or 8 modes.",
"In section REF , we also analyse how the results vary when the set of modes is changed to higher order, for which surface effects are much larger.",
"For each target, the uncertainties of the frequency values were taken to be $0.03\\mu \\mathrm {Hz}$ , similarly to the average error bars expected from datasets of the Kepler mission.",
"However, as we will see in the next section, most of the limitations of the determination does not come from the precision of the data, but from the low number of frequencies and systematic effects such as surface effects or the non-verification of the integral linear relations between relative frequency differences and corrections of thermodynamic quantities ."
],
[
"Calibrations using effective temperature and luminosity",
"First, we carried out inversions after having calibrated the models based solely on their effective temperatures $(T_{\\mathrm {eff}})$ and luminosities $(L)$ , using different physical ingredients than those used to build the 11 targets of Table REF .",
"In Table REF , we describe the properties of these reference models.",
"For some of the targets, the calibration was pushed so that both reference model and targets had nearly exactly the same $T_{eff}$ and $L$ , which means in turn that they have the same radius.",
"However, biases in the mass of the models (as can be seen when comparing Tables REF and REF ) ensure that the mean density is not the same for both target and reference models.",
"Moreover, strong changes in the physical ingredients have been applied such that the differences observed in individual frequencies are not only due to a mean density mismatch, but also to an inaccurate depiction of the internal structure of the targets by their reference model.",
"For some of the targets, we also computed inverted values using neighbouring models in the evolutionary sequence of the calibration, to demonstrate that the inversion was still efficient despite a radius mismatchA fact that has already been demonstrated in the test cases of and .. Other tests, not presented here, were also carried out using the FST formulation of convection and led to similar results.",
"Table: Properties of the reference models used for the inversion.Inversion results are illustrated in Figure REF for each pair of models, where we can see that for each test case, the inversion provides an estimate of the mean density within 1 per cent of the real value of the artificial target.",
"The reliability of the method is confirmed from the analysis of the value of the errors on the cross-term and averaging kernels, which remained small and of the same order of magnitude as what was observed in for main-sequence stars.",
"Tests were also conducted with the differential formulation based on the linear fit of the average large frequency separation as in and , as well as the so-called KBCD approach of , based on the surface correction law.",
"These methods are formally quite similar to the inversion, as they attempt to relate relative frequency differences to the mean density of a given target.",
"However, the first method is based on the fact that the average large frequency separation is simply a frequency combination.",
"From there, a differential form can be derived when comparing an observed target to a reference model and used to obtain a corrected mean density value.",
"The KBCD approach is based on Eq.",
"6 of , from which a differential form based on individual frequencies can also be derived and used to define a corrected mean density value from comparisons between observed data and a reference model.",
"We refer the reader to for more details and the mathematical expressions associated with these methods.",
"Both these methods showed unstable behaviours, in the sense that they could sometimes provide results of similar quality than those of the SOLA inversion, and sometimes provide much less accurate results, with differences to the target value of more than 4 per cent.",
"From an in-depth investigation, we can conclude that their accuracy, when good, is actually due to a systematic compensation of their various error contributions.",
"This result had also already been observed for main-sequence stars , .",
"Therefore, it is not surprising to observe a similar behaviour in these test cases using only radial modes.",
"Figure: Mean density inversion results for exercises between the targets of Table and the reference models from Table .In Figure REF , we illustrate the error contributions as defined in Eqs.",
"REF , and .",
"The first striking difference in the error contribution is the value of the cross-term error.",
"In , one can see that the cross-term error associated with $\\Gamma _{1}$ is very often one order of magnitude, if not more, lower than the averaging kernel and the residual error contribution.",
"Figure: Error contributions ϵ Avg \\epsilon _{Avg}, ϵ Cross \\epsilon _{Cross} and ϵ Res \\epsilon _{Res} as defined in Eqs.",
", and for each pair of target and reference models from Tables and .In these test cases, we can see that the cross-term error can sometimes become much more significant and even the dominant source of error.",
"The most striking cases being Targets 5, 6 and 7 in Figure REF , where the cross-term error clearly dominates all contributions.",
"This is a consequence of specific aspects of the inversions considered here.",
"The very low number of modes implies that the damping of the cross-term will be very difficult.",
"In turn, this implies that keeping a low cross-term contribution can only be achieved if the hypothesis that $\\frac{\\delta \\Gamma _{1}}{\\Gamma _{1}}$ is very small at all depths is satisfied.",
"This is, however, not always the case and is a consequence of the large radial extent of the helium ionization zones.",
"Consequently, the differences in $\\Gamma _{1}$ due to mismatches in the helium abundance will have a larger impact on the frequencies than in main-sequence stars, where their very narrow width implies that they do not have a strong contribution on the cross-term integral.",
"In addition to this effect, the density values, and in turn the averaging kernel term in this inversion, has a much lower value than on the main-sequence, as a consequence of the expansion of the star.",
"Looking back at Targets 5, 6 and 7 we find that these are the targets for which the differences both in mass and helium abundance are the largest of all our sample.",
"These differences naturally implied large values for $\\frac{\\delta \\Gamma _{1}}{\\Gamma _{1}}$ between these models and thus a higher cross-term contribution.",
"We illustrate this effect by plotting in Figure REF a comparison between the differences in $\\Gamma _{1}$ for Target 3, which has a low cross-term contribution and Target 5 for which $\\epsilon _{Cross}$ largely dominates.",
"These changes also imply that the trade-off parameters have to be properly adjusted depending on the proximity of the model with its target.",
"In these test cases, large variations of the helium abundance were observed and had to be damped by increasing the $\\beta $ parameter in the SOLA cost function.",
"Figure: Relative differences in adiabatic exponent Γ 1 \\Gamma _{1} for the third and fith test cases, illustrating the impact of Γ 1 \\Gamma _{1} on the cross-term contribution.",
"The relative differences in adiabatic exponent are here δΓ 1 Γ 1 =Γ 1 Tar -Γ 1 Ref Γ 1 Ref \\frac{\\delta \\Gamma _{1}}{\\Gamma _{1}}=\\frac{\\Gamma ^{\\mathrm {Tar}}_{1}-\\Gamma ^{\\mathrm {Ref}}_{1}}{\\Gamma ^{\\mathrm {Ref}}_{1}}, where “Tar” refers to the artificial target and “Ref” to the associated reference model.From Figure REF , it is however clear that the accuracy of the inversion does not stem from large compensation effects between the various error contributions.",
"Such a compensation would be seen if for example a large positive value of a few per cent was found for example for $\\epsilon _{\\mathrm {Avg}}$ in Figure REF and a large negative value of the same order of magnitude was found for $\\epsilon _{\\mathrm {Cross}}$ in the same inversion.",
"This would imply that the agreement between the inverted value and the target value is purely fortuitous and not representative of the real limitations of the inversion.",
"In Figure REF , we see that for most of the cases, the sum of the modulus of all the error contributions remains within 1 per cent and never exceeds $1.5$ per cent for the remaining cases.",
"As a comparison, the usual scaling law using the large frequency separation and a solar reference showed at best differences of $1.3$ per cent for Target 3, where the inversion shows differences of less than $0.1$ per cent with the observed value.",
"For the other targets, the accuracy of the scaling laws could be worse than 30 per cent, especially for the more massive targets.",
"Although one could apply corrections to the scaling laws to improve their accuracy, it is clear that the inversions have the advantage of using in the most optimal way all the seismic information of the frequency spectrum.",
"This will be further illustrated on real data in section .",
"Similarly, using the differential form of the $< \\Delta \\nu >$ scaling law defined in , who used another reference than the Sun, we showed that this formulation relied, as for main-sequence stars, on a compensation of its intrinsic errors.",
"This implies a much lower robustness of the method, especially if only a few modes of low radial order are observed.",
"One should also note that the test cases presented here did not use any seismic constraints to define the reference model for the inversion.",
"Hence, the frequency differences between target and reference could sometimes be very large, which can induce a lower stability and reliability of the inversion.",
"In practice, seismic constraints should be used to define the reference models to ensure an optimal result.",
"The effect of seismic constraints will be presented and discussed in the next section."
],
[
"Calibrations using seismic constraints",
"In section REF , we only used classical parameters to define our reference models, which is far from what would be done for stars for which individual frequencies have been determined.",
"In such cases, seismic modelling would be favoured, at least by using global parameters such as the average large frequency separation and the frequency at maximum power, $\\nu _{Max}$ .",
"If individual frequencies are observed, one might even want to fit directly these constraints to extract as much seismic information as possible.",
"While this seems an efficient approach, it should be kept in mind that directly fitting the frequencies can lead to underestimated uncertainties and strongly depends on the surface effects correctionsThis is particularly true for the main-sequence stars for which a large number of frequencies are observed with Kepler..",
"Both approaches were tested for five additional artificial targets, whose properties are given in Table REF .",
"The dataset used for both the forward modelling and the inversions contained 9 radial modes, from $n=5$ to $n=13$ .",
"The reference models were obtained through forward modelling using the AIMS pipeline.",
"The underlying grid of models used for the fitting was built using the Liège stellar evolution code (CLES) with a mass range in solar masses between $\\left[0.75,2.25 \\right]$ with a step of $0.02$ and a $\\left[ \\mathrm {Fe/H} \\right]$ range between $\\left[-0.75, 0.25\\right]$ with a step of $0.25$ .",
"The mixing-length parameter value is fixed by a solar calibration at a value of $1.691$ .",
"The effects of microscopic diffusion are not taken into account in the models.",
"Moreover, the grid is built with the GN93 abundance tables , the FreeEOS equation of state and the OPAL opacities , while the nuclear reaction rates from [1] are used.",
"The results of this modelling are shown in Table REF .",
"We denoted the reference models computed from the fitting of individual frequencies as $1.1$ , $2.1$ and so on, whereas models denoted as $1.2$ , $2.2$ , etc were built using the global seismic indicators, namely the average large frequency separation, $< \\Delta \\nu >$ , and the frequency of maximum power, $\\nu _{Max}$ .",
"Classical constraints such as $\\left[\\mathrm {Fe/H}\\right]$ and $T_{eff}$ were also used, where uncertainties of $0.1$ dex and 80K were respectively considered.",
"No surface correction was used when carrying out the fits, as only models using similar atmospheric models and adiabatic frequencies were used here.",
"These effects will be discussed separately, in section REF .",
"Table: Properties of the target models used for the inversion and AIMS modelling.The first conclusion that can be drawn is that the fit of the mean density is very good using both global seismic constraints and individual frequencies.",
"More specifically, the fit of the mean density is exact to numerical precision when the individual modes are used and in fact, since almost every frequency is fitted within its error bars, the inversion process is in this case useless.",
"Indeed, by definition, the inversion, based on the recombination of individual frequency differences, will not bring any additional information.",
"This is the reason why the Reference $1.1$ , $2.1$ , $3.1$ , $4.1$ and $5.1$ are not illustrated in Figure REF .",
"However, we can still see that the masses and radii of the artificial targets are not always properly reproduced.",
"This is due to the incorrect reproduction of the helium abundance, since the grid used for the AIMS modelling was built using a fixed enrichment law in heavy elements abundances and the targets were built without following any such approach.",
"This implies that the determination of stellar mass on the RGB still strongly relies on accurate determinations of stellar luminosities and/or radii, as provided by the Gaia mission.",
"Table: Properties of the reference models used for the inversion and AIMS modelling.Moreover, we will see in section that mean density inversions can still be useful in real observed cases.",
"First, because it is in practice nearly impossible to fit every individual mode for real data and the results still depend in any case from the adopted surface corrections.",
"Second, because the inversion can provide a useful verification step to the forward modelling process and provide a mean density value which can then be directly introduced in the cost function of second forward modelling step.",
"Figure: Inversion results for the artificial targets 1 to 5, using the reference models 1.21.2 to 5.25.2.",
"The abscissa refers to the target number.From Table REF , we can see that the fits using global seismic parameters can lead to less accurate results, even for the mean density.",
"In these cases, since the individual frequencies were not reproduced within their uncertainties, computing the seismic inversions could provide an improvement of the mean density value.",
"These results are illustrated in Figure REF and the associated error contributions are illustrated in Figure REF .",
"Figure: Error contributions ϵ Avg \\epsilon _{Avg}, ϵ Cross \\epsilon _{Cross} and ϵ Res \\epsilon _{Res} as defined in Eqs.",
", and for the inversion exercises using targets 1 to 5 and reference models 1.21.2 to 5.25.2.From Figure REF , we see that the SOLA method is able to provide a better determination of the mean density without large compensation of its errors.",
"A certain degree of compensation can be observed, but the modulus of each error contribution remains well within 1 per cent, ensuring that the inversion still provides additional information after the forward modelling process.",
"This confirms that the use of seismic inversions offers a gain in accuracy and tighter constraints on the mean density.",
"The inverted value can than be used directly as constraint in a second run of seismic forward modelling."
],
[
"Red clump stars",
"In addition to RGB targets, two red clump models of the same evolutionary sequence were also used to test the inversion techniques.",
"The properties of these targets are given in Table REF .",
"We first made attempts to carry out inversions for these artificial targets using a clump and asymptotic giant branch models as references.",
"These references were built using slightly different physical ingredients.",
"Reference 1 was calibrated to approximately reproduce both $T_{eff}$ and $\\nu _{Max}$ of the targets.",
"References 2 and 3 are two additional models tested with both Target 1 and 2.",
"They were just taken as additional test cases to see how far the inversion could be pushed once it was established that the star was in the helium burning phase.",
"Both reference models and artificial targets were computed using the MESA evolution code and their characteristics are summarized in Table REF .",
"Again, we used 9 radial oscillations as dataset for the inversion, namely the modes with $n=5$ to $n=13$ .",
"Table: Properties of the clump targets and references.The results of these exercises are illustrated in Figure REF , and the error contributions to the inversions are illustrated in Figure REF .",
"From these tests cases, it seems that mean density inversions can also be performed for clump stars even if only a small number of radial frequencies are available.",
"The typical errors of these inversions is then of around 1 per cent, which is much better than what can be achieved using the average large frequency separation.",
"Figure: Inversion results for the clump models 1 and 2.",
"The abscissa refers to the pair of target and reference models used for the inversion.While these tests demonstrate the efficiency of the method, they also rely on a fundamental hypothesis, which is that one was able to determine that the target belonged to the clump and not the RGB.",
"For stars with an observed period spacing, there is no ambiguity possible, but higher on the asymptotic giant branch and even for some clump stars, no period spacing can be determined.",
"For these specific cases, confusion can remain and the reference model can be very far from its actual target.",
"Figure: Error contributions ϵ Avg \\epsilon _{Avg}, ϵ Cross \\epsilon _{Cross} and ϵ Res \\epsilon _{Res} as defined in Eqs.",
", and for the inversion exercises using clump stars as artificial targets.",
"The abscissa refers to the pair of target and reference models used for the inversion.To simulate such effects, we used the AIMS software to fit the individual frequencies of the clump models using a grid of RGB models only.",
"As a result, we got reference models very far from the targets but which actually reproduced quite well (although not as well as for the tests on the RGB) the target frequencies.",
"These models are denoted Reference 4 and 5 and display for example non-physical ages and are completely off in terms of luminosity and $T_{eff}$ , which were not used in the fits.",
"Therefore, in a real case, these models could clearly be rejected, but to further test and break the inversion, we still kept them and attempted to determine the mean density of both Target 1 using Reference 4 and Target 2 using Reference 5.",
"These test cases are very enlightening as neither the fitting process using AIMS nor the inversion could provide a reliable estimate of the mean density.",
"This is illustrated in Figures REF and REF .",
"In Figure REF , we can see that the SOLA method is unable to provide an accurate fit of the target function and hence a reliable mean density value.",
"The only way for the inversion to work was to include the fundamental harmonics in the frequencies used for the inversion.",
"We have added this test case in Figures REF and REF to illustrate the effect of using the full set of modes.",
"This is not really surprising, as the fundamental mode is the one that carries most of the information on the deep layers.",
"Its value was indeed radically different from that of the corresponding modes of the artificial target, while all other modes could be fitted quite well.",
"This illustrates two things, first, that the fundamental radial mode can be used to distinguish between RGB and red clump stars, and second, that even if the frequencies seem well fitted, the mean-density of the star is not.",
"This is in strong contrast with the RGB case and demonstrates again the importance of providing as reliable as possible reference models when computing linear structural inversions.",
"Indeed, these test cases simply illustrate the well-known fact that inversions in asteroseismology are not fully model-independent and thus require a proper assessment of the model-dependency to avoid biased and hasty interpretations of their results."
],
[
"Importance of surface effects",
"As stated in section , surface effects are an important source of uncertainties for seismic inversions based on the integral relations for individual frequencies.",
"In recent years, various corrections have been proposed to take these effects into account in seismic modelling.",
"The most recent ones, tested with patched models, are those of and .",
"In this section, we test these corrections when applied to the inversion of the mean density of RGB stars, using the patched model I of as a target, computed with the CESTAM stellar evolution code , .",
"Frequencies for this model were computed in the adiabatic approximation using the Aarhus pulsation package and the MAD non-adiabatic oscillation code , , which takes into account a time-dependent treatment of convection including the effects of turbulent pressure and variations of the convective flux due to the oscillations.",
"In addition to the various corrections proposed in the literature, there are multiple ways of including these corrections in the SOLA inversion.",
"The classical approach is to mimic what has been done in helioseismology, by including an additional term in the SOLA cost function of Eq.",
"REF , which will then have a specific form depending on the surface correction chosen.",
"The parameters for the correction are free parameters of the inversion, to be determined alongside the structural corrections.",
"However, this approach might be considered suboptimal in asteroseismology, where the number of individual modes is very limited and where including the surface correction might lead to low quality fits of the averaging kernels.",
"This problem is especially true for the cases studied here, where one might have around 10 individual modes or less to extract the structural constraints.",
"Another approach is to correct the frequencies before carrying out the inversion, using the empirical law provided in or the coefficients from and obtained while carrying out the forward modellingThis approach might induce correlations between the frequencies.",
"In this study, since the coefficients of the empirical law of are fixed from non-seismic parameters, no correlation is introduced.",
"Similarly, the correction has been implemented in AIMS as an additional free parameter of the forward modelling process and used to correct the theoretical frequencies, which also avoids the introduction of additional correlations..",
"The inversion is then carried out assuming that the surface effects have been corrected and no surface term is then added to the SOLA cost function.",
"In what follows, we will provide illustrations of both approaches.",
"Reference models were obtained with the AIMS software , , using both adiabatic and non-adiabatic frequencies for the target patched model.",
"The reference models were built using various sets of constraints, using the Liège stellar evolution code and included an Eddington grey atmosphere.",
"A first reference model for the artificial target was determined using the effective temperature, the average large frequency separation of the radial modes determined using a linear regression and the frequency of maximum power, $\\nu _{Max}$ , derived from the scaling laws.",
"However, to see the impact of the surface effects when using individual modes, we also carried out a second fit of the artificial target using the effective temperature and the individual frequencies of the radial mode as constraints.",
"No surface correction was considered in either case, implying that both fits should be biased.",
"We considered low order modes, from $n=1$ to $n=15$ , which have low frequencies compared to $\\nu _{Max}$ and thus should not display extreme changes due to the upper layers.",
"This is confirmed in Figure REF where we compare the adiabatic and non-adiabatic frequencies from $n=1$ up to $n=20$ .",
"As expected, a clear trend with frequency is observed.",
"Figure: Relative differences between adiabatic and non-adiabatic frequencies in frequency plotted against frequency for the radial modes of the patched model considered as target for our exercise.The results using AIMS for the forward modelling are the following.",
"Using the mean large frequency separation, the difference of the mean density value is of $3.6$ per cent when adiabatic frequencies are considered.",
"If non-adiabatic frequencies are used in the modelling, then the difference between the target value and the value determined by AIMS is of $3.8$ per cent.",
"The fact that using the non-adiabatic frequencies only induces a small change of the results is due to the properties of the dataset, which contains low frequencies for which non-adiabatic effects are small.",
"If one uses individual frequency values as seismic constraints, the difference is reduced by a factor 3, going down to $1.1$ per cent if adiabatic frequencies are considered and to $1.3$ per cent if non-adiabatic frequencies are used.",
"We then tested the inversion technique using 4 reference models close to the models found with AIMS, to also assess the model dependency of the inversion technique.",
"The results are illustrated in Figure REF and the error contributions are given in Figure REF for each model and most of the surface effect corrections.",
"In Figure REF , the notation “prior” and “posterior” has been adopted when the correction was respectively applied on the frequencies beforehand, using the coefficients directly from the paper and when the correction was applied in the SOLA cost function directly in a linearized form.",
"The notation “empirical” was used to denote the case where the coefficients were derived from the empirical formula in $T_{\\mathrm {eff}}$ and $\\log g$ from their paper.",
"In Figure REF , we chose not to plot the error contributions of some of the test cases to avoid redundancy.",
"Figure: Inversion results using various surface effects corrections for each of the 4 models using patched model I from as a target and either adiabatic or non-adiabatic frequencies.A quick inspection of the results show that the inversion results are always well below the $3.7$ per cent agreement with the target value showed by the forward modelling process using only the average large frequency separation.",
"However, not all results are superior to the $1.1$ per cent differences from the use of individual radial modes.",
"To better understand what is happening here, we have to take a look at the individual error contributions and see which one is contributing to the biases in some of the inversions.",
"A first inspection shows that the best results are often obtained when no correction is applied, except for model 2 where compensation is found for all other inversions and the excellent agreement is thus fortuitious.",
"This is a consequence of the low frequency modes used in the set, for which all surface effect corrections seem to be overestimated and thus bias the results, we will see how these effects change when another set of modes is used for the inversion.",
"A second conclusion that can be drawn is that for most cases, applying the correction within the SOLA cost function is not the best option.",
"Only model 1 still shows good results with this implementation of the surface effect correction.",
"In all other cases, a significant increase in the averaging kernel error $\\epsilon _{\\mathrm {Avg}}$ is seen and the quality of the results is relatively poor for both the and the correction laws.",
"The increase of $\\epsilon _{\\mathrm {Avg}}$ is simply due to the fact that including the correction within the SOLA cost function will introduce a strong trend in the averaging kernel which depends on the form of the correction.",
"This trend then leads to a much less accurate fit of the target function of the inversion and thus to a much less accurate result.",
"Figure: Error contributions ϵ Avg \\epsilon _\\mathrm {{Avg}}, ϵ Cross \\epsilon _{\\mathrm {Cross}} and ϵ Res \\epsilon _{\\mathrm {Res}} as defined in Eqs.",
", and for each of the 4 models selected as a reference for the patched target for some test cases of the surface effects.",
"Each test case has its dedicated subplot to make the analysis easier.The other approach we proposed was to introduce the surface corrections using the empirical law proposed by .",
"Overall, we can see from Figure REF that this approach leads to larger residual errors, hence, this method does not seem to reduce significantly the surface effect for the set of modes considered.",
"However, it has the strong advantage of not reducing the fit of the target function.",
"The poor performance of the surface correction laws in this particular test case is a consequence the very low frequency of the modes, which results in an overestimate of the surface corrections for the very low $n$ modes which are the most useful to the inversion.",
"Also, it seems that biasing the correction law only slightly affects the inversion results.",
"From Figure REF , it appears that shifts of less than $0.1$ per cent of the inverted mean density values are induced by changing the $T_{\\mathrm {eff}}$ and $\\log g$ values in the empirical law of .",
"In this particular case, we shifted the values of the quantities by $45K$ and $0.1$ dex respectively.",
"Larger shifts would induce larger deviations but they would remain negligible in the total error budget of the inverted results.",
"From this analysis, it seems that the combined law does the best job at reducing the residual error and thus the contribution of surface effects.",
"However, this reduction of the residual error is made at the expense of a larger increase of the error from the fit of the averaging kernels when this approach is directly implemented in the SOLA cost function.",
"It thus seems that the optimal choice would be to be able to apply the correction before carrying out the inversion or defining some law with a similar form as in for this surface correction.",
"To further test the impact of surface effects, we take a look at the impact of changing the set of observed modes and using only modes of higher frequencies when non-adiabatic effects are taken into account.",
"The results are illustrated in Figure REF for Model 1 and 4.",
"For both models, we either considered not including any surface correction or using the empirical formula of to correct the frequencies before carrying out the inversion.",
"We started with the full set of non-adiabatic frequencies available for the target, thus with modes of $n=1$ up to $n=20$ and proceeded to eliminate some of the lowest order modes up to $n=7$ .",
"Figure: Inversion results for models 1 and 4 using patched model I from as target and varying the lowest nn in the set of modes used for the inversion.From Figure REF , one can see that the results strongly depend on the set of modes used to carry out the inversion and that, as expected, the lowest order modes are crucial to derive a good result.",
"One could be tempted to say that for model 1 the empirical law does an excellent job at correcting the frequencies.",
"However, looking at Figure REF , we can see that the excellent results for the empirical correction are simply due to a fortuitous compensation effect.",
"These test cases have also revealed a fundamental limitation of the seismic determination of the mean density using the radial oscillations of red giant stars.",
"It appears that without the fundamental mode, the accuracy of the inversion is at best of $1.5$ per cent and that if only high $n$ modes are available, the mean density cannot be determined within less than 1 per cent to the observed value.",
"However, if the fundamental mode is present, the accuracy of the inversion goes down to less than 1 per cent and is not strongly affected by the surface effects.",
"The importance of the fundamental mode can also be observed when just looking at the change of the quality of the fit of the averaging kernel in Figure REF .",
"The importance of this specific mode is in fact not a surprise, since it carries a lot of information about the mass distribution inside the star , , and thus is crucial to seismic methods.",
"We can see that for both models, if the set of observed modes is reduced to $n$ higher than 7, the quality of the inversion result is greatly reduced.",
"This is observed whether a surface correction is applied or not, since $\\epsilon _{\\mathrm {Avg}}$ quickly becomes the dominant source of errors.",
"We can also see that the residual error, $\\epsilon _{\\mathrm {Res}}$ , used here to quantify the surface effects contribution, rises if no correction is applied.",
"However, we have also noted that the empirical correction of used in this test case does not always reduce efficiently the contribution of the surface effects.",
"The correction could perhaps provide a better reduction of the surface effect but it should not be directly implemented in the SOLA cost function, as it then strongly reduces the quality of the fit of the target function.",
"Figure: Error contributions ϵ Avg \\epsilon _{Avg}, ϵ Cross \\epsilon _{Cross} and ϵ Res \\epsilon _{Res} as defined in Eqs.",
", and for models 1 and 4, varying the lowest nn in the set of observed frequencies.In addition to the seismic inversion of the mean density, we also tested the variational formulation of the scaling law for the average large frequency separation determined from a linear regression, presented in .",
"For all test cases presented here, this approach provided worse results than the SOLA inversion.",
"This is also seen in the AIMS fits, where using the average large frequency separation leads to a $3.7$ per cent error on the estimate of the mean density.",
"This approach did not provide good results and was found to be very sensitive to surface effects.",
"For example, the results significantly varied if the empirical correction of was biased.",
"We also found that the quality of the results strongly varied with the set of modes.",
"For example, if low $n$ modes are used, the impact of surface effects is strongly reduced but since the modes are far from being in the asymptotic regime, the accuracy remains very low.",
"Results obtained from the large frequency separation improve if higher order modes are taken, but are then affected by the surface effects.",
"In addition, we observed, as in , that the accuracy of this method relied on error compensations rather than a high-quality fit of the kernels.",
"This implies that surface effects, non-linear behaviour, or an inadequate verification of the asymptotic relations can quickly lead to disagreements with the real value as large as 2 or 3 per cent.",
"In fact, in a worst case scenario, using non-adiabatic frequencies, differences as high as 8 per cent were observed for the mean density determined from the average large frequency separation.",
"Overall, it seems that using the information of the individual modes offers much better constraints, since the fits obtained with AIMS could reach an agreement with the target value of around 1 per cent.",
"A mean density inversion can in such cases provide additional information and reach below the 1 per cent limit for the mode set considered here.",
"Depending on the set of modes, the accuracy of the inversion technique will vary, but so will the accuracy of the forward modelling technique.",
"We emphasize here that the 1 per cent differences of the forward modelling process were obtained for a set of modes where the fundamental mode was observed.",
"Additional tests on a larger sample of artificial targets and observed modes are thus required to provide better insights on the problem of surface effects and its impact on forward modelling approaches.",
"An additional comment should also be made regarding the way the surface effects are simulated in this study.",
"First, the modelling of the non-adiabatic effects is far from perfect and problems still remain in the physical representation of the coupling between convection and oscillations, as can be seen from the differences in linewidth and amplitudes between simulated non-adiabatic spectra and observations in .",
"This implies that the tests we carried out with non-adiabatic frequencies are only qualitative but can serve as a warning when using various surface correction laws which are derived from purely adiabatic computations.",
"In addition to numerical exercises on artificial targets, we also tested the mean density inversions on a few eclipsing binaries observed by Kepler, previously studied by and , namely KIC5786154, KIC7037405, KIC8410637, KIC8430105 and KIC9970396.",
"It should be noted that in the present study, we concern ourselves with the stability of the mean density inversions.",
"A full assessment of the accuracy of seismic masses and radii using individual frequencies would require a better assessment of the surface effect corrections, the importance of the helium abundance and of the atmospheric models used for the whole sample of eclipsing binaries available, which is beyond the scope of this study."
],
[
"Peak-bagging and determination of individual frequencies",
"Mode frequencies were estimated by performing bespoke mode fitting to each of the stars.",
"Using the full set of available Kepler photometric data, we computed the estimate of the frequency power spectrum following .",
"We located the modes of oscillation using visual inspection and determined the mode identification (i.e., selected the radial and quadrupole modes).",
"We also checked the consistency of the detection with existing red giant measurements following .",
"In order to determine the frequencies for the radial modes, we fitted each pair of radial and quadrupole modes with a sum of Lorentzians.",
"For full details of the mode fitting method, we refer to .",
"Moreover, because of the binary nature of the targets, we have not applied a frequency correction to the mode frequencies normally applied to remove the Doppler shift as a result of the line-of-sight velocity ."
],
[
"Forward modelling and inversion results",
"The forward modelling process was carried out with the AIMS software, using the same grid as in the numerical exercises on artificial targets.",
"Three separate runs were made to check for convergence and reliability of the process.",
"The modelling was performed using only $\\left[\\mathrm {Fe}/\\mathrm {H}\\right]$ and the individual frequencies using surface corrections from .",
"For KIC8430105 and KIC8410637, a two terms surface correction was considered, while a single term correction was considered for the other targets whose spectrum contained much less oscillation modes.",
"The masses and radii from this forward modelling process are given in Table REF alongside the values from and .",
"Again, we mention that they only consist in a preliminary modelling result, as other constraints such as effective temperature, luminosity and/or radius could change these values to a certain extent and that a more thorough study using various physical ingredients is required for a full assessment of the robustness of these masses and radii determinations.",
"It should thus be noted that the reference models considered here used Eddington grey atmospheres and hence do not reproduce the effective temperature of the RGB.",
"Table: Masses and radii for the Kepler eclipsing binaries from seismic forward modelling and from eclipses measurements ( EB -B\\mathrm {EB-B} denotes values from and EB -G\\mathrm {EB-G} from ).The inversion results for the mean density of each target are given in Table REF , where we considered three cases for each reference model.",
"First, we carried out the inversion without applying any surface correction to the frequencies, second, we applied the correction as determined by the forward modelling in AIMS, third, we applied the empirical correction on the frequencies.",
"All these corrections were applied before carrying out the inversion, as including them directly in the cost function of the SOLA method does not allow for an accurate reproduction of the target function of the inversion and leads to strong biases.",
"From Table REF , we can see that the surface effect corrections can induce variations of up to 1 per cent of the mean density of the star.",
"This result confirms the fact that the errors of seismic inversions cannot be reliably determined only by the propagation of the observed uncertainties, as other effects such as model-dependency and effects of surface corrections will dominate the uncertainties.",
"Similar tests were performed using a few models around the best reference determined by AIMS and the results remained within this 1 per cent interval, defining the total variation of the inversion results.",
"Therefore, taking into account potential further model-dependencies and uncertainties, we can consider that in these test cases, the mean density has been determined with a precision of $\\pm 1.5$ per cent, considering a conservative interval 3 times larger than that determined by the mean density inversion.",
"This also implies that within these very conservative uncertainties, all reference models agreed with the inversion results.",
"This is no surprise, as most of the frequencies were very well fitted by the forward modelling process, as can be illustrated by the echelle diagram computed for KIC8410637, plotted in Figure REF and showing the good agreement between theoretical and observed frequencies once surface effects corrections are included.",
"Figure: Echelle diagram of KIC8410637 illustrating the observed, theoretical frequencies (both corrected for surface effects using formula and uncorrected).Table: Inverted mean densities for the Kepler eclipsing binaries of this study.As such, this implies that seismic inversions of the mean density can in these cases play a first role of confirmation of the quality of the fit, but can also be included in a second step of forward modelling to determine better masses for red giants.",
"Indeed, using individual frequencies directly as constraints can lead to cost functions dominated by seismic information and unrealistically precise determinations of fundamental parameters of stars.",
"Hence, using the inverted mean density directly as a kind of observational constraint, as done for example for the 16Cyg binary system in , , alongside luminosities or radii determined by Gaia may lead to more accurate masses of red giants and clump stars.",
"To illustrate this, we show in Figure REF the mass values obtained from the inverted mean densities and radii values from the eclipses observations.",
"We selected the most precise determinations of radii between and .",
"This is not a major concern since the radii values agree very well with each other for the targets present in both studies.",
"We note that for the case of KIC7037405 and KIC9970396, the mean density inversions lead to an unambiguous correction which leads to a very good agreement with the dynamical mass of for KIC7037405.",
"Further investigations show that the mean density derived from the dynamical values of the mass and radius from differs by around $4 \\pm 2.5$ per cent with the ones determined in this study.",
"Similarly, the values determined from global seismic indices using the PARAM software also show discrepancies of around 2 per cent.",
"For this particular target, the closest value to the inverted one is found using the mass and radii values from the corrected scaling relations in Table 4 of , while the uncorrected scaling relations give a disagreement of 4 per cent.",
"As for KIC9970396, a slight disagreement remains and further modelling using tighter constraints is required to determine whether the slightly higher mass value found by seismology is due to model-dependencies or if the origin is to be found elsewhere.",
"In this case, the mean density value determined from the eclipses is in agreement within $1 \\sigma $ (for this case around 3 per cent) with the value determined here.",
"The value determined from the corrected scaling relations shows an already quite good agreement.",
"However, the mean density value determined from the uncorrected scaling relations shows a larger disagreement of approximately 6 per cent whereas the values determined from PARAM differ by approximately 3 per cent with the inverted results and significantly from the mean density value from the eclipses.",
"This seems to indicate that overall, using the entire information of the frequency spectrum leads to the best agreement with the dynamical values.",
"Figure: Masses and radii values obtained from eclipses observations, seismology, and combining the inverted mean densities using various surface corrections to the dynamical radii measures.We can see in this Figure that for all cases, using the inverted mean density and the dynamical radii further improve the results.",
"However, most of the improvement does not stem from the improvement of the mean density as the corrections remained quite small, but rather from the use of the radii values from the eclipses.",
"This implies that the use of classical constraints such as precise luminosities or radii will lead a major role in the determination of reliable masses for red giants and red clump stars."
],
[
"Conclusion",
"In this study, we have demonstrated the feasibility and robustness of seismic inversions of the mean density of red giant and red clump stars using only a few observed radial modes, providing an extension to the framework of the approach initially applied to main-sequence solar-like stars , .",
"We have started by introducing the approach to the inverse problem in Sect.",
"and applied it in extensive numerical tests using calibration techniques based on effective temperature and luminosity in Sect.",
"REF as well as seismic constraints REF .",
"We have analysed the possibility of carrying out inversions for the mean density both for red giant branch and clump stars, as well as trying to determine the mean density of a misidentified clump star.",
"In the last case, the inversion proved to be inaccurate and thus the method provided here is only valid for unambiguously identified red giant branch and clump stars.",
"This last point also proves the need for a reliable reference model before carrying out the inversion.",
"This weakness has also been observed for other numerical exercises were the so-called cross-term errors could contribute very significantly to the total error budget of the inversion and even dominate other errors.",
"This is radically different from inversions on the main-sequence and is due to both the small number of modes and the large radial extent of the ionization zones, leading to more widespread differences in $\\Gamma _{1}$ from one model to the other.",
"In addition to testing the use of seismic constraints, we also carried out in Sect.",
"REF inversions for an artificial target from including an atmospheric model from an averaged hydrodynamical simulation.",
"For this target, we used both adiabatic and non-adiabatic frequencies to carry out the inversion.",
"These tests illustrated the impact of surface effects and the importance of correcting them.",
"It should be noted that none of the current methods seemed to work perfectly, as error compensations were sometimes observed and that directly implementing the corrections as free parameters in the SOLA method provided inaccurate fits of the target function of the inversion.",
"In Sect.",
", we carried out mean density inversions on observed red giants in eclipsing binary systems from the studies of and .",
"We first showed that using individual frequencies, corrected for surface effects using the approach of , lead to a much better agreement in terms of mass and radius than simply using scaling relations or global seismic indices.",
"Furthermore, we showed that the mean density inversions could provide either further small corrections to the mean density obtained from forward modelling or an additional verification step.",
"Combining the inverted mean density to the radii determinations from eclipses provided an overall good agreement in terms of masses for these stars.",
"However, these test cases also demonstrate the importance of classical constraints for the modelling of red giants, since the most significant improvement came from using the dynamical radii values.",
"Indeed, the mean density of these stars was already very accurately determined through forward modelling.",
"A couple of conclusions can be drawn from this last point.",
"Firstly, that pure seismic modelling using only radial modes might lead to degeneracies since the frequencies will be mostly sensitive to the mean density of the star.",
"Secondly, directly using the individual frequencies might lead to underestimated uncertainties and the seismic constraints might dominate the classical constraints.",
"Therefore, using directly an inverted value for the mean density, alongside a precise and accurate value for the luminosity, the $\\left[\\mathrm {Fe/H}\\right]$ and the radius might provide a more direct and balanced approach to the modelling of red giant stars.",
"Other seismic constraints could also be used, such as the asymptotic period spacing or ratios of radial oscillation frequencies as is done for Cepheids and other classical pulsators.",
"The application of such approaches to an extended set of eclipsing binaries will provide a unique opportunity to test the reliability of seismic modelling and the importance of classical constraints.",
"In the near future, using such approaches alongside constraints from the second Gaia data release will help better understand the properties of red giants.",
"Providing more accurate masses is indeed crucial to determine the properties of various stellar populations in the Galaxy but also for example to pinpoint the properties of additional mixing at the base of the convective envelope, manifesting itself through the so-called RGB bump [2], , .",
"In stellar clusters, accurate masses could also be used to characterize mass loss on the red giant branch , one of the major issues in current stellar evolution models."
],
[
"Acknowledgements",
"AM and GB acknowledge support from the ERC Consolidator Grant funding scheme (project ASTEROCHRONOMETRY, G.A.",
"n. 772293).",
"We gratefully acknowledge the support of the UK Science and Technology Facilities Council (STFC).",
"S.J.A.J.S.",
"is funded by ARC grant for Concerted Research Actions, financed by the Wallonia-Brussels Federation.",
"TS acknowledges support from JSPS KAKENHI Grant Number 17J00631.",
"This article made use of an adapted version of InversionKit, a software developed in the context of the HELAS and SPACEINN networks, funded by the European Commissions's Sixth and Seventh Framework Programmes."
]
] | 1808.08391 |
[
[
"The Social Cost of Strategic Classification"
],
[
"Abstract Consequential decision-making typically incentivizes individuals to behave strategically, tailoring their behavior to the specifics of the decision rule.",
"A long line of work has therefore sought to counteract strategic behavior by designing more conservative decision boundaries in an effort to increase robustness to the effects of strategic covariate shift.",
"We show that these efforts benefit the institutional decision maker at the expense of the individuals being classified.",
"Introducing a notion of social burden, we prove that any increase in institutional utility necessarily leads to a corresponding increase in social burden.",
"Moreover, we show that the negative externalities of strategic classification can disproportionately harm disadvantaged groups in the population.",
"Our results highlight that strategy-robustness must be weighed against considerations of social welfare and fairness."
],
[
"Introduction",
"As machine learning increasingly supports consequential decision making, its vulnerability to manipulation and gaming is of growing concern.",
"When individuals learn to adapt their behavior to the specifics of a statistical decision rule, its original predictive power will deteriorate.",
"This widely observed empirical phenomenon, known as Campbell's Law or Goodhart's Law, is often summarized as: “Once a measure becomes a target, it ceases to be a good measure” [24].",
"Institutions using machine learning to make high-stakes decisions naturally wish to make their classifiers robust to strategic behavior.",
"A growing line of work has sought algorithms that achieve higher utility for the institution in settings where we anticipate a strategic response from the the classified individuals [10], [4], [13].",
"Broadly speaking, the resulting solution concepts correspond to more conservative decision boundaries that increase robustness to some form of distributional shift.",
"But there is a flip side to strategic classification.",
"As insitutional utility increases as a result of more cautious decision rules, honest individuals worthy of a positive classification outcome may face a higher bar for success.",
"The costs incurred by individuals as a consequence of strategic classification are by no means hypothetical, as the example of lending shows.",
"In the United States, credit scores are widely deployed to allocate credit.",
"However, even creditworthy individuals routinely engage in artificial practices intended to improve their credit scores, such as opening up a certain number of credit lines in a certain time period [9].",
"In this work, we study the tension between accuracy to the institution and impact to the individuals being classified.",
"We first introduce a general measure of the cost of strategic classification, which we call the social burden.",
"Informally, the social burden measures the expected cost that a positive individual needs to incur to be correctly classified correctly.",
"For a broad class of cost functions, we prove there exists an intrinsic trade-off between institutional accuracy and social burden: any increase in institutional accuracy comes at an increase in social burden.",
"Moreover, we precisely characterize this trade-off and show the commonly considered Stackelberg equilibrium solution achieves maximal institutional accuracy at the expense of maximal social burden.",
"Equipped with this generic trade-off result, we turn towards a more careful study of how the social burden of strategic classification impacts different subpopulations.",
"We find that the social burden can fall disproportionally on disadvantaged subpopulations, under two different notions by which one group can be disadvantaged relative to another group.",
"Furthermore, we show that as the institution improves its accuracy, it exacerbates the gap between the burden to an advantaged and disadvantaged group.",
"Finally, we illustrate these conditions and their consequences with a case study on FICO data.",
"In this paper, we make the following contributions: We prove a general result demonstrating the trade-off between institutional accuracy and individual utility in the strategic setting.",
"Our theoretical characterization is supplemented with examples illustrating when an institution would prefer to operate along different points in this trade-off curve.",
"We show fairness considerations inevitably arise in the strategic setting.",
"When individuals incur cost as a consequence of making a classifier robust to strategic behavior, we show the costs can disproportionally fall by disadvantaged subpopulations.",
"Furthermore, as the institution increases its robustness, it also increases the disparity between subpopulations.",
"Using FICO credit data as a case-study, we empirically validate our modeling assumptions and illustrate both the general trade-offs and fairness concerns involved with strategic classification in a concrete setting.",
"Reflecting on our results, we argue that the existing view of strategic classification has been instituition-centric, ignoring the social burden resulting from improved institutional utility.",
"Our framework makes it possible to select context-specific trade-offs between institutional and individual utility, leading to a richer space of solutions.",
"Another key insight is that discussions of strategy-robustness must go hand in hand with considerations of fairness and the real possibility that robustness-promoting mechanisms can have disparate impact in different segments of the population.",
"Throughout this work, we consider the binary classification setting.",
"Each individual has features $x \\in \\mathcal {X}$ and a label $y \\in \\mathcal {Y}= \\lbrace 0, 1\\rbrace $ .",
"The institution publishes a classifier $f : \\mathcal {X}\\rightarrow \\mathcal {Y}$ .",
"In the non-strategic setting, the institution maximizes the non-strategic utility, which is simply the classification accuracy of $f$ : $& \\mathcal {U}(f) = \\mathbb {P}(f(x) = y) \\,.$ In the strategic setting, the individual can modify their features, and the institution aims to preempt the individual's strategic manipulation.",
"In response to the institution's classifier $f$ , the individual can change her features $x$ to new features $x^{\\prime }$ .",
"However, modification incurs a cost given by $c : \\mathcal {X}\\times \\mathcal {X}\\rightarrow \\mathbb {R}_{\\ge 0}$ .",
"The individual then receives an individual utility $u_x(f, x^{\\prime }) = f(x^{\\prime })- c(x, x^{\\prime })$ , which trades off between the cost of manipulation $c(x, x^{\\prime })$ and the benefits of classification $f(x^{\\prime })$ .",
"The institution models the individual $x$ as maximizing their utility $u_x(f,x^{\\prime })$ and acting according to the best-response to the classifier $f$ : $\\Delta (x; f) = \\operatornamewithlimits{arg\\,max}_{x^{\\prime }} u_x(f, x^{\\prime }).$ When it is clear from context we will drop the dependence on $f$ and write the individual's best response as $\\Delta (x)$ .",
"Although $\\Delta (x)$ may not have a unique maximizer, it is assumed that the individual $x$ does not adapt her features if she is already accepted by the classifier, i.e.",
"$f(x) = 1$ , or if there is no maximizer $x^{\\prime }$ she can move to such that $f(x^{\\prime }) = 1$ .",
"In cases where the individual does adapt, let $x^{\\prime }$ be an arbitrary maximizer such that $f(x^{\\prime }) = 1$ .",
"In practice, it is unlikely individuals actually play best-response solutions, and we will discuss as appropriate the impact of deviations from best-response play.",
"Given this model, the institution aims to maximize the strategic utility, which measures accuracy after individual responses: $\\mathcal {U}_\\Delta (f) = \\mathbb {P}(f(\\Delta (x)) = y).$ For example, imagine that the institution is trying to rank pages on a social network.",
"Although the number of likes a page has may be predictive, it is also an easy feature to game.",
"Therefore, models with high strategic utility will assign low weight to this feature, even if it is useful in the static setting.",
"Henceforth, we will refer to the strategic utility as simply the institutional utility."
],
[
"Focusing purely on maximizing $\\mathcal {U}_\\Delta $ , as done in prior work, ignores the cost a classifier imposes on individuals [5], [14], [12].",
"To account for these costs, we define the individual burden of a classifier $f$ as the minimum cost an individual needs to incur in order to be classified positively: $b_f(x) = \\min _{f(x^{\\prime }) = 1} c(x, x^{\\prime })$ .",
"For positive individuals with $y=1$ , a high individual burden means the individual has to incur great cost to obtain the correct classification.",
"To quantity this cost, we introduce the social burden, defined as the expected individual burden of positive individuals.",
"Definition 2.1 (Social burden) The social burden of a classifier $f$ is defined as $\\mathcal {B}_{+}(f) = \\mathbb {E}\\left[b_f(x) \\mid y = 1 \\right]$ .",
"The social burden measures the expected cost that positive individuals would need to incur to be classified positively, regardless of whether the best response $\\Delta (x)$ indicates that they should adapt.",
"One could imagine other ways of measuring the impact on individuals, such as the expected utility of positive individuals, $\\mathbb {E}\\left[u_x(f, \\Delta (x)) \\mid y = 1\\right]$ , or the corresponding measure over all individuals, rather than only positive individuals.",
"Most of our results still hold for these alternative measures, and we relegate discussion about the choice of social burden to Section ."
],
[
"While there are many possible models for the cost function, we restrict our attention to a natural set of cost functions that we call outcome monotonic.",
"Outcome monotonic costs capture two intuitive properties: (1) Monotonically improving one's outcome requires monotonically increasing amounts of work, and (2) it is zero cost to worsen one's outcome.",
"This captures the intuition that, for example, it is harder to pay back loans than it is to go bankrupt.",
"Definition 2.2 (Outcome likelihood) The outcome likelihood of an individual $x$ is $\\ell (x) = \\mathbb {P}(Y = 1 \\mid X =x)$ .",
"We assume that all individuals have a positive outcome likelihood, i.e, $\\ell (x)> 0$ for all $x$ .",
"Definition 2.3 (Outcome Monotonic Cost) A cost function $c : \\mathcal {X}\\times \\mathcal {X}\\rightarrow \\mathbb {R}_{\\ge 0}$ is outcome monotonic if for any $x, x^{\\prime }, x^{*} \\in X$ the following properties hold.",
"Zero-cost to move to lower outcome likelihoods.",
"$c(x, x^{\\prime }) >0$ if and only if $\\ell (x^{\\prime }) > \\ell (x)$ .",
"Monotonicity in first argument.",
"$c(x, x^{*}) > c(x^{\\prime }, x^{*}) > 0$ if and only if $\\ell (x^*) > \\ell (x^{\\prime }) > \\ell (x)$ .",
"Monotonicity in second argument.",
"$c(x, x^{*}) > c(x, x^{\\prime }) >0$ if and only if $\\ell (x^*) > \\ell (x^{\\prime }) > \\ell (x)$ .",
"Under these assumptions, we can equivalently express the cost as a cost over outcome likelihoods, $c_L: \\ell (\\mathcal {X}) \\times \\ell (\\mathcal {X}) \\rightarrow \\mathbb {R}_{\\ge 0}$ , defined in the following lemma.",
"Lemma 2.1 When the cost function $c(x, x^{\\prime })$ is outcome monotonic, then it can be written as a cost function over outcome likelihoods $c_L(l, l^{\\prime })c(x, x^{\\prime })$ where $x, x^{\\prime } \\in \\mathcal {X}$ are any points such that $l =\\ell (x)$ and $l^{\\prime } = \\ell (x^{\\prime })$ .",
"The monotonicity assumptions imply that if $\\ell (x^*) = \\ell (x^{\\prime })$ , then $c(\\cdot , x^{\\prime }) = c(\\cdot , x^*)$ and $c(x^{\\prime }, \\cdot ) = c(x^*, \\cdot )$ .",
"Thus, $c_L(l, l^{\\prime }) c(x, x^{\\prime })$ is well-defined because any points $x$ and $x^{\\prime }$ such that $l = \\ell (x)$ and $l^{\\prime } = \\ell (x^{\\prime })$ yield the same value of $c(x,x^{\\prime })$ .",
"Throughout the paper, we will make use of the equivalent likelihood cost $c_L$ when a proof is more naturally expressed with $c_L$ , rather than with the underlying cost $c$ .",
"In this section, we characterize the inherent trade-offs between institutional utility and social burden in the strategic setting.",
"In particular, we show any classifier that improves institutional utility over the best classifier in the static setting causes a corresponding increase in social burden.",
"To prove this result we first show that any classifier can be represented as a threshold classifier that accepts all individuals with outcome likelihood greater than some threshold $\\tau \\in [0, 1]$ .",
"Then, we show increasing utility for the institution requires raising this threshold $\\tau $ , but that this always increases the social burden.",
"Equipped with this result, we show the (Pareto-optimal) set of classifiers that increase institutional utility in the strategic setting corresponds to an interval $I$ .",
"Each threshold $\\tau \\in I$ represents a particular trade-off between institutional utility and social burden.",
"Strategic classification corresponds to one extremum: the best strategic utility but the worst social burden.",
"The non-strategic utility corresponds to the other: doing nothing to prevent gaming.",
"Neither is likely to be the right trade-off in practical contexts.",
"Real domains will require a careful weighting of these two utilities, leading to a choice somewhere in between.",
"Thus, a main contribution of our work is exposing this interval."
],
[
"General Trade-Off",
"We now proceed to prove the trade-off between institutional utility and social burden.",
"Our first step is to show that in the strategic setting we can restrict attention to classifiers that threshold on the outcome likelihood (assuming the cost is outcome monotonic as in Definition REF ).",
"Definition 3.1 (Outcome threshold classifier) An outcome threshold classifier $f$ is a classifier of the form $f(x) =\\mathbb {I}\\lbrace \\ell (x) \\ge \\tau \\rbrace $ for $\\tau \\in [0, 1]$ .",
"In practice, the institution may not know the outcome likelihood $\\ell (x) =\\mathbb {P}(Y = 1 \\mid X = x)$ .",
"However, as shown in the next lemma, for any classifier that they do use, there is a threshold classifier with equivalent institutional utility and social burden.",
"Thus, we can restrict our theoretical analysis to only consider threshold classifiers.",
"Lemma 3.1 For any classifier $f$ there is an outcome threshold classifier $f^{\\prime }$ such that $\\mathcal {U}_\\Delta (f) = \\mathcal {U}_\\Delta (f^{\\prime })$ and $\\mathcal {B}_{+}(f) = \\mathcal {B}_{+}(f^{\\prime })$ .",
"Let $\\tau (f) = \\min _{x: f(x) = 1} \\ell (x)$ be the minimum outcome likelihood at which an individual is accepted by the classifier $f$ .",
"Then, let $f^{\\prime }(x) = \\mathbb {I}\\lbrace \\ell (x) \\ge \\tau (f)\\rbrace $ be the outcome threshold classifier that accepts all individuals above $\\tau (f)$ .",
"We will show that the institutional utility and social burden of $f$ and $f^{\\prime }$ are equal.",
"Since the cost function is outcome monotonic, it is the same cost to move to any point with the same outcome likelihood.",
"Furthermore, it is higher cost to move to points of higher likelihood, i.e, if $\\ell (x^{\\prime }) >\\ell (x^{*}) > \\ell (x)$ , then $c(x, x^{\\prime })> c(x, x^{*}) > 0$ .",
"Since individuals game optimally, when an individual changes her features in response to the classifier $f$ , she has no incentive to move to a point with likelihood higher than $\\tau (f)$ – that would just cost more.",
"Therefore, she will move to any point with likelihood $\\tau (f)$ to be accepted by $f$ and will incur the same cost, regardless of which point it is.",
"Thus, we can write the set of individuals accepted by $f$ , $\\mathcal {A}_{\\Delta }(f)$ , as $\\mathcal {A}_{\\Delta }(f) & = \\lbrace x \\mid f(\\Delta (x)) = 1\\rbrace \\\\& = \\lbrace x \\mid \\exists x^{\\prime } : f(x^{\\prime }) = 1, ~c(x, x^{\\prime }) \\le 1 \\rbrace \\\\& = \\lbrace x \\mid \\exists ~x^{\\prime }: \\ell (x^{\\prime }) = \\tau (f), ~c(x, x^{\\prime }) \\le 1 \\rbrace \\, .$ Since $\\tau (f) = \\tau (f^{\\prime })$ , the individuals accepted by $f$ and $f^{\\prime }$ are equal: $\\mathcal {A}_{\\Delta }(f) = \\mathcal {A}_{\\Delta }(f^{\\prime })$ .",
"Therefore, their institutional utilities $\\mathcal {U}_\\Delta (f)$ and $\\mathcal {U}_\\Delta (f^{\\prime })$ are equal.",
"We can similarly show that the social burdens of $f$ and $f^{\\prime }$ are also equal: $\\mathcal {B}_{+}(f) & = \\mathbb {E}\\left[\\min _{x^{\\prime }:f(x^{\\prime }) = 1} c(x, x^{\\prime }) \\mid y = 1 \\right] \\, , \\\\& = \\mathbb {E}\\left[c(x, x^{\\prime }) \\mid y = 1 \\right] \\quad \\text{for some }\\;x^{\\prime } : \\ell (x^{\\prime }) = \\tau (f), \\\\& = \\mathbb {E}\\left[c(x, x^{\\prime }) \\mid y = 1 \\right] \\quad \\text{for some }\\; x^{\\prime } :\\ell (x^{\\prime }) = \\tau (f^{\\prime }), \\\\& = \\mathbb {E}\\left[\\min _{x^{\\prime }:f^{\\prime }(x^{\\prime }) = 1} c(x, x^{\\prime }) \\mid y = 1 \\right] = \\mathcal {B}_{+}(f^{\\prime }) \\, .$ Since outcome threshold classifiers can represent all classifiers in the strategic setting, we will henceforth only consider outcome threshold classifiers.",
"Furthermore, we will overload notation and use $\\mathcal {U}_\\Delta (\\tau )$ and $\\mathcal {B}_{+}(\\tau )$ to refer to $\\mathcal {U}_\\Delta (f_\\tau )$ and $\\mathcal {B}_{+}(f_\\tau )$ where $f_\\tau (x) = \\mathbb {I}\\lbrace \\ell (x) \\ge \\tau \\rbrace $ is the outcome threshold classifier with threshold $\\tau $ .",
"Figure: The general shapes of the institution utility and social burden as afunction of the threshold the institution chooses.",
"The threshold t 0 t_0 is thenon-strategic optimal, while the threshold τ * \\tau ^{*} is the Stackelbergequilibrium.Figure REF illustrates how institutional utility and social burden change as the threshold of the classifier increases.",
"The institutional utility is quasiconcave, while the social burden is monotonically non-decreasing.",
"The next lemma provides a formal characterization of the shapes shown in Figure REF .",
"Theorem 3.1 The institutional utility $\\mathcal {U}_\\Delta (\\tau )$ is quasiconcave in $\\tau $ and has a maximum at a threshold $\\tau ^{*} \\ge \\tau _0$ where $\\tau _0=0.5$ is the threshold of the non-strategic optimal classifier.",
"The social burden $\\mathcal {B}_{+}(\\tau )$ is monotonically non-decreasing in $\\tau $ .",
"Furthermore, if $\\mathcal {U}_\\Delta (\\tau ) \\ne \\mathcal {U}_\\Delta (\\tau ^{\\prime })$ , then $\\mathcal {B}_{+}(\\tau ) \\ne \\mathcal {B}_{+}(\\tau ^{\\prime })$ .",
"Let $\\mathcal {A}_{\\Delta }(\\tau )$ and $\\mathcal {A}(\\tau )$ be the set of individuals accepted by $f$ in the strategic and non-strategic setting, respectively.",
"If $\\tau <\\tau _0$ , we have $\\mathcal {A}_{\\Delta }(\\tau ) \\supseteq \\mathcal {A}_{\\Delta }(\\tau _0) \\supseteq \\mathcal {A}(\\tau _0)$ .",
"Since $\\mathcal {A}(\\tau _0)$ is the optimal non-strategic acceptance region, any $x\\notin \\mathcal {A}(\\tau _0)$ has $\\ell (x) < 0.5$ , and one can increase $\\mathcal {U}_\\Delta $ by not accepting $x$ , which implies $\\mathcal {U}_\\Delta (\\tau ) \\le \\mathcal {U}_\\Delta (\\tau _0)$ .",
"Therefore, if a threshold $\\tau ^{*}$ is optimal for the institution, i.e.",
"if $\\mathcal {U}_\\Delta (\\tau ^{*}) = \\max _{\\tau } \\mathcal {U}_\\Delta (\\tau )$ , then $\\tau ^*\\ge \\tau _0$ .",
"Recall that a univariate function $f(z)$ is quasiconcave if there exists $z^*$ such that $f$ is non-decreasing for $z < z^*$ and is non-increasing for $z > z^*$ .",
"Let $\\tau ^{*}$ be as above.",
"For $\\tau _0 < \\tau _1 <\\tau ^{*}$ , we have $\\mathcal {A}_{\\Delta }(\\tau _0) \\supseteq \\mathcal {A}_{\\Delta }(\\tau _1)\\supseteq \\mathcal {A}_{\\Delta }(\\tau ^{*})$ .",
"Since $\\mathcal {U}_\\Delta (\\tau ^{*})$ is optimal, $\\mathcal {A}_{\\Delta }(\\tau ^{*})$ is the optimal strategic acceptance region, and thus $\\mathcal {U}_\\Delta (\\tau _0) \\le \\mathcal {U}_\\Delta (\\tau _1) \\le \\mathcal {U}_\\Delta (\\tau ^{*})$ .",
"Similarly, if $\\tau _1 > \\tau _0 > \\tau ^{*}$ we have that $\\mathcal {A}_{\\Delta }(\\tau ^*) \\supseteq \\mathcal {A}_{\\Delta }(\\tau _0) \\supseteq \\mathcal {A}_{\\Delta }(\\tau _1)$ , and thus $\\mathcal {U}_\\Delta (\\tau _1) \\le \\mathcal {U}_\\Delta (\\tau _0) \\le \\mathcal {U}_\\Delta (\\tau ^*)$ .",
"Therefore, $\\mathcal {U}_\\Delta (\\tau )$ is quasiconcave in $\\tau $ .",
"The individual burden $b_\\tau (x) = \\min _{\\ell (x^{\\prime }) \\ge \\tau } c(x, x^{\\prime })$ is monotonically non-decreasing in $\\tau $ .",
"Since the social burden $\\mathcal {B}_{+}(\\tau )$ is equal to $\\mathbb {E}[b_\\tau (x) \\mid y = 1]$ , the social burden is also monotonically non-decreasing.",
"Suppose $\\mathcal {U}_\\Delta (\\tau ) \\ne \\mathcal {U}_\\Delta (\\tau ^{\\prime })$ and without loss of generality let $\\tau < \\tau ^{\\prime }$ .",
"For all individuals $x$ , $b_{\\tau ^{\\prime }}(x) \\ge b_{\\tau }(x)$ .",
"If there is at least one individual $x$ such that $b_{\\tau ^{\\prime }}(x) > b_{\\tau }(x)$ , then $\\mathcal {B}_{+}(\\tau ^{\\prime }) > \\mathcal {B}_{+}(\\tau )$ .",
"But since $\\mathcal {U}_\\Delta (\\tau ) \\ne \\mathcal {U}_\\Delta (\\tau ^{\\prime })$ , there must exist an individual $x$ such that $x \\in \\mathcal {A}_{\\Delta }(\\tau ) / \\mathcal {A}_{\\Delta }(\\tau ^{\\prime })$ and $p(X = x \\mid Y = 1) > 0$ since $\\ell (x) > 0$ by assumption.",
"For this individual $b_{\\tau ^{\\prime }}(x) > b_{\\tau }(x)$ .",
"Therefore, $\\mathcal {U}_\\Delta (\\tau ) \\ne \\mathcal {U}_\\Delta (\\tau ^{\\prime })$ implies $\\mathcal {B}_{+}(\\tau ^{\\prime }) \\ne \\mathcal {B}_{+}(\\tau )$ .",
"As a corollary, if the institution increases its utility beyond that attainable by the optimal classifier in the non-strategic setting, then the institution also causes higher social burden.",
"Corollary 3.1 Let $\\tau $ be any threshold and $\\tau _0 = 0.5$ be the optimal threshold in the non-strategic setting.",
"If $\\mathcal {U}_\\Delta (\\tau ) > \\mathcal {U}_\\Delta (\\tau _0)$ , then $\\mathcal {B}_{+}(\\tau ) > \\mathcal {B}_{+}(\\tau _0)$ ."
],
[
"Choosing a Concrete Trade-off",
"The previous section shows increases in institutional utility come at a cost in terms of social burden and vice-versa.",
"This still leaves open the question: what is the concrete trade-off an institution should choose?",
"Theorem REF provides a precise characterization of the choices available to trade-off between institutional utility and social burden.",
"The baseline choice for the institution is to not account for strategic behavior and use the non-strategic optimum $\\tau _0$ .",
"Maximizing utility without regard to social burden leads the institution to choose $\\tau ^{*}$ .",
"In general, the interval $[\\tau _0,\\tau ^{*}]$ offers the set of trade-offs the institution considers.",
"Choosing $\\tau > \\tau _0$ can increase robustness at the price of increasing social burden.",
"Thresholds $\\tau >\\tau ^{*}$ are not Pareto-efficient and are not considered.",
"Much of the prior work in machine learning has focused exclusively on solutions corresponding to the thresholds at the extreme: $\\tau _0$ and $\\tau ^{*}$ .",
"The threshold $\\tau _0$ is the solution when strategic behavior is not accounted for.",
"The threshold $\\tau ^{*}$ is also known as the Stackelberg equilibrium and is the subject of recent work in strategic classification [5], [13], [12].",
"While using $\\tau ^{*}$ may be warranted in some cases, a proper accounting of social burden would lead institutions to choose classifiers somewhere between the extremes of $\\tau _0$ and $\\tau ^{*}$ .",
"The exact choice of $\\tau \\in [\\tau _0, \\tau ^{*}]$ is context-dependent and depends on balancing concerns between institutional and broader social interest.",
"We now highlight cases where using $\\tau _0$ or $\\tau ^{*}$ may be suboptimal, and using a threshold $\\tau \\in (\\tau _0, \\tau ^{*})$ that balances robustness with social burden is preferable.",
"Example 3.1 (Expensive features.)",
"If measuring a feature is costly for individuals and offers limited gains in predictive accuracy, an institution may choose to ignore the feature, even if it means giving up accuracy on the margin.",
"In an educational context, a university may decide to no longer require applicants to submit standardized test scores, which can cost applicants hundreds of dollars, if the corresponding improvement in admissions outcomes is very small [1].",
"Example 3.2 (Reducing social burden under resource constraints.)",
"Aid organizations increasingly use machine learning to determine where to allocate resources after natural disasters [17].",
"In these cases, positive individuals are precisely those people who are in need of aid and may experience very high costs to change their features.",
"Using thresholds with high social burden is therefore undesirable.",
"At the same time, aid organizations often face significant resource constraints.",
"False positives from individuals gaming the classifier ties up resources that could be better used elsewhere.",
"Consequently, using the non-strategic threshold is also undesirable.",
"The aid organization should choose a some threshold $\\tau $ with $\\tau _0 < \\tau <\\tau ^{*}$ that reflects these trade-offs.",
"Example 3.3 (Misspecification of agent model.)",
"Strategic classification models typically assume the individual optimally responds to the classifier $f$ .",
"In reality, individuals will not have perfect knowledge of the classifier $f$ when it is first deployed.",
"Instead, they may be able to learn about how the classifier works over time, and gradually improve their ability to game the classifier.",
"For example, self-published romance authors exchanged information in private chat groups about how to best game Amazon's book recommendation algorithms [18].",
"For the institution, it is difficult to a priori model the dynamics of how information about the classifier propagates.",
"A preferable solution may be to simply make the assumption that the individual can best respond to the classifier, but to only gradually increase the threshold from the non-strategic $\\tau _0$ to the Stackelberg optimal $\\tau ^*$ over time.",
"In fact, misspecification of the agent model (described above), is why [4] suggest the Stackelberg equilibrium is too conservative, and instead prefer to use Nash equilibrium strategies.",
"Complementary to their observation, we show that there is a more general reason Nash equilibria may be preferable.",
"Namely, that Nash equilibria have lower social burden than the Stackelberg solution.",
"As the following lemma shows, in our context, the set of Nash equilibria form an interval $[t_N,\\tau ^{*}]\\subset I$ for some $t_N \\ge t_0$ .",
"The proof is deferred to the appendix.",
"Lemma 3.2 Suppose the cost over likelihoods $c_L$ is continuous and $\\ell (\\mathcal {X}) = [0, 1]$ , i.e, all likelihoods have non-zero support.",
"Then, the set of Nash equilibrium strategies for the institution is $[\\tau _N, \\tau ^*]$ for some $\\tau _N \\ge \\tau _0$ where $\\tau _0 = 0.5$ is the non-strategic optimal threshold and $\\tau ^*$ is the Stackelberg equilibrium strategy.",
"The Stackelberg equilibrium requires the institution to choose $\\tau ^{*}$ , whereas Nash equilibria give the institution latitude to trade-off between institutional utility and social burden by choosing from the interval $[t_N,\\tau ^{*}]\\subset I$ .",
"This provides an additional argument in favor of Nash equilibria– institutions can still reason in terms of equilibria and achieve more favorable outcomes in terms of social burden."
],
[
"Fairness to Subpopulations",
"Our previous section showed that increased robustness in the face of strategic behavior comes at the price of additional social burden.",
"In this section, we show this social burden is not fairly distributed: when the individuals being classified are from latent subpopulations, say of race, gender, or socioeconomic status, the social burden can disproportionately fall on disadvantaged subpopulations.",
"Furthermore, we find that improving the institution's utility can exacerbate the gap between the social burden incurred by an advantaged and disadvantaged group.",
"Concretely, suppose each individual is from a subpopulation $g \\in \\lbrace a, b\\rbrace $ .",
"The social burden a classifier $f$ has on a group $g$ is the expected minimum cost required for a positive individual from group $g$ to be accepted: $\\mathcal {B}_{+, g}(f) = \\mathbb {E}\\left[\\min _{x^{\\prime }:f(x^{\\prime })=1} c(x, x^{\\prime }) \\mid Y = 1, G = g \\right]$ .",
"We can then define the social gap between groups $a$ and $b$ : Definition 4.1 (Social gap) The social gap $\\mathcal {G}(f)$ induced by a classifier $f$ is the difference in the social burden to group $b$ compared to $a$ : $\\mathcal {G}(f) =\\mathcal {B}_{+, b}(f) - \\mathcal {B}_{+, a}(f)$ .",
"The social gap is a measure of how much more costly it is for a positive individual from group $b$ to be accepted by the classifier than a positive individual from group $a$ .",
"For example, there is evidence that women need to attain higher educational qualifications than their male counterparts to receive the same salary [6].",
"A high social gap is alarming for two reasons.",
"First, even when two people from group $a$ and group $b$ are equally qualified, the individual from group $a$ may choose not to participate at all because of the cost she would need to endure to be accepted.",
"Secondly, if she does decide to participate, she may continue to be at a disadvantage after being accepted because of the additional cost she had to endure, e.g., repaying student loans.",
"Non-strategic classification can already induce a social gap between two groups, and strategic classification can exacerbate this gap.",
"We show this under two natural ways group $b$ may be disadvantaged.",
"In the first setting, the feature distributions of group $a$ and $b$ are such that a positive individual from group $b$ is less likely to be considered positive, compared to group $a$ .",
"In the second setting, individuals from group $b$ have a higher cost to adapt their features compared to group $a$ .",
"Under both of these conditions, any improvement the institution can make to its own strategic utility has the side effect of worsening (increasing) the social gap."
],
[
"Different Feature Distributions",
"In the first setting we analyze, the way groups $a$ and $b$ differ is through their distributions over features.",
"We say that group $b$ is disadvantaged if the features distributions are such that positive individuals from group $b$ are less likely to be considered positive than those from group $a$ .",
"Formally, this can be characterized as the following: Definition 4.2 (Disadvantaged in features) Let $L_{+, g} = \\ell (X) \\mid Y = 1, G = g$ be the outcome likelihood of a positive individual from group $g$ , and let $F_{+, g}$ be the cumulative distribution function of $L_{+, g}$ .",
"We say that group $b$ is disadvantaged in features if $F_{+, b}(l) > F_{+, a}(l)$ for all $l \\in (0, 1)$ .",
"In the economics literature, the relationship between $L_{+, a}$ and $L_{+, b}$ is referred to as strict first-order stochastic dominance [20].",
"Intuitively, that group $b$ is disadvantaged in features if and only if the distribution of $L_{+, a}$ can be transformed to the distribution of $L_{+, b}$ by transferring probability mass from higher values to lower values.",
"This definition captures the notion that the outcome likelihood of positive individuals from group $b$ is skewed lower than the outcome likelihood of positive individuals from $a$ .",
"In a case study on FICO credit scores in Section , we find the minority group (blacks) is disadvantaged in features compared to the majority group (whites) (see Figure REF ).",
"There are many reasons that a group could be disadvantaged in features.",
"Below, we go through a few potential causes.",
"Example 4.1 (Group membership explains away features) Even if two groups are equally likely to have positive individuals, i.e., $\\mathbb {P}(Y = 1 \\mid G = a) = \\mathbb {P}(Y = 1 \\mid G = b)$ , group $b$ can still be disadvantaged compared to group $a$ .",
"Consider the graph below.",
"Although the label $Y$ is independent of the group $G$ , the label $Y$ is not independent of the group $G$ once conditioned on the features $X$ because the group $G$ can provide an alternative reason for the observed features.",
"Figure: NO_CAPTION Concretely, let groups $a$ and $b$ be native and non-native speakers of english, $X$ be the number of grammatical errors on an individual's job application, and $Y$ be whether the individual is a qualified candidate.",
"Negative individuals ($Y = 0$ ) are less meticulous when filling out their application and more likely to have grammatical errors.",
"However, for individuals from group $b$ there is another explanation for having grammatical errors – being a non-native speaker.",
"Thus, positive individuals from group $b$ end up with lower outcome likelihoods than those from $a$ , even though they may be equally qualified.",
"Example 4.2 (Predicting base rates) Suppose the rate of positives in group $b$ is lower than that of group $a$ : $\\mathbb {P}(Y = 1 \\mid G = b) < \\mathbb {P}(Y = 1 \\mid G = a)$ .",
"If there is a feature in the dataset that can be used as a proxy for predicting the group, such as zip code or name for predicting race, then the outcome likelihoods of positive individuals from group $b$ can end up lower than those of positive individuals from group $a$ because the features are simply predicting the base rate of each group."
],
[
"We now state and prove the main result showing that the social gap increases as the institution increases its threshold for acceptance.",
"Before turning to the result, we introduce one technical requirement.",
"The likelihood condition is that $\\frac{\\partial c_L(l, \\tau )}{\\partial l}$ is monotonically non-increasing in $\\tau $ for $l, \\tau \\in [0, 1]$ .",
"When the cost function $c$ is outcome monotonic, the likelihood condition is satisfied for a broad class of differentiable likelihood cost functions $c_L$ , such as the following examples.",
"Differentiable separable cost functions of the form $c_L(l, l^{\\prime }) =\\max (c_2(l^{\\prime }) - c_1(l), 0)$ for $c_1, c_2 : [0, 1] \\rightarrow \\mathbb {R}_{\\ge 0}$ .",
"Differentiable shift-invariant cost functions of the form $c_L(l, l^{\\prime }) = {\\left\\lbrace \\begin{array}{ll}c_0(l^{\\prime } - l) & l < l^{\\prime } \\\\0 & l \\ge l^{\\prime }\\end{array}\\right.}",
"\\, ,$ for convex $c_0 : [0, 1] \\rightarrow \\mathbb {R}_{\\ge 0}$ .",
"Notably, any linear cost $c_L(l, l^{\\prime }) = \\max (\\alpha (l^{\\prime }-l), 0)$ where $\\alpha > 0$ satisfies the likelihood condition.",
"Under the likelihood condition, we now show that the social gap increases as the institution increases its threshold for acceptance.",
"Theorem 4.1 Let $\\tau \\in (0, 1]$ be the threshold of the classifier.",
"If group $b$ is disadvantaged in features compared to group $a$ , and $\\frac{\\partial c_L(l,\\tau )}{\\partial l}$ is monotonically non-increasing in $\\tau $ , then $\\mathcal {G}(\\tau )$ is positive and monotonically increasing over $\\tau $ .",
"By Lemma REF , any outcome monotonic cost function can be written as a cost over outcome likelihoods.",
"Therefore, the social burden can be written as $\\mathcal {B}_{+, g}(\\tau )&= \\mathbb {E}\\left[c_L(\\ell (x), \\tau ) \\mid Y = 1, G = g \\right]\\\\&= \\int _{0}^\\tau c_L(l, \\tau ) \\mathop {}\\!\\mathrm {d}F_{+, g}(l),$ where $F_{+, g}$ denotes the CDF of the group outcome likelihood $L_{+, g}$ .",
"Integrating by parts, we obtain a simple expression for $\\mathcal {B}_{+, g}(\\tau )$ : $\\mathcal {B}_{+, g}(\\tau )&= \\int _{0}^\\tau c_L(l, \\tau ) \\mathop {}\\!\\mathrm {d}F_{+, g}(l) \\\\&= [c_L(l, \\tau )F_{+, g}(l)]^{\\tau }_0 - \\int _{0}^\\tau \\frac{\\partial c_L(l, \\tau )}{\\partial l}F_{+, g}(l) \\mathop {}\\!\\mathrm {d}l \\\\&= - \\int _{0}^\\tau \\frac{\\partial c_L(l, \\tau )}{\\partial l}F_{+, g}(l) \\mathop {}\\!\\mathrm {d}l \\, ,$ where the last line follows because $c_L(\\tau , \\tau ) = 0$ and $F_{+, g}(0)=0$ .",
"This expression for $\\mathcal {B}_{+, g}(\\tau )$ allows us to conveniently write the social gap as $\\mathcal {G}(\\tau )= \\mathcal {B}_{+, b}(\\tau ) - \\mathcal {B}_{+, a}(\\tau )= \\int _{0}^\\tau \\frac{\\partial c_L(l, \\tau )}{\\partial l} (F_{+, a}(l)- F_{+, b}(l)) \\mathop {}\\!\\mathrm {d}l \\, .$ We now argue the social gap $\\mathcal {G}(\\tau )$ is positive.",
"By the monotonicity assumptions, $\\frac{\\partial c_L(l, \\tau )}{\\partial l} < 0$ for $l \\in (0,\\tau )$ .",
"Since group $b$ is disadvantaged in features, $F_{+, a}(l) -F_{+, b}(l) < 0$ for $l \\in (0, 1)$ .",
"Therefore, $\\mathcal {G}(\\tau ) > 0$ .",
"Now, we show $\\mathcal {G}(\\tau )$ is increasing in $\\tau $ .",
"Let $0 \\le \\tau < \\tau ^{\\prime }\\le 1$ .",
"Then, the difference in the social gap is given by $\\mathcal {G}(\\tau ^{\\prime }) - \\mathcal {G}(\\tau )&= \\int _0^{\\tau } \\frac{\\partial \\left(c_L(l, \\tau ^{\\prime }) -c_L(l, \\tau )\\right)}{\\partial l} (F_{+, a}(l)- F_{+, b}(l)) \\mathop {}\\!\\mathrm {d}l \\\\& + \\int _\\tau ^{\\tau ^{\\prime }} \\frac{\\partial c_L(l, \\tau ^{\\prime })}{\\partial l}(F_{+, a}(l)- F_{+, b}(l)) \\mathop {}\\!\\mathrm {d}l.$ Since group $b$ is disadvantaged in features, $(F_{+, a}(l)- F_{+, b}(l)) < 0$ for all $l$ .",
"By assumption, $\\frac{\\partial c_L(l, \\tau )}{\\partial l}$ is monotonically non-increasing in $\\tau $ , so the first term is non-negative.",
"Similarly, $\\frac{\\partial c_L(l, \\tau ^{\\prime })}{\\partial l} < 0$ by monotonicity, so the second term is positive.",
"Hence, $\\mathcal {G}(\\tau ^{\\prime }) - \\mathcal {G}(\\tau ) > 0$ , which establishes $\\mathcal {G}(\\tau )$ is monotonically increasing in $\\tau $ .",
"As a corollary, if the institution improves its utility beyond the non-strategic optimal classifier, then it also causes the social gap to increase.",
"Corollary 4.1 Suppose group $b$ is disadvantaged in features compared to group $a$ , and $\\frac{\\partial c_L(l, \\tau )}{\\partial l}$ is monotonically non-decreasing in $\\tau $ .",
"Let $\\tau \\in (0, 1]$ be a threshold and $\\tau _0=0.5$ be the optimal non-strategic threshold.",
"If $\\mathcal {U}_\\Delta (\\tau ) > \\mathcal {U}_\\Delta (\\tau _0)$ , then $\\mathcal {G}(\\tau )> \\mathcal {G}(\\tau _0)$ .",
"By Theorem REF , if $\\mathcal {U}_\\Delta (\\tau ) > \\mathcal {U}_\\Delta (\\tau _0)$ , then $\\tau > \\tau _0$ .",
"By Theorem REF , if $\\tau > \\tau _0$ , then $\\mathcal {G}(\\tau ) > \\mathcal {G}(\\tau _0)$ ."
],
[
"Different Costs",
"In Section REF , we showed that when two subpopulations have different feature distributions, the social burden can disproportionately fall on one group.",
"In this section, we show that even if the feature distributions of the two groups are exactly identical, the social burden can still disproportionately impact one group.",
"We have thus far assumed the existence of a cost function $c$ that is uniform across groups $a$ and $b$ .",
"For a variety of structural reasons, it is unlikely this assumption holds in practice.",
"Rather, it is often the case that different groups experience different costs for changing their features.",
"When the cost for group $b$ is systematically higher than the cost for group $a$ , we prove group $b$ incurs higher social burden than group $a$ .",
"Furthermore, if the institution improves its utility by increasing its threshold $\\tau $ , then as a side effect it also increases the social gap between group $b$ and $a$ (Theorem REF ).",
"Much of the prior work on fairness in classification focuses on preventing unfairness that can arise when different subpopulations have different distributions over features and labels , [14], [8].",
"Our result provides a reason to be concerned about the unfair impacts of a classifier even when two groups have identical initial distributions.",
"Namely, that it can be easier for one group to game the classifier than another.",
"Formally, we say that group $b$ is disadvantaged in cost compared to group $a$ if the following condition holds.",
"Definition 4.3 (Disadvantaged in cost) Let $c_g(x, x^{\\prime })$ be the cost for an individual from group $g$ to adapt their features from $x$ to $x^{\\prime }$ .",
"Group $b$ is disadvantaged in cost if $c_b(x, x^{\\prime }) = \\kappa c_a(x, x^{\\prime })$ for all $x, x^{\\prime } \\in X$ and some scalar $\\kappa > 1$ .",
"Next, we give a variety of example scenarios of when a group can be disadvantaged in cost.",
"Example 4.3 (Opportunity Costs) Many universities have adopted gender-neutral policies that stop the “tenure-clock” for a year for family-related reasons, e.g.",
"childbirth.",
"Ostensibly, no research is expected while the clock is stopped.",
"However, the adoption of gender-neutral clocks actually increased the gap between the percentage of men and women who received tenure [2].",
"The suggested cause is that women still shoulder more of the burden of bearing and caring for children, compared to men.",
"Men who stop their tenure clock are more productive during the period than women, who have a higher opportunity cost to doing research while raising a child.",
"Example 4.4 (Information Asymmetry) A large portion of high-achieving, low-income students do not apply to selective colleges, despite the fact that these colleges are typically less expensive for them because of the financial aid they would receive [15].",
"This phenomenon seems to be due to low-income students having less access to information about college .",
"Since low-income students have more barriers to gaining information about college, it is natural to assume that, compared to their wealthier peers, they have a higher cost to strategically manipulating their admission features.",
"Example 4.5 (Economic Differences) Consider a social media company that wishes to classify individuals as “influencers,” either to more widely disseminate their content or to identify promising accounts for online marketing campaigns.",
"Wealthy individuals can purchase followers or likes, whereas other groups have to increase these numbers organically [7].",
"Consequently, the costs to increasing one's popularity metric differs based on access to capital.",
"Finally, our main technical result shows that even when the distributions of groups $a$ and $b$ are identical, if group $b$ is disadvantaged in cost, then when the institution increases its threshold for acceptance, it also increases the social gap between the two groups.",
"Theorem 4.2 Suppose positive individuals from groups $a$ and $b$ have the same distribution over features, i.e, if $Z = \\left(X \\mid Y = 1\\right)$ , then $Z$ is independent of the group $G$ .",
"If group $b$ is disadvantaged in cost compared to group $a$ , then the social gap $\\mathcal {G}(\\tau )$ is non-negative and monotonically non-decreasing in the threshold $\\tau $ .",
"Since $X \\mid Y = 1$ is independent of $G$ , the social burden to a group $g$ can be written as $\\mathcal {B}_{+, g}(\\tau ) = \\int _{\\mathcal {X}} \\min _{x^{\\prime }:f_\\tau (x^{\\prime })=1} c_g(x, x^{\\prime })p(X = x\\mid Y = 1) \\mathop {}\\!\\mathrm {d}x$ where $f_\\tau $ is the outcome likelihood classifier with threshold $\\tau $ .",
"The social gap can then be expressed as $\\mathcal {G}(\\tau ) & = \\mathcal {B}_{+, b}(\\tau ) - \\mathcal {B}_{+, a}(\\tau ) \\\\& = \\int _{\\mathcal {X}}(\\kappa - 1) \\min _{x^{\\prime }:f_\\tau (x^{\\prime })=1} c_a(x, x^{\\prime }) ~p(X = x \\mid Y = 1) \\mathop {}\\!\\mathrm {d}x \\, \\\\& = (\\kappa - 1) \\mathcal {B}_{+, a}(\\tau ) .$ Since the group social burden $\\mathcal {B}_{+, a}(\\tau )$ is non-negative and monotonically non-decreasing, the social gap $\\mathcal {G}(\\tau )$ is also non-negative and monotonically non-decreasing."
],
[
"Case Study: FICO Credit Data",
"We illustrate the impact of strategic classification on different subpopulations in the context of credit scoring and lending.",
"FICO scores are widely used in the United States to predict credit worthiness.",
"The scores themselves are derived from a proprietary classifier that uses features that are susceptible to gaming and strategic manipulation, for instance the number of open bank accounts.",
"We use a sample of 301,536 FICO scores derived from TransUnion TransRisk scores [23] and preprocessed by [14].",
"The scores $X$ are normalized to lie between 0 and 100.",
"An individual's outcome is labeled as a default if she failed to pay a debt for at least 90 days on at least one account in the ensuing 18-24 month period.",
"Default events are labeled with $Y=0$ , and otherwise repayment is denoted with $Y=1$ .",
"The two subpopulations are given by race: $a=\\text{\\tt white}$ and $b=\\text{\\tt black}$ .",
"We assume the credit lending institution accepts individuals based on a threshold on the FICO score.",
"Using the normalized scale, a threshold of $\\tau =58$ is typically used to determine eligibility for prime rate loans [14].",
"Our results thus far have used thresholds on the outcome likelihood, rather than a score.",
"However, as shown in Figure REF , the outcome likelihood is monotonic in the FICO score.",
"Therefore, all our conditions and results can be validated using the score instead of the outcome likelihood."
],
[
"Different Feature Distributions",
"In Section REF , we studied the scenario where the distribution of outcome likelihoods $\\ell (X) = \\mathbb {P}(Y = 1 \\mid X)$ differed across subpopulations.",
"In particular, if the likelihoods of the positive individuals in group $B$ tend to be lower than the positive individuals in group $A$ , then increasing strategic robustness increases the social gap between $A$ and $B$ .",
"Interestingly, such a skew in score distributions exists in the FICO data.",
"Black borrowers who repay their loans tend to have lower FICO scores than white borrowers who repay their loans.",
"In terms of the corresponding score CDFs, for every score $x$ , $F_{+, \\tt black}(x) \\ge F_{+, \\tt white}(x)$ .",
"Figure REF demonstrates this observation.",
"Figure: Comparison of the distribution of FICO scores among black and whiteborrowers who repaid their loans.",
"Credit-worthy black individuals tendto have lower credit scores than credit-worthy white individuals.",
"Thecomparison of the corresponding CDFs demonstrates our“disadvantaged in features” assumption holds.When the score distribution among positive individuals is skewed, Theorem REF guarantees the social gap between groups is increasing in the threshold under a reasonable cost model.",
"Operationally, raising the loan threshold to protect against strategic behavior increases the relative burden on the black subgroup.",
"To demonstrate this empirically, we use a coarse linear cost model, $c(x, x^{\\prime }) = \\max (\\alpha (x^{\\prime }- x), 0)$ for some $\\alpha > 0$ .",
"Here, $\\alpha $ corresponds to the cost of raising one's FICO score one point.",
"Since the probability of repayment $\\mathbb {P}(Y=1\\mid x)$ is monotonically increasing in $x$ , the linear cost $c$ satisfies the requisite outcome monotonicity conditions.",
"Figure: Repayment probability as a function of credit score.",
"Crucially, theprobability of repayment ℙ(Y=1∣x)\\mathbb {P}(Y=1 \\mid x) is monotonically increasing inxx.In Figure REF , we compute $\\mathcal {G}(\\tau )$ as $\\tau $ varies from 0 to 100 for a range of different value of $\\alpha $ .",
"For any $\\alpha $ , the social utility gap is increasing in $\\tau $ .",
"Moreover, as $\\alpha $ becomes large, the rate of increase in the social gap grows large as well.",
"Figure: Impact of increasing the threshold τ\\tau on white and black creditapplicants.",
"When the cost to changing one's score α\\alpha is small,increases to the threshold have only a small effect on the social gap.However, as α\\alpha becomes large, even small increases to thethreshold can create large discrepancies in social burden between the twogroups."
],
[
"Different Cost Functions",
"In Section REF , we demonstrated when two subpopulations are identically distributed, but incur different costs for changing their features, there is a non-trivial social gap between the two.",
"In the context of the FICO scores, it is plausible that blacks are both disadvantaged in features and experience higher costs for changing their scores.",
"For instance, outstanding debt is an important component of FICO scores.",
"One way to reduce debt is to increase earnings.",
"However, a persistent black-white wage gap between the two subpopulations suggest increasing earnings is easier for group $a$ than group $b$ [11].",
"This setting is not strictly captured by our existing results, and we should expect the effects of both different costs functions and different feature distributions to compound and exacerbate the unfair impacts of strategic classification.",
"To illustrate this phenomenon, we again use a coarse linear cost model.",
"Suppose group A has cost $c_A(x, x^{\\prime }) = \\max \\lbrace \\alpha (x^{\\prime } - x), 0\\rbrace $ for some $\\alpha > 0$ , and group B has cost $c_B(x, x^{\\prime }) = \\max \\lbrace \\beta (x^{\\prime } - x), 0\\rbrace $ for some $\\beta \\ge \\alpha $ .",
"As in Section REF , group B is disadvantaged in cost provided the ratio $\\kappa = \\beta / \\alpha > 1$ .",
"In Figure REF , we show the social gap $\\mathcal {G}(\\tau )$ for various settings of $\\kappa $ .",
"The social gap is always increasing as a function of $\\tau $ , and the rate of increase grows large for even moderate values of $\\kappa $ .",
"When $\\kappa $ is large, even small increases in $\\tau $ can disproportionately increase the social burden for the disadvantaged subpopulation.",
"Figure: Impact of increasing the threshold τ\\tau on white and black creditapplicants, under the assumption that both groups incur different costsfor increasing their credit score.",
"As the ratio between the costsκ\\kappa increases, the social cost gap grows rapidlybetween the two groups."
],
[
"Strategic Classification",
"Prior work on strategic classification focuses solely on the institution, primarily aiming to create high-utility solutions for the institution.",
"Our work, on the other hand, studies the tradeoff between the institution's utility and the burden to the individuals being classified.",
"[13], [12], [5] give algorithms to compute the Stackelberg equilibrium, which corresponds to the extreme $\\tau ^{*}$ solution in our trade-off curves.",
"Although the Stackelberg equilibrium leads to maximal institutional utility, we show that it also causes high social burden.",
"We give several examples of when the high social burden induced by the Stackelberg equilibrium makes it an undesirable solution for the institution.",
"Rather than the Stackelberg equilibrium, others have also considered finding Nash equilibria of the game [4], [10].",
"[4] argue that since in practice people cannot optimally respond to the classifier, the Stackelberg solution tends to be too conservative, and thus a Nash equilibrium strategy is preferable.",
"Our work provides a complementary reason to prefer Nash equilibria over the Stackelberg solution.",
"Namely, for a broad class of cost functions, any Nash equilibrium that is not equal to the Stackelberg equilibrium places lower social burden on individuals.",
"Finally, we focus on the setting where individuals are merely “gaming” their features, i.e., they do not improve their true label by adapting their features.",
"However, if the classifier is able to incentivize strategic behavior that helps improve negative individuals, then the social burden placed on positive individuals may be considered acceptable.",
"[19] studies how to design classifiers that produce such incentives."
],
[
"Fairness",
"Our work studies how strategic classification results in differing impacts to different subpopulations and is complementary to the large body of work studying the differing impacts of classification [22], [3].",
"The prior work on classification is primarily concerned with preventing unfairness that can arise due to subpopulations having differing distributions over features or labels [14], , [8].",
"We show that in the strategic setting, a classifier can have differing impact due to the subpopulations having differing distributions or differing costs to adapting their features.",
"Therefore, when individuals are strategic, our work provides an additional reason to be concerned about the fairness of a classifier.",
"In particular, it can be easier for one group to game the classifier than another.",
"Furthermore, we show that if the institution modifies the classifier it uses to be more robust to strategic behavior, then it also as a side effect, increases the gap between the cost incurred by a disadvantaged subpopulation and an advantaged population.",
"Thus, strategic classification can exacerbate unfairness in classification.",
"Our work is also complementary to [21], who also analyze how the institution's utility trades-off with the impact to individuals.",
"They study the trade-off in the non-strategic setting and measure the impact of a classifier using a dynamics model of how individuals are affected by the classification they receive.",
"We study the tradeoff in the strategic setting and measure the impact of a classifier by the cost of the strategic behavior induced by the classifier.",
"In concurrent work, also study negative externalities of strategic classification.",
"In their model, they show that the Stackelberg equilibrium leads to only false negative errors on a disadvantaged population and false positives on the advantaged population.",
"Furthermore, they show that providing a cost subsidy for disadvantaged individuals can lead to worse outcomes for everyone."
],
[
"Discussion of Social Burden",
"To measure the impact of strategic classification on the individuals being classified, we introduced a measure of social burden, defined as the expected cost that positive individuals need to incur to be classified positively: $\\mathcal {B}_{+}(f) = \\mathbb {E}\\left[\\min _{f(x^{\\prime }) = 1} c(x, x^{\\prime }) \\mid y = 1 \\right]$ .",
"An alternative measure one might consider is the expected individual utility for the positives: $\\mathcal {S}_{+}(f) = \\mathbb {E}\\left[u_x(f, \\Delta (x)) \\mid y = 1\\right]$ , which we will denote the social utility.",
"We prefer social burden to social utility because it makes fewer assumptions about individual behavior.",
"Social utility measures the utility of the individual while assuming that they respond optimally and needs the assumption to hold to be a meaningful measure.",
"Social burden, on the other hand, applies irrespective of the different policies individuals may actually act according to.",
"Our analysis assumes the institution assumes individuals respond optimally, but we ourselves believe this to be a strong assumption to hold in practice, and would like our measure of impact on individuals to apply regardless.",
"Moreover, most of our results are agnostic to the specific choice of social cost measure.",
"The results in Section about the tradeoff between institutional utility and social burden all still hold.",
"Specifically, Theorem REF holds with social utility instead of social burden (and monotonically non-increasing instead of monotonically non-decreasing since lower utility is worse).",
"For our results in Section , i.e, Theorems REF and REF , there is still always a non-negative social gap (now defined as the difference in social utilities between the groups), but it is not necessarily true that the social gap increases as the institution's threshold increases.",
"While both social burden and social utility apply only to positive individuals, one could also use versions that integrate over all individuals: $\\mathcal {B}(f)=\\mathbb {E}\\left[\\min _{f(x^{\\prime }) = 1} c(x, x^{\\prime }) \\right]$ and $\\mathcal {S}(f) = \\mathbb {E}\\left[u_x(f, \\Delta (x))\\right]$ .",
"Our results for $\\mathcal {B}_{+}(f)$ go through for $\\mathcal {B}(f)$ , and the results for $\\mathcal {S}_{+}(f)$ go through for $\\mathcal {S}(f)$For the results in Section REF the disadvantaged in features condition defined in Definition REF should be modified to no longer condition on $Y=1$ ..",
"However, in many cases giving a positive classification (e.g.",
"a loan) to a negative individual (someone who will default) can result in a long-term negative impact to that individual [21].",
"In general, it is uncertain whether the reducing the costs incured by the negative individuals confers positive social benefits, and we do not incorporate these costs into our measure.",
"Overall, there are many potential measures that are complementary to our measure of social burden, but they all provide a similar takeaway.",
"Namely, that in the strategic setting, there is a tradeoff between institutional accuracy and individual impact that must be considered when making choices about strategy-robustness."
],
[
"Acknowledgements",
"This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No.",
"DGE 1752814.",
"Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation."
],
[
"Proof of Lemma ",
"The lemma follows by proving the following properties about the Nash equilibrium strategies for the institution.",
"The Stackelberg threshold $\\tau ^*$ is a Nash equilibrium strategy.",
"All Nash equilibrium strategies lie in the interval $[\\tau _0, \\tau ^*]$ .",
"If $\\tau _N$ is a Nash equilibrium strategy, then all $\\tau \\in [\\tau _N,\\tau ^*]$ are also Nash equilibrium strategies.",
"Together, the three properties imply that the set of institution equilibrium strategies is $[\\tau _N, \\tau ^*]$ for some $\\tau _N \\in [\\tau _0, \\tau ^*]$ .",
"Before proceeding, we first establish and recall a few definitions.",
"Let $\\Delta _{\\tau }(x)$ be the best response of individual $x$ to the threshold $\\tau $ .",
"Define the set of individuals accepted for threshold $\\tau $ and response $\\Delta _\\tau $ by $\\mathcal {A}_{\\Delta _\\tau }(\\tau ) = \\lbrace x : f_\\tau (\\Delta _\\tau (x)) =1\\rbrace $ .",
"Recall the outcome likelihood $\\ell (x) = \\mathbb {P}(Y = 1 \\mid X=x)$ .",
"Define the strategic outcome likelihood as $\\ell _{\\Delta _\\tau }(x^{\\prime }) = \\mathbb {P}(Y = 1 \\mid \\Delta _\\tau (X) = x^{\\prime })$ .",
"The outcome likelihood is the probability that an individual is positive given their true features, while the strategic outcome likelihood is the probability that an individual is positive given their gamed features.",
"For the pair $(\\tau , \\Delta _\\tau )$ to be a Nash equilibrium, $\\tau $ must be a best response to the individual's best response $\\Delta _\\tau (x)$ .",
"With knowledge of the individual's response, $x^{\\prime } = \\Delta _{\\tau }(x)$ , the institution's best response is to play a threshold $\\tau ^{\\prime }$ so that $x^{\\prime } \\in \\mathcal {A}_{\\Delta _\\tau }(\\tau ^{\\prime })$ iff $\\ell _{\\Delta _\\tau }(x^{\\prime }) \\ge 0.5$ .",
"Therefore, to show $(\\tau , \\Delta _\\tau )$ is a Nash equilibrium, we must show $\\tau = \\tau ^{\\prime }$ , i.e.",
"$x^{\\prime } \\in \\mathcal {A}_{\\Delta _\\tau }(\\tau )$ iff $\\ell _{\\Delta _\\tau }(x^{\\prime }) \\ge 0.5$ .",
"To verify the condition $x^{\\prime } \\in \\mathcal {A}_{\\Delta _\\tau }(\\tau )$ iff $\\ell _{\\Delta _\\tau }(x^{\\prime }) \\ge 0.5$ , there are three cases to consider.",
"If $\\ell (x) > \\tau $ , then $x \\in \\mathcal {A}_{\\Delta _\\tau }(\\tau )$ , $\\Delta _\\tau (x) =x$ , and $\\ell _{\\Delta _\\tau }(x) = \\ell (x)$ .",
"Therefore, it suffices to check $\\ell (x) \\ge 0.5$ .",
"If $\\ell (x) < \\tau $ and $c_L(\\ell (x), \\tau ) > 1$ , then $x \\notin \\mathcal {A}_{\\Delta _\\tau }(\\tau )$ and $\\Delta _\\tau (x) = x$ .",
"In this case, it suffices to check $\\ell (x) < 0.5$ .",
"If $\\ell (x) < \\tau $ and $c_L(\\ell (x), \\tau ) \\le 1$ , then $\\Delta _\\tau (x)\\in \\mathcal {A}_{\\Delta _\\tau }(\\tau )$ , but $x \\ne \\Delta _\\tau (x)$ , so we must directly verify $\\mathbb {P}(Y=1 \\mid c_L(\\ell (X), \\tau ) \\le 1, \\ell (X) \\le \\tau ) \\ge 0.5$ .",
"We now proceed to the proof.",
"First, we show the Stackelberg equilibrium $(\\tau ^{*}, \\Delta _{\\tau ^{*}})$ is a Nash equilibrium.",
"The Stackelberg threshold $\\tau ^{*}$ is the largest $\\tau ^{*}$ such that $c_L(0.5, \\tau ^*) \\le 1$ .",
"If $\\ell (x) > \\tau ^{*}$ , by monotonicity, $\\ell (x) \\ge 0.5$ .",
"If $\\ell (x) < \\tau ^{*}$ and $c_L(\\ell (x), \\tau ^{*}) > 1$ , then $\\ell (x) < 0.5$ by definition of $\\tau ^{*}$ .",
"Similarly, if $\\ell (x) <\\tau ^{*}$ and $c_L(\\ell (x), \\tau ^{*}) \\le 1$ , then $\\ell (x) \\ge 0.5$ , so trivially $\\mathbb {P}(Y=1 \\mid c_L(\\ell (X), \\tau ) \\le 1, \\ell (X) \\le \\tau )\\ge 0.5$ .",
"Hence, $(\\tau ^{*}, \\Delta _{\\tau ^{*}})$ is a Nash equilibrium.",
"Next, we show that all Nash strategies must lie in the interval $[\\tau _0,\\tau ^*]$ .",
"Suppose $\\tau < \\tau _0 = 0.5$ .",
"For all $x$ such that $\\ell (\\Delta _\\tau (x)) =\\tau $ , $\\ell (x) < 0.5$ .",
"Therefore, $\\mathbb {P}(Y=1 \\mid c_L(\\ell (X), \\tau ) \\le 1, \\ell (X) \\le \\tau ) < 0.5$ , so $\\tau $ cannot be a Nash equilibrium strategy for the institution.",
"Suppose $\\tau > \\tau ^*$ .",
"By definition, $\\tau ^{*}$ is the largest $\\tau $ such that $c_L(0.5, \\tau ) \\le 1$ .",
"Thus, if $\\tau > \\tau ^{*}$ , there exists $x$ with $\\ell (x) < \\tau ^{*}$ and $c_L(\\ell (x), \\tau ) > 1$ , but $\\ell (x) \\ge 0.5$ .",
"Hence, $\\tau $ cannot be a Nash strategy.",
"Finally, we show that if $\\tau _N$ is a Nash equilibrium strategy, then so is $\\tau $ for any $\\tau \\in [\\tau _N, \\tau ^*]$ .",
"We consider each of the three cases in turn.",
"Suppose $\\ell (x) > \\tau $ .",
"Then $\\ell (x) > \\tau > \\tau _N \\ge \\tau _0 =0.5$ .",
"Suppose $\\ell (x) < \\tau $ and $c_L(\\ell (x), \\tau ) > 1$ .",
"Since $\\tau \\le \\tau ^{*}$ , by monotonicity in the second argument, $c_L(0.5, \\tau ) \\le c_L(0.5, \\tau ^{*}) \\le 1$ .",
"By monotonicity in the first argument, since $c_L(0.5, \\tau )$ , if $c_L(\\ell (x), \\tau ) > 1$ , then it must be the case that $\\ell (x) < 0.5$ .",
"Suppose $\\ell (x) < \\tau $ and $c_L(\\ell (x), \\tau ) \\le 1$ .",
"Let $L(\\tau ) = \\lbrace l : c_L(l, \\tau ) \\le 1, l \\le \\tau \\rbrace $ be the set of outcome likelihoods that game to the threshold $\\tau $ and $l_\\tau = \\min _{l \\in L_\\tau } l$ be the minimum such outcome likelihood.",
"The points that game under the thresholds $\\tau _N$ and $\\tau $ form the intervals $[l_{\\tau _N}, \\tau _N]$ and $[l_\\tau , \\tau ]$ , respectively.",
"Since $l_{\\tau _N} \\le l_\\tau $ and $\\tau _N \\le \\tau $ , we have that $& \\mathbb {P}(Y = 1 \\mid c_L(\\ell (X), \\tau ) \\le 1, \\ell (X) \\le \\tau ) \\\\& = \\mathbb {P}(Y = 1 \\mid \\ell (X) \\in [l_{\\tau }, \\tau ]) \\\\& \\ge \\mathbb {P}(Y = 1 \\mid \\ell (X) \\in [l_{\\tau _N}, \\tau _N]) \\\\& = \\mathbb {P}(Y = 1 \\mid c_L(\\ell (X), \\tau _N) \\le 1,\\ell (X) \\le \\tau _N) \\ge 0.5$ where the last inequality holds because $t_N$ is a Nash strategy.",
"Since each of the three cases are satisfied, any $\\tau \\in [\\tau _N, \\tau ^{*}]$ is a Nash strategy.",
"We have now demonstrated each of the three properties outlined at the beginning, and the lemma follows."
]
] | 1808.08460 |
[
[
"On centro-affine curves and Backlund transformations of the KdV equation"
],
[
"Abstract We continue the study of the Korteweg-de Vries equation in terms of cento-affine curves, initiated by U. Pinkall.",
"A centro-affine curve is a closed parametric curve in the affine plane such that the determinant made by the position and the velocity vectors is identically 1.",
"The space of centro-affine curves is acted upon by the special linear group, and the quotient is identified with the space of Hill's equations with periodic solutions.",
"It is known that the space of centro-affine curves carries two pre-symplectic structures, and the KdV flow is identified with is a bi-Hamiltonian dynamical system therein.",
"We introduce a 1-parameter family of transformations on centro-affine curves, prove that they preserve both presymplectic structures, commute with the KdV flow, and share the integrals with it.",
"Furthermore, the transformation commute with each other (Bianchi permutability).",
"We also describe integrals of the KdV equation as arising from the monodromy of Riccati equations associated with centro-affine curves.",
"We are motivated by our work in progress (joint with M. Arnold, D. Fuchs, and I. Izmenstiev), concerning the cross-ratio dynamics on ideal polygons in the hyperbolic plane and hyperbolic space, whose continuous version is studied in the present note."
],
[
"A family of transformations on the space of curves",
"This note stems from [1] where we study the integrable dynamics of a 1-parameter family of correspondences on ideal polygons in the hyperbolic plane and hyperbolic space: two $n$ -gons $P=(p_1,p_2,\\ldots )$ and $Q=(q_1,q_2,\\ldots )$ in ${\\mathbb {RP}}^1$ or in ${\\mathbb {CP}}^1$ are in correspondence $PQ̰$ if $[p_i,p_{i+1},q_i,q_{i+1}]=c$ for all $i$ ; the constant $c$ is a parameter.",
"In the limit $n\\rightarrow \\infty $ , a polygon becomes a parameterized curve.",
"The ground field can be either ${\\mathbb {R}}$ of $; to fix ideas, choose $ R$.",
"Let ususe the following definition of cross-ratio to define our correspondence (other five definitions result in the change of the constant $ c$):\\begin{equation} [p_i,p_{i+1},q_i,q_{i+1}]=\\frac{(q_{i+1}-q_i)(p_{i+1}-p_i)}{(q_i-p_i)(q_{i+1}-p_{i+1})}=c.\\end{equation}$ We replace polygons by non-degenerate closed curves ${\\gamma }: {\\mathbb {R}}\\rightarrow {\\mathbb {RP}}^1$ with ${\\gamma }^{\\prime }(t) >0$ ; to be concrete, let the period be $\\pi $ : ${\\gamma }(t+\\pi )={\\gamma }(t)$ .",
"Also let us assume that the rotation number of the curve ${\\gamma }$ is 1, that is, ${\\gamma }: {\\mathbb {R}}/\\pi {\\mathbb {Z}}\\rightarrow {\\mathbb {RP}}^1$ is a diffeomorphism.",
"Denote the space of such curves by $\\widetilde{\\mathcal {C}}$ and let $\\mathcal {C}=\\widetilde{\\mathcal {C}}/\\operatorname{PSL}(2,{\\mathbb {R}})$ be the moduli space.",
"Then a continuous version of () is $ \\frac{{\\gamma }^{\\prime }(t){\\delta }^{\\prime }(t)}{({\\delta }(t)-{\\gamma }(t))^2}=c.$ Write ${\\gamma }$ to denote this relation on $\\widetilde{\\mathcal {C}}$ .",
"Since cross-ratio is Möbius invariant, we also have a relation on $\\mathcal {C}$ which we continue to denote by $.",
"Note that $ is a symmetric relation.",
"Lemma 1.1 For a generic curve ${\\gamma }$ , the relation ${\\gamma }$ is a (partially defined) 2-2 map $T_c: {\\gamma }\\mapsto {\\delta }$ ."
],
[
"Proof.",
"Given ${\\gamma }(t)$ , equation (REF ) is a Riccati equation on ${\\delta }(t)$ , its monodromy is a Möbius transformation (see, e.g., [5]) which has either two or no fixed points, unless it is the identity.",
"Over $, there are always two fixed points (possibly, coinciding), and over $ R$, we need to assume that they exist.",
"Then $ defines a 2-2 map.",
"$\\Box $ Thus, given ${\\gamma }$ , there are two choices of ${\\delta }=T_c ({\\gamma })$ .",
"Once a choice is made, one similarly has two choices for $T_c({\\delta })$ , but one of them is ${\\gamma }$ , so we choose the other one, and so on.",
"Hence the choice of ${\\delta }$ determines the map $T_c$ ; the other choice gives the inverse map $T_c^{-1}$ .",
"Following the standard procedure (see, e.g., [7]), lift a curve ${\\gamma }(t)$ from ${\\mathbb {RP}}^1$ to ${\\mathbb {R}}^2$ , normalizing the lift ${\\Gamma }(t)$ so that $[{\\Gamma },{\\Gamma }^{\\prime }]=1$ (here and elsewhere $[\\, , ]$ denotes the determinant made by two vectors).",
"Explicitly, ${\\Gamma }=(({\\gamma }^{\\prime })^{-1/2}, ({\\gamma }^{\\prime })^{-1/2} {\\gamma })$ .",
"Note the square root: the curve $-{\\Gamma }$ will do as well, the lift is defined up to the sign, and the action of $\\operatorname{PSL}(2,{\\mathbb {R}})$ is replaced by that of $\\operatorname{SL}(2,{\\mathbb {R}})$ .",
"We obtain centro-affine realizations of the spaces $\\widetilde{\\mathcal {C}}$ and $\\mathcal {C}$ .",
"The curve ${\\Gamma }$ satisfies a Hill equation $ {\\Gamma }^{\\prime \\prime }(t)=p(t){\\Gamma }(t)$ with a $\\pi $ -periodic potential $p(t)$ , and ${\\Gamma }(t+\\pi )=-{\\Gamma }(t)$ (the curve makes exactly half-rotation on $[0,\\pi ]$ ).",
"In geometric terms, the potential $p$ is the (negative) centro-affine curvature of the curve ${\\Gamma }$ .",
"In these terms, equation () becomes $\\frac{[{\\Delta }(t),{\\Gamma }(t)][{\\Delta }(t+{\\varepsilon }),{\\Gamma }(t+{\\varepsilon })]}{[{\\Delta }(t),{\\Delta }(t+{\\varepsilon })][{\\Gamma }(t),{\\Gamma }(t+{\\varepsilon })]} = {\\rm const},$ and, in the limit ${\\varepsilon }\\rightarrow 0$ , we obtain an analog of equation (REF ): $[{\\Gamma }(t),{\\Delta }(t)]^2=c^2.$ Break the symmetry between ${\\Gamma }$ and ${\\Delta }$ by taking square root: $ [{\\Gamma }(t),{\\Delta }(t)]=c.$ This defines a map on the lifted curves: $T_c({\\Gamma })={\\Delta }$ .",
"Note that $T_c({\\Delta })=-{\\Gamma }$ .",
"Lemma 1.2 $T_c: {\\Gamma }\\mapsto {\\Delta }$ is a (partially defined) 2-2 map."
],
[
"Proof.",
"Let us search for ${\\Delta }$ in the form ${\\Delta }= a {\\Gamma }+ b {\\Gamma }^{\\prime }$ , where $a$ and $b$ are $\\pi $ -periodic functions.",
"Then equation $[{\\Gamma },{\\Delta }]=c$ implies that $b(t)=c$ , ${\\Delta }=a{\\Gamma }+c{\\Gamma }^{\\prime }$ , and then ${\\Delta }^{\\prime }=(a^{\\prime }+cp){\\Gamma }+ a{\\Gamma }^{\\prime }$ .",
"The condition $[{\\Delta },{\\Delta }^{\\prime }]=1$ now implies $ a^{\\prime } = \\frac{a^2-1}{c} - cp.$ This is a Riccati equation on function $a(t)$ with periodic coefficients.",
"The monodromy of this equation is a Möbius transformation, hence it has two fixed points (always, if one works over $, and over $ R$ one needs to assume that it does), corresponding to two periodic solutions of (\\ref {Ric}).",
"Each solution defines a curve $$ with $ Tc()= $.$$\\bigskip $ As before, once a choice of a fixed point of the monodromy is made, the map becomes 1-1: of the two choices available for the next curve ${\\Delta }$ , one is extraneous because it takes one back to $-{\\Gamma }$ ."
],
[
"Two pre-symplectic forms and a bi-Hamiltonian structure",
"Starting with U. Pinkall [9], a number of recent papers were devoted to the study of the Korteweg-de Vries equation in terms of cento-affine curves [2], [3], [4], [11].",
"Let us present the relevant results.",
"A tangent vector at a cento-affine curve ${\\Gamma }$ is a vector field along ${\\Gamma }$ that can be written as a linear combination $h{\\Gamma }+f{\\Gamma }^{\\prime }$ where $h,f$ are $\\pi $ -periodic functions.",
"Lemma 2.1 The function $f$ is arbitrary, and $h=-\\frac{1}{2} f^{\\prime }.$"
],
[
"Proof.",
"If ${\\Gamma }+{\\varepsilon }v$ is a deformation of ${\\Gamma }$ , then $[{\\Gamma },v^{\\prime }]=[{\\Gamma }^{\\prime },v]$ because $[{\\Gamma },{\\Gamma }^{\\prime }]=1$ .",
"For $v=h{\\Gamma }+f{\\Gamma }^{\\prime }$ , this implies that $h=-\\frac{1}{2} f^{\\prime }.$ $\\Box $ Denote the tangent vectors by $U,V$ or $U_f,V_f$ , in the format $-\\frac{1}{2} f^{\\prime } {\\Gamma }+ f {\\Gamma }^{\\prime }$ .",
"The following pre-symplectic structure on space $\\widetilde{\\mathcal {C}}$ was introduced in [9].",
"Let $U,V$ be tangent vector fields along ${\\Gamma }$ ; define $\\omega (U,V) = \\int _{\\Gamma }[U,V]\\ dt,$ that is, $\\omega (U_f,V_g) = \\frac{1}{2}\\int _0^{\\pi } (fg^{\\prime }-f^{\\prime }g)dt.$ The kernel of $\\omega $ is spanned by the field ${\\Gamma }^{\\prime }$ , that is, by the reparameterizations $t \\mapsto t+$ const.",
"Pinkall observed that the Hamiltonian vector field of the function $\\int p\\ dt$ is $U_p$ , which induces the KdV evolution of the potential $p$ $\\dot{p} = -\\frac{1}{2} p^{\\prime \\prime \\prime } + 3p^{\\prime }p$ (the signs differ from those of Pinkall because he used the opposite sign for the potential of Hill's equation).",
"The second pre-symplectic structure was introduced in [4]: for tangent vector fields $U,V$ along ${\\Gamma }$ , let $\\Omega (U,V) = \\int _{\\Gamma }([U^{\\prime },V^{\\prime }] +p[U,V])\\ dt,$ that is, $\\Omega (U_f,V_g) = \\int _0^{\\pi } \\left[\\frac{1}{4}(f^{\\prime }g^{\\prime \\prime }-f^{\\prime \\prime }g^{\\prime }) + p(fg^{\\prime }-f^{\\prime }g)\\right]\\ dt.$ Concerning the kernel of $\\Omega $ , one has Lemma 2.2 ([4]) The kernel of $\\Omega $ is 3-dimensional, it is generated by the Killing vector fields $A({\\Gamma })$ with $A \\in \\operatorname{SL}(2)$ ."
],
[
"Proof.",
"One has $\\Omega (U,V) = \\int [pU-U^{\\prime \\prime },V]\\ dt.$ Hence $U$ is in the kernel if and only if $U^{\\prime \\prime }=pU$ , that is, $U(t)$ is $\\operatorname{SL}(2)$ -equivalent to ${\\Gamma }(t)$ .",
"$\\Box $ Thus the form $\\Omega $ descends on the moduli space $\\mathcal {C}$ as a symplectic form.",
"It is shown in [2], [11], [4] that the forms $\\omega $ and $\\Omega $ provide a bi-Hamiltonian structure on $\\widetilde{\\mathcal {C}}$ , corresponding to a pair of compatible Poisson brackets for the KdV equation.",
"Namely, let $X_0,X_1,\\ldots $ and $H_1,H_2,\\ldots $ be the vector fields and the Hamiltonians of the KdV hierarchy in terms of centro-affine curves: $X_0=U_1={\\Gamma }^{\\prime }, X_1=U_p=-\\frac{p^{\\prime }}{2} {\\Gamma }+ p {\\Gamma }^{\\prime },\\ldots , H_1 = \\int p dt, H_2 = \\frac{1}{2} \\int p^2 dt, \\ldots $ Then one has $ \\Omega (X_{j-1},\\cdot ) = d H_j = \\omega (X_j,\\cdot ),\\ j=1,2,\\ldots $ see [4]."
],
[
"The forms $\\omega $ and {{formula:79cb2ea3-45ed-4652-9d76-0855c9dff016}} on projective curves.",
"Let us calculate these forms in terms of the curves ${\\gamma }: {\\mathbb {R}}\\rightarrow {\\mathbb {RP}}^1$ .",
"In [1], the following differential 2-form on the space of polygons $(p_1,\\ldots ,p_n) \\subset {\\mathbb {RP}}^1$ was considered $\\omega ^{\\prime }=\\sum _i \\frac{dp_i \\wedge dp_{i+1}}{(p_{i+1}-p_i)^2},$ and it was proved that this form was $T_c$ -invariant.",
"In the continuous limit, a polygon becomes a curve ${\\gamma }(t)$ .",
"Let $u(t),v(t)$ be two vector fields along ${\\gamma }(t)$ , that is, two periodic functions.",
"Then, in the continuous limit, we obtain the form $\\omega ^{\\prime }(u,v) = \\int \\frac{uv^{\\prime }-u^{\\prime }v}{({\\gamma }^{\\prime })^2}\\ dt.$ Lemma 2.3 One has $\\omega =\\frac{1}{2} \\omega ^{\\prime }.$"
],
[
"Proof.",
"Since ${\\Gamma }_1=({\\gamma }^{\\prime })^{-1/2},\\ {\\Gamma }_2=({\\gamma }^{\\prime })^{-1/2} {\\gamma },$ one calculates the respective vector field along ${\\Gamma }$ : $U = \\left( -\\frac{1}{2} u^{\\prime } {\\Gamma }_1^3, -\\frac{1}{2} u^{\\prime } {\\Gamma }_1^2{\\Gamma }_2+u{\\Gamma }_1 \\right),$ and likewise for $V$ .",
"Then $[U,V]=\\frac{1}{2} {\\Gamma }_1^4 (uv^{\\prime }-u^{\\prime }v),$ and the result follows.",
"$\\Box $ By Lemma REF , the 2-form $\\Omega $ descends to the moduli space of projective curves, that is, to the space of Hill's equations.",
"This space is a coadjoint orbit of the Virasoro algebra, and $\\Omega $ coincides (up to a factor) with the celebrated Kirillov-Kostant-Souriau symplectic structure, see, e.g., [6], [7] for this material.",
"Namely, let ${\\gamma }$ be a curve in ${\\mathbb {RP}}^1$ , and let $u$ and $v$ be vector fields along ${\\gamma }$ .",
"The Kirillov-Kostant-Souriau symplectic form is given by the formula $\\Omega ^{\\prime }(u,v)=\\int \\frac{u^{\\prime \\prime }(t)v^{\\prime }(t)-u^{\\prime }(t)v^{\\prime \\prime }(t)}{{\\gamma }^{\\prime }(t)^2}\\ dt,$ see, e.g., [8].",
"Lemma 2.4 One has $\\Omega = -\\frac{1}{4} \\Omega ^{\\prime }.$"
],
[
"Proof.",
"As in the proof of Lemma REF , $U = \\left( -\\frac{1}{2} u^{\\prime } {\\Gamma }_1^3, -\\frac{1}{2} u^{\\prime } {\\Gamma }_1^2{\\Gamma }_2+u{\\Gamma }_1 \\right),$ and then $U^{\\prime }=\\left( -\\frac{1}{2} u^{\\prime \\prime }{\\Gamma }_1^3-\\frac{3}{2} u^{\\prime }{\\Gamma }_1^2{\\Gamma }_1^{\\prime }, -\\frac{1}{2} u^{\\prime \\prime }{\\Gamma }_1^2{\\Gamma }_2 - u^{\\prime }{\\Gamma }_1{\\Gamma }_2{\\Gamma }_1^{\\prime } -\\frac{1}{2} u^{\\prime }{\\Gamma }_1^2{\\Gamma }_2^{\\prime }+u^{\\prime }{\\Gamma }_1+u{\\Gamma }_1^{\\prime } \\right).$ Similar formulas hold for $V$ .",
"Now one computes, using the fact that ${\\Gamma }_2^{\\prime }{\\Gamma }_1-{\\Gamma }_1^{\\prime }{\\Gamma }_2=1$ , $[U^{\\prime },V^{\\prime }] = -\\frac{1}{4}{\\Gamma }_1^4(u^{\\prime \\prime }v^{\\prime }-u^{\\prime }v^{\\prime \\prime }) - \\frac{1}{2} {\\Gamma }_1^3{\\Gamma }_1^{\\prime } (u^{\\prime \\prime }v-uv^{\\prime \\prime }) - \\frac{3}{2} {\\Gamma }_1^2({\\Gamma }_1^{\\prime })^2 (u^{\\prime }v-uv^{\\prime }),$ and $p[U,V] = -\\frac{1}{2} p{\\Gamma }_1^4 (u^{\\prime }v-uv^{\\prime }) = -\\frac{1}{2} {\\Gamma }_1^3 {\\Gamma }_1^{\\prime \\prime } (u^{\\prime }v-uv^{\\prime }).$ Integrating by parts, $- \\int {\\Gamma }_1^3{\\Gamma }_1^{\\prime } (u^{\\prime \\prime }v-uv^{\\prime \\prime })\\ dt = \\int ({\\Gamma }_1^3{\\Gamma }_1^{\\prime \\prime } + 3 {\\Gamma }_1^2 ({\\Gamma }_1^{\\prime })^2 ) (u^{\\prime }v-uv^{\\prime })\\ dt,$ and collecting terms, $\\Omega (U,V) = \\int ([U^{\\prime },V^{\\prime }]+p[U,V])\\ dt = -\\frac{1}{4} \\int {\\Gamma }_1^4(u^{\\prime \\prime }v^{\\prime }-u^{\\prime }v^{\\prime \\prime })\\ dt = -\\frac{1}{4} \\Omega ^{\\prime }(u,v),$ as claimed.",
"$\\Box $"
],
[
"$T_c$ -invariance of the bi-Hamiltonian structure and complete integrability of the transformations {{formula:3af07986-d393-4da6-8b2f-8c43ecf82286}}",
"Let $T_c({\\Gamma })={\\Delta }$ with ${\\Delta }^{\\prime \\prime }=q {\\Delta }$ ; one can write ${\\Delta }(t)=a(t) {\\Gamma }(t) + c {\\Gamma }^{\\prime }(t)$ , where $a(t)$ is a periodic function.",
"Lemma 3.1 One has: ${\\Gamma }=a{\\Delta }-c{\\Delta }^{\\prime },\\ p+q=\\frac{2}{c^2} (a^2-1),\\ q-p=\\frac{2}{c} a^{\\prime }.$"
],
[
"Proof.",
"Since $[{\\Delta },-{\\Gamma }]=c$ , we can write $-{\\Gamma }=b{\\Delta }+c{\\Delta }^{\\prime }$ where $b(t)$ is a periodic function.",
"Substitute ${\\Delta }=a{\\Gamma }+c{\\Gamma }^{\\prime }$ in this equation to find that $b=-a$ .",
"We also have an analog of (REF ) for function $b(t)$ : $cb^{\\prime }=b^2-1-c^2q$ .",
"This implies the relations between $p$ and $q$ stated in the lemma.",
"$\\Box $ Let $T_c({\\Gamma })={\\Delta }$ , and let $U_f, V_g$ be two tangent vectors, at ${\\Gamma }$ and ${\\Delta }$ , respectively, related by the differential of $T_c$ .",
"Lemma 3.2 One has $ \\frac{c}{2} (f^{\\prime }+g^{\\prime }) = a(g-f),$ where the function $a(t)$ is as above."
],
[
"Proof.",
"One has $[U,{\\Delta }]+[{\\Gamma },V]=0$ , or $\\frac{c}{2} (f^{\\prime }+g^{\\prime }) = f[{\\Gamma }^{\\prime },{\\Delta }]+g[{\\Gamma },{\\Delta }^{\\prime }] = a(g-f),$ where the last equality makes use of ${\\Delta }= a {\\Gamma }+ c {\\Gamma }^{\\prime }$ and of $[{\\Gamma }^{\\prime },{\\Delta }]+[{\\Gamma },{\\Delta }^{\\prime }]=0$ .",
"$\\Box $ The following theorem is our main observation.",
"Theorem 1 The forms $\\omega $ and $\\Omega $ are invariant under the maps $T_c$ : $T_c^*(\\omega )=\\omega ,\\ T_c^*(\\Omega )=\\Omega .$"
],
[
"Proof.",
"Let $T_c({\\Gamma })={\\Delta }$ , and let $U_{f_i}, V_{g_i}, i=1,2$ , be two pairs of tangent vectors, at ${\\Gamma }$ and ${\\Delta }$ , respectively, related by the differential of $T_c$ .",
"One has $\\begin{split}\\int [(g_1^{\\prime } g_2-g_1g_2^{\\prime }) - (f_1^{\\prime } f_2-f_1f_2^{\\prime })] dt=\\int [(g_1^{\\prime } g_2-g_1g_2^{\\prime }) - (f_1^{\\prime } f_2-f_1f_2^{\\prime })& \\\\- (g_1^{\\prime }f_2+g_1f_2^{\\prime }) + (g_2^{\\prime }f_1+g_2f_1^{\\prime })]dt&\\\\=\\int [(f_1^{\\prime }+g_1^{\\prime })(g_2-f_2)- (f_2^{\\prime }+g_2^{\\prime })(g_1-f_1)]dt =0&,\\end{split}$ where the first equality follows from the fact that $g_1^{\\prime }f_2+g_1f_2^{\\prime }=(g_1f_2)^{\\prime }$ and $g_2^{\\prime }f_1+g_2f_1^{\\prime }=(g_2f_1)^{\\prime }$ , which integrates to zero, and the last equality follows from (REF ).",
"Thus $T_c^*(\\omega )=\\omega $ .",
"To prove that $T_c^*(\\Omega )=\\Omega $ , we argue similarly, although the computation is more involved.",
"Differentiate (REF ) to obtain $ \\frac{c}{2} (f^{\\prime \\prime }+g^{\\prime \\prime })=a^{\\prime }(g-f)+a(g^{\\prime }-f^{\\prime }).$ We want to show that the integral $ \\int \\left(\\frac{1}{4} (f_1^{\\prime }f_2^{\\prime \\prime }-f_1^{\\prime \\prime }f_2^{\\prime }) + p(f_1f_2^{\\prime }-f_1^{\\prime }f_2) - \\frac{1}{4} (g_1^{\\prime }g_2^{\\prime \\prime }-g_1^{\\prime \\prime }g_2^{\\prime }) -q(g_1g_2^{\\prime }-g_1^{\\prime }g_2)\\right) dt$ vanishes.",
"One has $f_1^{\\prime }f_2^{\\prime \\prime }-f_1^{\\prime \\prime }f_2^{\\prime } -g_1^{\\prime }g_2^{\\prime \\prime }+ g_1^{\\prime \\prime }g_2^{\\prime } = (f_1^{\\prime \\prime }+g_1^{\\prime \\prime })(g_2^{\\prime }-f_2^{\\prime }) - (f_2^{\\prime \\prime }+g_2^{\\prime \\prime })(g_1^{\\prime }-f_1^{\\prime }) +(f_2^{\\prime }g_1^{\\prime }-f_1^{\\prime }g_2^{\\prime })^{\\prime },$ hence $\\begin{split}&\\frac{1}{4} \\int (f_1^{\\prime }f_2^{\\prime \\prime }-f_1^{\\prime \\prime }f_2^{\\prime } -g_1^{\\prime }g_2^{\\prime \\prime }+ g_1^{\\prime \\prime }g_2^{\\prime }) dt \\\\= &\\frac{1}{2c} \\int \\lbrace [a^{\\prime }(g_1-f_1)+a(g_1^{\\prime }-f_1^{\\prime })](g_2^{\\prime }-f_2^{\\prime }) - [a^{\\prime }(g_2-f_2)+a(g_2^{\\prime }-f_2^{\\prime })](g_1^{\\prime }-f_1^{\\prime })\\rbrace dt\\\\= &\\int \\frac{a^{\\prime }}{2c} [(g_1-f_1)(g_2^{\\prime }-f_2^{\\prime }) - (g_2-f_2)(g_1^{\\prime }-f_1^{\\prime })] dt,\\end{split}$ where the first equality follows from (REF ).",
"Next we evaluate the remaining part of the integral (REF ), using Lemma REF : $\\begin{split}&\\int [p(f_1f_2^{\\prime }-f_1^{\\prime }f_2)-q(g_1g_2^{\\prime }-g_1^{\\prime }g_2)] dt \\\\= &\\int \\frac{a^2-1}{c^2}(f_1f_2^{\\prime }-f_1^{\\prime }f_2-g_1g_2^{\\prime }+g_1^{\\prime }g_2) dt - \\int \\frac{a^{\\prime }}{c} (f_1f_2-f_1^{\\prime }f_2+g_1g_2^{\\prime }-g_1^{\\prime }g_2) dt.\\end{split}$ Collecting the integrals together, we obtain $\\begin{split}&\\int \\frac{a^{\\prime }}{2c} [(f_1^{\\prime }+g_1^{\\prime })(f_2+g_2) - (f_2^{\\prime }+g_2^{\\prime })(f_1+g_1)] dt \\\\+ &\\int \\frac{a^2-1}{c^2}(f_1f_2^{\\prime }-f_1^{\\prime }f_2-g_1g_2^{\\prime }+g_1^{\\prime }g_2) dt \\\\= &\\int \\frac{2aa^{\\prime }}{c^2} (f_2g_1-f_1g_2) dt + \\int \\frac{a^2-1}{c^2}(f_1f_2^{\\prime }-f_1^{\\prime }f_2-g_1g_2^{\\prime }+g_1^{\\prime }g_2) dt,\\end{split}$ where the equality is due to (REF ).",
"Finally, notice that $(a^2-1)^{\\prime }=2aa^{\\prime }$ , and integrate by parts to obtain $\\begin{split}\\int \\frac{a^2-1}{c^2} [(f_1f_2^{\\prime }-f_1^{\\prime }f_2-g_1g_2^{\\prime }+g_1^{\\prime }g_2) - (f_2^{\\prime }g_1 + f_2g_1^{\\prime }-f_1^{\\prime }g_2-f_1g_2^{\\prime })] dt \\\\= \\int \\frac{a^2-1}{c^2} [(f_2^{\\prime }+g_2^{\\prime })(f_1-g_1)-(f_1^{\\prime }+g_1^{\\prime })(f_2-g_2)] dt =0,\\end{split}$ since the last integrand vanishes due to (REF ).",
"$\\Box $ Corollary 2 The maps $T_c$ commute with the KdV flows and preserve the KdV integrals."
],
[
"Proof.",
"One argues inductively using formulas (REF ): $\\Omega (X_{j-1},\\cdot ) = d H_j = \\omega (X_j,\\cdot ).$ If $T_c$ preserves $X_{j-1}$ then, since it also preserves $\\Omega $ , it preserves $dH_j$ .",
"If $T_c$ preserves $dH_j$ then, since it preserves $\\omega $ , it also preserves $X_j$ .",
"To start the induction, we check that $\\int p\\ dt$ is invariant: $\\int (q(t)-p(t))\\ dt = \\frac{2}{c} \\int a^{\\prime }(t)\\ dt =0$ due to Lemma REF .",
"Since $dH_j$ is preserved, it could be that $T_c$ changes $H_j$ by a constant.",
"To see that this constant is zero, let ${\\Gamma }$ be the circle $(\\cos t, \\sin t)$ .",
"Then ${\\Delta }$ differs from ${\\Gamma }$ by a parameter shift, and the values of the functions $H_j$ on ${\\Gamma }$ and ${\\Delta }$ are equal.",
"$\\Box $ Thus the transformations $T_c$ are symmetries of the Korteweg-de Vries equation.",
"Remark 3.3 The argument above is similar to the one given in [10] which concerned with the filament equation and the bicycle transformations as its symmetries."
],
[
"Additional integrals.",
"Let ${\\Gamma }=({\\Gamma }_1,{\\Gamma }_2)$ .",
"Consider the functions $ I= \\int {\\Gamma }_1^2\\ dt,\\ J=\\int {\\Gamma }_1{\\Gamma }_2\\ dt,\\ K= \\int {\\Gamma }_2^2\\ dt$ on the space of centro-affine curves.",
"Proposition 3.4 The functions $I,J,K$ are the Hamiltonians of the generator of the action of $\\operatorname{sl}(2,{\\mathbb {R}})$ on $\\widetilde{\\mathcal {C}}$ with respect to the 2-form $\\omega $ .",
"The function $IK-J^2$ is $\\operatorname{SL}(2,{\\mathbb {R}})$ -invariant."
],
[
"Proof.",
"The generators of $\\operatorname{sl}(2,{\\mathbb {R}})$ are the fields $({\\Gamma }_2,0),\\ ({\\Gamma }_1, -{\\Gamma }_2),\\ (0,{\\Gamma }_1).$ Let us consider the first one; the other ones are dealt with similarly.",
"We claim that $({\\Gamma }_2,0) = - V_{{\\Gamma }_2^2}.$ Indeed, $V_{{\\Gamma }_2^2} = -{\\Gamma }_2{\\Gamma }_2^{\\prime } {\\Gamma }+ {\\Gamma }_2^2 {\\Gamma }^{\\prime }.$ The first component of this vector is $-{\\Gamma }_2 ({\\Gamma }_2^{\\prime }{\\Gamma }_1 - {\\Gamma }_1^{\\prime }{\\Gamma }_2) = -{\\Gamma }_2$ , and the second component is $-{\\Gamma }_2^{\\prime }{\\Gamma }_2^2 + {\\Gamma }_2^{\\prime }{\\Gamma }_2^2=0$ .",
"Let $U_f$ be a test vector field.",
"Then $dK(U_f) = \\int {\\Gamma }_2\\left( {\\Gamma }_2^{\\prime }f - \\frac{1}{2} {\\Gamma }_2 f^{\\prime }\\right)\\ dt = 2 \\int {\\Gamma }_2{\\Gamma }_2^{\\prime }f\\ dt.$ On the other hand, $\\omega (U_f,U_{{\\Gamma }_2^2}) = \\int 2{\\Gamma }_2{\\Gamma }_2^{\\prime }f\\ dt,$ as needed.",
"As to $\\operatorname{sl}(2,{\\mathbb {R}})$ invariance of $IK-J^2$ , let us again check invariance under the field $({\\Gamma }_2,0)$ (the rest is similar).",
"Calculating mod ${\\varepsilon }^2$ , one has $\\begin{split}\\left(\\int ({\\Gamma }_1+{\\varepsilon }{\\Gamma }_2)^2\\ dt\\right) \\left(\\int {\\Gamma }_2^2\\ dt\\right) - \\left( \\int ({\\Gamma }_1+{\\varepsilon }{\\Gamma }_2){\\Gamma }_2\\ dt \\right)^2= IK-J^2& \\\\+ 2 {\\varepsilon }\\left[ \\left(\\int {\\Gamma }_1{\\Gamma }_2\\ dt\\right) \\left(\\int {\\Gamma }_2^2\\ dt\\right) - \\left(\\int {\\Gamma }_1{\\Gamma }_2\\ dt\\right) \\left(\\int {\\Gamma }_2^2\\ dt\\right)\\right]= IK-J^2&,\\end{split}$ as needed $\\Box $ Next we show that $I,J,K$ are integrals of the transformations $T_c$ .",
"Theorem 3 Let $T_c({\\Gamma })= {\\Delta }$ , then $I({\\Gamma })=I({\\Delta }), J({\\Gamma })=J({\\Delta }), K({\\Gamma })=K({\\Delta }).$"
],
[
"Proof.",
"Consider the case of $I$ ; the other two cases are similar.",
"We have ${\\Delta }=a{\\Gamma }+c{\\Gamma }^{\\prime }$ , and we want to show that $\\int {\\Delta }_1^2 = \\int {\\Gamma }_1^2$ .",
"Indeed, $\\begin{split}&\\int ({\\Delta }_1^2 - {\\Gamma }_1^2)\\ dt = \\int [(a^2-1){\\Gamma }_1^2+2ca{\\Gamma }_1{\\Gamma }_1^{\\prime }+c^2({\\Gamma }_1^{\\prime })^2]\\ dt\\\\&= \\int [(a^2-1-ca^{\\prime }) {\\Gamma }_1^2 + c^2({\\Gamma }_1^{\\prime })^2]\\ dt = c^2 \\int [p {\\Gamma }_1^2 + ({\\Gamma }_1^{\\prime })^2]\\ dt \\\\&= c^2 \\int [{\\Gamma }_1^{\\prime \\prime }{\\Gamma }_1 + ({\\Gamma }_1^{\\prime })^2]\\ dt = 0,\\end{split}$ where the second equality is integration by parts, the third is due to (REF ), the fourth is due to ${\\Gamma }^{\\prime \\prime }=p{\\Gamma }$ , and the last one is again integration by parts.",
"$\\Box $"
],
[
"Monodromy integrals and permutability",
"Now we describe an infinite collection of $\\operatorname{SL}(2,{\\mathbb {R}})$ -invariant integrals of the maps $T_c$ that arise from the monodromy of the Riccati equations.",
"Let $x$ be an affine coordinate on ${\\mathbb {RP}}^1$ .",
"The Lie algebra $\\operatorname{sl}(2,{\\mathbb {R}})$ is generated by the vector fields $ \\partial _x, x \\partial _x, x^2 \\partial _x.$ Introduce time-dependent vector fields, depending on ${\\gamma }(t)$ or ${\\delta }(t)$ , respectively, taking values in $\\operatorname{sl}(2,{\\mathbb {R}})$ for each $t$ : $\\xi _{\\gamma }= \\left(\\frac{{\\gamma }^2}{{\\gamma }^{\\prime }} -2\\frac{{\\gamma }}{{\\gamma }^{\\prime }}x + \\frac{1}{{\\gamma }^{\\prime }} x^2 \\right) \\partial _x,\\ \\ \\xi _{\\delta }= \\left(\\frac{{\\delta }^2}{{\\delta }^{\\prime }} -2\\frac{{\\delta }}{{\\delta }^{\\prime }}x + \\frac{1}{{\\delta }^{\\prime }} x^2 \\right) \\partial _x.$ Then equation (REF ) describes ${\\delta }$ as evolving under the field $c\\xi _{\\gamma }$ and, equivalently, ${\\gamma }$ as evolving under $c\\xi _{\\delta }$ .",
"Fix a (spectral) parameter $\\lambda $ , and consider the time-$\\pi $ flows of the fields $\\lambda \\xi _{\\gamma }$ and $\\lambda \\xi _{\\delta }$ , where ${\\gamma }$ and ${\\delta }$ are related by (REF ).",
"Denote these projective transformations of ${\\mathbb {RP}}^1$ by $\\Phi _{\\lambda ,{\\gamma }}$ and $\\Phi _{\\lambda ,{\\delta }}$ .",
"Theorem 4 For every $\\lambda $ , the maps $\\Phi _{\\lambda ,{\\gamma }}$ and $\\Phi _{\\lambda ,{\\delta }}$ are conjugate in $\\operatorname{PSL}(2,{\\mathbb {R}})$ .",
"It follows that the spectral invariants of $\\Phi _{\\lambda ,{\\gamma }}$ , say $\\rm Tr^2/\\det $ , as functions of $\\lambda $ , are integrals of the maps $T_c$ for all values of $c$ ."
],
[
"Proof.",
"Let ${\\gamma }$ and ${\\delta }$ satisfy (REF ).",
"Introduce a time-dependent matrix, also depending on parameter $\\mu $ : $A_{\\mu ,{\\gamma },{\\delta }} (t) = \\frac{1}{{\\gamma }(t)-{\\delta }(t)}\\begin{bmatrix}{\\gamma }(t)-\\mu {\\delta }(t),&{\\gamma }(t){\\delta }(t)(\\mu -1)\\\\1-\\mu ,&{\\gamma }(t)\\mu -{\\delta }(t)\\end{bmatrix}.$ We claim that if $\\lambda = c(1-\\mu )$ , then $A_{\\mu ,{\\gamma },{\\delta }} (t)$ conjugates the vector fields $\\lambda \\xi _{\\gamma }$ and $\\lambda \\xi _{\\delta }$ .",
"Namely, let ${\\varepsilon }$ be an infinitesimal parameter, and set $V_{\\gamma }(t,{\\varepsilon }) =\\begin{bmatrix}1- \\frac{{\\varepsilon }\\lambda {\\gamma }(t)}{{\\gamma }(t)^{\\prime }},& \\frac{{\\varepsilon }\\lambda {\\gamma }(t)^2}{{\\gamma }(t)^{\\prime }}\\\\-\\frac{{\\varepsilon }\\lambda }{{\\gamma }(t)^{\\prime }},&1+\\frac{{\\varepsilon }\\lambda {\\gamma }(t)}{{\\gamma }(t)^{\\prime }}\\end{bmatrix}$ This time-dependent Möbius transformation is the time-${\\varepsilon }$ flow of the vector field $\\lambda \\xi _{\\gamma }$ .",
"Then one has $V_{\\delta }(t,-{\\varepsilon }) A_{\\mu ,{\\gamma },{\\delta }} (t+{\\varepsilon }) V_{\\gamma }(t,{\\varepsilon }) = V_{\\delta }(t,{\\varepsilon }) A_{\\mu ,{\\gamma },{\\delta }} (t-{\\varepsilon }) V_{\\gamma }(t,- {\\varepsilon }) \\mod {{\\varepsilon }}^2,$ which is verified by a direct calculation or, in the limit ${\\varepsilon }\\rightarrow 0$ , $\\begin{bmatrix}\\frac{{\\delta }(t)}{{\\delta }^{\\prime }(t)},& \\frac{{\\delta }(t)^2}{{\\delta }^{\\prime }(t)}\\\\-\\frac{1}{{\\delta }^{\\prime }(t)},&\\frac{{\\delta }(t)}{{\\delta }^{\\prime }(t)}\\end{bmatrix} A_{\\mu ,{\\gamma },{\\delta }}(t)- A_{\\mu ,{\\gamma },{\\delta }}(t) \\begin{bmatrix}\\frac{{\\gamma }(t)}{{\\gamma }^{\\prime }(t)},& \\frac{{\\gamma }(t)^2}{{\\gamma }^{\\prime }(t)}\\\\-\\frac{1}{{\\gamma }^{\\prime }(t)},&\\frac{{\\gamma }(t)}{{\\gamma }^{\\prime }(t)}\\end{bmatrix}= \\frac{1}{\\lambda } A_{\\mu ,{\\gamma },{\\delta }}^{\\prime } (t).$ This equality implies that the vector fields $\\lambda \\xi _{\\gamma }$ and $\\lambda \\xi _{\\delta }$ are conjugate, and so are $\\Phi _{\\lambda ,{\\gamma }}$ and $\\Phi _{\\lambda ,{\\delta }}$ : $ \\Phi _{\\lambda ,{\\delta }} = A_{\\mu ,{\\gamma },{\\delta }} (0) \\Phi _{\\lambda ,{\\gamma }} A_{\\mu ,{\\gamma },{\\delta }}^{-1} (0),$ as needed.",
"$\\Box $ Remark 4.1 The above theorem is also a continuous analog of a result for ideal polygons in [1]."
],
[
"Bianchi permutability.",
"Let us show that the maps $T_c$ commute; the argument is similar to that given in [1] for ideal polygons.",
"Theorem 5 Let three closed curves satisfy ${\\gamma }\\stackrel{c_1}{\\sim }{\\gamma }_1$ and ${\\gamma }\\stackrel{c_2}{\\sim }{\\gamma }_2$ .",
"Then there exists a fourth curve ${\\gamma }_{12}$ such that ${\\gamma }_1 \\stackrel{c_2}{\\sim }{\\gamma }_{12}$ and ${\\gamma }_2 \\stackrel{c_1}{\\sim }{\\gamma }_{12}$ ."
],
[
"Proof.",
"We use (REF ), writing $A$ instead of $A(0)$ .",
"Since ${\\gamma }\\stackrel{c_1}{\\sim }{\\gamma }_1$ and ${\\gamma }\\stackrel{c_2}{\\sim }{\\gamma }_2$ , we have $\\Phi _{c_1,{\\gamma }} ({\\gamma }_1(0)) = {\\gamma }_1(0),\\ \\Phi _{c_2,{\\gamma }} ({\\gamma }_2(0)) = {\\gamma }_2(0).$ By (REF ), $\\Phi _{c_1,{\\gamma }_2} = A_{\\mu ,{\\gamma },{\\gamma }_2} \\Phi _{c_1,{\\gamma }} A_{\\mu ,{\\gamma },{\\gamma }_2}^{-1}, \\ \\Phi _{c_2,{\\gamma }_1} = A_{\\nu ,{\\gamma },{\\gamma }_1} \\Phi _{c_2,{\\gamma }} A_{\\nu ,{\\gamma },{\\gamma }_1}^{-1}$ with $ c_1=c_2(1-\\mu ), c_2=c_1(1-\\nu ).$ It follows that $\\Phi _{c_1,{\\gamma }_2} (A_{\\mu ,{\\gamma },{\\gamma }_2} ({\\gamma }_1(0))) = A_{\\mu ,{\\gamma },{\\gamma }_2} ({\\gamma }_1(0)),\\ \\Phi _{c_2,{\\gamma }_1} (A_{\\nu ,{\\gamma },{\\gamma }_1} ({\\gamma }_2(0))) = A_{\\nu ,{\\gamma },{\\gamma }_1} ({\\gamma }_2(0)).$ Thus we need to show that $ A_{\\mu ,{\\gamma },{\\gamma }_2} ({\\gamma }_1(0)) = A_{\\nu ,{\\gamma },{\\gamma }_1} ({\\gamma }_2(0)).$ This is indeed the case: (REF ) implies that $\\frac{1}{\\mu } + \\frac{1}{\\nu } =1$ , and then a calculation shows that $\\frac{1}{\\mu }\\begin{bmatrix}{\\gamma }-\\mu {\\gamma }_2,&{\\gamma }{\\gamma }_2(\\mu -1)\\\\1-\\mu ,&{\\gamma }\\mu -{\\gamma }_2\\end{bmatrix}\\begin{bmatrix}{\\gamma }_1\\\\1\\end{bmatrix}=\\frac{1}{\\nu }\\begin{bmatrix}{\\gamma }-\\nu {\\gamma }_1,&{\\gamma }{\\gamma }_1(\\nu -1)\\\\1-\\nu ,&{\\gamma }\\nu -{\\gamma }_1\\end{bmatrix}\\begin{bmatrix}{\\gamma }_2\\\\1\\end{bmatrix}, $ as needed.",
"$\\Box $ Remark 4.2 The above considerations can be extended to centro-affine twisted curves, that is, curves with monodromy, ${\\Gamma }(t+\\pi )=M({\\Gamma }(t))$ , where the monodromy $M\\in \\operatorname{SL}(2,{\\mathbb {R}})$ is not necessarily $-\\operatorname{Id}$ .",
"One can define the maps $T_c$ on twisted curves: given ${\\Gamma }$ , consider the respective $\\pi $ -periodic potential of the Hill equation $p(t)$ , find a $\\pi $ -periodic solution $a(t)$ to equation (REF ), and define ${\\Delta }=a{\\Gamma }+c{\\Gamma }^{\\prime }$ .",
"Then the monodromy of ${\\Delta }$ coincides with that of ${\\Gamma }$ .",
"At the level of Hill's equations, this is the map $p\\mapsto q$ .",
"We do not dwell on this extension here.",
"Acknowledgements.",
"It is a pleasure to acknowledge the stimulating discussions with A. Calini, A. Izosimov, I. Izmestiev, B. Khesin, and V. Ovsienko.",
"This work was supported by NSF grant DMS-1510055."
]
] | 1808.08454 |
[
[
"Autonomous Driving without a Burden: View from Outside with Elevated\n LiDAR"
],
[
"Abstract The current autonomous driving architecture places a heavy burden in signal processing for the graphics processing units (GPUs) in the car.",
"This directly translates into battery drain and lower energy efficiency, crucial factors in electric vehicles.",
"This is due to the high bit rate of the captured video and other sensing inputs, mainly due to Light Detection and Ranging (LiDAR) sensor at the top of the car which is an essential feature in autonomous vehicles.",
"LiDAR is needed to obtain a high precision map for the vehicle AI to make relevant decisions.",
"However, this is still a quite restricted view from the car.",
"This is the same even in the case of cars without a LiDAR such as Tesla.",
"The existing LiDARs and the cameras have limited horizontal and vertical fields of visions.",
"In all cases it can be argued that precision is lower, given the smaller map generated.",
"This also results in the accumulation of a large amount of data in the order of several TBs in a day, the storage of which becomes challenging.",
"If we are to reduce the effort for the processing units inside the car, we need to uplink the data to edge or an appropriately placed cloud.",
"However, the required data rates in the order of several Gbps are difficult to be met even with the advent of 5G.",
"Therefore, we propose to have a coordinated set of LiDAR's outside at an elevation which can provide an integrated view with a much larger field of vision (FoV) to a centralized decision making body which then sends the required control actions to the vehicles with a lower bit rate in the downlink and with the required latency.",
"The calculations we have based on industry standard equipment from several manufacturers show that this is not just a concept but a feasible system which can be implemented.The proposed system can play a supportive role with existing autonomous vehicle architecture and it is easily applicable in an urban area."
],
[
"INTRODUCTION",
"Autonomous vehicles (AVs) have become a hot topic and most of the automobile manufacturers put a lot of resources on the research.",
"Tesla is one of the leading companies which released commercialized self-driving cars while Uber, Google produced AVs for their applications.It is reported that 15% of the vehicles were equipped by the driver-assisted automated systems by 2015 and it will increase up to 50% to 60% by 2020 with a higher level of autonomy [1].",
"Light Detection and Ranging (LiDAR) sensor which is used to monitor the surroundings and create a High Definition (HD) point cloud for map generation, high resolution camera modules installed in the vehicle for its vision.",
"Artificial Intelligence and supercomputing / cloud computing capability are key enabling technologies of a self-driving car[2],[3].",
"The reason why AVs are becoming popular and captures the imagination of many people is because of the way it releases the humans from engaging in an otherwise laborious task more often than not such as driving to work or general transportation rather than taking a journey where driving is likely more enjoyable.",
"However, as everyone knows it creates driver fatigue in long journeys, especially at night time or when it becomes monotonous as in when we are trapped in long traffic jams with no end in sight.A significant majority of accidents are due to preventable human error.",
"The main disadvantages of the current AVs which will be elaborated further, are briefly as follows.",
"They result in an enormous amount of data gathering in the order of TBs through high bit rate video captures mainly coming from LiDAR with other sensors, cameras and RADARs contributing to that.",
"The signal processing burden for this is significant and needs many graphics processing units (GPUs) which in turn translates into a power consumption in the order of several kWs.",
"This is manifestly quite inefficient[4].",
"We therefore propose a new feasible system architecture supported by calculations based on industry standard sensors to overcome most of these deficiencies.",
"While the discussion is concentrated towards urban scenario it can be applied to others including rural areas with obvious adjustments.",
"Coupled with a communication system complemented also by the 5G now being standardized, which is shown to be capable of handling the required latencies and reliability, we are confident that this will usher in a new era in AVs and not just being restricted to that.",
"Appropriately configured, the same solution will cater to the needs of all automation use cases where mobility is a significant factor, such as in factory floors, autonomous harbors and in industrial robot applications.",
"The rest of the paper is organized as follows.",
"Section II provides a description of the current AV system architecture, and then in section III the details of our proposal are given.",
"The feasibility calculations are given in section IV and section V concludes the paper along with the possibilities for further investigations."
],
[
"CURRENT AUTONOMOUS VEHICLE SYSTEM ARCHITECTURE",
"Nowadays most modern cars are drive-by-wire (DBW) enabled, where electrical signals perform the vehicle functions traditionally achieved by the mechanical mechanisms[5].",
"This proves all the top automakers are on their way to level 5 autonomy where the human driver is not needed to drive under any condition (full automation in all conditions)[4].",
"However, currently, human driver should be ready within the vehicle to take over the control in case of an emergency or if an unpredictable event happens.",
"Various real-time sensors such as LiDAR, Sonar, Radar and cameras collaboratively sense environment for this level of autonomy.",
"LiDAR sensor which is fixed on the top of the vehicle generates a HD point cloud.",
"Identification of road signs and fog lines are carried out by the camera modules.",
"All the data collected from the vision-able sensors are processed within the vehicle to generate a high precision map.",
"Based on the map and remaining sensor data, path planning and required decisions are taken Using machine learning techniques and advance algorithms.",
"Decisions are executed by navigation system.",
"With the existing AV network architecture, there are a lot of design constraints to be addressed by the automobile manufacturers[4].",
"The system should able to respond to an incoming situation with low latency to avoid accidents.",
"According to the previous work the reaction time of an autonomous driving system is determined by frame rate and processing latency.",
"The fastest possible action of a human driver takes around 100 ms - 150 ms [4].",
"To bring a higher precision than a human driver, processing latency of the autonomous driving system is expected to be less than 100 ms. To achieve such a processing capability, 40 times powerful computer should be installed in each AV.",
"Prior map needs to be saved in the AV to guarantee secure navigation without depending on internet connectivity.",
"It is infeasible to request map from a cloud all the time.",
"To achieve a decimeter precision localization, these pre-stored maps are widely used.",
"Terabyte scale storage space is needed to store such a map in the vehicle.",
"It is found that entire map of the United States is about 41 TB[4].",
"High speed and highly reliable storage system with an enormous capacity should be set up in the AV.",
"The temperature outside the passenger cabin may go up to $105^{0}C$ .",
"A reliable cooling mechanism should be installed to control the temperature.",
"All these systems should withstand high impacts.",
"The shock absorbing capability will secure the systems in case of an accident.",
"All above mentioned constraints show the disadvantages of in-vehicle processing instead of road infrastructure based processing.",
"Power hungry components like highly powerful computers, storage engines and cooling mechanisms will degrade the fuel consumption.",
"It is reported that storage and cooling overhead resulting in driving range reduction as much as 11.5%.",
"In the mean time sensors like LiDAR and sonar are adding an unusual shape to the vehicle’s body which should be carefully evaluated from the customer satisfaction point of view.",
"One cannot easily forget the debacle of IRIDIUM satellite system based mobile phones, happened almost entirely owing to the lack of attention paid to the customer side.",
"In addition there were some unforeseen technological limitations at the design phase which surfaced only after implementation.",
"Due to the installation of storage and processing units, utilizable space of the vehicle is highly reduced and considerable weight is gained compared to a normal passenger car.",
"The ultimate Cost of the AV is very high due to all these additions.Spinning LiDAR is the costliest compenent of all, the price of it will be about USD 8000 where it is expected to be reduced by 50% in mass production[6].",
"Though top class companies used these AVs, customer attraction and their affordability will play a vital role in the future of AVs.",
"One must be cognizant of the fact that being too expensive and spectacular disasters caused the ultimate demise of Concorde.",
"These constraints make the way towards infrastructure based processing which will shift processing and computation capabilities to fixed infrastructure[7]."
],
[
"Communication Assisted Autonomous Driving",
"A typical communication architecture with cars is shown in Fig.",
"REF .",
"The relevant links are shown in V2X fashion.",
"Several publications mention and investigate sensor fusion to support driving through better sensing obtained through road side units (RSUs) equipped with sensors, e.g., placed at the elevation of lamp posts, providing a bird's eye view [8],[9].",
"These are essentially functioning in a support roles.",
"Figure: V2X CommunicationIt is found that the total amount of data collected from a HD map generation during one hour is about 1 TB.",
"Google's automated car collects a total of about 6 - 10 TB of sensor data in a day[10].",
"This huge data rate cannot be handled by the existing wireless communication standards (vehicle to infrastructure -V2I) which will lead to intensive computations within the car."
],
[
"THE PROPOSED ARCHITECTURE : ELEVATED LiDAR (ELiD) SYSTEM",
"Let us go back to the history of racing video games.",
"Initially the games were designed with a bird eye view which was very easier to handle because it gave an idea about hundred meters ahead.In order to be more realistic and to increase user experience, game designers changed the angle to driver's perspective with the advancement of technology which is same as the map generated by the AVs.",
"It is similar to the human driver's range of vision instead of the elevation.",
"Replacing the human driver with a set of sensors with a FoV which is almost similar to that of the driver is unlikely to improve the reliability.",
"In fact, there should be a third eye where AVs can monitor the environment, which we had in initial racing video games.",
"We propose an Elevated LiDAR (ELiD) system, which is based on this concept and is reliable due to its stationarity."
],
[
"Architecture Overview",
"LiDAR or high resolution cameras can be used for map generation due to its/their high reliability[11], which is important in a crowded city.",
"Here LiDAR is preferred over cameras for its high accuracy up to a few centimeters.",
"Two LiDAR sensors, the details of which will be given later, are combined in the ELiD and are mounted in an elevated position in road infrastructure to create a bird eye view.",
"It can generate an HD point cloud over the responsible road section.",
"Such a series of ELiDs can collectively generate an HD road map in a central location (CL).",
"Processing and decision making can be done at the CL and the required control and other data for the self-driving, can be sent back to the corresponding ELiD unit and then to AV as a downlink communication.",
"Elevated LiDAR is mounted on a high elevation, centered to the road.",
"ELiD Module with two stationary LiDAR sensors (not rotating) are angled towards relevant road sections.",
"Both sensor modules collaboratively cover their region up to the maximum accurate distance.",
"A wireless transceiver in the ELiD receives other supportive sensor data from moving vehicles and transmits decisions and commands required for the vehicle.",
"ELiDs and CL are connected by fiber.",
"ELiD is the key component of the system to monitor the environment up to a few centimeter precision and is responsible for establishing a reliable vehicle to infrastructure (V2I) communication with low latency.",
"A high speed backhaul connection will ensure low latency in data transmission and its capacity is in Terabyte scale realized by optical fiber[12].",
"Distance of the backhaul connection will be estimated, according to the latency constraints.",
"CL is responsible for the data processing and storing in a highly secured environment.",
"Object detection and tracking, localization, fusion and motion planning algorithms are performed on the supercomputers to meet a latency requirement lesser than 100 ms which is the expectation of the in-vehicle processing delay in AVs.",
"Real time maps can be generated from the stored data for multiple applications such as traffic predictions with more precision than existing applications."
],
[
"TECHNICAL FEASIBILITY OF THE PROPOSED SYSTEM",
"Technical feasibility can be proven by existing technologies.",
"We searched about various commercial vehicle mounted LiDARs which are rotating.",
"Rotation is the most critical factor in those.",
"Most of the leading LiDAR manufacturers put more focus on vehicle mounted LiDARs which are rotating at 300 to 900 rpm due to existing AV configuration[6].",
"These LiDARs become very expensive due to the actuators.",
"For the proposed system we use Velarray LiDAR sensor which is announced by the Velodyne LiDAR (Fig.",
"REF ) with the specifications given below[6], $120^{0}$ horizontal field-of-view $35^{0}$ verticle field-of-view 200m range for even low reflective objects Small form factor (125mm X 50mm X 55mm) Figure: Velodyne LiDARs Velarray Field of View Measurement (Photo: Business Wire)For our calculations we assume the following (Table I), which ultimately results in 20.8 m coverage width as shown in Fig.",
"REF -REF .",
"Table: AssumptionsFigure: ELiD RangeFigure: Top viewFigure: Side viewMapping the vertical FoV to width of the road, we calculated the elevation of the ELiD based on that as shown in Fig.",
"REF .",
"Horizontal FoV of the LiDAR will produce the maximum coverage by using maximum range.",
"The proposed ELiD made out of two LiDAR sensors covers two directions of the road to maximize the range as in Fig.",
"REF with $120^{0}$ hrozontal FoV of one LiDAR.",
"We carried out the same calculation for the commercially available OS-1 (16-64) rotating LiDAR [13] with the same set of assumptions.",
"The obtained results are tabulated in the Table II and illustrated in Fig.",
"REF .",
"We choose Velarray LiDAR for further investigation due to it's cost effectiveness and suitability.",
"Figure: System OverviewTable: ResultsAlthough the 33 m (Fig.",
"REF ) is a considerable height we can realize this using skyscrapers in an urban area.",
"The proposed architecture (Fig.",
"REF ) is very much suitable for an urban area since infrastructure is already available.",
"Usually optical fiber communication links are well established in urban areas.",
"Using those existing communication links, the system can carry data nearly 203 - 205 km within a millisecond[12].",
"Therefore, in the proposed architecture, we constrain the maximum length of the fiber length to 100 km to maintain total propagation delay below 1 ms.It is essentially same as the processing data within the ELiD.",
"In general a normal LiDAR is working with 100 Mbps output data rate[6].",
"According to our calculation LiDAR density is 5 sensors per km.",
"This will result a 50 Gbps data rate in the fiber under the assumption of 100 km maximum distance.",
"The number of CLs needed to cover a large geographical area will depend on the above mentioned values.",
"Finally the processed data can be transmitted from the corresponding ELiD to the vehicle using a V2I communication protocol.",
"Since the maximum transmission distance is nearly 200 m and the required capacity is small compared to a video streaming, this can be achieved easily.",
"This Idea can be extended to cover the rural areas by reducing the precision of the map which is not as critical as in a highly congested city areas where vehicle density is very high.",
"We can increase the elevation of the ELiD to degrade the precision and decrease the LiDAR density since this is the most expensive component.",
"The High speed fiber backhaul connection should be there as the right-of-way concept [14].",
"Obtaining the required elevation might be the hardest issue to be fixed, which can be sorted-out by a low cost balloon system or using an economical method for civil construction.",
"Proposed architecture has many advantages over the current architecture.",
"The requirement for intensive computational capability of the AVs can be removed from the vehicle and more powerful computations can be carried out at the CL with a low processing latency.",
"The resultant weight reduction, available space utilization and the shape of the vehicle will keep customer interest as same as for the non-AVs.",
"For the current AVs the customer should pay a large amount of money to buy one.",
"However our system can minimize all the costly overheads.",
"It is reported that the LiDAR module used in existing AV's is USD 8000 but velarray LiDAR will cost only a few hundreds of dollars [6].",
"In mass production this will be reduced further.",
"The power requirement is a crucial factor in current AVs.",
"With the removal of the power hungry components from the vehicle it will be able to perform efficiently.",
"If we assume, in a highly congested situation there can be 200 AVs within a 1 km (with 4 lanes assumed as in calculations).",
"All 200 vehicles will generate their own maps which will be an enormous amount of redundant data.",
"In the proposed system, this can be done using 5 LiDAR sensors which will reduce redundant data processing significantly.",
"Also, this has the capability to minimize emergency situations due to the availability of data about hundreds of meters ahead.",
"Since the stationary ELiD is well aware of the existing environment, road signs can be completely eliminated from the road infrastructure.",
"Not just AVs, ordinary vehicles can also benefit from the proposed system.",
"They can receive safety messages and alerts from the smart architecture.",
"Stored data can be used for other applications like traffic monitoring.",
"All the vehicles navigate through a well monitored system (Fig.",
"REF ) where even a speeding driver can receive a warning first or a speed ticket immediately as an Email, as one wishes.",
"There are some challenges which should be addressed properly in the proposed system.",
"With the height of the ELiD and the inclination, tall vehicles can obstruct the view at the edge of the coverage region.",
"This can be fixed by increasing the number of ELiD module in a given distance or by decreasing the elevation, which will reduce the ELiD range.",
"The resolution and the quality of the ELiD modules can be improved more by merging LiDAR sensing in real time.",
"[15] Figure: Graphical Representation of the System"
],
[
"Discussion",
"With this system, automobile manufactures can be released from the signal processing related research and they can focus more on the safety of the passenger.",
"Third party companies like telecommunication providers can design required system and they can operate it according to their policies and earn revenue with a one time investment.",
"Above of all,the standards should be developed in collaboration with automobile manufacturers.",
"System users have to pay a fee for using the ELiD system which will prevent spending more than USD 8000 when purchasing a vehicle.",
"The proposed system can greatly reduce the burden towards communication.",
"With any V2V or V2I standards ongoing [1], AVs will not be able to realize the required massive data rates needed as of now.",
"This system makes the real-time data available to the road infrastructure which is a difficult thing to be done due to uplink constraints.",
"The system can be developed further by addressing security issues and improving reliability.",
"Even with the existing technology of a self-driving car the proposed system can play a supporting role or can be considered as a fallback option to reduce in-vehicle processing and power consumption.",
"In tandem with the smart city concept this architecture will be suitable for city transportation and industry applications such as factories of future (FoF) concept, autonomous cargo handling etc."
],
[
"Related Work",
"In the article [5],The authors give their concept about infrastructure enabled autonomy (IEA) which also corroborates our proposal carried out independently.",
"They describe about the responsibility distribution of current AVs and non-AVs.",
"They propose a system based on infrastructure and give a high level description about how each component is responsible for the system.",
"They focus on distributing responsibilities and liabilities of existing AVs architecture.",
"They assess the “Blame” towards a component after re-distributing responsibilities.",
"They show that this will result in the accelerated deployment of AVs.",
"This is an evidence that our proposal also facilitates the same.",
"The authors in [7] point out that with suitably modified infrastructure systems will be able to deliver low cost in-vehicle technology.",
"In contrast we give a feasible solution which can be implemented anywhere with existing technologies.",
"We provide evidence to support our proposed architecture."
],
[
"CONCLUSIONS",
"We carried out a feasibility analysis of an autonomous vehicular network where the processing burden of the vehicle is transferred to an elevated LiDAR sensor based network architecture.",
"We have used industry standard sensors and available and soon to arrive 5G capabilities.",
"The effect of mobility and problems associated with limited FoV can be successfully addressed in this proposed configuration.",
"The centralized location where all the processing and decision making is done facilitates global perception and accurate decision making.",
"Importantly the impossible task of multi-gigabit uplink rate challenge which is required if the captured data is to be processed outside, is resolved due to this.",
"The current AVs face significant computational and energy challenges at present.",
"Thus the view from outside configuration allows better energy efficiency in electric cars and more reliable decision making.",
"It also paves the way to address similar concerns in other use case scenarios.",
"In addition the proposed network architecture poses several research directions.",
"These are discussed below.",
"One must keep in mind that the existing problems of LiDARs such as lack of visibility in a storm or in heavy snow warrant careful study."
],
[
"Research Problems for AVs",
"The following are the direct possibilities for further investigations.",
"Collaborative map generation in the centralized location.",
"Collating them in an efficient manner to result in a global map for the region is important.",
"Here current mapping methodologies may prove inadequate due to latency issues.",
"Development of application specific LiDAR.",
"In the calculations we have used specifications of a LiDAR still to be released and a commercially available LiDAR.",
"An application specific LiDAR module can improve the performance of the ELiD system.",
"Backhaul design.",
"Suitable capacities are needed in the fiber connections and focus is needed in the latency to determine positions for centralized location.",
"V2I communication link.",
"Resource allocation for this can be done centrally in a cloud RAN (C-RAN) configuration.",
"Network security.",
"As the proposed system will gather data and importantly will send control data for a multiple vehicles The security aspect needs to be fully investigated.",
"This is more important now that the entire system is connected for driving purpose."
],
[
"Research Problems for Other Applications with Mobility",
"The following are the implications for the following cases, connected with a proper communication link.",
"Factories where robots and AVs are used (Fig.",
"REF ).",
"Their vision can be complemented by outside elevated LiDARs or high precision cameras in a similar configuration as discussed in the paper.",
"Figure: ELiD system for a factory Autonomous harbors (Fig.",
"REF ).",
"Here there are many mobile sections which can be greatly facilitated by perception obtained outside.",
"Figure: ELiD system for a harbor Dangerous cases handled by autonomous robots, e.g., places where there is a chemical or otherwise poisonous environment where external LiDARS / cameras mounted on drones can be utilized to generate the required map and send only the relevant control data to the robot in a downlink.",
"Thus there are numerous possibilities in many areas."
],
[
"ACKNOWLEDGMENT",
"Discussions with colleagues working in a leading automobile manufacturer in the US greatly helped in obtaining relevant references and identifying problems faced by AVs.",
"Project 5G-Viima was submitted to Finnish Technology Agency proposing the use of ELiDs in factory floors.",
"This work has been financially supported in part by the 6Genesis (6G) Flagship project (grant 318927)."
]
] | 1808.08617 |
[
[
"An atomic receiver for AM and FM radio communication"
],
[
"Abstract Radio reception relies on antennas for the collection of electromagnetic fields carrying information, and receiver elements for demodulation and retrieval of the transmitted information.",
"Here we demonstrate an atom-based receiver for AM and FM microwave communication with a 3-dB bandwidth in the baseband of $\\sim$100~kHz that provides optical circuit-free field pickup, multi-band carrier capability, and inherently high field sensitivity.",
"The quantum receiver exploits field-sensitive cesium Rydberg vapors in a centimeter-sized glass cell, and quantum-optical readout of baseband signals modulated onto carriers with frequencies ranging over four octaves, from C-band to Q-band.",
"Receiver bandwidth, dynamic range and sideband suppression are characterized, and acquisition of audio waveforms of human vocals demonstrated.",
"The atomic radio receiver is a valuable receiver technology because it does not require antenna structures and is resilient against electromagnetic interference, while affording multi-band operation in a single compact receiving element."
],
[
"Introduction",
"Since its advent in the late nineteenth century, radio communication in the audio band has been an integral component of society and proven essential to the advancement of science and technology [1], [2], [3].",
"Ongoing challenges include information security, bandwidth increase by tapping the microwave- and mm-wave regimes [4], and resilience against electromagnetic interference (EMI).",
"In recent years, quantum technologies have become a topic of interest in this field, as they exploit quantum-mechanical phenomena to build devices for secure data transfer [5], [6], [7], realize small atom-based receivers that harness the high sensitivity of atoms [8], [9], [10], [11] and artificial quantum structures [12], [13], [14] to electromagnetic fields, and quantum receivers that employ the immunity of atoms to very intense fields [15], [16] for wire-free EMI-tolerant detector technology.",
"An emerging type of quantum sensor employs Rydberg atoms, which are atoms in highly excited atomic states [17], [18] that offer a giant electromagnetic response due to resonant frequency matching with radio-frequency and microwave radiation (RF), at carrier frequencies typically used in communications, ranging from tens of MHz into the THz range.",
"The atomic response arises from the strong electric-dipole coupling of the atoms to the electromagnetic field and can be retrieved quantum-optically using electromagnetically induced transparency (EIT) [19], a spectroscopic method that enables optical measurement of RF fields, without circuitry required within the atom-based RF field sensing structure.",
"The growing interest in Rydberg-atom-based field measurement includes the development of calibration-free, non-invasive sensors for static and time-varying electric [8], [9], [10], [11], [15], [20], [16], [21] fields, as well as for magnetic fields up to 1 Tesla [22], [23], [24].",
"Pulse and analog modulation of carrier waves allows the adaptation of atomic sensors to applications in communications tasks.",
"For instance, atom-based magnetic sensors have already been proposed for communication based on binary phase shift keying [25], and digital communication based on Rydberg atoms has recently been realized [26].",
"In this work, we demonstrate a Rydberg quantum sensor to receive, play back and record baseband signals in the audio range that are amplitude-modulated (AM) or frequency-modulated (FM) onto microwave carriers [27].",
"A single quantum detector is employed for multi-band AM and FM communication using carrier frequencies spanning more than four octaves, from the C-band to Q-band.",
"The atomic radio wave receiver (or “atomic radio”) operates by direct real-time optical detection of the atomic response to AM and FM baseband signals, precluding the need for traditional de-modulation and signal-conditioning electronics.",
"The small atomic vapor-cell sensor head replaces the function of traditional antennas, realizing a compact multi-band receiving element.",
"The absence of any circuitry within the receiver components exposed to the incident fields makes the atomic radio inherently EMI-tolerant.",
"In the presented work we develop the principle of operation of the Rydberg atomic radio.",
"We employ an experimental demonstration unit to characterize the baseband bandwidths and dynamic ranges for both AM and FM modulation, and we record audio samples of human vocals.",
"The demonstrated atomic radio exhibits good performance over the entire human audio band, with an upper baseband frequency limit exceeding 100 kHz.",
"We discuss the selection of laser operating points that maximize radio performance for given spectroscopic signatures, which depend on the atomic transitions and carrier waves used.",
"While the achieved dynamic range between 20 dB and 30 dB falls slightly short of radio standards, we see considerable room for improvement of our demonstration unit."
],
[
"Experimental setup",
"The experimental setup comprising of the transmitter system and quantum RF receiver is illustrated in Fig.",
"REF a.",
"The transmitter system consists of either an audio source collected by a microphone or a function generator that produces a baseband signal used to amplitude-modulate or frequency-modulate a microwave carrier.",
"The microwave signal, produced by a microwave signal generator, is fed into and emitted from a horn antenna, and is directed towards an atomic quantum RF receiver located several centimeters away.",
"The quantum receiver implements a 20 mm spectroscopic cell containing a cesium (Cs) vapor as the detector element.",
"The incident AM and FM transmission signals are detected by real-time optical readout of field-sensitive Rydberg states of the Cs vapor in the cell using electromagnetically induced transparency (EIT).",
"An energy-level diagram for the Rydberg-EIT optical readout on the Cs 47S$_{1/2}$ Rydberg state is shown in Fig.",
"REF b.",
"The EIT readout is performed using a probe beam with a wavelength of 852 nm focused to $330~\\mu $ m full-width-half-maximum (FWHM), that is counter-propagated and overlapped with a coupler beam with a wavelength of $\\sim $ 510 nm focused to $390~\\mu $ m FWHM.",
"The probe is frequency-stabilized to the Cs $6S_{1/2}~F=4\\rightarrow 6P_{3/2}~F^{\\prime }=5$ transition, and its absorption through the cell is monitored on a photodiode, while the coupling laser is set near the chosen $6P_{3/2}\\rightarrow Rydberg$ transition.",
"The coupler-beam wavelength is tuned to reach the desired Rydberg state.",
"Two experimental modes are then employed.",
"To analyze the Rydberg EIT lines and to identify coupler-laser frequency operating points suitable for radio reception, the coupler frequency is scanned linearly at a rate of a few Hz over a few hundred MHz across the selected Rydberg transition (Figures REF and REF ).",
"In radio reception mode, the coupler frequency is held fixed at an operating point near the center of a Rydberg resonance; this mode is suitable for real-time baseband detection (Figures REF , REF and REF ).",
"The probe photodiode signal is amplified by a preamplifier and then acquired by an oscilloscope, connected to an A-to-D converter for digital recording, or plugged directly into an audio speaker.",
"Figure: a Experimental setup.",
"b Energy-level diagram.",
"c Spectral readout of the Cs 47S 1/2 _{1/2} Rydberg line without microwave signal (black), with an un-modulated 37.4065 GHz carrier resonantly driving the Cs 47S 1/2 _{1/2} to 47P 1/2 _{1/2} Rydberg transition (blue), and with the carrier amplitude-modulated at baseband AM frequency 1 kHz and modulation depth ±25%\\pm 25\\% (red).",
"d Spectral readout of the Cs 30D 5/2 _{5/2} Rydberg line without microwave (black), with an un-modulated 29.458 GHz carrier resonantly driving the Cs 30D 5/2 _{5/2} to 31P 3/2 _{3/2} Rydberg transition (blue),and with the carrier frequency-modulated at baseband FM frequency 1 kHz and modulation deviation ±30\\pm 30 MHz (red)."
],
[
"Optical readout of AM or FM microwave carrier fields",
"Atoms with a quasi-free electron in a high-lying Rydberg state exhibit large polarizabilities and electric dipole moments that scale with principal quantum number $n$ as $\\sim n^7$ and $\\sim n^2$ , respectively, rendering them exquisitely sensitive to electric fields [17].",
"Figure REF c shows the spectroscopic EIT readout from the vapor-cell receiver of the Cs 47S$_{1/2}$ Rydberg line without RF applied (black curve) and with application of a 37.4065 GHz K$_{\\rm a}$ -band un-modulated carrier wave resonant with the Cs 47S$_{1/2}$ to 47P$_{1/2}$ Rydberg transition at a power of -5 dBm injected into the transmission line and horn (blue curve).",
"Application of the RF field causes the line to split symmetrically into a pair of Autler-Townes (AT) lines whose separation corresponds to the Rabi frequency $\\Omega $ of the Rydberg/RF-field interaction.",
"The measured splitting is directly proportional to the microwave carrier electric field strength and is given by the relation $E=\\hbar \\Omega /d$ , where $d$ is the transition dipole moment of the Rydberg transition.",
"From the observed splitting in Fig.",
"REF c, we measure a carrier field amplitude at the location of the receiver atoms of $E$ =5.9 V/m.",
"A Rydberg-EIT-AT spectrum taken with a microwave carrier that is AM-modulated by a sinusoidal 1 kHz baseband signal with $\\pm 25\\%$ modulation depth is overlaid in Fig.",
"REF c (red curve).",
"The laser detuning $\\Delta _{510}$ is scanned linearly at a rate of $\\sim $ 10 Hz across the atomic resonance.",
"The 1 kHz-AM of the carrier manifests in the optical sampling/readout in the form of periodic oscillations that occur while $\\Delta _{510}$ is slowly scanned across the EIT-AT resonance.",
"The oscillations result from the dependence of the Rydberg-EIT-AT spectrum on the RF electric field amplitude.",
"The AM causes a modulation of the AT-splitting of the spectrum and of the corresponding optical signal, which occurs rapidly and concurrently with the relatively slow scan of $\\Delta _{510}$ across the AT-split 47S$_{1/2}$ to 47P$_{1/2}$ resonance.",
"The vertical range of the AM-induced excursions of the optical signal provides a measure for the AM-sensitivity of the Rydberg-EIT-AT signal as a function of $\\Delta _{510}$ .",
"In the case displayed in Fig.",
"REF c (red curve), the detector exhibits a maximal AM response at $\\Delta _{510}=0$ , the center of the 47S$_{1/2}$ to 47P$_{1/2}$ Rydberg-EIT-AT resonance.",
"Hence, for the case in Fig.",
"REF c $\\Delta _{510}=0$ is chosen as the coupler-laser frequency operating point for real-time detection of AM baseband signals (black dot and dashed line in Fig.",
"REF c).",
"Rydberg states of atoms and their resonant AT responses to RF fields are, in addition to being dependent on RF amplitude, also susceptible to changes in RF field frequency [28].",
"In the weak-field limit of resonant AT, a change in the RF field frequency manifests in an asymmetry of the AT-line-pair signal strength as well as in a variation of the AT-splitting and the AT line positions.",
"The combination of these effects affords baseband signal detection in FM-modulated carriers via a similar spectroscopic readout as with AM.",
"Figure REF d shows the optical readout from the receiver of the 30D$_{5/2}$ atomic Rydberg line without RF applied (black curve), with an un-modulated 29.458 GHz K$_{\\rm a}$ -band 6.3-V/m carrier wave that is resonant with the Cs 30D$_{5/2}$ to 31P$_{3/2}$ Rydberg transition (blue curve), and with a 1 kHz sinusoidal baseband signal FM-modulated onto the carrier at a $\\pm $ 30 MHz maximal deviation (red curve).",
"For D to P transitions, dipole selection rules dictate that linearly-polarized microwave fields may only couple $\\Delta m_j =0$ Rydberg transitions.",
"As a result, the 30D$_{5/2}$ to 31P$_{3/2}$ $m_j$ =1/2 and 3/2 components are respectively coupled by the microwave field and exhibit similar AT splittings, whereas the 30D$_{5/2}$ $m_j=5/2$ level that is not coupled to another state remains unaffected by the RF and produces a line centered at $\\Delta _{\\rm 510}=0$ .",
"It follows that the desired coupler-laser operating frequency for FM baseband detection using resonant D to P Rydberg transitions is different from that when using S to P transitions.",
"This is apparent in Fig.",
"REF d, where we see that for D to P transitions the atomic response to the FM signal is maximal at the inner inflection points of the AT-split pair (black dot and dashed line in Fig.",
"REF d)."
],
[
"Dynamic range and harmonic response to AM and FM signals.",
"For application in communication systems, the baseband bandwidth and transient response behavior of the quantum receiver are useful performance metrics and benchmarks for comparison to traditional antenna-receivers.",
"In Fig.",
"REF we characterize the response and detection speed of the quantum receiver for an AM-modulated 37.4 GHz carrier resonantly driving the 47S$_{1/2}$ to 47P$_{1/2}$ transition (Fig.",
"REF b).",
"Here, we set $\\Delta _{510}$ =0 MHz at the field-free Rydberg resonance for maximum sensitivity to the AM field (Fig.",
"REF b) and monitor the optical 852 nm transmission through the receiver cell in real time.",
"Figure REF a shows the optical readout of a harmonic 1 kHz baseband signal for a series of AM modulation depths over a 5 ms interval.",
"The measured optical readout maintains a high baseband signal visibility and integrity over a range of modulation depths from 5% to 45%, indicating a well-behaved, linear atomic response over a wide dynamic range.",
"To quantify the noise characteristics of the receiver we compute the power spectrum for the received AM signal at $\\pm 25\\%$ modulation depth (Fig.",
"REF b).",
"The analysis reveals a single overtone at 2 kHz (second harmonic) that is 19.1 dB weaker than the fundamental.",
"The overtone is attributed to the mildly nonlinear dependence of the Rydberg-EIT-AT signal at $\\Delta _{510}$ =0 MHz on the carrier amplitude.",
"To characterize the receiver's transient response, we transmit and receive a signal that is AM-modulated with a 1 kHz square-wave pulse in the baseband, which contains strong higher-harmonic content at odd overtone frequencies (Figs.",
"REF c, d, e).",
"The rise time of the received pulse is measured to be a few microseconds, indicating an upper limit to the receiver AM bandwidth of a few hundred kHz.",
"The response time is limited by the bandwidth of the photodiode-amplifier used in our demonstration setup; the fundamental response time of the Rydberg-EIT-AT signal itself is shorter.",
"The power spectrum of the received square-wave pulse (Fig.",
"REF c) is observed to consist of odd higher harmonics at $(2k+1)f$ , with integer $k$ and fundamental AM frequency $f$ , with spectral power decreasing as $dB\\sim -2\\log (k)$ , as one would expect for an ideal square wave pulse (dashed line in Fig.",
"REF e).",
"Figure: Optical response to AM with harmonic and square-wave baseband signals on the 47S 1/2 _{1/2} to 47P 1/2 _{1/2} Rydberg transition.",
"a Real-time optical readout for a 1 kHz sinusoidal baseband signal with AM modulation depths of ±\\pm 5% (blue), ±\\pm 25% (purple), and ±\\pm 45% (black) at laser operating point Δ 510 \\Delta _{510}=0 MHz (see Fig. c).",
"b Power spectrum of the received AM 1 kHz signal at a modulation depth of ±\\pm 25%.",
"c Optical readout of a square pulse, d a zoom-in on its rising edge, and e its power spectrum.The harmonic response and dynamic range of the quantum receiver for both AM and FM transmission with $K_a$ -band microwave carriers using the resonant Rydberg transitions and coupler-frequency operating points shown in Figs.",
"REF c and d are presented in Figure REF .",
"In Figs.",
"REF a and c we plot the measured power spectral density at a selection of baseband modulation frequencies for AM and FM modulation, respectively; these figures reveal the baseband bandwidth of the Rydberg radio.",
"For both AM and FM, the measured power spectral densities indicate a 3-dB baseband bandwidth $\\gtrsim 100$ kHz.",
"In Figs.",
"REF b and d we plot the power of the signal received at 1 kHz modulation frequency as a function of AM modulation depth and FM deviation, respectively; these figures show the dynamic range (at 1 kHz).",
"For the experimental conditions used in the present work, including selected Rydberg transitions, carrier strengths, and coupler-laser-frequency operating points, the receiver exhibits a dynamic range between 20 and 30 dB, with sensitivities reaching AM modulation depths of a few percent and FM deviations of a few hundred kHz.",
"Figure: Power spectral density and dynamic range of the quantum receiver for a, b AM transmission on a 37.4 GHz carrier tuned to the 47S 1/2 _{1/2} to 47P 1/2 _{1/2} Rydberg transition with Δ 510 =0\\Delta _{510}=0 MHz and c, d FM transmission on a 29.548 GHz carrier tuned to the 30D 5/2 _{5/2} to 31P 3/2 _{3/2} Rydberg transition with Δ 510 ≈50\\Delta _{510}\\approx 50 MHz."
],
[
"Multi-band operation and audio recording",
"The wide selection of field-sensitive atomic Rydberg transitions that can be optically accessed using EIT allows a single vapor-cell quantum RF receiver to operate over an unprecedented range of carrier frequencies, from MHz to THz [10].",
"To demonstrate the multi-band capability, we employ the quantum receiver for detection of AM and FM audio on a C-band carrier by tuning the frequency of the coupling laser to target a suitable Rydberg state.",
"In Fig.",
"REF we show AT-EIT spectra centered on the 54D$_{5/2}$ Rydberg line with 1 kHz AM and FM baseband signals transmitted on a 4.5 GHz C-band carrier field resonant with the Cs 54D$_{5/2}$ to 55p$_{3/2}$ Rydberg transition for comparable series of AM modulation depths and FM deviations.",
"The dipole moment of the C-band transition used here is larger than those of the $K_a$ -band transitions, leading to a higher field sensitivity and a slightly wider baseband dynamic range in the C-band case.",
"The demonstrated robust performance in the C-band shows that the receiver covers multiple carrier bands, with the carrier frequency range spanning more than four octaves.",
"Figure: Optical spectral readout of a 1 kHz baseband signal at a 4.5 GHz C-band RF carrier frequency for FM (left) and AM (right) transmission using the resonant Cs 54D 5/2 _{5/2} to 55P 3/2 _{3/2} Rydberg transition.",
"Spectra corresponding to a series of FM deviations and AM modulation depths are shown, as indicated.",
"The spectra are centered to the RF-field-free Cs 54D 5/2 _{5/2} Rydberg line.Figure: Audio waveforms of human vocals recorded from received FM (a, b, c) and AM (d, e, f) audio transmission at a 4.5 GHz (C-band) carrier.Inspection of the strongly modulated spectra in Fig.",
"REF also reveals a difference in EIT-AT response for AM and FM detection that may arise near the peaks of the AT-line pairs.",
"For FM detection, the readout near the center of the AT peaks generally exhibits a minimum in the observable modulation.",
"This is due to a cancellation of the simultaneous change in relative AT-peak signal strength and the AT line shift.",
"For AM detection, on the other hand, the modulated AT-splitting at the peak leads to a doubling of the optically-detected modulation signal-frequency.",
"The identification of these features in the present demonstration supports the choice of coupler-frequency operating points away from the AT-split peak centers.",
"Nevertheless, these features may prove advantageous to realizing advanced operation modes of the quantum receiver.",
"For a demonstration of the full functionality of the quantum receiver in radio communications in the audio band, we implement our unit for AM and FM C-band transmission recording and playback of human vocals.",
"Recorded waveforms of a human voice singing the first verse of 'Mary had a little Lamb' transmitted via AM and FM on a 4.5 GHz carrier and received by the quantum sensor are shown in Fig.",
"REF .",
"Higher-resolution plots of equivalent sub-sections of the full original waveforms reveal details of the complex audio waveforms captured by the receiver, and variations in harmonic content and volume between the two separate AM and FM recordings are reproduced.",
"For audio files of recordings using the atomic receiver please contact the authors."
],
[
"Conclusion",
"An atom-based quantum AM and FM receiver with a 3-dB bandwidth in the baseband of $\\sim $ 100 kHz has been demonstrated.",
"The receiver is based on direct atom-optical detection and demodulation of AM and FM baseband signals on carriers ranging from C-band to Q-band, with a single quantum sensing element.",
"We have investigated the selection of laser operating points that maximize radio performance for the given spectroscopic signatures, which depend on the atomic transitions and carrier-wave fields used.",
"We have characterized the harmonic response and measured a dynamic range between 20 dB and 30 dB for our demonstration unit.",
"The presented quantum receiver is based on direct quantum-optical collection and demodulation of AM and FM radio waves and paves the way for radio communication applications inaccessible to existing antenna-receiver technology, such as multi-carrier-frequency operation combined with high field-sensitivity in a single, compact detector unit.",
"Further development of Rydberg-based quantum receivers towards higher-bandwidth data transfer over a wider range of carrier frequencies appears readily accessible.",
"Upon completion of this manuscript we became aware of related work on optical detection of modulated carrier signals using Rydberg atoms [29]."
],
[
"Acknowledgement",
"This work was supported by Rydberg Technologies."
]
] | 1808.08589 |
[
[
"Measure-valued solutions for the equations of polyconvex adiabatic\n thermoelasticity"
],
[
"Abstract For the system of polyconvex adiabatic thermoelasticity, we define a notion of dissipative measure-valued solution, which can be considered as the limit of a viscosity approximation.",
"We embed the system into a symmetrizable hyperbolic one in order to derive the relative entropy.",
"However, we base our analysis in the original variables, instead of the symmetric ones (in which the entropy is convex) and we prove measure- valued weak versus strong uniqueness using the averaged relative entropy inequality."
],
[
"Introduction",
"For systems of hyperbolic conservation laws, the class of measure-valued solutions [16] provides a notion of solvability vast enough to support a global existence theory.",
"These solutions usually arise as limits of converging sequences satisfying an approximating parabolic problem [12].",
"As these solutions are considered to be very weak, it is crucial to examine their stability properties with respect to classical solutions and to attempt that in their natural energy framework.",
"The relative entropy method of Dafermos [11], [10] and DiPerna [15] provides an analytical framework upon which one can examine such questions, and has been tested in a variety of contexts (e.g.",
"[5], [14], [20], [8], [18], [6]).",
"In this article, we derive (in the appendix) a framework of dissipative measure valued solutions for the system of adiabatic polyconvex thermoelasticity, motivated by approximating that system by the system of thermoviscoelasticity on the natural energy framework.",
"The relative entropy method is then used to show weak-strong uniqueness for polyconvex thermoelasticity in the class of measure-valued solutions.",
"The main novelty of this work is the derivation of the averaged relative entropy inequality with respect to a dissipative measure-valued solution.",
"This solution is defined by means of generalized Young measures, describing both oscillatory and concentration effects.",
"The analysis is based on the embedding of polyconvex thermoelasticity into an augmented, symmetrizable, hyperbolic system, [7].",
"However, the embedding cannot be used in a direct manner, and notably, instead of working with the extended variables, we base our analysis on the parent system in the original variables using the weak stability properties of some transport-stretching identities, which allow us to carry out the calculations by placing minimal regularity assumptions in the energy framework.",
"Consider the system of adiabatic thermoelasticity, $\\begin{split}\\partial _t F_{i\\alpha }&=\\partial _{\\alpha }v_i \\\\\\partial _t v_i&=\\partial _{\\alpha }\\Sigma _{i\\alpha }\\\\\\partial _t\\left(\\frac{1}{2}|v|^2+e\\right)&=\\partial _{\\alpha }(\\Sigma _{i\\alpha }v_i)+r\\end{split}$ describing the evolution of a thermomechanical process $\\big ( y(x,t) , \\theta (x,t) \\big ) \\in \\mathbb {R}^3\\times \\mathbb {R}^+$ with $(x,t)\\in \\mathbb {R}^3\\times \\mathbb {R}^+$ .",
"Here, $F \\in \\mathbb {M}^{3\\times 3}$ stands for the deformation gradient, $F = \\nabla y$ , while $v = \\partial _t y$ is the velocity of the motion $y$ and $\\theta $ is the temperature.",
"The condition $\\partial _{\\alpha }F_{i\\beta }=\\partial _{\\beta }F_{i\\alpha }, \\qquad i,\\alpha ,\\beta =1,2,3 \\, ,$ imposes that $F$ is a gradient and comes from equation (REF )$_1$ as an involution inherited from the initial data.",
"The stress is denoted as $\\Sigma _{i \\alpha }$ , the internal energy as $e$ and the radiative heat supply as $r$ .",
"The requirement of consistency with the Clausius-Duhem inequality imposes that the elastic stresses $\\Sigma $ , the entropy $\\eta $ and the internal energy $e$ are related to the free-energy function $\\psi $ via the constitutive theory $\\psi =\\psi (F,\\theta ),\\quad \\Sigma =\\frac{\\partial \\psi }{\\partial F},\\quad \\eta =-\\frac{\\partial \\psi }{\\partial \\theta },\\quad e=\\psi +\\theta \\eta \\, .$ Here, we work in the polyconvex regime where the free energy $\\psi $ factorizes as a uniformly convex function of the null-Lagrangian vector $\\Phi (F)$ and the temperature $\\theta ,$ namely $\\psi (F, \\theta ) = \\hat{\\psi }(\\Phi (F),\\theta )\\,,$ satisfying $\\hat{\\psi }_{\\xi \\xi } (\\xi , \\theta ) > 0 \\, , \\quad \\hat{\\psi }_{\\theta \\theta } (\\xi , \\theta ) < 0 \\, ,$ where $\\hat{\\psi }=\\hat{\\psi }(\\xi ,\\theta )$ is a strictly convex function on $\\mathbb {R}^{19}\\times \\mathbb {R}^+$ and $\\Phi (F)=(F,\\mathrm {cof} F,\\det F)\\in \\mathbb {M}^{3\\times 3}\\times \\mathbb {M}^{3\\times 3}\\times \\mathbb {R}.$ The main result of this article is the weak-strong uniqueness of polyconvex adiabatic thermoelasticity (REF )-(REF ) in the class of dissipative measure-valued solutions.",
"The advantage of the dissipative framework is that the averaged energy equation holds in its integrated form.",
"Even though this notion of solutions is generally considered to be very weak, not possessing detailed information, this result contributes to a long list of similar works [1], [17], [13], [14], [8] on hyperbolic systems of conservation laws, pointing out the importance of this framework in the analysis of such physical problems.",
"Unlike the case of scalar conservation laws [16], [22], where the theory of Young measures suffices to deal with nonlinearities and overcome oscillatory behaviors, when it comes to hyperbolic systems, one must take into account the formation of both oscillations and concentrations.",
"In our case, the concentration effects are described through a concentration measure, which appears in the energy equation since the Fundamental Lemma of Young measures cannot represent the weak limits of $\\frac{1}{2}|v|^2+e,$ due to lack of $L^1$ precompactness.",
"This is illustrated in Appendix .",
"Thus we turn our attention to the theory of generalized Young measures [1], [17], [5], [14], [8], [20] and apply the relative entropy formulation to compare a dissipative measure-valued solution to polyconvex thermoelasticity against a strong solution.",
"We organize this paper as follows: In Section , we define the notion of dissipative measure-valued solutions for polyconvex thermoelasticity.",
"This definition comes as a result of the limiting process we discuss in Appendix , starting from the associated viscous problem.",
"Section is dedicated to the study of the generated Young measure and the concentration measure, which is a well-defined, nonnegative Radon measure for a subsequence of approximate solutions coming from a uniform bound on the energy.",
"In Section we calculate the averaged relative entropy inequality (REF ) and in Section we use it to prove the main theorem on uniqueness of strong solutions in the class of measure-valued solutions.",
"The proof is heavily based on the estimates (REF ) and (REF )-(REF ) on the relative entropy, namely Lemmas REF , REF , which are stated and proved at the level of the original variables, instead of the extended ones.",
"As a result, we only assume quite minimal growth hypotheses on the constitutive functions, which guarantee all the necessary technical requirements for the dissipative measure-valued versus strong uniqueness to hold.",
"Additionally, the proof is carried on with respect to a dissipative solution which satisfies an averaged and integrated version of the energy equation, where the concentration measure appears.",
"This setting has the strong advantage that we need no artificial integrability restrictions on the energy equation.",
"Similar results are available for the incompressible Euler equations [5], for polyconvex elastodynamics [14], and for the isothermal gas dynamics system [20]."
],
[
"Measure-valued solutions for polyconvex adiabatic thermoelasticity",
"Consider the system of adiabatic thermoelasticity (REF ), (REF ) together with the entropy production identity $\\partial _t \\eta =\\frac{r}{\\theta } \\, ,$ under the constitutive theory (REF ) and the polyconvexity hypothesis (REF ).",
"To avoid unnecessary technicalities, henceforth we work in a domain $Q_T=\\mathbb {T}^d\\times [0,T)$ , where $\\mathbb {T}^d$ is the torus, $d=3$ and $T\\in [0,\\infty )$ .",
"In the polyconvex case, the Euler-Lagrange equation, $\\partial _{\\alpha }\\left( \\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }} (\\nabla y ) \\right)=0 \\, , \\quad B=1,\\dots ,19\\;,$ formulated for the vector of the minors $\\Phi (F)=(F,\\mathrm {cof} F,\\det F)\\in \\mathbb {M}^{3\\times 3}\\times \\mathbb {M}^{3\\times 3}\\times \\mathbb {R},$ holds for any motion $y(x,t)$ and together with the kinematic compatibility equation (REF )$_1$ and (REF ), allows to express $\\partial _t \\Phi ^B(F)$ as $\\partial _t \\Phi ^B(F)=\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)\\partial _{\\alpha }v_i=\\partial _{\\alpha }\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)v_i\\right),$ for any deformation gradient $F$ and velocity field $v.$ Additionally, the stress tensor $\\Sigma $ becomes $\\Sigma _{i\\alpha }=\\frac{\\partial \\psi }{\\partial F_{i\\alpha }}(F,\\theta )=\\frac{\\partial }{\\partial F_{i\\alpha }}\\left(\\hat{\\psi }(\\Phi (F),\\theta )\\right)=\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta )\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F).$ The first nine components of $\\Phi (F)$ are the components of $F,$ therefore (REF ) implies that we can express the entropy $\\eta $ and the internal energy $e$ with respect to the null-Lagrangian vector $\\Phi (F),$ namely $\\begin{split}\\eta (F,\\theta )&=-\\frac{\\partial \\psi }{\\partial \\theta }(F,\\theta )=-\\frac{\\partial \\hat{\\psi }}{\\partial \\theta }(\\Phi (F),\\theta ) = :\\hat{\\eta }( \\Phi (F) , \\theta ),\\\\e(F,\\theta )&=\\psi (F,\\theta )-\\theta \\frac{\\partial \\psi }{\\partial \\theta }(F,\\theta )=\\hat{\\psi }(\\Phi (F),\\theta )-\\theta \\frac{\\partial \\hat{\\psi }}{\\partial \\theta }(\\Phi (F),\\theta )= \\hat{e}(\\Phi (F),\\theta ) \\, ,\\end{split}$ where we have set $\\hat{\\eta }( \\xi , \\theta ) := -\\frac{\\partial \\hat{\\psi }}{\\partial \\theta }( \\xi ,\\theta ) \\, , \\quad \\hat{e}(\\xi ,\\theta ) := \\hat{\\psi }(\\xi ,\\theta )- \\theta \\frac{\\partial \\hat{\\psi }}{\\partial \\theta }(\\xi ,\\theta ) \\, .$ This allows to supplement the equations of polyconvex thermoelasticity (REF ), (REF ) with (REF ) and write $\\begin{split}\\partial _t \\Phi ^B(F)&=\\partial _{\\alpha }\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)v_i\\right)\\\\\\partial _t v_i&=\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta ) \\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)\\right)\\\\\\partial _t \\left(\\frac{1}{2}|v|^2+\\hat{e}(\\Phi (F),\\theta )\\right)&=\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta ) \\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F) v_i\\right)+r\\\\\\partial _{\\alpha }F_{i\\beta }&=\\partial _{\\beta }F_{i\\alpha }\\end{split}$ while the entropy production identity (REF ) becomes $\\partial _t \\hat{\\eta }(\\Phi (F),\\theta ) =\\frac{r}{\\theta }.$ This implies that $(\\xi = \\Phi (F) , v, \\theta )$ satisfies the augmented system $\\begin{split}\\partial _t \\xi ^B &=\\partial _{\\alpha }\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)v_i\\right)\\\\\\partial _t v_i&=\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\xi ,\\theta ) \\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)\\right)\\\\\\partial _t \\left(\\frac{1}{2}|v|^2+\\hat{e}(\\xi ,\\theta )\\right)&=\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\xi ,\\theta ) \\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F) v_i\\right)+r\\\\\\partial _{\\alpha }F_{i\\beta }&=\\partial _{\\beta }F_{i\\alpha }\\end{split}$ that consists of conservation laws in $\\mathbb {R}^{23}$ subject to the involution (REF )$_4$ .",
"The augmented system satisfies the entropy production identity $\\partial _t \\hat{\\eta }(\\xi ,\\theta ) =\\frac{r}{\\theta }$ and is thus symmetrizable; see [7] for further details.",
"The system (REF ) belongs to a general class of hyperbolic conservation laws of the form $\\partial _t A(U)+\\partial _{\\alpha }f_{\\alpha }(U)=0,$ $U:\\mathbb {R}^d\\times \\mathbb {R}^+\\rightarrow \\mathbb {R}^n,$ studied in [8].",
"Due to (REF ) it is symmetrizable, hyperbolic in the extended variables.",
"A general theory including a theorem establishing recovery of classical solutions from dissipative measure–valued solutions for hyperbolic systems endowed with a convex entropy, was developed in [8].",
"We note that since in the variables $(F,v,\\theta )$ system (REF ) is not equipped with a convex entropy, we cannot treat this problem as a direct application of the general setting developed in [8].",
"In [7], system (REF )–(REF ) was studied by augmenting it to (REF ) using the relative entropy method in order to prove convergence from thermoviscoelasticity to the system (REF )–(REF ).",
"The objective in the present paper is to prove a weak-strong uniqueness theorem in the context of measure-valued solutions.",
"This requires to work at the level of the original rather than the augmented system what presents various technical challenges.",
"Following the theory on generalized Young measures [1], [8], [17], we define a dissipative measure-valued solution to polyconvex thermoelasticity, which involves a parametrized Young measure $\\nu =\\nu _{(x,t)}$ describing the oscillatory behavior of the solution and a Radon measure $\\gamma \\in \\mathcal {M}^+(Q_T)$ describing concentration effects.",
"According to the analysis in Appendix , we can treat dissipative measure-valued solutions as limits of an approximating solution for the associated viscous problem, that satisfy an averaged and integrated energy equation.",
"The reason behind the formation of concentrations, lies with the fact that the energy function $(x,t)\\mapsto |v|^2 + e(F,\\theta ) $ is not weakly precompact in $L^1$ and thus, the Young measure representation fails.",
"Since the only uniform bound at one's disposal is on the energy, the way we construct these solutions corresponds to a minimal framework obtained from this natural bound, for viscosity approximations of the adiabatic thermoelasticity system.",
"The analysis in Appendix leads to the following definition: Definition 2.1 A dissipative measure valued solution to polyconvex thermoelasticity (REF ), (REF ) consists of a thermomechanical process $(y(t,x), \\theta (t,x)) : [0,T] \\times \\mathbb {T}^3 \\rightarrow \\mathbb {R}^3 \\times \\mathbb {R}^+$ , $y \\in W^{1,\\infty }(L^2(\\mathbb {T}^3)) \\cap L^\\infty (W^{1,p} (\\mathbb {T}^3)) \\, , \\quad \\theta \\in L^{\\infty }(L^{\\ell } (\\mathbb {T}^3) ) \\, ,$ a parametrized family of probability Young measures $\\nu =\\nu _{(x,t)\\in \\bar{Q}_T},$ with averages $F=\\left\\langle \\nu ,\\lambda _F \\right\\rangle ,\\quad v=\\left\\langle \\nu ,\\lambda _{v}\\right\\rangle , \\quad \\theta =\\left\\langle \\nu ,\\lambda _{\\theta }\\right\\rangle \\, ,$ and a nonnegative Radon measure $\\gamma \\in \\mathcal {M}^+(Q_T)$ , where $F=\\nabla y\\in L^{\\infty }(L^p), \\quad v=\\partial _t y\\in L^{\\infty }(L^2) \\, ,$ $\\Phi (F)=(F, \\mathrm {cof} F, \\det F)\\in L^{\\infty }(L^p)\\times L^{\\infty }(L^q)\\times L^{\\infty }(L^{\\rho }) \\, ,$ $p\\ge 4,$ $q \\ge 2$ , $\\rho > 1$ , $\\ell >1$ , which satisfy the averaged equations $\\begin{split}\\partial _t \\Phi ^B(F)&=\\partial _{\\alpha }\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)v_i\\right) \\\\\\partial _t \\left\\langle \\nu ,\\lambda _{v_i}\\right\\rangle &=\\partial _{\\alpha }\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _{F})\\right\\rangle \\\\\\partial _t \\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle &\\ge \\left\\langle \\nu ,\\frac{r}{\\lambda _{\\theta }}\\right\\rangle \\end{split}$ in the sense of distributions, together with the integrated form of the averaged energy equation, $\\begin{split}\\int & \\varphi (0) \\left(\\left\\langle \\nu ,\\frac{1}{2}|\\lambda _{v}|^2+\\hat{e}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle (x,0)\\:dx +\\gamma _0(dx) \\right)\\\\&+\\int _0^T\\int \\varphi ^{\\prime }(t) \\left(\\left\\langle \\nu ,\\frac{1}{2}|\\lambda _{v}|^2+\\hat{e}(\\Phi (\\lambda _{F}),\\lambda _{\\theta }) \\right\\rangle (x,t) \\, dx\\:dt +\\gamma (dx\\:dt) \\right) \\\\&=-\\int _0^T\\int \\left\\langle \\nu ,r\\right\\rangle \\varphi (t)\\:dx\\:dt,\\end{split}$ for all $\\varphi \\in C^1_c[0,T].$ In this definition, the first equation (REF )$_1$ holds in a classical weak sense under the regularity conditions (REF ),(REF ) placed on the motion and its derivatives for $p\\ge 4,$ $q \\ge 2,$ $\\rho , \\ell >1,$ as a consequence of the weak continuity of the null-Lagrangian vector $(F,\\mathrm {cof} F,\\det F)$ and the weak continuity of the transport-stretching identities $\\begin{aligned}\\partial _t F_{i\\alpha }&=\\partial _{\\alpha }v_i \\\\\\partial _t \\det F&=\\partial _\\alpha \\bigl ((\\mathrm {cof} F)_{i\\alpha }v_i\\bigr )\\\\\\partial _t(\\mathrm {cof} F)_{k \\gamma }&=\\partial _\\alpha (\\epsilon _{ijk}\\epsilon _{\\alpha \\beta \\gamma }F_{j\\beta }v_i).\\end{aligned}$ We summarize the corresponding results, taken out of [3] and [13], in the following lemma.",
"As the weak continuity property is important for the forthcoming analysis, we present the proof here for the reader's convenience.",
"Lemma 2.1 [3], [13] The followings hold true: $(i)$ For $y \\in W^{1,\\infty }(L^2(\\mathbb {T}^3)) \\cap L^\\infty (W^{1,p} (\\mathbb {T}^3))$ with $p \\ge 4$ , $F= \\nabla y$ and $v = \\partial _t y$ , the formulas (REF ) hold in the sense of distributions.",
"$(ii)$ Suppose the family $\\lbrace y^\\varepsilon \\rbrace _{\\varepsilon > 0}$ , where $y^{\\varepsilon } : [0,\\infty ) \\times \\mathbb {T}^3 \\rightarrow \\mathbb {R}^3$ satisfies $y^{\\varepsilon } \\; \\mbox{is unifomly bounded in } \\; W^{1,\\infty }(L^2(\\mathbb {T}^3)) \\cap L^\\infty (W^{1,p} (\\mathbb {T}^3)) \\, ,$ and let $v^\\varepsilon = \\partial _t y^\\varepsilon $ , $F^\\varepsilon = \\nabla y^\\varepsilon $ .",
"Then, along a subsequence, $(F^{\\varepsilon },\\mathrm {cof}F^{\\varepsilon },\\det F^{\\varepsilon })\\rightharpoonup (F, \\mathrm {cof}F,\\det F) ,\\:\\text{weakly in\\:}L^{\\infty }(L^p) \\times L^{\\infty }(L^q)\\times L^{\\infty }(L^{\\rho })$ with $p\\ge 2,\\:q\\ge \\frac{p}{p-1},\\:q\\ge \\frac{4}{3}$ , $\\rho > 1$ .",
"Moreover, if $p\\ge 4$ the identities (REF ) are weakly stable in the regularity class (REF ).",
"We note the formulas, for smooth maps, $\\begin{split}(\\mathrm {cof}F)_{i\\alpha }&=\\frac{1}{2}\\epsilon _{ijk}\\epsilon _{\\alpha \\beta \\gamma }F_{j\\beta }F_{k\\gamma },\\\\\\det F&=\\frac{1}{6}\\epsilon _{ijk}\\epsilon _{\\alpha \\beta \\gamma }F_{i\\alpha }F_{j\\beta }F_{k\\gamma }=\\frac{1}{3}(\\mathrm {cof} F)_{i \\alpha }F_{i \\alpha }\\end{split}$ and $\\partial _t \\det F&=\\partial _\\alpha \\bigl ((\\mathrm {cof} F)_{i\\alpha }v_i\\bigr )\\\\\\partial _t(\\mathrm {cof} F)_{k \\gamma }&=\\partial _\\alpha (\\epsilon _{ijk}\\epsilon _{\\alpha \\beta \\gamma }F_{j\\beta }v_i).$ Step $1.$ For $y \\in W^{1,\\infty }(L^2(\\mathbb {T}^3)) \\cap L^\\infty (W^{1,p} (\\mathbb {T}^3))$ , we extend $y$ to a function defined for all times, by putting $y(t,x)=y(0,x),$ for $t\\le 0.$ The extended $y$ belongs to the same regularity class.",
"Define the convolution (in space and time) $y_{\\epsilon }:=y\\star f_{\\epsilon },$ where $f_{\\epsilon }=\\varrho _{\\epsilon }(t)\\prod _{i=1}^{3}\\varrho _{\\epsilon }(x_i),$ $\\varrho _{\\epsilon }=\\frac{1}{\\epsilon }\\varrho \\left(\\frac{s}{\\epsilon }\\right),$ for $\\varrho \\in C_0^{\\infty }(\\mathbb {R})$ positive, $\\int \\rho (s) ds = 1$ .",
"Then $y_{\\epsilon }\\in C^{\\infty }( \\mathbb {R} \\times \\mathbb {T}^3)$ and such that for all $s<\\infty ,\\:T>0:$ $\\Vert \\partial _ty_{\\epsilon }-\\partial _ty\\Vert _{L^s([-T,T];L^2)}+\\Vert y_{\\epsilon }-y\\Vert _{L^{\\infty }([-T,T];W^{1,p})}\\rightarrow 0.$ Let $F_{\\epsilon }=\\nabla y_{\\epsilon }$ and $v_{\\epsilon }=\\partial _t y_{\\epsilon }.$ Since the cofactor matrix is bilinear in the components of $F,$ and the determinant is trilinear, it follows by repeated use of Hölder inequalities that for some numerical constant $C$ , $\\big \\Vert \\mathrm {cof}F_{\\epsilon } - \\mathrm {cof}F \\big \\Vert _{L^s ( L^{p/2} )}&\\le C \\big \\Vert F_{\\epsilon } - F \\big \\Vert _{L^{2s} ( L^{p} )} \\, \\big \\Vert |F_{\\epsilon }| + |F | \\big \\Vert _{L^{2s} ( L^{p} )}\\\\\\big \\Vert \\det F_{\\epsilon } - \\det F \\big \\Vert _{L^s ( L^{p/3} )}&\\le C \\big \\Vert F_{\\epsilon } - F \\big \\Vert _{L^{3s} ( L^{p} )} \\, \\left( \\big \\Vert |F_{\\epsilon }| + |F | \\big \\Vert _{L^{3s} ( L^{p} )} \\right)^2.$ We thus conclude: $\\mathrm {cof}F_{\\epsilon }\\rightarrow \\mathrm {cof}F \\quad \\text{in\\:\\:} L^s(L^{p/2}) \\, , \\quad \\det F_{\\epsilon }\\rightarrow \\det F \\quad \\text{in\\:\\:} L^s(L^{p/3}) \\, .$ Passing to the limit $\\epsilon \\rightarrow 0$ , for $p\\ge 4$ , in the formulas $\\partial _t(\\mathrm {cof}F_{\\epsilon })_{k\\gamma }& =\\partial _{\\alpha } \\big ( \\epsilon _{ijk}\\epsilon _{\\alpha \\beta \\gamma } (F_{\\epsilon })_{j\\beta }(v_{\\epsilon })_i \\big )\\\\\\partial _t(\\det F_{\\epsilon })&=\\partial _{\\alpha }((\\mathrm {cof}F_{\\epsilon })_{i \\alpha }(v_{\\epsilon })_i)$ we obtain (REF ) in the sense of distributions and complete the proof of (i).",
"Step $2.$ Let $\\lbrace y^\\varepsilon \\rbrace _{\\varepsilon > 0}$ be a family satisfying the uniform bound (REF ) and let $F^\\varepsilon =\\nabla y^{\\varepsilon }$ and $v^\\varepsilon =\\partial _t y^{\\varepsilon }.$ We adapt the proof of [3] suggesting to write the cofactor and the determinant in divergence form: $(\\mathrm {cof}F^\\varepsilon )_{i\\alpha }&=\\frac{1}{2}\\epsilon _{ijk}\\epsilon _{\\alpha \\beta \\gamma }F^\\varepsilon _{j\\beta }F^\\varepsilon _{k\\gamma }=\\frac{1}{2}\\partial _{\\beta }(\\epsilon _{ijk}\\epsilon _{\\alpha \\beta \\gamma }y^{\\varepsilon }_{j}F^\\varepsilon _{k\\gamma }),\\\\\\det F^\\varepsilon &=\\frac{1}{3}(\\mathrm {cof}F^\\varepsilon )_{i \\alpha }F^\\varepsilon _{i \\alpha }=\\frac{1}{3}\\partial _{\\alpha }(y^{\\varepsilon }_{i}(\\mathrm {cof}F^\\varepsilon )_{i \\alpha }).$ With $p\\ge 2,\\:q\\ge \\frac{p}{p-1},$ hypothesis (REF ) implies $y^{\\varepsilon } \\rightharpoonup y$ weakly in $W^{1,2}_{loc}([0,\\infty )\\times \\mathbb {T}^3)$ along subsequences and Rellich's theorem (for dimension $3+1$ ) implies $y^{\\varepsilon } \\rightarrow y$ strongly in $L^z_{loc}([0,\\infty )\\times \\mathbb {T}^3)$ for $z<4.$ Additionally, $F^\\varepsilon \\rightharpoonup F$ weak-$\\ast $ in $L^{\\infty }(L^p),$ for $p\\ge 2>\\frac{4}{3}$ the dual exponent to $4.$ Therefore, we can pass to the limit in the sense of distributions: $\\frac{1}{2}\\partial _{\\beta }(\\epsilon _{ijk}\\epsilon _{\\alpha \\beta \\gamma }y^{\\varepsilon }_{j}F^\\varepsilon _{k\\gamma })\\rightharpoonup \\frac{1}{2}\\partial _{\\beta }(\\epsilon _{ijk}\\epsilon _{\\alpha \\beta \\gamma }y_{j}F_{k\\gamma })=(\\mathrm {cof}F)_{i\\alpha }$ and similarly for the determinant $\\frac{1}{3}\\partial _{\\alpha }(y^{\\varepsilon }_{i}(\\mathrm {cof}F^\\varepsilon )_{i \\alpha })\\rightharpoonup \\frac{1}{3}\\partial _{\\alpha }(y_{i}(\\mathrm {cof}F)_{i \\alpha })=\\det F,$ since $\\mathrm {cof}F^\\varepsilon \\rightharpoonup \\mathrm {cof}F$ weak-$\\ast $ in $L^{\\infty }(L^q)$ , for $q>\\frac{4}{3}$ the dual exponent to $4.$ The distributional limits in (REF ) and (REF ) coincide with the limits in the weak-$\\ast $ topology.",
"Altogether we have $(\\mathrm {cof}F^\\varepsilon )_{i\\alpha }&\\rightharpoonup (\\mathrm {cof}F)_{i\\alpha },\\quad \\text{weak-$\\ast $ in\\:} L^{\\infty }(L^q),\\quad \\text{for\\:} q\\ge \\frac{p}{p-1},\\:q>\\frac{4}{3}\\\\\\det F^\\varepsilon &\\rightharpoonup \\det F,\\quad \\text{weak-$\\ast $ in\\:} L^{\\infty }(L^{\\rho }).$ Next, note that $y^\\varepsilon $ , $F^\\varepsilon $ , $v^\\varepsilon $ satisfy $\\partial _t (\\mathrm {cof}F^\\varepsilon )_{k \\gamma }&= \\partial _t \\partial _{\\alpha }( \\frac{1}{2}\\epsilon _{ijk}\\epsilon _{\\alpha \\beta \\gamma }y^{\\varepsilon }_{i}F^\\varepsilon _{j \\beta }) =\\partial _{\\alpha }( \\epsilon _{ijk}\\epsilon _{\\alpha \\beta \\gamma }F^\\varepsilon _{j \\beta } v^{\\varepsilon }_{i} ) , \\\\\\partial _t \\det F^\\varepsilon &= \\partial _t \\partial _\\alpha \\big ( \\frac{1}{3} y^{\\varepsilon }_{i}(\\mathrm {cof}F^\\varepsilon )_{i \\alpha } \\big )= \\partial _\\alpha \\big ( \\mathrm {cof} F^\\varepsilon _{i \\alpha } v_i^\\varepsilon \\big ) .$ Using the weak continuity properties of $\\mathrm {cof} F$ and $\\det F$ and that for $p \\ge 4$ equations (REF ) hold for functions $y$ of class (REF ), we conclude that equations (REF ) are weakly stable.",
"Remark 2.1 On the definition of the dissipative measure-valued solution: Combining the requirements of Lemma REF , with those of Lemmas REF and REF , we must assume the exponents $p\\ge 4,$ $q \\ge 2,$ $\\rho , \\ell >1$ .",
"Henceforth, we assume the measure $\\gamma _0=0,$ meaning that we consider initial data with no concentrations at time $t=0.$ Next, we highlight why we choose to work with the system in the physical variables $(\\Phi (F),v,\\theta )$ instead of the extended ones $(\\xi ,v,\\theta )$ : This allows to avoid imposing restrictive growth conditions on the constitutive functions with respect to the cofactor and the determinant derivatives.",
"From previous works in isothermal polyconvex elastodynamics (e.g.",
"[14]) or even in [7], it becomes evident that when considering the extended system, one has to impose growth condition on terms $\\frac{\\partial \\hat{\\psi }}{\\partial F}(\\xi ,\\theta ) \\, , \\quad \\frac{\\partial \\hat{\\psi }}{\\partial \\zeta }(\\xi ,\\theta ) \\frac{\\partial (\\mathrm {cof}F ) }{\\partial F} \\, , \\quad \\frac{\\partial \\hat{\\psi }}{\\partial w }(\\xi ,\\theta ) \\frac{\\partial (\\det F ) }{\\partial F}$ where $\\xi = (F, \\zeta , w)$ , in order to achieve representation of the associated weak limits via Young measures.",
"The resulting regularity class of functions is far too restrictive and in particular functions with general power-like behavior do not satisfy such assumptions and their weak-limits cannot be represented.",
"By contrast, if one works with the original variables, the growth hypotheses (REF )–(REF ) placed on $\\psi (F,\\theta )$ and $e(F,\\theta )$ , which are compatible with the constitutive theory, are also sufficient to allow representation of the corresponding weak limits.",
"The reasoning behind studying the integrated form of the averaged energy equation lies in the technical advantage that, one does not need to place any integrability condition on the right hand-side of the energy equation (REF )$_3$ , namely on the term $\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta ) \\, \\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)v_i \\, , $ since it appears as a divergence and its contribution integrates to zero."
],
[
"Young measures and concentration measures",
"We assume the following growth conditions on the constitutive functions $e(F,\\theta ),$ $\\psi (F,\\theta ),$ $\\eta (F,\\theta )$ and $\\Sigma (F,\\theta ):$ $c(|F|^p+\\theta ^{\\ell })-c\\le e(F,\\theta )\\le c(|F|^p+\\theta ^{\\ell })+c\\;,$ $c(|F|^p+\\theta ^{\\ell })-c\\le \\psi (F,\\theta )\\le c(|F|^p+\\theta ^{\\ell })+c\\;,$ $\\lim _{|F|^p+\\theta ^{\\ell }\\rightarrow \\infty }\\frac{|\\partial _{\\theta }\\psi (F,\\theta )|}{|F|^p+\\theta ^{\\ell }}=\\lim _{|F|^p+\\theta ^{\\ell }\\rightarrow \\infty }\\frac{|\\eta (F,\\theta )|}{|F|^p+\\theta ^{\\ell }}=0\\;,$ and $\\lim _{|F|^p+\\theta ^{\\ell }\\rightarrow \\infty }\\frac{|\\partial _{F}\\psi (F,\\theta )|}{|F|^p+\\theta ^{\\ell }}=\\lim _{|F|^p+\\theta ^{\\ell }\\rightarrow \\infty }\\frac{|\\Sigma (F,\\theta )|}{|F|^p+\\theta ^{\\ell }}=0$ -which are consistent with the constitutive theory (REF )- for some constant $c>0$ and $p\\ge 4,\\ell >1.$ As presented in the appendix, we consider measure-valued solutions as limits of approximations that satisfy the uniform bound $\\int _{\\mathbb {T}^d} \\hat{e}(\\Phi (F^\\varepsilon ),\\theta ^\\varepsilon )+\\frac{1}{2}|v^\\varepsilon |^2\\:dx\\overset{(\\ref {eta&e})}{=}\\int _{\\mathbb {T}^d} e(F^\\varepsilon ,\\theta ^\\varepsilon )+\\frac{1}{2}|v^\\varepsilon |^2\\:dx<C,$ coming from the energy conservation equation (REF )$_3,$ given that the radiative heat supply $r$ is a bounded function in $L^1(Q_T)$ .",
"The growth condition (REF ) in combination with (REF ) suggests that the functions $F^\\varepsilon \\in L^p,$ $v^\\varepsilon \\in L^2,$ $\\theta ^\\varepsilon \\in L^\\ell $ are all (uniformly) bounded in the respective spaces.",
"The approximating sequence $U^{\\varepsilon }=(F^\\varepsilon ,v^\\varepsilon ,\\theta ^\\varepsilon ),$ represents weak limits of the form $\\text{wk-}\\lim _{\\varepsilon \\rightarrow 0} f(F^\\varepsilon ,v^\\varepsilon ,\\theta ^\\varepsilon )=\\left\\langle \\nu ,f(F,v,\\theta )\\right\\rangle $ for all continuous functions $f=f(\\lambda _{F},\\lambda _{v},\\lambda _{\\theta })$ such that $\\lim _{|\\lambda _{F}|^p+|\\lambda _{v}|^2+\\lambda _{\\theta }^{\\ell }\\rightarrow \\infty }\\frac{|f(\\lambda _{F},\\lambda _{v},\\lambda _{\\theta })|}{|\\lambda _{F}|^p+|\\lambda _{v}|^2+\\lambda _{\\theta }^{\\ell }}=0\\;,$ where $\\lambda _{F}\\in \\mathbb {M}^{3\\times 3}$ , $\\lambda _{v}\\in \\mathbb {R}^3$ ,$\\lambda _{\\theta }\\in \\mathbb {R}^+.$ The generated Young measure $\\nu _{(x,t)}$ is associated with the motion $y:Q_T\\rightarrow \\mathbb {R}^3$ through $F$ and $v,$ by imposing that a.e.",
"$F=\\left\\langle \\nu ,\\lambda _{F}\\right\\rangle ,\\quad v=\\left\\langle \\nu ,\\lambda _{v}\\right\\rangle , \\quad \\theta =\\left\\langle \\nu ,\\lambda _{\\theta }\\right\\rangle $ and its action is well-defined for all functions $f$ that grow slower than the energy.",
"To take into account the formation of concentration effects, we introduce the concentration measure $\\gamma ,$ depending on the total energy.",
"This is a well-defined nonnegative Radon measure for a subsequence of $\\displaystyle e(F^\\varepsilon ,\\theta ^\\varepsilon )+\\frac{1}{2}|v^\\varepsilon |^2.$ To prove this claim, let us define the sets $&\\mathcal {F}_0=\\left\\lbrace h\\in C^b(\\mathbb {R}^d):\\;h^{\\infty }(z)=\\lim _{s\\rightarrow \\infty }h(sz)\\;\\text{exists and is continuous on}\\; S^{d-1} \\right\\rbrace \\\\&\\mathcal {F}_1=\\left\\lbrace g\\in C(\\mathbb {R}^d):\\;g(z)=h(z)(1+|z|),\\; h\\in \\mathcal {F}_0 \\right\\rbrace .$ Let $X$ be a locally compact Hausdorff space, where we define the set of all Radon measures $\\mathcal {M}(X)$ and all positive Radon measures $\\mathcal {M}^+(X),$ while $\\mathrm {Prob}(X)$ denotes all probability measures on $X.$ Let $\\Omega $ be any open subset of $\\mathbb {R}^d$ and fix a Radon measure $\\lambda $ on $\\Omega $ .",
"We denote by $\\mathcal {P}(\\lambda ;X)=L^{\\infty }_w(d\\lambda ;\\mathrm {Prob}(X))$ the parametrized families of probability measures $(\\nu _z)_{z\\in \\Omega }$ acting on $X$ which are weakly measurable with respect to $z\\in \\Omega .$ When $\\lambda $ is the Lebesgue measure, we use the notation $\\mathcal {P}(\\lambda ;X)=\\mathcal {P}(\\Omega ;X).$ The following theorem as it appears in [1], [17] uses the theory of generalized Young measures to describe weak limits of the form $\\lim _{n\\rightarrow \\infty }\\int _{\\Omega }\\phi (x)g(u_n(x))\\:dx,$ for $\\phi \\in C^0(\\Omega ),$ any bounded sequence $u_n$ in $L^1,$ and test functions $g$ such that $g(z)=\\bar{g}(z)(1+|z|), \\qquad \\bar{g}\\in C^b(\\mathbb {R}^d).$ $\\begin{split}&\\text{For $g\\in \\mathcal {F}_1,$ the $L^1$-recession function}\\;\\;g^{\\infty }(z)=\\lim _{s\\rightarrow \\infty }\\frac{g(sz)}{s}, \\;\\; z\\in S^{d-1},\\;\\; \\text{coincides}\\\\&\\text{with $h^{\\infty }(z),$ where $h\\in \\mathcal {F}_0$ and $g(z)=h(z)(1+|z|).$}\\end{split}$ Theorem 3.1 Let $\\lbrace u_n\\rbrace $ be bounded in $L^1(\\Omega ;\\mathbb {R}^d).$ There exists a subsequence $\\lbrace u_{n_{k}}\\rbrace $ , a nonnegative Radon measure $\\mu \\in \\mathcal {M}^+(\\Omega )$ and parametrized families of probability measures $\\nu \\in \\mathcal {P}(\\Omega ;\\mathbb {R}^d), \\qquad \\nu ^{\\infty }\\in \\mathcal {P}(\\lambda ;S^{d-1})$ such that $g(u_{n_{k}}) \\rightharpoonup \\left\\langle \\nu ,g\\right\\rangle +\\left\\langle \\nu ^{\\infty },g^{\\infty }\\right\\rangle \\mu \\quad \\text{weak-$\\ast $ in}\\:\\:\\mathcal {M}^+(\\Omega ),$ for any $g\\in \\mathcal {F}_1.$ Given that the only available bound for the approximate sequence $U^{\\varepsilon }$ is of the form $\\int _{\\mathbb {T}^d} f(F^\\varepsilon ,v^\\varepsilon ,\\theta ^\\varepsilon )\\:dx<C,$ we want to represent the weak limits wk-$\\ast $ $\\displaystyle \\lim _{\\varepsilon \\rightarrow 0}f(F^\\varepsilon ,v^\\varepsilon ,\\theta ^\\varepsilon ),$ for a continuous test function $f$ satisfying the growth condition $|f(F^\\varepsilon ,v^\\varepsilon ,\\theta ^\\varepsilon )|\\le C (1+|F|^p+|v|^2+\\theta ^{\\ell }).$ In order to apply Theorem REF , we perform the change of variables $(A,b,c)=(|F|^{p-1}F,|v|v,\\theta ^{\\ell })$ and define $f(F,v,\\theta ):=g(|F|^{p-1}F,|v|v,\\theta ^{\\ell }),$ imposing that the function $g$ grows like $|g(A,b,c)|\\le C (1+|A|+|b|+|c|).$ Then Theorem REF applies to represent the wk-$\\ast $ limits of $g$ : $g^{\\infty }(A,b,c)=\\lim _{s\\rightarrow \\infty }\\frac{g(sA,sb,sc)}{1+s(A,b,c)},$ for all $(A,b,c)\\in S^{d^2+d}\\cap \\lbrace c>0\\rbrace .$ Consequently, there exist a nonnegative Borel measure $M\\in \\mathcal {M}^+(\\bar{Q}_T)$ and probability measures $N\\in \\mathcal {P}(\\bar{Q}_T;\\mathbb {R}^{d^2+d+1}),$ $N^{\\infty }\\in \\mathcal {P}(\\bar{Q}_T;S^{d^2+d})$ which up to subsequence $g(A_n,b_n,c_n)\\rightharpoonup \\left\\langle N,g(\\lambda _{A},\\lambda _{b},\\lambda _{c})\\right\\rangle +\\left\\langle N^{\\infty },g^{\\infty }(\\lambda _{A},\\lambda _{b},\\lambda _{c})\\right\\rangle M.$ Then, property (REF ) implies that $f(F_n,v_n,\\theta _n)\\rightharpoonup \\left\\langle \\nu ,f(\\lambda _{F},\\lambda _{v},\\lambda _{\\theta })\\right\\rangle +\\left\\langle \\nu ^{\\infty },f^{\\infty }(\\lambda _{F},\\lambda _{v},\\lambda _{\\theta })\\right\\rangle M$ where $\\left\\langle \\nu ,f(\\lambda _{F},\\lambda _{v},\\lambda _{\\theta })\\right\\rangle =\\left\\langle N,g(|\\lambda _{F}|^{p-1}\\lambda _{F},|\\lambda _{v}|\\lambda _{v},\\lambda _{\\theta }^{\\ell })\\right\\rangle $ and $\\left\\langle \\nu ^{\\infty },f(\\lambda _{F},\\lambda _{v},\\lambda _{\\theta })\\right\\rangle =\\left\\langle N,g^{\\infty }(|\\lambda _{F}|^{p-1}\\lambda _{F},|\\lambda _{v}|\\lambda _{v},\\lambda _{\\theta }^{\\ell })\\right\\rangle .$ Therefore, given the bound (REF ) and assuming that the recession function $\\left(e(F,\\theta )+\\frac{1}{2}|v|^2\\right)^{\\infty }=\\lim _{s\\rightarrow \\infty }\\frac{e\\left(s^{1/p}F,s^{1/\\ell }\\theta \\right)+\\displaystyle \\frac{s}{2}|v|^2}{1+s(|F|^p,|v|^2,\\theta ^{\\ell })},$ exists and is continuous for all $(|F|^{p-1}F,|v|v,\\theta ^{\\ell })\\in S^{d^2+d}\\cap \\lbrace c>0\\rbrace ,$ we have that (along a subsequence) $\\text{wk-$\\ast $-}\\lim _{\\varepsilon \\rightarrow 0}\\left(\\hat{e}(\\Phi (F^\\varepsilon ),\\theta ^\\varepsilon )+\\frac{1}{2}|v^\\varepsilon |^2\\right)&=\\left\\langle \\nu ,e(\\lambda _{F},\\lambda _{\\theta })+\\frac{1}{2}|\\lambda _{v}|^2\\right\\rangle \\\\&\\quad +\\left\\langle \\nu ^{\\infty },\\left(e(\\lambda _{F},\\lambda _{\\theta })+\\frac{1}{2}|\\lambda _{v}|^2\\right)^{\\infty }\\right\\rangle M\\;,$ recalling (REF $)_2$ .",
"Then (REF ) implies that $\\left(e(\\lambda _{F},\\lambda _{\\theta })+\\frac{1}{2}|\\lambda _{v}|^2\\right)^{\\infty }>0,$ therefore $\\begin{split}\\gamma :=\\left\\langle \\nu ^{\\infty },\\left(\\frac{1}{2}|\\lambda _{v}|^2+e(\\lambda _{F},\\lambda _{\\theta })\\right)^{\\infty }\\right\\rangle M=\\left\\langle \\nu ^{\\infty },\\left(\\frac{1}{2}|\\lambda _{v}|^2+\\hat{e}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right)^{\\infty }\\right\\rangle M\\in \\mathcal {M}^+(\\bar{Q}_T).\\end{split}$"
],
[
"The averaged relative entropy inequality",
"The augmented system (REF ) belongs to a general class of hyperbolic systems of the form $\\partial _t A(U)+\\partial _{\\alpha }f_{\\alpha }(U)=0$ where $U=U(x,t)\\in \\mathbb {R}^n,$ is the unknown with $x\\in \\mathbb {R}^d,$ $t\\in \\mathbb {R}^+$ and $A,f_{\\alpha }:\\mathbb {R}^n\\rightarrow \\mathbb {R}^n$ are given smooth functions of $U.$ It is symmetrizable in the sense of Friedrichs and Lax [19], under appropriate hypotheses: The map $A(U)$ is globally invertible and there exists an entropy-entropy flux pair $(H,q)$ , i.e.",
"there exists a smooth multiplier $G(U):\\mathbb {R}^{n}\\rightarrow \\mathbb {R}^{n}$ such that $\\nabla H&=G\\cdot \\nabla A\\\\\\nabla q_\\alpha &=G\\cdot \\nabla f_\\alpha ,\\quad \\alpha =1,\\dots ,d.$ In our case $U=\\left( \\begin{array}{c} \\Phi (F)\\\\ v\\\\ \\theta \\end{array} \\right),\\;\\;A(U)=\\left( \\begin{array}{c} \\Phi (F)\\\\ v\\\\ \\frac{1}{2} |v|^2+\\hat{e}(\\Phi (F), \\theta ) \\end{array} \\right),\\;\\;f_\\alpha (U)= \\left( \\begin{array}{c} \\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)v_i\\\\ \\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta )\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)\\\\\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta )\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)v_i \\end{array} \\right)\\; ,$ while the (mathematical) entropy is given by $H(U)=-\\hat{\\eta }(\\Phi (F),\\theta ),$ the entropy flux $q_{\\alpha } = 0$ and the associated multiplier is $G(U)=\\frac{1}{\\theta }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta ),v,-1\\right)^T,\\quad B=1,\\dots ,19$ see [8], [7].",
"Consider a strong solution $(\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })^T \\in W^{1,\\infty }(Q_T)$ to (REF ) that satisfies the entropy identity (REF ) and a dissipative measure valued solution to (REF ), (REF ) according to Definition REF .",
"We write the difference of the weak form of equations (REF ), (REF ) and (REF ), (REF ) to obtain the following three integral identities $\\begin{split}\\int (\\Phi ^B(F)-\\Phi ^B(\\bar{F}))(x,0)&\\phi _1(x,0)\\:dx +\\int _0^T \\int (\\Phi ^B(F)-\\Phi ^B(\\bar{F}))\\partial _t\\phi _1(x,t)\\:dx\\:dt\\\\&=\\int _0^T \\int \\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)v_i-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\bar{v}_i\\right)\\partial _{\\alpha }\\phi _1(x,t)\\:dx\\:dt,\\end{split}$ $\\begin{split}\\int &(\\left\\langle \\nu ,\\lambda _{v_i}\\right\\rangle -\\bar{v}_i)(x,0)\\phi _2(x,0)\\:dx+\\int _0^T\\int (\\left\\langle \\nu ,\\lambda _{v_i}\\right\\rangle -\\bar{v}_i)\\partial _t\\phi _2(x,t)\\:dx\\:dt\\\\&=\\int _0^T\\int \\left(\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _F)\\right\\rangle -\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\partial _{\\alpha }\\phi _2(x,t)\\:dx\\:dt\\;,\\end{split}$ and $\\begin{split}\\int &\\left(\\left\\langle \\nu ,\\frac{1}{2}|\\lambda _{v}|^2+\\hat{e}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle -\\frac{1}{2}|\\bar{v}|^2-\\hat{e}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\!",
"(x,0)\\:\\phi _3(x,0)\\:dx \\\\&\\;+\\int _0^T \\int \\left\\lbrace \\left(\\left\\langle \\nu ,\\frac{1}{2}|\\lambda _{v}|^2+\\hat{e}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle -\\frac{1}{2}|\\bar{v}|^2-\\hat{e}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)+\\gamma \\right\\rbrace \\partial _t\\phi _3(x,t)\\:dx\\:dt\\\\&=-\\int _0^T \\int (\\left\\langle \\nu ,r\\right\\rangle -\\bar{r})\\phi _3(x,t)\\:dx\\:dt,\\end{split}$ for any $\\phi _i\\in C^1_c(Q_T)$ , $i=1, 2$ and $\\phi _3\\in C^1_c[0,T).$ Similarly, testing the difference of (REF ) and (REF )$_3$ against $\\phi _4\\in C^1_c(Q_T),$ with $\\phi _4\\ge 0$ , we have $\\begin{split}-\\int &(\\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle -\\hat{\\eta }(\\Phi (\\bar{F}),\\bar{\\theta }))(x,0)\\phi _4(x,0)\\:dx\\\\&\\quad -\\int _0^T\\int (\\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle -\\hat{\\eta }(\\Phi (\\bar{F}),\\bar{\\theta }))\\partial _t\\phi _4(x,t)\\:dx\\:dt\\\\ &\\ge \\int _0^T\\int \\left(\\left\\langle \\nu ,\\frac{r}{\\lambda _{\\theta }}\\right\\rangle -\\frac{\\bar{r}}{\\bar{\\theta }}\\right)\\phi _4(x,t)\\:dx\\:dt.\\end{split}$ We then choose $(\\phi _1,\\phi _2,\\phi _3)=-\\bar{\\theta }\\,G(\\bar{U})\\varphi (t)=( -\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta }) ,-\\bar{v},1)^T\\varphi (t)$ , for some $\\varphi \\in C_c^1[0,T]$ , thus (REF ), (REF ) and (REF ) become $\\begin{split}\\int &\\left(-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })(\\Phi ^B(F)-\\Phi ^B(\\bar{F}))\\right)(x,0)\\varphi (0)\\:dx \\\\&\\qquad +\\int _0^T\\int \\left(-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })(\\Phi ^B(F)-\\Phi ^B(\\bar{F}))\\right)\\varphi ^{\\prime }(t)\\:dx\\:dt\\\\&=\\int _0^T\\int \\left[\\partial _t\\Big (\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\Big )(\\Phi ^B(F)-\\Phi ^B(\\bar{F}))\\right.\\\\&\\qquad \\quad \\left.-\\partial _{\\alpha }\\Big (\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\Big )\\Big (\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)v_i-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\bar{v}_i\\Big )\\right]\\varphi (t) dxdt \\, ,\\end{split}$ $\\begin{split}\\int &(-\\bar{v_i}(\\left\\langle \\nu ,\\lambda _{v_i}\\right\\rangle -\\bar{v_i}))(x,0)\\varphi (0)\\:dx+\\int _0^T\\int -\\bar{v_i}(\\left\\langle \\nu ,\\lambda _{v_i}\\right\\rangle -\\bar{v_i})\\varphi ^{\\prime }(t)\\:dx\\:dt\\\\&=-\\int _0^T\\int \\left[-\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)(\\left\\langle \\nu ,\\lambda _{v_i}\\right\\rangle -\\bar{v}_i)\\right.\\\\&\\:\\:\\left.+\\partial _{\\alpha }\\bar{v_i}\\left(\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _F)\\right\\rangle -\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\right]\\varphi (t)\\:dx\\:dt\\;,\\end{split}$ and $\\begin{split}\\int &\\left(\\left\\langle \\nu ,\\frac{1}{2}|\\lambda _{v}|^2+\\hat{e}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle -\\frac{1}{2}|\\bar{v}|^2-\\hat{e}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\!",
"(x,0)\\;\\varphi (0)\\:dx \\\\&\\quad +\\int _0^T \\int \\left\\lbrace \\left(\\left\\langle \\nu ,\\frac{1}{2}|\\lambda _{v}|^2+\\hat{e}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle -\\frac{1}{2}|\\bar{v}|^2-\\hat{e}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)+\\gamma \\right\\rbrace \\varphi ^{\\prime }(t)\\:dx\\:dt\\\\&=-\\int _0^T \\int (\\left\\langle \\nu ,r\\right\\rangle -\\bar{r})\\varphi (t)\\:dx\\:dt.\\end{split}$ For inequality (REF ), we choose accordingly $\\phi _4:=\\bar{\\theta }\\varphi (t)\\ge 0$ , $\\varphi \\ge 0$ so that $\\begin{split}-\\int \\bar{\\theta }&(\\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle -\\hat{\\eta }(\\Phi (\\bar{F}),\\bar{\\theta }))(x,0)\\varphi (0)\\:dx\\\\&\\quad -\\int _0^T\\int \\bar{\\theta }(\\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle -\\hat{\\eta }(\\Phi (\\bar{F}),\\bar{\\theta }))\\varphi ^{\\prime }(t)\\:dx\\:dt\\\\&\\ge \\int _0^T\\int \\left[\\partial _t\\bar{\\theta }(\\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle -\\hat{\\eta }(\\Phi (\\bar{F}),\\bar{\\theta }))+\\bar{\\theta }\\left(\\left\\langle \\nu ,\\frac{r}{\\lambda _{\\theta }}\\right\\rangle -\\frac{\\bar{r}}{\\bar{\\theta }}\\right)\\right]\\varphi (t)\\:dx\\:dt.\\end{split}$ Adding together (REF ), (REF ), (REF ) and (REF ), we obtain the integral inequality $\\int &\\varphi (0)\\bigg [-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })(\\Phi ^B(F)-\\Phi ^B(\\bar{F}))(x,0)-\\langle \\nu ,\\bar{v}_i(\\lambda _{v_i}-\\bar{v}_i)\\rangle (x,0) \\nonumber \\\\&\\qquad +\\left\\langle \\nu ,\\frac{1}{2}|\\lambda _{v}|^2+\\hat{e}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\frac{1}{2}|\\bar{v}|^2-\\hat{e}(\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle (x,0) \\nonumber \\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad -\\bar{\\theta }\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\hat{\\eta }(\\Phi (\\bar{F}),\\bar{\\theta })\\rangle (x,0)\\bigg ]\\:dx \\nonumber \\\\&+\\int _{0}^{T}\\!\\!\\!\\!\\int \\varphi ^{\\prime }(t)\\bigg [-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })(\\Phi ^B(F)-\\Phi ^B(\\bar{F}))-\\langle \\nu ,\\bar{v}_i(\\lambda _{v_i}-\\bar{v}_i)\\rangle \\nonumber \\\\&\\qquad \\qquad +\\left\\langle \\nu ,\\frac{1}{2}|\\lambda _{v}|^2+\\hat{e}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\frac{1}{2}|\\bar{v}|^2-\\hat{e}(\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle \\nonumber \\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad -\\bar{\\theta }\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\hat{\\eta }(\\Phi (\\bar{F}),\\bar{\\theta })\\rangle +\\gamma \\bigg ]\\:dx\\:dt \\nonumber \\\\&\\!\\!\\!\\!\\!\\!\\ge -\\int _{0}^{T}\\!\\!\\!\\!\\int \\varphi (t)\\bigg [-\\partial _t\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\!\\!",
"(\\Phi ^B(F)-\\Phi ^B(\\bar{F})) \\nonumber \\\\&\\qquad \\qquad \\qquad +\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)v_i-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\bar{v}_i\\right) \\nonumber \\\\&\\qquad \\qquad \\qquad +\\partial _{\\alpha }\\bar{v}_i\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _F)-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right\\rangle \\nonumber \\\\&\\qquad \\qquad \\qquad -\\partial _t\\bar{\\theta }\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\hat{\\eta }(\\Phi (\\bar{F}),\\bar{\\theta })\\rangle \\nonumber \\\\&\\qquad \\qquad \\qquad -\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\langle \\nu ,(\\lambda _{v_i}-\\bar{v}_i)\\rangle -\\bar{\\theta }\\left\\langle \\nu ,\\frac{r}{\\lambda _{\\theta }}-\\frac{\\bar{r}}{\\bar{\\theta }}\\right\\rangle +\\langle \\nu ,r-\\bar{r}\\rangle \\bigg ]\\:dx\\:dt\\nonumber \\\\&\\!\\!\\!\\!\\!\\!=:-\\int _{0}^{T}\\int \\varphi (t)K(x,t)\\:dx\\:dt\\;.$ Using the entropy identity (REF ) and the null-Lagrangian property (REF ), the quantity $K(x,t)$ in the integrand on the right hand-side of (REF ) becomes $K&=-\\partial _t\\bar{\\theta }\\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle +\\partial _t\\Phi ^B(\\bar{F})\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle \\nonumber \\\\&-\\partial _t\\bar{\\theta }\\:\\frac{\\partial \\hat{\\eta }}{\\partial \\theta }(\\Phi (\\bar{F}),\\bar{\\theta })\\langle \\nu ,\\lambda _{\\theta }-\\bar{\\theta }\\rangle \\nonumber \\\\&-\\partial _t\\Phi ^B(\\bar{F})\\left(\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle -\\frac{\\partial ^2 \\hat{\\psi }}{\\partial \\xi ^B\\partial \\theta }(\\Phi (\\bar{F}),\\bar{\\theta })\\langle \\nu ,\\lambda _{\\theta }-\\bar{\\theta }\\rangle \\!\\right)\\nonumber \\\\&+\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\!\\!\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)v_i-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\bar{v}_i\\right)-\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\langle \\nu ,(\\lambda _{v_i}-\\bar{v}_i)\\rangle \\nonumber \\\\&+\\partial _{\\alpha }\\bar{v}_i\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _F)-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right\\rangle -\\bar{\\theta }\\left\\langle \\nu ,\\frac{r}{\\lambda _{\\theta }}-\\frac{\\bar{r}}{\\bar{\\theta }}\\right\\rangle +\\langle \\nu ,r-\\bar{r}\\rangle \\nonumber \\\\&\\!\\!\\!\\!\\!\\!=-\\partial _t\\bar{\\theta }\\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle +\\partial _t\\Phi ^B(\\bar{F})\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle -\\frac{\\bar{r}}{\\bar{\\theta }}\\left\\langle \\nu ,\\lambda _{\\theta }-\\bar{\\theta }\\right\\rangle \\nonumber \\\\&-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\partial _{\\alpha }\\bar{v}_i\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle \\nonumber \\\\&+\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)v_i-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\bar{v}_i\\right)\\nonumber \\\\&-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\partial _{\\alpha }\\!\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\langle \\nu ,(\\lambda _{v_i}-\\bar{v}_i)\\rangle \\nonumber \\\\&+\\partial _{\\alpha }\\bar{v}_i\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _F)-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right\\rangle \\nonumber \\\\&-\\bar{\\theta }\\left\\langle \\nu ,\\frac{r}{\\lambda _{\\theta }}-\\frac{\\bar{r}}{\\bar{\\theta }}\\right\\rangle +\\langle \\nu ,r-\\bar{r}\\rangle $ employing (REF )$_1$ and (REF ).",
"Here, we use the quantites $\\begin{split}\\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle &:=\\Bigg \\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\hat{\\eta }(\\Phi (\\bar{F}),\\bar{\\theta })\\\\&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\\frac{\\partial \\hat{\\eta }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })(\\Phi ^B(\\lambda _{F})-\\Phi ^B(\\bar{F}))-\\frac{\\partial \\hat{\\eta }}{\\partial \\theta }(\\Phi (\\bar{F}),\\bar{\\theta })(\\lambda _{\\theta }-\\bar{\\theta })\\Bigg \\rangle \\;,\\end{split}$ and $\\begin{split}\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle &:=\\Bigg \\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\\\&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!",
"!\\!\\!\\!\\!\\!\\!\\!-\\frac{\\partial ^2 \\hat{\\psi }}{\\partial \\xi ^B\\xi ^A}(\\Phi (\\bar{F}),\\bar{\\theta })(\\Phi ^B(\\lambda _{F})-\\Phi ^B(\\bar{F}))-\\frac{\\partial ^2 \\hat{\\psi }}{\\partial \\xi ^B\\partial \\theta }(\\Phi (\\bar{F}),\\bar{\\theta })(\\lambda _{\\theta }-\\bar{\\theta })\\Bigg \\rangle .\\end{split}$ Next, we rewrite the terms $&-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\partial _{\\alpha }\\bar{v}_i\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle \\nonumber \\\\&+\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)v_i-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\bar{v}_i\\right)\\nonumber \\\\&-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\partial _{\\alpha }\\!\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\langle \\nu ,(\\lambda _{v_i}-\\bar{v}_i)\\rangle \\nonumber \\\\&+\\partial _{\\alpha }\\bar{v}_i\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _F)-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right\\rangle \\nonumber \\\\&\\;=\\partial _{\\alpha }\\bar{v}_i\\left\\langle \\nu ,\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\right\\rangle \\nonumber \\\\&\\quad \\quad \\;\\;\\;+\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\!\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\!\\langle \\nu ,(\\lambda _{v_i}-\\bar{v}_i)\\rangle \\nonumber \\\\&\\quad \\quad \\;\\;\\;+\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\bar{v}_i\\right)\\;,$ since there holds $\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)=\\left\\langle \\nu ,\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _F)\\right\\rangle $ and because of the null-Lagrangian property (REF ).",
"Also, we observe that $-\\frac{\\bar{r}}{\\bar{\\theta }}\\left\\langle \\nu ,\\lambda _{\\theta }-\\bar{\\theta }\\right\\rangle -\\bar{\\theta }\\left\\langle \\nu ,\\frac{r}{\\lambda _{\\theta }}-\\frac{\\bar{r}}{\\bar{\\theta }}\\right\\rangle +\\langle \\nu ,r-\\bar{r}\\rangle =\\left\\langle \\nu ,\\left(\\frac{r}{\\lambda _{\\theta }}-\\frac{\\bar{r}}{\\bar{\\theta }}\\right)(\\lambda _{\\theta }-\\bar{\\theta })\\right\\rangle \\;.$ Finally, if we define the averaged quantity $\\begin{split}I(\\lambda _{U}&|\\bar{U})=I(\\Phi (\\lambda _{F}),\\lambda _{v},\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\\\&:=\\hat{\\psi }(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta })+(\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\hat{\\eta }(\\Phi (\\bar{F}),\\bar{\\theta })(\\lambda _{\\theta }-\\bar{\\theta })+\\frac{1}{2}|\\lambda _{v}-\\bar{v}|^2,\\end{split}$ for $\\hat{\\psi }(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta }):=&\\hat{\\psi }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\hat{\\psi }(\\Phi (\\bar{F}),\\bar{\\theta })-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })(\\Phi ^B(\\lambda _{F})-\\Phi ^B(\\bar{F}))\\\\&-\\frac{\\partial \\hat{\\psi }}{\\partial \\theta }(\\Phi (\\bar{F}),\\bar{\\theta })(\\lambda _{\\theta }-\\bar{\\theta }),$ and then combine (REF ),(REF ),(REF ) and (REF ), we arrive at the relative entropy inequality $\\begin{split}&\\int \\varphi (0)[\\left\\langle \\nu ,I(\\lambda _{U_0}|\\bar{U_0})\\right\\rangle \\:dx]+\\int _{0}^{T}\\int \\varphi ^{\\prime }(t)\\left[\\left\\langle \\nu ,I(\\lambda _{U}|\\bar{U})\\right\\rangle \\:dx\\:dt+\\gamma (dx\\,dt)\\right]\\\\&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\ge -\\int _{0}^{T}\\int \\varphi (t)\\left[-\\partial _t\\bar{\\theta }\\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle +\\partial _t\\Phi ^B(\\bar{F})\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle \\right.\\\\&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\left.+\\partial _{\\alpha }\\bar{v}_i\\left\\langle \\nu ,\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\right\\rangle \\right.\\\\&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\left.+\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\!\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\!\\langle \\nu ,(\\lambda _{v_i}-\\bar{v}_i)\\rangle +\\left\\langle \\nu ,\\left(\\frac{r}{\\lambda _{\\theta }}-\\frac{\\bar{r}}{\\bar{\\theta }}\\right)(\\lambda _{\\theta }-\\bar{\\theta })\\right\\rangle \\right]\\:dx\\,dt.\\end{split}$ We note that the last term in (REF ) vanishes when we substitute into the integral relation (REF )."
],
[
"Uniqueness of smooth solutions in the class of dissipative measure-valued solutions",
"In this section, we state and prove the main theorem on dissipative measure-valued versus strong uniqueness.",
"Before we proceed with the proof, we show some useful estimates on the terms appearing in (REF ), which yield the relative entropy as a “metric” measuring the distance between the two solutions.",
"These bounds are obtained by using the convexity of the free energy function in the compact domain and the growth conditions (REF )-(REF ) placed on the constitutive functions in the original variables, in the unbounded domain.",
"Lemma 5.1 Assume that $(\\bar{F},\\bar{v},\\bar{\\theta })$ are defined in the compact set $\\Gamma _{M,\\delta }:=\\left\\lbrace (\\bar{F},\\bar{v},\\bar{\\theta }): |\\bar{F}|\\le M, |\\bar{v}|\\le M, \\;\\; 0<\\delta \\le \\bar{\\theta }\\le M\\right\\rbrace $ for some positive constants $M$ and $\\delta $ and let $\\hat{\\psi }=\\hat{e}-\\theta \\hat{\\eta }\\in C^2(\\mathbb {R}^{19}\\times [0,\\infty )).$ Assuming the growth conditions (REF )-(REF ) and $p>3,\\:\\ell >1,$ then there exist $R=R(M,\\delta )$ and constants $K_1=K_1(M,\\delta ,c)>0,\\:K_2=K_2(M,\\delta ,c)>0$ such that $\\begin{split}I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\ge \\begin{dcases}\\frac{K_1}{2}(|F|^p+\\theta ^{\\ell }+|v|^2), \\qquad \\qquad \\qquad \\quad \\quad \\quad |F|^p\\!+\\theta ^{\\ell }\\!+|v|^2>R\\\\K_2(|\\Phi (F)-\\Phi (\\bar{F})|^2\\!+\\!|\\theta -\\bar{\\theta }|^2\\!+\\!|v-\\bar{v}|^2), \\quad |F|^p\\!+\\theta ^{\\ell }\\!+|v|^2\\le R\\end{dcases}\\end{split}$ for all $(\\bar{F},\\bar{v},\\bar{\\theta })\\in \\Gamma _{M,\\delta }$ .",
"Let $(\\bar{F},\\bar{v},\\bar{\\theta })\\in \\Gamma _{M,\\delta }$ and choose $r=r(M):=M^p+M^{\\ell }+M^2$ for which $\\Gamma _{M,\\delta }\\subset B_r=\\lbrace (F,v,\\theta ):\\; |F|^p+\\theta ^{\\ell }+|v|^2\\le r\\rbrace $ .",
"Taking under consideration (REF ), we can write $I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })$ in the form $I&(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\\\&=\\hat{e}(\\Phi (F),\\theta )-\\hat{\\psi }(\\Phi (\\bar{F}),\\bar{\\theta })-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi }(\\Phi (\\bar{F}),\\bar{\\theta })\\cdot (\\Phi (F)-\\Phi (\\bar{F}))-\\bar{\\theta }\\hat{\\eta }(\\Phi (F),\\theta )+\\frac{1}{2}|v-\\bar{v}|^2\\\\&=e(F,\\theta )-\\psi (\\bar{F},\\bar{\\theta })-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi }(\\Phi (\\bar{F}),\\bar{\\theta })\\cdot (\\Phi (F)-\\Phi (\\bar{F}))-\\bar{\\theta }\\eta (F,\\theta )+\\frac{1}{2}|v-\\bar{v}|^2\\:.$ Using (REF ) and (REF ) we have $I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\ge c(|F|^p+\\theta ^{\\ell }) -c_1 -c_2|F|^3 -c_3|\\eta (F,\\theta )| +\\frac{1}{2}|v|^2 -c_4|v|.$ Selecting now $R$ sufficiently large such that $R>r(M)+1$ and for $|F|^p+\\theta ^{\\ell }+|v|^2>R$ we have $I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })&\\ge \\min \\left\\lbrace c,\\frac{1}{2}\\right\\rbrace (|F|^p+\\theta ^{\\ell }+|v|^2)-c_3|\\eta (F,\\theta )|-c_5\\\\&\\ge \\frac{1}{2}\\min \\left\\lbrace c,\\frac{1}{2}\\right\\rbrace (|F|^p+\\theta ^{\\ell }+|v|^2)$ and (REF ) is established within the region $|F|^p+\\theta ^{\\ell }+|v|^2>R$ .",
"In the complementary region $|F|^p+\\theta ^{\\ell }+|v|^2\\le R,$ observe that $(F,\\mathrm {cof}F,\\det F,v,\\theta )$ takes values in the set $D:=\\left\\lbrace (\\Phi (F),v,\\theta ): |F|\\le R^{1/p}, |\\mathrm {cof}F|\\le CR^{2/p}, |\\det F|\\le CR^{3/p}, |v|\\le R^{1/2}, 0<\\theta \\le R^{1/\\ell }\\right\\rbrace ,$ for some constant $C.$ We use the convexity of the entropy $\\tilde{H}(V)$ in the symmetric variables $V:=A(U)=\\left(\\xi ,v,\\frac{|v|^2}{2}+\\hat{e}(\\xi ,\\theta )\\right)^T:$ $\\frac{1}{\\bar{\\theta }}I(\\xi ,v,\\theta |\\bar{\\xi },\\bar{v},\\bar{\\theta })&=\\tilde{H}(A(U)|A(\\bar{U}))\\\\&=\\tilde{H}(A(U))-\\tilde{H}(A(\\bar{U}))-\\tilde{H}_V(A(\\bar{U}))(A(U)-A(\\bar{U}))\\\\&\\ge \\min _{V^*\\in D^*}\\lbrace \\tilde{H}_{VV} (V^*)\\rbrace |A(U)-A(\\bar{U})|^2,$ since $\\tilde{H}(V)$ is convex in $V$ and $D^*$ is the compact domain determined by the map $V=A(U)$ and the set $D$ defined above.",
"Moreover, using the invertibility at the map $U\\mapsto A(U)$ $|U-\\bar{U}|&=\\left|\\int _0^1\\frac{d}{d\\tau }[A^{-1}(\\tau A(U)+(1-\\tau )A(\\bar{U}))]\\:d\\tau \\right|\\\\&\\le \\left|\\int _0^1\\nabla _V(A^{-1})(\\tau A(U)+(1-\\tau )A(\\bar{U}))\\:d\\tau \\right|\\:|A(U)-A(\\bar{U})|\\\\&\\le C^{\\prime } |A(U)-A(\\bar{U})|,$ where $C^{\\prime }=\\sup _{\\begin{array}{c}U\\in B_R \\\\ \\bar{U}\\in \\Gamma _{M,\\delta }\\end{array}}\\left|\\int _0^1\\nabla _V(A^{-1})(\\tau A(U)+(1-\\tau )A(\\bar{U}))\\:d\\tau \\right|\\:|A(U)-A(\\bar{U})|<\\infty .$ Therefore $I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\ge \\frac{K_2}{C^{\\prime }}|U-\\bar{U}|^2,$ for $K_2:=\\displaystyle \\delta \\min \\lbrace \\tilde{H}_{VV}( V^*)\\rbrace >0$ and the proof is complete.",
"Lemma 5.2 Under the assumptions of Lemma REF and the additional growth hypothesis $\\left|\\frac{\\partial \\hat{\\psi }}{\\partial \\xi }(\\xi ,\\theta )\\right|\\le c\\:|\\hat{\\psi }(\\xi ,\\theta )|,\\qquad \\forall \\,\\xi ,\\,\\theta $ for some positive constant $c,$ the following bounds hold true: There exist constants $C_1,C_2,C_3,C_4>0$ such that $\\left|\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta )-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\right|\\le C_1 I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\;,$ $\\left|\\frac{\\partial \\hat{\\psi }}{\\partial \\xi }(\\Phi (F),\\theta |\\Phi (\\bar{F}),\\bar{\\theta })\\right|\\le C_2 I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\;,$ $|\\hat{\\eta }(\\Phi (F),\\theta |\\Phi (\\bar{F}),\\bar{\\theta })|\\le C_3 I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\;,$ and $\\left|\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)(v_i-\\bar{v}_i)\\right|\\le C_4 I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })$ for all $(\\bar{F},\\bar{v},\\bar{\\theta })\\in \\Gamma _{M,\\delta }$ .",
"There exist constants $K_1^{\\prime },\\:K_2^{\\prime }$ and $R>0$ sufficiently large such that $\\begin{split}I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\ge \\begin{dcases}\\frac{K_1^{\\prime }}{4}(|F-\\bar{F}|^p+|\\theta -\\bar{\\theta }|^{\\ell }+|v-\\bar{v}|^2),\\quad \\quad \\quad |F|^p\\!+\\theta ^{\\ell }\\!+|v|^2>R\\\\K_2^{\\prime }(|\\Phi (F)-\\Phi (\\bar{F})|^2\\!+\\!|\\theta -\\bar{\\theta }|^2\\!+\\!|v-\\bar{v}|^2), \\quad |F|^p\\!+\\theta ^{\\ell }\\!+|v|^2\\le R\\end{dcases}\\end{split}$ for all $(\\bar{F},\\bar{v},\\bar{\\theta })\\in \\Gamma _{M,\\delta }$ .",
"We divide the proof into 5 steps.",
"Step $1.$ To prove (REF ), we use (REF ) to obtain $&\\left|\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta )-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\right|\\\\&=\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta )\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta )\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\\\&\\qquad +\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\\\&=\\Sigma (F,\\theta )-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta )\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})+\\Sigma (\\bar{F},\\bar{\\theta }).$ For $|F|^p+\\theta ^{\\ell }+|v|^2>R$ and $(\\bar{F},\\bar{v},\\bar{\\theta })\\in \\Gamma _{M,\\delta },$ using (REF ), (REF ), (REF ) and Young's inequality we have $&\\left|\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta )-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\right|\\\\&\\qquad \\qquad \\qquad \\le |\\Sigma (F,\\theta )|+c_1\\left|\\frac{\\partial \\Phi ^B}{\\partial F}(F)\\right|+c_2\\left|\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta )\\right|+c_3\\\\&\\qquad \\qquad \\qquad \\le c_4|\\psi (F,\\theta )|+c_5(|F|^2+|F|+1)+c_6|\\hat{\\psi }(\\Phi (F),\\theta )|+c_3\\\\&\\qquad \\qquad \\qquad \\le c_7(|F|^p+\\theta ^{\\ell })+c_8.$ Selecting now $R$ large enough, so that $c_8<c(|F|^p\\!+\\theta ^{\\ell }\\!+|v|^2)$ for $p>3,\\:\\ell >1,$ we conclude that $\\left|\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta )-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\right|\\le C_1^{\\prime } I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })$ by Lemma REF .",
"In the region $|F|^p+|v|^2+\\theta ^{\\ell }\\le R$ , we have that $(\\Phi (F),v,\\theta ),(\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\in D$ so that $&\\left|\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F),\\theta )-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\right|\\\\&\\qquad \\qquad \\qquad \\le \\max _{D}\\left|\\nabla ^2_{\\xi }\\hat{\\psi }(\\xi ,\\theta )\\right| \\left|\\Phi (F)-\\Phi (\\bar{F})\\right| \\left|\\frac{\\partial \\Phi }{\\partial F}(F)-\\frac{\\partial \\Phi }{\\partial F}(\\bar{F})\\right|\\\\&\\qquad \\qquad \\qquad \\le c_1 \\left|\\Phi (F)-\\Phi (\\bar{F})\\right|^2\\\\&\\qquad \\qquad \\qquad \\le C_1^{\\prime \\prime } I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })$ using again Lemma REF .",
"Choosing now $C_1=\\max \\lbrace C_1^{\\prime },C_1^{\\prime \\prime }\\rbrace ,$ estimate (REF ) follows.",
"Step $2.$ Using Young's inequality and (REF ), it follows $\\left|\\frac{\\partial \\hat{\\psi }}{\\partial \\xi }(\\Phi (F),\\theta |\\Phi (\\bar{F}),\\bar{\\theta })\\right|&\\le \\left|\\frac{\\partial \\hat{\\psi }}{\\partial \\xi }(\\Phi (F),\\theta )\\right|+c_1+c_2|\\Phi (F)|+c_3|\\theta |\\\\&\\le c_4 |\\hat{\\psi }(\\Phi (F),\\theta )|+c_2(|F|^3+|F|^2+|F|)+c_3|\\theta |+c_1\\\\&\\le c_5(|F|^p+\\theta ^{\\ell })+c_6\\;.$ Choosing again $R$ large enough, such that $R>r(M)+1$ there holds $\\left|\\frac{\\partial \\hat{\\psi }}{\\partial \\xi }(\\Phi (F),\\theta |\\Phi (\\bar{F}),\\bar{\\theta })\\right|&\\le c_7(|F|^p+\\theta ^{\\ell }+|v|^2)\\\\&\\le C_2^{\\prime } I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })$ by (REF ) and for $|F|^p+\\theta ^{\\ell }+|v|^2>R$ and $(\\bar{F},\\bar{v},\\bar{\\theta })\\in \\Gamma _{M,\\delta }.$ In the complementary region $|F|^p+\\theta ^{\\ell }+|v|^2\\le R,$ there holds $(\\Phi (F),v,\\theta ),(\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\in D$ , therefore $\\left|\\frac{\\partial \\hat{\\psi }}{\\partial \\xi }(\\Phi (F),\\theta |\\Phi (\\bar{F}),\\bar{\\theta })\\right|&\\le \\max _{D}\\left|\\nabla _{(\\xi ,\\theta )}^2\\hat{\\psi }(\\xi ,\\theta )\\right| (|\\Phi (F)-\\Phi (\\bar{F})|^2+|\\theta -\\bar{\\theta }|^2)\\\\&\\le C_2^{\\prime \\prime } I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })$ again by (REF ).",
"Choosing $C_2=\\max \\lbrace C_2^{\\prime },C_2^{\\prime \\prime }\\rbrace ,$ the proof of (REF ) is complete.",
"Step $3.$ We proceed in a similar manner as in Step $2.$ to prove (REF ).",
"First we study the region $|F|^p+|v|^2+\\theta ^{\\ell } >R$ and we use growth assumption (REF ) and relation (REF ) to get $&\\lim _{|F|^p+\\theta ^{\\ell }\\rightarrow \\infty }\\frac{|\\hat{\\eta }(\\Phi (F),\\theta |\\Phi (\\bar{F}),\\bar{\\theta })|}{|F|^p+\\theta ^{\\ell }}=\\lim _{|F|^p+\\theta ^{\\ell }\\rightarrow \\infty }\\frac{|\\hat{\\eta }(\\Phi (F),\\theta )|}{|F|^p+\\theta ^{\\ell }}=\\lim _{|F|^p+\\theta ^{\\ell }\\rightarrow \\infty }\\frac{|\\eta (F,\\theta )|}{|F|^p+\\theta ^{\\ell }}=0.$ So immediately we deduce $|\\hat{\\eta }(\\Phi (F),\\theta |\\Phi (\\bar{F}),\\bar{\\theta })|\\le C_3^{\\prime } I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta }),$ for $R$ large enough.",
"On the complementary region $|F|^p+|v|^2+\\theta ^{\\ell }\\le R,$ $|\\hat{\\eta }(\\Phi (F),\\theta |\\Phi (\\bar{F}),\\bar{\\theta })|&\\le \\max _{D}|\\nabla _{(\\xi ,\\theta )}^2\\hat{\\eta }(\\Phi (F),\\theta )|(|\\Phi (F)-\\Phi (\\bar{F})|^2+|\\theta -\\bar{\\theta }|^2)\\\\&\\le C_3^{\\prime \\prime } I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\;,$ by (REF ).",
"Choosing $C_3=\\max \\lbrace C_3^{\\prime },C_3^{\\prime \\prime }\\rbrace ,$ the proof of (REF ) is complete.",
"Step $4.$ Similarly, for (REF ), when $|F|^p+\\theta ^{\\ell } +|v|^2>R$ and $(\\bar{F},\\bar{v},\\bar{\\theta })\\in \\Gamma _{M,\\delta }$ , we have $\\left|\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)(v_i-\\bar{v}_i)\\right|\\le c_1|\\Phi (F)|^2+ c_2|v|^2 +c_3$ and choosing appropriately the radius $R,$ proceeding as before, it follows $\\left|\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)(v_i-\\bar{v}_i)\\right|&\\le c_4(|F|^p+\\theta ^{\\ell }+|v|^2)\\\\&\\le C_4^{\\prime } I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta }),$ where we use again (REF ) and $p>3.$ Then, for $|F|^p+\\theta ^{\\ell }+|v|^2\\le R$ and for all $(\\Phi (F),v,\\theta ),$ $(\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\in D$ , we also get $\\left|\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)(v_i-\\bar{v}_i)\\right|&\\le \\ \\tfrac{1}{2} \\left| \\frac{\\partial \\Phi ^B}{\\partial F}(F)-\\frac{\\partial \\Phi ^B}{\\partial F}(\\bar{F}) \\right|^2+ \\tfrac{1}{2} |v-\\bar{v}|^2\\\\&\\le c_1 (|\\Phi (F)-\\Phi (\\bar{F})|^2+|v-\\bar{v}|^2)\\\\&\\le C_4^{\\prime \\prime } I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })$ by (REF ).",
"Choosing $C_4=\\max \\lbrace C_4^{\\prime },C_4^{\\prime \\prime }\\rbrace ,$ estimate (REF ) follows.",
"Step $5.$ Since $(\\bar{F},\\bar{v},\\bar{\\theta })\\in \\Gamma _{M,\\delta }\\subset B_r$ -for sufficiently large $R$ - there holds $|F-\\bar{F}|^p&+|\\theta -\\bar{\\theta }|^{\\ell }+|v-\\bar{v}|^2\\le (|F|+M)^p+(\\theta +M)^{\\ell }+(|v|+M)^2\\;$ and $\\lim _{|F|^p+\\theta ^{\\ell }+|v|^2\\rightarrow \\infty }\\frac{(|F|+M)^p+(\\theta +M)^{\\ell }+(|v|+M)^2}{|F|^p+\\theta ^{\\ell }+|v|^2}=1\\;.$ Thus, we may select $R$ such that $|F-\\bar{F}|^p+|\\theta -\\bar{\\theta }|^{\\ell }+|v-\\bar{v}|^2&\\le 2(|F|^p+\\theta ^{\\ell }+|v|^2+1)\\\\&\\le C(|F|^p+\\theta ^{\\ell }+|v|^2)$ when $|F|^p+\\theta ^{\\ell }+|v|^2 \\ge R.$ Thus (REF ) follows from (REF ).",
"This concludes the proof.",
"We now consider a dissipative measure-valued solution for polyconvex thermoelasticity as defined in Definition REF .",
"Using the averaged relative entropy inequality (REF ), we prove that in the presence of a classical solution, given that the associated Young measure is initially a Dirac mass, the dissipative measure-valued solution must coincide with the classical one.",
"Theorem 5.1 Let $\\bar{U}$ be a Lipschitz bounded solution of (REF ),(REF ) with initial data $\\bar{U}^0$ and $(\\nu ,\\gamma ,U)$ be a dissipative measure-valued solution satisfying (REF ),(REF ), with initial data $U^0,$ both under the constitutive assumptions (REF ) and such that $r(x,t)=\\bar{r}(x,t)=0$ .",
"Suppose that $\\nabla _{\\xi }^2\\hat{\\psi }(\\Phi (F),\\theta )>0$ and $\\hat{\\eta }_{\\theta }(\\Phi (F),\\theta )>0$ and the growth conditions (REF ), (REF ), (REF ), (REF ), (REF ) hold for $p\\ge 4,$ and $\\ell > 1.$ If $\\bar{U}\\in \\Gamma _{M,\\delta },$ for some positive constants $M,\\delta $ and $\\bar{U}\\in W^{1,\\infty }(Q_T)$ , whenever $\\nu _{(0,x)}=\\delta _{\\bar{U}^0}(x)$ and $\\gamma _0=0$ we have that $\\nu =\\delta _{\\bar{U}}$ and $U=\\bar{U}$ a.e.",
"on $Q_T.$ Let $\\lbrace \\varphi _n\\rbrace $ be a sequence of monotone decreasing functions such that $\\varphi _n\\ge 0,$ for all $n\\in \\mathbb {N},$ converging as $n \\rightarrow \\infty $ to the Lipschitz function $\\varphi (\\tau )=\\begin{dcases}1 & 0\\le \\tau \\le t\\\\\\frac{t-\\tau }{\\varepsilon }+1 & t\\le \\tau \\le t+\\varepsilon \\\\0 & \\tau \\ge t+\\varepsilon \\end{dcases}$ for some $\\varepsilon >0.$ Writing the relative entropy inequality (REF ) for $r(x,t)=\\bar{r}(x,t)=0,$ tested against the functions $\\varphi _n$ we have $\\begin{split}\\int &\\varphi _n(0)\\left\\langle \\nu ,I(\\lambda _{U_0}|\\bar{U_0})\\right\\rangle \\:dx+\\int _{0}^{t}\\int \\varphi ^{\\prime }_n(\\tau )\\left[\\left\\langle \\nu ,I(\\lambda _{U}|\\bar{U})\\right\\rangle \\:dx\\:d\\tau +\\gamma (dx\\,d\\tau )\\right]\\\\&\\ge -\\int _{0}^{t}\\int \\varphi _n(\\tau )\\left[-\\partial _t\\bar{\\theta }\\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle +\\partial _t\\Phi ^B(\\bar{F})\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle \\right.\\\\&\\quad \\left.+\\partial _{\\alpha }\\bar{v}_i\\left\\langle \\nu ,\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\right\\rangle \\right.\\\\&\\quad \\left.+\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\!\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\!\\langle \\nu ,(\\lambda _{v_i}-\\bar{v}_i)\\rangle \\right]\\:dx\\,d\\tau .\\end{split}$ Passing to the limit as $n\\rightarrow \\infty $ we get $\\int &\\left\\langle \\nu ,I(\\Phi (\\lambda _{F}),\\lambda _{v},\\lambda _{\\theta }| \\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\right\\rangle (x,0)\\:dx\\\\&\\quad -\\frac{1}{\\varepsilon }\\int _{t}^{t+\\varepsilon }\\int \\left[\\left\\langle \\nu ,I(\\Phi (\\lambda _{F}),\\lambda _{v},\\lambda _{\\theta }| \\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\right\\rangle \\:dx\\:d\\tau +\\gamma (dxd\\tau )\\right] \\\\&\\ge - \\int _0^{t+\\varepsilon }\\!\\!\\!\\int \\!\\left[-\\partial _t\\bar{\\theta }\\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle +\\partial _t\\Phi ^B(\\bar{F})\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle \\right.\\\\&\\quad \\left.+\\partial _{\\alpha }\\bar{v}_i\\left\\langle \\nu ,\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\right\\rangle \\right.\\\\&\\quad \\left.+\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\!\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\!\\langle \\nu ,(\\lambda _{v_i}-\\bar{v}_i)\\rangle \\right]\\:dx\\,d\\tau .$ Passing now to the limit as $\\varepsilon \\rightarrow 0^{+}$ and using the fact that $\\gamma \\ge 0$ in combination with the estimates (REF ), (REF ), (REF ) and (REF ), we arrive at $\\int \\left\\langle \\nu ,I(\\Phi (\\lambda _{F}),\\lambda _{v},\\lambda _{\\theta }| \\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\right\\rangle \\:dx\\:dt\\le C &\\int _0^t\\int \\left\\langle \\nu ,I(\\Phi (\\lambda _{F}),\\lambda _{v},\\lambda _{\\theta }| \\Phi (\\bar{F}) ,\\bar{v},\\bar{\\theta })\\right\\rangle \\:dx\\:d\\tau \\\\&+\\int \\left\\langle \\nu ,I(\\Phi (\\lambda _{F}),\\lambda _{v},\\lambda _{\\theta }| \\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\right\\rangle (x,0)\\:dx$ for $t\\in (0,T).$ Note that the constant $C$ depends only on the smooth bounded solution $\\bar{U}.$ Then Gronwall's inequality implies $\\int \\left\\langle \\nu ,I(\\Phi (\\lambda _{F}),\\lambda _{v},\\lambda _{\\theta }| \\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\right\\rangle \\:dx\\:dt\\le C_1 e^{C_2t}\\int \\left\\langle \\nu ,I(\\Phi (\\lambda _{F}),\\lambda _{v},\\lambda _{\\theta }| \\Phi (\\bar{F})\\bar{v},\\bar{\\theta })\\right\\rangle (x,0)\\:dx$ and the proof is complete by (REF ).",
"An extension of Theorem REF holds in case we assume $r(x,t)=\\bar{r}(x,t)\\ne 0.$ For this purpose, we need the additional assumption $\\mathrm {supp}\\:\\nu \\subset \\:\\mathbb {R}^{19}\\times \\mathbb {R}^3\\times [\\underline{\\delta },\\infty ).$ to control the terms that arise from the radiative heat supply in (REF ).",
"We first prove the following lemma: Lemma 5.3 Suppose that $r(x,t)=\\bar{r}(x,t)\\in L^{\\infty }(Q_T)$ and that $\\mathrm {supp}\\:\\nu \\subset \\:\\mathbb {R}^{19}\\times \\mathbb {R}^3\\times [\\underline{\\delta },\\infty ),$ for some small positive constant $\\underline{\\delta }.$ Then there exists a constant $C_5>0$ such that $\\left|\\left\\langle \\nu ,\\left(\\frac{r}{\\lambda _{\\theta }}-\\frac{\\bar{r}}{\\bar{\\theta }}\\right)(\\lambda _{\\theta }-\\bar{\\theta })\\right\\rangle \\right|\\le C_5 \\left\\langle \\nu ,I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\right\\rangle $ for all $(\\bar{F},\\bar{v},\\bar{\\theta })\\in \\Gamma _{M,\\delta }$ .",
"Assume first that $|F|^p +\\theta ^{\\ell }+|v|^2>R.$ Then $\\left|\\left\\langle \\nu ,\\left(\\frac{r}{\\lambda _{\\theta }}-\\frac{\\bar{r}}{\\bar{\\theta }}\\right)(\\lambda _{\\theta }-\\bar{\\theta })\\right\\rangle \\right|&=\\left|\\int \\left(\\frac{\\bar{r}}{\\lambda _{\\theta }}-\\frac{\\bar{r}}{\\bar{\\theta }}\\right)(\\lambda _{\\theta }-\\bar{\\theta })\\:d\\nu \\right|\\\\&\\le \\Vert \\bar{r}\\Vert _{L^{\\infty }}\\left|\\int \\frac{(\\lambda _{\\theta }-\\bar{\\theta })^2}{\\lambda _{\\theta }\\bar{\\theta }}\\:d\\nu \\right|\\\\&=\\Vert \\bar{r}\\Vert _{L^{\\infty }} \\left|\\int \\left(\\frac{\\lambda _{\\theta }}{\\bar{\\theta }}-2+\\frac{\\bar{\\theta }}{\\lambda _{\\theta }}\\right)\\:d\\nu \\right|\\\\&\\le \\Vert \\bar{r}\\Vert _{L^{\\infty }} \\left(\\left|\\left\\langle \\nu ,\\frac{\\lambda _{\\theta }}{\\bar{\\theta }}\\right\\rangle \\right|+\\left|\\left\\langle \\nu ,\\frac{\\bar{\\theta }}{\\lambda _{\\theta }}\\right\\rangle \\right|+c_1\\right)\\\\&\\le c_2\\left|\\left\\langle \\nu ,\\lambda _{\\theta }\\right\\rangle \\right|+C_3\\left|\\left\\langle \\nu ,1\\right\\rangle \\right|+c_4\\\\&\\le c_5 (|\\theta |+1)\\;.$ Choosing $R$ sufficiently large, we get for $\\ell >1$ $\\left|\\left\\langle \\nu ,\\left(\\frac{r}{\\lambda _{\\theta }}-\\frac{\\bar{r}}{\\bar{\\theta }}\\right)(\\lambda _{\\theta }-\\bar{\\theta })\\right\\rangle \\right|&\\le c_6 (|F|^p+|\\theta |^{\\ell }+|v|^2)\\\\&\\le C_5^{\\prime }\\left\\langle \\nu ,I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\right\\rangle ,$ where, the last inequality holds because of Lemma REF and the constant $C_5^{\\prime }$ depends on $\\bar{r},\\:\\delta ,\\:M$ and $\\underline{\\delta }.$ Now, similarly, if $|F|^p +\\theta ^{\\ell }+|v|^2\\le R,$ we have $\\left|\\left\\langle \\nu ,\\left(\\frac{r}{\\lambda _{\\theta }}-\\frac{\\bar{r}}{\\bar{\\theta }}\\right)(\\lambda _{\\theta }-\\bar{\\theta })\\right\\rangle \\right|&\\le \\Vert \\bar{r}\\Vert _{L^{\\infty }}\\int \\frac{|\\lambda _{\\theta }-\\bar{\\theta }|^2}{|\\lambda _{\\theta }\\bar{\\theta }|}\\:d\\nu \\\\&\\le C_1 \\int |\\lambda _{\\theta }-\\bar{\\theta }|^2\\:d\\nu \\\\&\\le C_5^{\\prime \\prime }\\left\\langle \\nu ,I(\\Phi (F),v,\\theta |\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\right\\rangle \\;,$ again by estimate (REF ).",
"By choosing $C_5=\\max \\lbrace C_5^{\\prime }(\\bar{r},\\delta ,M,\\underline{\\delta }),C_5^{\\prime \\prime }(\\bar{r},\\delta ,\\underline{\\delta })\\rbrace ,$ the proof is complete.",
"Then Theorem REF extends to : Theorem 5.2 Let $\\bar{U}$ be a Lipschitz bounded solution of (REF ),(REF ) with initial data $\\bar{U}^0$ and $(\\nu ,\\gamma ,U)$ be a dissipative measure-valued solution satisfying (REF ),(REF ), with initial data $U^0,$ both under the constitutive assumptions (REF ) and such that $r(x,t)=\\bar{r}(x,t)$ .",
"Assume also that there exists a small constant $\\underline{\\delta }>0$ such that (REF ) holds true.",
"Suppose that $\\nabla _{\\xi }^2\\hat{\\psi }(\\Phi (F),\\theta )>0$ and $\\hat{\\eta }_{\\theta }(\\Phi (F),\\theta )>0$ and the growth conditions (REF ), (REF ), (REF ), (REF ), (REF ) hold for $p\\ge 4,$ and $\\ell > 1.$ If $\\bar{U}\\in \\Gamma _{M,\\delta },$ for some positive constants $M,\\delta $ and $\\bar{U}\\in W^{1,\\infty }(Q_T)$ , whenever $\\nu _{(0,x)}=\\delta _{\\bar{U}^0}(x),$ and $\\gamma _0=0$ we have that $\\nu =\\delta _{\\bar{U}}$ and $U=\\bar{U}$ a.e.",
"on $Q_T.$ The proof is a simple variant of the one for Theorem REF .",
"Assuming the sequence $\\lbrace \\varphi _n\\rbrace $ as before, the relative entropy inequality (REF ) becomes $\\begin{split}\\int &\\varphi _n(0)\\left\\langle \\nu ,I(\\lambda _{U_0}|\\bar{U_0})\\right\\rangle \\:dx+\\int _{0}^{t}\\int \\varphi ^{\\prime }_n(\\tau )\\left[\\left\\langle \\nu ,I(\\lambda _{U}|\\bar{U})\\right\\rangle \\:dx\\:d\\tau +\\gamma (dx\\,d\\tau )\\right]\\\\&\\ge -\\int _{0}^{t}\\int \\varphi _n(\\tau )\\left[-\\partial _t\\bar{\\theta }\\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle +\\partial _t\\Phi ^B(\\bar{F})\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{\\theta })\\right\\rangle \\right.\\\\&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\left.+\\partial _{\\alpha }\\bar{v}_i\\left\\langle \\nu ,\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })-\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\right\\rangle \\right.\\\\&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\left.+\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\bar{F}),\\bar{\\theta })\\right)\\!\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)-\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\bar{F})\\right)\\!\\langle \\nu ,(\\lambda _{v_i}-\\bar{v}_i)\\rangle +\\left\\langle \\nu ,\\left(\\frac{r}{\\lambda _{\\theta }}-\\frac{\\bar{r}}{\\bar{\\theta }}\\right)(\\lambda _{\\theta }-\\bar{\\theta })\\right\\rangle \\right]\\:dx\\,d\\tau .\\end{split}$ Passing to the limit as $n\\rightarrow \\infty $ and then as $\\varepsilon \\rightarrow 0^{+}$ we obtain $\\int \\left\\langle \\nu ,I(\\Phi (\\lambda _{F}),\\lambda _{v},\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\right\\rangle \\:dx\\:dt\\le C &\\int _0^t\\int \\left\\langle \\nu ,I(\\Phi (\\lambda _{F}),\\lambda _{v},\\lambda _{\\theta }|\\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\right\\rangle \\:dx\\:d\\tau \\\\&+\\int \\left\\langle \\nu ,I(\\Phi (\\lambda _{F}),\\lambda _{v},\\lambda _{\\theta }| \\Phi (\\bar{F}),\\bar{v},\\bar{\\theta })\\right\\rangle (x,0)\\:dx$ for $t\\in (0,T).$ Here, we used that $\\gamma \\ge 0$ and the estimates (REF ), (REF ), (REF ), (REF ) and (REF ), so that constant $C$ depends on the smooth bounded solution $\\bar{U}$ and $\\underline{\\delta }.$ By virtue of Gronwall's inequality and (REF ), we conclude the proof.",
"Let us note that, as Lemma REF indicates, one needs to assume (REF ) in order to be able to bound from below the averaged temperature $\\left\\langle \\nu ,\\lambda _{\\theta }\\right\\rangle $ and achieve estimate (REF ).",
"Though it could be considered as a rather mild assumption, it is interesting that all the estimates in Lemmas REF and REF that involve the averaged temperature, do not require (REF ) to hold.",
"This is because the averaged temperature is involved only through the constitutive functions $\\hat{\\psi },\\:\\hat{e}$ and $\\hat{\\eta }$ which we assume to be smooth enough, i.e.",
"$\\hat{\\psi }=\\hat{e}-\\theta \\hat{\\eta }\\in C^2$ , and therefore we avoid any loss of smoothness as the temperature approaches zero."
],
[
"The natural bounds of viscous approximation for polyconvex thermoelasticity",
"Since measure-valued solutions usually occur as limits of an approximating problem, consider the system of polyconvex thermoelasticity with Newtonian viscosity and Fourier heat conduction $\\begin{split}\\partial _t \\Phi ^B(F^{\\mu ,k})&=\\partial _{\\alpha }\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F^{\\mu ,k})v_i^{\\mu ,k}\\right)\\\\\\partial _t v^{\\mu ,k}_i&=\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})\\frac{\\partial \\Phi ^B}{F_{i\\alpha }}(F^{\\mu ,k})\\right)+\\partial _{\\alpha }(\\mu \\partial _{\\alpha }v_i^{\\mu ,k})\\\\\\partial _t \\left(\\frac{1}{2}|v^{\\mu ,k}|^2+\\hat{e}(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})\\right)&=\\partial _{\\alpha }\\left(\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})\\frac{\\partial \\Phi ^B}{F_{i\\alpha }}(F^{\\mu ,k}) v_i^{\\mu ,k}\\right)+\\\\&\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad +\\partial _{\\alpha }(\\mu v_i^{\\mu ,k}\\partial _{\\alpha }v_i^{\\mu ,k}+k\\partial _{\\alpha }\\theta ^{\\mu ,k})+r\\\\\\partial _t\\hat{\\eta }(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})&=\\partial _{\\alpha }\\left(k\\frac{\\nabla \\theta ^{\\mu ,k}}{\\theta ^{\\mu ,k}}\\right)+k\\frac{|\\nabla \\theta ^{\\mu ,k}|^2}{(\\theta ^{\\mu ,k})^2}+\\mu \\frac{|\\nabla v^{\\mu ,k}|^2}{\\theta ^{\\mu ,k}}+\\frac{r}{\\theta ^{\\mu ,k}}\\end{split}$ where $\\frac{\\partial \\hat{\\psi }}{\\partial F_{i\\alpha }}(\\Phi (F),\\theta )=\\frac{\\partial \\psi }{\\partial F_{i\\alpha }}(\\Phi (F),\\theta )\\frac{\\partial \\psi }{\\partial \\zeta ^{k\\gamma }}(\\Phi (F),\\theta )\\frac{\\partial (\\mathrm {cof}F)_{k\\gamma }}{\\partial F_{i\\alpha }}+\\frac{\\partial \\psi }{\\partial w}(\\Phi (F),\\theta )\\frac{\\partial (\\det F)}{\\partial F_{i\\alpha }}.$ Suppose first that the energy radiation $r\\equiv 0.$ The viscosity and heat conduction coefficients are assumed to satisfy the condition $\\begin{split}|\\mu (F,\\theta )\\theta |&<\\mu _0|e(F,\\theta )|,\\\\|k(F,\\theta )|&<k_0|e(F,\\theta )|,\\\\\\end{split}$ for some constants $\\mu _0,\\: k_0>0$ , we are going to examine how we can obtain measure-valued solutions in the limit as $\\mu _0\\rightarrow 0$ and $k_0\\rightarrow 0.$ We work in a periodic domain in space $Q_T=\\mathbb {T}^d\\times [0,T),$ for $T\\in [0,\\infty )$ and $d=3.$ We impose the growth conditions $c(|F|^p+\\theta ^{\\ell })-c\\le e(F,\\theta )\\le c(|F|^p+\\theta ^{\\ell })+c.$ $c(|F|^p+\\theta ^{\\ell })-c\\le \\psi (F,\\theta )\\le c(|F|^p+\\theta ^{\\ell })+c\\;,$ $\\lim _{|F|^p+\\theta ^{\\ell }\\rightarrow \\infty }\\frac{|\\partial _{\\theta }\\psi (F,\\theta )|}{|F|^p+\\theta ^{\\ell }}=\\lim _{|F|^p+\\theta ^{\\ell }\\rightarrow \\infty }\\frac{|\\eta (F,\\theta )|}{|F|^p+\\theta ^{\\ell }}=0\\;,$ and $\\lim _{|F|^p+\\theta ^{\\ell }\\rightarrow \\infty }\\frac{|\\partial _{F}\\psi (F,\\theta )|}{|F|^p+\\theta ^{\\ell }}=\\lim _{|F|^p+\\theta ^{\\ell }\\rightarrow \\infty }\\frac{|\\Sigma (F,\\theta )|}{|F|^p+\\theta ^{\\ell }}=0\\;,$ for some constant $c>0$ and $p,\\ell >1.$ Integrating the energy equation (REF )$_3$ in $Q_T$ we get $\\int \\left(\\frac{1}{2}|v^{\\mu ,k}|^2+\\hat{e}(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})\\right)\\:dx\\le \\int \\left(\\frac{1}{2}|v_0^{\\mu ,k}|^2+\\hat{e}(\\Phi (F_0^{\\mu ,k}),\\theta _0^{\\mu ,k})\\right)\\:dx\\le C_1$ so that $\\sup _{0<t<T}\\int \\left(\\frac{1}{2}|v^{\\mu ,k}|^2+\\hat{e}(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})\\right)\\:dx\\le C<\\infty .$ Therefore (REF ) implies that $\\int \\left(|F^{\\mu ,k}|^p+\\frac{1}{2}|v^{\\mu ,k}|^2+|\\theta ^{\\mu ,k}|^{\\ell }\\right)\\:dx\\le C<\\infty $ and then the functions $(F^{\\mu ,k},v^{\\mu ,k},\\theta ^{\\mu ,k})$ are all bounded in the spaces: $F^{\\mu ,k}\\in L^{\\infty }(L^p),\\quad v^{\\mu ,k}\\in L^{\\infty }(L^2), \\quad \\theta ^{\\mu ,k}\\in L^{\\infty }(L^{\\ell })$ and weakly converging to the averages $F^{\\mu ,k}\\rightharpoonup \\langle \\nu ,\\lambda _{F}\\rangle , \\quad \\text{weak-$\\ast $ in\\:} L^{\\infty } (L^p ) \\;,\\\\v^{\\mu ,k}\\rightharpoonup \\langle \\nu ,\\lambda _{v}\\rangle , \\quad \\text{weak-$\\ast $ in\\:} L^{\\infty }( L^2 ) \\;,\\\\\\theta ^{\\mu ,k}\\rightharpoonup \\langle \\nu ,\\lambda _{\\theta }\\rangle , \\quad \\text{weak-$\\ast $ in\\:} L^{\\infty }(L^{\\ell } ) \\, .$ Integrating now (REF )$_4$ ($r\\equiv 0$ ), in $Q_T$ we obtain $\\int \\hat{\\eta }(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})\\:dx-\\int \\hat{\\eta }(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})(x,0)\\:dx=\\int _0^T\\!\\!\\!\\!\\int \\left(k\\frac{|\\nabla \\theta ^{\\mu ,k}|^2}{(\\theta ^{\\mu ,k})^2}+\\mu \\frac{|\\nabla v^{\\mu ,k}|^2}{\\theta ^{\\mu ,k}}\\right)\\:dxdt.$ Then (REF ) and (REF ) immediately imply that $\\hat{\\eta }(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})\\in L^{\\infty }(L^1)$ while $0<\\int _0^T\\int \\left(k\\frac{|\\nabla \\theta ^{\\mu ,k}|^2}{(\\theta ^{\\mu ,k})^2}+\\mu \\frac{|\\nabla v^{\\mu ,k}|^2}{\\theta ^{\\mu ,k}}\\right)\\:dxdt\\le C.$ Now, let us consider the first equation in (REF ).",
"We employ Lemma REF , in order to pass to the limit in the minors and the identities (REF ).",
"It follows that (REF )$_1$ holds in the classical weak sense for motions with regularity as in (REF ) and $(\\Phi (F),v,\\theta )=(F,\\zeta ,w,v,\\theta )\\in L^{\\infty }(L^p)\\times L^{\\infty }(L^q)\\times L^{\\infty }(L^{\\rho })\\times L^{\\infty }(L^2)\\times L^{\\infty }(L^{\\ell })$ with $p\\ge 4,\\:q\\ge 2, \\rho , \\ell >1.$ To pass to the limit in the second equation (REF )$_2,$ we use the Theorem of Ball [2] on representation via Young measures in the $L^p$ setting: Lemma 1.1 Let $f^{\\epsilon }:\\bar{Q}_T\\rightarrow {\\mathbb {R}^m}$ be a bounded function in $L^p.$ Then for all $F:\\mathbb {R}^m\\rightarrow \\mathbb {R}$ which are continuous and such that $F(f^{\\epsilon })$ is $L^1$ weakly precompact, there holds (along a subsequence) $F(f^{\\epsilon })\\rightharpoonup \\langle \\nu ,F\\rangle , \\quad \\text{weakly in\\:} L^1(Q_T).$ If $f^{\\epsilon }:\\bar{Q}_T\\rightarrow {\\mathbb {R}^m}$ is uniformly bounded in $Q_T,$ then for all continuous $F:\\mathbb {R}^m\\rightarrow \\mathbb {R}$ there holds (along a subsequence) $F(f^{\\epsilon })\\rightharpoonup \\langle \\nu ,F\\rangle ,\\quad \\text{weak-$\\ast $ in\\:} L^{\\infty }(Q_T).$ Note that the assumption $\\lbrace F(f^{\\epsilon })\\rbrace $ is $L^1$ weakly precompact cannot be dropped or just replaced by $L^1$ boundness, because in such case concentrations might develop.",
"We are going to examine each term separately, in order to obtain (along a non-relabeled subsequence) $\\partial _t \\left\\langle \\nu ,\\lambda _{v_i}\\right\\rangle &=\\partial _{\\alpha }\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _{F})\\right\\rangle $ in the sense of distributions, which means $\\int \\left\\langle \\nu ,\\lambda _{v_i}\\right\\rangle (x,0)\\phi (x,0)\\:dx&+\\int _0^T\\int \\left\\langle \\nu ,\\lambda _{v_i}\\right\\rangle \\partial _t\\phi \\:dxdt\\\\&=\\int _0^T\\int \\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _{F})\\right\\rangle \\partial _{\\alpha }\\phi \\:dxdt$ for $\\phi (x,t)\\in C_c^1(Q_T).$ Observe that (REF ), (REF ) combined with (REF ) yield that the term $\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})\\frac{\\partial \\Phi ^B}{F_{i\\alpha }}(F^{\\mu ,k})$ is representable with respect to the Young measure $\\nu _{(x,t)\\in \\bar{Q}_T}$ so that $\\frac{\\partial \\hat{\\psi }}{\\partial F_{i\\alpha }}(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})\\rightharpoonup \\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial F_{i\\alpha }}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle , \\quad \\text{weakly in\\:} L^1.$ Now, we examine the limit of the diffusion term $\\mathrm {div}(\\mu \\nabla v^{\\mu ,k})$ as $\\mu _0\\rightarrow 0$ .",
"First, we write $\\lim _{\\mu _0\\rightarrow 0}\\left|\\int _0^T\\!\\!\\int \\mathrm {div}(\\mu \\nabla v^{\\mu ,k})\\phi \\:dxdt\\right|&=\\lim _{\\mu _0\\rightarrow 0}\\int _0^T\\!\\!\\int \\left|\\mu \\nabla v^{\\mu ,k}\\cdot \\nabla \\phi \\right|\\:dxdt,$ while using Hölder's inequality: $\\begin{split}\\lim _{\\mu _0\\rightarrow 0}\\int _0^T\\!\\!\\int &\\left|\\mu \\nabla v^{\\mu ,k}\\cdot \\nabla \\phi \\right|\\:dxdt\\\\&\\le \\lim _{\\mu _0\\rightarrow 0}\\left[\\left(\\int _0^T\\!\\!\\!\\int |\\mu |\\frac{|\\nabla v^{\\mu ,k}|^2}{\\theta ^{\\mu ,k}}\\:dxdt\\right)^{1/2}\\!\\!\\!\\!\\left(\\int _0^T\\!\\!\\int |\\mu \\,\\theta ^{\\mu ,k}|\\, |\\nabla \\phi |^2\\:dxdt\\right)^{1/2}\\right]\\\\&\\le \\lim _{\\mu _0\\rightarrow 0} C\\mu _0=0\\end{split}$ for $\\phi (x,t)\\in C_c^1(Q_T)$ , by virtue of (REF )$_1$ , (REF ) and (REF ).",
"Next, we move to the entropy identity (REF )$_4,$ with $r\\equiv 0.$ In the limit, we aim to have $-\\int \\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle (x,0)\\phi (x,0)\\:dx&-\\int _0^T\\int \\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle \\partial _t\\phi \\:dxdt\\ge 0,$ for $\\phi (x,t)\\in C_c^1(Q_T),\\:\\phi \\ge 0.$ Growth condition (REF ) combined with (REF )$_1$ allows to use again the Fundamental Lemma on Young measures to get $ \\hat{\\eta }(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})\\rightharpoonup \\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle , \\quad \\text{weakly in\\:} L^1,$ The diffusion term $\\mathrm {div}\\left(k\\frac{\\nabla \\theta ^{\\mu ,k}}{\\theta ^{\\mu ,k}}\\right)$ can be treated exactly as in (REF ), using again Hölder's inequality, (REF )$_2$ , (REF ) and (REF ): $\\begin{split}\\lim _{k_0\\rightarrow 0}&\\left|\\int _0^T\\!\\!\\!\\int \\mathrm {div}\\left(k\\frac{\\nabla \\theta ^{\\mu ,k}}{\\theta ^{\\mu ,k}}\\right)\\phi \\:dxdt\\right|\\\\&=\\lim _{k_0\\rightarrow 0}\\left|\\int _0^T\\!\\!\\!\\int k\\frac{\\nabla \\theta ^{\\mu ,k}}{\\theta ^{\\mu ,k}}\\cdot \\nabla \\phi \\:dxdt\\right|\\\\&\\le \\lim _{k_0\\rightarrow 0}\\left[\\left(\\int _0^T\\!\\!\\!\\int |k|\\,\\frac{|\\nabla \\theta ^{\\mu ,k}|^2}{(\\theta ^{\\mu ,k})^2}\\:dxdt\\right)^{1/2}\\!\\!\\!\\!\\left(\\int _0^T\\!\\!\\!\\int |k|\\,|\\nabla \\phi |^2\\:dxdt\\right)^{1/2}\\right]\\\\&\\le \\lim _{k_0\\rightarrow 0} Ck_0=0\\end{split}$ where $\\phi (x,t)\\in C_c^1(Q_T),\\:\\phi \\ge 0.$ On account of (REF ), we apply the Banach-Alaoglou Theorem to the sequence $\\left|\\sqrt{k}\\frac{\\nabla \\theta ^{\\mu ,k}}{\\theta ^{\\mu ,k}}\\right|^2$ to obtain a sequential weak-$\\ast $ limit $\\sigma _k\\in \\mathcal {M}^+(\\bar{Q}_T)$ $\\sigma _k(\\phi )=\\int _0^T\\int \\phi \\:d\\sigma _k=\\lim _{k_0\\rightarrow 0}\\int _0^T\\int \\phi \\left|\\sqrt{k}\\frac{\\nabla \\theta ^{\\mu ,k}}{\\theta ^{\\mu ,k}}\\right|^2\\:dxdt,\\quad \\forall \\:\\phi \\in C(\\bar{Q}_T),\\:\\phi \\ge 0.$ Similarly, the term $\\mu \\frac{|\\nabla v^{\\mu ,k}|^2}{\\theta ^{\\mu ,k}}\\rightharpoonup \\sigma _{\\mu },\\quad \\text{weak-$\\ast $ in $\\mathcal {M}^+(\\bar{Q}_T),$}$ as $\\mu _0\\rightarrow 0,$ because of (REF ).",
"In summary, in the limit as $k_0,\\mu _0\\rightarrow 0,$ it holds (along a subsequence) $\\partial _t \\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle =\\sigma _{\\mu }+\\sigma _k\\ge 0$ in the sense of distributions.",
"Next we consider the energy equation (REF )$_3$ .",
"The function $(x,t)\\mapsto \\big ( \\tfrac{1}{2}|v^{\\mu ,k}|^2 + \\hat{e} (\\Phi (F^{\\mu ,k}), \\theta ^{\\mu ,k} ) \\big ) dxdt$ is weakly precompact in the space of nonnegative Radon measures $\\mathcal {M}^+(\\bar{Q}_T),$ but not weakly precompact in $L^1,$ therefore the Young measure representation fails.",
"To capture the resulting formation of concentrations, we introduce the concentration measure $\\gamma (dx,dt),$ which is a non-negative Radon measure in $Q_T,$ for a subsequence of $\\frac{1}{2}|v^{\\mu ,k}|^2+\\hat{e}(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k}),$ according to the analysis in Section 3.",
"As we aim to construct dissipative solutions, in the limit, we consider an integrated averaged energy identity, tested against $\\varphi =\\varphi (t)$ in $C^1_c[0,T],$ that does not depend on the spatial variable; as a result, all the flux terms in (REF )$_3$ vanish.",
"Having this in mind, the energy equation becomes $\\int \\varphi (0) \\Big ( \\langle \\nu ,\\frac{1}{2}|\\lambda _{v}|^2+&\\hat{e}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\rangle (x,0) \\:dx+\\gamma _0(dx) \\Big )\\\\&+\\int _0^T \\!\\!\\int \\varphi ^{\\prime }(t)\\left(\\left\\langle \\nu ,\\frac{1}{2}|\\lambda _{v}|^2+\\hat{e}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle \\:dx\\:dt+\\gamma (dx\\:dt)\\right) =0.$ for all $\\varphi =\\varphi (t)$ in $C^1_c[0,T]$ .",
"Altogether, we conclude: $\\partial _t \\Phi ^B(F)&=\\partial _{\\alpha }\\left(\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(F)v_i\\right) \\\\\\partial _t \\left\\langle \\nu ,\\lambda _{v_i}\\right\\rangle &=\\partial _{\\alpha }\\left\\langle \\nu ,\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\frac{\\partial \\Phi ^B}{\\partial F_{i\\alpha }}(\\lambda _{F})\\right\\rangle \\\\\\partial _t \\left\\langle \\nu ,\\hat{\\eta }(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle & \\ge 0$ and $\\int \\varphi (0) \\Big ( \\langle \\nu ,\\frac{1}{2}|\\lambda _{v}|^2+&\\hat{e}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\rangle (x,0) \\:dx+\\gamma _0(dx) \\Big )\\\\&+\\int _0^T \\!\\!\\int \\varphi ^{\\prime }(t)\\left(\\left\\langle \\nu ,\\frac{1}{2}|\\lambda _{v}|^2+\\hat{e}(\\Phi (\\lambda _{F}),\\lambda _{\\theta })\\right\\rangle \\:dx\\:dt+\\gamma (dx\\:dt)\\right) =0.$ The above analysis indicates the relevance of Definition REF of dissipative measure valued solution stated in Section 2.",
"Remark 1.1 Testing the energy equation (REF )$_3$ against a test function $\\varphi =\\varphi (x,t)\\in C^1_c(Q_T)$ , yields a different notion of measure-valued solution in the limit, the so-called entropy measure-valued solution.",
"In that case, additional assumptions are required.",
"First, one should represent the term $\\frac{\\partial \\hat{\\psi }}{\\partial \\xi ^B}(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})\\frac{\\partial \\Phi ^B}{F_{i\\alpha }}(F^{\\mu ,k}) v_i^{\\mu ,k}$ in the flux, which requires growth conditions on $\\Sigma _{i \\alpha }(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})v_i^{\\mu ,k}$ .",
"Second, to treat the terms $\\text{div}(\\mu v_i^{\\mu ,k}\\partial _{\\alpha }v_i^{\\mu ,k})$ and $\\text{div}(k\\partial _{\\alpha }\\theta ^{\\mu ,k})$ the additional uniform bounds $|\\mu (F,\\theta ) \\theta |\\le C\\mu _0 \\, \\quad \\mbox{and} \\quad |k(F,\\theta ) \\theta |\\le k_0|e(F ,\\theta ) | \\, ,$ on the diffusion coefficients are needed to pass to the limit.",
"Remark 1.2 Let us return to the notion of dissipative measure-valued solution but now with $r\\ne 0.$ To establish the limit as $\\mu _0$ , $k_0\\rightarrow 0^+$ , we need to represent the terms $r$ and $\\displaystyle \\frac{r}{\\theta ^{\\mu ,k}},$ as they appear on the right-hand side of (REF )$_3$ and (REF )$_4$ respectively.",
"Then (REF ) and (REF ) become $\\int \\left(\\frac{1}{2}|v^{\\mu ,k}|^2+\\hat{e}(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})\\right)\\:dx&\\le \\int _0^T\\int |{r}|\\:dxdt+\\int \\left(\\frac{1}{2}|v_0^{\\mu ,k}|^2+\\hat{e}(\\Phi (F_0^{\\mu ,k}),\\theta _0^{\\mu ,k})\\right)\\:dx$ and $\\int \\hat{\\eta }(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})\\:dx-\\int \\hat{\\eta }(\\Phi (F^{\\mu ,k}),\\theta ^{\\mu ,k})(x,0)\\:dx&=\\int _0^T\\!\\!\\!\\!\\int \\left(k\\frac{|\\nabla \\theta ^{\\mu ,k}|^2}{(\\theta ^{\\mu ,k})^2}+\\mu \\frac{|\\nabla v^{\\mu ,k}|^2}{\\theta ^{\\mu ,k}}\\right)\\:dxdt\\\\&+\\int _0^T\\int \\frac{r}{\\theta ^{\\mu ,k}}\\:dxdt$ respectively.",
"In turn, to have the bounds (REF ) and (REF ), the previous analysis to pass to the limit $(\\mu _0,k_0)\\rightarrow (0,0)$ suggests to require $r\\in L^{\\infty }(Q_T)\\,\\,\\text{ and}\\quad 0<\\bar{\\delta }\\le \\theta ^{\\mu ,k}$ for some $\\bar{\\delta }>0$ , so that both $r$ and $\\,\\displaystyle \\frac{r}{\\theta ^{\\mu ,k}}\\in L^1(Q_T)$ uniformly in $\\mu $ and $k$ .",
"Hence, as $\\mu _0$ , $k_0\\rightarrow 0$ , we get $&r\\rightharpoonup r\\left\\langle \\nu ,1\\right\\rangle ,\\\\\\frac{r}{\\theta ^{\\mu ,k}}\\rightharpoonup &\\left\\langle \\nu ,\\frac{r}{\\lambda _{\\theta }}\\right\\rangle =r\\left\\langle \\nu ,\\frac{1}{\\lambda _{\\theta }}\\right\\rangle ,$ weak-$\\ast $ in $L^{\\infty }(\\bar{Q}_T).$ In summary, the viscosity limit produces a dissipative measure valued solution assuming (REF ) and the growth estimates (REF )–(REF ) for $r\\equiv 0$ and in addition, (REF ) for $r\\ne 0$ ."
],
[
"Acknowledgments",
"This project has received funding from the European Union's Horizon 2020 programme under the Marie Sklodowska-Curie grant agreement No 642768.",
"Christoforou was partially supported by the Internal grant SBLawsMechGeom #21036 from University of Cyprus."
]
] | 1808.08382 |
[
[
"Label and Sample: Efficient Training of Vehicle Object Detector from\n Sparsely Labeled Data"
],
[
"Abstract Self-driving vehicle vision systems must deal with an extremely broad and challenging set of scenes.",
"They can potentially exploit an enormous amount of training data collected from vehicles in the field, but the volumes are too large to train offline naively.",
"Not all training instances are equally valuable though, and importance sampling can be used to prioritize which training images to collect.",
"This approach assumes that objects in images are labeled with high accuracy.",
"To generate accurate labels in the field, we exploit the spatio-temporal coherence of vehicle video.",
"We use a near-to-far labeling strategy by first labeling large, close objects in the video, and tracking them back in time to induce labels on small distant presentations of those objects.",
"In this paper we demonstrate the feasibility of this approach in several steps.",
"First, we note that an optimal subset (relative to all the objects encountered and labeled) of labeled objects in images can be obtained by importance sampling using gradients of the recognition network.",
"Next we show that these gradients can be approximated with very low error using the loss function, which is already available when the CNN is running inference.",
"Then, we generalize these results to objects in a larger scene using an object detection system.",
"Finally, we describe a self-labeling scheme using object tracking.",
"Objects are tracked back in time (near-to-far) and labels of near objects are used to check accuracy of those objects in the far field.",
"We then evaluate the accuracy of models trained on importance sampled data vs models trained on complete data."
],
[
"Introduction",
"Autonomous driving is receiving enormous development effort with many companies predicting large-scale commercial deployment in 2-3 years [35].",
"One of the most important features of autonomous driving vehicles is the ability to interpret the surroundings and perform complex perception task such as the detection and recognition of lanes, roads, pedestrians, vehicles, and traffic signs [27].",
"Recently, the growth of Convolutional Neural Networks (CNNs), and large labeled data sets [8], [11] have led to tremendous progress in object detection and recognition [14], [15], [26], [20].",
"It is now possible to detect objects with high accuracy [26], [25], [20].",
"Figure: Overview of our method.",
"Given a sequence ofvideo frame inputs, the object detection network firstdetects objects in a frame, the forward pass step.",
"Thenthe loss of the images are calculated.",
"Both detectionresults and loss will be sent to object trackerwhich keeps a list of active objects.",
"The importancesampler determines whether to save the detection ornot based on the loss.",
"Furthermore, the near-to-farlabeling step checks the accuracy of objects that arein the far field using the classification result ofnear field objects, where we believe near field objectsare larger and the classification is more accurate.Videos collected in-vehicle have a great potential to improve model quality (through offline training) but at the scales achievable in a few years (billions to trillions of hours of video), training on all the data is completely impractical.",
"Nor is it desirable - most images contain “typical\" content and objects that are recognized with good accuracy.",
"These models contribute little to the final model.",
"But it is the less common “interesting\" images that are most important for training (i.e.",
"images containing objects that are mis-classified, or classified with low confidence).",
"To benefit from these images, it's still important to have accurate label information.",
"The images of distance objects in isolation are not good for this purpose - by definition they contain objects which are difficult to label automatically.",
"But we can use particular characteristics of vehicle video: namely that most of the time (vehicle moving forward), objects gradually grow and become clearer and easier to recognize.",
"We exploit this coherence by tracking objects back in time and using near, high-reliability labels to label more distant objects.",
"We demonstrate this process on a hand-labeled dataset [13] which only has a small fraction of total frames labeled and also has short length video clips that can be used for object tracking.",
"We show that we can extend the labelled data using near-to-far tracking strategy, and importance sampling can be used to refine the automatically labeled dataset to improve its quality.",
"To automate far-object tracking, we need both single-image object detection and between-image tracking.",
"While these two modules can be used separately, we designed a strategy to nicely combine them together.",
"Specifically we used a Faster-RCNN object detector [26] and Kalman filtering to track objects.",
"We use predicted object positions from tracking to augment the Faster-RCNN's region proposals, and then use the Faster-RCNN's bounding box regression to correct the object position estimates.",
"The result is that we can track and persist object layers much further into the distance where it might be hard for Faster-RCNN to give accurate object region proposal.",
"The automatically-generated labels are then used to compute the importance of each image.",
"As shown in [2], an optimal model is obtained when images are importance sampled using the norm of the back-propagated gradient magnitude for the image.",
"While computing the full back-propagated gradients in vehicle video systems would be very expensive, we can actually use the loss function as a surrogate for gradient as it is easier to obtain.",
"We further show in our experiments that the loss function can be a approximate of the gradient norm.",
"Importance sampling for training data filtering is also described in [7].",
"Contributions.",
"Starting with a sparsely labeled video dataset, we combine object tracking and object detection to generate new labeled data, and then use importance sampling for data reduction.",
"The contributions of this paper are: 1) We show that near-to-far object tracking can produce useful training data augmentation.",
"2) Empirically, gradient norm can be approximated by loss function and last layer gradient norm in deep neural networks.",
"3) Importance sampling produces large reductions in training data and training time and modest loss of accuracy."
],
[
"Related Work",
"Semi-Supervised Data Labeling.",
"With the large amount of sparsely labeled image datasets, some work has been done in the field of semi-supervised object detection and data labeling [34], [31], [18], [6].",
"These work try to learn a set of similar attributes for image classes [31], [6] to label new datasets, or to cluster similar images [18], or perform transfer learning to recognize similar types of objects [34].",
"However, these work are typically suitable for image dataset where they process images individually and do not consider the temporal continuity of video dataset for semi-supervised learning.",
"Semi-supervised learning for video dataset is also described in [19], [24], [33].",
"While their performance is good, they assume labeling the salient object in a video and thus do not apply well to multi-object detection or tracking case.",
"A large body of work has also been done in the field of tracking-by-detection [3], [23], [16], [17], [5].",
"However, they either assume the possibility of negative data being sampled from around the object or they do not use the special characteristic of driving video that objects in the near field are easier to be detected than objects in the far field.",
"In addition, these naive combinations of tracking and detection may introduce additional noise in labeling images.",
"Also, tracking tends to drift away in the long run and the related data association is also very challenging [3].",
"In the work of [21], they proposed to use semi-supervised learning to label large video datasets by tracking multiple objects in videos.",
"However, their application scenario is not driving video dataset and the object they detect only include cars.",
"In addition, their focus is on short term tracking of objects and they do not require short tracklets to be associated with each other.",
"Therefore, the applicability of their method is limited especially when there are multiple categories of objects in a scene, since ignoring the data association part would be problematic if the goal is to label multiple categories of objects.",
"In our work, we do consider the problem of data association and we use object tracker's prediction as the region proposal for object detector to provide more accurate bounding box annotation.",
"However, similar to the work of [21], we do not perform long-run tracking to prevent the tracker from drifting away.",
"We also use near-to-far labeling to help correct the detector's classification results.",
"Importance Sampling.",
"Importance sampling is a well-known technique used for reducing the variance when estimating properties of a particular distribution while only having samples generated from another distribution [22], [28].",
"The work of [38] studied the problem of improving traditional stochastic optimization with importance sampling, where they improved the convergence rate of prox-SMD [9], [10] and prox-SDCA [29] by reducing the stochastic variance using importance sampling.",
"The work of [2] improves over [38] by incorporating stochastic gradient descent and deep neural networks.",
"Also there are some work in using importance sampling for minibatch SGD [7], where they proposed to use importance sampling to do data sampling in minibatch SGD and this can improve the convergence rate of SGD.",
"The idea of hard negative example mining is also highly related to our work.",
"As shown in [30] where they presented an approach to perform efficient object detection training by training on an optimally sampled bounding boxes according to their gradient.",
"As for self-driving vehicles' vision system training, we typically do not know the ground truth distribution of the data, which are the images or video data captured by cameras.",
"Thus, importance sampling will be very useful in estimating the properties of the data from a data-driven sampling scheme.",
"The work of [36] and [12] proposed to use importance sampling for visual tracking, but their focus was not on reducing the training data amount and creating labeled data using visual tracking.",
"In our work, we use importance sampling to obtain an optimal set of data so that our training efficiency is high as we train on the most informative data.",
"The information that each image carries is characterized by their detection loss, which is reasonably suitable in our case as images with high loss are usually images that are difficult for the current detector."
],
[
"Methods",
"Our approach for creating labeled data and performing data reduction by importance sampling can be divided into two parts.",
"First of all, based on the sparsely labeled image frames, we initialize object tracker by incorporating Kalman-filter algorithm [37] and use the tracker to predict bounding box of objects in the previous (since we predict back in time) frame.",
"We then use the prediction as region proposal input and send this to object detection module.",
"Based on the region proposal received, the object detection module trained on sparsely labeled data will do a bounding box regression to get the final bounding box and detection loss.",
"The object tracker further matches new detections to existing trackers or create new trackers if the new detection cannot match any of the existing trackers.",
"The near-to-far labeling module will double check object detection results within each tracker to use the classification results of objects in the near field which are more accurate to check the results of objects in the far field.",
"The bounding box produced by the object detection module are used as labels for those unlabeled video frames.",
"The detection loss will be used as the sampling weights for importance sampling.",
"Secondly, based on the detection loss recorded in the first part, the importance sampler will sample an optimal subset of labeled images and these selected labeled images will be used as the training data to train a new object detector.",
"The system architecture is shown in figure REF .",
"Here, we first describe the framework of semi-supervised data labeling followed by the data reduction using the importance sampling framework.",
"[t] Description:Update object bounding box label for a single track-back-in-time step.",
"$\\mathcal {T}$ : the list of active trackers; $l_{\\epsilon }$ : threshold of detection loss, save a detection if the detection loss is larger than this threshold; $\\mathcal {K}$ : Kalman filter; $\\mathcal {D}$ : object detector; $n$ : limit of steps to retain a tracker; $\\mathcal {R}$ (collections of detections to be saved, a detection is a bounding box containing state information $(x,y,s,r, label)$ where $label$ is the class of the object) Initialize: $\\mathcal {R}=\\varnothing $ , $\\mathcal {R}^{\\prime }=\\varnothing $ , $\\mathcal {P}_m=\\varnothing $ , $\\mathcal {P}_m$ is the matched pairs of detection and tracker.",
"$\\forall $ $T$ $\\in $ $\\mathcal {T}$ /* Tracking back in time */ $s$ = predict_state($T$ , $\\mathcal {K}$ ); /* Use $s$ (prediction of RoI) as region proposal */ $d$ = get_new_detection($s$ , $\\mathcal {D}$ ); % Refer to section 3.1 $\\mathcal {R}^{\\prime }$ = $\\mathcal {R}^{\\prime }\\bigcup d$ ; $\\forall $ $d, T$ $\\in $ $\\mathcal {R}^{\\prime }, \\mathcal {T}$ Match $d$ , $T$ $d,T$ can match $\\mathcal {P}_m = \\mathcal {P}_m\\bigcup (d,T)$ ; $\\mathcal {R}^{\\prime }_{um}$ = $\\lbrace d; d\\in \\mathcal {R}^{\\prime }\\quad \\&\\quad d\\notin \\mathcal {P}_m\\rbrace $ (get unmatched $d$ ); $\\mathcal {T}_{um}$ = $\\lbrace T; T\\in \\mathcal {T}\\quad \\&\\quad T\\notin \\mathcal {P}_m\\rbrace $ (get unmatched $T$ ); $\\forall $ $(d, T)\\in \\mathcal {P}_m$ Update tracker $T$ using $d$ using Kalman Filter; Using historical records in $trk$ to check the accuracy of this new detection $d$ ; $\\mathcal {R} = \\mathcal {R}\\bigcup d$ ; $\\forall $ unmatched $d\\in \\mathcal {R}^{\\prime }_{um}$ $T$ = init_new_tracker($d$ ); $\\mathcal {R}$ =$\\mathcal {R}\\bigcup d$ ; $\\forall $ unmatched $T\\in \\mathcal {T}_{um}$ $T$ has not been updated for more than $n$ times Remove $T$ from $\\mathcal {T}$ $\\forall $ d $\\in \\mathcal {R}$ $d$ has loss $> l_{\\epsilon }$ mark this $d$ to be saved return $\\mathcal {R}$ Object Tracking and Labeling Algorithm [t] Description: Match detections with trackers.",
"$\\mathcal {R}^{\\prime }$ : object detection bounding boxes, $d=[x_1, y_1, x_2, y_2]$ ; $\\mathcal {T}$ : trackers; Matched detection and tracker pairs $\\mathcal {P}_m$ ; Unmatched detections $\\mathcal {R}^{\\prime }_{um}$ ; Unmatched Trackers $\\mathcal {T}_{um}$ ; len($\\mathcal {T}$ ) == 0 $\\mathcal {P}_m=\\varnothing $ ; $\\mathcal {R}^{\\prime }_{um}=\\mathcal {R}^{\\prime }$ ; $\\mathcal {T}_{um}=\\varnothing $ ; return $\\mathcal {P}_m, \\mathcal {R}^{\\prime }_{um},\\mathcal {T}_{um}$ .",
"$M$ = An all-zeros matrix of size $[len(\\mathcal {R}^{\\prime }), len(\\mathcal {T})]$ .",
"$i$ from 1 to len($\\mathcal {R}^{\\prime }$ ) $j$ from 1 to len($\\mathcal {T}$ ) $M[i,j]$ = IntersectionOverUnion($\\mathcal {R}^{\\prime }[i], \\mathcal {T}[j]$ ) $M2$ = linear_assignment($M[i,j]$ ); $i$ from 1 to len($\\mathcal {R}^{\\prime }$ ) $i\\notin M2[:,0]$ $\\mathcal {R}_{um}^{\\prime } = \\mathcal {R}_{um}^{\\prime }\\bigcup \\mathcal {R}^{\\prime }[i]$ $j$ from 1 to len($\\mathcal {T}$ ) $j\\notin M2[:,1]$ $\\mathcal {T}_{um} = \\mathcal {T}_{um}\\bigcup \\mathcal {T}[j]$ $i,j \\in M2$ $\\mathcal {P}_m = \\mathcal {P}_m\\bigcup (\\mathcal {R}^{\\prime }[i], \\mathcal {T}[j])$ .",
"return $\\mathcal {P}_m, \\mathcal {R}^{\\prime }_{um},\\mathcal {T}_{um}$ .",
"Match Detections with Trackers"
],
[
"Semi-supervised Data Labeling",
"Object Tracking.",
"Starting with a few sparsely annotated video frames, we first trained an object detection network using Faster-RCNN [26].",
"By using Kalman filter [37], we initialize object trackers with the ground truth labeled frames.",
"The specific object tracking framework we use is similar to that of [1], where the state of the tracker includes 7 parameters, namely, the center position of the bounding box $x,y$ , the scale $s$ (the area of the bounding box) and aspect ratio $r$ of the bounding box ( the ratio of the width over the height of the bounding box), and the rate of change of the center position $v_x, v_y$ and scale $v_s$ of the bounding box.",
"We follow the assumption in [1] that the aspect ratio of the bounding box does not change over time.",
"The measurement is just the first four parameters of the state vector.",
"$\\text{state} = [x, y, s, r, v_x, v_y, v_s] \\\\$ $\\text{measurement} = [x, y, s, r]$ We always use ground truth labeled bounding box to initialize object trackers, and the tracking is done from the near field to the far field, which means the video is played in the opposite direction as it was collected, so that at the very beginning, the camera is close to the labeled objects and at the very end the camera is far away from the the object.",
"Therefore, it is reasonably to believe that the classification and detection results for objects in the near field are more reliable while there is more noise in the detection results for object in the far field.",
"Prediction as Region Proposal.",
"After the trackers are initialized with ground truth bounding box, based on the principle of a Kalman filter, predictions of bounding boxes of the objects being tracked will be calculated.",
"These predictions will be used as a hint for the object detection network to produce new bounding boxes in the next frame.",
"The network we used for object detection is Faster-RCNN [26], which is composed of a region proposal network (RPN) and object detection network Fast-RCNN [15].",
"Usually the RPN will be used as the region proposer, however, as we already have the prior information of where the object might be, we can directly use this information to help the object detection network avoid uncertainty in region proposal.",
"This part corresponds to the get_new_detection method in algorithm .",
"Matching Tracker with Detections.",
"Given the predictions sent by the object tracker, the object detection network will produces a set of candidate bounding boxes in the next frame and the object tracker will try to match the existing trackers and the new detections using linear assignment.",
"We also use intersection over Union (IoU) to filter out detection-tracker pairs that do not have IoU values higher than a pre-defined threshold.",
"After finishing detection-tracker matching, the state of valid trackers will be updated, and trackers that remain inactive (not being updated) for a certain steps will be removed from the trackers list.",
"Now we finished one step of object tracking and labeling.",
"The bounding boxes produced by object detection network will be used as labels for those unlabeled video frames.",
"The more detailed algorithm description for one step of tracking and labeling is shown in algorithm .",
"Matching trackers with detections is further described in algorithm .",
"The tracker is a class containing state of the current object being tracked and methods for updating object's state given ground truth state of the object.",
"A detailed implementation of the tracker class can be found in [4].",
"Near-to-Far Labeling.",
"Another key ingredient of our approach is the near-to-far labeling scheme.",
"Consider the case that we are tracking an object from far to near field.",
"When the image is far away from our current location, the object could be very small or blurred in the image, which makes it very difficult to be correctly classified.",
"As the object approaches the vehicle, the detection network has a higher confidence to correctly classify this object.",
"As we trust object detection results in the near field, if object detection results of the same object being tracked in the far field differ from that in the near field, we can use the detection results in the near field to correct that.",
"To do this, we restrict object tracker's initialization only to ground truth bounding boxes so as to avoid the additional noise introduced by imperfect object detection network.",
"In case the classification of objects in the far field diverges, we use the detection result of the same tracker in the near field to correct that.",
"Examples of near-to-far labeling are shown in figure REF ."
],
[
"Sampling an Optimal Subset of Images",
"Inspired by the idea of importance sampling [2], we can select an optimal subset of the data by sampling the data according to importance sampling probability distribution so that the variance of the sampled data is minimized under an expected size of sampled data.",
"Here, the sampling distribution is proportional to the object detection loss of each image.",
"Images with higher loss obtain more importance as they provide more useful information for accurate object detection.",
"In our case, we are interested in estimating the expectation of $f(x)$ based on a distribution $p(x)$ , where $f(x)$ is the detection loss of each image, $p(x)$ denotes the image distribution and $x$ denotes a particular image with an object detection loss.",
"The problem is expressed by the following equation, $\\int {p(x)f(x)\\text{d}x} = \\mathbb {E}_{p(x)}[f(x)] \\approx \\frac{1}{N} \\sum _{n=1}^Nf(x_n),$ where $x_n \\sim p(x)$ .",
"However, usually we do not know the ground truth distribution of the data $p(x)$ , so we rely on a sampling proposal $q(x)$ to to unbiasedly estimate this expectation, with the requirement that $q(x) > 0$ whenever $p(x) > 0$ .",
"This is commonly known as importance sampling: $\\int {p(x)f(x)}dx = \\mathbb {E}_{p(x)}[f(x)] = \\mathbb {E}_{q(x)}[\\frac{p(x)}{q(x)}f(x)].$ It has been proved in [2] that the variance of this estimation can be minimized when we have, $q^{*}(x) \\propto p(x)|f(x)|.$ Defining $\\tilde{q^{*}}(x_i)$ as the unnormalized optimal probability weight of image $x_i$ , it is obvious that images with a larger detection loss should have a larger weight.",
"Although we do not know $p(x)$ , we have access to a dataset $\\mathcal {D} = \\lbrace x_n\\rbrace _{n=1}^{N}$ sampled from $p(x)$ .",
"Therefore, we can obtain $q^{*}(x)$ by associating the unnormalized probability weight $\\tilde{q^{*}}(x_n) = |f(x_n)|$ to every $x_n \\in \\mathcal {D}$ , and to sample from $q^{*}(x)$ we just need to normalize these weights: $q^{*}(x_n) = \\frac{\\tilde{q^{*}}(x_n)}{\\sum _{i=1}^N \\tilde{q^{*}}(x_i)}= \\frac{|f(x_n)|}{\\sum _{i=1}^N|f(x_i)|}$ where $f(x_i)$ is the loss of input $x_i$ .",
"To reduce the total number of data instances used for estimating $\\mathbb {E}_{p(x)}[f(x)]$ , we draw $M$ samples from the whole $N$ data instances ($M << N$ ) based on a multinomial distribution where $(q^{*}(x_1),...,q^{*}(x_N))$ are the parameters of this multinomial distribution.",
"Based on the discussion above, we obtained an estimation of $\\mathbb {E}_{p(x)}[f(x)]$ which has least variance compared to all cases where we draw $M$ samples from the entire $N$ data set.",
"We further provide some prove in the appendix.",
"Figure: Bounding box generated by using strategies mentioned in experiment 1(left 1 and 2) and experiment 2(right 1 and 2)Table: Object detection average precision (%) on KITTI dataset usingdifferent models.",
"Ground Truth: results of model trained on groundtruth labeled data (comes from KITTI).",
"New labeled and ground truth:model trained on both new labeled data and ground truth data, correspondingto experiment 1.",
"SampledNL and GT: model trained on importance sampled new labeled data and groundtruth data, corresponding to experiment 2.",
"Only NL: model trainedonly on new labeled data, corresponding to experiment 3.Figure: Plot of gradient Frobenius norm of last layerin VGG 16 versusthe gradient Frobenius norm of fully-connected layer 7 (FC7),fully-connected layer 6(FC6), Convolutional Layer 5 (Conv5)and Convolutional Layer 1 (Conv1).Figure: Examples of near-to-far labeling.",
"These images are from the KITTI Benchmark data set .",
"The labeling results are obtained from pre-trained Faster-RCNN model.",
"The bounding box shows the detected objects being tracked.",
"Near field object detection results are used to check the accuracy of the detection results of objects in the far field.",
"The first image labeling results from left to right: motorbike, car, car (ground truth).",
"As the vehicle approaches the object, it becomes clearer and no longer hidden by the pole.",
"The second image labeling results from left to right: bus, car, train (ground truth).",
"The object looks like a bus in the far field, but is classified as train when it's in the near field considering it's on railroad.",
"The third image labeling results from left to right: train, car, car (ground truth).",
"The object looks like a train in the far field, but is classified as car when it's in the near field.",
"Fourth image labeling results from left to right: bus, car, car (ground truth).",
"At first sight, the car is blurred and hidden by other objects, then it became more clearer that this is a car."
],
[
"Measuring Variance Reduction Efficiency",
"Once we get the sampling distribution $q^{*}(x_i)$ , we then perform the importance sampling.",
"Images with a higher detection loss will get higher likelihood to be sampled.",
"We, further, measure how efficient that we estimate the detection loss distribution.",
"Since the goal of using importance sampling approach here is to reduce the variance while estimating properties of the data from a subset of the data.",
"To show that the expectation of loss estimated from the sampled images have close variance with loss variance estimated from all images, we computed a relative variance value.",
"This value is the ratio of whole data set detection loss variance over sampled images' detection loss variance.",
"Suppose the data set is $\\mathcal {D}= \\lbrace x_n\\rbrace _{n=1}^N$ , and we can get detection loss $g(x_i)$ given individual input $x_i$ .",
"In order to calculate the relative variance more easily, we will first normalize $g(x)$ .",
"Then, we define the sampling probability of image $x_i$ when we expect to sample M out of N images ($M < N$ ) as, $ q(x_i) = \\min \\bigg [1, \\frac{M|g(x_i)|}{\\sum _{i=1}^N|g(x_i)|}\\bigg ]$ taking the minimum compared with 1 is to ensure that the probability of sampling image $x_i$ can not be larger than 1, which happens when $\\frac{M|g(x_i)|}{\\sum _{i=1}^N|g(x_i)|}$ is saturated.",
"Note that, when the sampling probability is 1, we should sample this image.",
"With the scaled sampling weight $\\frac{M|g(x_i)|}{\\sum _{i=1}^N|g(x_i)|}$ , we change $M$ so that we can get different numbers of images out of the entire image date.",
"Typically, choosing a $M$ such that the sample gradient norm variance is close to whole data gradient norm variance.",
"Since the data are in the discrete space, the relative variance is defined as, $\\begin{split}{\\rm R}& = \\frac{\\sum _{i=1}^N|g(x_i)|^2}{\\sum _{i=1}^N|g(x_i)|^2/q(x_i)}.\\\\\\end{split}$ Figure: Relative Variance Evaluation Results"
],
[
"Experiments",
"Our framework has several major contributions.",
"First of all, we proposed to use object tracker's prediction as the region proposal input for the object detection network to detect objects.",
"Secondly, we proposed to use near-to-far labeling to help correct labels that may not be correct.",
"Thirdly, we use importance sampling to select an optimal subset of images to remove images with less reliable labels and obtain a smaller but more informative set of data.",
"We designed several comparative experiments to show the impact of our contribution."
],
[
"Comparative Experiments",
"Datasets.",
"To show that our algorithm is able to scale to a relatively large video dataset, we choose the KITTI benchmark dataset [13] which contains hundreds of autonomous driving video clips, and each of the video clips lasts about 10 to 30 seconds.",
"The data set is fairly rich as it contains high-resolution color and grayscale video frames captured in many kinds of driving environments: city, residential, road, campus, person, etc.",
"The KITTI dataset also contains a set of sparsely labeled image frames for object detection purposes.",
"The number of images with ground truth bounding box labeling we used in our experiment is 7481, while the total number of images is around 40000.",
"Categories of objects being labeled include cars, pedestrians, vans, trams, cyclist, truck, person sitting, and so on.",
"For simplicity, we choose 3 categories from them to detect, which include cars, pedestrians, and cyclist.",
"We manually and randomly divide the dataset into the training, validation and test data set.",
"The training dataset contains 4206 images, the validation dataset contains 1404 images, and the test data set contains 1871 images.",
"Experiment 0.",
"We first trained a basic object detection network based on the ground truth labeled data using the Faster-RCNN [26] object detection network.",
"As for details of training, we used pre-trained Faster-RCNN model with VGG16 network [32] trained on PASCAL VOC 2007 dataset [11], and then finetuned with KITTI dataset.",
"The number of training iterations is 300k with the initial learning rate of 0.01 and decay every 30k iterations.",
"Experiment 1.",
"The first experiment is our labeling by tracking approach using semi-supervised learning.",
"In this experiment, we use the ground truth labeled bounding boxes to initialize object trackers.",
"Since images in the KITTI dataset are sparsely labeled with unlabeled images between labeled images in the original video sequence, we use the labeled data as a guidance to label images without ground truth labeling.",
"It is useful to notice that, in this case, the object detection network does not use RPN to generate region proposals.",
"Instead, it takes the object tracker's prediction of bounding box in the next frame as region proposal and then perform bounding box regression to generate optimal bounding box for the object being tracked.",
"In other words, only ground truth labeled images can be used to initialize object tracker, which is based on our assumption that objects in the near field provide more accurate information and we only predict bounding boxes based on reliable information instead of relying on some random detection.",
"We used both ground truth data from KITTI combined with new labeled data to train the object detector.",
"The training setting is the same as in experiment 0.",
"Experiment 2.",
"In this experiment, we adopt the approach we take in experiment 1 and we further combine it with importance sampling.",
"As images labeled using the approach in experiment 1 may still contain redundant information such as images that are already easy for the network to process, so we use importance sampling to select an optimal set of images that are more informative.",
"We choose to sample 60% of the data ( which consists of both ground truth and new labeled data) in experiment 1 using the importance sampling method mentioned in previous section.",
"As shown in figure REF , 60% of data corresponds to around 0.90 sampling efficiency, which is reasonably high.",
"The training setting is also the same as in experiment 0.",
"Experiment 3.",
"We further remove the ground truth data which comes from KITTI and only used newly labeled data using to train an object detector with the same training setting as in experiment 0.",
"Evaluation of Accuracy.",
"We trained Faster-RCNN object detection networks using data mentioned in experiment 1, 2, and 3 respectively, all using the same training configurations as we did in experiment 0.",
"We evaluate the performance of models in experiment 0,1,2,and 3 by testing the models on a held out test dataset of 1871 images.",
"The average precision is evaluated on the 3 categories of objects mentioned before."
],
[
"Results and Analysis",
"Loss as a Approximation for Gradient First, we show our finding that the gradient of the network we used has some linear correlation between different layers as shown in figure REF .",
"Therefore, we can use last layer gradient (as it is easier to obtain) as a approximation of total gradients.",
"On the other hand, we also show in figure REF that loss can be used as a approximation for the total gradient norm.",
"Therefore, we can also use loss to approximate gradient and use it as sampling weight for different object bounding box labels.",
"Figure: Plot of gradient Frobenius norm of entire networkweight (not include bias term) VS the loss of individual data point.Qualitative Results for Bounding Box Generation.",
"As mentioned in experiment description, we use two different strategies to generate bounding boxes using Faster-RCNN.",
"The first strategy uses region proposal network to generate bounding boxes, and the second strategy uses object tracker's prediction as region proposals.",
"We show some qualitative results of bounding boxes generated by the two methods in figure REF .",
"Quantitative Results for Object Detection.",
"The accuracy of models trained on experiment 0,1,2,and 3 are evaluated on a test data set of 1871 images.",
"The average precision (AP) on 3 categories of objects and the mAPs are reported in table REF .",
"The results show the average precision for different categories of objects with different degrees of difficulty.",
"With the ground truth data, the model shows the best performance, which is not a surprise since labels generated by tracking may introduce noise that harms the performance of the detector.",
"However, after filtering the data by importance sampling, we can obtain better detection accuracy using the same training setting, which means importance sampling has helped to reduce data volume and makes it easier to train a model to convergence.",
"Relative Variance Results We use the relative variance mentioned in section REF to measure how good we estimate the image detection loss distribution.",
"The result is shown here REF .",
"From the plot, we can see that by scaling the importance sampling weight as mentioned in REF , we are able to keep high sampling efficiency (0.90) with 60 % of the original labeled data being sampled.",
"This curve will be useful for determining how much data to sample given the desired sampling efficiency."
],
[
"Conclusion",
"We proposed a framework of automatically generating object bounding box labels for large volume driving video dataset with sparse labels.",
"Our work generates object bounding boxes on the new labeled data by employing a near-to-far labeling strategy, a combination of object tracker's prediction and object detection network and the importance sampling scheme.",
"Our experiments show that with our semi-supervised learning framework, we are able to annotate driving video dataset with bounding box labels and improve the accuracy of object detection with the new labeled data using importance sampling."
],
[
"Introduction",
"We provide proof of the importance sampling framework and their optimality in this supplementary material.",
"We also provide detailed explanations for the measurement of relative variance and the meaning of relative variance."
],
[
"Importance Sampling Framework Proof",
"The importance sampling algorithm is used for data reduction.",
"It is also used for the selection of an optimal subset of data from the original labeled dataset with minimal variance.",
"In the paper, we stated that by using a reference proposal distribution $q^{*}(x) \\propto p(x)|f(x)|$ we can get an estimation of the expectation of $f(x)$ with the least variance.",
"We now provide the proof.",
"In importance sampling, the expectation of $f(x)$ is estimated by using $\\mathbb {E}_{p(x)}[f(x)] = \\mathbb {E}_{q(x)}[f(x)p(x)/q(x)]$ .",
"We require that $q(x) >0$ whenever $f(x)p(x)\\ne 0$ .",
"It is thus easily to verify that this estimation is unbiased.",
"Suppose that $\\mathbb {E}_{p(x)}[f(x)]$ is defined on $x\\in A$ while $\\mathbb {E}_{q(x)}[f(x)p(x)/q(x)]$ is defined on $x\\in B$ .",
"We have $A = \\lbrace x| p(x) > 0\\rbrace $ and $B = \\lbrace x | q(x) > 0 \\rbrace $ .",
"So that we have for $x\\in A\\cap B^c$ , $f(x) = 0$ and for $x \\in A^c\\cap B$ , $p(x) = 0$ .",
"That is to say, for $x \\in A\\cap B^c$ and $x \\in A^c\\cap B$ , we have $f(x)p(x) = 0$ .",
"So the expectation of $f(x)$ can be written as, $\\begin{split}\\mathbb {E}_{q(x)}[\\frac{p(x)f(x)}{q(x)}]= &\\int _{B}\\frac{f(x)p(x)}{q(x)}q(x)\\text{d}x \\\\= & \\int _{A}f(x)p(x)\\text{d}x +\\int _{B\\cap A^c}f(x)p(x)\\text{d}x\\\\& -\\int _{A\\cap B^c}f(x)p(x)\\text{d}x \\\\= &\\int _{A} f(x)p(x)\\text{d}x \\\\= & \\mathbb {E}_{p(x)}[f(x)]\\end{split}$ Then we prove that when sampling distribution $q(x) \\propto p(x)|f(x)|$ , we can obtain the minimal variance in the estimation of the expectation.",
"Let $\\mathbb {E}_{p(x)}[f(x)] = \\mu $ , and let, ${\\mu }_q = \\frac{1}{n}\\sum _{i=1}^n \\frac{f(x_i)p(x_i)}{q(x_i)}$ given samples $x_i$ are sampled from $q(x)$ .",
"Then the variance of $\\mu _q$ is, $\\begin{split}\\mathrm {Var}({\\mu }_q)& = \\frac{1}{n}\\mathrm {Var}\\bigg (\\frac{f(x_0)p(x_0)}{q(x_0)}\\bigg ) \\\\& = \\frac{1}{n}\\bigg (\\int (f(x)p(x))^2/q(x) \\text{d}x - \\mu ^2\\bigg )\\\\\\end{split}$ By choosing $q^{*}(x) = |f(x)|p(x)/\\mathbb {E}_p(|f(x)|)$ , and let $q(x)$ be any density function that is positive given $f(x)p(x)\\ne 0$ .",
"We have, $\\begin{split}\\mathrm {\\mathrm {Var}}({\\mu }^{*}_q) & = \\frac{1}{n}\\bigg (\\int \\frac{(f(x)p(x))^2}{q^{*}(x)}\\text{d}x - \\mu ^2\\bigg ) \\\\& = \\frac{1}{n}\\bigg (\\int \\frac{(f(x)p(x))^2}{|f(x)|p(x)/\\mathbb {E}_p(|f(x)|)}\\text{d}x - \\mu ^2\\bigg ) \\\\& = \\frac{1}{n}\\bigg (\\mathbb {E}_p(|f(x)|)^2 - \\mu ^2\\bigg ) \\\\& = \\frac{1}{n}\\bigg (\\mathbb {E}_q(|f(x)|p(x)/q(x))^2 - \\mu ^2\\bigg ) \\\\& \\le \\frac{1}{n}\\bigg (\\mathbb {E}_q(f(x)^2p(x)^2/q(x)^2) - \\mu ^2\\bigg ) \\\\& = \\mathrm {\\mathrm {Var}}({\\mu }_q)\\end{split}$ The last inequality is the Cauchy-Schwarz inequality.",
"Therefore, we show that by choosing sampling distribution $q(x) \\propto p(x)|f(x)|$ and sampling data according to the normalized $q_{normalized}(x_i) = q(x_i)/\\sum {q(x_i)}$ , we can obtain the minimal variance estimation.",
"In the case where $p(x)$ is not known directly, but we have a dataset sampled from $p(x)$ , we can use $q(x_i) = |f(x_i)|/\\sum _i{|f(x_i)|}$ as the sampling weight."
],
[
"Measuring the Efficiency of Sampling",
"We define the efficiency as the ratio between the original data variance and the sampled data variance.",
"To make things simpler, suppose we want to estimate the expectation of $f(x)$ , we first normalize $f(x)$ and obtain $g(x) = (f(x) - \\overline{f(x)})/\\sigma {[f(x)]}$ , where $\\overline{f(x)}$ and $\\sigma {[f(x)]}$ are the mean and standard deviation of $f(x)$ .",
"Now we use importance sampling to estimate the expectation of $g(x)$ under $p(x)$ by using proposal distribution $q(x)$ .",
"We sample $M$ images out of a total $N$ images, the probability of a particular image $x_i$ being sampled is, $s(x_i) = \\min \\bigg [{1, \\frac{M|g(x_i)|}{\\sum _i^{N}{|g(x_i)|}}}\\bigg ]$ As mentioned in the paper, we take the minimum compared with 1 to ensure that the probability is always no more than 1.",
"Obviously, $\\sum _{i=1}^N s(x_i)>1$ since $s(x_i)$ describes the probability of a particular image $x_i$ being selected.",
"We further define $q(x_i) = s(x_i)/N$ which has an upper bound of $1/N$ .",
"Therefore, it is easy to see that $\\sum _{i=1}^Nq(x_i) \\le 1$ .",
"To get $M$ images, we select images according to their sampling probabilities $s(x_i)$ .",
"The expectation of $g(x)$ based on the sampled images is, $\\mathbb {E}_{q(x)}[g(x)p(x)/q(x)] =\\frac{1}{N} \\sum _{i=1}^Ng(x_i)p(x_i)/q(x_i)$ where $x_i \\sim q(x)$ .",
"On the other hand, if we sample the entire dataset and get $N$ images, then $s(x_i) = 1$ and $q(x_i) = 1/N$ , the expectation will be, $\\mathbb {E}_{q(x)}[g(x)p(x)/q(x)] = \\frac{1}{N}\\sum _{i=1}^Ng(x_i)p(x_i)*N$ which is just $\\mathbb {E}_{p(x)}[g(x)]$ .",
"It is no harm to assume $p(x)$ is a uniform distribution since we consider it to be unknown.",
"In the case where we sample the entire dataset, $s(x_i)= 1$ , $p(x_i) = 1/N$ , and $\\sum _{i=1}^Ng(x_i)=0$ , then the variance of $g(x)$ by sampling the entire dataset is, $\\begin{split}\\mathrm {Var}_{q}\\bigg [\\frac{g(x)p(x)}{q(x)}\\bigg ] = & \\mathbb {E}_{q}\\bigg [\\big (\\frac{g(x)p(x)}{q(x)}\\big )^2\\bigg ] -\\bigg (\\mathbb {E}_{q}\\bigg [\\frac{g(x)p(x)}{q(x)}\\bigg ]\\bigg )^2 \\\\= & \\sum _{i=1}^N\\bigg [\\frac{g(x_i)p(x_i)}{s(x_i)/N}\\bigg ]^2\\frac{s(x_i)}{N} \\\\& - \\bigg (\\sum _{i=1}^N\\bigg [\\frac{g(x_i)p(x_i)}{s(x_i)/N}\\bigg ]\\frac{s(x_i)}{N}\\bigg )^2 \\\\= & \\sum _{i=1}^N\\bigg [\\frac{g(x_i)/N}{1/N}\\bigg ]^2\\frac{1}{N} \\\\& - \\bigg (\\sum _{i=1}^Ng(x_i)/N\\bigg )^2 \\\\= & \\sum _{i=1}^Ng^2(x_i)/N\\end{split}$ In the case where we sample $M$ images out of $N$ images, $s(x_i) \\le 1$ , $q(x_i) = 1/N$ , and $\\sum _{i=1}^Ng(x_i)=0$ , then the variance of $g(x)$ by sampling $M$ images out of $N$ images is, $\\begin{split}\\mathrm {Var}_{q(x)}\\bigg [\\frac{g(x)p(x)}{q(x)}\\bigg ] = & \\sum _{i=1}^N\\bigg [\\frac{g(x_i)p(x_i)}{s(x_i)/N}\\bigg ]^2\\frac{s(x_i)}{N} \\\\& - \\bigg (\\sum _{i=1}^N\\bigg [\\frac{g(x_i)p(x_i)}{s(x_i)/N}\\bigg ]\\frac{s(x_i)}{N}\\bigg )^2 \\\\= & \\sum _{i=1}^N\\bigg [\\frac{g(x_i)/N}{s(x_i)/N}\\bigg ]^2\\frac{s(x_i)}{N} \\\\& - \\bigg (\\sum _{i=1}^Ng(x_i)/N\\bigg )^2 \\\\= & \\sum _{i=1}^N\\frac{g^2(x_i)}{s(x_i)*N}\\end{split}$ The efficiency is defined as the ratio between REF and REF , which is, $\\begin{split}R = \\frac{\\sum _{i=1}^Ng^2(x_i)}{\\sum _{i=1}^Ng^2(x_i)/s(x_i)}\\end{split}$ which is the same as the efficiency (relative variance) defined in the main paper.",
"Obviously, since $s(x_i) \\le 1$ , this ratio will always be no larger than 1.",
"If we sample all the data, which means $s(x_i) = 1$ , then we can obtain a sampling efficiency of 1.",
"To simplify the calculation of $R$ , We can further express $\\sum _{i=1}^Ng^2(x_i)/s(x_i)$ as, $\\begin{split}\\sum _{i=1}^N\\frac{g^2(x_i)}{s(x_i)} & = \\sum _{j=1}^k\\frac{\\sum _{i=1}^N|g(x_i)|}{M|g(x_j)|}|g(x_j)|^2 + \\sum _{j=k+1}^N|g(x_j)|^2 \\\\& = \\frac{\\sum _{i=1}^N|g(x_i)|}{M}(\\sum _{j=1}^k|g(x_j)|) + \\sum _{j=k+1}^N|g(x_j)|^2 \\\\\\end{split}$ where $s(x_1)$ , $s(x_2)$ , $\\cdots $ , $s(x_k)$ are smaller than 1 and $s(x_{k+1})$ , $\\cdots $ , $s(x_N)$ are equal to 1."
]
] | 1808.08603 |
[
[
"Diagnostics of ionized gas in galaxies with the \"BPT--radial velocity\n dispersion\" relation"
],
[
"Abstract In order to study the state of gas in galaxies, diagrams of the relation of optical emission line fluxes are used allowing one to separate main ionization sources: young stars in the H II regions, active galactic nuclei, and shock waves.",
"In the intermediate cases, when the contributions of radiation from OB stars and from shock waves mix, identification becomes uncertain, and the issue remains unresolved on what determines the observed state of the diffuse ionized gas (DIG) including the one on large distances from the galactic plane.",
"Adding of an extra parameter - the gas line-of-sight velocity dispersion - to classical diagnostic diagrams helps to find a solution.",
"In the present paper, we analyze the observed data for several nearby galaxies: for UGC 10043 with the galactic wind, for the star forming dwarf galaxies VII Zw 403 and Mrk 35, for the galaxy Arp 212 with a polar ring.",
"The data on the velocity dispersion are obtained at the 6-m SAO RAS telescope with the Fabry-Perot scanning interferometer, the information on the relation of main emission-line fluxes - from the published results of the integral-field spectroscopy (the CALIFA survey and the MPFS spectrograph).",
"A positive correlation between the radial velocity dispersion and the contribution of shock excitation to gas ionization are observed.",
"In particular, in studying Arp 212, \"BPT-sigma relation\" allowed us to confirm the assumption on a direct collision of gaseous clouds on the inclined orbits with the main disk of the galaxy."
],
[
"Introduction",
"Diagrams of ratios of optical emission-line fluxes are widely used for diagnostics of gas-ionization sources in galaxies.",
"In the classic work by [6] the two-dimension diagram of line fluxes of [O III]$\\lambda 5007$ /H$\\beta $ and [N II]$\\lambda 6583$ /H$\\alpha $ was suggested for separation of objects with different ionization sources.",
"The method became popular due to the use of measurements of lines bright in the visible range that are close in wavelengths and, consequently, with a weak dependence of their intensity ratio on the interstellar extinction.",
"Later, this method was extended by adding the relations [S II]/H$\\alpha $Hereinafter, for short we will designate [O I]$\\lambda 6300$ as [O I], [O III]$\\lambda 5007$ as [O III], [N II]$\\lambda 6583$ as [N II], and [S II]$\\lambda 6717$ +[S II]$\\lambda 6731$ as [S II].",
"and [O I]/H$\\alpha $ [31], [16] as the second parameter.",
"All the mentioned diagrams are frequently called in the literature the “BPT diagrams” after the authors of the method.",
"Using them, it is possible to distinguish the regions, where the largest contribution to gas ionization is made by young massive stars (hereinafter, the H II type) and the regions of dominating hard ionizing radiation of the active galactic nucleus (AGN).",
"At the same time, the regions ionized by shock waves, the asymptotic giant branch (AGB) stars or nuclei of galaxies of the LINER type mix in the diagrams Low-Ionization Narrow Emission-line Region in which the shock ionization of gas can be associated both with a burst of star formation and with a weak nuclear activity..",
"Various variants of the demarcation lines were suggested [27], [13], but it is often problematic to separate the contribution of ionizing sources with the soft spectrum.",
"Addition of one more parameter – the velocity dispersion of the ionized gas along the line of sight ($\\sigma $ ) – to the classic diagnostic diagrams allows one to escape uncertainty in the cases, when the increase of $\\sigma $ indicates the increase of turbulent velocities of gas beyond the front of a shock wave.",
"However, to estimate $\\sigma $ reliably, the spectral resolution is necessary that is noticeably better than that usually required for measuring the fluxes of radial velocities of separate spectral lines.",
"Thus, until recently, the dependence of the relation of line fluxes characterizing the shock ionization from $\\sigma $ was rarely considered and mainly for the objects with $\\sigma >100$ –$200\\,\\mbox{km s$^{-1}$}$ such as galaxies with intense star formation [27], [13].",
"Such an approach has not previously been used to study the ionization of diffuse gas in dwarf galaxies, around separate star-forming regions, or at some distance from the plane of the galactic disk.",
"There is a discussion about the sources of ionization of this diffuse ionized gas (DIG) in galaxies whose role is assigned to an old stellar population, leakage of Lyman quanta from H II regions, and also possibly to shock fronts caused by star formation processes [14], [11].",
"The most effective methods for studying the extended low-brightness structures in galaxies are panoramic spectroscopy also called integral-field, or 3D.",
"In a recent paper based on the results of the SDSS MaNGA survey by [32], it was concluded that DIG is associated mainly with the evolved stellar population (AGB stars, etc.).",
"At the same time, in Section 6.2 of the paper cited, it was noted that shock waves can be the cause of the observed increase in the flux ratio of the forbidden and Balmer lines.",
"It is difficult to verify, since the spectral resolution of the MaNGA survey is about two times poorer than that required to be able see the effects of moderate shock waves (with a velocity of less than 500 km s$^{-1}$) in the observed kinematics of the ionized gas.",
"Unfortunately, most of the available observed data on spectrophotometry and kinematics of the gas of nearby galaxies are obtained with the spectral resolution $FWHM>5$ Å which corresponds to values greater than 100 km s$^{-1}$ in terms of radial velocity dispersion or greater than 230 km s$^{-1}$ in terms of the $FWHM$ in the H$\\alpha $ line.",
"Observations with such resolution are a compulsory compromise in the study of low-surface-brightness objects.",
"In the SAMI survey of galaxies [13] with the 3D spectroscopy at the 3.9-m Anglo-Australian Telescope (AAT), the “line ratio–velocity dispersion” diagrams were built for the galaxies with active star formation.",
"A positive correlation of the ionized gas $\\sigma $ with a characteristic emission lines ratios was noticed, which was interpreted as an increase of shock waves contribution with velocities of about 200-300 km s$^{-1}$ accompanying a burst of star formation.",
"The spectral resolution of the SAMI survey is greater than that of MaNGA and equals $R\\approx 4500$ .",
"A significant limitation of these two most massive today 3D spectroscopy surveys of galaxies is rather low spatial resolution (more than 1 kpc).",
"It is related to the fact that the field of view of integral field unit (IFU) is small and is about 15 in SAMI [9] and 12–32 in SDSS MaNGA [7].",
"In these surveys, relatively distant ($z>0.01$ ) galaxies are studied.",
"At the same time, the largest contribution to the kinematics of interstellar medium from motion due to supernovae and winds of young stars in star-forming regions is made on considerably smaller spatial scales (from tens to hundreds of parsec).",
"Consequently, any observed manifestations of shock fronts in star-forming regions become unevident, when averaging over a scale of one kpc or more.",
"The examples of decreasing of peak velocity dispersion of the ionized gas in dwarf galaxies with degradation of a spatial resolution are presented in [25].",
"[30] considered the same effect in simulations of multiple supernova explosions.",
"Therefore, for observational studies of the relation between an ionization state of gas and dispersion of its radial velocities in galaxies without an active nucleus and with a moderate star formation rate, 3D spectroscopic data are required simultaneously with a considerably high spectral and spatial resolution.",
"In this paper, we consider this relation for several nearby galaxies using a combination of two spectroscopic methods with similar spatial resolution and quite a large field of view.",
"Velocity dispersion map are derived from the observations with a scanning Fabry-Perot interferometer (FPI) at the 6-m SAO RAS telescope.",
"Information on the main emission lines ratios are taken from open data on the integral-field spectroscopy with low spectral resolution.",
"In order to show the relation between the velocity dispersion and the lines ratios characterizing the ionization state, we use various methods through our paper: coloring in BPT diagrams, the “$\\sigma $ –line-flux relation” diagrams, and “$\\sigma $ –distance from the H II/AGN demarcation line.” As a general name for the dependencies under study, we use the term “BPT relation–$\\sigma $ ”.",
"Classical BPT–diagrams are two-dimensional plots, where the axes represent relations of line fluxes.",
"The inclusion of the velocity dispersion in the analysis is equivalent to the transition to three-dimensional plots, where a coordinate axis $\\sigma $ is added to each diagram.",
"The more familiar two-dimensional plots given in our paper and in the above papers are a projection of the BPT–$\\sigma $ common relation to the selected plane.",
"Table: Characteristics of the galaxies under study and parameters of their observations with different methodsWe considered the sample of of galaxies with data on a ionized gas state obtained with two 3D spectroscopy methods.",
"When compiling the sample, we prepared a list of nearby galaxies for which, based on observations with the scanning FPI at the 6-m telescope of SAO RAS, the fields of velocity dispersion of ionized gas in the H$\\alpha $ or [N II] emission lines were constructed.",
"In total, this is about 60 objects that were observed in 2002–2015; most of them are presented in [26].",
"For each galaxy in the list, we checked the presence of data cubes in open sources obtained with the integral-field spectroscopy.",
"For three galaxies: Arp 212, Mrk 35, and UGC 10043 such data were obtained within the framework of the CALIFA survey [29].",
"We used the third data release of CALIFA; the spectra are available on the project websitehttp://califa.caha.es/.",
"Let us notice that UGC 10043 was observed with the 6-m telescope specially on request of the CALIFA team.",
"The results were presented in our collaborative paper [20] and triggered our interest for further study of the BPT–$\\sigma $ relation.",
"Nevertheless, to keep homogeneity, we repeated the analysis of the UGC 10043 data in this paper using the same methods as for other galaxies.",
"For the galaxy VII Zw 403, there was a data cube obtained by combining several fields of the MPFS spectrograph in observations at the 6-m telescope of SAO RAS and published by [4].",
"Table REF briefly presents the objects under study (accepted distance $D$ and absolute magnitude $M_B$ based on the NED data) and on the data used (observation instruments, $\\Delta \\lambda $ –spectral range or a selected line, $\\delta \\lambda $ –spectral resolution in terms of $FWHM$ , $\\theta $ –angular resolution).",
"Figure REF shows the images of the sample galaxies in the $r$ filter and in the emission lines, and velocity dispersion fields $\\sigma $ of the ionized gas constructed from the scanning FPI data.",
"It also shows the field of view of the spectrographs used.",
"Figure: The images of the studied galaxies.",
"The left-hand columnshows the SDSS rr-filter imagesfrom , for UGC 6456 the image in theRR filter is given from .",
"Themiddle column gives the images in the [N II]λ6583\\lambda 6583 line(UGC 10043) or Hα\\alpha line (for others) from the observed data withthe FPI at the 6-m BTA telescope.",
"The right-hand column shows thevelocity dispersion of the ionized gas; the scale in km s -1 ^{-1}.",
"Imagesize is 90 '' ×90 '' 90^{\\prime \\prime }\\times 90^{\\prime \\prime }.",
"The arrangement of the fields of view ofintegral-field spectrographs is shown: the CALIFA survey and theMPFS mosaic (for UGC 6456).",
"The velocity dispersion fields aretaken from and (for UGC 10043) withoutcorrection for the thermal line broadening."
],
[
"Low-Resolution Spectra",
"For UGC 10043, Mrk 35, and Arp 212, we used the CALIFA data obtained at the 3.5-m telescope of the Calar Alto observatory in the mode of the integral-field spectroscopy of the PPAK wide field [15] of the PMAS spectrograph [28].",
"The array of PPAK optical fibers comprises 331 elements of the 27 diameter collected in the $74^{\\prime \\prime }\\times 64^{\\prime \\prime }$ hexagonal field.",
"We used the cubes obtained in the low-resolution mode covering the entire visible range (grating V500, $R\\sim 850$ ).",
"The reduces data are presented in the form of cubes extrapolated from a hexagonal grid to a square grid with a spatial element size (spaxel) of $1^{\\prime \\prime }$ .",
"The galaxy VII Zw 403 was observed with the MPFS multislit spectrograph [1] at the SAO RAS 6-m telescope.",
"An array of square lenses combined with fiber optics provided a field of view of $16\\times 16$ elements with a scale of $1^{\\prime \\prime }$ per lens.",
"The data cube presented in [4] is a mosaic of the size of $49^{\\prime \\prime }\\times 31^{\\prime \\prime }$ comprising seven MPFS fields.",
"The spectral range was 4250–7200 Å; the resolution was 8 Å.",
"For the analysis, we used the data cubes with the $2\\times 2$ binning to a scale of $2^{\\prime \\prime }$ per element (see next Section REF ).",
"Approximation of the lines in the spectra was carried out by the one-component Gaussian function.",
"The line fluxes ratios were measured only from the spectra in which $S/N>2$ for each emission line.",
"The local continuum level near each line was taken into account."
],
[
"Kinematics of the Ionized Gas",
"The archival observations with the scanning Fabry-Perot interferometer installed at the SCORPIO [2] and SCORPIO-2 [3] focal reducers in the primary focus of the 6-m telescope were used to create the velocity dispersion maps.",
"The emission line (H$\\alpha $ or [N II]$\\lambda 6583$ ) was selected with the a narrow-band filter with a bandwidth $\\sim 20$ Å. UGC 10043 was observed with the IFP186 interferometer providing the spectral resolution $FWHM=1.7$ Å.",
"In the study of other galaxies, IFP501 with the resolution $FWHM=0.8$ Å was used.",
"In observations of VII Zw 403, the image scale was 056 px$^{-1}$ with the field of view of 48, for other galaxies: 07 px$^{-1}$ and 61, respectively.",
"The result of the reduction of the set of interferograms was a data cube, where each pixel contained the spectrum of the selected emission line consisting of 36–40 channels.",
"The details of data reduction and observational logs were published earlier (see references in Section ).",
"Since the instrumental contour of the interferometer is well described with the Lorentz profile, the observed profiles of emission lines were approximated by the Voigt function — the convolution of the Gauss and Lorentz functions [24].",
"It is assumed that the initial (without any instrumental broadening) profile of an emission line is satisfactorily described by a Gaussian, which is a good approximation for observations of H II regions, with the exception of individual peculiar cases (expanding envelopes, neighborhood of Wolf-Rayet stars, etc.",
"[25], [11]).",
"Based on the results of approximation, we built the monochromatic images in this line, the distribution of radial velocities of the ionized gas, and the radial velocity dispersion maps free from the instrumental broadening of the spectral line profile [24].",
"The $\\sigma $ maps in Fig.",
"REF are shown with the original sampling of the FPI images (06–07) which is better than in the data used in the integral-field spectroscopy ($1^{\\prime \\prime }$ /spaxel), while the angular resolution $\\theta $ of both data sets is similar (see Table REF ).",
"To account for this effect, first we interpolated the FPI cube to a coarser grid corresponding to the CALIFA or MPFS data.",
"The accuracy of coincidence of both data sets was controlled from the images in the emission lines and continuum and was better than 02–05.",
"In order to detect the emission lines in the low-surface-brightness regions in the optimum way, we performed the pixel binning of $2\\times 2$ for both combined data sets.",
"This procedure also reduced possible errors of a small difference between the angular resolution of the FPI data and integral-field spectroscopy.",
"Therefore, all the measurements presented below are performed from the cubes with the $2^{\\prime \\prime }$ element size.",
"After combining and binning in the FPI cubes, we built the maps $\\sigma $ .",
"In the next section, these maps were used for direct per-pixel comparison with the low-resolution spectra.",
"We used masking to highlight the points with the ratio $S/N\\ge 3$ in the maps $\\sigma $ .",
"Let us note that, as distinct from [26], we did not correct the velocity dispersion maps for thermal line broadening.",
"Figure: Upper row: BPT diagrams for UGC 10043.",
"The ionized gasline-of-sight velocity dispersion in the given pixel according tothe map shown bottom right is colored.",
"The lines separating theH II regions, the objects with the combined ionization typeactive Seyfert galaxies, and LINER are taken from.",
"The other diagrams: thedependence between the velocity dispersion and the emission linesratios.Figure: UGC 10043.",
"Dependence of σ\\sigma on the distanceof the point to the demarcation curve in the BPT diagramseparating the H II regions and regions with other ionizationmechanisms ."
],
[
"Galactic Wind in UGC 10043",
"UGC 10043 is an edge-on spiral galaxy.",
"Observations in the H$\\alpha $ and [N II] lines carried out with the HST [22] have shown signs of star formation in the galactic core, as well as an extended emission structure that is perpendicular to the disk and is the result of the galactic wind influence.",
"[20] presented diagnostic diagrams for the central part of the galaxy according to the CALIFA survey.",
"Some of the points on the diagram relating to the central region of star formation turned out to be located in the region characteristic of photoionization by young stars, while others fell into the region typical of shock excitation.",
"Within the framework of a shock excitation model, a wind velocity was constrained: no greater than 400 km s$^{-1}$.",
"Analyzing the gas velocity field in the [N II] line constructed with a scanning FPI allowed us to obtain a more strict limitation on the galactic wind velocity: less than 250 km s$^{-1}$in accordance with the gas shock excitation model.",
"In the same paper [20], it has been shown that there is a distinct relation BPT–$\\sigma $ in the wind nebula of UGC 10043.",
"As it is shown in the diagnostic diagrams presented in Fig.",
"REF (the upper row), the regions with shock excitation of emission lines in the wind nebula are characterized by a higher velocity dispersion as compared to the regions dominated by photoionization.",
"In this case, there is a positive correlation between the relations of line fluxes of [S II] to H$\\alpha $ , of [N II] to H$\\alpha $ , of [O I] to H$\\alpha $ and $\\sigma $ (see Fig.",
"REF ).",
"At the same time, negative correlation is observed between the ratio of the sulfur doublet lines ([S II]6731/[S II]6717) and $\\sigma $ .",
"This means that a higher velocity dispersion is characteristic of the diffuse gas with a lower electron density $n_e$ .",
"We tried to quantify the BPT–$\\sigma $ relation.",
"For each point in the diagrams of the lines ratios, it is possible to determine the minimum distance to the curve that bounds the H II-type ionization region from [16] (in the case of the [N II]/H$\\alpha $ –[O III]/H$\\beta $ diagram, this is the boundary between the Comp and AGN regions in our figures).",
"We marked this distance as $\\rho $ and determined it so that negative values of $\\rho $ corresponded to the shift of the points from the demarcation line to the side corresponding to photoionization by young stars, and positive–towards other ionization mechanisms.",
"Figure REF shows the examples of relations involved this parameter.",
"For brevity and convenience of reading, we have designated the value $\\rho $ for the [N II]/H$\\alpha $ –[O III]/H$\\beta $ diagrams as $\\rho ([{\\rm N\\,II}])$ , for the [S II]/H$\\alpha $ –[O III]/H$\\beta $ diagrams as $\\rho ([{\\rm S\\,II}])$ , and for the [O I]/H$\\alpha $ –[O III]/H$\\beta $ diagrams as $\\rho ([{\\rm O\\,I}])$ .",
"It can be seen that in all the cases presented, the increase in the velocity dispersion along the line of sight correlates with the distance from the region characteristic of ionization by young stars in the BPT diagram.",
"Figure: Same as in Fig.",
", forVII Zw 403.Figure: Diagrams similar to those in Fig.",
", forVII Zw 403.Figure: Same as in Fig.",
", for Mrk 35.Figure: Diagrams similar to those Fig.",
", for Mrk 35.Figure: Same as in Fig.",
", for Arp 212.Figure: Diagrams similar to those Fig.",
", for Arp 212."
],
[
"Mrk 35 and VII Zw 403—Dwarf Galaxies with a Burst of Star Formation",
"VII Zw 403 is one of the nearest blue compact dwarf galaxies with several episodes of recent star formation.",
"The current outburst is located in the central kpc, where several compact OB-stars associations are identified and the associated H II shells that are immersed in the diffuse ionized gas [10].",
"The fields of velocities and velocity dispersions of the ionized gas in this galaxy were previously considered in [21], [25], [26], where a sufficiently quiet kinematics of the gas with a low level of peculiar velocities was noticed.",
"The value $\\sigma $ is in the range of 15–40 km s$^{-1}$.",
"In the BPT diagrams, most points are located in the region of photoionization (see Fig.",
"REF ).",
"A certain number of points with higher dispersion are found near the separation curve.",
"Along with this, the expanding H II shells associated with bright star formation regions are characterized by a smaller value of $\\sigma \\sim 20$ km s$^{-1}$.",
"One of the regions with a higher $\\sigma $ is located between these two shells.",
"Others are located on the periphery of the ionized gas disk [25].",
"In the “lines ratio–velocity dispersion” diagrams, there are no noticeably significant correlations (Fig.",
"REF ).",
"Therefore, we can conclude that the contribution of shock excitation to gas ionization in this galaxy is negligible and even at the boundaries of the expanding shells it is noticeably inferior to photoionization (the H II type).",
"This is also indicated by the absence of significant correlation between $\\rho $ and $\\sigma $ in Fig.",
"REF .",
"Mrk 35 is another example of a blue compact galaxy.",
"The ongoing star formation here is concentrated in several bright compact regions.",
"Star-forming regions near the optical center of the galaxy form a bar-like structure, where the population of Wolf–Rayet stars is observed [8].",
"The radial velocity dispersion of the ionized gas in the galaxy reaches about 70 km s$^{-1}$, whereas in the central regions it lies within the range of 20–35 km s$^{-1}$.",
"The highest dispersion of radial velocities is observed in the gas located between three central regions of star formation.",
"In the “arms”, the dispersion is several times lower in comparison with the central regions; and as a whole does not exceed 20 km s$^{-1}$.",
"In the BPT diagrams (see Fig.",
"REF ), the points corresponding to the regions with the ongoing star formation are located in the region of photoionization.",
"The outer parts of the galaxy, characterized by low surface brightness and high dispersion of radial velocities, appear near the separation curves which suggests a certain contribution of shock waves to the gas ionization in these regions.",
"As well as in UGC 10043, the sulfur lines ratio in Fig.",
"REF demonstrates the anticorrelation.",
"The $\\sigma $ –$\\rho $ diagrams show a positive correlation between the distance to the model curve and the velocity dispersion (Fig.",
"REF )."
],
[
"Arp 212 – a Polar Ring Galaxy",
"Arp 212 is a peculiar galaxy in which two rotating gas subsystems that are kinematically different have been discovered: an internal disk of the 3.5-kpc size and outer H II regions whose orbits are inclined at a significant angle to the stellar disk [23].",
"The observed picture was explained in the assumption that the gas (mostly neutral) in the outer regions of the galaxy is located in a wide ring of a diameter of about 20 kpc rotating in the plane almost orthogonal to the disk.",
"As the radii of the gaseous-cloud orbits decrease, their inclination angle decreases too; and at a radius of 2–3 kpc, the gas from the ring begins to fall out onto the plane of the galaxy inducing a burst of star formation.",
"It is the region with the highest observed velocity dispersion reaching 80–100 km s$^{-1}$(see Fig.",
"REF ).",
"The points belonging to this collision region of the gas subsystems are shifted in the BPT diagrams (Fig.",
"REF ) from the regions dominated by photoionization towards the dominance of shock ionization.",
"At the same time, photoionization clearly dominates in the central region of the galaxy.",
"As well as in UGC 10043 and Mrk 35, there is a positive correlation between $\\sigma $ and $\\rho $ for all the BPT diagrams (Fig.",
"REF ).",
"It is important to notice that in all three galaxies this dependence can be observed for the velocity dispersion above 30–40 km s$^{-1}$and practically disappears for smaller $\\sigma $ .",
"In other words, the correlation between $\\sigma $ and $\\rho $ manifests itself in the presence of shock excitation in the diffuse gas (DIG) and disappears in the H II regions characterized by a low level of turbulent motions.",
"This can also be confirmed by the absence of distinct $\\sigma $ –$\\rho $ correlations in the galaxy VII Zw 403, where in all the points $\\sigma <40$ km s$^{-1}$.",
"As distinct from UGC 10043 and Mrk 35, the relation of sulfur lines ratio in Arp 212 does not show any pronounced dependence on $\\sigma $ which agrees with the assumption that high velocity dispersion is observed not only in DIG with low electron density but also in a denser medium of colliding gaseous clouds."
],
[
"Conclusion",
"For an observational study of the relation between turbulent motions of the ionized gas in nearby galaxies and the state of its ionization, it is required to have panoramic spectroscopy data together with a large field of view and quite high spectral resolution.",
"Since it is necessary to observe the low-surface-brightness region with an angular resolution of about $1^{\\prime \\prime }$ , then an optical telescope of a large ($D>3$ –5 m) diameter is needed.",
"All these requirements are implemented together probably only in the unique MUSE instrument at the 8-m VLT telescope [5].",
"Our idea is to combine the ionization ionized gas velocity dispersion maps obtained in the observations with the scanning FPI and the panoramic spectrophotometry data for low-spectral-resolution galaxies.",
"The observed line-of-sight velocity dispersion characterizing the turbulent motions of the ionized gas can be due to various causes such as virial motions in the galaxy's gravitational potential, the effect of expanding shells on the gas, or, more generally, energy injected into the interstellar medium by star-forming processes [26], [19].",
"Various factors influence the value of line flux relations with different excitation mechanisms.",
"Observational information fusion makes it possible, in certain cases, to draw unambiguous conclusions about contribution of shock waves to the gas ionization in low-surface-brightness regions.",
"From the lines flux ratio in the conical nebula in UGC 10043 only, it cannot be definitely concluded what leads the growth of the relative intensity of the forbidden lines: ionization by shock waves from the central burst of star formation or the old stellar population of the thick disk, in which it is located.",
"Additional information on the gas kinematics allows to say that there is a galactic wind.",
"For Arp 212, our approach allowed to confirm the previously assumption in [23] on the direct collision of gaseous clouds on inclined orbits with the main disk of the galaxy generating shock fronts.",
"Thus, the use of the BPT–$\\sigma $ diagram together with the classical diagnostic methods based on lines ratios helps us better understand of ionization of the galactic interstellar medium in each specific case.",
"The only galaxy in which we did not find a correlation between $\\sigma $ and characteristic line flux relations (or $\\rho $ parameter) is VII Zw 403.",
"The ongoing star-formation rate here is the lowest in our sample [21].",
"Apparently, for this reason, the contribution of shock waves to gas ionization is practically invisible.",
"We plan to conduct further expansion of the sample of the objects under study in two ways.",
"The first is new observations with a high spectral resolution of galaxies, for which there are the CALIFA survey data already, with the scanning FPI.",
"The second is the creation of images in the emission lines of galaxies, for which we already have maps of the velocity dispersion of the ionized gas.",
"Here it is proposed to use a tunable–filter photometer, the first observations with which are already being conducted by our teamhttps://www.sao.ru/Doc-en/Events/2017/Moiseev/moiseev_eng.html .",
"The work was supported by the Russian Science Foundation (project No.",
"17-12-01335 “Ionized gas in galactic disks and beyond the optical radius.” The paper used the survey data provided by the Calar Alto Legacy Integral Field Area (CALIFA) survey ( http://califa.caha.es/) based on the observations collected at the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto operated jointly with the Max-Planck-Institut für Astronomie and the Instituto de Astrofísica de Andalucía (CSIC).",
"This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.",
"The authors are grateful to Alexandrina Smirnova and the reviewer for constructive comments."
]
] | 1808.08525 |
[
[
"xFitter 2.0.0: Heavy quark matching scales: Unifying the FFNS and VFNS"
],
[
"Abstract xFitter is an open-source package that provides a framework for the determination of the parton distribution functions (PDFs) of the proton for many different kinds of analyses in Quantum Chromodynamics (QCD).",
"It incorporates experimental data from a wide range of experiments including fixed-target, Tevatron, HERA, and LHC.",
"xFitter version 2.0.0 has recently been released, and offers an expanded set of tools and options.",
"The new xFitter 2.0.0 program links to the APFEL code which has implemented generalized matching conditions that enable the switch from $N_F$ to $N_F+1$ active flavors at an arbitrary matching scale $\\mu_m$.",
"This enables us to generalize the transition between a FFNS and a VFNS and essentially vary continuously between the two schemes; in this sense the matching scale $\\mu_m$ allows us to unify the FFNS and VFNS in a common framework.",
"This paper provides a brief overview of xFitter with emphasis of these new features."
],
[
"Introduction",
"The Parton Distribution Functions (PDFs) are the essential components that allow us to make theoretical predictions for experimental measurements of protons and hadrons.",
"The precision of the PDF analysis has advanced tremendously in recent years, and these studies are now performed with very high precision at NLO and NNLO in perturbation theory.",
"The xFitter projectxFitter can be downloaded from www.xFitter.org.",
"An overview of the program can be found in Ref. [1].",
"is an open source QCD fit framework that can perform PDF fits, assess the impact of new data, compare existing PDF sets, and perform a variety of other tasks [1].",
"The modular structure of xFitter allows for interfaces to a variety of external programs including: QCDNUM [4], APFEL [2], LHAPDF [5], APPLGRID [6], APFELGRID [7], FastNLO [8] and HATHOR [9].",
"A schematic of the modular structure is illustrated in Fig.",
"REF .",
"An overview of the recent xFitter updates and available tutorials is available in Ref. [10].",
"In this short report we will focus on the implementation of a generalized heavy quark matching scale $\\mu _m$ and the implications for PDF fits.A more extensive report of these features can be found in Ref.",
"[3]."
],
[
"The VFNS and FFNS",
"The inclusion of heavy quarks $Q=\\lbrace c,b,...\\rbrace $ into the PDF framework has been a formidable challenge.",
"In the Fixed Flavor Number Scheme (FFNS), the heavy quark is excluded from the PDF parton-model framework; here, the heavy quark $Q$ must be produced explicitly such as in the process $\\gamma g \\rightarrow Q \\bar{Q}$ .",
"In contrast, in the Variable Flavor Number Scheme (VFNS), the heavy quark is included as a parton in the PDF at scales above the $\\mu _m$ matching scale;Details on the distinction between the matching and transition scales can be found in Ref.",
"[11] thus, we have the option of exciting a heavy quark $Q$ from within the proton, e.g.",
"$\\gamma Q \\rightarrow Q g$ .",
"Both the FFNS and VFNS, as traditionally implemented, have advantages and disadvantages.",
"The FFNS has the simplicity of avoiding an $N_F$ flavor threshold in the PDFs, but at large energy scales (such as at the LHC) the heavy quarks $Q=\\lbrace c,b,...\\rbrace $ are treated differently from the light quarks.",
"Conversely, the VFNS has the advantage that it resums the heavy quark contributions using the DGLAP evolution and treats all the quarks on an equal footing at large energy scales; however, the VFNS can have some delicate cancellations when the heavy quark matching scale $\\mu _m$ is similar to the heavy quark mass $m_H$ .",
"Traditionally in most implementations of the VFNS, the heavy quark matching scale was chosen equal to the heavy quark mass $\\mu _m=m_H$ for a number of reasons as outlined in Ref.",
"[3], [11].",
"The new xFitter 2.0.0 program does not impose $\\mu _m=m_H$ , and has the flexibility to choose any value for the matching scale $\\mu _m$ ; thus, the difficulties of the traditional VFNS implementation with $\\mu _m=m_H$ are avoided.",
"In a general sense, the variable matching scale allows us to interpolate continuously between the traditional VFNS (with $\\mu _m=m_H$ ) and the FFNS (with $\\mu _m\\rightarrow \\infty $ ).",
"This situation is summarized diagrammatically in Fig.",
"REF .",
"In Fig.",
"REF -a), we see the traditional choice where the matching scale $\\mu _m$ is set equal to the heavy quark mass $m_H$ .",
"In Fig.",
"REF -b), we remove the $\\mu _m=m_H$ constraint and allow $\\mu _m$ to take an arbitrary values.",
"This is the new flexibility provided by xFitter 2.0.0."
],
[
"Boundary Conditions",
"One of the key steps for implementing the variable heavy quark matching scales is the correct boundary conditions between the $N_F$ and $N_F+1$ active flavors.",
"These boundary conditions are displayed in Fig.",
"REF for the case of the bottom quark PDF.",
"At NLO, if we match exactly at the bottom quark mass $\\mu _b=m_b$ , we findThis accidental cancellation for $\\overline{MS}$ at NLO was, in part, the reason for the traditional VFNS choice $\\mu _m=m_H$ .",
"$f_b(x,\\mu =m_b)=0$ .",
"For values $\\mu _b \\ne m_b$ , the boundary conditions are determined by the NLO contributions from the DGLAP evolution kernels, and this is displayed in Fig.",
"REF -a).",
"These contributions are driven by the $\\ln (\\mu /m_b)$ terms which are negative for $\\mu <m_b$ .",
"At large $\\mu $ scales, we observe the differences due to the choice of different boundary conditions; this is due to the (un-resummed) higher order ${\\cal O}(\\alpha _S^2)$ terms which are not included at NLO.",
"In Fig.",
"REF -b) we display the NNLO matching conditions.",
"In this case we find $x f_b(x,\\mu ) \\ne 0$ for $\\mu _b=m_b$ .",
"At this order, we have included terms of one higher order in $\\alpha _S$ compared to the previous case, and we see this tremendously reduces the variation of $x f_b(x,\\mu )$ for different choices of the matching scale $\\mu _m$ .",
"This behavior is crucial as the choice of the heavy quark matching scale amounts to a scheme choice, and the resulting physics observables should be insensitive up to the corresponding order of perturbation theory."
],
[
"Scheme Independence",
"We can further illustrate the insensitivity of the physical observables to the choice of the heavy quark matching scale $\\mu _m$ by examining the structure function $F_2^b(x,Q)$ displayed in Fig.",
"REF .",
"In Fig.",
"REF -a) we compute $F_2^b(x,Q)$ at NLO for a choice of $\\mu _m$ values; at large energy scales $Q\\sim 32\\, $ GeV we observe a large dependence on the choice of $\\mu _m$ .",
"In contrast, at NNLO in Fig.",
"REF -b) the variation of $F_2^b(x,Q)$ is significantly reduced.",
"Thus, the inclusion of the ${\\cal O}(\\alpha _S^2)$ NNLO contributions yields a result for the physical $F_2^b(x,Q)$ which is very stable w.r.t.",
"$\\mu _m$ .",
"Therefore, the NNLO implementation of the heavy quark matching scale in xFitter 2.0.0 has eliminated many of the difficulties previously encountered with the NLO VFNS with the traditional choice of $\\mu _m=m_H$ .",
"Figure: We displayF 2 b (x,Q)F_2^b(x,Q) for different choicesof the matching scalesμ m ={m b /2,m b ,2m b }\\mu _{m}=\\lbrace m_b/2,m_b,2m_b \\rbrace (indicated by the vertical lines)computed atNLO (Fig.-a)and NNLO (Fig.-b).Here, we have chosen μ=Q\\mu =Q.For details on the FONNL calculation see Ref.",
"."
],
[
"Impact on Fits",
"To facilitate comparisons of the NLO and NNLO results, Fig.",
"REF displays the ratio $\\chi ^{2}/\\chi ^{2}_0$ for charm (on the left) and bottom (on the right) where $\\chi ^{2}_0$ is the value of the $\\chi ^2$ at $\\mu _m=m_H$ .",
"By plotting $\\chi ^{2}/\\chi ^{2}_0$ , we can better compare the fractional variation of $\\chi ^2$ across the matching scale values.See Ref.",
"[3] for the full details of the fit.",
"At NLO for the case of charm, the optimal heavy quark matching scale for $\\mu _c$ is in the general range $\\mu _c\\sim m_c$ .",
"For lower scales ($\\mu _c \\ll m_c$ ), $\\alpha _S(\\mu )$ is large and the charm PDFs are negative.",
"For higher scales ($\\mu _c \\gg m_c$ ), $\\chi ^2/\\chi ^2_0$ increases.",
"At NNLO for the case of charm, the $\\chi ^2/\\chi ^2_0$ variation is greatly reduced ($\\sim 2\\%$ ), and there is minimal sensitivity to the $\\mu _c$ scale in this range.",
"For the case of bottom, the the $\\chi ^2/\\chi ^2_0$ variation is very mild ($\\sim 1\\%$ ) for both NLO and NNLO; hence, the physics results are relatively insensitive to the particular choice of the heavy quark matching scale $\\mu _b$ .",
"While the detailed characteristics of the above fits will depend on specifics of the analysis, there are two general patterns which emerge: i) the $\\chi ^2$ variation of the NNLO results are generally reduced compared to the NLO results, and ii) the relative $\\chi ^2$ variation across the bottom transition is reduced compared to the charm transition.",
"For example, although the global $\\chi ^2$ can be modified by different choices of data sets and weight factors, these general properties persist across separate data sets.",
"[3] Additionally, there are a variety of prescriptions for computing the heavy flavor contributions; these primarily differ in how the higher order contributions are organized.",
"As a cross check, we performed a NLO fit using the FONNL-A scheme; while the absolute value of $\\chi ^2$ differed, the above general properties persisted.",
"The net result is that we can now quantify the theoretical uncertainty associated with the transition between different $N_F$ sub-schemes.",
"In practical applications, if we choose $\\mu _{c}\\sim m_c$ , the impact of the $N_F=3$ to $N_F=4$ transition is reduced as this is often below the minimum kinematic cuts of the analysis (e.g.",
"$Q_{min}^2$ and $W_{min}^2$ ).",
"Conversely, the $N_F=4$ to $N_F=5$ transition is more likely to fall in the region of fitted data; hence, it is useful to quantify the uncertainty associated with the $\\mu _b$ choice.",
"Figure: The ratio (χ 2 /χ 0 2 \\chi ^2/\\chi ^2_0) of total χ 2 \\chi ^2 values(all data sets combined)as a function of the a) charm and b) bottom matching scale μ c,b \\mu _{c,b} in GeV.χ 0 2 \\chi _{0}^{2} is the χ 2 \\chi ^{2} value for μ m \\mu _{m} equal to the quark mass.The triangles (blue blue▴\\blacktriangle ) are NLOand the diamonds (red red⧫\\blacklozenge ) are NNLO.The fits are from Ref.",
"."
],
[
"Conclusion",
"The xFitter 2.0.0 program is a versatile, flexible, modular, and comprehensive tool that can facilitate analyses of the experimental data and theoretical calculations.",
"In this study we have examined the impact of the heavy flavor matching scales $\\mu _m$ on a PDF fit to the combined HERA data set.",
"These observations can be useful when performing fits.",
"While charm has a larger $\\chi ^{2}$ variation (especially at NLO), the charm quark mass $m_c\\sim 1.45$ GeV lies in a region which is generally excluded by cuts in $Q^2$ and/or $W^2$ .",
"On the contrary, the $\\chi ^{2}$ variation for the bottom quark is relatively small at both NLO and NNLO.",
"Since the bottom quark mass $m_b\\sim 4.5$ GeV is in a region where there is abundance of precision HERA data, this flexibility allows us to shift the heavy flavor threshold (and the requisite discontinuities) away from any particular data set.",
"Functionally, this means that we can analyze the HERA data using an $N_F=4$ flavor scheme up to relatively large $\\mu $ scales, and then perform the appropriate NNLO matching (with the associated constants and log terms) so that we can analyze the high-scale LHC data in the $N_F=5$ or even $N_F=6$ scheme.",
"These variable heavy flavor matching scales $\\mu _m$ allow us to generalize the transition between a FFNS and a VFNS, and provides a theoretical “laboratory” which can quantitatively test proposed implementations.",
"In conclusion, we find that the ability to vary the heavy flavor matching scales $\\mu _m$ , not only provides new insights into the intricacies of QCD, but also has practical advantages for PDF fits.",
"1.0"
]
] | 1808.08623 |
[
[
"Convenient Antiderivatives For Differential Linear Categories"
],
[
"Abstract Differential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic.",
"A differential category is said to have antiderivatives if a natural transformation $\\mathsf{K}$, which all differential categories have, is a natural isomorphism.",
"Differential categories with antiderivatives come equipped with a canonical integration operator such that generalizations of the Fundamental Theorems of Calculus hold.",
"In this paper, we show that Blute, Ehrhard, and Tasson's differential category of convenient vector spaces has antiderivatives.",
"To help prove this result, we show that a differential linear category -- which is a differential category with a monoidal coalgebra modality -- has antiderivatives if and only if one can integrate over the monoidal unit and such that the Fundamental Theorems of Calculus hold.",
"We also show that generalizations of the relational model (which are biproduct completions of complete semirings) are also differential linear categories with antiderivatives."
],
[
"Introduction",
"In single-variable calculus, the relationship between differentiation and integration is captured by the two Fundamental Theorems of Calculus, which in particular relates antiderivatives to definite integrals.",
"The First Fundamental Theorem of Calculus states that the bounded integral of a smooth function is one of its antiderivatives, that is, the derivative of the integral of a function is equal to the original function: $\\frac{{\\sf d} (\\int _a^t f(u)~{\\sf d}u)}{{\\sf d}t}(x)=f(x)$ While the Second Fundamental Theorem of Calculus directly relates the derivative and the Riemann integral in the following way: for any differential function $f: \\mathbb {R} \\mathrel {\\scriptsize }\\scriptsize $$$ R$, the Riemann integral of its derivative over an interval $ [a,b]$ is given by the difference at the endpoints of $ f$:{\\begin{@align*}{1}{-1}\\int _a^b\\frac{{\\sf d} f(t)}{{\\sf d}t}(s)~{\\sf d}s=f(b)-f(a)\\end{@align*}}The generalization of the Second Fundamental Theorem of Calculus to the multivariable setting is given by the Fundamental Theorem of Line Integrals (also sometimes known as the Gradient Theorem) which, as the names suggest, relates line integration to the gradient.Given a vector field $ F: Rn"
]
] | 1808.08513 |
[
[
"Rate-Splitting for Multi-Antenna Non-Orthogonal Unicast and Multicast\n Transmission: Spectral and Energy Efficiency Analysis"
],
[
"Abstract In a Non-Orthogonal Unicast and Multicast (NOUM) transmission system, a multicast stream intended to all the receivers is superimposed in the power domain on the unicast streams.",
"One layer of Successive Interference Cancellation (SIC) is required at each receiver to remove the multicast stream before decoding its intended unicast stream.",
"In this paper, we first show that a linearly-precoded 1-layer Rate-Splitting (RS) strategy at the transmitter can efficiently exploit this existing SIC receiver architecture.",
"We further propose multi-layer transmission strategies based on the generalized RS and power-domain Non-Orthogonal Multiple Access (NOMA).",
"Two different objectives are studied for the design of the precoders, namely, maximizing the Weighted Sum Rate (WSR) of the unicast messages and maximizing the system Energy Efficiency (EE), both subject to Quality of Service (QoS) rate requirements of all the messages and a sum power constraint.",
"A Weighted Minimum Mean Square Error (WMMSE)-based algorithm and a Successive Convex Approximation (SCA)-based algorithm are proposed to solve the WSR and EE problems, respectively.",
"Numerical results show that the proposed RS-assisted NOUM transmission strategies are more spectrally and energy efficient than the conventional Multi-User Linear-Precoding (MU-LP), Orthogonal Multiple Access (OMA) and power-domain NOMA in a wide range of user deployments (with a diversity of channel directions, channel strengths and qualities of channel state information at the transmitter) and network loads (underloaded and overloaded regimes).",
"It is superior for the downlink multi-antenna NOUM transmission."
],
[
"Introduction",
"Two essential services, namely, unicast where each message is intended for a single user and multicast where each message is intended for multiple users, are commonly supported in wireless networks.",
"Advanced wireless devices continue to strive for higher data rates of unicast services.",
"Recently, the demands for multicast services, such as media streaming, mobile TV have been growing exponentially.",
"Motivated by the scarcity of the radio resources in the Fifth Generation (5G), researchers have focused on Non-Orthogonal Unicast and Multicast (NOUM) transmission [2], [3], [4], [5], [6], [7], [8] where the unicast and multicast services are enabled in the same time-frequency resource blocks.",
"Such a transmission also finds applications as Layered Division Multiplexing (LDM) in the digital TV standard ATSC 3.0 [9] and recent interest for 5G in the 3rd Generation Partnership Project (3GPP) on concurrent delivery of both unicast and multicast services to users and efficient multiplexing of multicast and unicast in time and frequency domains [10].",
"LDM has been shown to achieve a higher spectral efficiency than Time Division Multiplexing (TDM)/Frequency Division Multiplexing (FDM) in [11].",
"From an information-theoretic perspective, Superposition Coding (SC) combined with Dirty Paper Coding (DPC) is first investigated in [12] and further proved in [13] to achieve the capacity region of the two-user NOUM transmission system.",
"Due to the high computational burden of implementing DPC, Multi-User Linear Precoding (MU–LP) becomes the most attractive alternative to simplify the transmitter design.",
"At the transmitter, the multicast stream intended for all users and the independent unicast streams are linearly precoded and superimposed before being sent to the users.",
"At each user, the multicast stream is first decoded and removed using Successive Interference Cancellation (SIC) and then the intended unicast stream is decoded by fully treating any residual interference as noise.",
"Such MU–LP-assisted NOUM has been studied previously with the objective of minimizing the transmit power [5], [6], maximizing the Weighted Sum Rate (WSR) [7] or the Energy Efficiency (EE) [8].",
"The benefit of MU–LP-assisted transmission is to exploit all spatial multiplexing gains of a multi-antenna Broadcast Channel (BC) with perfect Channel State Information at the Transmitter (CSIT).",
"However, MU–LP is mainly suited to the underloaded regime (where the number of streams is smaller than the number of transmit antennas).",
"It is sensitive to the user channel orthogonality and strengths, and does not optimally exploit the multiplexing gain of a multi-antenna BC with imperfect CSIT [14].",
"Moreover, the presence of SIC at the receivers is not exploited to manage the interference among the unicast streams, but only to separate the multicast stream from the unicast streams.",
"In this paper, we resolve the above limitations of conventional MU–LP-assisted NOUM by resorting to linearly-precoded Rate-Splitting (RS) approaches.",
"Rate-Splitting was originally developed for the two-user single-antenna Interference Channel (IC) [15] and has recently been introduced in [16] as a promising multi-user multi-antenna non-orthogonal transmission strategy to tackle numerous problems faced by modern Multiple Input Multiple Output (MIMO) wireless networks.",
"Uniquely, RS enables to partially decode the interference and partially treat the interference as noise.",
"This allows RS to explore a more general and powerful transmission framework, namely, Rate-Splitting Multiple Access (RSMA) for downlink multi-antenna systems that contains MU–LP and power-domain Non-Orthogonal Multiple Access (NOMA) as special cases, and provides room for rate and Quality of Service (QoS) enhancements [14].",
"Though originally introduced for the two-user Single Input Single Output (SISO) IC, RS has recently appeared as an underpinning communication-theoretic strategy to tackle modern interference-related problems and has been successfully investigated in several multi-antenna broadcast channel settings, namely, unicast-only transmission with perfect CSIT [14], [17], [18], [19], [20] and imperfect CSIT [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], as well as (multigroup) multicast-only transmission [35], [36].",
"With RS, each stream is split at the transmitter into a common part and a private part.",
"The common parts are jointly encoded into one common stream to be decoded by all users while the private parts are independently encoded into the private streams to be decoded by the intended users.",
"Upon decoding the common stream and the private stream, a user can reconstruct its original message.",
"Due to the superimposed transmission of the common and private streams, RS can be viewed mathematically as a NOUM system.",
"Hence, RS was termed joint multicasting and broadcasting in [37].",
"Though both the common stream in the RS-assisted transmission and the conventional multicast stream are decoded by multiple users, they are transmitted with different intentions.",
"The multicast stream contains a single message intended for all those users (because users are genuinely interested in the same message).",
"On the other hand, the common stream in RS contains parts of the unicast messages of a subset of users, is intended to that subset of users, and is transmitted for interference management purposes.",
"All of the existing works on RS only considered unicast-only or multicast-only transmissions.",
"The benefits of RS in NOUM transmissions have not been investigated yet.",
"Motivated by the benefits of RS in the unicast-only and multicast-only transmissions as well as the limitations of conventional MU–LP-assisted NOUM, we study the application of RS in the NOUM transmission in this paper.",
"The contributions of the paper are summarized as follows.",
"First, we propose a 1-layer RS-assisted NOUM transmission strategy and design the precoder to maximize WSR and EE, respectively.",
"By splitting the unicast streams into common and private parts and encoding the common parts along with the multicast message into a super-common stream to be decoded by all users, the SIC in 1-layer RS is used for the dual purpose of separating the unicast and multicast streams as well as managing the interference among the unicast streams.",
"The key benefit of 1-layer RS in the NOUM transmission is the fact that 1-layer RS does not lead to any complexity increase for the receivers compared to conventional MU–LP-assisted NOUM since one layer of SIC is required to separate multicast stream from unicast streams.",
"This contrasts with unicast-only and muticast-only transmissions where 1-layer RS was found beneficial over MU–LP in [25], [14], [17] but at the cost of a receiver complexity increase due to the need of SIC for RS to operate.",
"To the best of our knowledge, this is the first work that applies RS to NOUM transmissions.",
"Second, besides the 1-layer RS NOUM transmission strategy that incorporates a single layer of SIC, we further propose multi-layer SIC-assisted NOUM transmission strategies based on the generalized RS and power-domain NOMA (referred to simply as NOMA in the rest of the paper).",
"NOMA relies on SC at the transmitter and SIC at the receivers (SC–SIC)[38].",
"It forces some users to fully decode and cancel the interference created by other users.",
"Two NOMA-assisted NOUM transmission strategies are proposed, namely, `SC–SIC' and `SC–SIC per group'.",
"To the best of our knowledge, this has not been investigated in the literature of multi-user multi-antenna NOUM transmissions.",
"Comparing with 1-layer RS, the proposed generalized RS allows the number of layers of the common streams to be increased with the number of served users.",
"Thanks to its ability of partially decoding interference and partially treating interference as noise, the generalized RS model proposed in this work is a more general framework of multi-user multi-antenna NOUM transmission that encompasses MU–LP and NOMA as special cases.",
"Third, we study the WSR and EE maximization problems subject to the QoS rate requirements and a sum power constraint for all investigated NOUM strategies.",
"Two optimization frameworks are proposed to solve the WSR and EE maximization problems based on the Weighted Minimum Mean Square Error (WMMSE) and Successive Convex Approximation (SCA) algorithms, respectively.",
"The effectiveness of the proposed algorithms is verified in the numerical results.",
"Fourth, we show through numerical results that the proposed 1-layer RS-assisted NOUM transmission strategy is more spectrally and energy efficient than the existing MU–LP-assisted transmission in a wide range of user deployments (with a diversity of channel directions, channel strengths and qualities of channel state information at the transmitter) and network loads (underloaded and overloaded regimes).",
"Importantly, applying 1-layer RS to NOUM boosts WSR and EE of the system but maintains the same receiver complexity as MU–LP.",
"Hence, the performance gain comes at no additional cost for the receivers since one layer of SIC is required to separate unicast and multicast streams in the conventional MU–LP-assisted NOUM.",
"In other words, 1-layer RS makes a better use of the existing SIC architecture.",
"Comparing with the proposed NOMA-assisted NOUM, 1-layer RS achieves a more robust WSR and EE performance in a wide range of user deployments and network loads while its receiver complexity is much lower.",
"Fifth, we show that the WSR and EE performance of the proposed generalized RS is always equal to or larger than that of MU–LP and NOMA in the realm of NOUM transmissions.",
"It is also more robust to the variation of user deployments, CSIT inaccuracy and network loads.",
"As a consequence, the generalized RS is less sensitive to user pairing and therefore does not require complex user scheduling.",
"The generalized RS requires a higher encoding and decoding complexity than MU–LP and NOMA since multiple common streams are required to be encoded on top of the private streams.",
"The observations in this paper confirm the superiority of RS over MU–LP, Orthogonal Multiple Access (OMA) where the unicast stream is only intended for a single user, and NOMA in NOUM transmissions, and complement our previous findings in [14], [17], [35], [25] that have shown the superiority of RS in unicast-only and multicast-only transmissions.",
"The rest of the paper is organized as follows.",
"Section introduces the system and power model.",
"Section reviews the conventional MU–LP-assisted NOUM and the proposed 1-layer RS strategy.",
"Section specifies the proposed generalized RS and NOMA-assisted NOUM.",
"Section discusses the optimization frameworks to solve the WSR and EE problems.",
"Section and illustrate numerical results of WSR and EE.",
"Section concludes the paper."
],
[
"System Model and Power Model",
"Consider a BS equipped with $N_t$ antennas serving $K$ single-antenna users in the user set $\\mathcal {K}=\\lbrace 1,\\ldots ,K\\rbrace $ .",
"In each time frame, user-$k, \\forall k\\in \\mathcal {K}$ requires a dedicated unicast message $W_k$ and a multicast message $W_0$ .",
"At the BS, the multicast message $W_0$ intended for all users and the $K$ unicast messages $W_1,\\ldots ,W_K$ are encoded into the data stream vector $\\mathbf {s}$ and linearly precoded using the precoder $\\mathbf {P}$ .",
"The transmit signal vector $\\mathbf {x}=\\mathbf {P}\\mathbf {s}$ is subject to the power constraint $\\mathbb {E}\\lbrace ||\\mathbf {x}||^2\\rbrace \\le P_{t}$ .",
"Assuming that $\\mathbb {E}\\lbrace \\mathbf {{s}}\\mathbf {{s}}^H\\rbrace =\\mathbf {I}$ , we have $\\mathrm {tr}(\\mathbf {P}\\mathbf {P}^{H})\\le P_{t}$ .",
"The signal received at user-$k$ is $y_{k}=\\mathbf {{h}}_{k}^{H}\\mathbf {{x}}+n_{k},$ where $\\mathbf {{h}}_{k}\\in \\mathbb {C}^{N_{t}\\times 1}$ is the channel between the BS and user-$k$ , it is assumed to be perfectly known at the transmitter and receivers.",
"The imperfect CSIT scenario will be discussed in the proposed algorithm and numerical results.",
"The received noise $n_{k}$ is modeled as a complex Gaussian random variable with zero mean and variance $\\sigma _{n,k}^{2}$ .",
"Without loss of generality, we assume the noise variances are equal to one ($\\sigma _{n,k}^{2}=1,\\forall k\\in \\mathcal {K}$ ).",
"Hence, the transmit Signal-to-Noise Ratio (SNR) is equal to the transmit power consumption.",
"In this work, the total power consumption at the BS is [39] $P_{\\textrm {total}}=\\frac{1}{\\eta }\\mathrm {tr}\\left(\\mathbf {P}\\mathbf {P}^{H}\\right)+P_{\\textrm {cir}},\\vspace{-4.2679pt}$ where $\\eta \\in [0,1]$ is the power amplifier efficiency.",
"$P_{\\textrm {cir}}=N_tP_{\\textrm {dyn}}+P_{\\textrm {sta}}$ is the circuit power consumption of the BS, where $P_{\\textrm {dyn}}$ is the dynamic power consumption of one active radio frequency chain and $P_{\\textrm {sta}}$ is the static power consumption of the cooling systems, power supply and so on.",
"$\\eta $ and $P_{\\textrm {sta}}$ are assumed to be fixed for simplicity.",
"Figure: KK-user one-layer SIC-based multi-antenna NOUM transmission model"
],
[
"One-layer SIC-based transmission",
"In this section, we focus on the NOUM transmission model that only requires one layer of SIC at each receiver.",
"We first introduce the baseline MU–LP-assisted strategy followed by the proposed 1-layer RS-assisted NOUM transmission model."
],
[
"MU–LP",
"The conventional MU–LP-assisted NOUM transmission model is illustrated in Fig.",
"REF .",
"The multicast message $W_0$ and the unicast messages $W_1,\\ldots ,W_K$ are independently encoded into the data streams $s_0,s_1,\\ldots ,s_K$ .",
"The stream vector $\\mathbf {s}=[s_0,s_1,\\ldots ,s_K]^{T}$ is precoded using the precoder $\\mathbf {P}=[\\mathbf {p}_0,\\mathbf {p}_1,\\ldots ,\\mathbf {p}_K]$ , where $\\mathbf {p}_0,\\mathbf {p}_k\\in \\mathbb {C}^{N_t\\times 1}$ are the respective precoders of the multicast stream $s_0$ and the unicast stream $s_k$ .",
"The resulting transmit signal $\\mathbf {x}\\in \\mathbb {C}^{N_t\\times 1}$ is $\\mathbf {x}=\\mathbf {P}\\mathbf {{s}}=\\underbrace{\\mathbf {p}_{0}s_{0}}_{{\\text{multicast stream}}}+\\underbrace{\\sum _{k\\in \\mathcal {K}}\\mathbf {p}_{k}s_{k}}_{{\\text{unicast streams}}}.\\vspace{-5.69054pt}$ The signal received at user-$k$ becomes $\\begin{aligned}y_{k}&=\\underbrace{\\mathbf {{h}}_{k}^{H}\\mathbf {p}_{0}s_{0}}_{{\\text{intended multicast stream}}}+\\underbrace{\\mathbf {{h}}_{k}^{H}\\mathbf {p}_{k}s_{k}}_{{\\text{intended unicast stream}}}\\\\&+\\underbrace{\\sum _{j\\in \\mathcal {K},j\\ne k}\\mathbf {{h}}_{k}^{H}\\mathbf {p}_{j}s_{j}}_{{\\text{interference among unicast streams}}}+\\underbrace{n_{k}}_{{\\text{noise}}}.\\end{aligned}$ Each user-$k,\\forall k\\in \\mathcal {K}$ decodes the multicast stream $s_0$ and the intended unicast stream $s_k$ under the assistance of one SIC.",
"The decoding order of $s_0$ and $s_k$ can be optimized for each instantaneous channel condition.",
"The decoding order follows the rule that the data stream intended for more users has a higher decoding priority [5], [7].",
"Hence, we assume that the multicast stream is decoded first and removed from the received signal using SIC before decoding the unicast streams at all users.",
"This assumption will be applied to all the transmission strategies proposed in the rest of the paper.",
"The multicast stream $s_0$ is decoded by treating the signal of all unicast streams as noise.",
"The Signal-to-Interference-plus-Noise Ratio (SINR) of decoding $s_0$ at user-$k$ is $\\gamma _{k,0}=\\frac{|\\mathbf {{h}}_{k}^{H}\\mathbf {{p}}_{0}|^{2}}{\\sum _{j\\in \\mathcal {K}}|\\mathbf {{h}}_{k}^{H}\\mathbf {{p}}_{j}|^{2}+1}.\\vspace{-0.56905pt}$ Once $s_{0}$ is successfully decoded and subtracted from the original received signal $y_k$ , user-$k$ decodes the intended unicast stream $s_{k}$ by treating the interference from the unicast streams of other users as noise.",
"The SINR of decoding $s_{k}$ at user-$k$ is $\\gamma _{k}=\\frac{|\\mathbf {{h}}_{k}^{H}\\mathbf {{p}}_{k}|^{2}}{\\sum _{j\\in \\mathcal {K},j\\ne k}|\\mathbf {{h}}_{k}^{H}\\mathbf {{p}}_{j}|^{2}+1}.\\vspace{-0.56905pt}$ The corresponding achievable rates of decoding $s_{0}$ and $s_{k}$ at user-$k$ are $R_{k,0}=\\log _{2}\\left(1+\\gamma _{k,0}\\right)$ , $R_{k}=\\log _{2}\\left(1+\\gamma _{k}\\right)$ .",
"As $s_0$ is decoded by all users, to ensure that $s_{0}$ is successfully decoded by all users, the corresponding code-rate should not exceed the rate achievable by the weakest receiver [23], [35], which is given by $R_{0}=\\min \\left\\lbrace R_{1,0},\\ldots ,R_{K,0}\\right\\rbrace .\\vspace{-5.69054pt}$ Two different objectives are studied for the design of the precoders: 1) Weighted sum rate maximization problem: To investigate the spectral efficiency, we study the problem of maximizing the WSR of the unicast messages while the QoS rate constraints of all messages and the power constraint of the BS should be met.",
"For a given weight vector $\\mathbf {u}=[u_1,\\ldots ,u_K]$ , the WSR maximization problem in the $K$ -user MU–LP-assisted NOUM is [left=WSRMU–LP]align P kKukRk s.t.",
"RkRkth, kK, Rk,0R0th,kK, tr(PPH)Pt, where Constraint (REF ) is the QoS rate requirement of each unicast message.",
"$R_k^{th}$ is the rate lower bound of the unicast message $W_k$ .",
"Constraint (REF ) ensures that each user decodes the multicast message $W_0$ with a rate larger than or equal to $R_0^{th}$ .",
"2) Energy efficiency maximization problem: To investigate the EE of MU–LP, we maximize the WSR of all the messages divided by the sum power of the transmitter.",
"For a given weight vector $\\mathbf {u}_{tot}=[u_0,u_1,\\ldots ,u_K]$ of all the messages, the EE maximization problem of MU–LP is $\\textrm {EE}_{\\textrm {MU--LP}}\\begin{dcases}\\max _{\\mathbf {{P}}}\\,\\,&\\frac{u_{0}R_{0}+\\sum \\limits _{k\\in \\mathcal {K}}u_{k}R_{k} }{\\frac{1}{\\eta }\\mathrm {tr}(\\mathbf {P}\\mathbf {P}^{H})+P_{\\textrm {cir}}} \\\\\\mbox{s.t.",
"}\\quad & \\textrm {(\\ref {c1_mulp})--(\\ref {c3_mulp})}.\\end{dcases}$ Remark 1: Recall that MU–LP does not require any SIC at each user in the unicast-only transmission.",
"In comparison, one layer of SIC is necessary at each user to remove the multicast stream before decoding the intended unicast stream in the MU–LP-assisted NOUM transmission.",
"The SIC is used for the purpose of separating the unicast and multicast streams."
],
[
"1-layer RS",
"The proposed $K$ -user 1-layer RS-assisted NOUM transmission model is illustrated in Fig.",
"REF .",
"The unicast message $W_k$ intended for user-$k$ , $\\forall k\\in \\mathcal {K}$ is split into a common sub-message $W_{k,c}$ and a private sub-message $W_{k,p}$ .",
"The private sub-messages $W_{1,p},\\ldots ,W_{K,p}$ of the unicast messages are independently encoded into the private streams ${s}_1, \\ldots ,{s}_K$ while the common sub-messages $W_{1,c},\\ldots ,W_{K,c}$ of the unicast messages are jointly encoded with the multicast message $W_0$ into a super-common stream ${s}_0$ required to be decoded by all users.",
"Different from the common stream ${s}_0$ in MU–LP that only includes the multicast meesage, the super-common stream ${s}_0$ in 1-layer RS includes the whole multicast message as well as parts of the unicast messages.",
"Following the transmission procedure in MU–LP, the formed stream vector $\\mathbf {s}$ is linearly precoded and broadcast to the users.",
"The super-common stream and private streams are decoded using one layer of SIC in a similar way as decoding the multicast stream and the unicast streams in the MU–LP-assisted NOUM transmission with higher decoding priority given to the super-common stream.",
"Since $R_{0}$ is now shared by the achievable rates of transmitting the multicast message $W_0$ and the common sub-messages $W_{1,c},\\ldots ,W_{K,c}$ of the unicast messages, it is equal to $C_{0}+\\sum _{k\\in \\mathcal {K}}C_{k,0}=R_{0},$ where $C_0$ is the portion of $R_{0}$ transmitting $W_{0}$ and $C_{k,0}$ is the user-$k$ 's portion of $R_{0}$ transmitting $W_{k,c}$ .",
"The portions of rate allocated to $W_{0}$ and $W_{1,c},\\ldots ,W_{K,c}$ will be optimized by solving the optimization problems formulated in this section.",
"In the proposed 1-layer RS-assisted NOUM transmission, the achievable rate of each unicast message contains two parts.",
"One part is $C_{k,0}$ transmitted via $W_{k,c}$ encoded in the super-common stream $s_0$ .",
"The other part is $R_k$ transmitted via $W_{k,p}$ encoded in the private stream $s_k$ .",
"Hence, the achievable rate of transmitting the unicast message $W_k$ of user-$k$ is $R_{k,tot}=C_{k,0}+R_{k}, \\forall k\\in \\mathcal {K}.$ The corresponding WSR and EE maximization problems are given by 1) Weighted sum rate maximization problem: The WSR maximization problem in the $K$ -user 1-layer RS-assisted NOUM transmission for a given $\\mathbf {u}$ is [left=WSR1-layer RS]alignP, c kKukRk,tot s.t.",
"Ck,0+RkRkth, k K C0R0th C0+jKCj,0Rk,0,kK Ck,00,kK tr(PPH)Pt where $\\mathbf {c}=[C_0,C_{1,0},\\ldots ,C_{K,0}]$ is the common rate vector required to be optimized with the precoder $\\mathbf {P}.$ When $C_{k,0}=0, \\forall k\\in \\mathcal {K}$ , Problem $\\textrm {WSR}_{\\textrm {1-layer RS}}$ reduces to Problem $\\textrm {WSR}_{\\textrm {MU--LP}}$ .",
"Hence, the proposed RS model always achieves the same or superior performance to MU–LP.",
"Constraint (REF ) ensures the super-common stream can be successfully decoded by all users.",
"Constraints (REF ) and (REF ) are the QoS rate constraints of all messages.",
"2) Energy efficiency maximization problem: The EE maximization problem of 1-layer RS for a given $\\mathbf {u}_{tot}$ is $\\textrm {EE}_{\\textrm {1-layer RS}}\\begin{dcases}\\max _{\\mathbf {c},\\mathbf {{P}}}\\,\\,&\\frac{u_{0}C_{0}+\\sum _{k\\in \\mathcal {K}}u_kR_{k,tot}}{\\frac{1}{\\eta }\\mathrm {tr}(\\mathbf {P}\\mathbf {P}^{H})+P_{\\textrm {cir}}} \\\\\\mbox{s.t.",
"}\\quad & \\textrm {(\\ref {c1_rs})--(\\ref {c5_rs})}.\\end{dcases}$ Remark 2: Similarly to the $K$ -user 1-layer RS-assisted unicast-only transmission discussed in [14], one layer of SIC is required at each user in the $K$ -user 1-layer RS-assisted NOUM transmission.",
"In contrast with the MU–LP-assisted NOUM, the SIC of 1-layer RS-assisted NOUM transmission is used for separating the unicast and multicast streams as well as better managing the multi-user interference among the unicast streams.",
"The presence of SIC is therefore better exploited in the 1-layer RS-assisted NOUM than in the MU–LP-assisted NOUM."
],
[
"Multi-layer SIC-based transmission",
"To further enhance the system spectral and energy efficiencies, the co-channel interference among unicast streams can be better managed by introducing multiple layers of SIC at each receiver to decode part of the interference.",
"There are two multi-layer SIC-based transmission strategies, namely, RSMA and NOMA-based transmission.",
"In the unicast-only transmission, it has been shown in [14], [17] that NOMA achieves better spectral and energy efficiency than MU–LP when the user channels are aligned and there is certain channel strength difference among users.",
"The generalized RS-based RSMA bridges MU–LP and NOMA and achieves a better spectrum efficiency [14].",
"In this section, both RSMA and NOMA strategies are applied to the NOUM transmission.",
"To simplify the explanation, we focus on the three-user case ($\\mathcal {K}=\\lbrace 1,2,3\\rbrace $ ) for all multi-layer SIC transmission strategies.",
"It can be extended to solve the $K$ -user problem."
],
[
"Generalized rate-splitting",
" Figure: Three-user generalized RS-assisted multi-antenna NOUM transmission modelDifferent from the 1-layer RS transmission model introduced in Section REF where the unicast message of each user is split into two parts, the unicast message of each user is split into four different parts in the three-user generalized RS transmission model.",
"For user-1, the unicast message $W_1$ is split into sub-messages $\\lbrace W_{1}^{123}$ , $W_{1}^{12}$ , $W_{1}^{13}$ , $W_{1}^{1}\\rbrace $ .",
"The unicast messages of user-2 and user-3 are split into sub-messages $\\lbrace W_{2}^{123}, W_{2}^{12},W_{2}^{23},W_{2}^{2}\\rbrace $ and $\\lbrace W_{3}^{123}, W_{3}^{13},W_{3}^{23},W_{3}^{3}\\rbrace $ , respectively.",
"The superscript of each sub-message represents a group of users.",
"The sub-messages with the same superscript are encoded together into a common stream intended for the users within that specific user group.",
"Sub-messages $W_{1}^{123},W_{2}^{123},W_{3}^{123}$ are jointly encoded with the multicast message $W_0$ into the super-common stream $s_{0}$ intended for all the three users.",
"Sub-messages $W_{1}^{12}, W_{2}^{12}$ are encoded together into the partial-common stream $s_{12}$ intended for user-1 and user-2 only.",
"Similary, we obtain the partial-common streams $s_{13}$ and $s_{23}$ encoded by $W_{1}^{13}, W_{3}^{13}$ and $W_{2}^{23}, W_{3}^{23}$ , respectively.",
"Sub-messages $W_1^1, W_2^2, W_3^3$ are respectively encoded into the private streams $s_1, s_2,s_3$ for a single user only.",
"The intention of splitting each unicast message into different sub-messages and reuniting the sub-messages is to enable each user the capability of dynamic interference management.",
"For example, when user-1 decodes $s_{0}$ , it not only decodes the intended multicast message $W_{0}$ and the intended unicast sub-message $W_{1}^{123}$ but also partially decodes the interference resulting from sub-messages $W_{2}^{123}$ and $W_{3}^{123}$ .",
"The encoded data streams $\\mathbf {{s}}=[s_{0},s_{12},s_{13},s_{23},s_{1},s_{2},s_{3}]^T$ are precoded via the precoder $\\mathbf {{P}}=[\\mathbf {{p}}_{0},\\mathbf {{p}}_{12},\\mathbf {{p}}_{13},\\mathbf {{p}}_{23},\\mathbf {{p}}_{1},\\mathbf {{p}}_{2},\\mathbf {{p}}_{3}]$ and then broadcast to the users.",
"The transmit signal $\\mathbf {x}\\in \\mathbb {C}^{N_t\\times 1}$ is $\\mathbf {x}=\\underbrace{\\mathbf {p}_{0}s_{0}}_{\\text{super-common stream}}+\\underbrace{\\sum _{i\\in \\lbrace 12,13,23\\rbrace }\\mathbf {p}_{i}s_{i}}_{\\text{partial-common streams}}+\\underbrace{\\sum _{k\\in \\mathcal {K}}\\mathbf {p}_{k}s_{k}}_{\\text{private streams}} .\\vspace{-0.56905pt}$ At user sides, each user decodes the data streams that carry its intended sub-messages using SIC.",
"The decoding procedure starts from the super-common stream to the partial-common streams and then progresses downwards to the private streams.",
"At user-1, the data streams $s_{0}, s_{12}, s_{13}, s_{1}$ are decoded using SIC.",
"Similarly, user-2 and user-3 decode the data streams $s_{0}, s_{12}, s_{23}, s_{2}$ and $s_{0}, s_{13}, s_{23}, s_{3}$ , respectively.",
"As $s_{12},s_{13},s_{23}$ are all intended for two users, the decoding order needs to be optimized together with the precoder $\\mathbf {P}$ .",
"The decoding order of all streams intended for two users is denoted by $\\pi _2$ .",
"For instance, when the decoding order is $\\pi _2=12\\rightarrow 13\\rightarrow 23$ , $s_{12}$ will be decoded before $s_{13}$ and $s_{13}$ will be decoded before $s_{23}$ at all users.",
"Since user-1 only decodes the partial-common streams $s_{12}$ and $s_{13}$ , the corresponding decoding order at user-1 is denoted by $\\pi _{2,1}=12\\rightarrow 13$ .",
"We further use $s_{\\pi _{2,k}(i)}$ to represent the $i$ th data stream to be decoded at user-$k$ based on the decoding order $\\pi _2$ .",
"When the decoding order at user-1 is $\\pi _{2,1}=12\\rightarrow 13$ , we have $s_{\\pi _{2,1}(1)}=s_{12}$ and $s_{\\pi _{2,1}(2)}=s_{13}$ .",
"The proposed three-user generalized RS-assisted NOUM transmission model with the decoding order $\\pi _2=12\\rightarrow 13\\rightarrow 23$ is illustrated in Fig.",
"REF .",
"The SINRs of decoding the streams $s_{0}, s_{\\pi _{2,1}(1)}, s_{\\pi _{2,1}(2)}, s_{1}$ using SIC at user-1 are respectively given by $\\gamma _{1}^{0}=\\frac{\\left|\\mathbf {{h}}_{1}^{H}\\mathbf {{p}}_{0}\\right|^{2}}{\\sum _{i\\in \\lbrace 12,13,23\\rbrace }\\left|\\mathbf {{h}}_{1}^{H}\\mathbf {{p}}_{i}\\right|^{2}+\\sum _{k=1}^3\\left|\\mathbf {{h}}_{1}^{H}\\mathbf {{p}}_{k}\\right|^{2}+1},$ $\\gamma _{1}^{\\pi _{2,1}{(1)}}=\\frac{\\left|\\mathbf {{h}}_{1}^{H}\\mathbf {{p}}_{\\pi _{2,1}{(1)}}\\right|^{2}}{\\left|\\mathbf {{h}}_{1}^{H}\\mathbf {{p}}_{\\pi _{2,1}{(2)}}\\right|^{2}+\\left|\\mathbf {{h}}_{1}^{H}\\mathbf {{p}}_{23}\\right|^{2}+\\sum _{k=1}^3\\left|\\mathbf {{h}}_{1}^{H}\\mathbf {{p}}_{k}\\right|^{2}+1},$ $\\gamma _{1}^{\\pi _{2,1}{(2)}}=\\frac{\\left|\\mathbf {{h}}_{1}^{H}\\mathbf {{p}}_{\\pi _{2,1}{(2)}}\\right|^{2}}{\\left|\\mathbf {{h}}_{1}^{H}\\mathbf {{p}}_{23}\\right|^{2}+\\sum _{k=1}^3\\left|\\mathbf {{h}}_{1}^{H}\\mathbf {{p}}_{k}\\right|^{2}+1},$ $\\gamma _{1}=\\frac{\\left|\\mathbf {{h}}_{1}^{H}\\mathbf {{p}}_1\\right|^{2}}{\\left|\\mathbf {{h}}_{1}^{H}\\mathbf {{p}}_{23}\\right|^{2}+\\sum _{k=2}^3\\left|\\mathbf {{h}}_{1}^{H}\\mathbf {{p}}_{k}\\right|^{2}+1}.$ The resulting achievable rates of decoding the intended streams at user-1 are calculated by $R_{1}^{i}=\\log _{2}\\left(1+\\gamma _{1}^{i}\\right), \\forall i\\in \\lbrace 0,12,13,1\\rbrace $ .",
"By using the same method, we could obtain the individual rates of decoding the intended streams at user-2 and user-3, respectively.",
"To ensure that the streams are decodable by the corresponding groups of users, the transmission common rates should not exceed $R_{0}=\\min \\left\\lbrace R_{1}^{0},R_{2}^{0},R_{3}^{0}\\right\\rbrace ,R_{12}=\\min \\left\\lbrace R_{1}^{12},R_{2}^{12}\\right\\rbrace ,R_{13}=\\min \\left\\lbrace R_{1}^{13},R_{3}^{13}\\right\\rbrace ,R_{23}=\\min \\left\\lbrace R_{2}^{23},R_{3}^{23}\\right\\rbrace $ .",
"Following the above RS structure, the rate of each common stream is split for the corresponding groups of users.",
"Let $C_0$ be the portion of $R_0$ transmitting $W_0$ and $C_k^i$ be the portions of rate $R_i$ allocated to user-$k$ for the transmission of the sub-message $W_{k}^i$ , we have $C_0+\\sum _{k\\in \\lbrace 1,2,3\\rbrace }C_{k}^{123}=R_{0}$ , $\\sum _{k\\in \\lbrace 1,2\\rbrace }C_k^{12}=R_{12}$ , $\\sum _{k\\in \\lbrace 1,3\\rbrace }C_k^{13}=R_{13}$ , $\\sum _{k\\in \\lbrace 2,3\\rbrace }C_k^{23}=R_{23}$ .",
"Hence, the individual rate of transmitting the unicast message of each user is the summation of the portions of rate in the intended common streams, which is given by $R_{k,tot}=\\sum _{i_k} C_{k}^{i_k}+R_{k}$ , where $i_1\\in \\lbrace 0,12,13\\rbrace $ , $i_2\\in \\lbrace 0,12,23\\rbrace $ and $i_3\\in \\lbrace 0,13,23\\rbrace $ .",
"The corresponding WSR and EE maximization problems are given by 1) Weighted sum rate maximization problem: The WSR maximization problem in the three-user generalized RS-assisted NOUM transmission for a given $\\mathbf {u}$ is [left=WSRgeneral RS]alignP, c, kKukRk,tot s.t.",
"Rk,totRkth, kK C0R0th C0+jKCj123Rk0,kK C112+C212 Rk12,k{1,2} C113+C313 Rk13,k{1,3} C223+C323 Rk23,k{2,3} c0 tr(PPH)Pt where $\\mathbf {c}=[C_0, C_1^{123},C_2^{123},C_3^{123},C_1^{12},C_2^{12},C_1^{13},C_3^{13},C_2^{23},$ $C_3^{23}]$ is the common rate vector.",
"When there is zero common rate allocated to the sub-messages intended for two users, i.e., $C_k^{i}=0,\\forall i\\in \\lbrace 12,13,23\\rbrace , k\\in \\mathcal {K}$ , Problem $\\textrm {WSR}_{\\textrm {general RS}}$ reduces to Problem $\\textrm {WSR}_{\\textrm {1-layer RS}}$ .",
"When $C_k^{i}=0,\\forall i\\in \\lbrace 123,12,13,23\\rbrace , k\\in \\mathcal {K}$ , Problem $\\textrm {WSR}_{\\textrm {general RS}}$ reduces to $\\textrm {WSR}_{\\textrm {MU--LP}}$ .",
"Hence, the proposed generalized RS model always achieves the same or superior performance to 1-layer RS and MU–LP.",
"Constraints (REF )–(REF ) ensures all common streams are decodable by the intended users.",
"Constraints (REF ) and (REF ) are the QoS rate constraints.",
"2) Energy efficiency maximization problem: The EE maximization problem of the generalized RS for a given $\\mathbf {u}_{tot}$ is $\\textrm {EE}_{\\textrm {general RS}}\\begin{dcases}\\max _{\\mathbf {{P}}, \\mathbf {c},\\pi }\\,\\,&\\frac{u_{0}C_{0}+\\sum _{k\\in \\mathcal {K}}u_kR_{k,tot}}{\\frac{1}{\\eta }\\mathrm {tr}(\\mathbf {P}\\mathbf {P}^{H})+P_{\\textrm {cir}}} \\\\\\mbox{s.t.",
"}\\quad & \\textrm {(\\ref {c1_grs})--(\\ref {c8_grs})}.\\end{dcases}\\vspace{-2.84526pt}$ Remark 3: The proposed generalized RS-based NOUM is a super-strategy of the 1-layer RS-based NOUM proposed in Section REF .",
"As more layers of SIC are required at each user to decode the partial-common streams, the receiver complexity of the proposed generalized RS-based NOUM increases with the number of served users $K$ .",
"In comparison, the receiver complexity of 1-layer RS does not depend on $K$ and is much lower especially when $K$ is large."
],
[
"NOMA",
"There are two main strategies in the multi-antenna NOMA, namely, `SC–SIC' and `SC–SIC per group' [14].",
"Both are applied in the NOUM transmission.",
"Comparing with the SC–SIC-assisted unicast-only transmission, the main difference in the SC–SIC-assisted NOUM transmission is that the multicast message $W_0$ is jointly encoded with the unicast message to be decoded first into a common stream $s_0$ .",
"At user sides, each user first decodes $s_0$ with the highest priority.",
"Then the users carry on decoding the unicast streams according to the decoding order $\\pi $ .",
"The proposed three-user SC–SIC-assisted NOUM transmission model with the decoding order $\\pi =1\\rightarrow 2\\rightarrow 3$ is illustrated in Fig.",
"REF .",
"The first layer of SIC is used for two different purposes.",
"It is used not only to decode the multi-user interference among the unicast streams, but also to separate the unicast and multicast streams.",
"The decoding order is required to be optimized with the precoder for both WSR and EE optimization problem.",
"Figure: Three-user NOMA-assisted multi-antenna NOUM transmission modelTable: Qualitative comparison of the complexity of different strategies for NOUMIn the three-user SC–SIC per group-assisted NOUM transmission, the users are separated into two different groups.",
"Users within each group are served using SC–SIC while the users across the groups are served using MU–LP[14].",
"As the inter-group interference is mitigated using MU–LP, none of the unicast messages can be encoded with the multicast message $W_0$ .",
"One more layer of SIC is required to separate the multicast and unicast streams in the SC–SIC per group-assisted NOUM transmission.",
"Each user first decodes the multicast stream with the highest priority.",
"A three-user example is illustrated in Fig.",
"REF .",
"The users are divided into two different groups with user-1 in group 1 while user-2 and user-3 in group 2.",
"As there are two users in group 2, the decoding order is required to be optimized.",
"The decoding order of the unicast messages for the users in group 2 is denoted by $\\pi _2$ .",
"In Fig.",
"REF , the decoding order in group 2 is fixed to $\\pi _2=2\\rightarrow 3$ .",
"Due to the page limitation, the detailed NOMA strategies are not specified.",
"If the readers fully understand Fig.",
"REF as well as the application of NOMA in the unicast-only transmission discussed in Section 3.2 of [14], the system model of `SC–SIC' and `SC–SIC per group' in the NOUM transmission will be easily traced out.",
"Remark 4: Following [14], both the proposed two NOMA-assisted NOUM transmission strategies are sub-strategies of the generalized RS-assisted NOUM.",
"The transmitter complexity of SC–SIC per group-assisted NOUM is higher than the SC–SIC-assisted and the generalized RS-assisted NOUM since the decoding order and user grouping are required to be optimized together with the precoder.",
"A qualitative comparison of the complexity of all the strategies is illustrated in Table REF , where the total number of user grouping methods to be considered in SC–SIC per group is $\\sum _{k=1}^KS(K,k)$ .",
"$S(K,k)$ is the number of ways of partitioning a set of $K$ elements into $k$ nonempty sets which is known as a Stirling set number [40].",
"It is computed from the sum $S(K,k)=\\frac{1}{k!",
"}\\sum _{i=0}^k(-1)^i\\binom{k}{i}(k-i)^K$ .",
"From Table REF , we obtain that 1-layer RS has the simplest scheduler complexity while maintaining the same low encoder and receiver complexity as MU–LP.",
"Since the generalized RS has the highest encoder and receiver complexity while SC–SIC per group has the highest scheduler complexity, both strategies are preferred to be applied to the scenarios when $K$ is small so as to achieve a better tradeoff between the performance improvement and transmitter/receiver complexity."
],
[
"Optimization Frameworks",
" In this section, we specify the optimization frameworks proposed to solve the WSR and EE maximization problems, respectively."
],
[
"WMMSE-based AO algorithm for WSR problems",
"The WMMSE algorithm to solve the sum rate maximization problem in RS without a multicast message is proposed in [25].",
"It is extended to solve the WSR maximization problems in this work.",
"We firstly explain the procedure to solve the Problem $\\textrm {WSR}_{\\textrm {1-layer RS}}$ and then specify how the WSR problem of MU–LP, the generalized RS and NOMA can be solved correspondingly.",
"Considering 1-layer RS, user-$k$ decodes the super-common stream $s_{0}$ and the private stream $s_{k}$ sequentially using one layer of SIC.",
"$s_{0}$ and $s_{k}$ are respectively estimated using the equalizers $g_{k,0}$ and $g_{k}$ .",
"Once $s_{0}$ is successfully decoded by $\\hat{s}_{0}=g_{k,0}y_k$ and removed from $y_k$ , $s_{k}$ is decoded by $\\hat{s}_{k}=g_{k}(y_k-\\mathbf {h}_k^H\\mathbf {{p}}_{0}\\hat{s}_{0})$ .",
"The Mean Square Errors (MSEs) of decoding $s_{0}$ and $s_{k}$ are calculated as $\\begin{aligned}&\\varepsilon _{k,0}\\triangleq \\mathbb {E}\\lbrace |\\hat{s}_{k,0}-s_{k,0}|^{2}\\rbrace =|g_{k,0}|^2T_{k,0}-2\\Re \\lbrace g_{k,0}\\mathbf {h}_k^H\\mathbf {p}_{0}\\rbrace +1,\\\\&\\varepsilon _{k}\\triangleq \\mathbb {E}\\lbrace |\\hat{s}_{k}-s_{k}|^{2}\\rbrace =|g_{k}|^2T_k-2\\Re \\lbrace g_{k}\\mathbf {h}_k^H\\mathbf {p}_k\\rbrace +1,\\end{aligned}\\vspace{-2.84526pt}$ where $T_{k,0}\\triangleq |\\mathbf {h}_k^H\\mathbf {p}_{0}|^2+\\sum _{j\\in \\mathcal {K}}|\\mathbf {h}_k^H\\mathbf {p}_{j}|^2+1$ and $T_{k}\\triangleq T_{k,0}-|\\mathbf {h}_k^H\\mathbf {p}_{0}|^2$ .",
"By solving $\\frac{\\partial \\varepsilon _{k,0}}{\\partial g_{k,0}}=0$ and $\\frac{\\partial \\varepsilon _{k}}{\\partial g_{k}}=0$ , the optimum MMSE equalizers are given by $\\begin{aligned}&g_{k,0}^{\\mathrm {MMSE}}=\\mathbf {p}_{0}^H\\mathbf {h}_k{T}_{k,0}^{-1},\\,\\,g_{k}^{\\mathrm {MMSE}}=\\mathbf {p}_k^H\\mathbf {h}_k{T}_{k}^{-1}.\\end{aligned}\\vspace{-2.84526pt}$ Substituting (REF ) into (REF ), the MMSEs become $\\varepsilon _{k,0}^{\\textrm {MMSE}} ={({T}_{k,0}-|\\mathbf {h}_k^H\\mathbf {p}_{k}|^2)}/{{T}_{k,0}}$ and $ \\varepsilon _{k}^{\\textrm {MMSE}} ={(T_{k}-|\\mathbf {h}_k^H\\mathbf {p}_{k}|^2)}/{T_{k}}.$ Then the SINRs of $s_0$ and $s_k$ can be transformed to $\\gamma _{k,0}={1}/{\\varepsilon _{k,0}^{\\textrm {MMSE}}}-1$ and $\\gamma _{k}={1}/{\\varepsilon _{k}^{\\textrm {MMSE}}}-1$ .",
"The rates become $R_{k,0}=-\\log _{2}(\\varepsilon _{k,0}^{\\textrm {MMSE}})$ and $R_{k}=-\\log _{2}(\\varepsilon _{k}^{\\textrm {MMSE}})$ .",
"By introducing the positive weights ($w_{k,0},w_{k}$ ), the WMSEs of decoding $s_0$ and $s_k$ at user-$k$ are defined as $\\xi _{k,0}\\triangleq w_{k,0}\\varepsilon _{k,0}-\\log _{2}(w_{k,0}),\\,\\,\\xi _{k}\\triangleq w_{k}\\varepsilon _{k}-\\log _{2}(w_{k}).\\vspace{-2.84526pt}$ Then the Rate-WMMSE relationships are established as $\\begin{aligned}\\xi _{k,0}^{\\textrm {MMSE}}&\\triangleq \\min _{w_{k,0},g_{k,0}}\\xi _{k,0}=1-R_{k,0},\\\\\\xi _{k}^{\\textrm {MMSE}}&\\triangleq \\min _{w_{k},g_{k}}\\xi _{k}=1-R_{k}.\\end{aligned}\\vspace{-4.2679pt}$ where $\\xi _{k,0}^{\\textrm {MMSE}}$ and $\\xi _{k}^{\\textrm {MMSE}}$ are obtained by substituting the optimum MMSE equalizers $g_{k,0}^*$ , $g_{k}^*$ and the optimum MMSE weights $w_{k,0}^*$ , $w_{k}^*$ back to the WMSEs.",
"The optimum MMSE equalizers and MMSE weights are $g_{k,0}^*=g_{k,0}^{\\textrm {MMSE}}$ and $g_{k}^*=g_{k}^{\\textrm {MMSE}}$ , respectively $w_{k,0}^*=w_{k,0}^{\\textrm {MMSE}}\\triangleq (\\varepsilon _{k,0}^{\\textrm {MMSE}})^{-1}$ and $w_{k}^*=w_{k}^{\\textrm {MMSE}}\\triangleq (\\varepsilon _{k}^{\\textrm {MMSE}})^{-1}$ .",
"They are derived by checking the first order optimality conditions.",
"Based on the Rate-WMMSE relationships in (REF ), Problem (REF ) is equivalently transformed into the WMMSE problem $&\\min _{\\mathbf {{P}}, \\mathbf {x},\\mathbf {w},\\mathbf {g}} \\sum _{k\\in \\mathcal {K}}u_{k}\\xi _{k,tot} \\\\\\mbox{s.t.",
"}\\quad &X_{k,0}+ \\xi _{k,0}\\le 1-R_k^{th}, \\forall k\\in \\mathcal {K}\\\\&X_{0}+\\sum _{j\\in \\mathcal {K}}X_{j,0}+1\\ge \\xi _{k,0}, \\forall k\\in \\mathcal {K}\\\\& X_{0}\\le -R_0^{th}\\\\& X_{k,0}\\le 0, \\forall k\\in \\mathcal {K}\\\\& \\text{tr}(\\mathbf {P}\\mathbf {P}^{H})\\le P_{t}$ where $\\mathbf {x}=[X_{0},X_{1,0},\\ldots ,X_{K,0}]$ is the transformation of the common rate $\\mathbf {c}$ .",
"The MMSE weights and equalizers are $\\mathbf {w}=[w_{1,0},\\ldots ,w_{K,0},w_{1},\\ldots ,w_{K}]$ and $\\mathbf {g}=[g_{1,0},\\ldots ,g_{K,0},g_{1},\\ldots ,g_{K}]$ , respectively $\\xi _{k,tot}=X_{k,0}+\\xi _{k},\\forall k\\in \\mathcal {K}$ .",
"Denote $\\mathbf {w}^{\\mathrm {MMSE}}$ and $\\mathbf {g}^{\\mathrm {MMSE}}$ as two vectors formed by the corresponding optimum MMSE equalizers and weights obtained by minimizing (REF ) with respect to $\\mathbf {w}$ and $\\mathbf {g}$ , respectively.",
"$(\\mathbf {w}^{\\mathrm {MMSE}}, \\mathbf {g}^{\\mathrm {MMSE}})$ satisfies the Karush-Kuhn-Tucker (KKT) conditions of Problem (REF ).",
"Based on (REF ) and the common rate transformation $\\mathbf {c}=-\\mathbf {x}$ , Problem (REF ) can be transformed to Problem (REF ).",
"The solution given by ($\\mathbf {P}^*,\\mathbf {c}^*=-\\mathbf {x}^*$ ) meets the KKT optimality conditions of (REF ) for any point ($\\mathbf {P}^*,\\mathbf {x}^*,\\mathbf {w}^*,\\mathbf {g}^*$ ) satisfying the KKT optimality conditions of (REF ).",
"Hence, (REF ) and (REF ) are equivalent.",
"Though the joint optimization of ($\\mathbf {P},\\mathbf {x},\\mathbf {w},\\mathbf {g}$ ) in (REF ) is still non-convex, (REF ) is convex in each block of $(\\mathbf {P},\\mathbf {x})$ , $\\mathbf {w}$ , $\\mathbf {g}$ by fixing the other two blocks.",
"The block-wise convexity of (REF ) motivates us to use the Alternating Optimization (AO) algorithm to solve the problem.",
"Algorithm REF specifies the detailed steps of AO.",
"$(\\mathbf {w},\\mathbf {g})$ and $(\\mathbf {P},\\mathbf {x})$ are updated iteratively until the convergence of the WSR.",
"$\\mathrm {WSR}^{[n]}$ is the WSR calculated based on the updated $(\\mathbf {P},\\mathbf {x})$ at iteration $[n]$ .",
"The convergence of the AO algorithm is guaranteed [25] since $\\mathrm {WSR}^{[n]}$ is increasing with $n$ and it is bounded above for a given power constraint.",
"Note that the initialization of $\\mathbf {P}$ will influence the point of convergence due to the non-convexity of the problem.",
"[t!]",
"Initialize: $n\\leftarrow 0$ , $\\mathbf {P}^{[n]}$ , $\\mathrm {WSR}^{[n]}$ $|\\mathrm {WSR}^{[n]}-\\mathrm {WSR}^{[n-1]}|\\le \\epsilon $ $n\\leftarrow n+1$ $\\mathbf {P}^{[n-1]}\\leftarrow \\mathbf {P}$ $\\mathbf {w}\\leftarrow \\mathbf {w}^{\\mathrm {MMSE}}(\\mathbf {P}^{[n-1]})$ ; $\\mathbf {g}\\leftarrow \\mathbf {g}^{\\mathrm {MMSE}}(\\mathbf {P}^{[n-1]})$ update $(\\mathbf {x},\\mathbf {P})$ by solving (REF ) using the updated $\\mathbf {w}, \\mathbf {g}$ ; WMMSE-based AO algorithm When CSIT is imperfect, the sampling-based method proposed in [25] is adopted to approximate the average rate over the CSIT error distribution for a given channel state estimate.",
"The precoders are designed to maximize the average rate by using the optimization framework described above.",
"The WSR maximization problem of MU–LP, the generalized RS and NOMA are solved by respectively reformulating them into the equivalent WMMSE problem and using the corresponding AO algorithms to solve them."
],
[
"SCA-based algorithm for EE problems",
"The SCA-based algorithm to solve the two-user EE maximization problem of RS without individual QoS rate constraints in the unicast-only transmission is proposed in [17].",
"It is extended to solve the EE maximization problems in the NOUM transmission in this work.",
"We firstly explain the procedure to solve the Problem $\\textrm {EE}_{\\textrm {1-layer RS}}$ and then specify how the EE problem of MU–LP, the generalized RS and NOMA are solved correspondingly.",
"Comparing with the EE optimization problem (9) in [17], the main difference of Problem (REF ) in the NOUM transmission lies in the introduced QoS rate, Constraints (REF ) and the multicast rate $C_0$ in (REF ), (REF ), (REF ).",
"Similar as [17], we first use scalar variables $\\omega ^2$ , $z$ and $t$ , respectively to represent the WSR, total power consumption and EE metric, then Problem (REF ) is equivalently transformed into $\\max _{\\mathbf {c},\\mathbf {{P}},\\omega , z,t}&\\quad \\,\\, t\\\\ \\mbox{s.t.",
"}\\quad &\\frac{\\omega ^{2}}{z}\\ge t \\\\ &u_{0}C_{0}+\\sum _{k\\in \\mathcal {K}}u_k\\left(C_{k,0}+R_{k}\\right)\\ge \\omega ^{2} \\\\ &z\\ge \\frac{1}{\\eta }\\mathrm {tr}(\\mathbf {P}\\mathbf {P}^{H})+P_{\\textrm {cir}} \\\\ &\\textrm {(\\ref {c1_rs}) -- (\\ref {c5_rs})}$ The equivalence between (REF ) and (REF ) is established since (REF )–() hold with equality at optimum.",
"By introducing variables $\\mathbf {\\alpha }=[\\alpha _1,\\ldots ,\\alpha _K]^T$ , Constraints (REF ) and () become [left= (REF ) , ()]align Ck,0+kRkth, k K u0C0+kKuk(Ck,0+k)2 Rkk, k K By adding variables $\\mathbf {\\vartheta }=[\\vartheta _1,\\ldots , \\vartheta _K]^T$ , Constraint (REF ) is transformed into [left=(REF )]align k2k, k K 1+kk , k K By further introducing $\\mathbf {\\beta }=[\\beta _1,\\ldots ,\\beta _K]^T$ to represent the interference plus noise at each user to decode its private steam, Constraint (REF ) is transformed into [left=(REF )]align |hkHpk|2kk-1 , k {1,2} kjk|hkHpj|2+1 , k {1,2} Therefore, Constraints (REF ) and () are equivalent to the Constraints $\\textrm {(\\ref {c1_rs transform})}, \\textrm {(\\ref {con: rate non convex})},\\textrm {(\\ref {con: v and a})},\\textrm {(\\ref {con: 1+SINR})}$ .",
"The same method is used to transform Constraint (REF ).",
"By introducing variable sets $\\mathbf {\\alpha }_0=[\\alpha _{1,0},\\ldots ,\\alpha _{K,0}]^H$ , $\\mathbf {\\vartheta }_0=[\\vartheta _{1,0},\\ldots , \\vartheta _{K,0}]^T$ , $\\mathbf {\\beta }_{0}=[\\beta _{1,0},\\ldots ,\\beta _{K,0}]^T$ , (REF ) becomes [left=(REF )]align C0+jKCj,0k,0,kK k,02k,0 , k K |hkHp0|2k,0k,0-1 , k K k,0jK|hkHpj|2+1 Table: Computational complexity comparison of the algorithms using different strategiesHence, Problem (REF ) is equivalently transformed into $ \\begin{aligned}\\max _{\\begin{array}{c}\\mathbf {c},\\mathbf {{P}}, \\omega , z,t,\\\\ \\mathbf {\\alpha }_{0},\\mathbf {\\alpha },\\mathbf {\\vartheta }_{0},\\mathbf {\\vartheta },\\mathbf {\\beta }_{0},\\mathbf {\\beta }\\end{array}}&\\quad \\,\\, t\\\\ \\mbox{s.t.",
"}\\quad &\\quad \\textrm {(\\ref {c2_rs})},\\textrm {(\\ref {c4_rs})},\\textrm {(\\ref {c5_rs})},\\textrm {(\\ref {EE RSMA transform constraint 0})},\\textrm {(\\ref {EE RSMA transform constraint 2})} \\\\&\\quad \\textrm {(\\ref {c1_rs transform})}, \\textrm {(\\ref {con: rate non convex})},\\textrm {(\\ref {con: v and a})},\\textrm {(\\ref {con: 1+SINR})}, \\textrm {(\\ref {con: common})}\\end{aligned}$ However, Constraints (REF ), (REF ) and (REF ) are non-convex.",
"Linear approximation methods adopted in [17] are used to approximate the non-convex part of the constraints in each iteration.",
"Left side of (REF ) is approximated at the point ($\\omega ^{[n]},z^{[n]}$ ) of the $n$ th iteration by $\\frac{\\omega ^{2}}{z}\\ge \\frac{2\\omega ^{[n]}}{z^{[n]}}\\omega -(\\frac{\\omega ^{[n]}}{z^{[n]}})^{2}z\\triangleq \\Omega ^{[n]}(\\omega ,z)$ .",
"The left side of (REF ) is approximated at the point ($\\mathbf {p}_k^{[n]},\\beta _{k}^{[n]}$ ) as ${\\left|\\mathbf {{h}}_{k}^{H}\\mathbf {{p}}_{k}\\right|^{2}}/{\\beta _{k}}\\ge {2\\mathrm {Re}((\\mathbf {{p}}_{k}^{[n]})^{H}\\mathbf {{h}}_{k}\\mathbf {{h}}_{k}^{H}\\mathbf {{p}}_{k})}/{\\beta _{k}^{[n]}}-({|\\mathbf {{h}}_{k}^{H}\\mathbf {{p}}_{k}^{[n]}|}/{\\beta _{k}^{[n]}})^{2}\\beta _{k}\\triangleq \\Psi _{k}^{[n]}(\\mathbf {{p}}_{k},\\beta _{k})$ .",
"Similarly, the left side of (REF ) is approximated at the point ($\\mathbf {p}_0^{[n]},\\beta _{k,0}^{[n]}$ ) by $\\Psi _{k,0}^{[n]}(\\mathbf {{p}}_{0},\\beta _{k,0})={2\\mathrm {Re}((\\mathbf {{p}}_{0}^{[n]})^{H}\\mathbf {{h}}_{k}\\mathbf {{h}}_{k}^{H}\\mathbf {{p}}_{0})}/{\\beta _{k,0}^{[n]}}-({|\\mathbf {{h}}_{k}^{H}\\mathbf {{p}}_{0}^{[n]}|}/{\\beta _{k,0}^{[n]}})^{2}\\beta _{k,0}$ .",
"Based on the above approximations, Problem (REF ) is approximated at iteration $n$ as $\\begin{aligned}\\max _{\\begin{array}{c}\\mathbf {c},\\mathbf {{P}}, \\omega , z,t,\\\\ \\mathbf {\\alpha }_{0},\\mathbf {\\alpha },\\mathbf {\\vartheta }_{0},\\mathbf {\\vartheta },\\mathbf {\\beta }_{0},\\mathbf {\\beta }\\end{array}}&\\quad \\,\\, t\\\\ \\mbox{s.t.",
"}\\quad &\\quad \\Omega ^{[n]}(\\omega ,z)\\ge t \\\\&\\quad \\Psi _{k}^{[n]}(\\mathbf {{p}}_{k},\\beta _{k}) \\ge \\vartheta _{k}-1, \\forall k \\in \\mathcal {K}\\\\&\\quad \\Psi _{k,0}^{[n]}(\\mathbf {{p}}_{0},\\beta _{k,0})\\ge \\vartheta _{k,0}-1, \\forall k \\in \\mathcal {K}\\\\&\\quad \\textrm {(\\ref {c2_rs})},\\textrm {(\\ref {c4_rs})},\\textrm {(\\ref {c5_rs})},\\textrm {(\\ref {c1_rs transform})},\\textrm {(\\ref {con: rate non convex})},\\textrm {(\\ref {con: v and a})}, \\\\&\\quad \\textrm {(\\ref {con: private noise interference})},\\textrm {(\\ref {con: common rate})},\\textrm {(\\ref {con: common SINR})},\\textrm {(\\ref {con: common noise interference})}\\end{aligned}$ Problem (REF ) is convex and can be solved using CVX in Matlab [41].",
"The details of the SCA-based algorithm is specified in Algorithm REF .",
"In each iteration $[n]$ , the approximate Problem (REF ) defined around the solution of iteration $[n-1]$ is solved.",
"[h!]",
"Initialize: $n\\leftarrow 0$ , $t^{[n]},\\omega ^{[n]}, z^{[n]}$ , $\\mathbf {P}^{[n]}, \\mathbf {\\beta }_0^{[n]}, \\mathbf {\\beta }^{[n]}$ $|t^{[n]}-t^{[n-1]}|<\\epsilon $ $n\\leftarrow n+1$ Solve (REF ) using $\\omega ^{[n-1]}$ , $z^{[n-1]}$ , $\\mathbf {P}^{[n-1]}$ , $\\mathbf {\\beta }_0^{[n-1]}$ , $\\mathbf {\\beta }^{[n-1]}$ and denote the optimal values as $\\omega ^*$ , $z^*$ , $\\mathbf {P}^*$ , $\\mathbf {\\beta }_0^*$ , $\\mathbf {\\beta }^*$ Update $t^{[n]}\\leftarrow t^*$ , $\\omega ^{[n]}\\leftarrow \\omega ^*$ , $z^{[n]}\\leftarrow z^*$ , $\\mathbf {P}^{[n]}\\leftarrow \\mathbf {P}^*$ , $\\mathbf {\\beta }_0^{[n]}\\leftarrow \\mathbf {\\beta }_0^*$ , $\\mathbf {\\beta }^{[n]}\\leftarrow \\mathbf {\\beta }^*$ SCA-based beamforming algorithm Initialization: The precoder $\\mathbf {P}^{[0]}$ is initialized by finding the feasible beamformer satisfying the constraints (REF )–(REF ).",
"We assume in the initialization that $ C_{0}=R_0$ , $C_{k,0}=0,\\forall k \\in \\mathcal {K}$ .",
"The non-convex rate constraints are relaxed based on the convex relaxations introduced in [7].",
"After relaxation, the feasibility problem becomes a Second Order Cone Problem (SOCP) and can be solved by the standard solvers in MatLab.",
"$\\omega ^{[0]}$ , $z^{[0]}$ , $\\beta _k^{[0]}$ and $\\beta _{k,0}^{[0]}$ are initialized by respectively replacing the inequalities of (), (), (REF ) and (REF ) with equalities.",
"Convergence Analysis: The solution of Problem (REF ) in iteration $[n]$ is also a feasible solution of the problem in iteration $[n+1]$ since the approximated Problem (REF ) in iteration $[n+1]$ is defined around the solution of iteration $[n]$ .",
"Therefore, the EE objective $t^{[n+1]}$ is larger than or equal to $t^{[n]}$ .",
"Algorithm REF generates a nondecreasing sequence of objective values.",
"Moreover, the EE objective $t$ is bounded above by the transmit power constraint.",
"Hence, Algorithm REF is guaranteed to converge while the global optimality of the achieved solution can not be guaranteed.",
"The EE maximization problem of MU–LP, the generalzied RS and NOMA are solved by respectively approximating them using the above transformation and approximation, which are then solved iteratively by the corresponding SCA-based beamforming algorithms as well."
],
[
"Computational complexity analysis",
"The computational complexity of both Algorithm REF and Algorithm REF for all strategies are illustrated in Table REF under the assumption that $N_t\\ge K$ .",
"At each iteration of Algorithm REF , the MMSE equalizers and weights $(\\mathbf {w},\\mathbf {g})$ are updated with complexity $\\mathcal {O}(K^2N_t)$ for MU–LP and 1-layer RS-assisted strategies.",
"The complexity of the generalized RS to update the equalizers and weights is $\\mathcal {O}(2^KK^2N_t)$ .",
"Both SC–SIC and SC–SIC per group strategies require complexity $\\mathcal {O}(K^3N_t)$ to update the MMSE equalizers and weights.",
"The precoders and common rate vector $(\\mathbf {P},\\mathbf {x})$ are then updated by solving the SOCP problem.",
"Each SOCP is solved by using interior-point method with computational complexity $\\mathcal {O}([X]^{3.5})$ , where $X$ is the total number of variables in the equivalent SOCP problem [42].",
"For each strategy, the number of variables in the SOCP problem is given by $X_{\\textrm {MU--LP}}=KN_t+N_t$ , $X_{\\textrm {1-layer RS}}=KN_t+N_t+K+1$ , $X_{\\textrm {SC--SIC}}=KN_t+2$ , $X_{\\textrm {SC--SIC per group}}=KN_t+N_t$ , $X_{\\textrm {Generalized RS}}=2^KN_t+2^{K-1}K+1-K$ .",
"The total number of iterations required for the convergence is $\\mathcal {O}(\\log (\\epsilon ^{-1}))$ , where $\\epsilon $ the convergence tolerance of Algorithm 1.",
"As specified in Table REF , SC–SIC, SC–SIC per group and the generalized RS have high scheduling complexity since Algorithm REF is required to be repeated for all possible decoding order and user grouping at the scheduler.",
"At each iteration of Algorithm REF , the approximated SOCP problem is solved.",
"Though additional variables $\\mathbf {\\alpha }_{0}$ , $\\mathbf {\\alpha }$ , $\\mathbf {\\vartheta }_{0}$ , $\\mathbf {\\vartheta }$ , $\\mathbf {\\beta }_{0}$ , $\\mathbf {\\beta }$ are introduced for convex relaxation, the main complexity still comes from the precoder design.",
"Algorithm REF is also required to be repeated for all possible decoding order and user grouping.",
"Therefore, Algorithm REF and Algorithm REF has the same worst-case computational complexity approximation."
],
[
"Numerical Results of WSR problem",
"In this section, we evaluate the WSR of all the transmission strategies in various user deployments and network loads.",
"Besides the typical underloaded scenarios appearing in MU-MIMO and massive MIMO, we also investigate overloaded scenarios.",
"Overloaded regimes, described as the scenarios where the number of served users exceeds the number of transmitting antennas, are becoming more important due to the growing demands for ultra-high connectivity [43], [35], [44].",
"Applications of overloaded scenarios can also be found in multibeam satellite systems where each beam carries the messages of multiple users, forming a multicast group [45], as well as in NOMA [38], [44], and coded caching [46]."
],
[
"Two-user deployments",
"When $K=2$ , the generalized RS model reduces to the 1-layer RS model.",
"Hence, we use `RS' to represent both strategies.",
"RS is still a more general strategy that encompasses MU–LP and SC–SIC-based NOUM strategies as special cases.",
"We compare MU–LP, RS and SC–SIC-based NOUM strategies.",
"The OMA transmission is considered as the baseline in which a multicast stream is transmitted for both users while the superimposed unicast stream is only intended for a single user.",
"This user decodes the multicast and unicast streams by using SIC while the other user only decodes the multicast stream.",
"The receiver complexities of MU–LP, RS and SC–SIC-assisted strategies are the same when $K=2$ .",
"Only one layer of SIC is required.",
"We assume the BS has four or two antennas ($N_t=2, 4$ ) and serves two single-antenna users.",
"The initialization of precoders follows the methods used in [14], [25].",
"SNR is fixed to 20 dB.",
"The boundary of the rate region is the set of achievable points calculated by solving the WSR maximization problem with various weights assigned to users.",
"The weight of user-1 is fixed to $u_1=1$ for each weight of user-2 in $u_2 \\in 10^{[-3, -1,-0.95,\\cdots ,0.95,1, 3]}$ as used in [14].",
"To investigate the largest achievable rate region of the unicast messages, the rate constraints of the unicast messages are set to 0 in all strategies $R_k^{th}=0,\\forall k\\in \\lbrace 1,2\\rbrace $ .",
"We first consider the channel model when $\\mathbf {h}_k$ has independent and identically distributed (i.i.d.)",
"complex Gaussian entries, i.e., $\\mathcal {CN}(0,\\sigma _{k}^2)$ .",
"Fig.",
"REF shows the rate region comparison of different strategies averaged over 100 random channel realizations and $\\sigma _{1}^2=1$ .",
"When $\\sigma _{2}^2=1$ (subfigure (a) and (c)), SC–SIC performs worst as there is no disparity of averaged channel strength.",
"In contrast, MU–LP achieves a rate region close to RS.",
"However, as the number of transmit antenna decreases, the rate region gap between MU–LP and RS becomes more obvious.",
"When $\\sigma _{2}^2=0.09$ (subfigure (b) and (d)), the average channel strength disparity between the users is 10 dB.",
"The rate region of SC–SIC comes closer to RS while that of MU–LP becomes worse.",
"RS bridges MU–LP and SC–SIC as well and achieves a better rate region.",
"In all subfigures, the rate region of OMA is the worst as it is a line segment between the two extremity points of the two users' achievable rate since the unicast rate is dedicatedly allocated to a single user in OMA.",
"The points along the line segment is achieved by time-sharing.",
"RS exhibits a clear rate region improvement over OMA.",
"We further investigate specific channel realizations to get some insights into the influence of user angle and channel strength disparity on the system performance.",
"Following the two-user deployment in [14], the channels of the users are realized as $\\mathbf {h}_1=\\left[1, 1, 1, 1\\right]^H,\\mathbf {h}_2=\\gamma \\times \\left[1,e^{j\\theta },e^{j2\\theta },e^{j3\\theta }\\right]^H.$ $\\gamma $ controls the channel strength difference between the users.",
"$\\gamma =1$ and $\\gamma =0.3$ represent equal channel strength and 10 dB channel strength difference, respectively.",
"For each $\\gamma $ , we consider $\\theta \\in \\left[\\frac{\\pi }{9},\\frac{2\\pi }{9},\\frac{\\pi }{3},\\frac{4\\pi }{9}\\right]$ .",
"The user channels are sufficiently aligned when $0<\\theta <\\frac{\\pi }{9}$ while the channels are sufficiently orthogonal when $\\frac{4\\pi }{9}<\\theta <\\frac{\\pi }{2}$ .",
"Figure: Rate region comparison of different strategies in perfect CSIT, γ=1\\gamma =1, R 0 th =0.5R_0^{th}=0.5 bit/s/Hz.Figure: Rate region comparison of different strategies in perfect CSIT, γ=0.3\\gamma =0.3, R 0 th =0.5R_0^{th}=0.5 bit/s/Hz.Figure: Rate region comparison of different strategies in perfect CSIT, γ=1\\gamma =1, R 0 th =1.5R_0^{th}=1.5 bit/s/Hz.Fig.",
"REF –REF show the achievable rate region comparison of different strategies in perfect CSIT.",
"In all figures, the rate region of RS is confirmed to be equal to or larger than that of SC–SIC and MU–LP.",
"RS performs well for all investigated channel strength disparities as well as angles between the user channels.",
"In contrast, SC–SIC and MU–LP are sensitive to the channel strength disparities and channel angles.",
"In each figure, RS exhibits a clear rate region improvement over MU–LP when the user channels are closely aligned.",
"When the users have similar channel strengths or (semi-)orthogonal channel angles, the performance of SC–SIC is much worse than RS.",
"Comparing with MU–LP and SC–SIC, RS is more robust to a wide range of channel strength difference and channel angles among users.",
"This WSR gain comes at no additional cost for the receivers since one layer of SIC is required for MU–LP and SC–SIC in the two-user deployments.",
"As the multicast rate constraint $R_0^{th}$ increases, the rate region of each strategy decreases.",
"This can be observed by comparing the corresponding subfigures of Fig.",
"REF and Fig.",
"REF .",
"However, the rate region gaps among the three strategies decrease when $R_0^{th}$ increases since a larger portion of the power is used for transmitting the multicast stream via the super-common stream.",
"RS achieves a better unicast rate region than MU–LP and SC–SIC when a larger portion of the transmit power is allocated to the unicast streams.",
"Adequate power allocation for the unicast streams allows RS to better determine the level of the interference to decode and treat as noise."
],
[
"Imperfect CSIT",
"When CSIT is imperfect, the estimated channels of user-1 and user-2 are realized as $\\nonumber \\widehat{\\mathbf {h}}_{1}=\\left[1,1,1,1\\right]^H$ and $\\nonumber \\widehat{\\mathbf {h}}_2=\\gamma \\times \\left[1, e^{j\\theta }, e^{j2\\theta }, e^{j3\\theta }\\right]^H$ , respectively.",
"The precoders are initialized and designed using the estimated channels $\\widehat{\\mathbf {h}}_{1},\\widehat{\\mathbf {h}}_{2}$ and the same methods as stated in [25], [14].",
"The real channel realization is obtained as $\\mathbf {h}_k=\\widehat{\\mathbf {h}}_{k}+\\widetilde{\\mathbf {h}}_{k},\\forall k\\in \\lbrace 1,2\\rbrace $ , where $\\widetilde{\\mathbf {h}}_{k}$ is the estimation error of user-$k$ with independent and identically distributed (i.i.d.)",
"complex Gaussian entries drawn from $\\mathcal {CN}(0,\\sigma _{e,k}^2)$ .",
"The error covariances of user-1 and user-2 are $\\sigma _{e,1}^2=P_t^{-0.6}$ and $\\sigma _{e,2}^2=\\gamma P_t^{-0.6}$ , respectively.",
"Other unspecified parameters remain consistent with perfect CSIT results.",
"After generating 1000 different channel error samples for each user, each point in the rate region is the average rate over the resulting 1000 channels.",
"Note that the average rate is a short-term (instantaneous) measure that captures the expected performance over the CSIT error distribution for a given channel state estimate.",
"Readers are referred to [25] for more details of the channel model when CSIT is imperfect.",
"Figure: Rate region comparison of different strategies in imperfect CSIT, γ=1\\gamma =1, R 0 th =0.5R_0^{th}=0.5 bit/s/Hz.Fig.",
"REF shows the results when $R_0^{th}=0.5$ bit/s/Hz in imperfect CSIT, $\\gamma =1$ .",
"Comparing the corresponding figures of perfect and imperfect CSIT (Fig.",
"REF and Fig.",
"REF ), we observe that the rate region gap between RS and MU–LP increases in imperfect CSIT.",
"RS is more robust to a wide range of CSIT inaccuracy, channel strength difference and channel angles among users.",
"The transmit scheduler of RS is simpler as it copes with any user deployment scenarios.",
"RS always outperforms MU–LP and SC–SIC.",
"In the three-user deployments, we compare MU–LP, SC–SIC, SC–SIC per group, 1-layer RS and the generalized RS transmission strategies.",
"In the SC–SIC per group, the grouping method and decoding order are required to be jointly optimized with the precoder in order to maximize the WSR, which results in very high computational burden at the BS as the number of user increases.",
"To reduce the complexity, we consider a fixed grouping method.",
"We assume user-1 is in group-1 while user-2 and user-3 are in group-2.",
"The decoding order will be optimized together with the precoder."
],
[
"Perfect CSIT",
"Following the precoder initialization and channel realizations for three-user deployments in [14], we consider specific channel realizations given by $\\mathbf {h}_1=\\left[1, 1, 1, 1\\right]^H$ , $\\mathbf {h}_2=\\gamma _1\\times [1, e^{j\\theta _1},e^{j2\\theta _1}, e^{j3\\theta _1}]^H$ , $\\mathbf {h}_3=\\gamma _2\\times [1, e^{j\\theta _2}, e^{j2\\theta _2}, e^{j3\\theta _2}]^H$ for the underloaded three-user deployments ($N_t=4$ ).",
"For the overloaded three-user deployments ($N_t=2$ ), the channels are realized as $\\mathbf {h}_1=\\left[1, 1\\right]^H$ , $\\mathbf {h}_2=\\gamma _1\\times [1, e^{j\\theta _1}]^H$ , $\\mathbf {h}_3=\\gamma _2\\times [1, e^{j\\theta _2}]^H.$ $\\gamma _1, \\gamma _2$ and $\\theta _1,\\theta _2$ are control variables.",
"We assume user-1 and user-2 have equal channel strength ($\\gamma _1=1$ ) and there is a 10 dB channel strength difference between user-1/user-2 and user-3 ($\\gamma _2=0.3$ ).",
"For the given set of $\\gamma _1,\\gamma _2$ , $\\theta _1$ adopts value from $\\theta _1=\\left[\\frac{\\pi }{9},\\frac{2\\pi }{9},\\frac{\\pi }{3},\\frac{4\\pi }{9}\\right]$ and $\\theta _2=2\\theta _1$ .",
"The weights of the users are assumed to be equal to $u_1=u_2=u_3=1$ .",
"The QoS rate requirements of the multicast and unicast messages are assumed to be equal and the rate threshold is increasing with SNR.",
"For $\\mathrm {SNR}=[0,5,10,15,20,25,30]$ dBs, the corresponding rate constraint vector of message-$j$ is $\\mathbf {r}_j^{th}=[0.005, 0.01, 0.05, 0.15, 0.3, 0.4, 0.4]$ bit/s/Hz, $\\forall j\\in \\lbrace 0,1,2,3\\rbrace $ .",
"Figure: WSR versus SNR comparison of different strategies for underloaded three-user deployment in perfect CSIT, γ 1 =1,γ 2 =0.3\\gamma _1=1,\\gamma _2=0.3, N t =4N_t=4.Figure: WSR versus SNR comparison of different strategies for overloaded three-user deployment in perfect CSIT, γ 1 =1,γ 2 =0.3\\gamma _1=1,\\gamma _2=0.3, N t =2N_t=2.Fig.",
"REF and Fig.",
"REF show the results of WSR versus SNR comparison of different strategies in perfect CSIT for the underloaded and overloaded three-user deployments, respectively.",
"RS exhibits a clear WSR gain over 1-layer RS, MU–LP, SC–SIC, SC–SIC per group in both figures.",
"1-layer RS achieves a more stable performance than MU–LP, SC–SIC, SC–SIC per group as the channel strength disparity and channel angles among users changes.",
"The WSR performance of MU–LP deteriorates as the channel angles among users become smaller (aligned) or the network loads become overloaded.",
"In contrast, the WSR performance of SC–SIC deteriorates as the channel angles among users become larger or the network load becomes underloaded.",
"SC–SIC per group compensates the shortcomings of SC–SIC.",
"It achieves a better performance than SC–SIC for orthogonal channels or underloaded network loads as it allows the inter-group interference to be treated as noise.",
"Thanks to the ability of partially decoding the interference and partially treating the interference as noise, RS and 1-layer RS are less sensitive to the user channel orthogonality as well as the network loads.",
"Considering the trade-off between performance and complexity, 1-layer RS is the best choice since it has the lowest receiver complexity and a more robust performance over various user deployments and network loads.",
"Figure: Convergence of the proposed two algorithms with different transmission strategies, θ 1 =2π 9\\theta _1=\\frac{2\\pi }{9}, θ 2 =4π 9\\theta _2=\\frac{4\\pi }{9}, N t =2N_t=2.The convergence rates of all the considered transmission strategies for a specific channel realization are analyzed in Fig.",
"REF a.",
"The rate constraints of all messages are equal to the corresponding value in $\\mathbf {r}_j^{th}$ for a given SNR (i.e.",
"when SNR = 5 dB, $R_j^{th}=0.01$ bit/s/Hz, $\\forall j\\in \\lbrace 0,1,2,3\\rbrace $ ).",
"As the decoding orders in RS, SC–SIC and SC–SIC per group are required to be optimized with the precoders, the convergence rate of the optimal decoding order that achieves the highest WSR for the corresponding transmission strategy is illustrated in Fig.",
"REF a.",
"For various SNR values, only a few iterations are required for each strategy to converge.",
"Our proposed WMMSE algorithm solves the WSR problem efficiently.",
"The Convex-Concave Procedure (CCP) algorithm proposed in [7] can be adopted to solve the WSR maximization problem by transforming the non-convex SINR constraints into a set of Difference of Convex (DC) constraints and approximated using the first-order Taylor expansion.",
"However, due to the individual QoS rate constraint in the investigated WSR maximization problem, additional variables representing the SINR of users' unicast and multicast streams are introduced, which enlarge the dimension of variables in the SOCP problem to be solved in each iteration.",
"The convergence speed of using CCP-based algorithm is therefore slower.",
"Fig.",
"REF shows the convergence comparison of CCP and WMMSE-based algorithms using 1-layer RS and MU–LP.",
"For both algorithms, the initialization of precoders $\\mathbf {{P}}$ and the channel model are the same as discussed in Section REF .",
"For the CCP-based algorithm, $\\rho ,\\,\\rho _{0}$ are initialized by $2^{R_{k}^{th}}-1$ and $2^{R_{0}^{th}}-1$ , respectively.",
"We could draw the conclusion that the WMMSE-based algorithm converge faster than the CCP-based algorithm and both algorithms achieve almost the same performance.",
"Figure: Convergence comparison of CCP and WMMSE-based algorithms, γ=1\\gamma =1, θ=2π 9\\theta =\\frac{2\\pi }{9}, R 0 th =R k th =0.1R_0^{th}=R_k^{th}=0.1 bit/s/Hz.Figure: WSR versus CSIT inaccuracy comparison of different strategies over 100 random channel realizations."
],
[
"Imperfect CSIT",
"When CSIT is imperfect, we first investigate random channel realizations.",
"The channel of each user has i.i.d.",
"complex Gaussian entries.",
"Fig.",
"REF illustrates the WSR comparison of the generalized RS, 1-layer RS and MU–LP strategies averaged over 100 random channel realizations where $N_t=4$ , $u_1=u_2=u_3=1$ , $R_0^{th} =R_k^{th}=0.2$ bit/s/Hz and SNR = 20 dBs.",
"The inaccuracy of the channel is controlled by the error covariance defined as $\\sigma _{e,1}^2=\\sigma _{e,2}^2=\\sigma _{e,3}^2=P_t^{-\\tau }$ .",
"$\\tau =0$ represents a fixed quality with respect to SNR, e.g.",
"a constant number of feedback bits, and $\\tau =1$ corresponds to perfect CSIT in the DoF sense [25].",
"We assume there is a group of 20 candidate users in the system and only $K=3$ active users are selected.",
"For MU–LP, the User Scheduling (US) algorithm based on channel correlation proposed in [47] is adopted.",
"Its worst-case computational complexity is $\\mathcal {O}(N_t^3K)$ .",
"As RS-based strategies suit to all channel angles, the three users with best channel strength are selected.",
"The computational complexity of such US algorithm is $\\mathcal {O}(K)$ .",
"No User Scheduling (NUS) baseline schemes MU–LP: NUS, 1-layer RS: NUS, and RS: NUS where users are randomly selected are illustrated as well.",
"We observe from Fig.",
"REF that the WSR gap between 1-layer RS: NUS (1-layer RS: US) and MU–LP: NUS (MU–LP: US) increases as $\\tau $ decreases.",
"RS is more robust to the inaccuracy of CSIT.",
"Comparing the performance when US is considered, 1-layer RS outperforms MU–LP but it uses a simpler scheduling algorithm.",
"The generalized RS and 1-layer RS without US outperform MU–LP with US when $\\tau $ ranges from 0 to 0.3.",
"Therefore, RS-assisted strategies achieves non-negligible gains over MU–LP no matter whether US is considered or not.",
"When considering specific channel realizations, the precoder initialization and channel realizations follow the methods discussed in the two-user deployment of Section REF .",
"Readers are also referred to Appendix E in [14] for more details.",
"Other unspecified parameters remain consistent with the perfect CSIT scenarios of Section REF .",
"Figure: WSR versus SNR comparison of different strategies for overloaded three-user deployment in imperfect CSIT, γ 1 =1,γ 2 =0.3\\gamma _1=1,\\gamma _2=0.3, N t =2N_t=2.Fig.",
"REF shows the results of WSR versus SNR comparison in the overloaded three-user deployment with imperfect CSIT.",
"Comparing Fig.",
"REF and Fig.",
"REF , the WSR gap between RS and SC–SIC per group/MU–LP is enlarged when the CSIT becomes imperfect.",
"Though 1-layer RS has the lowest receiver complexity, it achieves a better WSR than SC–SIC, SC–SIC per group and MU–LP."
],
[
"Numerical Results of EE problem",
"In this section, we evaluate the EE performance of all the transmission strategies in various user deployments and network loads."
],
[
"Two-user deployments",
"Same as the numerical results of WSR problem, we compare MU–LP, RS and SC–SIC-assisted NOUM transmission strategies in the two-user deployments.",
"We first consider the scenarios when the channel of each user $\\mathbf {h}_k$ has i.i.d complex Gaussian entries with a certain variance, i.e., $\\mathcal {CN}(0,\\sigma _k^2)$ .",
"The variance of entries of $\\mathbf {h}_1$ is fixed to 1 ($\\sigma _1^2=1$ ) while the variance of entries of $\\mathbf {h}_2$ is varied ($\\sigma _2^2=1,0.09$ ).",
"The BS is equipped with two or four antennas and serves two single-antenna users.",
"Following the simulation parameters used in [17], the static power consumption is $P_{\\textrm {sta}}=30$ dBm and the dynamic power consumption is $P_{\\textrm {dyn}}=27$ dBm.",
"The power amplifier efficiency is $\\eta =0.35$ .",
"The weights allocated to the streams are equal to one, i.e., $u_0=u_1=u_2=1$ .",
"Figure: Energy Efficiency versus R 0 th R_0^{th} comparison of different strategies for two-user deployment in perfect CSIT, averaged over 100 random channels.",
"R 1 th =R 2 th =0.5R_1^{th}=R_2^{th}=0.5 bit/s/Hz, SNR = 10 dB.Fig.",
"REF shows the results of EE versus the multicast rate requirement $R_0^{th}$ comparison of three transmission strategies for the two-user deployment with perfect CSIT.",
"The proposed RS-assisted NOUM transmission outperforms SC–SIC and MU–LP in all considered user deployments.",
"Comparing subfigure (a) and (c), we observe that the EE gap between RS and MU–LP increases as the number of transmit antenna decreases.",
"MU–LP achieves a better EE performance in the underloaded regime.",
"In contrast, SC–SIC performs better in the overloaded regime.",
"Such observation of the EE performance is consistent with that of the WSR performance."
],
[
"Specific channel realizations",
"The specific channel realizations and relevant simulation parameters specified in Section REF are considered here.",
"In order to investigate the EE region achieved by the unicast streams, the rate allocated to the multicast stream is fixed at $R_0^{th}$ , i.e., $C_0=R_0^{th}$ .",
"In the following results, we assume $R_0^{th}=0.5$ bit/s/Hz and $u_0=1$ .",
"SNR is fixed at 10 dB and the transmitter is equipped with four tansmit antennas ($N_t=4$ ).",
"The unspecified parameters remain the same as in the random channel realization section.",
"The EE metric of each unicast stream is defined as the achievable unicast rate divided by the sum power.",
"The individual EE of user-$k$ is $\\textrm {EE}_k={R_{k,tot}}/{(\\frac{1}{\\eta }\\mathrm {tr}(\\mathbf {P}\\mathbf {P}^{H})+P_{\\textrm {cir}})},\\forall k\\in \\lbrace 1,2\\rbrace .$ Figure: Energy Efficiency region comparison of different strategies for two-user deployment in perfect CSIT, γ=1\\gamma =1.Figure: Energy Efficiency region comparison of different strategies for two-user deployment in perfect CSIT, γ=0.3\\gamma =0.3.Fig.",
"REF and Fig.",
"REF illustrate the EE region of different strategies for the two-user deployment in perfect CSIT, $\\gamma =1$ and $\\gamma =0.3$ , respectively.",
"The EE region of RS is always larger than or equal to the EE region of MU–LP or SC–SIC in both figures.",
"The EE performance of MU–LP is superior when the user channels are sufficiently aligned.",
"In contrast, the EE performance of SC–SIC is superior when there is a 10 dB channel strength difference or the user channels are aligned.",
"Comparing with the EE regions of the unicast-only transmission illustrated in [17], the EE region improvement of RS in Fig.",
"REF and Fig.",
"REF is not obvious due to the introduced multicast stream.",
"As discussed in Section REF , the overall optimization space is reduced since part of transmit power is allocated to the multicast stream so as to meet the multicast rate requirement.",
"Same as the discussion of Fig.",
"REF , the EE region of OMA is a line segment between the two corner points of the users' achievable EE.",
"Therefore, the EE region of OMA is the worst and RS achieves a much better EE region improvement over OMA.",
"Figure: Energy Efficiency versus P dyn P_{\\textrm {dyn}} comparison of different strategies for underloaded three-user deployment in perfect CSIT.",
"N t =4N_t=4.",
"In the three-user deployment, we focus on the specific channel realizations and the influence of different $P_{\\textrm {dyn}}$ values on the EE performance is further investigated.",
"Following the three-user WSR analysis, we compare the proposed 1-layer RS, generalized RS, SC–SIC, SC–SIC per group with MU–LP described in previous sections.",
"The specific channel model specified in Section REF is used in this section.",
"In the following results, the QoS rate constraints of the multicast and unicast messages are assumed to be equal to 0.1 bit/s/Hz, i.e., $R_0^{th}=R_1^{th}=R_2^{th}=R_3^{th}=0.1$ bit/s/Hz.",
"The weights allocated to the streams are equal to one, i.e., $u_0=u_1=u_2=u_3=1$ .",
"SNR is fixed to 10 dB.",
"The channel strength disparities are fixed to $\\gamma _1=1,\\gamma _2=0.3$ .",
"Figure: Energy Efficiency versus P dyn P_{\\textrm {dyn}} comparison of different strategies for overloaded three-user deployment in perfect CSIT.",
"N t =2N_t=2.Fig.",
"REF and Fig.",
"REF illustrate the EE versus $P_{\\textrm {dyn}}$ comparison of different strategies for underloaded and overloaded three-user deployments with perfect CSIT, respectively.",
"In both figures, the generalized RS always outperforms all other strategies.",
"Though MU–LP and the proposed 1-layer RS have the lowest receiver complexity, the EE performance of 1-layer RS outperforms MU–LP in all figures.",
"It achieves a better EE performance than SC–SIC per group in most simulated user deployments and network loads.",
"1-layer RS also outperforms SC–SIC when the user channels are sufficiently orthogonal.",
"We conclude that 1-layer RS provides more robust EE performance than MU–LP, SC–SIC and SC–SIC per group towards different user deployments and network loads.",
"The EE convergence of all considered transmission strategies for a specific channel realization is analyzed in Fig.",
"REF b.",
"For various dynamic power values $P_{\\textrm {dyn}}$ , a few iterations are required for each strategy to converge.",
"Both MU–LP and 1-layer RS-assisted transmission strategies use Algorithm REF just once to complete the optimization procedure.",
"In contrast, Algorithm REF is required to be repeated for each decoding order of RS/SC–SIC/SC–SIC per group-assisted strategies, which results in much higher computational burden at the transmitter especially when the number of served users is large.",
"The proposed 1-layer RS-assisted NOUM transmission achieves an excellent tradeoff between EE performance and complexity."
],
[
"Conclusions",
"To conclude, we initiate the study of rate-splitting in NOUM transmission by proposing a 1-layer RS and generalized RS-assisted transmission strategies.",
"We also propose two NOMA-assisted transmission strategies, namely, `SC–SIC' and `SC–SIC per group'.",
"The precoders of all the strategies are designed by maximizing the WSR/EE subject to the sum power constraint and the QoS rate requirements of all messages.",
"Two low-complexity WMMSE-based and SCA-based optimization frameworks are proposed to solve the WSR and EE maximization problems, respectively.",
"Numerical results show that the proposed generalized RS-assisted strategy softly bridges and outperforms MU–LP, OMA and NOMA in a wide range of user deployments (with a diversity of channel directions, channel strengths and qualities of channel state information at the transmitter) and network loads (underloaded and overloaded regimes).",
"It is a more general and powerful transmission strategy that encompasses MU–LP, OMA and NOMA as special cases.",
"The proposed 1-layer RS-assisted strategy gets most of the performance benefits of the multi-layer (generalized) RS at a much lower complexity, and is more spectrally efficient and energy efficient than the existing MU–LP-assisted strategy in various user deployments and network loads.",
"It also achieves a more robust WSR and EE performance than the proposed NOMA-assisted strategies.",
"Most importantly, the high-quality performance of 1-layer RS comes without any increase in the receiver complexity compared with MU–LP and the receiver complexity of 1-layer RS is much lower than the proposed NOMA-based strategies.",
"The one layer SIC in RS is used for the dual purpose of separating the unicast and multicast streams as well as better managing the multi-user unicast interference.",
"Hence, the presence of SIC is better exploited in the proposed 1-layer RS-based strategy.",
"Yijie Mao is a postdoctoral research associate with the Communications and Signal Processing Group, Department of the Electrical and Electronic Engineering at the Imperial College London.",
"Her research interests include MIMO, rate-splitting and NOMA for 5G and beyond.",
"Bruno Clerckx is a Reader, the Head of the Wireless Communications and Signal Processing Lab, and the Deputy Head of the Communications and Signal Processing Group, within the Electrical and Electronic Engineering Department, Imperial College London, London, U.K. His area of expertise is communication theory and signal processing for wireless networks.",
"Victor O.K.",
"Li is Chair of Information Engineering and Cheng Yu-Tung Professor in Sustainable Development at the Department of Electrical and Electronic Engineering at the University of Hong Kong.",
"His research interests include big data, AI, optimization techniques, and interdisciplinary clean energy and environment studies."
]
] | 1808.08325 |
[
[
"Motivic Gau{\\ss}-Bonnet formulas"
],
[
"Abstract The apparatus of motivic stable homotopy theory provides a notion of Euler characteristic for smooth projective varieties, valued in the Grothendieck-Witt ring of the base field.",
"Previous work of the first author and recent work of D\\'eglise-Jin-Khan establishes a \"Gau\\ss-Bonnet formula\" relating this Euler characteristic to pushforwards of Euler classes in motivic cohomology theories.",
"In this paper, we apply this formula to SL-oriented motivic cohomology theories to obtain explicit characterizations of this Euler characteristic.",
"The main new input is a unicity result for pushforward maps in SL-oriented theories, identifying these maps concretely in examples of interest."
],
[
"Introduction",
"Let $k$ be a field and let $X$ be a smooth projective $k$ -scheme.",
"Let ${\\operatorname{SH}}(k)$ denote the motivic stable homotopy category over $k$ ; recall that this comes equipped with the structure of a symmetric monoidal category, whose tensor product we denote $\\wedge _k$ and whose unit object (the motivic sphere spectrum) we denote $1_k$ .",
"Our starting point in this paper is the following fact, which shall be reviewed in §, and which goes back to the categorical notion of Euler characteristic introduced by Dold-Puppe [14].",
"Proposition 1.1 The infinite suspension spectrum $\\Sigma ^\\infty _X̰_+ \\in {\\operatorname{SH}}(k)$ is dualizable.",
"In particular, we can associate to $X$ a natural Euler characteristic $\\chi (X/k) \\in {\\rm End}_{{\\operatorname{SH}}(k)}(1_k)$ , defined as the composition $1_k \\xrightarrow{} \\Sigma ^\\infty _X̰_+ \\wedge _k (\\Sigma ^\\infty _X̰_+)^\\vee \\xrightarrow{} (\\Sigma ^\\infty _X̰_+)^\\vee \\wedge _k \\Sigma ^\\infty _X̰_+ \\xrightarrow{} 1_k,$ where the maps $\\delta $ and $\\epsilon $ are the coevaluation and evaluation that comprise the duality, and $\\tau $ is the symmetry isomorphism.",
"For $k$ perfect, a theorem of Morel identifies ${\\rm End}_{{\\operatorname{SH}}(k)}(1_k)$ with the Grothendieck-Witt group ${\\operatorname{GW}}(k)$ , i.e.",
"the Grothendieck group of $k$ -vector spaces equipped with a nondegenerate symmetric bilinear form.",
"Hence, we may think of the Euler characteristic $\\chi (X/k)$ as a class in ${\\operatorname{GW}}(k)$ .",
"It is then natural to wonder whether there is an explicit interpretation of this Euler characteristic in terms of symmetric bilinear forms.",
"An intuitive speculation is that the Euler characteristic should be given by the value of some cohomology theory on $X$ , equipped with an intersection pairing.",
"One of the main results in this paper is to make precise and confirm this speculation.",
"To state the result, we recall that classes in ${\\operatorname{GW}}(k)$ can be represented not just by nondegenerate symmetric bilinear forms on $k$ -vector spaces, but also by nondegenerate symmetric bilinear forms on perfect complexes over $k$ (see §REF for a review of what this means).",
"With this in mind, we give the following explicit interpretation of the Euler characteristic $\\chi (X/k)$ .",
"Construction 1.2 Suppose that $X$ is of pure dimension $d$ .",
"Then we have the Hodge cohomology groups ${\\operatorname{H}}^i(X;\\Omega ^j_{X/k})$ for $0 \\le i,j \\le d$ and the canonical trace map ${\\rm Tr}: {\\operatorname{H}}^d(X;\\Omega ^d_{X/k}) \\rightarrow k.$ We define a perfect complex of $k$ -vector spaces (with zero differential), $\\mathrm {Hdg}(X/k) := \\bigoplus _{i,j=0}^d {\\operatorname{H}}^i(X,\\Omega ^j_{X/k})[j-i],$ and the trace map defines a nondegenerate symmetric bilinear form on $\\mathrm {Hdg}(X/k)$ via the pairings ${\\operatorname{H}}^i(X,\\Omega ^j_{X/k}) \\otimes _k {\\operatorname{H}}^{d-i}(X,\\Omega ^{d-j}_{X/k}) \\xrightarrow{} {\\operatorname{H}}^d(X,\\Omega ^d_{X/k}) \\xrightarrow{} k,$ where the first map denotes the cup product (that this is indeed a nondegenerate symmetric bilinear form will be shown in §REF ).",
"We thus obtain a Grothendieck-Witt class $(\\mathrm {Hdg}(X/k),{\\rm Tr}) \\in {\\operatorname{GW}}(k)$ .",
"This construction extends in an evident manner to the case that $X$ is not necessarily of pure dimension.",
"The following formula for $\\chi (X/k)$ was proposed by J-P. Serre.private communication to ML, 28.07.2017 Theorem 1.3 Assume that $k$ is a perfect field of characteristic different from two.",
"Then $\\chi (X/k) = (\\mathrm {Hdg}(X/k), {\\rm Tr}) \\in {\\operatorname{GW}}(k)$ .",
"We in fact prove a more general result over a base-scheme $B$ ; see Theorem REF and Corollary REF for details.",
"In the case $k={\\mathbb {R}}$ , a class in ${\\operatorname{GW}}(k)$ is determined by two ${\\mathbb {Z}}$ -valued invariants, rank and signature, and Theorem REF reproves the following known result (see [1] and [24]).",
"Corollary 1.4 Suppose that $k={\\mathbb {R}}$ and $X$ is of even pure dimension $2n$ .",
"Then the symmetric bilinear form ${\\operatorname{H}}^n(X,\\Omega ^n_{X/{\\mathbb {R}}}) \\times {\\operatorname{H}}^n(X,\\Omega ^n_{X/{\\mathbb {R}}}) \\xrightarrow{} {\\operatorname{H}}^{2n}(X,\\Omega ^{2n}_{X/{\\mathbb {R}}}) \\xrightarrow{} {\\mathbb {R}}$ has signature equal to $\\chi ^\\mathrm {top}(X({\\mathbb {R}}))$ , the classical Euler characteristic of the real points of $X$ in the analytic topology.",
"In particular, we have $|\\chi ^\\mathrm {top}(X({\\mathbb {R}}))| \\le \\operatorname{dim}_{\\mathbb {R}}{\\operatorname{H}}^n(X,\\Omega ^n_{X/{\\mathbb {R}}}).$ This is Corollary REF in the text.",
"Besides giving an explicit formula for the rather abstractly defined $\\chi (X/k)$ , Theorem REF opens the way to computing $\\chi (X/k)$ in the situation that $X$ is a twisted form of another $k$ -scheme $Y$ , namely by twisting the symmetric bilinear form $(\\mathrm {Hdg}(Y/k), {\\rm Tr})$ by the descent data for $X$ .",
"This cannot be done with the class $\\chi (Y/k)\\in {\\operatorname{GW}}(k)$ , as ${\\operatorname{GW}}(-)$ does not satisfy Galois descent.",
"This is discussed in detail in §REF .",
"Let us now explain our methods for proving Theorem REF .",
"The idea is to use the theory of Euler classes in motivic cohomology theories.",
"More specifically, our focus is on cohomology theories represented by $\\operatorname{SL}$ -oriented motivic ring spectra; recall that this refers to a commutative monoid object ${\\mathcal {E}}$ in ${\\operatorname{SH}}(k)$ equipped with a compatible system of Thom classes for oriented vector bundles (where an orientation is a specified trivialization of the determinant line bundle).",
"The example of interest for proving Theorem REF is hermitian K-theory; other examples of interest include Chow-Witt theory, ordinary motivic cohomology, and algebraic K-theory (the last two are actually $\\operatorname{GL}$ -oriented, meaning they have Thom classes for all vector bundles).",
"The assumption that $k$ has characteristic different from two in Theorem REF arises from this use of hermitian K-theory, which at present is only known to satisfy the properties we need when ${\\rm char}(k) \\ne 2$ ; we do not know of any counter-examples to our formula for $\\chi (X/k)$ for $k$ of characteristic two.",
"Given an $\\operatorname{SL}$ -oriented motivic ring spectrum ${\\mathcal {E}}\\in {\\operatorname{SH}}(k)$ , one may define certain pushforward maps in twisted ${\\mathcal {E}}$ -cohomology.",
"Namely, if $Y$ and $Z$ are smooth quasi-projective $k$ -schemes, $f : Z \\rightarrow Y$ is a proper morphism of relative dimension $d \\in {\\mathbb {Z}}$ , and $L$ is a line bundle on $Y$ , then there is a pushforward map $f_* : {\\mathcal {E}}^{a,b}(Z;\\omega _{Z/k} \\otimes f^*L) \\rightarrow {\\mathcal {E}}^{a-2d,b-d}(Y;\\omega _{Y/k}\\otimes L),$ where $\\omega _{-/k}$ denotes the canonical bundle.",
"This is defined abstractly via the six-functor formalism for motivic stable homotopy theory.",
"We note two key examples of these pushforwards, assuming our smooth projective variety $X$ is of pure dimension $d$ for simplicity: the structural morphism $\\pi : X \\rightarrow {\\rm Spec\\,}(k)$ gives a pushforward map $\\pi _* : {\\mathcal {E}}^{2d,d}(X,\\omega _{X/k}) \\rightarrow {\\mathcal {E}}^{0,0}({\\rm Spec\\,}k);$ given a vector bundle $p : V \\rightarrow X$ , the zero section $s : X \\hookrightarrow V$ gives a pushforward map $s_* : {\\mathcal {E}}^{0,0}(X) \\rightarrow {\\mathcal {E}}^{2d,d}(V;p^*\\operatorname{det}^{-1}(V)).$ The first should be thought of as a kind of integration map.",
"The second allows us to define the Euler class of a vector bundle $V \\rightarrow X$ , $e^{\\mathcal {E}}(V) := s^*s_*(1) \\in {\\mathcal {E}}^{2d,d}(X;\\operatorname{det}^{-1}(V)),$ where $s$ again denotes the zero-section, and $1 \\in {\\mathcal {E}}^{0,0}(X)$ denotes the unit element.",
"Using the above notions, we may state the following motivic version of the classical Gauß-Bonnet formula equating the Euler characteristic with the integral of the Euler class of the tangent bundle; this result is a fairly immediate consequence of [26]: Theorem 1.5 (Motivic Gauß-Bonnet) Let ${\\mathcal {E}}$ be an $\\operatorname{SL}$ -oriented motivic ring spectrum in ${\\operatorname{SH}}(k)$ .",
"Let $u : 1_k \\rightarrow {\\mathcal {E}}$ denote the unit map, inducing the map $u_* : {\\operatorname{GW}}(k) \\simeq 1_k^{0,0}({\\rm Spec\\,}k) \\rightarrow {\\mathcal {E}}^{0,0}({\\rm Spec\\,}k)$ .",
"Then we have $u_*(\\chi (X/k)) = \\pi _*(e^{\\mathcal {E}}(T_{X/k})) \\in {\\mathcal {E}}^{0,0}({\\rm Spec\\,}k).$ A general motivic Gauß-Bonnet formula is also proven in [13], which implies the above formula by applying the unit map.",
"Our method is somewhat different from [13] in that we replace their general theory of Euler classes with the more special version for $\\operatorname{SL}$ -oriented theories used here; see Theorem REF below for our general statement of this result.",
"As stated above, we deduce Theorem REF from Theorem REF by considering the example of hermitian K-theory, ${\\mathcal {E}}= {\\operatorname{BO}}$ .",
"In this case, the map $u_* : {\\operatorname{GW}}(k) \\rightarrow {\\operatorname{BO}}^{0,0}({\\rm Spec\\,}k)$ is an isomorphism.",
"The deduction requires an explicit understanding of both the Euler class $e^{\\mathcal {E}}(T_{X/k})$ and the pushforward $\\pi _*$ in hermitian K-theory; the former is fairly straightforward, but the latter requires new input.",
"What we do is identify the abstractly defined projective pushforward maps in hermitian K-theory with the concrete ones defined in terms of pushforward of sheaves and Grothendieck-Serre duality.",
"This comparison follows from a uniqueness result we prove for pushforward maps in an $\\operatorname{SL}$ -oriented theory ${\\mathcal {E}}$ , characterizing them, under certain further hypotheses on ${\\mathcal {E}}$ , in terms of their behavior in the case of the inclusion of the zero-section of a vector bundle (which is governed by Thom isomorphisms).",
"We leave the detailed statement of this result to the body of the paper (see Theorem REF ), as it would take too long to spell out here.",
"Remark 1.6 The recent paper of Bachmann–Wickelgren [5] discusses results closely related to those discussed here.",
"For example, they identify the abstract pushforward maps in hermitian K-theory with those defined by Grothendieck-Serre duality in the case of a finite syntomic morphism (as opposed to the case of a smooth and proper morphism between smooth schemes addressed here).",
"Moreover, combining their identifications of various Euler classes with our motivic Gauss-Bonnet formula, one may recover Theorem REF above."
],
[
"Outline",
"The paper is organized as follows.",
"In §, we review the basic framework of motivic homotopy theory, as well as relevant aspects of the dualizability result Proposition REF .",
"In §, we review basic facts about $\\operatorname{SL}$ -oriented motivic ring spectra.",
"In §, we describe the abstractly defined pushforwards in the twisted cohomology theory arising from an $\\operatorname{SL}$ -oriented motivic ring spectrum.",
"In §, we prove the general Gauß-Bonnet formula for $\\operatorname{SL}$ -oriented motivic ring spectra.",
"In §, we axiomatize the features of the twisted cohomology theory arising from an $\\operatorname{SL}$ -oriented motivic ring spectrum.",
"In §, we use these axiomatics to prove our unicity/comparison theorem characterizing the pushforward maps in $\\operatorname{SL}$ -oriented theories.",
"And finally, in §, we apply the previous results in specific examples of $\\operatorname{SL}$ -oriented theories to obtain various concrete consequences, in particular proving Theorem REF .",
"Both authors thank the referees for their detailed comments and suggestions, which helped us correct a number of ambiguities in an earlier version and have greatly improved the exposition.",
"AR: I would like to thank my PhD advisor, Søren Galatius, for many suggestions and ideas that led to my thinking about the questions discussed in this paper.",
"I would also like to thank Tony Feng and Jesse Silliman for helpful conversations.",
"My work has been supported by an NSF Graduate Research Fellowship.",
"ML: I would like to thank J-P. Serre for proposing the formula Theorem REF for the motivic Euler characteristic and J-P. Serre and Eva Bayer-Fluckiger for an enlightening correspondence on computations in the case of geometrically rational surfaces.",
"My work has been supported by the DFG through the SFB Transregio 45 and the SPP 1786."
],
[
"Duality and Euler characteristics",
"In this section, we review the strong dualizability of smooth projective schemes as objects of the stable motivic homotopy category, which supplies a notion of Euler characteristic for these schemes.",
"We additionally recall a result from [26] that gives an alternative characterization of this Euler characteristic."
],
[
"Preliminaries",
"Let us first recall the basic framework of stable motivic homotopy theory, which will be used throughout the paper.",
"Notation 2.1 Throughout the paper, we let $B$ denote a noetherian separated base scheme of finite Krull dimension.",
"Let $\\mathrm {Sch}_B$ denote the category of quasi-projective $B$ -schemes, that is, $B$ -schemes $X \\rightarrow B$ that admit a closed immersion $i : X \\hookrightarrow U$ over $B$ , with $U$ an open subscheme of ${\\mathbb {P}}^N_B$ for some $N$ .",
"Let $\\mathrm {Sch}_B^{\\text{pr}}$ denote the subcategory of $\\mathrm {Sch}_B$ with the same objects as $\\mathrm {Sch}_B$ but with morphisms the proper morphisms.",
"Let ${\\mathrm {Sm}}_B$ denote the full subcategory of $\\mathrm {Sch}_B$ with objects the smooth (quasi-projective) $B$ -schemes.",
"(The same notation will be used when working over schemes other than $B$ .)",
"For $X$ a $B$ -scheme, we will usually denote the structure morphism by $\\pi _X:X\\rightarrow B$ .",
"Notation 2.2 Given $X \\in \\mathrm {Sch}_B$ , we let ${\\operatorname{SH}}(X)$ denote the stable motivic homotopy category over $X$ .",
"We will rely on the six-functor formalism for this construction, as established in [4], [20].",
"In particular, for each morphism $f : Y \\rightarrow X$ in $\\mathrm {Sch}_B$ , one has the adjoint pairs of functors ${{\\operatorname{SH}}(X)@<3pt>[r]^{f^*}&{\\operatorname{SH}}(Y)@<3pt>[l]^{f_*}}\\quad \\text{and}\\quad {{\\operatorname{SH}}(Y)@<3pt>[r]^{f_!}&{\\operatorname{SH}}(X)@<3pt>[l]^{f^!",
"}};$ natural isomorphisms $(gf)^*\\simeq f^*g^*$ , $(gf)^!\\simeq f^!g^!$ , $(gf)_*\\simeq g_*f_*$ , $(gf)_!\\simeq g_!f_!$ for composable morphisms, with the usual associativity; a natural transformation $\\eta ^f_{!",
"*}:f_!\\rightarrow f_*$ , which is an isomorphism if $f$ is proper.",
"There are various base-change morphisms, which we will recall as needed.",
"In addition, for $f$ smooth, there is a further adjoint pair ${{\\operatorname{SH}}(Y)@<3pt>[r]^{f_\\sharp }&{\\operatorname{SH}}(X)@<3pt>[l]^{f^*}}.$ There is also the symmetric monoidal structure on ${\\operatorname{SH}}(X)$ ; we denote the tensor product by $\\wedge _X$ and the unit by $1_X\\in {\\operatorname{SH}}(X)$ .",
"For $f$ a closed immersion, we have the adjoint pair $f_*\\dashv f^!$ arising from a corresponding adjoint pair in the unstable setting, so we will take $f_!=f_*$ with $\\eta ^f_{!",
"*}={\\operatorname{id}}$ .",
"Similarly, if $f$ is an open immersion, we have a canonical isomorphism of adjoint pairs $(f_\\#\\dashv f^*)\\cong (f_!\\dashv f^!",
")$ , so we take $f_!=f_\\#$ and $f^!=f^*$ .",
"We also have the unstable motivic homotopy category ${\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)$ , which we recall is a localization of the category ${\\mathrm {Spc}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)$ of presheaves of pointed simplicial sets on ${\\mathrm {Sm}}_X$ .",
"For $Y\\rightarrow X$ in ${\\mathrm {Sm}}_X$ , we write $Y_+$ for the presheaf represented by the $X$ -scheme $Y\\amalg X\\rightarrow X$ , that is, the presheaf $Z\\mapsto {\\rm Hom}_{{\\mathrm {Sm}}_X}(X, Y)_+$ (where here $(-)_+$ denotes addition of a disjoint basepoint to a set and we regard a set as a constant simplicial set).",
"The category ${\\mathrm {Spc}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)$ has a canonical symmetric monoidal structure with unit object $X_+$ , and ${\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)$ inherits this structure via the localization functor.",
"Finally, we have the infinite suspension functor $\\Sigma ^\\infty _{\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)\\rightarrow {\\operatorname{SH}}(X)$ , which is canonically symmetric monoidal, so that in particular we have a canonical identification $1_X \\simeq \\Sigma ^\\infty _X̰_+$ .",
"For $f : Y \\rightarrow X$ a smooth morphism in $\\mathrm {Sch}_B$ , the adjoint pair $f_\\sharp \\dashv f^*$ mentioned above for stable motivic homotopy categories arises from an adjoint pair in the unstable setting, ${{\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(Y)@<3pt>[r]^{f_\\sharp }&{\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)@<3pt>[l]^{f^*}},$ where the left adjoint is induced by the functor $f_\\#:{\\mathrm {Spc}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(Y)\\rightarrow {\\mathrm {Spc}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)$ obtained as the left Kan extension of the functor $f_\\#:{\\mathrm {Sm}}_Y\\rightarrow {\\mathrm {Spc}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)$ sending $p:W\\rightarrow Y$ to $(f\\circ p:W\\rightarrow X)_+$ .",
"We often write $-/X$ for the functor $\\Sigma ^\\infty _{\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)\\rightarrow {\\operatorname{SH}}(X)$ , and if $\\pi _X:X\\rightarrow B$ is smooth, then we write $-/B$ for the functor $\\pi _{X\\#}\\circ \\Sigma ^\\infty _{\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)\\rightarrow {\\operatorname{SH}}(B)$ .",
"We use the same notation to denote the precompositions of these functors with the functor $(-)_+ : {\\mathrm {Sm}}_X \\rightarrow {\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)$ .",
"Remark 2.3 For $p : E \\rightarrow X$ an affine space bundle in $\\mathrm {Sch}_B$ (that is, a torsor in the Zariski topology for a vector bundle), the composition $p_\\sharp \\circ p^*$ is an autoequivalence of ${\\operatorname{SH}}(X)$ (this is a formulation of homotopy invariance).",
"Notation 2.4 (The localization triangle) Let $j:U\\rightarrow X$ be an open immersion in $\\mathrm {Sch}_B$ with (reduced) complement $i:Z\\rightarrow X$ .",
"We have the localization distinguished triangle of endofunctors on ${\\operatorname{SH}}(X)$ $j_!j^!\\rightarrow {\\operatorname{id}}_{{\\operatorname{SH}}(X)}\\rightarrow i_*i^*\\rightarrow j_!j^!",
"[1],$ where the morphism $j_!j^!\\rightarrow {\\operatorname{id}}_{{\\operatorname{SH}}(X)}$ is the counit of the adjunction $j_!\\dashv j^!$ and the morphism ${\\operatorname{id}}_{{\\operatorname{SH}}(X)}\\rightarrow i_*i^*$ is the unit of the adjunction $i^*\\dashv i_*$ .",
"Moreover $i_*=i_!$ , $j_!=j_\\#$ and $j^!=j^*$ .",
"We often write $X_Z/X$ for $i_*(1_Z)\\in {\\operatorname{SH}}(X)$ .",
"With this notation (and that of Notation REF ), applying (REF ) to $1_X$ gives us the distinguished triangle in ${\\operatorname{SH}}(X)$ , $U/X\\xrightarrow{} X/X\\rightarrow X_Z/X\\rightarrow U/X[1];$ in other words, we have a canonical isomorphism $X_Z/X \\simeq \\Sigma ^\\infty _T(X/U)$ ; accordingly, we often write $X_Z$ for the quotient presheaf $X/U$ in ${\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)$ .",
"Notation 2.5 (Suspension and Thom spaces) Let $p : V \\rightarrow X$ be a vector bundle over some $ X \\in \\mathrm {Sch}_B$ , with zero-section $s : X \\hookrightarrow V$ .",
"We have the endofunctors $\\Sigma ^{-V}, \\Sigma ^V:{\\operatorname{SH}}(X)\\rightarrow {\\operatorname{SH}}(X)$ defined by $\\Sigma ^{-V}:=s^!p^*$ and $\\Sigma ^V:=p_\\sharp \\circ s_*$ .",
"These are in fact inverse autoequivalences.",
"The endofunctor $\\Sigma ^V$ can also be written in terms of Thom spaces.",
"Setting $0_V := s(X) \\subset V$ , the Thom space of the vector bundle is defined as ${\\operatorname{Th}}_X(V) := V/(V \\setminus 0_V) \\in {\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X).$ To lighten the notation, we often write ${\\operatorname{Th}}_X(V)$ for ${\\operatorname{Th}}_X(V)/X=\\Sigma ^\\infty _{\\operatorname{Th}}_X(V)) \\in {\\operatorname{SH}}(X)$ , when the context makes the meaning clear.",
"For $\\pi _X:X\\rightarrow B$ in ${\\mathrm {Sm}}_B$ , we set ${\\operatorname{Th}}(V):=\\pi _{X\\#}({\\operatorname{Th}}_X(V))$ in ${\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(B)$ , and we similarly write ${\\operatorname{Th}}(V)$ for ${\\operatorname{Th}}(V)/B \\simeq \\pi _{X\\#}({\\operatorname{Th}}_X(V)/X)\\in {\\operatorname{SH}}(B)$ when appropriate.",
"To see the relation between the Thom space ${\\operatorname{Th}}_X(V)$ and the suspension functor $\\Sigma ^V$ , consider the localization distinguished triangle $j_!j^!\\rightarrow {\\operatorname{id}}_V\\rightarrow s_*s^*\\rightarrow j_!j^!",
"[1],$ where $s$ still denotes the zero section and $j$ denotes the open complement $V \\setminus 0_V \\hookrightarrow V$ .",
"Evaluating the sequence at $1_V$ and noting that $j_!j^!=j_\\#j^*$ and $s_*=s_!$ , we obtain an identification between ${\\operatorname{Th}}_X(V)\\in {\\operatorname{SH}}(X)$ and $p_\\#s_*(1_X)=\\Sigma ^V(1_X)$ .",
"Consequently, we have identifications $\\pi _{V\\#}(s_*(1_X))\\simeq \\pi _{X\\#}(\\Sigma ^V(1_X))\\simeq {\\operatorname{Th}}(V)$ .",
"In parallel, we shall write ${\\operatorname{Th}}_X(-V)$ for $\\Sigma ^{-V}(1_X)$ and ${\\operatorname{Th}}(-V)$ for $\\pi _{X\\#}\\Sigma ^{-V}(1_X)$ .",
"With these notational conventions, there are canonical natural isomorphisms $\\Sigma ^V(-)\\simeq {\\operatorname{Th}}_X(V)\\wedge _X(-), \\quad \\Sigma ^{-V}(-)\\simeq {\\operatorname{Th}}_X(-V)\\wedge _X(-).$ Remark 2.6 Let ${\\operatorname{perf}}(X)_{\\mathrm {iso}}$ denote the subcategory of isomorphisms in the perfect derived category ${\\operatorname{perf}}(X)$ and let ${\\mathcal {K}}(X)$ denote the groupoid associated to the K-theory space of $X$ .",
"Then the assignment $V \\mapsto \\Sigma ^V$ extends to a functor $\\Sigma ^{(-)} :{\\operatorname{perf}}(X)_{\\mathrm {iso}} \\rightarrow {\\operatorname{Aut}}({\\operatorname{SH}}(X)),$ and moreover factors through the canonical functor ${\\operatorname{perf}}(X)_{\\mathrm {iso}} \\rightarrow {\\mathcal {K}}(X)$ , so that a distinguished triangle $E^{\\prime }\\rightarrow E\\rightarrow E^{\\prime \\prime }\\rightarrow E^{\\prime }[1]$ in ${\\operatorname{perf}}(X)$ determines a natural isomorphism $\\Sigma ^{E^{\\prime }}\\circ \\Sigma ^{E^{\\prime \\prime }}\\simeq \\Sigma ^E$ .",
"See for example [42] for a proof of this last statement in the special case concerning the functor ${\\operatorname{Th}}_X(-)=\\Sigma ^{(-)}(1_X) :{\\operatorname{perf}}(X)_{\\mathrm {iso}} \\rightarrow {\\operatorname{SH}}(X)$ (which in fact implies the general statement by (REF )).",
"In this context, for an integer $n$ , we sometimes write $n$ for the trivial bundle of virtual rank $n$ ; for example, $\\Sigma ^{n+E} \\simeq \\Sigma ^n_{{\\mathbb {P}}^1}\\circ \\Sigma ^E$ for $E$ in ${\\operatorname{perf}}(X)$ .",
"Remark 2.7 (Atiyah duality) For $f:Y\\rightarrow X$ a smooth morphism in $\\mathrm {Sch}_B$ with relative tangent bundle $T_f\\rightarrow Y$ , there are canonical natural isomorphisms $f_!",
"\\simeq f_\\sharp \\circ \\Sigma ^{-T_f}, \\quad f^!",
"\\simeq \\Sigma ^{T_f} \\circ f^*$ (see [20]).",
"In addition, for $X \\in \\mathrm {Sch}_B$ and $V\\rightarrow X$ a vector bundle, there are canonical natural isomorphisms $f_\\sharp \\circ \\Sigma ^{\\pm f^*V}\\simeq \\Sigma ^{\\pm V}\\circ f_\\sharp ,\\quad f^*\\circ \\Sigma ^{\\pm V}\\simeq \\Sigma ^{\\pm V}\\circ f^*,$ the latter valid for an arbitrary morphism $f:Y\\rightarrow X$ in ${\\mathrm {Sm}}_B$ .",
"Moreover, for $f:Y\\rightarrow X$ an arbitrary morphism in $\\mathrm {Sch}_B$ and $V\\rightarrow X$ a vector bundle, there is a canonical natural isomorphism $f^!\\circ \\Sigma ^{\\pm V}\\simeq \\Sigma ^{\\pm f^*V}\\circ f^!.$ (see the beginning of [20]).",
"Finally, for $f:Y\\rightarrow X$ a regular embedding in $\\mathrm {Sch}_B$ with normal bundle $N_f\\rightarrow Y$ , there is a canonical natural isomorphism $f^!\\simeq \\Sigma ^{-N_f}\\circ f^*.$ In fact, the isomorphism $f^!",
"\\simeq \\Sigma ^{T_f} \\circ f^*$ for smooth $f$ extends to the case of an lci morphism, as follows.",
"For $f:Y\\rightarrow X$ an lci morphism in $\\mathrm {Sch}_B$ , we factor $f$ as $f=p\\circ i$ with $i:Y\\rightarrow Z$ a regular embedding and $p:Z\\rightarrow X$ a smooth morphism (both in $\\mathrm {Sch}_B$ ).",
"This gives the relative virtual normal bundle $\\nu _f:=[N_i]-[i^*T_{Z/X}]$ in ${\\mathcal {K}}(Y)$ , independent up to canonical isomorphism on the choice of the factorization.",
"Thus, we have the canonically defined suspension automorphism $\\Sigma ^{-\\nu _f}$ and a canonical natural isomorphism $f^!=i^!\\circ p^!\\simeq \\Sigma ^{-N_i}\\circ i^*\\circ \\Sigma ^{T_{Z/X}}\\circ p^*\\simeq \\Sigma ^{-\\nu _f}\\circ f^*.$ One can then construct a left adjoint $f_\\sharp $ to $f^*$ by setting $f_\\sharp :=f_!\\circ \\Sigma ^{\\nu _f}.$ The functoriality $(gf)^*=f^*\\circ g^*$ gives rise to the functoriality on the adjoints $(gf)_\\sharp =g_\\sharp \\circ f_\\sharp $ for composable lci morphisms.",
"Notation 2.8 Let $\\pi : Z \\rightarrow X$ be a morphism in $\\mathrm {Sch}_B$ .",
"For $Y \\in {\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(Z)$ , we set $Y/X_\\mathrm {B.M.",
"}:= \\pi _!",
"(\\Sigma ^\\infty _Y̰) \\in {\\operatorname{SH}}(X),$ and for $Y \\in {\\mathrm {Sm}}_Z$ , we make the abbreviation $Y/X_\\mathrm {B.M.",
"}$ for $Y_+/X_\\mathrm {B.M.",
"}= \\pi _!",
"(\\Sigma ^\\infty _Y̰_+)$ .",
"In particular, we by definition have $Z/X_\\mathrm {B.M.",
"}= \\pi _!",
"(1_Z) \\in {\\operatorname{SH}}(X)$ .",
"Furthermore, for $i:W \\hookrightarrow Z$ the inclusion of a reduced closed subscheme, we set $Z_W/X_\\mathrm {B.M.",
"}:=\\pi _!",
"(i_*(1_W)) .$ We observe two facts about this object.",
"Firstly, letting $\\pi ^{\\prime }$ denote the composite $\\pi \\circ i : W\\rightarrow X$ , the isomorphism $\\pi ^{\\prime }_!=\\pi _!\\circ i_*$ determines an isomorphism $Z_W/X_\\mathrm {B.M.",
"}\\simeq W/X_\\mathrm {B.M.",
"}$ .",
"Secondly, let $j:U:=Z\\setminus W\\rightarrow Z$ denote the open complement of $W$ and consider the localization distinguished triangle $j_!j^!\\rightarrow {\\operatorname{id}}_{{\\operatorname{SH}}(Z)}\\rightarrow i_*i^*.$ Then the identities $j^!= j^*$ and $i_*=i_!$ give a canonical distinguished triangle in ${\\operatorname{SH}}(X)$ , $U/X_\\mathrm {B.M.",
"}\\rightarrow Z/X_\\mathrm {B.M.",
"}\\rightarrow Z_W/X_\\mathrm {B.M.",
"}\\rightarrow U/X_\\mathrm {B.M.",
"}[1].$ Remark 2.9 The assignment $Z \\mapsto Z/X_\\mathrm {B.M.",
"}$ extends to a functor $(-)/X_\\mathrm {B.M.",
"}: (\\mathrm {Sch}_X^{\\text{pr}})^{\\text{\\rm op}}\\rightarrow {\\operatorname{SH}}(X).$ This is described in a number of places, for example [27]; we recall the construction for the reader's convenience, referring to loc.",
"cit.",
"for details.",
"Let $g : Z \\rightarrow Y$ be a proper morphism in $\\mathrm {Sch}_X$ .",
"Let $\\pi _Y : Y \\rightarrow X$ and $\\pi _Z : Z \\rightarrow X$ denote the structural morphisms.",
"As mentioned in Notation REF , we have a natural isomorphism $\\eta ^g_{!",
"*} : g_!",
"\\simeq g_*$ .",
"We may then define a natural transformation $g^* : \\pi _{Y!}",
"\\rightarrow \\pi _{Z!",
"}g^*$ as the composition $\\pi _{Y!}",
"\\xrightarrow{} \\pi _{Y!",
"}g_*g^* \\xrightarrow{}\\pi _{Y!",
"}g_!g^* \\simeq \\pi _{Z!",
"}g^*,$ where $u_g:{\\operatorname{id}}_{{\\operatorname{SH}}(Y)}\\rightarrow g_*g^*$ is the unit of the adjunction.",
"Evaluating this natural transformation at $1_Y$ gives a map $g^* = g/X_\\mathrm {B.M.",
"}: Y/X_\\mathrm {B.M.",
"}\\rightarrow Z/X_\\mathrm {B.M.",
"}$ , and it follows directly from the definitions that this construction satisfies $(gh)^*=h^*g^*$ for composable proper morphisms $g$ , $h$ .",
"This establishes the claimed functoriality."
],
[
"Duality for smooth projective schemes",
"Let $({\\mathcal {C}}, \\otimes , 1, \\mu , \\tau )$ be a symmetric monoidal category.",
"Recall that the the dual of an object $x$ of ${\\mathcal {C}}$ is a triple $(y, \\delta , \\epsilon )$ with $\\delta _x:1\\rightarrow x\\otimes y$ and $\\epsilon _x:y\\otimes x\\rightarrow 1$ maps such that the two compositions $x\\xrightarrow{}1\\otimes x\\xrightarrow{}x\\otimes y\\otimes x\\xrightarrow{}x\\otimes 1\\xrightarrow{}x$ and $y\\xrightarrow{} y\\otimes 1\\xrightarrow{}y\\otimes x\\otimes y\\xrightarrow{}1\\otimes y\\xrightarrow{} y$ are equal to the respective identity maps (this notion goes back to [14]; see [30] for details).",
"In this subsection, we recall from [20] the construction of the dual of a smooth projective scheme in the stable motivic homotopy category.",
"Remark 2.10 Let $({\\mathcal {C}}, \\otimes , 1, \\mu , \\tau )$ be a symmetric monoidal category and let $x \\in {\\mathcal {C}}$ .",
"If a triple $(y,\\delta ,\\epsilon )$ satisfying the above definition of the dual exists, then it is unique up to unique isomorphism.",
"We often omit specific mention of $\\delta $ and $\\epsilon $ and denote the dual object $y$ by $x^\\vee $ .",
"When $x$ admits a dual $(x^\\vee ,\\delta ,\\epsilon )$ , then it is immediate that $x^\\vee $ is also dualizable, with dual $(x, \\tau \\circ \\delta , \\tau \\circ \\epsilon )$ , so that $x$ is canonically isomorphic to $(x^\\vee )^\\vee $ .",
"Sending $x \\mapsto x^\\vee $ extends to a contravariant functor $(-)^\\vee $ on the full subcategory of ${\\mathcal {C}}$ consisting of those objects $x$ that admit a dual with canonical isomorphism, and the above determines a natural isomorphism $((-)^\\vee )^\\vee \\simeq {\\operatorname{id}}$ .",
"For $f:x\\rightarrow z$ a morphism of dualizable objects in ${\\mathcal {C}}$ , the dual morphism $f^\\vee :z^\\vee \\rightarrow x^\\vee $ is the composition $z^\\vee \\xrightarrow{}z^\\vee \\otimes 1\\xrightarrow{}z^\\vee \\otimes x\\otimes x^\\vee \\xrightarrow{}z^\\vee \\otimes z\\otimes x^\\vee \\xrightarrow{}1\\otimes x^\\vee \\xrightarrow{}x^\\vee .$ Lemma 2.11 Let $\\pi _X : X \\rightarrow B$ be an object of ${\\mathrm {Sm}}_B$ .",
"View $X\\times _BX$ as a $X$ -scheme via the projection $p_2 : X\\times _BX \\rightarrow X$ onto the second factor.",
"Then there are canonical isomorphisms $\\pi _X^*(X/B_\\mathrm {B.M.})",
"\\simeq p_{2\\sharp } \\Sigma ^{-p_1^* T_{X/B}}(1_{X\\times _BX}) \\simeq X\\times _BX/X_\\mathrm {B.M.",
"}$ in ${\\operatorname{SH}}(X)$ and a canonical isomorphism $\\pi _{X\\sharp }(X\\times _BX/X_\\mathrm {B.M.",
"})\\simeq X/B_\\mathrm {B.M.",
"}\\wedge _BX/B$ in ${\\operatorname{SH}}(B)$ .",
"Consider the commutative square ${X\\times _BX[r]^-{p_2}[d]_{p_1}&X[d]^{\\pi _X}\\\\X[r]_-{\\pi _X}&B}$ This gives us the canonical isomorphism $T_{X\\times _BX/X}\\simeq p_1^*T_{X/B}.$ We have the identities and canonical isomorphisms $\\pi _X^*(X/B_\\mathrm {B.M.",
"})=\\pi _X^*\\pi _{X!",
"}(1_X)\\simeq \\pi _X^*\\pi _{X\\sharp }\\Sigma ^{-T_{X/B}}(1_X)\\operatornamewithlimits{{\\simeq }}^{\\text{(base change)}} p_{2\\sharp }p_1^*\\Sigma ^{-T_{X/B}}(1_X)\\\\\\simeq p_{2\\sharp }\\Sigma ^{-p_1^*T_{X/B}}(p_1^*1_X)=p_{2\\sharp }\\Sigma ^{-p_1^*T_{X/B}}(1_{X\\times _BX})$ which gives us the first isomorphism in ${\\operatorname{SH}}(X)$ .",
"The second follows from $p_{2\\sharp } \\Sigma ^{-p_1^* T_{X/B}}(1_{X\\times _BX})\\simeq p_{2\\sharp } \\Sigma ^{- T_{X\\times _BX/X}}(1_{X\\times _BX})\\simeq p_{2!",
"}(1_{X\\times _BX})=X\\times _BX/X_\\mathrm {B.M.",
"}.$ Finally, to give the isomorphism in ${\\operatorname{SH}}(B)$ , we have $\\pi _{X\\sharp }(X\\times _BX/X_\\mathrm {B.M.",
"})\\simeq \\pi _{X\\sharp }(p_{2\\sharp }(\\Sigma ^{-T_{X\\times _BX/X}}(1_{X\\times _BX}))\\simeq \\pi _{X\\sharp }p_{1\\sharp }(\\Sigma ^{-T_{X\\times _BX/X}}(1_{X\\times _BX}))\\\\\\simeq \\pi _{X\\sharp }p_{1\\sharp }(\\Sigma ^{-p_1^*T_{X/B}}(1_{X\\times _BX}))\\simeq \\pi _{X\\sharp }\\Sigma ^{-T_{X/B}}p_{1\\sharp }(1_{X\\times _BX})\\simeq \\pi _{X!",
"}p_{1\\sharp }p_2^*(1_X)\\\\\\operatornamewithlimits{{\\simeq }}^{\\text{(base change)}}\\pi _{X!",
"}\\pi _X^*\\pi _{X\\sharp }(1_X)\\simeq \\pi _{X!",
"}(1_X\\wedge _X\\pi _X^*(X/B))\\\\\\operatornamewithlimits{{\\simeq }}^{\\text{(projection formula)}}\\pi _{X!",
"}(1_X)\\wedge _B (X/B)=X/B_\\mathrm {B.M.",
"}\\wedge _B X/B.$ The base change isomorphisms follow from [20] and the projection formula is [20].",
"Construction 2.12 Let $\\pi _X : X \\rightarrow B$ be an object of ${\\mathrm {Sm}}_B$ that is proper (i.e.",
"a smooth projective scheme over $B$ ).",
"We recall (from Hoyois [20], but see also the constructions of Riou [43] and Ayoub [4]) the construction of a duality between the objects $X/B$ and $X/B_\\mathrm {B.M.",
"}$ in ${\\operatorname{SH}}(B)$ .",
"We first construct the coevaluation map $\\delta _{X/B} : 1_B \\rightarrow X/B\\wedge _BX/B_\\mathrm {B.M.",
"}$ .",
"Applying the functoriality of $(-)/B_\\mathrm {B.M.",
"}$ from Remark REF to the proper map $\\pi _X$ gives the map $\\pi _X^* : 1_B = B/B_\\mathrm {B.M.",
"}\\rightarrow X/B_\\mathrm {B.M.",
"}= \\pi _{X!",
"}(1_X) \\simeq \\pi _{X\\sharp }(\\Sigma ^{-T_{X/B}}(1_X))$ in ${\\operatorname{SH}}(B)$ , and the diagonal $\\Delta _{X/B} : X \\rightarrow X\\times _BX$ induces via the functoriality of $(-)/X:{\\mathrm {Sm}}_X\\rightarrow {\\operatorname{SH}}(X)$ the map $\\Delta _{X/B*}:=\\Delta _{X/B}/X : 1_X = X/X \\rightarrow X\\times _BX/X$ in ${\\operatorname{SH}}(X)$ .",
"We may put together these two maps to obtain the composition $1_B \\xrightarrow{} \\pi _{X\\sharp }(\\Sigma ^{-T_{X/B}}(1_X)) \\xrightarrow{} \\pi _{X\\sharp }\\Sigma ^{-T_{X/B}} (X\\times _BX/X),$ Using the identification $\\pi _{X\\sharp }\\Sigma ^{-T_{X/B}}(X\\times _BX/X) \\simeq X/B\\wedge _BX/B_\\mathrm {B.M.",
"}$ from Lemma REF , this gives the desired map $\\delta _{X/B}$ .",
"We now construct the evaluation map $\\epsilon _{X/B} : X/B_\\mathrm {B.M.",
"}\\wedge _B X/B \\rightarrow 1_B$ (which does not require properness of $\\pi _X$ ).",
"Here we apply the functoriality of $(-)/X_\\mathrm {B.M.",
"}$ to the proper map $\\Delta _{X/B}$ to obtain the map $\\Delta _{X/B}^* : X\\times _BX/X_\\mathrm {B.M.",
"}\\rightarrow X/X_\\mathrm {B.M.",
"}= 1_X$ in ${\\operatorname{SH}}(X)$ , and the functoriality of $(-)/B$ to the map $\\pi _X$ to obtain the map $\\pi _{X*} : X/B \\rightarrow B/B = 1_B.$ Putting these together yields the composition $\\pi _{X\\sharp }(X\\times _BX/X_\\mathrm {B.M.",
"})\\xrightarrow{} \\pi _{X/B\\sharp }(1_X) =X/B \\xrightarrow{} 1_B.$ Now using the identification $X/B_\\mathrm {B.M.",
"}\\wedge _BX/B \\simeq \\pi _{X\\sharp }(X\\times _BX/X_\\mathrm {B.M.",
"})$ from Lemma REF , we get the desired map $\\epsilon _{X/B}$ .",
"It is shown in [20] that the triple $(X/B_\\mathrm {B.M.",
"}, \\delta _{X/B}, \\epsilon _{X/B})$ is the dual of $X/B$ in ${\\operatorname{SH}}(B)$ .",
"Using our notation for Thom spaces, we have $X/B_\\mathrm {B.M.",
"}=\\pi _{X!",
"}(1_X)\\simeq \\pi _{X\\#}\\circ \\Sigma ^{-T_{X/B}}(1_X)={\\operatorname{Th}}_X(-T_{X/B})/B$ the coevaluation map is $\\delta _{X/B}:1_B\\rightarrow X/B\\wedge _B{\\operatorname{Th}}_X(-T_{X/B})/B$ and the evaluation map is $\\epsilon _{X/B}:{\\operatorname{Th}}_X(-T_{X/B})/B\\wedge _BX/B\\rightarrow 1_B$ The dualizability of $X/B$ as above allows one to define an Euler characteristic for smooth projective schemes: Definition 2.13 For $X \\rightarrow B$ in ${\\mathrm {Sm}}_B$ and proper, the Euler characteristic $\\chi (X/B)\\in {\\rm End}_{{\\operatorname{SH}}(B)}(1_B)$ is the composition $1_B\\xrightarrow{}X/B\\wedge _B{\\operatorname{Th}}_X(-T_{X/B})/B\\xrightarrow{}{\\operatorname{Th}}_X(-T_{X/B})/B\\wedge _BX/B\\xrightarrow{} 1_B,$ where $\\tau $ denotes the symmetry isomorphism, and $\\delta _{X/B}$ and $\\epsilon _{X/B}$ are as in Construction REF .",
"To finish this section, we give an alternative characterization of the Euler characteristic $\\chi (X/B)$ just defined (Lemma REF below).",
"We proved this result in the case $B={\\rm Spec\\,}k$ for $k$ a field in [26]; the proof in this more general setting is exactly the same.",
"Construction 2.14 We consider $X\\times _BX$ as a smooth $X$ -scheme via the projection $p_2$ .",
"The diagonal $\\Delta _X:X\\rightarrow X\\times _BX$ gives a section to $p_2$ .",
"Let $q:X\\times _BX\\rightarrow X\\times _BX/(X\\times _BX\\setminus \\Delta _X(X))$ be the quotient map.",
"We have the Morel-Voevodsky purity isomorphism [34], which gives the isomorphism in ${\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)$ $mv_\\Delta :X\\times _BX/(X\\times _BX\\setminus \\Delta _X(X))\\xrightarrow{}{\\operatorname{Th}}_X(N_{\\Delta _X})$ Composing the 0-section $s_N:X\\rightarrow N_{\\Delta _X} $ with the quotient map $N_{\\Delta _X}\\rightarrow {\\operatorname{Th}}(N_{\\Delta _X})$ defines $\\bar{s}_N:X\\rightarrow {\\operatorname{Th}}_X(N_{\\Delta _X})$ .",
"It follows from the proof of the purity isomorphism that we have the commutative diagram in ${\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)$ ${X\\times _BX_+[r]^-q& X\\times _BX/(X\\times _BX\\setminus \\Delta _X(X))[r]^-{mv_\\Delta }&{\\operatorname{Th}}_X(N_{\\Delta _X})\\\\&X_+[u]^{q\\circ \\Delta _X}[ur]_{\\bar{s}_N}[ul]^{\\Delta _X}}$ Finally, we have the isomorphism $N_{\\Delta _X}\\simeq T_{X/B}$ of vector bundles over $X$ furnished by the composition $T_{X/B}\\xrightarrow{}\\Delta _X^*p_2^*T_{X/B}\\xrightarrow{}\\Delta _X^*(p_1^*T_{X/B}\\oplus p_2^*T_{X/B})\\simeq \\Delta _X^*T_{X\\times _BX/B}\\xrightarrow{}N_{\\Delta _X}$ where $\\pi : \\Delta _X^*T_{X\\times _BX/B}\\rightarrow N_{\\Delta _X}$ is the canonical projection.",
"Putting these maps together gives us the composition in ${\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)$ $X_+\\xrightarrow{}X\\times _BX\\xrightarrow{}X\\times _BX/(X\\times _BX\\setminus \\Delta _X(X))\\xrightarrow{}{\\operatorname{Th}}_X(N_{\\Delta _X})\\xrightarrow{} {\\operatorname{Th}}_X(T_{X/B});$ the commutativity of the diagram (REF ) shows that this composition is equal to the map $\\bar{s}_{T_{X/B}}:X_+\\rightarrow {\\operatorname{Th}}_X(T_{X/B})$ , induced as for $\\bar{s}_{N_{\\Delta _X}}$ by the zero-section $s_{T_{X/B}}:X\\rightarrow T_{X/B}$ .",
"Applying $\\Sigma ^{-T_{X/B}}\\Sigma ^\\infty _{{\\mathbb {P}}^1}(-)$ and the isomorphism $\\Sigma ^{-T_{X/B}}{\\operatorname{Th}}(T_{X/B})\\simeq \\Sigma ^{-T_{X/B}}\\circ \\Sigma ^{T_{X/B}}(1_X)\\simeq 1_X$ gives the morphism $\\tilde{\\beta }_{X/B}:\\Sigma ^{-T_{X/B}}(1_X)\\rightarrow 1_X$ in ${\\operatorname{SH}}(X)$ .",
"Finally, applying $\\pi _{X\\#}$ gives the morphism $\\beta _{X/B}:{\\operatorname{Th}}(-T_{X/B})\\rightarrow X/B$ in ${\\operatorname{SH}}(B)$ .",
"Summarizing our construction, $\\beta _{X/B}$ is given by applying $\\pi _{X\\#}$ to the composition in ${\\operatorname{SH}}(X)$ $\\Sigma ^{-T_{X/B}}(1_X)\\xrightarrow{}\\Sigma ^{-T_{X/B}}(X\\times _BX/X)\\xrightarrow{}\\\\\\Sigma ^{-T_{X/B}}([X\\times _BX/(X\\times _BX\\setminus \\Delta _X)]/X)\\xrightarrow{}\\\\\\Sigma ^{-T_{X/B}}({\\operatorname{Th}}_X(N_{\\Delta _X}))\\simeq \\Sigma ^{-T_{X/B}}({\\operatorname{Th}}_X(T_{X/B}))\\\\\\simeq \\Sigma ^{-T_{X/B}}\\circ \\Sigma ^{T_{X/B}}(1_X)\\simeq 1_X$ or equivalently, to the composition $\\Sigma ^{-T_{X/B}}(1_X)\\xrightarrow{}\\Sigma ^{-T_{X/B}}({\\operatorname{Th}}_X(T_{X/B}))\\simeq \\Sigma ^{-T_{X/B}}\\circ \\Sigma ^{T_{X/B}}(1_X)\\simeq 1_X$ Lemma 2.15 For $X$ smooth and proper over $B$ , $\\chi (X/B)$ is equal to the composition $1_B \\xrightarrow{}{\\operatorname{Th}}(-T_{X/B})\\xrightarrow{}X/B\\xrightarrow{}1_B$ Let $p_i:X\\times _BX\\rightarrow X$ , $i=1,2$ , be the projections.",
"The map $\\mathrm {ev}_X:X/B^\\vee \\wedge _BX/B\\rightarrow 1_B$ is the composition $X/B^\\vee \\wedge _B X/B={\\operatorname{Th}}(-T_{X/k})\\wedge _BX/B\\\\={\\operatorname{Th}}(-p_1^*T_{X/k})\\xrightarrow{} {\\operatorname{Th}}(-p_1^*T_{X/B})/ {\\operatorname{Th}}(-j^*p_1^*T_{X/B})\\simeq {\\operatorname{Th}}( -T_{X/B}\\oplus N_{\\Delta _X})\\\\\\simeq {\\operatorname{Th}}(-T_{X/B}\\oplus T_{X/B})\\simeq X/B\\xrightarrow{} 1_B.$ From our description of $\\beta _{X/B}$ as $\\pi _{X\\#}$ applied to the composition (REF ), we see that $\\pi _{X*}\\circ \\beta _{X/B}=\\mathrm {ev}_{X/B}\\circ {\\operatorname{Th}}(\\Delta _X)$ .",
"Also, $\\pi _X^*=\\pi _X^\\vee $ and $\\pi _X^\\vee $ is given by $1_B\\xrightarrow{}X/B\\wedge _B X/B^\\vee ={\\operatorname{Th}}(-p_2^*T_{X/B})\\xrightarrow{}{\\operatorname{Th}}(-T_{X/B})$ It follows from the construction of the map $\\delta _{X/B}$ described above that $\\delta _{X/B}={\\operatorname{Th}}(\\Delta _X)\\circ \\pi _X^*.$ This gives us the commutative diagram ${40pt}{&X/B\\wedge _BX/B^\\vee [r]^{\\tau _{X, X^\\vee }}@{=}[d]&X/B^\\vee \\wedge _BX/B[dr]^{\\mathrm {ev}_X}\\\\1_B[ur]^-{\\delta _{X/B}}[dr]_{\\pi _X^*}&{\\operatorname{Th}}(-p_2^*T_{X/B})[r]^{\\tau _{X, X^\\vee }}&{\\operatorname{Th}}(-p_1^*T_{X/B})@{=}[u]&1_B\\\\&{\\operatorname{Th}}(-T_{X/B})[ur]_-{{\\operatorname{Th}}(\\Delta _X)}[u]_{{\\operatorname{Th}}(\\Delta _X)} [r]_{\\beta _{X/B}}&X/B[ur]_{\\pi _{X*}}}$"
],
[
"$\\operatorname{SL}$ -oriented motivic spectra",
"In this section, we discuss the definition and basic features of $\\operatorname{SL}$ -oriented motivic spectra.",
"This is treated by Ananyevskiy in [2] in the context of the motivic stable homotopy category ${\\operatorname{SH}}(k)$ for $k$ a field.",
"Essentially all of the constructions in op.",
"cit.",
"go through without change in the setting of a separated noetherian base-scheme $B$ of finite Krull dimension; the most one needs to do is replace a few arguments that rely on Jouanolou covers with some properties coming out of the six-functor formalism.",
"We will recall and suitably extend Ananyevskiy's treatment here without any claim of originality.",
"Definition 3.1 A motivic commutative ring spectrum in ${\\operatorname{SH}}(B)$ is a triple $({\\mathcal {E}}, \\mu , u)$ with ${\\mathcal {E}}$ in ${\\operatorname{SH}}(B)$ , and $\\mu :{\\mathcal {E}}\\wedge _B{\\mathcal {E}}\\rightarrow {\\mathcal {E}}$ , $u:1_B\\rightarrow {\\mathcal {E}}$ morphisms in ${\\operatorname{SH}}(B)$ , defining a commutative monoid object in the symmetric monoidal category ${\\operatorname{SH}}(B)$ .",
"We usually drop the explicit mention of the multiplication $\\mu $ and unit $u$ unless these are needed.",
"Notation 3.2 Recall that, given a motivic spectrum ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ , we have a notion of ${\\mathcal {E}}$ -cohomology ${\\mathcal {E}}^{**}(X)$ for $X \\in {\\operatorname{H}}(B)$ : this is the bigraded abelian group defined by ${\\mathcal {E}}^{a,b}(X) := {\\rm Hom}_{{\\operatorname{SH}}(B)}(\\Sigma ^\\infty _X̰_+, \\mathrm {S}^{a,b} \\wedge {\\mathcal {E}})$ for $a,b \\in {\\mathbb {Z}}$ , where $\\mathrm {S}^{a,b} \\in {\\operatorname{SH}}(B)$ denotes the usual bigraded stable motivic sphere.",
"This of course specializes to give the ${\\mathcal {E}}$ -cohomology of objects $X \\in {\\mathrm {Sm}}_B$ .",
"It also specializes to give ${\\mathcal {E}}$ -cohomology with supports ${\\mathcal {E}}^{**}_Z(X)$ for $X \\in {\\mathrm {Sm}}_B$ and $Z \\subseteq X$ a closed subset: namely, we define ${\\mathcal {E}}^{a,b}_Z(X):={\\mathcal {E}}^{a,b}(X/(X\\setminus Z)).$ For instance, if $V\\rightarrow X$ is a vector bundle, then ${\\mathcal {E}}^{a,b}_{0_V}(V)={\\mathcal {E}}^{a,b}({\\operatorname{Th}}(V))$ , where $0_V\\subset V$ is the image of the zero-section.",
"We further define ${\\mathcal {E}}^{a,b}_Z({\\operatorname{Th}}(V)) := {\\mathcal {E}}^{a,b}_{0_V \\cap q^{-1}(Z)}(V)$ for $Z \\subseteq X$ a closed subset and $q : V \\rightarrow X$ a vector bundle.",
"A commutative ring spectrum structure on ${\\mathcal {E}}$ determines a natural cup product structure on ${\\mathcal {E}}$ -cohomology and ${\\mathcal {E}}$ -cohomology with supports in the usual manner; we refrain from spelling it out in detail here.",
"Notation 3.3 For a scheme $X$ and a rank $r$ vector bundle $V\\rightarrow X$ , we let $\\operatorname{det}V\\rightarrow X$ denote the determinant line bundle, defined by $\\operatorname{det}V := \\Lambda ^rV$ .",
"Given two vector bundles $V_1\\rightarrow X$ and $V_2\\rightarrow X$ , we have a canonical isomorphism $\\alpha _{V_1, V_2}:\\operatorname{det}(V_1\\oplus V_2)\\rightarrow (\\operatorname{det}V_1) \\otimes _{{\\mathcal {O}}_X} (\\operatorname{det}V_2),$ characterized by requiring that, for a local basis of sections $s_1^1,\\ldots , s_n^1$ of $V_1$ and $s_1^2,\\ldots , s_m^2$ of $V_2$ , we have $\\alpha _{V_1, V_2}(((s_1^1,0)\\wedge \\ldots \\wedge (s_n^1,0)\\wedge (0, s_1^2)\\wedge \\ldots \\wedge (0, s_m^2)))= (s_1^1\\wedge \\ldots \\wedge s_n^1)\\otimes (s_1^2\\wedge \\ldots \\wedge s_m^2).$ This extends to a canonical natural isomorphism $\\alpha _E:\\operatorname{det}V \\rightarrow (\\operatorname{det}V_1) \\otimes _{{\\mathcal {O}}_X} (\\operatorname{det}V_2)$ for each exact sequence $E$ of vector bundles on $X$ , $0\\rightarrow V_1\\rightarrow V\\rightarrow V_2\\rightarrow 0;$ one can define $\\alpha _E$ by choosing local splittings and noting that the resulting isomorphism is independent of the choice of splitting.",
"Definition 3.4 An $\\operatorname{SL}$ -orientation of a motivic commutative ring spectrum ${\\mathcal {E}}$ in ${\\operatorname{SH}}(B)$ is an assignment of elements ${\\operatorname{th}}_{V,\\theta }\\in {\\mathcal {E}}^{2r,r}({\\operatorname{Th}}(V))$ for each pair $(V,\\theta )$ consisting of a rank $r$ vector bundle $V\\rightarrow X$ (for any $r \\ge 0$ ) with $X\\in {\\mathrm {Sm}}_B$ and an isomorphism $\\theta :\\operatorname{det}V\\rightarrow {\\mathcal {O}}_X$ of line bundles, satisfying the following axioms: Naturality: Let $(V \\rightarrow X, \\theta : \\operatorname{det}V \\rightarrow {\\mathcal {O}}_X)$ be as above and let $f:Y\\rightarrow X$ be a morphism in ${\\mathrm {Sm}}_B$ .",
"Consider the vector bundle $f^*V\\rightarrow Y$ and isomorphism $f^*\\theta :\\operatorname{det}f^*V\\simeq f^*\\operatorname{det}V\\rightarrow {\\mathcal {O}}_Y$ .",
"Then we have $f^*({\\operatorname{th}}_{V,\\theta })={\\operatorname{th}}_{f^*V, f^*\\theta }$ .",
"Products: Let $(V_1\\rightarrow X,\\theta _1:\\operatorname{det}V_1 \\rightarrow {\\mathcal {O}}_X)$ and $(V_2\\rightarrow X,\\theta _2:\\operatorname{det}V_2 \\rightarrow {\\mathcal {O}}_X)$ be two pairs as above.",
"Consider the vector bundle $V_1 \\oplus V_2 \\rightarrow X$ and the isomorphism $\\theta _1\\wedge \\theta _2:\\operatorname{det}(V_1\\oplus V_2)\\rightarrow {\\mathcal {O}}_X$ defined by $\\theta _1\\wedge \\theta _2 := (\\theta _1\\otimes \\theta _2)\\circ \\alpha _{V_1, V_2}.$ Then ${\\operatorname{th}}_{V_1\\oplus V_2, \\theta _1\\wedge \\theta _2}={\\operatorname{th}}_{V_1,\\theta _1}\\cup {\\operatorname{th}}_{V_2,\\theta _2}$ .",
"Normalization: For $X \\in {\\mathrm {Sm}}_B$ , consider the trivial vector bundle $V={\\mathcal {O}}_X$ and the identity isomorphism $\\theta : V \\rightarrow {\\mathcal {O}}_X$ .",
"Then, under the canonical identification ${\\operatorname{Th}}(V) \\simeq \\Sigma _X̰_+$ , the element ${\\operatorname{th}}_{V,\\theta }\\in {\\mathcal {E}}^{2,1}({\\operatorname{Th}}(V))$ is the image of the unit $1 \\in {\\mathcal {E}}^{0,0}(B)$ under the composition ${\\mathcal {E}}^{0,0}(B)\\xrightarrow{}{\\mathcal {E}}^{0,0}(X)\\operatornamewithlimits{{\\simeq }}^{\\text{suspension}}{\\mathcal {E}}^{2,1}(\\Sigma _X̰_+).$ An $\\operatorname{SL}$ -oriented motivic spectrum is a pair $({\\mathcal {E}}, {\\operatorname{th}}_{(-)})$ with ${\\mathcal {E}}$ a motivic commutative ring spectrum and ${\\operatorname{th}}_{(-)}$ an $\\operatorname{SL}$ -orientation on ${\\mathcal {E}}$ .",
"Variant 3.5 A $\\operatorname{GL}$ -orientation, or simply orientation, on a motivic commutative ring spectrum ${\\mathcal {E}}$ is an assignment $(V\\rightarrow X)\\mapsto {\\operatorname{th}}_V\\in {\\mathcal {E}}^{2r, r}({\\operatorname{Th}}(V))$ , where $V\\rightarrow X$ is a rank $r$ vector bundle (for any $r \\ge 0$ ) on $X\\in {\\mathrm {Sm}}_B$ , satisfying the evident modifications of the axioms (i)–(iii) in Definition REF , i.e.",
"omitting the conditions on the determinant line bundle.",
"The pair $({\\mathcal {E}}, {\\operatorname{th}}_{(-)})$ is called a $\\operatorname{GL}$ -oriented motivic spectrum, or more simply, an oriented motivic spectrum.",
"For the remainder of the section, we fix an $\\operatorname{SL}$ -oriented motivic spectrum ${\\mathcal {E}}$ .",
"Let us first observe that the $\\operatorname{SL}$ -orientation determines Thom isomorphisms in ${\\mathcal {E}}$ -cohomology for oriented vector bundles: Lemma 3.6 Let $({\\mathcal {E}}, {\\operatorname{th}}_{(-)})$ be an $\\operatorname{SL}$ -oriented motivic spectrum and let $q:V\\rightarrow X$ be a rank $r$ vector bundle on $X \\in {\\mathrm {Sm}}_B$ with an isomorphism $\\theta :\\operatorname{det}V\\rightarrow {\\mathcal {O}}_X$ .",
"Then sending $x\\in {\\mathcal {E}}^{a, b}_Z(X)$ to $q^*(x)\\cup {\\operatorname{th}}_{V,\\theta }$ defines an isomorphism $\\vartheta _{V,\\theta }:{\\mathcal {E}}^{a, b}_Z(X)\\rightarrow {\\mathcal {E}}^{a+2r, b+r}_Z({\\operatorname{Th}}(V))$ natural in $(X, V, \\theta )$ .",
"The naturality of the maps $\\vartheta _{V,\\theta }$ follows from the naturality of the Thom classes, i.e.",
"property (i) in their definition.",
"It follows from properties (i)–(iii) of the Thom class that for $V=\\oplus _{i=1}^r{\\mathcal {O}}_Xe_i$ and $\\theta :\\operatorname{det}V\\rightarrow {\\mathcal {O}}_X$ the canonical isomorphism given by $\\theta (e_1\\wedge \\ldots \\wedge e_n)=1$ , the map $\\vartheta _{V,\\theta }$ is the suspension isomorphism ${\\mathcal {E}}^{a, b}_Z(X)\\simeq {\\mathcal {E}}^{a+2r, b+r}(\\Sigma ^r_TX_+/\\Sigma ^r_T(X\\setminus Z)_+)\\simeq {\\mathcal {E}}^{a+2r, b+r}_Z({\\operatorname{Th}}(V)).$ The naturality of the maps $\\vartheta _{V,\\theta }$ allow one to use a Mayer-Vietoris sequence for a trivializing open cover of $X$ for $V$ to show that $\\vartheta _{V,\\theta }$ is an isomorphism in general.",
"The above Thom isomorphism may be extended to vector bundles without orientation by introducing twists to ${\\mathcal {E}}$ -cohomology, as follows.",
"Definition 3.7 For $X\\in {\\mathrm {Sm}}_B$ and $q:L\\rightarrow X$ a line bundle, we define the twisted ${\\mathcal {E}}$ -cohomology ${\\mathcal {E}}^{**}(X;L)$ by ${\\mathcal {E}}^{a,b}(X;L):={\\mathcal {E}}^{a+2, b+1}({\\operatorname{Th}}(L))={\\mathcal {E}}^{a+2, b+1}_{0_L}(L).$ We also have a version with supports: given in addition a closed subset $Z \\subseteq X$ , we define ${\\mathcal {E}}^{a,b}_Z(X;L):={\\mathcal {E}}^{a+2, b+1}_{0_L\\cap q^{-1}(Z)}(L).$ Finally, when we also have a vector bundle $V \\rightarrow X$ , we similarly define ${\\mathcal {E}}^{a,b}({\\operatorname{Th}}(V);L) := {\\mathcal {E}}^{a,b}_{0_V}(V;q^*L), \\quad {\\mathcal {E}}^{a,b}_Z({\\operatorname{Th}}(V);L) := {\\mathcal {E}}^{a,b}_{0_V\\cap q^{-1}(Z)}(V;q^*L);$ these definitions may be rewritten a bit, e.g.",
"for the former we have ${\\mathcal {E}}^{a,b}({\\operatorname{Th}}(V);L) = {\\mathcal {E}}^{a,b}_{0_V}(V;q^*L) = {\\mathcal {E}}^{a+2,b+1}_{0_{V \\oplus L}}(V\\oplus L) = {\\mathcal {E}}^{a+2,b+1}({\\operatorname{Th}}(V\\oplus L)).$ Remark 3.8 The product structure on ${\\mathcal {E}}$ -cohomology extends to a product structure on twisted ${\\mathcal {E}}$ -cohomology: namely, for $X \\in {\\mathrm {Sm}}_B$ and $L,M$ two line bundles on $X$ , combining the cup product $\\cup :{\\mathcal {E}}^{a+2,b+1}_{0_L}(L)\\otimes {\\mathcal {E}}^{c+2,d+1}_{0_M}(M)\\rightarrow {\\mathcal {E}}^{a+c+4, b+d+2}_{0_{L\\oplus M}}(L\\oplus M),$ the canonical isomorphism $\\alpha _{L, M}:\\operatorname{det}(L\\oplus M)\\rightarrow L\\otimes M$ , and the Thom isomorphism $\\vartheta _{L\\oplus M}$ , we obtain a map $\\cup :{\\mathcal {E}}^{a,b}(X;L)\\otimes {\\mathcal {E}}^{c,d}(X;M)\\rightarrow {\\mathcal {E}}^{a+c, b+d}(X;L\\otimes M),$ as well as a version with supports.",
"Remark 3.9 The definitions of twisted ${\\mathcal {E}}$ -cohomology given in Definition REF are instances of a more general definition.",
"Namely, for $X \\in {\\mathrm {Sm}}_B$ , $L \\rightarrow X$ a line bundle, and any $T \\in {\\operatorname{SH}}(B)$ equipped with an identification $T \\simeq \\pi _{X\\sharp }(T^{\\prime })$ for some $T^{\\prime } \\in {\\operatorname{SH}}(X)$ , we may define the twisted ${\\mathcal {E}}$ -cohomology ${\\mathcal {E}}^{**}(T;L)$ by ${\\mathcal {E}}^{a,b}(T;L) := {\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^L T^{\\prime }, \\mathrm {S}^{a+2,b+1} \\wedge \\pi _X^*{\\mathcal {E}}).$ Of course, this notation is abusive since ${\\mathcal {E}}^{**}(T;L)$ really depends on $T^{\\prime } \\in {\\operatorname{SH}}(X)$ .",
"This construction recovers the notions introduced in Definition REF as follows: We have ${\\mathcal {E}}^{**}(X;L) \\simeq {\\mathcal {E}}^{**}(T;L)$ for $T = \\Sigma ^\\infty _X̰_+ \\simeq \\pi _{X\\sharp }(1_X)$ .",
"We have ${\\mathcal {E}}^{**}_Z(X;L) \\simeq {\\mathcal {E}}^{**}(T;L)$ for $T = X_Z/B \\simeq \\pi _{X\\sharp }(X_Z/X)$ .",
"We have ${\\mathcal {E}}^{**}({\\operatorname{Th}}(V);L) \\simeq {\\mathcal {E}}^{**}(T;L)$ for $T = {\\operatorname{Th}}(V) \\simeq \\pi _{X\\sharp }({\\operatorname{Th}}_X(V))$ .",
"The extra generality will be invoked later on (see Lemma REF ) for the case $T = X_Z/B_\\mathrm {B.M.",
"}= \\pi _{X!",
"}(X_Z/X_\\mathrm {B.M.})",
"\\operatornamewithlimits{{\\simeq }}\\pi _{X\\sharp }(\\Sigma ^{-T_{X/B}} X_Z/X_\\mathrm {B.M.",
"}),$ where the last identification is by Atiyah duality (Remark REF ).",
"The following construction is due to Ananyevskiy [2].",
"Construction 3.10 Let $X \\in {\\mathrm {Sm}}_B$ , let $q:V\\rightarrow X$ be a rank $r$ vector bundle, and let $p:L\\rightarrow X$ denote the determinant bundle $\\operatorname{det}V$ ; let $p^{\\prime }:L^{-1}\\rightarrow X$ denote the inverse of $L$ .",
"Then we have canonical isomorphisms $\\alpha _{V, L^{-1}}:\\operatorname{det}(V\\oplus L^{-1})\\rightarrow {\\mathcal {O}}_X,\\quad \\alpha _{L^{-1}\\oplus L}:\\operatorname{det}(L^{-1}\\oplus L)\\rightarrow {\\mathcal {O}}_X.$ Consider now the two pullback bundles $p_V:q^*(L^{-1}\\oplus L)\\rightarrow V, \\quad q_L\\oplus p^{\\prime }_L:p^*(V\\oplus L^{-1})\\rightarrow L,$ which inherit trivializations of their determinants.",
"For $({\\mathcal {E}}, {\\operatorname{th}}_{(-)})$ an $\\operatorname{SL}$ -oriented motivic spectrum, Lemma REF gives us isomorphisms $\\vartheta _{q^*(L^{-1}\\oplus L), q^*\\alpha _{L^{-1}\\oplus L}}:{\\mathcal {E}}^{a+2r,b+r}_{0_V}(V)\\rightarrow {\\mathcal {E}}^{a+2r+2,b+r+1}_{q^{-1}(0_{L^{-1}}\\oplus 0_L)\\cap p_V^{-1}(0_V)}(q^*(L^{-1}\\oplus L))$ and $\\vartheta _{p^*(V\\oplus L^{-1}), p^*\\alpha _{V\\oplus L^{-1}}}:{\\mathcal {E}}^{a+2,b+1}_{0_L}(L)\\rightarrow {\\mathcal {E}}^{a+2r+2,b+r+1}_{p^{-1}(0_V\\oplus 0_{L^{-1}})\\cap p_L^{\\prime -1}(0_V)}(p^*(V\\oplus L^{-1})).$ However, as $X$ -schemes $q^*(L^{-1}\\oplus L)$ and $p^*(V\\oplus L^{-1})$ are both canonically isomorphic to $V\\oplus L^{-1}\\oplus L\\rightarrow X$ , and via this isomorphism, the closed subsets $q^{-1}(0_{L^{-1}}\\oplus 0_L)\\cap p_V^{-1}(0_V)$ and $p^{-1}(0_V\\oplus 0_{L^{-1}})\\cap p_L^{\\prime -1}(0_V)$ are both equal to $0_{V\\oplus L^{-1}\\oplus L}$ .",
"We thus have a canonical isomorphism $\\phi : {\\mathcal {E}}^{a+2r+2,b+r+1}_{q^{-1}(0_{L^{-1}}\\oplus 0_L)\\cap p_V^{-1}(0_V)}(q^*(L^{-1}\\oplus L))\\rightarrow {\\mathcal {E}}^{a+2r+2,b+r+1}_{p^{-1}(0_V\\oplus 0_{L^{-1}})\\cap p_L^{\\prime -1}(0_V)}(p^*(V\\oplus L^{-1}))$ Combining all of the above the, we obtain the Thom isomorphism $\\vartheta _V:{\\mathcal {E}}^{a,b}(X;L)\\rightarrow {\\mathcal {E}}^{a+2r, b+r}({\\operatorname{Th}}(V)),$ defined by $\\vartheta _V:=\\vartheta _{q^*(L^{-1}\\oplus L), q^*\\alpha _{L^{-1}\\oplus L}}^{-1}\\circ \\phi \\circ \\vartheta _{p^*(V\\oplus L^{-1}), p^*\\alpha _{V\\oplus L^{-1}}}.$ This construction extends to cohomology with supports in an evident manner.",
"We next discuss the Thom classes in twisted ${\\mathcal {E}}$ -cohomology governing this more general Thom isomorphism.",
"Definition 3.11 Let $X \\in {\\mathrm {Sm}}_B$ and let $q:V\\rightarrow X$ be a rank $r$ vector bundle.",
"The canonical Thom class ${\\operatorname{th}}_V \\in {\\mathcal {E}}^{2r, r}({\\operatorname{Th}}(V);\\operatorname{det}^{-1} V)$ is defined as follows.",
"As noted in Definition REF , we have ${\\mathcal {E}}^{2r, r}({\\operatorname{Th}}(V);\\operatorname{det}^{-1} V) = {\\mathcal {E}}^{2r+2, r+1}({\\operatorname{Th}}(V\\oplus \\operatorname{det}^{-1}V)).$ The isomorphism $\\alpha := \\alpha _{V,\\operatorname{det}^{-1}V}:\\operatorname{det}(V\\oplus \\operatorname{det}^{-1}V)\\rightarrow {\\mathcal {O}}_X$ gives us the Thom class ${\\operatorname{th}}_{V\\oplus \\operatorname{det}^{-1}V, \\alpha }\\in {\\mathcal {E}}^{2r+2, r+1}({\\operatorname{Th}}(V\\oplus \\operatorname{det}^{-1}V))$ , and we define ${\\operatorname{th}}_V$ to be the corresponding element of ${\\mathcal {E}}^{2r, r}({\\operatorname{Th}}(V);\\operatorname{det}^{-1} V)$ under the above identification.",
"Remark 3.12 Let $X$ and $V$ be as in Definition REF .",
"Let $\\pi _X:X\\rightarrow B$ denote the structure morphism.",
"Then the Thom class ${\\operatorname{th}}_V$ is an element of ${\\mathcal {E}}^{2r, r}({\\operatorname{Th}}(V);\\operatorname{det}^{-1} V)&={\\mathcal {E}}^{2r+2, r+1}(\\pi _{X\\sharp }(\\Sigma ^{V\\oplus \\operatorname{det}^{-1}(V)}(1_X))\\\\&= {\\rm Hom}_{{\\operatorname{SH}}(B)}(\\pi _{X\\sharp }(\\Sigma ^{V\\oplus \\operatorname{det}^{-1}(V)}(1_X), \\mathrm {S}^{2r+2, r+1}\\wedge {\\mathcal {E}}) \\\\&\\simeq {\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{V\\oplus \\operatorname{det}^{-1}(V)}(1_X), \\mathrm {S}^{2r+2, r+1}\\wedge \\pi _X^*{\\mathcal {E}}).",
"\\\\&\\simeq {\\rm Hom}_{{\\operatorname{SH}}(X)}(1_X, \\mathrm {S}^{2r+2, r+1}\\wedge \\Sigma ^{-(V\\oplus \\operatorname{det}^{-1}(V))} \\pi _X^*{\\mathcal {E}}).$ Thus, via the multiplication on ${\\mathcal {E}}$ , the class ${\\operatorname{th}}_V$ induces a map $\\times {\\operatorname{th}}_V :\\pi _X^*{\\mathcal {E}}\\rightarrow \\mathrm {S}^{2r+2, r+1}\\wedge \\Sigma ^{-(V\\oplus \\operatorname{det}^{-1}(V))} \\pi _X^*{\\mathcal {E}}$ in ${\\operatorname{SH}}(X)$ .",
"Lemma 3.13 The map $\\times {\\operatorname{th}}_V$ defined above is an isomorphism in ${\\operatorname{SH}}(X)$ .",
"For each object $x\\in {\\operatorname{SH}}(X)$ , $\\times {\\operatorname{th}}_V$ induces a map $\\vartheta _{V,x}:{\\rm Hom}_{{\\operatorname{SH}}(X)}(x, \\pi _X^*{\\mathcal {E}})\\rightarrow {\\rm Hom}_{{\\operatorname{SH}}(X)}(x,\\mathrm {S}^{2r+2, r+1}\\wedge \\Sigma ^{-(V\\oplus \\operatorname{det}^{-1}(V))} \\pi _X^*{\\mathcal {E}})$ By the Yoneda lemma, it suffices to show that $\\vartheta _{V,x}$ is an isomorphism for all $x\\in {\\operatorname{SH}}(X)$ .",
"The collection of objects $x$ for which $\\vartheta _{V,x}$ is an isomorphism is closed under arbitrary direct sums, hence is a localizing subcategory of ${\\operatorname{SH}}(X)$ .",
"The objects $x=\\mathrm {S}^{a,b} \\wedge 1_X$ are contained in this subcategory, since for these objects the map $\\vartheta _{V,x}$ identifies with the Thom isomorphism $\\theta _{V\\oplus \\operatorname{det}^{-1}(V),\\alpha }$ (where $\\alpha $ is as in Definition REF ).",
"For $p:Y\\rightarrow X$ in ${\\mathrm {Sm}}_X$ , applying $p^*$ and using the adjunction with $p_\\sharp $ shows that furthermore $\\vartheta _{V,x}$ is an isomorphism for $x=\\mathrm {S}^{a,b} \\wedge Y/X$ .",
"As ${\\operatorname{SH}}(X)$ is generated as a localizing subcategory by the objects $\\mathrm {S}^{a,b} \\wedge Y/X$ , this proves the lemma.",
"Definition 3.14 For $X\\in {\\mathrm {Sm}}_B$ and $V\\rightarrow X$ a rank $r$ vector bundle, we define $\\vartheta ^{\\mathcal {E}}_V:\\Sigma ^{1-\\operatorname{det}V}\\pi _X^*{\\mathcal {E}}\\rightarrow \\Sigma ^{r-V}\\pi _X^*{\\mathcal {E}}$ to be the composition of isomorphisms $\\Sigma ^{1-\\operatorname{det}V}\\pi _X^*{\\mathcal {E}}\\xrightarrow{}\\Sigma ^{\\operatorname{det}^{-1}V-1}\\pi _X^*{\\mathcal {E}}\\xrightarrow{}\\Sigma ^{r-V}\\pi _X^*{\\mathcal {E}}.$ Remark 3.15 Let $X \\in {\\mathrm {Sm}}_B$ and let $V\\rightarrow X$ be a rank $r$ vector bundle.",
"Under the identification ${\\operatorname{Th}}(V)=\\pi _{X\\#}\\Sigma ^V(1_X)$ and the isomorphisms ${\\mathcal {E}}^{a,b}(X,\\operatorname{det}V)\\simeq {\\rm Hom}_{{\\operatorname{SH}}(X)}(1_X, \\mathrm {S}^{a+2,b+1}\\wedge \\Sigma ^{-\\operatorname{det}V}\\pi _X^*{\\mathcal {E}})$ and ${\\mathcal {E}}^{2r+a, r+b}({\\operatorname{Th}}(V))\\simeq {\\rm Hom}_{{\\operatorname{SH}}(X)}(1_X, \\mathrm {S}^{2r+a, r+b} \\wedge \\Sigma ^{-V}\\pi _X^*{\\mathcal {E}})$ the Thom isomorphism $\\vartheta _V:{\\mathcal {E}}^{a,b}(X,\\operatorname{det}V)\\rightarrow {\\mathcal {E}}^{2r+a, r+b}({\\operatorname{Th}}(V))$ is the map induced by $\\vartheta _V^{\\mathcal {E}}$ .",
"Remark 3.16 Let $X \\in {\\mathrm {Sm}}_B$ and let $V\\rightarrow X$ and $W\\rightarrow X$ be vector bundles of respective ranks $r_V$ and $r_W$ .",
"We have have the multiplication map ${\\mathcal {E}}^{a,b}({\\operatorname{Th}}(V), \\operatorname{det}^{-1}V)\\otimes {\\mathcal {E}}^{c,d}({\\operatorname{Th}}(W),\\operatorname{det}^{-1}W)\\rightarrow {\\mathcal {E}}^{a+c, b+d}({\\operatorname{Th}}(V\\oplus W),\\operatorname{det}^{-1}(V\\oplus W))$ induced by the isomorphism $V/(V\\setminus \\lbrace 0_V\\rbrace )\\wedge _XW/(W\\setminus \\lbrace 0_W\\rbrace )\\rightarrow (V\\oplus W)/(V\\oplus W\\setminus \\lbrace 0_{V\\oplus W}\\rbrace )$ in ${\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)$ and our canonical isomorphism $\\operatorname{det}(V)\\otimes \\operatorname{det}(W)\\simeq \\operatorname{det}(V\\oplus W)$ .",
"The multiplicative property of the Thom classes (Definition REF (ii)) implies a similar multiplicativity for the canonical Thom classes: ${\\operatorname{th}}_{V\\oplus W}={\\operatorname{th}}_V\\cup {\\operatorname{th}}_W.$ This then implies, roughly speaking, that $\\text{``}\\ \\vartheta _{W\\oplus V}^{\\mathcal {E}}=\\vartheta _W^{\\mathcal {E}}\\circ \\vartheta _V^{\\mathcal {E}}.\\ \\text{''}$ More precisely, after using properties of $\\Sigma ^{(-)}$ to make the necessary identifications, the following diagram commutes: ${\\Sigma ^{1-\\operatorname{det}(W\\oplus V)}\\pi _X^*{\\mathcal {E}}[rr]^{\\vartheta _{\\operatorname{det}W\\oplus \\operatorname{det}V}^{\\mathcal {E}}}[d]_{\\vartheta _{W\\oplus V}^{\\mathcal {E}}}&&\\Sigma ^{2-(\\operatorname{det}W\\oplus \\operatorname{det}V)}\\pi _X^*{\\mathcal {E}}[r]^\\sim &\\Sigma ^{1-\\operatorname{det}W}\\Sigma ^{1-\\operatorname{det}V}\\pi _X^*{\\mathcal {E}}[d]^{\\Sigma ^{1-\\operatorname{det}W}\\vartheta _E^{\\mathcal {E}}}\\\\\\Sigma ^{r_W+r_V-(W\\oplus V)}\\pi _X^*{\\mathcal {E}}&&\\Sigma ^{r_V-V}\\Sigma ^{1-\\operatorname{det}W}\\pi _X^*{\\mathcal {E}}[ll]^{\\Sigma ^{r_V-V}\\vartheta ^{\\mathcal {E}}_W}&\\Sigma ^{1-\\operatorname{det}W}\\Sigma ^{r_V-V}\\pi _X^*{\\mathcal {E}}[l]^\\sim }$ Remark 3.17 Using Remark REF , the definition of $\\vartheta ^{\\mathcal {E}}_{-}$ extends to virtual bundles by setting $\\vartheta ^{\\mathcal {E}}_{V-W}:=\\vartheta _V^{\\mathcal {E}}\\circ (\\vartheta _W^{\\mathcal {E}})^{-1} =(\\vartheta _W^{\\mathcal {E}})^{-1}\\circ \\vartheta _V^{\\mathcal {E}}$ (with these identities understood as in Remark REF ), giving the isomorphism $\\vartheta ^{\\mathcal {E}}_{V-W}:\\Sigma ^{1-\\operatorname{det}V\\otimes \\operatorname{det}^{-1} W}\\pi _X^*{\\mathcal {E}}\\xrightarrow{}\\Sigma ^{r_V-r_W-V+W}\\pi _X^*{\\mathcal {E}}.$ Remark REF then extends directly to virtual bundles.",
"If we have an exact sequence $0\\rightarrow V^{\\prime }\\rightarrow V\\rightarrow V^{\\prime \\prime }\\rightarrow 0$ , then the corresponding isomorphisms $\\Sigma ^V\\simeq \\Sigma ^{V^{\\prime }\\oplus V^{\\prime \\prime }}$ and $\\operatorname{det}V\\simeq \\operatorname{det}(V^{\\prime }\\oplus V^{\\prime \\prime })$ transform $\\vartheta _V^{\\mathcal {E}}$ to $\\vartheta _{V^{\\prime }\\oplus V^{\\prime \\prime }}^{\\mathcal {E}}$ .",
"Remark 3.18 If the $\\operatorname{SL}$ -orientation on ${\\mathcal {E}}$ extends to a $\\operatorname{GL}$ -orientation, then all the results of this section for $\\operatorname{SL}$ -oriented theories hold for ${\\mathcal {E}}$ in simplified form: we can omit all the twisting by line bundles and replace $\\Sigma ^{1-\\operatorname{det}V}\\pi _X^*{\\mathcal {E}}$ with $\\pi _X^*{\\mathcal {E}}$ using the Thom class ${\\operatorname{th}}_{\\operatorname{det}V}$ to define an isomorphism $\\Sigma _{\\pi _X^*{\\mathcal {E}}\\simeq \\Sigma ^{\\operatorname{det}V}\\pi _X^*{\\mathcal {E}}.",
"}$ We close this section with one last result about twisted ${\\mathcal {E}}$ -cohomology in the $\\operatorname{SL}$ -oriented setting.",
"Proposition 3.19 Let $X \\in {\\mathrm {Sm}}_B$ , let $L, M$ be two line bundles on $X$ , and let $Z\\subseteq X$ a closed subset.",
"Then there is a natural isomorphism $\\psi _{L,M}:{\\mathcal {E}}^{*,*}_Z(X;L)\\rightarrow {\\mathcal {E}}^{*,*}_Z(X;L\\otimes M^{\\otimes 2}).$ Let $s:X\\rightarrow L\\oplus M$ be the zero-section.",
"We have the Thom isomorphisms ${\\mathcal {E}}^{*,*}_Z(X;L)\\simeq {\\mathcal {E}}^{*+4,*+2}_{s(Z)}(L\\oplus M;M^{-1}),\\quad {\\mathcal {E}}^{*,*}_Z(X;L\\otimes M^{\\otimes 2})\\simeq {\\mathcal {E}}^{*+4,*+2}_{s(Z)}(L\\oplus M;M).$ Replacing $L \\oplus M$ by $X$ and $M$ by $L$ , this reduces us to showing that there is a natural isomorphism $\\psi _L:{\\mathcal {E}}^{*,*}_Z(X;L)\\rightarrow {\\mathcal {E}}^{*,*}_Z(X;L^{-1})$ For this, we follow the proof of [2].",
"We have the morphism of $X$ -schemes $L\\oplus L^{-1}=L\\times _XL^{-1}\\xrightarrow{} X\\times _B{\\mathbb {A}}^1$ defined by $\\mu (x, y)=x\\cdot y$ ; let $Y:=\\mu ^{-1}(X\\times 1)$ .",
"Setting $L_0:=L\\setminus 0_L$ and $L^{-1}_0:=L^{-1}\\setminus 0_L$ , we see that $Y$ is a closed subscheme of $L\\times _XL^{-1}$ projecting isomorphically to $L_0$ via $p_1$ and isomorphically to $L_0^{-1}$ via $p_2$ .",
"Consider the commutative diagram ${Y[r][d]_{p_1}& L\\times _XL^{-1}[d]_{p_1}[r]& L\\times _XL^{-1}/Y[d]^{\\bar{p}_1}\\\\L_0[r] & L[r] & L/L_0}$ whose rows are cofiber sequences.",
"As the first two vertical maps are isomorphisms in ${\\operatorname{H}}(B)$ , the map $\\bar{p}_1$ induces an isomorphism $\\bar{p}_1/B: (L\\times _BL^{-1}/Y)/B\\rightarrow {\\operatorname{Th}}(L)/B$ in ${\\operatorname{SH}}(B)$ .",
"Similarly, we have the isomorphism $\\bar{p}_2/B:(L\\times _BL^{-1}/Y)/B\\rightarrow {\\operatorname{Th}}(L^{-1})/B$ in ${\\operatorname{SH}}(B)$ .",
"Replacing $X$ with $U:=X\\setminus Z$ , we have the isomorphisms $\\bar{p}_{1U}/B: (L\\times _BL^{-1}\\times _XU/Y\\times _XU)/B\\rightarrow {\\operatorname{Th}}(L\\times _XU)/B.$ and $\\bar{p}_{2U}/B: (L\\times _BL^{-1}\\times _XU/Y\\times _XU)/B\\rightarrow {\\operatorname{Th}}(L^{-1}\\times _XU)/B$ It follows that the arrows in following diagram are isomorphisms in ${\\operatorname{SH}}(B)$ after applying $-/B$ : ${&{\\operatorname{Th}}(L)/{\\operatorname{Th}}(L\\times _XU)\\\\\\hbox{t}o 150pt{\\left(L\\times _XL^{-1}/Y\\right)/\\left(L\\times _XL^{-1}\\times _XU/Y\\times _XU\\right)\\hss }[ur]^{\\overline{\\bar{p}}_1}[dr]_{\\overline{\\bar{p}}_2}\\\\&{\\operatorname{Th}}(L^{-1})/{\\operatorname{Th}}(L^{-1}\\times _XU)}$ Finally, applying ${\\rm Hom}_{{\\operatorname{SH}}(B)}(-, \\Sigma ^{*+2,*+1}{\\mathcal {E}})$ gives the desired isomorphism $\\psi _L:{\\mathcal {E}}^{*,*}_Z(X;L)\\rightarrow {\\mathcal {E}}^{*,*}_Z(X;L^{-1}).",
"$"
],
[
"Projective pushforward in twisted cohomology",
"In this section, we describe how one gets projective pushforward maps in twisted ${\\mathcal {E}}$ -cohomology for ${\\mathcal {E}}$ an $\\operatorname{SL}$ -oriented motivic spectrum.",
"We rely on the six-functor formalism.",
"This is a bit different from the treatment of projective pushforward given by Ananyevskiy in [2]: in that treatment, one relies on the factorization of an arbitrary projective morphism $Y\\rightarrow X$ into a closed immersion $Y\\rightarrow X\\times {\\mathbb {P}}^N$ followed by a projection $X\\times {\\mathbb {P}}^N\\rightarrow X$ .",
"This factorization property will however reappear in our treatment when we discuss the uniqueness of the pushforward maps in §.",
"We continue to work over a noetherian separated base scheme $B$ of finite Krull dimension.",
"Lemma 4.1 Let $s:Y\\rightarrow X$ be a section of a smooth morphism $p:X\\rightarrow Y$ and let $\\eta :{\\operatorname{id}}\\rightarrow s_*s^*$ , $\\epsilon :s^*s_*\\rightarrow {\\operatorname{id}}$ be unit and counit of adjunction.",
"Then the composition ${\\operatorname{id}}_{{\\operatorname{SH}}(Y)}\\xrightarrow{} p_!\\circ s_!= p_!\\circ s_*\\xrightarrow{}s^*\\circ s_*\\xrightarrow{}{\\operatorname{id}}_{{\\operatorname{SH}}(Y)}$ is the identity.",
"Here the morphism $s^*:p_!\\rightarrow s^*$ is the one constructed in Remark REF .",
"The functor $s^*s_*$ is an equivalence [20].",
"As a general property of adjoint functors, $s_*\\epsilon s^*\\circ \\eta s_*s^*={\\operatorname{id}}_{s_*s^*}$ (see, e.g., [29]), hence $s_*\\epsilon s^*s_*\\circ \\eta s_*s^*s_*={\\operatorname{id}}_{s_*s^*s_*}$ and thus $s_*\\epsilon \\circ \\eta s_*={\\operatorname{id}}_{s_*}$ .",
"The result follows from the commutative diagram ${&{\\operatorname{id}}[d]^\\wr \\\\p_{X!",
"}\\circ s_*[d]_{p_{X!",
"}\\eta s_*}[dr]^{{\\operatorname{id}}}@/_60pt/[ddd]_{s^*}&p_{X!",
"}\\circ s_!",
"@{=}[l] @{=}[d]\\\\p_{X!",
"}\\circ s_*s^*s_*[r]^{p_{X!}",
"s_*\\epsilon }@{=}[d] &p_{X!",
"}\\circ s_*@{=}[d]\\\\p_{X!",
"}\\circ s_!\\circ s^*\\circ s_*[d]_\\wr [r]^{p_{X!}",
"s_!\\epsilon }&p_{X!",
"}\\circ s_!",
"[d]^\\wr \\\\s^*\\circ s_*[r]_\\epsilon &{\\operatorname{id}}}$ Remark 4.2 Let $s:Y\\rightarrow V$ be the zero-section of a vector bundle $p:V\\rightarrow Y$ .",
"Then the canonical isomorphism ${\\operatorname{id}}_{SH(Y)}\\simeq p_!s_*$ is equal to the composition ${\\operatorname{id}}_{{\\operatorname{SH}}(Y)}\\simeq \\Sigma ^{-V}\\Sigma ^V=\\Sigma ^{-V}p_\\#s_*\\simeq p_\\#\\Sigma ^{-p^*V}s_*\\simeq p_!s_*$ Indeed $\\Sigma ^{-V}\\Sigma ^V\\rightarrow {\\operatorname{id}}_{{\\operatorname{SH}}(Y)}$ is the counit of the adjunction $\\Sigma ^{-V}\\dashv \\Sigma ^V$ , corresponding to ${\\operatorname{id}}:\\Sigma ^V\\rightarrow \\Sigma ^V$ , $\\Sigma ^{-V}p_\\#s_*\\rightarrow {\\operatorname{id}}_{{\\operatorname{SH}}(Y)}$ is the counit of the adjunction $\\Sigma ^{-V}=s^!p^*\\dashv p_\\#s_*$ .",
"The functors $\\Sigma ^{-V}$ and $p_\\#$ are both left adjoints, so $\\Sigma ^{-V}p_\\#$ is left adjoint to $s_*$ and the counit of the adjunction $\\Sigma ^{-V}p_\\#\\dashv p^*\\Sigma ^V\\simeq \\Sigma ^{p^*V}p^*\\simeq p^!\\simeq s_*$ is the same as that of $\\Sigma ^{-V}\\dashv p_\\#s_*$ .",
"Composing with the isomorphism $p_!\\simeq p_\\#\\Sigma ^{-p^*V}\\simeq \\Sigma ^{-V}p_\\#$ , we see that the counit of the adjunction $p_!\\dashv s_*$ is induced from that of $\\Sigma ^{-V}p_\\#\\dashv s_*$ , and thus agrees with the canonical isomorphism $p_!s_*\\simeq {\\operatorname{id}}_{{\\operatorname{SH}}(Y)}$ .",
"Lemma 4.3 Let $({\\mathcal {E}}, {\\operatorname{th}}_{(-)})$ be an $\\operatorname{SL}$ -oriented motivic spectrum in ${\\operatorname{SH}}(B)$ .",
"Suppose given $X\\in {\\mathrm {Sm}}_B$ of dimension $d_X$ over $B$ , $i:Z\\rightarrow X$ a closed subset, and $p:L\\rightarrow X$ a line bundle.",
"Then the isomorphism $\\vartheta _{L-T_{X/B}}^{\\mathcal {E}}:\\Sigma ^{1-\\operatorname{det}(L-T_{X/B})}\\pi _X^*{\\mathcal {E}}\\rightarrow \\Sigma ^{r_{L-T_{X/B}}-L+T_{X/B}}\\pi _X^*{\\mathcal {E}}$ induces an isomorphism $\\rho _{X, Z, L}:{\\mathcal {E}}^{a,b}_Z(X; \\omega _{X/B}\\otimes L)\\simeq {\\mathcal {E}}^{a-2d_X,b-d_X}(X_Z/B_\\mathrm {B.M.",
"}; L),$ where the right-hand side is as defined in Remark REF .",
"We have $\\operatorname{det}(L-T_{X/B})=\\operatorname{det}^{-1}(T_{X/B})\\otimes L=\\omega _{X/B}\\otimes L$ and $r_{L-T_{X/B}}=1-d_X$ .",
"Moreover, we have canonical isomorphisms ${\\mathcal {E}}^{a,b}_Z(X; \\omega _{X/B}\\otimes L)\\simeq {\\rm Hom}_{{\\operatorname{SH}}(X)}(i_*(1_Z), S^{a,b}\\wedge \\Sigma ^{1-\\omega _{X/B}\\otimes L}\\pi _X^*{\\mathcal {E}})$ and ${\\mathcal {E}}^{a-2d_X,b-d_X}(X_Z/B_\\mathrm {B.M.",
"}; L)\\simeq {\\rm Hom}_{{\\operatorname{SH}}(X)}(i_*(1_Z), S^{a,b}\\wedge \\Sigma ^{(1-d_X)+T_{X/B}-L}\\pi _X^*{\\mathcal {E}}).$ Finally, $\\vartheta _{L-T_{X/B}}^{\\mathcal {E}}$ induces the isomorphism $S^{a,b}\\wedge \\vartheta _{L-T_{X/B}}^{\\mathcal {E}}:S^{a,b}\\wedge \\Sigma ^{1-\\omega _{X/B}\\otimes L}\\pi _X^*{\\mathcal {E}}\\rightarrow S^{a,b}\\wedge \\Sigma ^{(1-d_X)+T_{X/B}-L}\\pi _X^*{\\mathcal {E}}$ which completes the proof.",
"Using the isomorphisms $\\rho _{X, Z,L}$ , we make the following definition.",
"Definition 4.4 Let $({\\mathcal {E}}, {\\operatorname{th}}_{(-)})$ be an $\\operatorname{SL}$ -oriented motivic spectrum in ${\\operatorname{SH}}(B)$ , let $f:Y\\rightarrow X$ be a proper morphism of relative dimension $d$ in ${\\mathrm {Sm}}_B$ , let $L\\rightarrow X$ be a line bundle, and let $Z\\subset X$ be a closed subset.",
"Define $f_*:{\\mathcal {E}}^{a,b}_{f^{-1}(Z)}(Y, \\omega _{Y/B}\\otimes f^*L)\\rightarrow {\\mathcal {E}}^{a-2d,b-d}_Z(X, \\omega _{X/B}\\otimes L)$ to be the unique map making the diagram ${{\\mathcal {E}}^{a-2d_Y,b-d_Y}(Y_{f^{-1}(Z)}/B_\\mathrm {B.M.",
"}; f^*L)[r]^-{(f^*)^*}&{\\mathcal {E}}^{a-2d_Y,b-d_Y}(X_Z/B_\\mathrm {B.M.",
"}; L)\\\\{\\mathcal {E}}^{a,b}_{f^{-1}(Z)}(Y, \\omega _{Y/B}\\otimes f^*L)[u]^{\\rho _{Y, f^{-1}Z, f^*L}}[r]_{f_*}&{\\mathcal {E}}^{a-2d,b-d}_Z(X, \\omega _{X/B}\\otimes L)[u]_{\\rho _{X, Z, L}}}$ commute.",
"Let $p:V\\rightarrow Y$ be a rank $r$ vector bundle on some $Y\\in {\\mathrm {Sm}}_B$ , with 0-section $s:Y\\rightarrow V$ .",
"Letting $L=\\operatorname{det}V$ , the exact sequence $0\\rightarrow p^*V\\rightarrow T_{V/B}\\xrightarrow{}p^*T_{Y/B}\\rightarrow 0$ gives the canonical isomorphism $\\omega _{V/B}\\simeq p^*(\\operatorname{det}^{-1} V\\otimes \\omega _{Y/B})$ , or $p^*\\operatorname{det}^{-1} V\\simeq \\omega _{V/B}\\otimes \\omega ^{-1}_{Y/B}$ .",
"Letting $({\\mathcal {E}}, {\\operatorname{th}})$ be an $\\operatorname{SL}$ -oriented motivic spectrum, we have the pushforward map $s_*:{\\mathcal {E}}^{a,b}(Y)\\rightarrow {\\mathcal {E}}^{ a+2r, b+r}(V, p^*\\operatorname{det}^{-1} V),$ and the version with supports, $s_*:{\\mathcal {E}}^{a,b}(Y)={\\mathcal {E}}^{a,b}_Y(Y)\\rightarrow {\\mathcal {E}}^{ a+2r, b+r}_{0_V}(V, p^*\\operatorname{det}^{-1} V).$ Lemma 4.5 Let $1_Y^{\\mathcal {E}}\\in {\\mathcal {E}}^{0,0}(Y)$ be the unit $\\pi _{Y/B}^*(u)$ .",
"Then $s_*(1_Y^{\\mathcal {E}})={\\operatorname{th}}_V \\in {\\mathcal {E}}^{2r, r}_{0_V}(V, p^*\\operatorname{det}^{-1} V).$ As consequence, $s_*(1_Y^{\\mathcal {E}})$ in ${\\mathcal {E}}^{2r, r}(V, p^*\\operatorname{det}^{-1} V)$ is the image of ${\\operatorname{th}}_V$ under the “forget supports” map ${\\mathcal {E}}^{2r, r}_{0_V}(V, p^*\\operatorname{det}^{-1} V)\\rightarrow {\\mathcal {E}}^{2r, r}(V, p^*\\operatorname{det}^{-1} V).$ The exact sequence $0\\rightarrow p^*V\\rightarrow T_{V/B}\\xrightarrow{}p^*T_{Y/B}\\rightarrow 0$ gives us the isomorphism $\\omega _{V/B}^{-1}\\otimes \\operatorname{det}^{-1} V \\simeq p^*\\omega _{Y/B}^{-1}.$ Keeping this in mind, we have the following commutative diagram defining $s_*$ : ${{\\mathcal {E}}^{-2d_Y,-d_Y}(Y/B_\\mathrm {B.M.",
"}; \\omega _{Y/B}^{-1})[r]^-{(s^*)^*}&{\\mathcal {E}}^{-2d_Y,-d_Y}_{0_V}(V/B_\\mathrm {B.M.",
"}, p^*\\omega _{Y/B}^{-1})\\\\{\\mathcal {E}}^{0,0}_Y(Y)[u]^{\\rho _{Y,Y, s^*p^*\\omega _{Y/B}^{-1}}}[r]_{s_*}[rd]_{s_*}&{\\mathcal {E}}^{2r,r}_{0_V}(V, \\operatorname{det}^{-1} V)[u]_{\\rho _{V, 0_V, p^*\\omega _{Y/B}^{-1}}}[d]^{\\vbox {\\tiny forget\\\\supports}}\\\\&{\\mathcal {E}}^{2r,r}(V,\\operatorname{det}^{-1} V )}$ Here the lower $s_*$ is the one we are considering and the upper $s_*$ is the map with supports.",
"Thus, we need to show that the upper $s_*$ satisfies $s_*(1_Y^{\\mathcal {E}})={\\operatorname{th}}_V$ .",
"We will be using the isomorphisms $\\\\&{\\mathcal {E}}^{-2d_Y,-d_Y}(Y/B_\\mathrm {B.M.",
"}; \\omega _{Y/B}^{-1})\\simeq {\\rm Hom}_{{\\operatorname{SH}}(Y)}(1_Y, \\Sigma ^{1-d_Y+T_{Y/B}-\\omega _{Y/B}^{-1}}\\pi _Y^*{\\mathcal {E}}),\\\\&{\\mathcal {E}}^{-2d_Y,-d_Y}_{0_V}(V/B_\\mathrm {B.M.",
"}, p^*\\omega _{Y/B}^{-1})\\simeq {\\rm Hom}_{{\\operatorname{SH}}(V)}(s_*(1_Y), \\Sigma ^{1-d_Y+T_{V/B}- p^*\\omega _{Y/B}^{-1}}\\pi _V^*{\\mathcal {E}}),\\\\&{\\mathcal {E}}^{2r,r}_{0_V}(V,\\operatorname{det}^{-1} V )\\simeq {\\rm Hom}_{{\\operatorname{SH}}(V)}(s_*(1_V), \\Sigma ^{r+1-\\operatorname{det}^{-1}V}\\pi _V^*{\\mathcal {E}}).$ By Lemma REF , the composition $\\pi _{Y!",
"}\\xrightarrow{}\\pi _{V!",
"}\\circ s_!\\xrightarrow{}\\pi _{V!",
"}\\circ s_*\\xrightarrow{}\\pi _{Y!",
"}$ is the identity.",
"Evaluating at $1_Y$ gives the commutative diagram ${Y_\\mathrm {B.M.",
"}[r]^-{\\phi }_--\\sim @{=}[d]&\\pi _{V!",
"}(s_*(1_Y))[dl]^{s^*}@{=}[r]&V_{0_V}/B_{\\mathrm {B.M.",
"}}\\\\Y_\\mathrm {B.M.",
"}}$ and the isomorphisms $\\phi $ induces the isomorphism $\\phi ^*:{\\mathcal {E}}^{-2d_Y, -d_Y}_{0_V}(V/B_\\mathrm {B.M.",
"}, p^*\\omega _{Y/B}^{-1})\\rightarrow {\\mathcal {E}}^{-2d_Y, -d_Y}(Y_\\mathrm {B.M.",
"}, \\omega _{Y/B}^{-1})$ The isomorphism $\\rho _{V, 0_V, p^*\\omega _{Y/B}^{-1}}$ is the map induced on ${\\rm Hom}_{{\\operatorname{SH}}(V)}(s_*(1_Y),-)$ by the isomorphism $\\vartheta _{p^*\\omega ^{-1}_{Y/B}- T_{V/B}}:\\Sigma ^{1-\\operatorname{det}^{-1} V}\\pi _V^*{\\mathcal {E}}\\rightarrow \\Sigma ^{1-d_V+T_{V/B}-p^*\\omega ^{-1}_Y/B}\\pi _V^*{\\mathcal {E}}.$ We have the isomorphisms $\\Sigma ^{r-V}\\vartheta _{-V}: \\Sigma ^{r+1-V-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}}\\rightarrow \\pi _Y^*{\\mathcal {E}}$ $\\Sigma ^{r-V}\\vartheta _{\\omega ^{-1}_{Y/B}-T_{Y/B}-V}:\\Sigma ^{r+1-V-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}}\\rightarrow \\Sigma ^{r+1-d_Y+T_{Y/B}-\\omega _{Y/B}}\\pi _Y^*{\\mathcal {E}}$ and $\\vartheta _{\\omega _{Y/B}-T_{T/B}}:\\pi _Y^*{\\mathcal {E}}\\rightarrow \\Sigma ^{1-d_Y+T_{Y/B}-\\omega _{Y/B}}\\pi _Y^*{\\mathcal {E}}.$ The first one induces an isomorphism $\\rho _{-V}:{\\rm Hom}_{{\\operatorname{SH}}(Y)}(1_Y, \\Sigma ^{r+1-V-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}})\\rightarrow {\\mathcal {E}}^{0,0}(Y)\\ $ the second an isomorphism $\\rho _{T_{Y/B}+V-\\omega _{Y/B}^{-1}}:{\\rm Hom}_{{\\operatorname{SH}}(Y)}(1_Y, \\Sigma ^{r+1-V-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}})\\rightarrow {\\mathcal {E}}^{-2d_Y, -d_Y}(Y/B_\\mathrm {B.M.",
"}, \\omega ^{-1}_{Y/B})$ while the third induces the isomorphism $\\rho _{Y,Y, \\omega _{Y/B}^{-1}}:{\\mathcal {E}}^{0,0}(Y)\\rightarrow {\\mathcal {E}}^{-2d_TY, -d_Y}(Y/B_\\mathrm {B.M.",
"}, \\omega ^{-1}_{Y/B})$ Altogether these maps and isomorphisms gives the diagram of isomorphisms ${5pt}{{\\mathcal {E}}^{-2d_Y, -d_Y}_{0_V}(V/B_\\mathrm {B.M.",
"}, p^*\\omega _{Y/B}^{-1})[r]^{\\phi ^*}& {\\mathcal {E}}^{-2d_Y, -d_Y}(Y/B_\\mathrm {B.M.",
"}, \\omega ^{-1}_{Y/B})\\\\&&{\\mathcal {E}}^{0,0}(Y)[ul]_-{\\rho _{Y,Y,\\omega _{Y/B}^{-1}}}\\\\{\\mathcal {E}}_{0_V}^{2r,r}(V, \\operatorname{det}^{-1}V)[uu]^{\\rho _{V, 0_V, \\operatorname{det}^{-1}V}}[r]_-\\psi &{\\rm Hom}_{{\\operatorname{SH}}(Y)}(1_Y, \\Sigma ^{r+1-V-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}})[uu]^{\\rho _{T_{Y/B}+V-\\omega _{Y/B}}}[ur]_-{\\rho _{-V}}}$ The triangle commutes by the functoriality of $\\vartheta _{-}$ , as expressed by Remark REF .",
"To see that square commutes, we have the diagram ${10pt}{\\Sigma ^{1-d+T_V-p^*\\omega _{Y/B}^{-1}}\\pi _V^*{\\mathcal {E}}[r]^-\\sim &\\Sigma ^{1-d+p^*(T_Y+V-\\omega _{Y/B}^{-1})}\\pi _V^*{\\mathcal {E}}[r]^-\\sim &p^*\\Sigma ^{1-d+T_Y+V-\\omega _{Y/B}^{-1}}\\pi _Y^*{\\mathcal {E}}\\\\\\Sigma ^{r+1-p^*\\operatorname{det}^{-1}V}\\pi _V^*{\\mathcal {E}}@{=}[r][u]_{\\Sigma ^r_{\\vartheta _{p^*\\omega _{Y/B}^{-1}-T_{V/B}}}&\\Sigma ^{r+1-p^*\\operatorname{det}^{-1}V}\\pi _V^*{\\mathcal {E}}[r]_\\sim [u]_{\\Sigma ^r_{\\vartheta _{p^*(\\omega _{Y/B}^{-1}-T_{Y/B}-V)}}&p^*\\Sigma ^{r+1-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}}[u]_{p^*\\Sigma ^r_{\\vartheta _{\\omega _{Y/B}^{-1}-T_{Y/B}-V}}}The first square commutes using the exact sequence 0\\rightarrow p^*V\\rightarrow T_{V/B}\\rightarrow p^*T_{Y/B}\\rightarrow 0 and the second by the naturality of \\vartheta _{-}.",
"Applying the adjunction p_\\#\\dashv p^*, the identity p_\\#s^*=\\Sigma ^V and applying \\Sigma ^{-V} to yield the isomorphism [\\Sigma ^Vx, y]_{{\\operatorname{SH}}(Y)}\\simeq [x,\\Sigma ^Vy]_{{\\operatorname{SH}}(Y)}, applying {\\rm Hom}_{{\\operatorname{SH}}(V)}(s_*(1_Y), -) to the last map gives the commutative square{{\\rm Hom}_{{\\operatorname{SH}}(V)}(s_*(1_Y), p^*\\Sigma ^{1-d+T_Y+V-\\omega _{Y/B}^{-1}}\\pi _Y^*{\\mathcal {E}})[dr]^\\sim \\\\&\\hspace{-50.0pt}{\\rm Hom}_{{\\operatorname{SH}}(Y)}(1_Y, \\Sigma ^{1-d+T_Y-\\omega _{Y/B}^{-1}}\\pi _Y^*{\\mathcal {E}})\\\\{\\rm Hom}_{{\\operatorname{SH}}(V)}(s_*(1_Y), p^*\\Sigma ^{r+1-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}})[uu]^{p^*\\Sigma ^r_{{\\mathbb {P}}^1}\\vartheta _{\\omega _{Y/B}^{-1}-T_{Y/B}-V*}}[rd]_\\sim \\\\&\\hspace{-50.0pt}{\\rm Hom}_{{\\operatorname{SH}}(Y)}(1_Y,\\Sigma ^{r+1-V-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}})[uu]_{\\Sigma ^{r-V}\\vartheta _{\\omega _{Y/B}^{-1}-T_{Y/B}-V*}}}Applying {\\rm Hom}_{{\\operatorname{SH}}(V)}(s_*(1_Y), -) to the first diagram, putting these two diagrams together and using the isomorphisms (\\ref {align:Isos}) and Remark~\\ref {rem:Pushforward} gives the commutativity of the square in (\\ref {eqn:CommDiagr}).",
"}It follows from the commutativity of (\\ref {eqn:Compat2}) that \\phi ^*\\circ (s^*)^*\\circ \\rho _{Y,Y,\\omega _{Y/B}^{-1}}=\\rho _{Y,Y,\\omega _{Y/B}^{-1}}.",
"The commutativity of (\\ref {eqn:Compat1}) and (\\ref {eqn:CommDiagr}) then shows that \\rho _{-V}\\circ \\psi \\circ s_*={\\operatorname{id}}.",
"By Remark~\\ref {rem:CanonicalThom}, and Remark~\\ref {rem:MultThom} the map \\rho _{-V}\\circ \\psi is the inverse of the canonical Thom isomorphism \\vartheta _V:{\\mathcal {E}}^{0,0}(Y)\\rightarrow {\\mathcal {E}}^{2r, r}_{0_V}(V, \\operatorname{det}^{-1}V).",
"Thus s_*=\\vartheta _V so s_*(1^{\\mathcal {E}}_Y)={\\operatorname{th}}_V.",
"}}\\begin{remark}If we have a \\operatorname{GL}-orientation on {\\mathcal {E}}, we have functorial pushforward mapsf_*:{\\mathcal {E}}^{a,b}_W(Y)\\rightarrow {\\mathcal {E}}^{a-2d, b-d}_Z(X)for f:Y\\rightarrow X a projective morphism in {\\mathrm {Sm}}_B, of relative dimension d, with W\\subset Y, Z\\subset X closed subsets with f(W)\\subset Z.",
"All the results of this section hold in the oriented context after deleting the twist by line bundles.",
"This follows from Remark~\\ref {rem:Oriented}.\\end{remark}$ Motivic Gauß-Bonnet Definition 5.1 Let $p:V\\rightarrow X$ be a rank $r$ vector bundle on some $X\\in {\\mathrm {Sm}}_B$ , and let ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ be an $\\operatorname{SL}$ -oriented motivic ring spectrum.",
"The Euler class $e^{\\mathcal {E}}(V)\\in {\\mathcal {E}}^{2r,r}(X,\\operatorname{det}^{-1}V)$ is defined as $e^{\\mathcal {E}}(V):=s^*s_*(1^{\\mathcal {E}}_X);\\quad 1^{\\mathcal {E}}_X\\in {\\mathcal {E}}^{0,0}(X)\\text{ the unit.",
"}$ Remark 5.2 By Lemma REF , $e^{\\mathcal {E}}(V):=s^*s_*(1^{\\mathcal {E}}_X)=\\bar{s}^*{\\operatorname{th}}_V$ , where $\\bar{s}:X\\rightarrow {\\operatorname{Th}}_X(V)$ is the map induced by $s$ .",
"Theorem 5.3 (Motivic Gauß-Bonnet) Let ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ be an $\\operatorname{SL}$ -oriented motivic ring spectrum, $\\pi _X:X\\rightarrow B$ a smooth and projective $B$ -scheme, let $u_{\\mathcal {E}}:1_B\\rightarrow {\\mathcal {E}}$ be the unit map.",
"Then $\\pi _{X/B*}(e^{\\mathcal {E}}(T_{X/B}))=u_{{\\mathcal {E}}*}(\\chi (X/B))\\in {\\mathcal {E}}^{0,0}(B).$ We have the canonical Thom isomorphism $\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}: {\\mathcal {E}}^{a,b}(X;\\omega _{X/B})\\rightarrow {\\mathcal {E}}^{a-2\\operatorname{dim}X, b-\\operatorname{dim}X}({\\operatorname{Th}}(-T_{X/B})).$ By Lemma REF , it suffices to show that the map $\\beta _{X/B}^*:{\\mathcal {E}}^{0,0}(X)\\rightarrow {\\mathcal {E}}^{0,0}({\\operatorname{Th}}(-T_{X/B}))$ sends $1^{\\mathcal {E}}_X$ to $\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}(e^{\\mathcal {E}}(T_{X/B}))$ ; by Remark REF , this is the same as $\\vartheta _{-T_{X/B}}(\\bar{s}^*{\\operatorname{th}}_{T_{X/B}})$ , where $\\bar{s}:X_+\\rightarrow {\\operatorname{Th}}(T_{X/B})$ is the map induced by the zero-section $s:X\\rightarrow T_{X/B}$ .",
"We use our description of $\\beta _{X/B}$ as $\\pi _{X\\#}$ applied to the composition (REF ).",
"Applying ${\\rm Hom}_{{\\operatorname{SH}}(X)}(-,\\pi _X^*{\\mathcal {E}})$ to $\\beta _{X/B}$ and using the adjunction ${\\rm Hom}_{{\\operatorname{SH}}(B)}(\\pi _{X\\#}(-), {\\mathcal {E}})\\simeq {\\rm Hom}_{{\\operatorname{SH}}(X)}(-,\\pi _X^*{\\mathcal {E}})$ , $ \\beta _{X/B}^*$ is given by the composition ${\\mathcal {E}}^{0,0}(X){[r]^a_\\sim &} {\\rm Hom}_{{\\operatorname{SH}}(X)}(1_X, \\pi _X^*{\\mathcal {E}}){[r]^b_\\sim &}{\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{-T_{X/B}}\\circ \\Sigma ^{T_{X/B}}(1_X), \\pi _X^*{\\mathcal {E}})\\\\{[r]^c_\\sim &}{\\rm Hom}_{{\\operatorname{SH}}(X)}({\\operatorname{Th}}_X(T_{X/B})), \\Sigma ^{T_{X/B}}\\pi _X^*{\\mathcal {E}})\\\\\\xrightarrow{}{\\rm Hom}_{{\\operatorname{SH}}(X)}(1_X, \\Sigma ^{T_{X/B}}\\pi _X^*{\\mathcal {E}})\\simeq {\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{-T_{X/B}}({\\operatorname{Th}}_X(T_{X/B})),\\pi _X^*{\\mathcal {E}})$ where the isomorphisms $a,b, c$ are the canonical ones.",
"The functoriality of the canonical Thom isomorphisms gives us the commutative diagram ${{\\mathcal {E}}^{0,0}(X)[r]^-{\\vartheta _{T_{X/B}}}[d]^a_\\wr &{\\mathcal {E}}^{2d_X, d_X}({\\operatorname{Th}}(T_{X/B}), \\omega _{X/B})[dd]^{\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}}\\\\{\\rm Hom}_{{\\operatorname{SH}}(X)}(1_X, \\pi _X^*{\\mathcal {E}})[d]^b_\\wr &\\\\{\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{-T_{X/B}}\\Sigma ^{T_{X/B}}(1_X),\\pi _X^*{\\mathcal {E}})[r]_-c^-\\sim &{\\rm Hom}_{{\\operatorname{SH}}(X)}({\\operatorname{Th}}_X(T_{X/B}),\\Sigma ^{T_{X/B}}\\pi _X^*{\\mathcal {E}})}$ Thus $(c\\circ b\\circ a)(1^{\\mathcal {E}}_X)= \\vartheta ^{\\mathcal {E}}_{-T_{X/B}}({\\operatorname{th}}_{T_{X/B}}).$ Applying $\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}$ as above gives us the commutative diagram ${40pt}{{\\mathcal {E}}^{2d_X, d_X}({\\operatorname{Th}}(T_{X/B}), \\omega _{X/B}) [r]^-{\\bar{s}^*}[d]_{\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}}&{\\mathcal {E}}^{2d_X, d_X}(X, \\omega _{X/B})[d]^{\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}}\\\\{\\rm Hom}_{{\\operatorname{SH}}(X)}({\\operatorname{Th}}_X(T_{X/B}), \\Sigma ^{T_{X/B}}\\pi _X^*{\\mathcal {E}})[r]_-{\\bar{s}^*}[d]_\\wr &{\\rm Hom}_{{\\operatorname{SH}}(X)}(1_X, \\Sigma ^{T_{X/B}}\\pi _X^*{\\mathcal {E}})[d]^\\wr \\\\{\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{-T_{X/B}}{\\operatorname{Th}}_X(T_{X/B}), \\pi _X^*{\\mathcal {E}})[r]_-{\\Sigma ^{-T_{X/B}}(\\bar{s}^*)}&{\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{-T_{X/B}}(1_X), \\pi _X^*{\\mathcal {E}})}$ and thus $\\beta _{X/B}^*(1^{\\mathcal {E}}_X)=\\vartheta _{-T_{X/B}}(\\bar{s}^*({\\operatorname{th}}_{T_{X/B}}))=\\vartheta _{-T_{X/B}}(e^{\\mathcal {E}}(T_{X/B})),$ as desired.",
"$\\operatorname{SL}$ -oriented cohomology theories Our ultimate goal is to apply the Gauß-Bonnet theorem of § when projective pushforwards are defined on a representable cohomology theory in some concrete manner, not necessarily relying on the six-functor formalism.",
"For this, we need a suitable axiomatization for such theories; we use a modification of the axioms of Panin-Smirnov [38], [39].",
"As before, our base-scheme $B$ is a noetherian, separated scheme of finite Krull dimension.",
"Definition 6.1 We let ${\\mathrm {Sm}\\text{-}\\mathrm {L}}_B$ denote the category of triples $(X, Z, L)$ with $X$ in ${\\mathrm {Sm}}_B$ , $Z\\subset X$ a closed subset and $L\\rightarrow X$ a line bundle.",
"A morphism $(f,\\tilde{f}):(X, Z, L)\\rightarrow (Y, W, M)$ is a morphism $f:X\\rightarrow Y$ with $Z\\supset f^{-1}(W)$ , together with an isomorphism of line bundles $\\tilde{f}:L\\rightarrow f^*M$ .",
"We let ${\\mathrm {Sm}\\text{-}\\mathrm {L}}^{\\text{pr}}_B$ denote the category with the same objects as ${\\mathrm {Sm}\\text{-}\\mathrm {L}}_B$ , but with morphisms $(f; \\tilde{f}):(X,Z, L)\\rightarrow (Y, W, M)$ a proper morphism $f:X\\rightarrow Y$ in ${\\mathrm {Sm}}_B$ , with $f(Z)\\subset W$ , and $\\tilde{f}:L\\rightarrow f^*M$ an isomorphism of line bundles.",
"Definition 6.2 An $\\operatorname{SL}$ -oriented cohomology theory on ${\\mathrm {Sm}}_B$ consists of the following data: A functor $H^{*,*}:{\\mathrm {Sm}\\text{-}\\mathrm {L}}_B^{\\text{\\rm op}}\\rightarrow \\mathrm {BiGr}{\\mathrm {Ab}}$ , $(X, Z, L)\\mapsto H^{*,*}_Z(X;L)$ ; we often write $f^*$ for $H^{*,*}(f,\\tilde{f})$ .",
"A functor $H_{*,*}:{\\mathrm {Sm}\\text{-}\\mathrm {L}}^{\\text{pr}}_B\\rightarrow {\\operatorname{\\rm Gr}}{\\mathrm {Ab}}$ , $(X, Z, L)\\mapsto H_{*,*}^Z(X,L)$ ; we often write $f_*$ for $H_{*,*}(f,\\tilde{f})$ .",
"Natural isomorphisms, for $X$ of dimension $d_X$ $H^{2d_X-n, d_X-m}_Z(X,\\omega _{X/B}\\otimes L)\\xrightarrow{} H_{n,m}^Z(X, L).$ An element $1\\in H^{0,0}_B(B;{\\mathcal {O}}_B)$ .",
"For $x:=(X, Z, L), y:=(Y, W, M)$ in ${\\mathrm {Sm}\\text{-}\\mathrm {L}}_B$ , a bigraded cup product map $\\cup _{x,y}:H^{*,*}_Z(X,L)\\otimes H^{*,*}_W(Y,M)\\rightarrow H^{*,*}_{Z\\times W}(X\\times _BY, p_1^*L\\otimes p_2^*M)$ For $Z\\subset W$ closed subsets of an $X\\in {\\mathrm {Sm}}_B$ a bigraded boundary map $\\delta _{X, W, Z}^{*,*}:H^{*, *}_{Z\\setminus W}(X\\setminus W;j_W^*L)\\rightarrow H^{*+1,*}_W(X, L)$ We write $H^{*,*}(X,L)$ for $H^{*,*}_X(X,L)$ and $H^{*,*}_Z(X)$ for $H^{*,*}_Z(X,{\\mathcal {O}}_X)$ ; we use the analogous notation for $H_{*,*}$ .",
"We write $\\cup $ for $\\cup _{x,y}$ and $\\delta $ for $\\delta _{X,Z,L}$ when the context makes the meaning clear.",
"For $f:Y\\rightarrow X$ a proper map of relative dimension $d$ in ${\\mathrm {Sm}}_B$ , with $Z\\subset X$ , $W\\subset Y$ closed subsets with $f(W)\\subset Z$ and $L\\rightarrow X$ a line bundle, combining D2 and D3 gives us pushforward maps $f_*:H^{*, *}_W(Y, \\omega _{Y/B}\\otimes f^*L)\\rightarrow H^{*-2d, *-d}_Z(X, \\omega _{X/B}\\otimes L)$ defined as the composition $H^{*, *}_W(Y, \\omega _{Y/B}\\otimes f^*L)\\xrightarrow{}H_{2d_Y-*, d_Y-*}^W(Y, f^*L)\\\\\\xrightarrow{}H_{2d_Y-*, d_Y-*}^Z(X, L)\\xrightarrow{}H^{*-2d, *-d}_Z(X, \\omega _{X/B}\\otimes L).$ These data are required to satisfy the following axioms: $H^{*,*}$ and $H_{*,*}$ are additive: $H^{*,*}$ transforms disjoint unions to products and $H_{*,*}$ transforms disjoint unions to coproducts; in particular, $H^{*,*}_Z(\\emptyset ,L)=0$ and $H_{*,*}^Z(\\emptyset , L)=0$ .",
"Let ${Y^{\\prime }[d]_{f^{\\prime }}[r]^-{g^{\\prime }}&Y[d]^f\\\\X^{\\prime }[r]_-g&X}$ be a cartesian diagram in $\\mathrm {Sch}_B$ , with $X, Y, X^{\\prime }, Y^{\\prime }$ in ${\\mathrm {Sm}}_B$ (sometimes called a transverse cartesian diagram in ${\\mathrm {Sm}}_B$ ) and with $f, f^{\\prime }$ proper of relative dimension $d$ .",
"This gives us the isomorphism $f^{\\prime *}\\omega _{X^{\\prime }/X}\\simeq \\omega _{Y^{\\prime }/Y}.$ Let $Z\\subset X$ be a closed subset, let $W\\subset Y$ be a closed subset with $f(W)\\subset Z$ , let $Z^{\\prime }=g^{-1}(Z)$ , $W^{\\prime }=g^{\\prime -1}(W)$ .",
"Let $L\\rightarrow X$ be a line bundle on $X$ and let $L^{\\prime }=g^{\\prime *}(L)$ .",
"Then the diagram ${H^{*, *}_{W^{\\prime }}(Y^{\\prime }, \\omega _{Y^{\\prime }/B}\\otimes \\omega _{Y^{\\prime }/Y}^{-1}\\otimes g^{\\prime *} L^{\\prime })[d]^{ f^{\\prime }_*}&[l]_-{g^{\\prime *}}H^{*, *}_W(Y, \\omega _{Y/B}\\otimes f^*L)[d]^{ f_*}\\\\H^{*-2d, *-d}_{Z^{\\prime }}(X^{\\prime }, \\omega _{X^{\\prime }/B}\\otimes \\omega _{X^{\\prime }/X}^{-1}\\otimes L^{\\prime })&[l]^-{g^*}H^{*-2d, *-d}_Z(X, \\omega _{X/B}\\otimes L)}$ commutes.",
"For $Z\\subset W$ closed subsets of an $X\\in {\\mathrm {Sm}}_B$ , let $U=X\\setminus Z$ with inclusion $j:U\\rightarrow X$ .",
"For $L\\rightarrow X$ a line bundle, this gives us the morphisms $({\\operatorname{id}}, {\\operatorname{id}}):(X, W, L)\\rightarrow (X, Z, L)$ and $(j, {\\operatorname{id}}):(U, W\\setminus Z, j^*L)\\rightarrow (X, W, L)$ .",
"Then the sequence $\\ldots \\xrightarrow{}H^{*, *}_Z(X, L)\\rightarrow H^{*,*}_W(X, L)\\\\\\xrightarrow{} H^{*,*}_{W\\setminus Z}(U, j^*L)\\xrightarrow{}H^{*+1, *}_Z(X, L)\\rightarrow \\ldots $ is exact.",
"Moreover, the maps $\\delta _{Z, W, X}$ are natural with respect to the pullback maps $g^*$ and the proper pushforward maps $f_*$ .",
"Let $i:Y\\rightarrow X$ be a closed immersion in ${\\mathrm {Sm}}_B$ , let $W\\subset Y$ be a closed subset, $L\\rightarrow X$ a line bundle.",
"Let $Z=i(W)$ , giving the morphism $(i, {\\operatorname{id}}):(Y, W, i^*L)\\rightarrow (X, Z, L)$ in ${\\mathrm {Sm}\\text{-}\\mathrm {L}}^{\\text{pr}}_B$ .",
"Then $i_*:H_{*,*}^W(Y, i^*L)\\rightarrow H_{*,*}^Z(X, L)$ is an isomorphism.",
"The cup products $\\cup $ of D4 are associative with unit 1.",
"The maps $f^*$ and $f_*$ are compatible with cup products: $(f\\times g)^*(\\alpha \\cup _{x,y}\\beta )=f^*(\\alpha )\\cup _{x,y} g^*(\\beta )$ .",
"Moreover, using the isomorphisms of D3, the cup products induce products $\\cup ^{x,y}$ on $H_{*,*}$ and one has $(f\\times g)_*(\\alpha \\cup ^{x,y}\\beta )=f_*(\\alpha )\\cup ^{x,y} g_*(\\beta )$ .",
"Finally, the boundary maps $\\delta _{Z, W, X}$ are module morphism: retaining the notation of D4, for $\\alpha \\in H^{*, *}_{Z\\setminus W}(X\\setminus W;j_W^*L)$ and $\\beta \\in H^{*, *}_{T}(Y, M)$ , we have $\\delta _{X\\times Y, Z\\times T, W\\times T}(\\alpha \\cup \\beta )=\\delta _{X, Z, W}(\\alpha )\\cup \\beta .$ Let $i:Y\\rightarrow X$ be a closed immersion in ${\\mathrm {Sm}}_B$ of codimension $c$ , $\\pi _Y:Y\\rightarrow B$ the structure map.",
"Let $1^H_Y\\in H^{0,0}(Y)$ be the element $\\pi _Y^*(1)$ .",
"Then $\\vartheta (i):=\\alpha _{X, Y}(i_*(1^H_Y))\\in H^{2c, c}_Y(X, \\operatorname{det}^{-1} N_i)$ is central, that is, for each $(U, T, M)\\in {\\mathrm {Sm}\\text{-}\\mathrm {L}}_B$ , and each $\\beta \\in H^{*,*}_T(U, M)$ , we have $\\tau ^*(\\beta \\cup \\vartheta (i))=\\vartheta (i)\\cup \\beta $ where $\\tau :X\\times _BU\\rightarrow U\\times _BX$ is the symmetry isomorphism.",
"Let $(f,{\\operatorname{id}}):(Y, W, f^*L)\\rightarrow (X, Z, L)$ be a morphism in ${\\mathrm {Sm}\\text{-}\\mathrm {L}}_B$ .",
"Suppose that the induced map $f:Y_W/B\\rightarrow X_Z/B$ is an isomorphism in ${\\operatorname{SH}}(B)$ .",
"Then $f^*:H^{*,*}_Z(X, L)\\rightarrow H^{*,*}_W(Y, f^*L)$ is an isomorphism.",
"Remark 6.3 It may seem strange that the proper pushforward maps respect products in the sense of (A5); one might rather expect a projection formula.",
"However, (A5) asks that the proper pushforward maps respect external products, not cup products, and in fact, having the pushforward and pullback maps respect products as in (A5) implies the projection formula, as one sees by considering the commutative pentagon associated to a proper morphism $f:Y\\rightarrow X$ in ${\\mathrm {Sm}}_B$ of relative dimension $d$ : ${Y[dd]_f@/^40pt/[drr]^{\\gamma _f:=(f\\times {\\operatorname{id}}_X)\\circ \\Delta _Y}[r]_-{\\Delta _Y}&Y\\times _BY[dr]_{f\\times {\\operatorname{id}}_Y}\\\\&&X\\times _BY[dl]^{{\\operatorname{id}}_X\\times f}\\\\X[r]_-{\\Delta _X}&X\\times _BX}$ Note that the square ${Y[r]^-{\\gamma _f}[d]_f&X\\times _BY[d]^{{\\operatorname{id}}_X\\times f}\\\\X[r]_-{\\Delta _X}&X\\times _BX}$ is transverse cartesian.",
"If we have closed subsets $Z\\subset X$ , $W\\subset Y$ with $f(W)\\subset Z$ , and line bundle $L\\rightarrow X$ , the pentagon diagram induces the diagram in cohomology ${15pt}{H^{*,*}_W(Y,\\omega _{Y/B}\\otimes f^*L)Y[dd]_{f_*}&[l]^-{\\Delta _Y^*}H^{*,*}_{W\\times W}(Y\\times _BY, \\omega _{Y/B}\\otimes f^*L)\\\\\\ &&\\hspace{-43.0pt}H^{*,*}_{Z\\times W}(X\\times _BY, \\omega _{Y/B}\\boxtimes L)@/_55pt/[ull]_{\\gamma _f^*}[ul]_{\\ \\ (f\\times {\\operatorname{id}}_Y)^*}[dl]^{\\ \\ ({\\operatorname{id}}_X\\times f)_*}\\\\H^{*-2d, *-d}_Z(X, \\omega _{X/B}\\otimes L)&H^{*-2d, *-d}_{Z\\times Z}(X\\times _BX, \\omega _{X/B}\\otimes L)[l]^-{\\Delta ^*_X}}$ Take $\\alpha \\in H^{a,b}_Z(X, M)$ , $\\beta \\in H^{c,d}_W(Y, \\omega _{Y/B}\\otimes f^*(L\\otimes M^{-1}))$ .",
"By functoriality of $(-)^*$ and (A5) for $(-)^*$ we have $\\gamma _f^*(\\alpha \\cup _{X,Y} \\beta )=f^*(\\alpha )\\cup _Y\\beta $ and by (A2) and (A5) for $(-)_*$ we have $f_*(f^*(\\alpha )\\cup _Y\\beta )=\\Delta _X^*({\\operatorname{id}}_X\\times f)_*(\\alpha \\cup _{X,Y}\\beta )=\\alpha \\cup _X f_*(\\beta ).$ Similarly, in the presence of (A2) and (A5) for $(-)^*$ , functoriality for $(-)^*$ and $(-)_*$ and the projection formula implies (A5) for $(-)_*$ .",
"Definition 6.4 A twisted cohomology theory on ${\\mathrm {Sm}}_B$ is given by the data D1, D4, D5 above, satisfying the parts of the axioms A1, A3-A7 that only involve $H^{*,*}$ .",
"Given an $\\operatorname{SL}$ -oriented cohomology theory $(H^{*,*}, H_{*,*}, \\ldots )$ on ${\\mathrm {Sm}}_B$ , one has the underlying twisted cohomology theory $(H^{*,*}, \\ldots )$ by forgetting the proper pushforward maps.",
"Example 6.5 The primary example of an $\\operatorname{SL}$ -oriented cohomology theory on ${\\mathrm {Sm}}_B$ is the one induced by an $\\operatorname{SL}$ -oriented motivic spectrum ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ : $(X,Z,L)\\mapsto {\\mathcal {E}}^{*,*}_Z(X;L).$ One defines, for $X\\in {\\mathrm {Sm}}_B$ of dimension $d_X$ over $B$ , ${\\mathcal {E}}_{m,n}^Z(X;L):={\\mathcal {E}}^{2d_X-m, d_X-n}_Z(X;\\omega _{X/B}\\otimes L);$ we extend the definition to arbitrary $X\\in {\\mathrm {Sm}}_B$ by taking the sum over the connected components of $X$ and write this also as ${\\mathcal {E}}^{2d_X-m, d_X-n}_Z(X;\\omega _{X/B}\\otimes L)$ by considering $d_X$ as a locally constant functor on $X$ .",
"The pushforward maps for a proper morphism of relative dimension $d$ , $f:Y\\rightarrow X$ , closed subsets $W\\subset Y$ , $Z\\subset X$ with $f(W)\\subset Z$ and line bundle $L\\rightarrow X$ are given by the pushforward $f_*:{\\mathcal {E}}^{2d_Y-m,2d_Y-n}_W(Y;\\omega _{Y/B}\\otimes f^*L)\\rightarrow {\\mathcal {E}}^{2d_X-m,2d_X-n}_Z(X;\\omega _{X/B}\\otimes L).$ Comparison isomorphisms We recall the element $\\eta \\in {\\rm Hom}_{{\\operatorname{SH}}(B)}(1_B, \\mathrm {S}^{-1,-1}\\wedge 1_B)$ induced by the map of $B$ -schemes $\\eta :{\\mathbb {A}}^2\\setminus \\lbrace 0\\rbrace \\rightarrow {\\mathbb {P}}^1$ , $\\eta (a,b)=(a:b)$ .",
"As every ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ is a module for $1_B$ , we have the map $\\times \\eta :{\\mathcal {E}}\\rightarrow \\mathrm {S}^{-1,-1}\\wedge {\\mathcal {E}}$ for each $x\\in {\\operatorname{SH}}(B)$ .",
"We say that $\\eta $ acts invertibly on ${\\mathcal {E}}$ if $\\times \\eta $ is an isomorphism in ${\\operatorname{SH}}(B)$ .",
"We consider the following situation: fix an $\\operatorname{SL}$ -oriented motivic spectrum ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ .",
"This gives us the twisted cohomology theory ${\\mathcal {E}}^{*,*}$ underlying the oriented cohomology defined by ${\\mathcal {E}}$ .",
"Let $({\\mathcal {E}}^{*,*}, \\tilde{{\\mathcal {E}}}_{*,*})$ be an extension of ${\\mathcal {E}}^{*,*}$ to an oriented cohomology theory on ${\\mathrm {Sm}}_B$ , in other words, we define new pushforward maps $\\hat{f}_*:{\\mathcal {E}}^{*,*}_W(Y, \\omega _{Y/B}\\otimes f^*L)\\rightarrow {\\mathcal {E}}^{*-2d,*-d}_Z(X, \\omega _{X/B}\\otimes L)$ The main result of this section is a comparison theorem.",
"Before stating the result we recall the decomposition of ${\\operatorname{SH}}(B)[1/2]$ into plus and minus parts.",
"We have the involution $\\tau :1_B\\rightarrow 1_B$ induced by the symmetry isomorphism $\\tau :{\\mathbb {P}}^1\\wedge {\\mathbb {P}}^1\\rightarrow {\\mathbb {P}}^1\\wedge {\\mathbb {P}}^1$ .",
"In ${\\operatorname{SH}}(B)[1/2]$ , this gives us the idempotents $({\\operatorname{id}}+\\tau )/2$ , $({\\operatorname{id}}-\\tau )/2$ , and so decomposes ${\\operatorname{SH}}(B)[1/2]$ into +1 and -1 “eigenspaces” for $\\tau $ : ${\\operatorname{SH}}(B)[1/2]={\\operatorname{SH}}(B)^+\\times {\\operatorname{SH}}(B)^-$ We decompose ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)[1/2]$ as ${\\mathcal {E}}={\\mathcal {E}}_+\\oplus {\\mathcal {E}}_-$ .",
"Theorem 7.1 Suppose the pushforward maps $f_*, \\hat{f}_*:{\\mathcal {E}}^{*,*}_W(Y, \\omega _{Y/B}\\otimes f^*L)\\rightarrow {\\mathcal {E}}^{*-2d,*-d}_Z(X, \\omega _{X/B}\\otimes L)$ agree for $W$ , $Z$ , $L$ , $X=V$ a vector bundle over $Y$ and $f:Y\\rightarrow V$ the zero-section.",
"Suppose in addition that one of the following conditions holds: the $\\operatorname{SL}$ -orientation of ${\\mathcal {E}}$ extends to a $\\operatorname{GL}$ -orientation; $\\eta $ acts invertibly on ${\\mathcal {E}}$ ; 2 acts invertibly on ${\\mathcal {E}}$ and ${\\mathcal {E}}_+^{-1,0}(U)=0$ for affine $U$ in ${\\mathrm {Sm}}_B$ .",
"Then $f_*=\\hat{f}_*$ for all $X, Y, Z, W, L, f$ for which the pushforward is defined.",
"By the standard argument of deformation to the normal cone, it follows that $f_*=\\hat{f}_*$ for all $f:Y\\rightarrow X$ a closed immersion, $Z, W, L$ .",
"As every proper map in ${\\mathrm {Sm}}_B$ is projective, $f$ admits a factorization $f=p\\circ i$ , with $i:Y\\rightarrow X\\times _B{\\mathbb {P}}^N$ a closed immersion and $p:X\\times _B{\\mathbb {P}}^N\\rightarrow X$ the projection.",
"By functoriality of the pushforward maps, it suffices to check that $p_*=\\hat{p}_*$ .",
"In case (i), this follows from the uniqueness assertion in [38].",
"Indeed, the cohomology theory associated to a $\\operatorname{GL}$ -oriented motivic spectrum ${\\mathcal {E}}$ satisfies the axioms of Panin-Smirnov and the associated Thom isomorphisms give rise to an “orientation” in the sense of [38], so we may apply the results cited.",
"We note that in [38] the base-scheme is ${\\rm Spec\\,}k$ , $k$ a field, so loc.",
"cit.",
"does not immediately apply to our setting of a more general base-scheme; we say a few words about the extension of this result to our base-scheme $B$ .",
"As a proper map $f:Y\\rightarrow X$ in ${\\mathrm {Sm}}_B$ is projective, one factors $f$ as $f=p\\circ i$ , with $i:Y\\rightarrow {\\mathbb {P}}^n_X$ a closed immersion and $p:{\\mathbb {P}}^n_X\\rightarrow X$ the projection.",
"The uniqueness for a closed immersion in ${\\mathrm {Sm}}_B$ reduces to the case of the zero-section of a vector bundle by the usual method of deformation to the normal bundle, and as the pushforward by the zero-section of our two theories are the same by assumption, we have agreement in the case of a closed immersion.",
"For the projection $p$ , the proof of [38] relies on [37], where for $p$ , using the projective bundle formula, the key point is to show that both pushforwards have the same value on the unit $1_{{\\mathbb {P}}^n_X}\\in {\\mathcal {E}}^{0,0}({\\mathbb {P}}^n_X)$ .",
"The proof of this relies on the formula for the pushforward of $1_{{\\mathbb {P}}^n_X}$ under the diagonal $\\Delta _{{\\mathbb {P}}^n_X}:{\\mathbb {P}}^n_X\\rightarrow {\\mathbb {P}}^n_X\\times _X{\\mathbb {P}}^n_X$ given by [37].",
"As $\\Delta _{{\\mathbb {P}}^n_X}$ is a closed immersion, the two pushforwards under $\\Delta _{{\\mathbb {P}}^n_X}$ agree, and the proof of the formula in [37] uses only formal properties of pushforward and pullback as expressed in our axioms, plus the projective bundle formula.",
"This latter in turn relies only on properties of the Thom class of ${\\mathcal {O}}(-1)$ and localization with respect to ${\\mathbb {A}}^m_X\\subset {\\mathbb {P}}^m_X$ , and thus we may use [37] in our more general setting.",
"The argument that the pushforward of $1_{{\\mathbb {P}}^n_X}$ under $p$ can be recovered from the formula for the pushforward of $1_{{\\mathbb {P}}^n_X}$ under $\\Delta _{{\\mathbb {P}}^n_X}$ is elementary and formal, and only uses the restriction of the two theories to ${\\mathrm {Sm}\\text{-}\\mathrm {L}}_X$ , and not the fact that these restrictions come from theories over $k$ .",
"Thus, the argument used in the proof of [37] may be used to prove our result in case (i).",
"In case (ii), we use Lemma REF below.",
"Indeed, if $N$ is odd, we may apply the closed immersion $X\\times _B{\\mathbb {P}}^N\\rightarrow X\\times _B{\\mathbb {P}}^{N+1}$ as a hyperplane, so we reduce to the case $N$ even, in which case both $p_*$ and $\\hat{p}_*$ are inverse to the map $i_*$ , where $i:X\\rightarrow X\\times _B{\\mathbb {P}}^N$ is the section associated to the point $(1:0:\\ldots :0)$ of ${\\mathbb {P}}^N$ .",
"In case (iii) we may work in the category ${\\operatorname{SH}}(B)[1/2]$ .",
"We decompose ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)[1/2]$ as ${\\mathcal {E}}={\\mathcal {E}}_+\\oplus {\\mathcal {E}}_-$ and similarly decompose the pushforward maps $f_*$ and $\\hat{f}_*$ .",
"By Lemma REF , $\\eta $ acts invertibly on ${\\operatorname{SH}}(B)^-$ and the projection of $\\eta $ to ${\\operatorname{SH}}(B)^+$ is zero.",
"By Lemma REF below, the $\\operatorname{SL}$ -orientation of ${\\mathcal {E}}$ induces an $\\operatorname{SL}$ -orientation on the projection ${\\mathcal {E}}^+$ that extends to a $\\operatorname{GL}$ -orientation.",
"By (i), this implies that $f_*^+=\\hat{f}_*^+$ .",
"By (ii), $f_*^-=\\hat{f}_*^-$ , so $f_*=\\hat{f}_*$ .",
"Lemma 7.2 ([2]) Let ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ be an $\\operatorname{SL}$ -oriented motivic spectrum on which $\\eta $ acts invertibly.",
"Let $0\\in {\\mathbb {P}}^N({\\mathbb {Z}})$ be the point $(1:0\\ldots :0)$ .",
"For $X\\in {\\mathrm {Sm}}_B$ , $L\\rightarrow X$ a line bundle and $Z\\subset X$ a closed subset, the pushforward map $i_*:{\\mathcal {E}}^{*-2N,*-N}_Z(X, \\omega _{X/B}\\otimes L)\\rightarrow {\\mathcal {E}}^{*,*}_{p^{-1}(Z)}(X\\times _B{\\mathbb {P}}^N, \\omega _{{\\mathbb {P}}^N/B}\\otimes p^*L)$ is an isomorphism.",
"Using a Mayer-Vietoris sequence, we see that the statement is local on $X$ for the Zariski topology, so we may assume that $L={\\mathcal {O}}_X$ .",
"If we prove the statement for the pair $(X,X)$ and $(X\\setminus Z, X\\setminus Z)$ the local cohomology sequence gives the result for $(X,Z)$ , thus we may assume that $Z=X$ , and we reduce to showing that $i_*:{\\mathcal {E}}^{*-2N,*-N}(X, \\omega _{X/B})\\rightarrow {\\mathcal {E}}^{*,*}(X\\times _B{\\mathbb {P}}^N, \\omega _{{\\mathbb {P}}^N/B})$ is an isomorphism.",
"This is [2] in case $B={\\rm Spec\\,}k$ , $k$ a field.",
"The proof over a general base-scheme is essentially the same, we say a few words about this generalization.",
"Most of the results that are used in the proof of loc.",
"cit.",
"are are already proved in the required generality here, for example, the Thom isomorphism (REF ) of Construction REF generalizes Ananyevskiy's construction [2] from $B={\\rm Spec\\,}k$ to general $B$ .",
"The proof of [2] relies also on [2], which in our setting reduces to the fact that for $X\\in \\mathrm {Sch}_B$ , and $u\\in \\Gamma (X,{\\mathcal {O}}_X^\\times )$ a unit, the automorphism of $X\\times {\\mathbb {P}}^1$ sending $(x,(t_0: t_1))$ to $(x, (ut_0, u^{-1}t_1)$ induces the identity on $X_+\\wedge {\\mathbb {P}}^1/X$ in ${\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)$ .",
"This follows by identifying ${\\mathbb {P}}^1_X$ with ${\\mathbb {P}}({\\mathbb {A}}^2_X)$ and noting that the diagonal matrix with entries $u, u^{-1}$ is an elementary matrix in $\\operatorname{GL}_2(\\Gamma (X,{\\mathcal {O}}_X))$ .",
"Lemma 7.3 Suppose that ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ is $\\operatorname{SL}$ oriented and that ${\\mathcal {E}}^{-1,0}(U)=0$ for all affine $U$ in ${\\mathrm {Sm}}_B$ .",
"Then the induced $\\operatorname{SL}$ orientation on ${\\mathcal {E}}_+\\in {\\operatorname{SH}}(B)_+$ extends to a $\\operatorname{GL}$ orientation.",
"Let $u\\in \\Gamma (X, {\\mathcal {O}}_X^\\times )$ be a unit on some $X\\in {\\mathrm {Sm}}_B$ .",
"Then the map $\\times u:X\\times _B{\\mathbb {P}}^1\\rightarrow X\\times _B{\\mathbb {P}}^1;\\quad (x, [t_0:t_1])\\mapsto (x, [ut_0:t_1])$ induces the identity on $\\mathrm {S}^{2,1}\\wedge X/B$ in ${\\operatorname{SH}}(B)_+$ .",
"Indeed, let $[u]:X/B\\rightarrow X/B\\wedge {\\mathbb {G}}_m$ be the map induced by $u:X\\rightarrow {\\mathbb {G}}_m$ .",
"The argument given by Morel [32], that $\\times u/B={\\operatorname{id}}+\\eta [u]$ in case $B={\\rm Spec\\,}k$ , $k$ a field, is perfectly valid over a general base-scheme: this only uses the fact that for ${\\mathcal {X}}$ and ${\\mathcal {Y}}$ pointed spaces over $B$ , one has $\\Sigma ^\\infty _{\\mathrm {S}^1}{\\mathcal {X}}\\times _B{\\mathcal {Y}}\\simeq \\Sigma ^\\infty _{\\mathrm {S}^1}{\\mathcal {X}}\\oplus \\Sigma ^\\infty _{\\mathrm {S}^1}{\\mathcal {Y}}\\oplus \\Sigma ^\\infty _{\\mathrm {S}^1}({\\mathcal {X}}\\wedge {\\mathcal {Y}})$ and that the map $\\times _u:\\mathrm {S}^1\\wedge {\\mathbb {G}}_m\\wedge X_+\\rightarrow \\mathrm {S}^1\\wedge {\\mathbb {G}}_m\\wedge X_+$ is the $\\mathrm {S}^1$ -suspension of the composition $\\mathrm {S}^1\\wedge {\\mathbb {G}}_m\\wedge X_+\\xrightarrow{}\\mathrm {S}^1\\wedge ({\\mathbb {G}}_m\\times {\\mathbb {G}}_m)\\wedge X_+\\xrightarrow{}\\mathrm {S}^1\\wedge {\\mathbb {G}}_m\\wedge X_+$ where $\\mu :{\\mathbb {G}}_m\\times {\\mathbb {G}}_m\\rightarrow {\\mathbb {G}}_m$ is the multiplication.",
"As $\\eta $ goes to zero in ${\\operatorname{SH}}(B)_+$ , it follows that $\\times u/B={\\operatorname{id}}$ in ${\\operatorname{SH}}(B)_+$ .",
"Now take $g\\in \\Gamma (X, \\operatorname{GL}_n({\\mathcal {O}}_X))$ , let $u=\\operatorname{det}g$ , let $m_u\\in \\Gamma (X, \\operatorname{GL}_n({\\mathcal {O}}_X))$ be the diagonal matrix with entries $u, 1,\\ldots , 1$ and let $h=m_u^{-1}\\cdot g\\in \\Gamma (X,\\operatorname{SL}_n({\\mathcal {O}}_X))$ .",
"We have ${\\operatorname{Th}}_X({\\mathcal {O}}_X^n)=({\\mathbb {P}}^1)^{\\wedge n}\\wedge X_+.$ Since ${\\mathcal {E}}$ is $\\operatorname{SL}$ -oriented, the map ${\\operatorname{Th}}(h):{\\operatorname{Th}}_X({\\mathcal {O}}_X^n)\\rightarrow {\\operatorname{Th}}_X({\\mathcal {O}}_X^n)$ induces the identity on ${\\mathcal {E}}^{**}$ and thus ${\\operatorname{Th}}(g)^*={\\operatorname{Th}}(m_u)^*:{\\mathcal {E}}^{*,*}_+({\\operatorname{Th}}_X({\\mathcal {O}}_X^n))\\rightarrow {\\mathcal {E}}^{*,*}_+({\\operatorname{Th}}_X({\\mathcal {O}}_X^n))$ But as ${\\operatorname{Th}}(m_u)=(\\times u)\\wedge {\\operatorname{id}}$ , our previous computation shows that ${\\operatorname{Th}}(m_u)^*={\\operatorname{id}}$ .",
"Now let $V\\rightarrow X$ be a rank $r$ vector bundle on some $X\\in {\\mathrm {Sm}}_B$ , choose a trivializing affine open cover ${\\mathcal {U}}=\\lbrace U_i\\rbrace $ of $X$ and let $\\phi _i:V_{|U_i}\\rightarrow U_i\\times {\\mathbb {A}}^r$ be a local framing.",
"We have the suspension isomorphism ${\\operatorname{Th}}(V_{U_i})\\simeq {\\operatorname{Th}}(U_i\\times {\\mathbb {A}}^r)=\\Sigma _r U_{i+}$ giving the isomorphism $\\theta _i:{\\mathcal {E}}_+^{a,b}(U_i)\\rightarrow {\\mathcal {E}}^{2r+a, r+b}_{+0_{V_{|U_i}}}(V_{|U_i}).$ Since $\\operatorname{GL}_r({\\mathcal {O}}_{U_i})$ acts trivially on ${\\mathcal {E}}_+^{**}({\\operatorname{Th}}(U_i\\times {\\mathbb {A}}^r))$ , the isomorphism $\\theta _i$ is independent of the choice of framing $\\phi _i$ .",
"In addition, the assumption ${\\mathcal {E}}^{-1,0}(U_i\\cap U_j)=0$ implies ${\\mathcal {E}}^{2r-1, r}_{+0_{V_{|U_i\\cap U_j}}}(V_{|U_i\\cap U_j})=0$ for all $i,j$ .",
"By Mayer-Vietoris, the sections $\\theta _i(1_{U_i})\\in {\\mathcal {E}}^{2r, r}_{+0_{V_{|U_i}}}(V_{|U_i})$ uniquely extend to an element $\\theta _V\\in {\\mathcal {E}}^{2r, r}_{+0_V}(V)$ The independence of the $\\theta _i$ on the choice of framing and the uniqueness of the extension readily implies the functoriality of $\\theta _V$ and similarly implies the product formula $\\theta _{V\\oplus W}=p_1^*\\theta _V\\cup p_2^*\\theta _W$ .",
"By construction, $\\theta _V$ is the suspension of the unit over $U_i$ , another application of independence of the choice of framing and the uniqueness of the extension shows that this is the case over every open subset $U\\subset X$ for which $V_{|U}$ is the trivial bundle.",
"Finally, the independence and uniqueness shows that $V\\mapsto \\theta _V$ is an extension of the $\\operatorname{SL}$ orientation on ${\\mathcal {E}}_+$ induced by that of ${\\mathcal {E}}$ .",
"Lemma 7.4 For $u\\in \\Gamma (X,{\\mathcal {O}}_X^\\times )$ we have $[u]\\eta =\\eta [u]:\\Sigma ^\\infty _X̰_+\\rightarrow \\Sigma ^\\infty _X̰_+$ We use the decomposition $\\Sigma ^\\infty _X̰_+\\wedge {\\mathbb {G}}_m\\times {\\mathbb {G}}_m=\\Sigma ^\\infty _X̰_+\\wedge {\\mathbb {G}}_m\\oplus \\Sigma ^\\infty _X̰_+\\wedge {\\mathbb {G}}_m\\oplus \\Sigma ^\\infty _X̰_+\\wedge {\\mathbb {G}}_m\\wedge {\\mathbb {G}}_m$ Via this, $\\eta $ is the map $[s]\\wedge [t]\\mapsto [st]-[s]-[t]$ so $\\eta [u]$ sends $[t]$ to $[ut]-[u]-[t]$ and ${\\operatorname{id}}_{{\\mathbb {G}}_m}\\wedge \\eta [u]$ sends $[s]\\wedge [t]$ to $[s]\\wedge [ut]-[s]\\wedge [u]-[s]\\wedge [t]$ , so $[u]\\eta $ is given by $[s]\\wedge [t]\\mapsto [st]-[s]-[t]\\mapsto [u]\\wedge [st]-[u]\\wedge [s]-[u]\\wedge [t].$ We have the automorphism $\\xi $ of ${\\mathbb {G}}_m^{\\wedge 3}$ sending $[u]\\wedge [s]\\wedge [t]$ to $[s]\\wedge [t]\\wedge [u]$ .",
"We have the isomorphism in ${\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(B)$ , $\\Sigma _{S_1}^6{\\mathbb {G}}_m^{\\wedge 3}\\simeq {\\mathbb {A}}^3/{\\mathbb {A}}^3\\setminus \\lbrace 0\\rbrace $ , via which $\\Sigma _{S_1}^6\\xi $ is induced by the linear map $(u,s,t)\\mapsto (s, t, u)$ .",
"As this latter linear map has matrix in the standard basis a product of elementary matrices, $\\Sigma _{\\mathrm {S}^1}^6\\xi $ is ${\\mathbb {A}}^1$ -homotopic to the identity, so after stabilizing, ${\\operatorname{id}}_{{\\mathbb {G}}_m}\\wedge \\eta [u]$ is the map $[s]\\wedge [t]\\mapsto [s]\\wedge [t]\\wedge [u]\\mapsto [u]\\wedge [s]\\wedge [t]\\mapsto [u]\\wedge [st]-[u]\\wedge [s]-[u]\\wedge [t]=[u]\\eta ([s]\\wedge [t]).$ Lemma 7.5 The projection $\\eta _-$ of $\\eta $ to ${\\operatorname{SH}}(B)_-$ is an isomorphism and the projection $\\eta _+$ of $\\eta $ to ${\\operatorname{SH}}(B)_+$ is zero.",
"Morel proves this in [32] in the case of a field, but the proof works in general.",
"In some detail, the map $\\tau $ is the map on ${\\mathbb {A}}^2/({\\mathbb {A}}^2\\setminus \\lbrace 0\\rbrace )$ induced by the linear map $(x, y)\\mapsto (y,x)$ .",
"The matrix identity $\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}\\cdot \\begin{pmatrix}1&0\\\\-1&1\\end{pmatrix}\\cdot \\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}\\cdot \\begin{pmatrix}-1&0\\\\0&1\\end{pmatrix}$ shows that the maps $(x, y)\\rightarrow (y,x)$ and $(x,y)\\mapsto (-x, y)$ are ${\\mathbb {A}}^1$ -homotopic.",
"By the arguments in Lemma REF , this latter map induces the map $1+\\eta [-1]=1+[-1]\\eta $ in ${\\operatorname{SH}}(B)$ , giving the identity $(1+\\eta [-1])_-=(1+[-1]\\eta )_-=-{\\operatorname{id}}\\Rightarrow \\eta \\cdot (-[-1]/2)= (-[-1]/2)\\cdot \\eta ={\\operatorname{id}}_{{\\operatorname{SH}}(B)_-}$ For $\\eta _+$ , the projector to ${\\operatorname{SH}}(B)_+$ is given by the idempotent $(1/2)(\\tau +1)=(1/2)(2+\\eta [-1])$ , so $\\eta _+=(1/2)\\eta \\cdot (2+\\eta [-1])$ .",
"Since the map $\\tau :{\\mathbb {P}}^1\\wedge {\\mathbb {P}}^1\\rightarrow {\\mathbb {P}}^1\\wedge {\\mathbb {P}}^1$ is $1+\\eta [-1]$ and ${\\mathbb {P}}^1=\\mathrm {S}^1\\wedge {\\mathbb {G}}_m$ , the symmetry $\\epsilon :{\\mathbb {G}}_m\\wedge {\\mathbb {G}}_m\\rightarrow {\\mathbb {G}}_m\\wedge {\\mathbb {G}}_m$ is $-(1+\\eta [-1])$ .",
"From our formula for $\\eta ([s]\\wedge [t])$ we see that $\\eta \\epsilon =\\eta $ which gives $\\eta \\cdot (2+\\eta [-1])=0$ .",
"Applications In this section, we apply the motivic Gauß-Bonnet formula of § and the comparison results of § to various specific $\\operatorname{SL}$ -oriented cohomology theories, and thereby make computations of the motivic Euler characteristic $\\chi (X/B)$ in different contexts.",
"Motivic cohomology and cohomology of the Milnor K-theory sheaves We work over the base-scheme $B={\\rm Spec\\,}k$ , with $k$ a perfect field.",
"In ${\\operatorname{SH}}(k)$ we have the motivic cohomology spectrum ${\\operatorname{H}}{\\mathbb {Z}}$ representing Voevodsky's motivic cohomology (see e.g.",
"[28] for a construction valid in arbitrary characteristic).",
"By [49], there is a natural isomorphism ${\\operatorname{H}}{\\mathbb {Z}}^{a,b}(X)\\simeq {\\rm CH}^b(X, 2b-a)$ for $X\\in {\\mathrm {Sm}}_B$ , where ${\\rm CH}^b(X, 2b-a)$ is Bloch's higher Chow group [9].",
"${\\operatorname{H}}{\\mathbb {Z}}$ admits a localization sequence: for $i:Z\\rightarrow X$ a closed immersion of codimension $d$ in ${\\mathrm {Sm}}_k$ , there is a canonical isomorphism ${\\operatorname{H}}{\\mathbb {Z}}^{a,b}_Z(X)\\simeq {\\operatorname{H}}{\\mathbb {Z}}^{a-2d, b-d}(Z)$ See for example [10].",
"In particular, for $p:V\\rightarrow X$ a rank $r$ vector bundle over $X\\in {\\mathrm {Sm}}_k$ , we have the isomorphism ${\\operatorname{H}}{\\mathbb {Z}}^{2r, r}_{0_V}(V)\\simeq {\\operatorname{H}}{\\mathbb {Z}}^{0,0}(X)$ which gives us Thom classes $\\vartheta _V^{{\\operatorname{H}}{\\mathbb {Z}}}\\in {\\operatorname{H}}{\\mathbb {Z}}^{2r, r}_{0_V}(V)$ corresponding to the unit $1^{{\\operatorname{H}}{\\mathbb {Z}}}_X\\in {\\operatorname{H}}{\\mathbb {Z}}^{0,0}(X)$ .",
"Thus ${\\operatorname{H}}{\\mathbb {Z}}$ is a $\\operatorname{GL}$ -oriented motivic spectrum.",
"Let $X$ be a smooth projective $k$ -scheme of dimension $n$ over $k$ .",
"For a class $x\\in {\\operatorname{H}}{\\mathbb {Z}}^{2n, n}(X)$ , the isomorphism ${\\operatorname{H}}{\\mathbb {Z}}^{2n,n}(X)\\simeq {\\rm CH}^n(X,0)={\\rm CH}^n(X)$ allows one to represent $x$ as the class of a 0-cycle $\\tilde{x}=\\sum _in_i p_i$ , with the $p_i$ closed points of $X$ .",
"One has the degree $\\deg _k(p_i):=[k(p_i):k]$ and extending by linearity gives the degree $\\deg _k(\\tilde{x})$ , which one shows passes to rational equivalence to define a degree map $\\deg _k: {\\operatorname{H}}{\\mathbb {Z}}^{2n, n}(X)\\rightarrow {\\rm CH}^0({\\rm Spec\\,}k)={\\mathbb {Z}}.$ As a $\\operatorname{GL}$ -oriented theory, ${\\operatorname{H}}{\\mathbb {Z}}$ has Chern classes for vector bundles: $c_r(V)\\in {\\operatorname{H}}{\\mathbb {Z}}^{2r, r}(X)$ for $V\\rightarrow X$ a vector bundle over some $X\\in {\\mathrm {Sm}}_k$ and $r \\ge 0$ .",
"Theorem 8.1 Let $X\\in {\\mathrm {Sm}}_k$ be projective of dimension $d_X$ .",
"Then $u^{{\\operatorname{H}}{\\mathbb {Z}}}(\\chi (X/k))=\\deg _k(c_{d_X}(T_{X/k})).$ One has well-defined pushforward maps on ${\\rm CH}^*(-,*)$ for projective morphisms (see e.g.",
"[9]).",
"Via the isomorphism $ {\\operatorname{H}}{\\mathbb {Z}}^{a,b}(X)\\simeq {\\rm CH}^b(X, 2b-a)$ [49], this gives pushforward maps $\\hat{f}_*$ on ${\\operatorname{H}}{\\mathbb {Z}}^{*,*}$ for $f:Y\\rightarrow X$ a projective morphism in ${\\mathrm {Sm}}_B$ (see [9] for details), making $(X, Z)\\mapsto {\\operatorname{H}}{\\mathbb {Z}}^{*,*}_Z(X)$ a $\\operatorname{GL}$ -oriented cohomology theory on ${\\mathrm {Sm}}_k$ .",
"In addition, for $\\pi _X:X\\rightarrow {\\rm Spec\\,}k$ in ${\\mathrm {Sm}}_k$ projective of dimension $n$ , the map $\\hat{\\pi }_{X*}:{\\operatorname{H}}{\\mathbb {Z}}^{2n, n}(X)\\rightarrow {\\rm CH}^0({\\rm Spec\\,}k)={\\mathbb {Z}}$ is $\\deg _k$ , and for $i:Y\\rightarrow X$ a closed immersion, the map $\\hat{i}_*$ is given by the localization theorem, which readily implies that $\\hat{i}_*=i_*$ .",
"By our comparison theorem REF , which here really reduces to the theorem of Panin-Smirnov, it follows that $\\hat{f}_*=f_*$ for all projective $f$ .",
"Finally, one has $c_{d_X}=s^*s_*(1^{{\\operatorname{H}}{\\mathbb {Z}}}_X)=e^{{\\operatorname{H}}{\\mathbb {Z}}}(V)$ ([18]), so applying the motivic Gauß-Bonnet theorem REF gives the statement.",
"One can obtain the same result by using the cohomology of the Milnor K-theory sheaves as a bigraded cohomology theory.",
"The homotopy t-structure on ${\\operatorname{SH}}(k)$ has heart the abelian category of homotopy modules $\\operatorname{\\Pi _*}(k)$ (see [32] and [33] for details); we let ${\\operatorname{H}}_0:{\\operatorname{SH}}(k)\\rightarrow \\operatorname{\\Pi _*}(k)$ be the associated functor.",
"The fact that ${\\operatorname{H}}{\\mathbb {Z}}^{n,n}({\\rm Spec\\,}F) \\simeq \\mathrm {K}^\\mathrm {M}_n(F)$ for $F$ a field [35], [47] says that ${\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}$ is canonically isomorphic to the homotopy module $(\\mathrm {K}^\\mathrm {M}_n)_n$ , which is in fact a cycle module in the sense of Rost [44].",
"This gives us the isomorphism ${\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}^{a,b}(X)\\simeq {\\operatorname{H}}^a(X, {\\mathcal {K}}^\\mathrm {M}_b).$ The isomorphism $H^n(X, {\\mathcal {K}}^\\mathrm {M}_n)\\simeq {\\rm CH}^n(X)$ (a special case of Rost's formula for the Chow groups of a cycle module, [44]) gives us as above Thom classes $\\vartheta ^{\\mathrm {K}^\\mathrm {M}_*}(V)\\in {\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}^{2r,r}_{0_V}(V)$ , giving ${\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}$ a $\\operatorname{GL}$ -orientation.",
"As for ${\\operatorname{H}}{\\mathbb {Z}}^{*,*}$ , one has explicitly defined pushforward maps on ${\\operatorname{H}}^*(-, {\\mathcal {K}}^M_*)$ which give ${\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}^{*,*}$ the structure of a $\\operatorname{GL}$ -oriented cohomology theory on ${\\mathrm {Sm}}_k$ and for which the pushforward map for the zero-section of a vector bundle is given by the Thom isomorphism.",
"Since the pushforward on ${\\operatorname{H}}^n(X, {\\mathcal {K}}^\\mathrm {M}_n)$ agrees with the classical pushforward on ${\\rm CH}^n$ , we deduce the following using the same proof as for Theorem REF .",
"Theorem 8.2 Let $X\\in {\\mathrm {Sm}}_k$ be projective of dimension $d_X$ .",
"Then $u^{{\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}}(\\chi (X/k))=\\deg _k(c_{d_X}(T_{X/k})) \\text{ in }{\\rm CH}^0({\\rm Spec\\,}k)={\\mathbb {Z}}.$ Algebraic K-theory We now let $B$ be any regular separated base-scheme of finite Krull dimension.",
"Algebraic K-theory on ${\\mathrm {Sm}}_B$ is represented by the motivic commutative ring spectrum ${\\operatorname{KGL}}\\in {\\operatorname{SH}}(B)$ (see [48]).",
"Just as for ${\\operatorname{H}}{\\mathbb {Z}}$ , the purity theorem ${\\operatorname{KGL}}_Z^{a,b}(X)\\simeq {\\operatorname{KGL}}^{a-2c, b-c}(Z)$ for $i:Z\\rightarrow X$ a closed immersion of codimension $d$ in ${\\mathrm {Sm}}_B$ (a consequence of Quillen's localization sequence for algebraic K-theory [41]) gives Thom class $\\vartheta ^{\\operatorname{KGL}}(V)\\in {\\operatorname{KGL}}^{2r,r}_{0_V}(V)$ for $V\\rightarrow X$ a rank $r$ vector bundle over $X\\in {\\mathrm {Sm}}_B$ , and makes ${\\operatorname{KGL}}$ a $\\operatorname{GL}$ -oriented motivic spectrum.",
"Explicitly, ${\\operatorname{KGL}}$ represents Quillen K-theory on ${\\mathrm {Sm}}_B$ via ${\\operatorname{KGL}}^{a,b}\\simeq \\mathrm {K}_{2b-a}$ and the Thom class for a rank $r$ vector bundle $p:V\\rightarrow X$ is represented by the Koszul complex ${\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})$ .",
"Here $\\text{can}:p^*V^\\vee \\rightarrow {\\mathcal {O}}_V$ is the dual of the tautological section ${\\mathcal {O}}_V\\rightarrow p^*V$ and ${\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})$ is the complex whose terms are given by ${\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})^{-r}=\\Lambda ^rp^*V^\\vee $ and whose differential $\\Lambda ^rp^*V^\\vee \\rightarrow \\Lambda ^{r-1}p^*V^\\vee $ is given with respect to a local framing of $V^\\vee $ by $d(e_{i_1}\\wedge \\ldots \\wedge e_{i_r})=\\sum _{j=1}^r(-1)^{j-1}\\text{can}(e_{i_j})\\cdot e_{i_1}\\wedge \\ldots \\wedge \\widehat{e_{i_j}}\\wedge \\ldots \\wedge e_{i_r}.$ This complex is a locally free resolution of $s_*({\\mathcal {O}}_X)$ , where $s:X\\rightarrow V$ is the zero-section.",
"Thus, by the identification of ${\\operatorname{KGL}}^{2r, r}_{0_V}(V)$ with the Grothendieck group of the triangulated category of perfect complexes on $V$ with support contained in $0_V$ , ${\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})$ gives rise to a class $[{\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})]\\in {\\operatorname{KGL}}^{2r, r}_{0_V}(V)$ which maps to $1_X$ under the purity isomorphism ${\\operatorname{KGL}}^{2r, r}_{0_V}(V)\\simeq {\\operatorname{KGL}}^{0,0}(X)$ , so that we indeed have $[{\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})] = \\vartheta ^{\\operatorname{KGL}}(V)$ .",
"Just as for motivic cohomology, one has explicit pushforward maps in K-theory given by Quillen's localization and devissage theorems identifying, for $X \\in {\\mathrm {Sm}}_B$ and $Z \\subseteq X$ a closed subscheme, the K-theory with support $\\mathrm {K}^Z(X)$ with the K-theory of the abelian category of coherent sheaves $\\operatorname{Coh}_Z$ on $Z$ , denoted $\\mathrm {G}(Z)$ .",
"For a projective morphism $f:Y\\rightarrow X$ , one has the pushforward map $\\hat{f}_*:\\mathrm {G}(Y)\\rightarrow \\mathrm {G}(X)$ defined by using a suitable subcategory of $\\operatorname{Coh}_Y$ on which $f_*$ is exact.",
"On $\\mathrm {K}_0$ , this recovers the usual formula $\\hat{f}_*([{\\mathcal {F}}])=\\sum _{j=0}^{\\operatorname{dim}Y}(-1)^j[\\mathrm {R}^jf_*({\\mathcal {F}})]$ for ${\\mathcal {F}}\\in \\operatorname{Coh}_Y$ .",
"Via the isomorphisms ${\\operatorname{KGL}}_Z^{a,b}(X)\\simeq \\mathrm {G}_{2b-a}(Z)$ , this gives pushforward maps $\\hat{f}_*$ for ${\\operatorname{KGL}}^{*,*}$ , defining a $\\operatorname{GL}$ -oriented cohomology theory on ${\\mathrm {Sm}}_B$ .",
"For $s:X\\rightarrow V$ the zero-section of a vector bundle, $\\hat{s}_*$ agrees with the pushforward $s_*$ using the Thom isomorphism/localization theorem, hence by our comparison theorem (again really the theorem of Panin-Smirnov), we have $\\hat{f}_*=f_*$ for all projective $f$ .",
"Theorem 8.3 Let $\\pi _X:X\\rightarrow B$ be a smooth projective morphism with $B$ a regular separated scheme of finite Krull dimension.",
"Then $u^{\\operatorname{KGL}}(\\chi (X/B))=\\sum _{j=0}^{\\operatorname{dim}_BX}\\sum _{i=0}^{\\operatorname{dim}_BX} (-1)^{i+j}[\\mathrm {R}^j\\pi _{X*}\\Omega _{X/B}^i]\\in \\mathrm {K}_0(B)={\\operatorname{KGL}}^{0,0}(B).$ Let $p:T_{X/B}\\rightarrow X$ denote the relative tangent bundle and let $s:X\\rightarrow T_{X/B}$ denote the zero-section.",
"We have $e^{\\operatorname{KGL}}(T_{X/B})=s^*({\\operatorname{th}}(T_{X/B}))=s^*({\\operatorname{Kos}}_{T_{X/B}}(p^*T_{X/B}^\\vee , \\text{can})).$ Since $T_{X/B}^\\vee =\\Omega _{X/B}$ , and $s^*(\\text{can})$ is the zero-map, it follows that, in $\\mathrm {K}_0(X)$ , $s^*({\\operatorname{Kos}}_{T_{X/B}}(p^*T_{X/B}^\\vee , \\text{can}))=\\sum _{i=0}^{\\operatorname{dim}_BX}(-1)^i[\\Omega _{X/B}^i],$ and thus $\\pi _{X*}(e^{\\operatorname{KGL}}(T_{X/B}))=\\sum _{j=0}^{\\operatorname{dim}_BX}\\sum _{i=0}^{\\operatorname{dim}_BX} (-1)^{i+j}[\\mathrm {R}^j\\pi _{X*}\\Omega _{X/B}^i].$ We conclude by applying the motivic Gauß-Bonnet theorem.",
"Milnor-Witt cohomology and Chow-Witt groups In this case, we again work over a perfect base-field $k$ .",
"The Milnor-Witt sheaves ${\\mathcal {K}}^\\mathrm {MW}_*$ constructed by Morel ([32], [33]) give rise to an $\\operatorname{SL}$ -oriented theory as follows.",
"Morel describes an isomorphism of ${\\mathcal {K}}^\\mathrm {MW}_0$ with the sheafification ${\\mathcal {GW}}$ of the Grothendieck-Witt rings[33] defines an isomorphism ${\\operatorname{GW}}(F)\\rightarrow \\mathrm {K}^\\mathrm {MW}(F)$ , $F$ a field.",
"[33] defines ${\\mathcal {K}}^\\mathrm {MW}_*$ as an unramified sheaf and it follows from [36] that ${\\mathcal {GW}}$ is an unramified sheaf.",
"From this it is not difficult to show that the isomorphism ${\\operatorname{GW}}(F)\\rightarrow \\mathrm {K}^\\mathrm {MW}(F)$ for fields extends to an isomorphism of sheaves.",
"; the map of sheaves of abelian groups ${\\mathbb {G}}_m\\rightarrow {\\mathcal {GW}}^\\times $ sending a unit $u$ to the one-dimensional form $\\langle u\\rangle $ allows one to define, for $L\\rightarrow X$ a line bundle, a twisted version ${\\mathcal {K}}^\\mathrm {MW}_*(L):={\\mathcal {K}}^\\mathrm {MW}_*\\times _{{\\mathbb {G}}_m}L^\\times $ as a Nisnevich sheaf on $X\\in {\\mathrm {Sm}}_k$ (see [33] or [11]).",
"One may use the Rost-Schmid complex for ${\\mathcal {K}}^\\mathrm {MW}_*(L)$ [33] to compute ${\\operatorname{H}}^*_Z(X, {\\mathcal {K}}^\\mathrm {MW}_*(L))$ for $Z\\subseteq X$ a closed subset, which gives a purity theorem: for $i:Z\\rightarrow X$ a codimension $d$ closed immersion in ${\\mathrm {Sm}}_k$ and $L\\rightarrow X$ a line bundle, there is a canonical isomorphism ${\\operatorname{H}}^*_Z(X, {\\mathcal {K}}^\\mathrm {MW}_*(L))\\simeq {\\operatorname{H}}^{*-d}(Z, {\\mathcal {K}}^\\mathrm {MW}_{*-d}(i^*L\\otimes \\operatorname{det}N_i)),$ where $N_i\\rightarrow Z$ is the normal bundle of $i$ .",
"Applying this to the zero-section of a rank $r$ vector bundle $p:V\\rightarrow X$ gives the isomorphism ${\\operatorname{H}}^0(X, {\\mathcal {GW}})\\simeq {\\operatorname{H}}^r_{0_V}(V, {\\mathcal {K}}^\\mathrm {MW}_r(p^*\\operatorname{det}^{-1} V));$ in particular, given an isomorphism $\\phi :\\operatorname{det}V\\rightarrow {\\mathcal {O}}_X$ , we obtain a Thom class $\\theta _{V, \\phi }\\in {\\operatorname{H}}^r_{0_V}(V, {\\mathcal {K}}^\\mathrm {MW}_r)$ corresponding to the unit section $1_X\\in {\\operatorname{H}}^0(X, {\\mathcal {GW}})$ .",
"On the other hand, Morel's computation of the 0th graded homotopy sheaf of the sphere spectrum ([32], [33]) gives an identification ${\\operatorname{H}}_0(1_k)\\simeq ({\\mathcal {K}}^\\mathrm {MW}_n)_{n\\in {\\mathbb {Z}}}$ in $\\operatorname{\\Pi _*}(k)$ , which then gives the natural isomorphism ${\\operatorname{H}}_0(1_k)^{a+b, b}_Z(X)\\simeq {\\operatorname{H}}^a_Z(X, {\\mathcal {K}}^\\mathrm {MW}_b).$ This is moreover compatible with twisting by a line bundle $p:L\\rightarrow X$ , on the ${\\operatorname{H}}_0(1_k)$ side using the Thom space construction ${\\operatorname{H}}_0(1_k)^{*, *}_Z(X;L):={\\operatorname{H}}_0(1_k)^{*+2, *+1}_Z(L)$ and on the Milnor-Witt cohomology side using the twisted Milnor-Witt sheaves.",
"To see this, note that the “untwisted” isomorphism gives us an isomorphism ${\\operatorname{H}}_0(1_k)^{a+b+2, b+1}_{Z}({\\operatorname{Th}}(L))\\simeq {\\operatorname{H}}_0(1_k)^{a+b+2, b+1}_{0_L\\cap p^{-1}(Z)} (L)\\simeq {\\operatorname{H}}^{a+1}_{0_L\\cap p^{-1}(Z)}(X, {\\mathcal {K}}^\\mathrm {MW}_{b+1}),$ so it suffices to identify the right-hand side with ${\\operatorname{H}}^a_Z(X,{\\mathcal {K}}^\\mathrm {MW}_b(L))$ .",
"For $Y\\in {\\mathrm {Sm}}_k$ and line bundle $M\\rightarrow Y$ , the Rost-Schmid complex for ${\\mathcal {K}}^\\mathrm {MW}_m(M)$ consists in degree $a$ of sums of terms of twisted Milnor-Witt groups of the form $\\mathrm {K}^\\mathrm {MW}_{m-a}(k(y); \\Lambda ^a(\\mathfrak {m}_y/\\mathfrak {m}_y^2)^\\vee \\otimes M)$ , for $y$ a codimension $a$ point of $Y$ and $\\mathfrak {m}_y\\subset {\\mathcal {O}}_{Y,y}$ the maximal ideal.",
"To compute cohomology with supports in $W\\subseteq Y$ , one restricts to those $y\\in W$ .",
"If we now take $Y=L$ and $M$ the trivial bundle, with supports in $p^{-1}(Z)\\cap 0_L$ and $m=b+1$ , and compare with $Y=X$ , with supports in $Z$ with $m=b$ , the term for $y\\in Z$ , of codimension $a+1$ on $L$ is $\\mathrm {K}^\\mathrm {MW}_{b-a}(k(y); \\Lambda ^a(\\mathfrak {m}_y/\\mathfrak {m}_y^2)^\\vee \\otimes L)$ while the term for $y\\in Z$ , of codimension $a$ on $X$ is $\\mathrm {K}^\\mathrm {MW}_{b-a}(k(y); \\Lambda ^a(\\mathfrak {m}_y/\\mathfrak {m}_y^2)^\\vee )$ , where $\\mathfrak {m}_y$ is the maximal ideal in ${\\mathcal {O}}_{X,y}$ in both cases.",
"This gives the desired identification ${\\operatorname{H}}^{a+1}_{0_L\\cap p^{-1}(Z)}(X, {\\mathcal {K}}^\\mathrm {MW}_{b+1})\\simeq {\\operatorname{H}}^{a}_{Z}(X, {\\mathcal {K}}^\\mathrm {MW}_{b}(L)).$ The purity isomorphism (REF ) is a special case of this construction.",
"The Thom class $\\theta _{V, \\phi }\\in {\\operatorname{H}}^r_{0_V}(V, {\\mathcal {K}}^\\mathrm {MW}_r)$ gives the Thom class $\\theta _{V, \\phi }\\in {\\operatorname{H}}_0(1_k)^{2r, r}_{0_V}(V),$ making ${\\operatorname{H}}_0(1_k)$ an $\\operatorname{SL}$ -oriented theory (see e.g [26]).",
"The resulting canonical Thom class ${\\operatorname{th}}_V\\in {\\operatorname{H}}_0(1_k)^{2r, r}_{0_V}(V;\\operatorname{det}^{-1}V)={\\operatorname{H}}^r_{0_V}(V, {\\mathcal {K}}^\\mathrm {MW}_r(\\operatorname{det}^{-1} V))$ agrees with the image of $1_X\\in {\\operatorname{H}}^0(X, {\\mathcal {GW}})$ under the Rost-Schmid isomorphism (REF ).",
"Let $\\pi _X:X\\rightarrow {\\rm Spec\\,}k$ be smooth and projective over $k$ of dimension $d$ .",
"Using the Rost-Schmid complex for the twisted homotopy module one has generators for ${\\operatorname{H}}^d(X, {\\mathcal {K}}^\\mathrm {MW}_d(\\omega _{X/k}))$ as formal sums $\\tilde{x}=\\sum _i\\alpha _i\\cdot p_i$ , with $\\alpha _i\\in {\\operatorname{GW}}(k(p_i))$ and $p_i\\in X$ closed points.",
"Since $k$ is perfect, the finite extension $k(p_i)/k$ is separable and one can define $\\widetilde{\\deg }_k(\\tilde{x}):=\\sum _i{\\rm Tr}_{k(p_i)/k}\\alpha _i\\in {\\operatorname{GW}}(k)$ where ${\\rm Tr}_{k(p_i)/k}:{\\operatorname{GW}}(k(p_i))\\rightarrow {\\operatorname{GW}}(k)$ is the transfer induced by the usual trace map ${\\rm Tr}_{k(p_i)/k}:k(p_i)\\rightarrow k$ (see for example [11] ).",
"It is shown in [11] that this descends to a map $\\widetilde{\\deg }_k:{\\operatorname{H}}_0^{2d,d}(X;\\omega _{X/k})={\\operatorname{H}}^d(X, {\\mathcal {K}}^\\mathrm {MW}_d(\\omega _{X/k}))\\rightarrow {\\operatorname{H}}_0^{0,0}({\\rm Spec\\,}k)={\\operatorname{GW}}(k)$ See also [21], which identifies this map with one induced by the Scharlau trace.",
"The methods of this paper give a new proof of the result given in [26]: Theorem 8.4 Let $k$ be a perfect field of characteristic different from two.",
"For $\\pi _X:X\\rightarrow {\\rm Spec\\,}k$ smooth and projective over $k$ , we have $\\chi (X/k)=\\widetilde{\\deg }_k(e^{{\\operatorname{H}}_0(1_k)}(T_{X/k})).$ Under Morel's isomorphism ${\\rm End}_{{\\operatorname{SH}}(k)}(1_X)\\simeq {\\operatorname{GW}}(k)$ ([32], [33]) and the isomorphism ${\\operatorname{H}}^0({\\rm Spec\\,}k, {\\mathcal {K}}^\\mathrm {MW}_0)\\simeq {\\operatorname{GW}}(k)$ , the unit map $u^{{\\operatorname{H}}_0(1_k)}:1_k\\rightarrow {\\operatorname{H}}_0(1_k)$ induces the identity map on $\\pi _{0,0}$ .",
"Using this, the proof of the claim is essentially the same as the other Gauß-Bonnet theorems we have discussed, but with a bit of extra work since we are no longer in the GL-oriented case.",
"Fasel [15] has defined pushforward maps $\\hat{f}_*:{\\operatorname{H}}^a_W(X, {\\mathcal {K}}^\\mathrm {MW}_b(\\omega _{X/k}\\otimes f^*L))\\rightarrow {\\operatorname{H}}^{a-d}_Z(Y, {\\mathcal {K}}^\\mathrm {MW}_{b-d}(L))$ for each projective morphism $f:X\\rightarrow Y$ in ${\\mathrm {Sm}}_k$ of relative dimension $d$ , line bundle $L\\rightarrow Y$ , and closed subsets $Z\\subseteq Y$ , $W\\subseteq X$ with $f(W)\\subseteq Z$ .",
"In the case of the structure map $\\pi _X:X\\rightarrow {\\rm Spec\\,}k$ , the pushforward $\\tilde{\\pi }_{X*}:{\\operatorname{H}}^d(X, {\\mathcal {K}}^\\mathrm {MW}_d(\\omega _{X/k}))\\rightarrow {\\operatorname{H}}^0({\\rm Spec\\,}k, {\\mathcal {K}}^\\mathrm {MW}_0)={\\operatorname{GW}}(k)$ is the map $\\widetilde{\\deg }_k$ .",
"For $s:X\\rightarrow V$ the zero-section of a vector bundle, $\\hat{s}_*$ is the Thom isomorphism $s_*$ .",
"Thus, if we pass to the $\\eta $ -inverted theory, ${\\operatorname{H}}_0(1_X)_\\eta :={\\operatorname{H}}_0(1_k)[\\eta ^{-1}]$ , our comparison theorem REF says that $\\hat{f}_{\\eta *}=f_{\\eta *}$ for all projective morphisms $f$ in ${\\mathrm {Sm}}_k$ .",
"We have ${\\mathcal {K}}^\\mathrm {MW}_*[\\eta ^{-1}]\\simeq {\\mathcal {W}}$ , the sheaf of Witt rings, and the map ${\\mathcal {K}}^\\mathrm {MW}_0={\\mathcal {GW}}\\rightarrow {\\mathcal {K}}^\\mathrm {MW}_*[\\eta ^{-1}]\\simeq {\\mathcal {W}}$ is the canonical map $q:{\\mathcal {GW}}\\rightarrow {\\mathcal {W}}$ realizing ${\\mathcal {W}}$ as the quotient of ${\\mathcal {GW}}$ by the subgroup generated by the hyperbolic form.",
"Thus, applying our motivic Gauß-Bonnet theorem gives the identity $q(\\chi (X/k))=q(\\widetilde{\\deg }_k(e^{{\\operatorname{H}}_0(1_k)}(T_{X/k})))\\text{ in }{\\operatorname{W}}(k).$ To lift this to an equality in ${\\operatorname{GW}}(k)$ and thereby complete the proof, we use that the map $({\\operatorname{\\text{rnk}}}, q): {\\mathcal {GW}}\\rightarrow {\\mathbb {Z}}\\times {\\mathcal {W}}$ is injective, together with the fact that we can recover the rank by applying ${\\operatorname{H}}_0$ to the unit map $1_k\\rightarrow {\\operatorname{H}}{\\mathbb {Z}}$ and using Theorem REF .",
"Hermitian K-theory and Witt theory We again let our base-scheme $B$ be a regular noetherian separated base-scheme of finite Krull dimension, but now assume that 2 invertible on $B$ .",
"Our goal in this subsection is to explain how the description of the “rank” of $\\chi (X/B)$ given by Theorem REF can be refined to give a formula for $\\chi (X/k)$ itself in terms of Hodge cohomology by using hermitian K-theory.",
"By work of Panin-Walter [40], Schlichting [45], and Schlichting-Tripathi [46], hermitian K-theory ${\\operatorname{KO}}^{[*]}_*(-)$ is represented by a motivic commutative ring spectrum ${\\operatorname{BO}}\\in {\\operatorname{SH}}(B)$ (we use the notation of [3]).",
"Panin-Walter give ${\\operatorname{BO}}$ an $\\operatorname{SL}$ -orientation.",
"${\\operatorname{BO}}$ -theory also represents particular cases of Schlichting's Grothendieck-Witt groups [45], via functorial isomorphisms ${\\operatorname{BO}}^{2r, r}(X;L)\\simeq {\\operatorname{KO}}^{[r]}_0(X, L):={\\operatorname{GW}}({\\operatorname{perf}}(X), L[r], \\text{can}),$ where $L\\rightarrow X$ is a line bundle and ${\\operatorname{GW}}({\\operatorname{perf}}(X), L[r], \\text{can})$ is the Grothendieck-Witt group of $L[r]$ -valued symmetric bilinear forms on ${\\operatorname{perf}}(X)$; we recall a version of the definition here.",
"Definition 8.5 Let $L\\rightarrow X$ be a line bundle and let $r \\in {\\mathbb {Z}}$ .",
"An $L[r]$ -valued symmetric bilinear form on $C\\in {\\operatorname{perf}}(X)$ is a map $\\phi :C\\otimes ^{\\mathrm {L}}C\\rightarrow L[r]$ in ${\\operatorname{perf}}(X)$ which satisfies the following conditions.",
"$\\phi $ is non-degenerate: the induced map $C\\rightarrow {\\mathcal {RH}om}(C, L[n])$ is an isomorphism in ${\\operatorname{perf}}(X)$ .",
"$\\phi $ is symmetric: $\\phi \\circ \\tau =\\phi $ , where $\\tau :C\\otimes ^{\\mathrm {L}}C\\rightarrow C\\otimes ^{\\mathrm {L}}C$ is the commutativity isomorphism.",
"(Note that we are assuming non-degeneracy in the definition but leaving this out of the terminology for the sake of brevity.)",
"Similar to the case of algebraic K-theory discussed in §REF , for a rank $r$ vector bundle $p:V\\rightarrow X$ , the Thom class $\\theta ^{\\operatorname{BO}}_V\\in {\\operatorname{BO}}^{2r,r}(V; p^*\\operatorname{det}^{-1}V)$ is given by the Koszul complex ${\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )$ , where the symmetric bilinear form $\\phi _V:{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )\\otimes {\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )\\rightarrow p^*\\operatorname{det}^{-1}V[r]=\\Lambda ^rV^\\vee [r]$ is given by the usual exterior product $-\\wedge -: \\Lambda ^iV^\\vee \\otimes \\Lambda ^{r-i}V^\\vee \\rightarrow \\Lambda ^rV^\\vee .$ Moreover, there are isomorphisms for $i<0$ ${\\operatorname{BO}}^{2r-i, r}(X;L)\\simeq {\\operatorname{W}}^{r-i}({\\operatorname{perf}}(X), L[r], \\text{can})$ where ${\\operatorname{W}}^{r-i}({\\operatorname{perf}}(X), L[r], \\text{can})$ is Balmer's triangulated Witt group.",
"In the case $B = {\\rm Spec\\,}k$ for $k$ a field of characteristic different from two, Ananyevskiy [3] shows that this isomorphism induces an isomorphism of $\\eta $ -inverted hermitian K-theory with Witt-theory, ${\\operatorname{BO}}[\\eta ^{-1}]^{*,*}\\simeq {\\operatorname{W}}^*[\\eta ,\\eta ^{-1}],$ where one gives $\\eta $ bidegree $(-1,-1)$ and an element $\\alpha \\eta ^n$ with $\\alpha \\in {\\operatorname{W}}^m$ has bidegree $(m-n,-n)$ ; the same proof works over out general base $B$ (with assumptions as at the beginning of this subsection).",
"For $f:Y\\rightarrow X$ a proper map of relative dimension $d_f$ in ${\\mathrm {Sm}}_B$ and $L$ a line bundle on $X$ , we follow Calmès-Hornbostel [12] in defining a pushforward map $\\hat{f}_*:{\\operatorname{BO}}^{2r,r}(Y, \\omega _{Y/B}\\otimes f^*L)\\rightarrow {\\operatorname{BO}}^{2r-2d_f,r-d_f}(X, \\omega _{X/B}\\otimes L)$ by Grothendieck-Serre duality.",
"In op.",
"cit., this is worked out for the $\\eta $ -inverted theory ${\\operatorname{BO}}_\\eta $ when the base is a field; however, the same construction works for ${\\operatorname{BO}}$ over the general base-scheme $B$ and goes as follows.",
"For $r \\ge 0$ , given an $L[r]$ -valued symmetric bilinear form $\\phi :C\\otimes ^{\\mathrm {L}}C\\rightarrow \\omega _{Y/B}\\otimes f^*L[r]$ , we have the corresponding isomorphism $\\tilde{\\phi }:C\\rightarrow {\\mathcal {RH}om}(C, \\omega _{Y/B}\\otimes f^*L[r]) \\simeq {\\mathcal {RH}om}(C, \\omega _{Y/X}\\otimes f^*(\\omega _{X/B}\\otimes L[r]))$ Grothendieck-Serre duality gives the isomorphism $\\mathrm {R}f_*{\\mathcal {RH}om}(C, \\omega _{Y/X}\\otimes f^*(\\omega _{X/B}\\otimes L[r])){[r]^{\\psi }_\\sim &}{\\mathcal {RH}om}(\\mathrm {R}f_*C, \\omega _{X/B}\\otimes L[r-d_f]).$ Composing these, we obtain the isomorphism $\\psi \\circ \\tilde{\\phi }:\\mathrm {R}f_*C\\rightarrow {\\mathcal {RH}om}(\\mathrm {R}f_*C, \\omega _{X/B}\\otimes L[r-d_f]),$ corresponding to the (nondegenerate) bilinear form $\\mathrm {R}f_*(\\phi ):\\mathrm {R}f_*C\\otimes ^{\\mathrm {L}}\\mathrm {R}f_*C\\rightarrow \\omega _{X/B}\\otimes L[r-d_f],$ which one can show is symmetric.",
"We explicitly define the above pushforward map by setting $\\hat{f}_*(C, \\phi ) := (\\mathrm {R}f_*C, \\mathrm {R}f_*(\\phi ))$ .",
"Applying this in the situation that $f=\\pi _X:X\\rightarrow B$ is a smooth and proper $B$ -scheme of relative dimension $d_X$ , we may obtain the formula $\\hat{\\pi }_{X*}(e^{\\operatorname{BO}}(T_{X/B}))=(\\oplus _{i,j=0}^{\\operatorname{dim}_BX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i], {\\rm Tr}),$ where ${\\rm Tr}:(\\oplus _{i,j=0}^{\\operatorname{dim}_kX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i])\\otimes (\\oplus _{i,j=0}^{\\operatorname{dim}_kX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i])\\rightarrow {\\mathcal {O}}_B$ is the symmetric bilinear form in ${\\operatorname{perf}}(B)$ determined by the pairings $(\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)\\otimes (\\mathrm {R}^{d_X-i}\\pi _{X*}\\Omega _{X/B}^{d_X-j})\\xrightarrow{}\\mathrm {R}^{d_X}\\pi _{X*}\\Omega _{X/B}^{d_X} \\xrightarrow{}{\\mathcal {O}}_B.$ Indeed, if $s:X\\rightarrow T_{X/B}$ denotes the zero-section, we have $e^{\\operatorname{BO}}(T_{X/B})=s^*({\\operatorname{Kos}}(T_{X/B}), \\phi )=(\\oplus _{j=0}^{d_X}\\Omega ^j_{X/B}[j],s^*\\phi )$ with $s^*\\phi $ determined by the product maps $\\Omega ^j_{X/B}[j]\\otimes \\Omega ^{d_X-j}_{X/B}[d_X-j]\\rightarrow \\omega _{X/B}[d_X],$ and thus $\\hat{\\pi }_{X*}(e^{\\operatorname{BO}}(T_{X/B}))$ is $\\oplus _{i,j=0}^{\\operatorname{dim}_kX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i]$ with the symmetric bilinear form ${\\rm Tr}$ as described above.",
"Passing to the $\\eta $ -inverted theory ${\\operatorname{BO}}_\\eta $ , our comparison theorem REF gives $q\\circ \\hat{\\pi }_{X*}=q\\circ \\pi _{X*}$ as maps ${\\operatorname{BO}}_\\eta ^{2d_X, d_X}(X,\\omega _{X/B})\\rightarrow {\\operatorname{BO}}_\\eta (B)$ .",
"We check that the conditions of the comparison theorem hold just as we did for algebraic K-theory.",
"Firstly, as mentioned above, the $\\operatorname{SL}$ -orientation for ${\\operatorname{BO}}$ defined by Panin-Walter can be described as follows: the Thom class for an oriented vector bundle ($p:V\\rightarrow X$ , $\\rho :{\\mathcal {O}}_X\\xrightarrow{}\\operatorname{det}V$ ) is given by the Koszul complex ${\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )$ equipped with the symmetric bilinear form $\\phi _V$ defined by the product in the exterior algebra followed by the isomorphism $p^*\\rho ^\\vee :p^*\\operatorname{det}^{-1}V\\rightarrow {\\mathcal {O}}_V$ .",
"On the other hand, the Calmès-Hornbostel pushforward for the zero-section $s:X\\rightarrow V$ of a rank $r$ vector bundle $p:V\\rightarrow X$ with isomorphism $\\rho :{\\mathcal {O}}_X\\xrightarrow{} \\operatorname{det}V$ is as described above, sending a symmetric bilinear form $\\psi :C\\otimes ^{\\mathrm {L}}C\\rightarrow \\omega _{X/B} \\otimes s^*L[n]$ to the symmetric bilinear form $\\mathrm {R}s_*(\\psi ): \\mathrm {R}s_*C \\otimes ^{\\mathrm {L}}\\mathrm {R}s_*C\\rightarrow \\omega _{V/B} \\otimes L[n+r].$ Since $s$ is finite, $\\mathrm {R}s_*C \\simeq C$ , which in ${\\operatorname{perf}}(V)$ is canonically isomorphic to $p^*C\\otimes _{{\\mathcal {O}}_V}{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )$ .",
"We may thus view $\\mathrm {R}s_*(\\psi )$ instead as a map $[p^*C\\otimes _{{\\mathcal {O}}_V}{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )] \\otimes ^{\\mathrm {L}}[p^*C\\otimes _{{\\mathcal {O}}_V}{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )] \\rightarrow \\omega _{V/B} \\otimes L[n+r];$ tracing through its definition, one finds that this map is given by the composition $[p^*C\\otimes _{{\\mathcal {O}}_V}{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )] \\otimes ^{\\mathrm {L}}[p^*C\\otimes _{{\\mathcal {O}}_V}{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )] \\\\\\xrightarrow{}[p^*C\\otimes ^{\\mathrm {L}}p^*C]\\otimes [{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )\\otimes {\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )]\\\\\\xrightarrow{} \\omega _{X/B} \\otimes L[n] \\otimes p^*\\operatorname{det}^{-1}V[r] \\xrightarrow{} \\omega _{V/B} \\otimes L[n+r].$ As this is exactly $p^*(C, \\phi ) \\otimes {\\operatorname{th}}_{V,\\rho }$ , we see that the Calmès-Hornbostel pushforward for $s$ is the same as that defined by the Panin-Walter $\\operatorname{SL}$ -orientation on ${\\operatorname{BO}}_\\eta $ , which verifies the hypothesis in our Theorem REF .",
"Having verified this, we can prove our main result.",
"Theorem 8.6 Let $B$ be a regular noetherian separated scheme of finite Krull dimension with 2 invertible in $\\Gamma (B, {\\mathcal {O}}_B)$ .",
"Let $X$ be a smooth projective $B$ -scheme.",
"Then: We have $u^{{\\operatorname{BO}}_\\eta }(\\chi (X/B))=(\\oplus _{i,j=0}^{\\operatorname{dim}_kX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i], {\\rm Tr})$ in ${\\operatorname{BO}}_\\eta ^{0,0}(B) \\simeq {\\operatorname{W}}({\\operatorname{perf}}(B)) \\simeq {\\operatorname{W}}(B)$ .",
"Let $f:{\\operatorname{GW}}(B)\\rightarrow \\mathrm {K}_0(B)$ denote the forgetful map discarding the symmetric bilinear form.",
"Suppose that the map $(f, q):{\\operatorname{GW}}(B)\\rightarrow \\mathrm {K}_0(B)\\times W(B)$ is injective (this is the case if for example $B$ is the spectrum of a local ring).",
"Then $u^{{\\operatorname{BO}}}(\\chi (X/B))=(\\oplus _{i,j=0}^{\\operatorname{dim}_BX}{\\operatorname{H}}^i(X, \\Omega _{X/B}^j)[j-i], {\\rm Tr})$ in ${\\operatorname{GW}}({\\operatorname{perf}}(B)) \\simeq {\\operatorname{GW}}(B)$ .",
"Suppose $B$ is in ${\\mathrm {Sm}}_k$ for $k$ a perfect field.",
"Then the image $\\widetilde{\\chi (X/B)}$ of $\\chi (X/B)$ in $\\pi _{0,0}(1_B)(B) \\simeq {\\operatorname{H}}^0(B, {\\mathcal {GW}})$ is given by $\\widetilde{\\chi (X/B)}=(\\oplus _{i,j=0}^{\\operatorname{dim}_BX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i], {\\rm Tr}) \\in {\\operatorname{H}}^0(B, {\\mathcal {GW}}).$ In particular, if $B={\\rm Spec\\,}k$ , then $\\chi (X/k)=(\\oplus _{i,j=0}^{\\operatorname{dim}_BX}{\\operatorname{H}}^i(X, \\Omega _{X/k}^j)[j-i], {\\rm Tr})\\in {\\operatorname{GW}}(k).$ The first statement follows from our comparison theorem REF , as detailed above, together with the motivic Gauß-Bonnet theorem REF .",
"Statement (2) follows from (1) and our result for algebraic K-theory, Theorem REF .",
"Finally, (3) follows from (2), after we check that the unit map $u^{\\operatorname{BO}}$ induces the identity map on ${\\operatorname{GW}}(k)$ via ${\\operatorname{GW}}(k){[r]^{(\\hbox{\\tiny Morel})}_\\sim &} 1_k^{0,0}({\\rm Spec\\,}k){[r]^{u^{\\operatorname{BO}}}_\\sim &} {\\operatorname{BO}}^{0,0}({\\rm Spec\\,}k)\\simeq {\\operatorname{KO}}^{[0]}_0(k)={\\operatorname{GW}}(k),$ where the first isomorphism arises from Morel's theorem [32], [33] identifying $1_k^{0,0}({\\rm Spec\\,}k)$ with ${\\operatorname{GW}}(k)$ .",
"The one dimensional forms $\\langle \\lambda \\rangle \\in {\\operatorname{GW}}(k)$ , $\\lambda \\in k^\\times $ , generate ${\\operatorname{GW}}(k)$ , and via Morel's isomorphism $\\langle \\lambda \\rangle $ maps to the automorphism of $1_k$ induced by the automorphism $\\phi _\\lambda :{\\mathbb {P}}^1_k\\rightarrow {\\mathbb {P}}^1_k$ , $\\phi _\\lambda ((x_0:x_1))= (x_0:\\lambda \\cdot x_1)$ .",
"By [3], the image of $\\phi _\\lambda $ under the unit map $u^{\\operatorname{BO}}$ is also $\\langle \\lambda \\rangle $ , after the canonical identification ${\\operatorname{BO}}^{0,0}({\\rm Spec\\,}k)\\simeq {\\operatorname{KO}}^{[0]}_0(k) \\simeq {\\operatorname{GW}}(k)$ .",
"Corollary 8.7 Let $k$ be a perfect field of characteristic different from two.",
"Let $H \\in {\\operatorname{GW}}(k)$ denote the class of the hyperbolic form $x^2-y^2$ .",
"Let $X$ be a smooth and projective $k$ -scheme.",
"Suppose $X$ has odd dimension $2n-1$ .",
"Let $m := \\sum _{i+j<2n-1}(-1)^{i+j}\\operatorname{dim}_k{\\operatorname{H}}^i(X,\\Omega _{X/k}^j)- \\sum _{\\hbox{t}o 30pt{\\vbox {\\tiny 0\\le i< j\\\\i+j= 2n-1}}} \\operatorname{dim}_k{\\operatorname{H}}^i(X,\\Omega _{X/k}^j).$ Then $\\chi (X/k)=m\\cdot H \\in {\\operatorname{GW}}(k)$ .",
"Assume $X$ has even dimension $2n$ .",
"Let $m := \\sum _{i+j<2n}(-1)^{i+j}\\operatorname{dim}_k{\\operatorname{H}}^i(X,\\Omega _{X/k}^j)+ \\sum _{\\hbox{t}o 30pt{\\vbox {\\tiny 0\\le i< j\\\\i+j= 2n}}} \\operatorname{dim}_k{\\operatorname{H}}^i(X,\\Omega _{X/k}^j)$ and let $Q$ be the symmetric bilinear form ${\\operatorname{H}}^n(X, \\Omega ^n_{X/k})\\times {\\operatorname{H}}^n(X, \\Omega ^n_{X/k}) \\xrightarrow{}{\\operatorname{H}}^{2n}(X, \\Omega ^{2n}_{X/k})\\xrightarrow{} k.$ Then $\\chi (X/k)=m\\cdot H+Q \\in {\\operatorname{GW}}(k)$ .",
"For $V$ a finite dimensional $k$ -vector space and $n \\in {\\mathbb {Z}}$ , we have the symmetric bilinear form in ${\\operatorname{perf}}(k)$ $h_n: (V[n]\\oplus V^\\vee [-n]) \\otimes (V[n]\\oplus V^\\vee [-n]) \\rightarrow k$ whose restriction to $V[n]\\otimes V^\\vee [-n]$ is the canonical pairing of $V[n]$ with $V^\\vee [-n]\\simeq V[n]^\\vee $ , and $(-1)^n$ times this pairing on $V^\\vee [-n]\\otimes V[n]$ .",
"The corresponding class of $h_n$ in ${\\operatorname{GW}}(k)$ is $(-1)^n$ times the class of $h_0$ , as $(V[n]\\oplus V^\\vee [-n], h_n)$ is the image of the class of $V[n]$ in $\\mathrm {K}_0(k)$ under the hyperbolic map $H:\\mathrm {K}_0(-)\\rightarrow {\\operatorname{KO}}^{[0]}_0(-)$ (see e.g.",
"[50]), and $[V[n]]=(-1)^n[V[0]]$ in $\\mathrm {K}_0({\\operatorname{perf}}(k))\\simeq \\mathrm {K}_0(k)$ .",
"With this in mind, we may deduce the claim from the formula $\\chi (X/k)=(\\oplus _{i,j=0}^{\\operatorname{dim}_BX}{\\operatorname{H}}^i(X, \\Omega _{X/k}^j)[j-i], {\\rm Tr})$ of Theorem REF .",
"Indeed, in the case $\\operatorname{dim}X=2n-1$ , the symmetric bilinear form ${\\rm Tr}$ is the sum of the “hyperbolic” forms as above on ${\\operatorname{H}}^i(X,\\Omega _{X/k}^j)[j-i]\\oplus {\\operatorname{H}}^{2n-1-i}(X,\\Omega _{X/k}^{2n-1-j})[i-j]$ for $i+j<2n-1$ , or $0\\le i<j$ and $ i+j= 2n-1$ ; and the argument in the even dimensional case is the same, except that one has the remaining factor coming from the symmetric pairing on ${\\operatorname{H}}^n(X, \\Omega ^n_{X/k})$ .",
"The next result was obtained in 1974 independently by Abelson [1] and Kharlamov [24] using an argument of Milnor's relying on the Lefschetz fixed point theorem.",
"Corollary 8.8 Let $k$ be a field equipped with an embedding $\\sigma : k \\hookrightarrow {\\mathbb {R}}$ .",
"Let $X$ be a smooth projective $k$ -scheme of even dimension $2n$ .",
"Then $|\\chi ^\\mathrm {top}(X({\\mathbb {R}}))|\\le \\operatorname{dim}_k {\\operatorname{H}}^n(X, \\Omega ^n_{X/k}).$ We know that $\\chi ^\\mathrm {top}(X({\\mathbb {R}}))$ is the signature of $\\sigma _*(\\chi (X/k)) \\in {\\operatorname{GW}}({\\mathbb {R}})$ (see [26]).",
"The description of $\\chi (X/k)$ given by Corollary REF gives the desired inequality $|\\operatorname{sig}\\sigma _*(\\chi (X/k))|\\le \\operatorname{dim}_k {\\operatorname{H}}^n(X, \\Omega _{X/k}).",
"$ Remark 8.9 Let $k$ be a perfect field of characteristic different from two.",
"The formula for the Euler characteristic given in Theorem REF shows that the invariant $\\chi (X/k)$ is “motivic” in the following sense.",
"Let $X$ and $Y$ be smooth projective $k$ -schemes of respective even dimensions $2n$ and $2m$ and let $\\alpha :X\\dasharrow Y$ be a correspondence with $k$ -coefficients of degree $n$ , that is, an element $\\alpha \\in {\\rm CH}^{m+n}(X\\times Y)_k$ .",
"The correspondence $\\alpha $ induces the map of $k$ -vector spaces $\\alpha ^*:{\\operatorname{H}}^m(Y,\\Omega ^m_{Y/k})\\rightarrow {\\operatorname{H}}^n(X, \\Omega ^n_{X/k})$ .",
"Suppose that $\\alpha ^*$ is an isomorphism and is compatible with the trace pairings on ${\\operatorname{H}}^m(Y,\\Omega ^m_{Y/k})$ and ${\\operatorname{H}}^n(X, \\Omega ^n_{X/k})$ appearing in Corollary REF .",
"Then $\\chi (X/k)=\\chi (Y/k)$ in the Witt ring ${\\operatorname{W}}(k)$ .",
"For instance, supposing $k$ has characteristic zero, if the motives of $X$ and $Y$ (for homological equivalence with respect to de Rham cohomology) have a Künneth decomposition $h(X) \\simeq \\oplus _{i=0}^{2n}h^i(X)\\langle i\\rangle ,\\quad h(Y) \\simeq \\oplus _{i=0}^{2m}h^i(Y)\\langle i\\rangle $ and $\\alpha $ induces an isomorphism $\\alpha ^*:h^m(Y)\\langle m\\rangle \\rightarrow h^n(X)\\langle n\\rangle $ , compatible with the respective intersection products $h^m(Y)\\langle m\\rangle \\otimes h^m(Y)\\langle m\\rangle \\rightarrow h^{2m}(Y)\\langle 2m\\rangle \\xrightarrow{}h^0(k) \\simeq {\\mathbb {Q}},$ $h^n(X)\\langle n\\rangle \\otimes h^n(X)\\langle n\\rangle \\rightarrow h^{2n}(X)\\langle 2n\\rangle \\xrightarrow{}h^0(k) \\simeq {\\mathbb {Q}},$ then $\\chi (Y/k)=\\chi (X/k)$ in ${\\operatorname{W}}(k)$ .",
"Presumably, merely having an isomorphism of motives $h^n(X)\\langle n\\rangle \\simeq h^m(Y)\\langle m\\rangle $ would not suffice to yield $\\chi (Y/k)=\\chi (X/k)$ in ${\\operatorname{W}}(k)$ , but we do not have an example.",
"Descent for the motivic Euler characteristic Let $k$ be a perfect field of characteristic different from two.",
"With the explicit formula for $\\chi (X/k)$ given by Theorem REF , we may find $\\chi (X/k)$ for forms $X$ of some $k$ -scheme $X_0$ by the usual twisting construction; this works for all manners of descent but we confine ourselves to the case of Galois descent here.",
"Let $X_0, X$ be smooth projective $k$ -schemes of even dimension $2n$ .",
"Let $K$ be a finite Galois extension field of $k$ with Galois group $G$ .",
"Let $X_K:=X\\times _kK$ , $X_{0K}:=X_0\\times _kK$ , and suppose we have an isomorphism $\\phi :X\\times _kK\\rightarrow X_0\\times _kK$ .",
"This gives us the cocycle $\\lbrace \\psi _\\sigma \\in {\\operatorname{Aut}}_K(X_0\\times _kK)\\rbrace _{\\sigma \\in G}$ where $\\psi _\\sigma :=\\phi ^\\sigma \\circ \\phi ^{-1}$ .",
"Letting $&b_0:{\\operatorname{H}}^n(X_0,\\Omega ^n_{X_0/k})\\times {\\operatorname{H}}^n(X_0,\\Omega ^n_{X_0/k})\\rightarrow k,\\\\&b:{\\operatorname{H}}^n(X,\\Omega ^n_{X/k})\\times {\\operatorname{H}}^n(X,\\Omega ^n_{X/k})\\rightarrow k$ denote the respective symmetric bilinear forms ${\\rm Tr}(x\\cup y)$ , the isomorphism $\\phi $ induces an isometry $\\phi ^*:({\\operatorname{H}}^n(X_{0K},\\Omega ^n_{X_{0K}/K}), b_{0K})\\rightarrow ({\\operatorname{H}}^n(X_{K},\\Omega ^n_{X_{K}/K}), b_{K}),$ and the cocycle $\\lbrace \\psi _\\sigma \\rbrace _{\\sigma \\in G}$ determines a cocycle $\\lbrace (\\psi _\\sigma ^*)^{-1} \\in \\mathrm {O}(b_0)(K)\\rbrace _{\\sigma \\in G}$ .",
"Twisting by the latter cocycle allows one to recovers $b$ from $b_0$ ; explicitly, this works as follows.",
"Firstly, as usual, one recovers the $k$ -vector space ${\\operatorname{H}}^n(X,\\Omega ^n_{X/k})$ from the $K$ -vector space ${\\operatorname{H}}^n(X_{0K},\\Omega ^n_{X_{0K}/K})$ as the $G$ -invariants for the map $x\\mapsto \\psi _\\sigma ^{*-1}(x^\\sigma )$ .",
"Secondly, letting $A\\in \\operatorname{GL}({\\operatorname{H}}^n(X_{0K},\\Omega ^n_{X_{0K}/K}))$ be a change of basis matrix comparing the $k$ -forms ${\\operatorname{H}}^n(X_0,\\Omega ^n_{X_0/k})\\subset {\\operatorname{H}}^n(X_{0K},\\Omega ^n_{X_{0K}/K})$ and ${\\operatorname{H}}^n(X,\\Omega ^n_{X/k})\\subset {\\operatorname{H}}^n(X_{0K},\\Omega ^n_{X_{0K}/K})$ , we recover $b$ (up to $k$ -isometry) as $b(x,y)=b_0(Ax, Ay)=:b_0^A(x,y).$ Having performed this twist at the level of symmetric bilinear forms, we may now pass Grothendieck-Witt classes to describe the Euler characteristic of $X$ : namely, Corollary REF (2) gives $\\chi (X_0/k)=[b_0+m\\cdot H],\\quad \\chi (X/k)=[b_0^A+m\\cdot H]$ in ${\\operatorname{GW}}(k)$ .",
"Remark 8.10 In the case of a smooth projective surface $S$ with $p_g(S)=0$ , over a characteristic zero field $k$ , the twisting construction reduces to a computation involving ${\\rm CH}^1(S_{\\bar{k}})/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ as a ${\\rm Gal}(k)$ -module; here $\\bar{k}$ is the algebraic closure of $k$ and $\\sim _{\\operatorname{num}}$ is numerical equivalence.",
"Indeed, the assumption $p_g(S)=0$ implies that the cycle class map in Hodge cohomology ${\\operatorname{cyc}}^\\mathrm {Hdg}:{\\rm CH}^1(S_{\\bar{k}})\\rightarrow {\\operatorname{H}}^1(S_{\\bar{k}}, \\Omega ^1_{S/\\bar{k}})$ induces an isomorphism ${{\\rm CH}^1(S_{\\bar{k}})/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}} \\otimes _{\\mathbb {Z}}\\bar{k}\\xrightarrow{} {\\operatorname{H}}^1(S_{\\bar{k}}, \\Omega ^1_{S/\\bar{k}}) \\simeq {\\operatorname{H}}^1(S, \\Omega ^1_{S/k})\\otimes _k\\bar{k}$ and the cycle class map ${\\operatorname{cyc}}^\\mathrm {Hdg}$ transforms the intersection product on ${\\rm CH}^1(S_{\\bar{k}})/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ to the quadratic form $b_0$ on ${\\operatorname{H}}^1(S_{\\bar{k}}, \\Omega ^1_{S/\\bar{k}})$ , induced by cup product and the trace map.",
"Thus, our quadratic form $b$ on ${\\operatorname{H}}^1(S, \\Omega ^1_{S/k})$ is equivalent to the one gotten by twisting the $\\bar{k}$ -linear extension of the intersection product on ${\\rm CH}^1(S_{\\bar{k}})/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ by the natural Galois action.",
"Analogous comments hold for a “geometrically singular” variety, by which we mean a smooth projective $k$ -scheme $X$ of dimension $2n$ such that ${\\operatorname{H}}^n(X_{\\bar{k}},\\Omega ^n_{X_{\\bar{k}}/\\bar{k}})$ is spanned by cycle classes, where we replace ${\\rm CH}^1/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ with ${\\rm CH}^n/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ .",
"For example, one could take a K3 surface with Picard rank 20 over $\\bar{k}$ or a cubic fourfold $X$ with ${\\operatorname{H}}^2(X_{\\bar{k}},\\Omega ^2_{X_{\\bar{k}}/k})\\simeq \\bar{k}^{21}$ spanned by algebraic cycles.",
"Examples 8.11 As as simple example, take $S$ to be a quadric surface in ${\\mathbb {P}}^3_k$ defined by a degree two homogeneous form $q(X_0,\\ldots , X_3)$ ; we may assume that $q$ is a diagonal form, $q(X_0,\\ldots , X_3)=a_0X_0^2+\\sum _{i=1}^3a_iX_i^2\\\\=a_0(X_0+\\sqrt{-a_1/a_0}X_1)(X_0-\\sqrt{-a_1}X_1)\\\\+a_2(X_2-\\sqrt{-a_3/a_2}X_3)(X_2+\\sqrt{-a_3/a_2}X_3).$ This trivializes ${\\rm CH}^1(S)$ over $K:=k(\\sqrt{-a_0a_1}, \\sqrt{-a_2a_3})$ , namely ${\\rm CH}^1(S)={\\mathbb {Z}}\\ell _1\\oplus {\\mathbb {Z}}\\ell _2$ with $\\ell _1$ defined by $(X_0-\\sqrt{-a_1}X_1)=(X_2-\\sqrt{-a_3/a_2}X_3)=0$ and $\\ell _2$ defined by $(X_0-\\sqrt{-a_1}X_1)=(X_2+\\sqrt{-a_3/a_2}X_3)=0$ .",
"Embedding ${\\rm Gal}(K/k)\\subset {\\rm Gal}(k(\\sqrt{-a_0a_1})/k)\\times {\\rm Gal}(k(\\sqrt{-a_2a_3})/k)=\\langle \\sigma _1\\rangle \\times \\langle \\sigma _2\\rangle $ , the Galois action is given by $\\sigma _1(\\ell _1, \\ell _2)=\\sigma _2(\\ell _1, \\ell _2)= (\\ell _2, \\ell _1)$ .",
"A Galois-invariant basis is thus given by $((\\ell _1+\\ell _2), \\sqrt{a_0a_1a_2a_3}(\\ell _1-\\ell _2))$ , and the intersection form in this basis has matrix $\\begin{pmatrix}2&0\\\\0&-2a_0a_1a_2a_3\\end{pmatrix}.$ In other words, $\\chi (S/k)=\\langle 2\\rangle +\\langle -2a_0a_1a_2a_3\\rangle $ .",
"Suppose $S$ is the blowup of ${\\mathbb {P}}^2_k$ along a 0-dimensional closed subscheme $Z\\subset {\\mathbb {P}}^2_k$ , with $Z$ étale over $k$ .",
"Let $\\ell $ denote the class of a line in ${\\rm CH}^1({\\mathbb {P}}^2)$ .",
"Writing $Z_{\\bar{k}}=\\lbrace p_1,\\ldots , p_r\\rbrace $ , we have ${\\rm CH}^1(S_{\\bar{k}}) \\simeq {\\mathbb {Z}}\\cdot \\ell \\oplus (\\oplus _{i=1}^r{\\mathbb {Z}}\\cdot p_r)$ , with the evident Galois action and with intersection form the diagonal matrix $(1, -1,\\ldots , -1)$ .",
"It is then easy to show that the twisted quadratic form $\\chi (S/k)$ is $\\langle 1\\rangle -{\\rm Tr}_{Z/k}(\\langle 1\\rangle )$ .",
"These last two examples have been computed by different methods before: (1) is a special case of [26] and (2) is a special case of [26].",
"Here is a more interesting example.",
"Example 8.12 Let $\\pi :S\\rightarrow C$ be a a conic bundle over a smooth projective curve $C$ , all defined over $k$ ; we assume for simplicity that $k\\subset .",
"Let $ ZC$ be the degeneracy locus of $$: that is, $ Z$ is the reduced proper closed subscheme of $ C$ over which $$ is not smooth.",
"For each geometric point $ z$ of $ Z$, the fiber $ -1(z)$ is isomorphic to two distinct lines in $ P2$: $ -1(z)=zz'$.",
"There is a ``double section^{\\prime \\prime } $ DS$ with $ DC$ a finite degree two morphism, and with $ Dz=1=Dz'$ for all $ zZ(k)$.$ Over $\\bar{k}$ , the bundle $S$ is isomorphic to the blow-up of a ${\\mathbb {P}}^1$ -bundle $\\bar{S}_{\\bar{k}}\\rightarrow C_{\\bar{k}}$ along a finite set $Z^{\\prime }\\subset \\bar{S}_{\\bar{k}}$ with $Z^{\\prime }\\xrightarrow{} Z_{\\bar{k}}$ via $\\pi $ .",
"Suppose $Z_{\\bar{k}}=\\lbrace z_1,\\ldots z_r\\rbrace $ .",
"If we fix a closed point $c_0\\in C\\setminus Z$ of degree $d$ over $k$ , we have the following basis for ${\\rm CH}^1(S_{\\bar{k}})_{\\mathbb {Q}}/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ : $\\ell _{z_1}-\\ell _{z_1}^{\\prime },\\ldots , \\ell _{z_r}-\\ell _{z_r}^{\\prime }, D, \\pi ^{-1}(c_0).$ We have the finite degree two extension $p:\\tilde{Z}\\rightarrow Z$ , where for each $z\\in Z$ , $p^{-1}(z)$ corresponds to the pair of lines $\\ell _z, \\ell _z^{\\prime }$ .",
"Let $L:=k(\\lbrace z_1,\\ldots , z_r\\rbrace )\\supset k$ and let $G:={\\operatorname{Aut}}(L/k)$ .",
"Writing $k(\\tilde{Z})=k(Z)(\\sqrt{\\delta })$ for some $\\delta \\in {\\mathcal {O}}_Z^\\times $ , we have a basis of ${\\rm CH}^1(S_L)_{\\mathbb {Q}}/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ given by $v_1,\\ldots , v_r, D, \\pi ^{-1}(c_0),$ with $v_i:=\\sqrt{d}(\\ell _{z_i}-\\ell _{z_i}^{\\prime })$ .",
"The intersection form on $\\langle v_1,\\ldots , v_r\\rangle $ is the diagonal matrix $(-4\\delta (z_1),\\ldots , -4\\delta (z_r))$ , the subspaces $\\langle v_1,\\ldots , v_r\\rangle $ and $\\langle D, \\pi ^{-1}(c_0)\\rangle $ are perpendicular and $\\langle D, \\pi ^{-1}(c_0)\\rangle $ is hyperbolic.",
"Moreover, the automorphism group ${\\operatorname{Aut}}(L/k)$ acts on $\\langle v_1,\\ldots , v_r\\rangle $ just as it does on $\\langle z_1,\\ldots , z_r\\rangle $ .",
"From this it follows that the twisted intersection form $b$ is given by $b=H-{\\rm Tr}_{k(Z)/k}(\\langle \\delta \\rangle ),$ and hence $\\chi (S/k)=m\\cdot H-{\\rm Tr}_{k(Z)/k}(\\langle \\delta \\rangle )$ with $m=2-\\operatorname{dim}_k{\\operatorname{H}}^0(S,\\Omega ^1_{S/k})-\\operatorname{dim}_k{\\operatorname{H}}^1(S,{\\mathcal {O}}_X)=\\operatorname{dim}_{\\mathbb {Q}}{\\operatorname{H}}^0(S^{\\rm an},{\\mathbb {Q}})-\\operatorname{dim}_{\\mathbb {Q}}{\\operatorname{H}}^1(S^{\\rm an},{\\mathbb {Q}})+1,$ where $S^{\\rm an}$ is the complex manifold associated to $S_.$ As a particular example, we may take $S$ to be a cubic surface $V\\subset {\\mathbb {P}}^3_k$ with a line $\\ell $ .",
"Projection from $\\ell $ realizes $V$ as a conic bundle $\\pi :V\\rightarrow {\\mathbb {P}}^1_k$ , with degeneracy locus $Z\\subset {\\mathbb {P}}^1_k$ a reduced closed subscheme of degree 5 over $k$ .",
"The above implies that the symmetric bilinear form $b_V$ is given by $b_V=H-{\\rm Tr}_{Z/k}(\\langle \\delta \\rangle )$ and computes $\\chi (S/k) = 2H-{\\rm Tr}_{Z/k}(\\langle \\delta \\rangle )$ .",
"Remark 8.13 In [8], Bayer-Fluckiger and Serre consider the finite $k$ -scheme $W$ representing the 27 lines on a cubic surface $V$ and compute the trace form ${\\rm Tr}_{W/k}(\\langle 1\\rangle )$ in [8].",
"They identify their form $q_{6,V}$ with the trace form on ${\\operatorname{H}}^1(V,\\Omega _{V/k})$ and show that ${\\rm Tr}_{W/k}(\\langle 1\\rangle )=\\lambda ^2b_V + (\\langle -1\\rangle -\\langle 2\\rangle )b_V+ 7 -\\langle -2\\rangle .$ In this section, we describe how one gets projective pushforward maps in twisted ${\\mathcal {E}}$ -cohomology for ${\\mathcal {E}}$ an $\\operatorname{SL}$ -oriented motivic spectrum.",
"We rely on the six-functor formalism.",
"This is a bit different from the treatment of projective pushforward given by Ananyevskiy in [2]: in that treatment, one relies on the factorization of an arbitrary projective morphism $Y\\rightarrow X$ into a closed immersion $Y\\rightarrow X\\times {\\mathbb {P}}^N$ followed by a projection $X\\times {\\mathbb {P}}^N\\rightarrow X$ .",
"This factorization property will however reappear in our treatment when we discuss the uniqueness of the pushforward maps in §.",
"We continue to work over a noetherian separated base scheme $B$ of finite Krull dimension.",
"Lemma 4.1 Let $s:Y\\rightarrow X$ be a section of a smooth morphism $p:X\\rightarrow Y$ and let $\\eta :{\\operatorname{id}}\\rightarrow s_*s^*$ , $\\epsilon :s^*s_*\\rightarrow {\\operatorname{id}}$ be unit and counit of adjunction.",
"Then the composition ${\\operatorname{id}}_{{\\operatorname{SH}}(Y)}\\xrightarrow{} p_!\\circ s_!= p_!\\circ s_*\\xrightarrow{}s^*\\circ s_*\\xrightarrow{}{\\operatorname{id}}_{{\\operatorname{SH}}(Y)}$ is the identity.",
"Here the morphism $s^*:p_!\\rightarrow s^*$ is the one constructed in Remark REF .",
"The functor $s^*s_*$ is an equivalence [20].",
"As a general property of adjoint functors, $s_*\\epsilon s^*\\circ \\eta s_*s^*={\\operatorname{id}}_{s_*s^*}$ (see, e.g., [29]), hence $s_*\\epsilon s^*s_*\\circ \\eta s_*s^*s_*={\\operatorname{id}}_{s_*s^*s_*}$ and thus $s_*\\epsilon \\circ \\eta s_*={\\operatorname{id}}_{s_*}$ .",
"The result follows from the commutative diagram ${&{\\operatorname{id}}[d]^\\wr \\\\p_{X!",
"}\\circ s_*[d]_{p_{X!",
"}\\eta s_*}[dr]^{{\\operatorname{id}}}@/_60pt/[ddd]_{s^*}&p_{X!",
"}\\circ s_!",
"@{=}[l] @{=}[d]\\\\p_{X!",
"}\\circ s_*s^*s_*[r]^{p_{X!}",
"s_*\\epsilon }@{=}[d] &p_{X!",
"}\\circ s_*@{=}[d]\\\\p_{X!",
"}\\circ s_!\\circ s^*\\circ s_*[d]_\\wr [r]^{p_{X!}",
"s_!\\epsilon }&p_{X!",
"}\\circ s_!",
"[d]^\\wr \\\\s^*\\circ s_*[r]_\\epsilon &{\\operatorname{id}}}$ Remark 4.2 Let $s:Y\\rightarrow V$ be the zero-section of a vector bundle $p:V\\rightarrow Y$ .",
"Then the canonical isomorphism ${\\operatorname{id}}_{SH(Y)}\\simeq p_!s_*$ is equal to the composition ${\\operatorname{id}}_{{\\operatorname{SH}}(Y)}\\simeq \\Sigma ^{-V}\\Sigma ^V=\\Sigma ^{-V}p_\\#s_*\\simeq p_\\#\\Sigma ^{-p^*V}s_*\\simeq p_!s_*$ Indeed $\\Sigma ^{-V}\\Sigma ^V\\rightarrow {\\operatorname{id}}_{{\\operatorname{SH}}(Y)}$ is the counit of the adjunction $\\Sigma ^{-V}\\dashv \\Sigma ^V$ , corresponding to ${\\operatorname{id}}:\\Sigma ^V\\rightarrow \\Sigma ^V$ , $\\Sigma ^{-V}p_\\#s_*\\rightarrow {\\operatorname{id}}_{{\\operatorname{SH}}(Y)}$ is the counit of the adjunction $\\Sigma ^{-V}=s^!p^*\\dashv p_\\#s_*$ .",
"The functors $\\Sigma ^{-V}$ and $p_\\#$ are both left adjoints, so $\\Sigma ^{-V}p_\\#$ is left adjoint to $s_*$ and the counit of the adjunction $\\Sigma ^{-V}p_\\#\\dashv p^*\\Sigma ^V\\simeq \\Sigma ^{p^*V}p^*\\simeq p^!\\simeq s_*$ is the same as that of $\\Sigma ^{-V}\\dashv p_\\#s_*$ .",
"Composing with the isomorphism $p_!\\simeq p_\\#\\Sigma ^{-p^*V}\\simeq \\Sigma ^{-V}p_\\#$ , we see that the counit of the adjunction $p_!\\dashv s_*$ is induced from that of $\\Sigma ^{-V}p_\\#\\dashv s_*$ , and thus agrees with the canonical isomorphism $p_!s_*\\simeq {\\operatorname{id}}_{{\\operatorname{SH}}(Y)}$ .",
"Lemma 4.3 Let $({\\mathcal {E}}, {\\operatorname{th}}_{(-)})$ be an $\\operatorname{SL}$ -oriented motivic spectrum in ${\\operatorname{SH}}(B)$ .",
"Suppose given $X\\in {\\mathrm {Sm}}_B$ of dimension $d_X$ over $B$ , $i:Z\\rightarrow X$ a closed subset, and $p:L\\rightarrow X$ a line bundle.",
"Then the isomorphism $\\vartheta _{L-T_{X/B}}^{\\mathcal {E}}:\\Sigma ^{1-\\operatorname{det}(L-T_{X/B})}\\pi _X^*{\\mathcal {E}}\\rightarrow \\Sigma ^{r_{L-T_{X/B}}-L+T_{X/B}}\\pi _X^*{\\mathcal {E}}$ induces an isomorphism $\\rho _{X, Z, L}:{\\mathcal {E}}^{a,b}_Z(X; \\omega _{X/B}\\otimes L)\\simeq {\\mathcal {E}}^{a-2d_X,b-d_X}(X_Z/B_\\mathrm {B.M.",
"}; L),$ where the right-hand side is as defined in Remark REF .",
"We have $\\operatorname{det}(L-T_{X/B})=\\operatorname{det}^{-1}(T_{X/B})\\otimes L=\\omega _{X/B}\\otimes L$ and $r_{L-T_{X/B}}=1-d_X$ .",
"Moreover, we have canonical isomorphisms ${\\mathcal {E}}^{a,b}_Z(X; \\omega _{X/B}\\otimes L)\\simeq {\\rm Hom}_{{\\operatorname{SH}}(X)}(i_*(1_Z), S^{a,b}\\wedge \\Sigma ^{1-\\omega _{X/B}\\otimes L}\\pi _X^*{\\mathcal {E}})$ and ${\\mathcal {E}}^{a-2d_X,b-d_X}(X_Z/B_\\mathrm {B.M.",
"}; L)\\simeq {\\rm Hom}_{{\\operatorname{SH}}(X)}(i_*(1_Z), S^{a,b}\\wedge \\Sigma ^{(1-d_X)+T_{X/B}-L}\\pi _X^*{\\mathcal {E}}).$ Finally, $\\vartheta _{L-T_{X/B}}^{\\mathcal {E}}$ induces the isomorphism $S^{a,b}\\wedge \\vartheta _{L-T_{X/B}}^{\\mathcal {E}}:S^{a,b}\\wedge \\Sigma ^{1-\\omega _{X/B}\\otimes L}\\pi _X^*{\\mathcal {E}}\\rightarrow S^{a,b}\\wedge \\Sigma ^{(1-d_X)+T_{X/B}-L}\\pi _X^*{\\mathcal {E}}$ which completes the proof.",
"Using the isomorphisms $\\rho _{X, Z,L}$ , we make the following definition.",
"Definition 4.4 Let $({\\mathcal {E}}, {\\operatorname{th}}_{(-)})$ be an $\\operatorname{SL}$ -oriented motivic spectrum in ${\\operatorname{SH}}(B)$ , let $f:Y\\rightarrow X$ be a proper morphism of relative dimension $d$ in ${\\mathrm {Sm}}_B$ , let $L\\rightarrow X$ be a line bundle, and let $Z\\subset X$ be a closed subset.",
"Define $f_*:{\\mathcal {E}}^{a,b}_{f^{-1}(Z)}(Y, \\omega _{Y/B}\\otimes f^*L)\\rightarrow {\\mathcal {E}}^{a-2d,b-d}_Z(X, \\omega _{X/B}\\otimes L)$ to be the unique map making the diagram ${{\\mathcal {E}}^{a-2d_Y,b-d_Y}(Y_{f^{-1}(Z)}/B_\\mathrm {B.M.",
"}; f^*L)[r]^-{(f^*)^*}&{\\mathcal {E}}^{a-2d_Y,b-d_Y}(X_Z/B_\\mathrm {B.M.",
"}; L)\\\\{\\mathcal {E}}^{a,b}_{f^{-1}(Z)}(Y, \\omega _{Y/B}\\otimes f^*L)[u]^{\\rho _{Y, f^{-1}Z, f^*L}}[r]_{f_*}&{\\mathcal {E}}^{a-2d,b-d}_Z(X, \\omega _{X/B}\\otimes L)[u]_{\\rho _{X, Z, L}}}$ commute.",
"Let $p:V\\rightarrow Y$ be a rank $r$ vector bundle on some $Y\\in {\\mathrm {Sm}}_B$ , with 0-section $s:Y\\rightarrow V$ .",
"Letting $L=\\operatorname{det}V$ , the exact sequence $0\\rightarrow p^*V\\rightarrow T_{V/B}\\xrightarrow{}p^*T_{Y/B}\\rightarrow 0$ gives the canonical isomorphism $\\omega _{V/B}\\simeq p^*(\\operatorname{det}^{-1} V\\otimes \\omega _{Y/B})$ , or $p^*\\operatorname{det}^{-1} V\\simeq \\omega _{V/B}\\otimes \\omega ^{-1}_{Y/B}$ .",
"Letting $({\\mathcal {E}}, {\\operatorname{th}})$ be an $\\operatorname{SL}$ -oriented motivic spectrum, we have the pushforward map $s_*:{\\mathcal {E}}^{a,b}(Y)\\rightarrow {\\mathcal {E}}^{ a+2r, b+r}(V, p^*\\operatorname{det}^{-1} V),$ and the version with supports, $s_*:{\\mathcal {E}}^{a,b}(Y)={\\mathcal {E}}^{a,b}_Y(Y)\\rightarrow {\\mathcal {E}}^{ a+2r, b+r}_{0_V}(V, p^*\\operatorname{det}^{-1} V).$ Lemma 4.5 Let $1_Y^{\\mathcal {E}}\\in {\\mathcal {E}}^{0,0}(Y)$ be the unit $\\pi _{Y/B}^*(u)$ .",
"Then $s_*(1_Y^{\\mathcal {E}})={\\operatorname{th}}_V \\in {\\mathcal {E}}^{2r, r}_{0_V}(V, p^*\\operatorname{det}^{-1} V).$ As consequence, $s_*(1_Y^{\\mathcal {E}})$ in ${\\mathcal {E}}^{2r, r}(V, p^*\\operatorname{det}^{-1} V)$ is the image of ${\\operatorname{th}}_V$ under the “forget supports” map ${\\mathcal {E}}^{2r, r}_{0_V}(V, p^*\\operatorname{det}^{-1} V)\\rightarrow {\\mathcal {E}}^{2r, r}(V, p^*\\operatorname{det}^{-1} V).$ The exact sequence $0\\rightarrow p^*V\\rightarrow T_{V/B}\\xrightarrow{}p^*T_{Y/B}\\rightarrow 0$ gives us the isomorphism $\\omega _{V/B}^{-1}\\otimes \\operatorname{det}^{-1} V \\simeq p^*\\omega _{Y/B}^{-1}.$ Keeping this in mind, we have the following commutative diagram defining $s_*$ : ${{\\mathcal {E}}^{-2d_Y,-d_Y}(Y/B_\\mathrm {B.M.",
"}; \\omega _{Y/B}^{-1})[r]^-{(s^*)^*}&{\\mathcal {E}}^{-2d_Y,-d_Y}_{0_V}(V/B_\\mathrm {B.M.",
"}, p^*\\omega _{Y/B}^{-1})\\\\{\\mathcal {E}}^{0,0}_Y(Y)[u]^{\\rho _{Y,Y, s^*p^*\\omega _{Y/B}^{-1}}}[r]_{s_*}[rd]_{s_*}&{\\mathcal {E}}^{2r,r}_{0_V}(V, \\operatorname{det}^{-1} V)[u]_{\\rho _{V, 0_V, p^*\\omega _{Y/B}^{-1}}}[d]^{\\vbox {\\tiny forget\\\\supports}}\\\\&{\\mathcal {E}}^{2r,r}(V,\\operatorname{det}^{-1} V )}$ Here the lower $s_*$ is the one we are considering and the upper $s_*$ is the map with supports.",
"Thus, we need to show that the upper $s_*$ satisfies $s_*(1_Y^{\\mathcal {E}})={\\operatorname{th}}_V$ .",
"We will be using the isomorphisms $\\\\&{\\mathcal {E}}^{-2d_Y,-d_Y}(Y/B_\\mathrm {B.M.",
"}; \\omega _{Y/B}^{-1})\\simeq {\\rm Hom}_{{\\operatorname{SH}}(Y)}(1_Y, \\Sigma ^{1-d_Y+T_{Y/B}-\\omega _{Y/B}^{-1}}\\pi _Y^*{\\mathcal {E}}),\\\\&{\\mathcal {E}}^{-2d_Y,-d_Y}_{0_V}(V/B_\\mathrm {B.M.",
"}, p^*\\omega _{Y/B}^{-1})\\simeq {\\rm Hom}_{{\\operatorname{SH}}(V)}(s_*(1_Y), \\Sigma ^{1-d_Y+T_{V/B}- p^*\\omega _{Y/B}^{-1}}\\pi _V^*{\\mathcal {E}}),\\\\&{\\mathcal {E}}^{2r,r}_{0_V}(V,\\operatorname{det}^{-1} V )\\simeq {\\rm Hom}_{{\\operatorname{SH}}(V)}(s_*(1_V), \\Sigma ^{r+1-\\operatorname{det}^{-1}V}\\pi _V^*{\\mathcal {E}}).$ By Lemma REF , the composition $\\pi _{Y!",
"}\\xrightarrow{}\\pi _{V!",
"}\\circ s_!\\xrightarrow{}\\pi _{V!",
"}\\circ s_*\\xrightarrow{}\\pi _{Y!",
"}$ is the identity.",
"Evaluating at $1_Y$ gives the commutative diagram ${Y_\\mathrm {B.M.",
"}[r]^-{\\phi }_--\\sim @{=}[d]&\\pi _{V!",
"}(s_*(1_Y))[dl]^{s^*}@{=}[r]&V_{0_V}/B_{\\mathrm {B.M.",
"}}\\\\Y_\\mathrm {B.M.",
"}}$ and the isomorphisms $\\phi $ induces the isomorphism $\\phi ^*:{\\mathcal {E}}^{-2d_Y, -d_Y}_{0_V}(V/B_\\mathrm {B.M.",
"}, p^*\\omega _{Y/B}^{-1})\\rightarrow {\\mathcal {E}}^{-2d_Y, -d_Y}(Y_\\mathrm {B.M.",
"}, \\omega _{Y/B}^{-1})$ The isomorphism $\\rho _{V, 0_V, p^*\\omega _{Y/B}^{-1}}$ is the map induced on ${\\rm Hom}_{{\\operatorname{SH}}(V)}(s_*(1_Y),-)$ by the isomorphism $\\vartheta _{p^*\\omega ^{-1}_{Y/B}- T_{V/B}}:\\Sigma ^{1-\\operatorname{det}^{-1} V}\\pi _V^*{\\mathcal {E}}\\rightarrow \\Sigma ^{1-d_V+T_{V/B}-p^*\\omega ^{-1}_Y/B}\\pi _V^*{\\mathcal {E}}.$ We have the isomorphisms $\\Sigma ^{r-V}\\vartheta _{-V}: \\Sigma ^{r+1-V-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}}\\rightarrow \\pi _Y^*{\\mathcal {E}}$ $\\Sigma ^{r-V}\\vartheta _{\\omega ^{-1}_{Y/B}-T_{Y/B}-V}:\\Sigma ^{r+1-V-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}}\\rightarrow \\Sigma ^{r+1-d_Y+T_{Y/B}-\\omega _{Y/B}}\\pi _Y^*{\\mathcal {E}}$ and $\\vartheta _{\\omega _{Y/B}-T_{T/B}}:\\pi _Y^*{\\mathcal {E}}\\rightarrow \\Sigma ^{1-d_Y+T_{Y/B}-\\omega _{Y/B}}\\pi _Y^*{\\mathcal {E}}.$ The first one induces an isomorphism $\\rho _{-V}:{\\rm Hom}_{{\\operatorname{SH}}(Y)}(1_Y, \\Sigma ^{r+1-V-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}})\\rightarrow {\\mathcal {E}}^{0,0}(Y)\\ $ the second an isomorphism $\\rho _{T_{Y/B}+V-\\omega _{Y/B}^{-1}}:{\\rm Hom}_{{\\operatorname{SH}}(Y)}(1_Y, \\Sigma ^{r+1-V-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}})\\rightarrow {\\mathcal {E}}^{-2d_Y, -d_Y}(Y/B_\\mathrm {B.M.",
"}, \\omega ^{-1}_{Y/B})$ while the third induces the isomorphism $\\rho _{Y,Y, \\omega _{Y/B}^{-1}}:{\\mathcal {E}}^{0,0}(Y)\\rightarrow {\\mathcal {E}}^{-2d_TY, -d_Y}(Y/B_\\mathrm {B.M.",
"}, \\omega ^{-1}_{Y/B})$ Altogether these maps and isomorphisms gives the diagram of isomorphisms ${5pt}{{\\mathcal {E}}^{-2d_Y, -d_Y}_{0_V}(V/B_\\mathrm {B.M.",
"}, p^*\\omega _{Y/B}^{-1})[r]^{\\phi ^*}& {\\mathcal {E}}^{-2d_Y, -d_Y}(Y/B_\\mathrm {B.M.",
"}, \\omega ^{-1}_{Y/B})\\\\&&{\\mathcal {E}}^{0,0}(Y)[ul]_-{\\rho _{Y,Y,\\omega _{Y/B}^{-1}}}\\\\{\\mathcal {E}}_{0_V}^{2r,r}(V, \\operatorname{det}^{-1}V)[uu]^{\\rho _{V, 0_V, \\operatorname{det}^{-1}V}}[r]_-\\psi &{\\rm Hom}_{{\\operatorname{SH}}(Y)}(1_Y, \\Sigma ^{r+1-V-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}})[uu]^{\\rho _{T_{Y/B}+V-\\omega _{Y/B}}}[ur]_-{\\rho _{-V}}}$ The triangle commutes by the functoriality of $\\vartheta _{-}$ , as expressed by Remark REF .",
"To see that square commutes, we have the diagram ${10pt}{\\Sigma ^{1-d+T_V-p^*\\omega _{Y/B}^{-1}}\\pi _V^*{\\mathcal {E}}[r]^-\\sim &\\Sigma ^{1-d+p^*(T_Y+V-\\omega _{Y/B}^{-1})}\\pi _V^*{\\mathcal {E}}[r]^-\\sim &p^*\\Sigma ^{1-d+T_Y+V-\\omega _{Y/B}^{-1}}\\pi _Y^*{\\mathcal {E}}\\\\\\Sigma ^{r+1-p^*\\operatorname{det}^{-1}V}\\pi _V^*{\\mathcal {E}}@{=}[r][u]_{\\Sigma ^r_{\\vartheta _{p^*\\omega _{Y/B}^{-1}-T_{V/B}}}&\\Sigma ^{r+1-p^*\\operatorname{det}^{-1}V}\\pi _V^*{\\mathcal {E}}[r]_\\sim [u]_{\\Sigma ^r_{\\vartheta _{p^*(\\omega _{Y/B}^{-1}-T_{Y/B}-V)}}&p^*\\Sigma ^{r+1-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}}[u]_{p^*\\Sigma ^r_{\\vartheta _{\\omega _{Y/B}^{-1}-T_{Y/B}-V}}}The first square commutes using the exact sequence 0\\rightarrow p^*V\\rightarrow T_{V/B}\\rightarrow p^*T_{Y/B}\\rightarrow 0 and the second by the naturality of \\vartheta _{-}.",
"Applying the adjunction p_\\#\\dashv p^*, the identity p_\\#s^*=\\Sigma ^V and applying \\Sigma ^{-V} to yield the isomorphism [\\Sigma ^Vx, y]_{{\\operatorname{SH}}(Y)}\\simeq [x,\\Sigma ^Vy]_{{\\operatorname{SH}}(Y)}, applying {\\rm Hom}_{{\\operatorname{SH}}(V)}(s_*(1_Y), -) to the last map gives the commutative square{{\\rm Hom}_{{\\operatorname{SH}}(V)}(s_*(1_Y), p^*\\Sigma ^{1-d+T_Y+V-\\omega _{Y/B}^{-1}}\\pi _Y^*{\\mathcal {E}})[dr]^\\sim \\\\&\\hspace{-50.0pt}{\\rm Hom}_{{\\operatorname{SH}}(Y)}(1_Y, \\Sigma ^{1-d+T_Y-\\omega _{Y/B}^{-1}}\\pi _Y^*{\\mathcal {E}})\\\\{\\rm Hom}_{{\\operatorname{SH}}(V)}(s_*(1_Y), p^*\\Sigma ^{r+1-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}})[uu]^{p^*\\Sigma ^r_{{\\mathbb {P}}^1}\\vartheta _{\\omega _{Y/B}^{-1}-T_{Y/B}-V*}}[rd]_\\sim \\\\&\\hspace{-50.0pt}{\\rm Hom}_{{\\operatorname{SH}}(Y)}(1_Y,\\Sigma ^{r+1-V-\\operatorname{det}^{-1}V}\\pi _Y^*{\\mathcal {E}})[uu]_{\\Sigma ^{r-V}\\vartheta _{\\omega _{Y/B}^{-1}-T_{Y/B}-V*}}}Applying {\\rm Hom}_{{\\operatorname{SH}}(V)}(s_*(1_Y), -) to the first diagram, putting these two diagrams together and using the isomorphisms (\\ref {align:Isos}) and Remark~\\ref {rem:Pushforward} gives the commutativity of the square in (\\ref {eqn:CommDiagr}).",
"}It follows from the commutativity of (\\ref {eqn:Compat2}) that \\phi ^*\\circ (s^*)^*\\circ \\rho _{Y,Y,\\omega _{Y/B}^{-1}}=\\rho _{Y,Y,\\omega _{Y/B}^{-1}}.",
"The commutativity of (\\ref {eqn:Compat1}) and (\\ref {eqn:CommDiagr}) then shows that \\rho _{-V}\\circ \\psi \\circ s_*={\\operatorname{id}}.",
"By Remark~\\ref {rem:CanonicalThom}, and Remark~\\ref {rem:MultThom} the map \\rho _{-V}\\circ \\psi is the inverse of the canonical Thom isomorphism \\vartheta _V:{\\mathcal {E}}^{0,0}(Y)\\rightarrow {\\mathcal {E}}^{2r, r}_{0_V}(V, \\operatorname{det}^{-1}V).",
"Thus s_*=\\vartheta _V so s_*(1^{\\mathcal {E}}_Y)={\\operatorname{th}}_V.",
"}}\\begin{remark}If we have a \\operatorname{GL}-orientation on {\\mathcal {E}}, we have functorial pushforward mapsf_*:{\\mathcal {E}}^{a,b}_W(Y)\\rightarrow {\\mathcal {E}}^{a-2d, b-d}_Z(X)for f:Y\\rightarrow X a projective morphism in {\\mathrm {Sm}}_B, of relative dimension d, with W\\subset Y, Z\\subset X closed subsets with f(W)\\subset Z.",
"All the results of this section hold in the oriented context after deleting the twist by line bundles.",
"This follows from Remark~\\ref {rem:Oriented}.\\end{remark}$ Motivic Gauß-Bonnet Definition 5.1 Let $p:V\\rightarrow X$ be a rank $r$ vector bundle on some $X\\in {\\mathrm {Sm}}_B$ , and let ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ be an $\\operatorname{SL}$ -oriented motivic ring spectrum.",
"The Euler class $e^{\\mathcal {E}}(V)\\in {\\mathcal {E}}^{2r,r}(X,\\operatorname{det}^{-1}V)$ is defined as $e^{\\mathcal {E}}(V):=s^*s_*(1^{\\mathcal {E}}_X);\\quad 1^{\\mathcal {E}}_X\\in {\\mathcal {E}}^{0,0}(X)\\text{ the unit.",
"}$ Remark 5.2 By Lemma REF , $e^{\\mathcal {E}}(V):=s^*s_*(1^{\\mathcal {E}}_X)=\\bar{s}^*{\\operatorname{th}}_V$ , where $\\bar{s}:X\\rightarrow {\\operatorname{Th}}_X(V)$ is the map induced by $s$ .",
"Theorem 5.3 (Motivic Gauß-Bonnet) Let ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ be an $\\operatorname{SL}$ -oriented motivic ring spectrum, $\\pi _X:X\\rightarrow B$ a smooth and projective $B$ -scheme, let $u_{\\mathcal {E}}:1_B\\rightarrow {\\mathcal {E}}$ be the unit map.",
"Then $\\pi _{X/B*}(e^{\\mathcal {E}}(T_{X/B}))=u_{{\\mathcal {E}}*}(\\chi (X/B))\\in {\\mathcal {E}}^{0,0}(B).$ We have the canonical Thom isomorphism $\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}: {\\mathcal {E}}^{a,b}(X;\\omega _{X/B})\\rightarrow {\\mathcal {E}}^{a-2\\operatorname{dim}X, b-\\operatorname{dim}X}({\\operatorname{Th}}(-T_{X/B})).$ By Lemma REF , it suffices to show that the map $\\beta _{X/B}^*:{\\mathcal {E}}^{0,0}(X)\\rightarrow {\\mathcal {E}}^{0,0}({\\operatorname{Th}}(-T_{X/B}))$ sends $1^{\\mathcal {E}}_X$ to $\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}(e^{\\mathcal {E}}(T_{X/B}))$ ; by Remark REF , this is the same as $\\vartheta _{-T_{X/B}}(\\bar{s}^*{\\operatorname{th}}_{T_{X/B}})$ , where $\\bar{s}:X_+\\rightarrow {\\operatorname{Th}}(T_{X/B})$ is the map induced by the zero-section $s:X\\rightarrow T_{X/B}$ .",
"We use our description of $\\beta _{X/B}$ as $\\pi _{X\\#}$ applied to the composition (REF ).",
"Applying ${\\rm Hom}_{{\\operatorname{SH}}(X)}(-,\\pi _X^*{\\mathcal {E}})$ to $\\beta _{X/B}$ and using the adjunction ${\\rm Hom}_{{\\operatorname{SH}}(B)}(\\pi _{X\\#}(-), {\\mathcal {E}})\\simeq {\\rm Hom}_{{\\operatorname{SH}}(X)}(-,\\pi _X^*{\\mathcal {E}})$ , $ \\beta _{X/B}^*$ is given by the composition ${\\mathcal {E}}^{0,0}(X){[r]^a_\\sim &} {\\rm Hom}_{{\\operatorname{SH}}(X)}(1_X, \\pi _X^*{\\mathcal {E}}){[r]^b_\\sim &}{\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{-T_{X/B}}\\circ \\Sigma ^{T_{X/B}}(1_X), \\pi _X^*{\\mathcal {E}})\\\\{[r]^c_\\sim &}{\\rm Hom}_{{\\operatorname{SH}}(X)}({\\operatorname{Th}}_X(T_{X/B})), \\Sigma ^{T_{X/B}}\\pi _X^*{\\mathcal {E}})\\\\\\xrightarrow{}{\\rm Hom}_{{\\operatorname{SH}}(X)}(1_X, \\Sigma ^{T_{X/B}}\\pi _X^*{\\mathcal {E}})\\simeq {\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{-T_{X/B}}({\\operatorname{Th}}_X(T_{X/B})),\\pi _X^*{\\mathcal {E}})$ where the isomorphisms $a,b, c$ are the canonical ones.",
"The functoriality of the canonical Thom isomorphisms gives us the commutative diagram ${{\\mathcal {E}}^{0,0}(X)[r]^-{\\vartheta _{T_{X/B}}}[d]^a_\\wr &{\\mathcal {E}}^{2d_X, d_X}({\\operatorname{Th}}(T_{X/B}), \\omega _{X/B})[dd]^{\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}}\\\\{\\rm Hom}_{{\\operatorname{SH}}(X)}(1_X, \\pi _X^*{\\mathcal {E}})[d]^b_\\wr &\\\\{\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{-T_{X/B}}\\Sigma ^{T_{X/B}}(1_X),\\pi _X^*{\\mathcal {E}})[r]_-c^-\\sim &{\\rm Hom}_{{\\operatorname{SH}}(X)}({\\operatorname{Th}}_X(T_{X/B}),\\Sigma ^{T_{X/B}}\\pi _X^*{\\mathcal {E}})}$ Thus $(c\\circ b\\circ a)(1^{\\mathcal {E}}_X)= \\vartheta ^{\\mathcal {E}}_{-T_{X/B}}({\\operatorname{th}}_{T_{X/B}}).$ Applying $\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}$ as above gives us the commutative diagram ${40pt}{{\\mathcal {E}}^{2d_X, d_X}({\\operatorname{Th}}(T_{X/B}), \\omega _{X/B}) [r]^-{\\bar{s}^*}[d]_{\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}}&{\\mathcal {E}}^{2d_X, d_X}(X, \\omega _{X/B})[d]^{\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}}\\\\{\\rm Hom}_{{\\operatorname{SH}}(X)}({\\operatorname{Th}}_X(T_{X/B}), \\Sigma ^{T_{X/B}}\\pi _X^*{\\mathcal {E}})[r]_-{\\bar{s}^*}[d]_\\wr &{\\rm Hom}_{{\\operatorname{SH}}(X)}(1_X, \\Sigma ^{T_{X/B}}\\pi _X^*{\\mathcal {E}})[d]^\\wr \\\\{\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{-T_{X/B}}{\\operatorname{Th}}_X(T_{X/B}), \\pi _X^*{\\mathcal {E}})[r]_-{\\Sigma ^{-T_{X/B}}(\\bar{s}^*)}&{\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{-T_{X/B}}(1_X), \\pi _X^*{\\mathcal {E}})}$ and thus $\\beta _{X/B}^*(1^{\\mathcal {E}}_X)=\\vartheta _{-T_{X/B}}(\\bar{s}^*({\\operatorname{th}}_{T_{X/B}}))=\\vartheta _{-T_{X/B}}(e^{\\mathcal {E}}(T_{X/B})),$ as desired.",
"$\\operatorname{SL}$ -oriented cohomology theories Our ultimate goal is to apply the Gauß-Bonnet theorem of § when projective pushforwards are defined on a representable cohomology theory in some concrete manner, not necessarily relying on the six-functor formalism.",
"For this, we need a suitable axiomatization for such theories; we use a modification of the axioms of Panin-Smirnov [38], [39].",
"As before, our base-scheme $B$ is a noetherian, separated scheme of finite Krull dimension.",
"Definition 6.1 We let ${\\mathrm {Sm}\\text{-}\\mathrm {L}}_B$ denote the category of triples $(X, Z, L)$ with $X$ in ${\\mathrm {Sm}}_B$ , $Z\\subset X$ a closed subset and $L\\rightarrow X$ a line bundle.",
"A morphism $(f,\\tilde{f}):(X, Z, L)\\rightarrow (Y, W, M)$ is a morphism $f:X\\rightarrow Y$ with $Z\\supset f^{-1}(W)$ , together with an isomorphism of line bundles $\\tilde{f}:L\\rightarrow f^*M$ .",
"We let ${\\mathrm {Sm}\\text{-}\\mathrm {L}}^{\\text{pr}}_B$ denote the category with the same objects as ${\\mathrm {Sm}\\text{-}\\mathrm {L}}_B$ , but with morphisms $(f; \\tilde{f}):(X,Z, L)\\rightarrow (Y, W, M)$ a proper morphism $f:X\\rightarrow Y$ in ${\\mathrm {Sm}}_B$ , with $f(Z)\\subset W$ , and $\\tilde{f}:L\\rightarrow f^*M$ an isomorphism of line bundles.",
"Definition 6.2 An $\\operatorname{SL}$ -oriented cohomology theory on ${\\mathrm {Sm}}_B$ consists of the following data: A functor $H^{*,*}:{\\mathrm {Sm}\\text{-}\\mathrm {L}}_B^{\\text{\\rm op}}\\rightarrow \\mathrm {BiGr}{\\mathrm {Ab}}$ , $(X, Z, L)\\mapsto H^{*,*}_Z(X;L)$ ; we often write $f^*$ for $H^{*,*}(f,\\tilde{f})$ .",
"A functor $H_{*,*}:{\\mathrm {Sm}\\text{-}\\mathrm {L}}^{\\text{pr}}_B\\rightarrow {\\operatorname{\\rm Gr}}{\\mathrm {Ab}}$ , $(X, Z, L)\\mapsto H_{*,*}^Z(X,L)$ ; we often write $f_*$ for $H_{*,*}(f,\\tilde{f})$ .",
"Natural isomorphisms, for $X$ of dimension $d_X$ $H^{2d_X-n, d_X-m}_Z(X,\\omega _{X/B}\\otimes L)\\xrightarrow{} H_{n,m}^Z(X, L).$ An element $1\\in H^{0,0}_B(B;{\\mathcal {O}}_B)$ .",
"For $x:=(X, Z, L), y:=(Y, W, M)$ in ${\\mathrm {Sm}\\text{-}\\mathrm {L}}_B$ , a bigraded cup product map $\\cup _{x,y}:H^{*,*}_Z(X,L)\\otimes H^{*,*}_W(Y,M)\\rightarrow H^{*,*}_{Z\\times W}(X\\times _BY, p_1^*L\\otimes p_2^*M)$ For $Z\\subset W$ closed subsets of an $X\\in {\\mathrm {Sm}}_B$ a bigraded boundary map $\\delta _{X, W, Z}^{*,*}:H^{*, *}_{Z\\setminus W}(X\\setminus W;j_W^*L)\\rightarrow H^{*+1,*}_W(X, L)$ We write $H^{*,*}(X,L)$ for $H^{*,*}_X(X,L)$ and $H^{*,*}_Z(X)$ for $H^{*,*}_Z(X,{\\mathcal {O}}_X)$ ; we use the analogous notation for $H_{*,*}$ .",
"We write $\\cup $ for $\\cup _{x,y}$ and $\\delta $ for $\\delta _{X,Z,L}$ when the context makes the meaning clear.",
"For $f:Y\\rightarrow X$ a proper map of relative dimension $d$ in ${\\mathrm {Sm}}_B$ , with $Z\\subset X$ , $W\\subset Y$ closed subsets with $f(W)\\subset Z$ and $L\\rightarrow X$ a line bundle, combining D2 and D3 gives us pushforward maps $f_*:H^{*, *}_W(Y, \\omega _{Y/B}\\otimes f^*L)\\rightarrow H^{*-2d, *-d}_Z(X, \\omega _{X/B}\\otimes L)$ defined as the composition $H^{*, *}_W(Y, \\omega _{Y/B}\\otimes f^*L)\\xrightarrow{}H_{2d_Y-*, d_Y-*}^W(Y, f^*L)\\\\\\xrightarrow{}H_{2d_Y-*, d_Y-*}^Z(X, L)\\xrightarrow{}H^{*-2d, *-d}_Z(X, \\omega _{X/B}\\otimes L).$ These data are required to satisfy the following axioms: $H^{*,*}$ and $H_{*,*}$ are additive: $H^{*,*}$ transforms disjoint unions to products and $H_{*,*}$ transforms disjoint unions to coproducts; in particular, $H^{*,*}_Z(\\emptyset ,L)=0$ and $H_{*,*}^Z(\\emptyset , L)=0$ .",
"Let ${Y^{\\prime }[d]_{f^{\\prime }}[r]^-{g^{\\prime }}&Y[d]^f\\\\X^{\\prime }[r]_-g&X}$ be a cartesian diagram in $\\mathrm {Sch}_B$ , with $X, Y, X^{\\prime }, Y^{\\prime }$ in ${\\mathrm {Sm}}_B$ (sometimes called a transverse cartesian diagram in ${\\mathrm {Sm}}_B$ ) and with $f, f^{\\prime }$ proper of relative dimension $d$ .",
"This gives us the isomorphism $f^{\\prime *}\\omega _{X^{\\prime }/X}\\simeq \\omega _{Y^{\\prime }/Y}.$ Let $Z\\subset X$ be a closed subset, let $W\\subset Y$ be a closed subset with $f(W)\\subset Z$ , let $Z^{\\prime }=g^{-1}(Z)$ , $W^{\\prime }=g^{\\prime -1}(W)$ .",
"Let $L\\rightarrow X$ be a line bundle on $X$ and let $L^{\\prime }=g^{\\prime *}(L)$ .",
"Then the diagram ${H^{*, *}_{W^{\\prime }}(Y^{\\prime }, \\omega _{Y^{\\prime }/B}\\otimes \\omega _{Y^{\\prime }/Y}^{-1}\\otimes g^{\\prime *} L^{\\prime })[d]^{ f^{\\prime }_*}&[l]_-{g^{\\prime *}}H^{*, *}_W(Y, \\omega _{Y/B}\\otimes f^*L)[d]^{ f_*}\\\\H^{*-2d, *-d}_{Z^{\\prime }}(X^{\\prime }, \\omega _{X^{\\prime }/B}\\otimes \\omega _{X^{\\prime }/X}^{-1}\\otimes L^{\\prime })&[l]^-{g^*}H^{*-2d, *-d}_Z(X, \\omega _{X/B}\\otimes L)}$ commutes.",
"For $Z\\subset W$ closed subsets of an $X\\in {\\mathrm {Sm}}_B$ , let $U=X\\setminus Z$ with inclusion $j:U\\rightarrow X$ .",
"For $L\\rightarrow X$ a line bundle, this gives us the morphisms $({\\operatorname{id}}, {\\operatorname{id}}):(X, W, L)\\rightarrow (X, Z, L)$ and $(j, {\\operatorname{id}}):(U, W\\setminus Z, j^*L)\\rightarrow (X, W, L)$ .",
"Then the sequence $\\ldots \\xrightarrow{}H^{*, *}_Z(X, L)\\rightarrow H^{*,*}_W(X, L)\\\\\\xrightarrow{} H^{*,*}_{W\\setminus Z}(U, j^*L)\\xrightarrow{}H^{*+1, *}_Z(X, L)\\rightarrow \\ldots $ is exact.",
"Moreover, the maps $\\delta _{Z, W, X}$ are natural with respect to the pullback maps $g^*$ and the proper pushforward maps $f_*$ .",
"Let $i:Y\\rightarrow X$ be a closed immersion in ${\\mathrm {Sm}}_B$ , let $W\\subset Y$ be a closed subset, $L\\rightarrow X$ a line bundle.",
"Let $Z=i(W)$ , giving the morphism $(i, {\\operatorname{id}}):(Y, W, i^*L)\\rightarrow (X, Z, L)$ in ${\\mathrm {Sm}\\text{-}\\mathrm {L}}^{\\text{pr}}_B$ .",
"Then $i_*:H_{*,*}^W(Y, i^*L)\\rightarrow H_{*,*}^Z(X, L)$ is an isomorphism.",
"The cup products $\\cup $ of D4 are associative with unit 1.",
"The maps $f^*$ and $f_*$ are compatible with cup products: $(f\\times g)^*(\\alpha \\cup _{x,y}\\beta )=f^*(\\alpha )\\cup _{x,y} g^*(\\beta )$ .",
"Moreover, using the isomorphisms of D3, the cup products induce products $\\cup ^{x,y}$ on $H_{*,*}$ and one has $(f\\times g)_*(\\alpha \\cup ^{x,y}\\beta )=f_*(\\alpha )\\cup ^{x,y} g_*(\\beta )$ .",
"Finally, the boundary maps $\\delta _{Z, W, X}$ are module morphism: retaining the notation of D4, for $\\alpha \\in H^{*, *}_{Z\\setminus W}(X\\setminus W;j_W^*L)$ and $\\beta \\in H^{*, *}_{T}(Y, M)$ , we have $\\delta _{X\\times Y, Z\\times T, W\\times T}(\\alpha \\cup \\beta )=\\delta _{X, Z, W}(\\alpha )\\cup \\beta .$ Let $i:Y\\rightarrow X$ be a closed immersion in ${\\mathrm {Sm}}_B$ of codimension $c$ , $\\pi _Y:Y\\rightarrow B$ the structure map.",
"Let $1^H_Y\\in H^{0,0}(Y)$ be the element $\\pi _Y^*(1)$ .",
"Then $\\vartheta (i):=\\alpha _{X, Y}(i_*(1^H_Y))\\in H^{2c, c}_Y(X, \\operatorname{det}^{-1} N_i)$ is central, that is, for each $(U, T, M)\\in {\\mathrm {Sm}\\text{-}\\mathrm {L}}_B$ , and each $\\beta \\in H^{*,*}_T(U, M)$ , we have $\\tau ^*(\\beta \\cup \\vartheta (i))=\\vartheta (i)\\cup \\beta $ where $\\tau :X\\times _BU\\rightarrow U\\times _BX$ is the symmetry isomorphism.",
"Let $(f,{\\operatorname{id}}):(Y, W, f^*L)\\rightarrow (X, Z, L)$ be a morphism in ${\\mathrm {Sm}\\text{-}\\mathrm {L}}_B$ .",
"Suppose that the induced map $f:Y_W/B\\rightarrow X_Z/B$ is an isomorphism in ${\\operatorname{SH}}(B)$ .",
"Then $f^*:H^{*,*}_Z(X, L)\\rightarrow H^{*,*}_W(Y, f^*L)$ is an isomorphism.",
"Remark 6.3 It may seem strange that the proper pushforward maps respect products in the sense of (A5); one might rather expect a projection formula.",
"However, (A5) asks that the proper pushforward maps respect external products, not cup products, and in fact, having the pushforward and pullback maps respect products as in (A5) implies the projection formula, as one sees by considering the commutative pentagon associated to a proper morphism $f:Y\\rightarrow X$ in ${\\mathrm {Sm}}_B$ of relative dimension $d$ : ${Y[dd]_f@/^40pt/[drr]^{\\gamma _f:=(f\\times {\\operatorname{id}}_X)\\circ \\Delta _Y}[r]_-{\\Delta _Y}&Y\\times _BY[dr]_{f\\times {\\operatorname{id}}_Y}\\\\&&X\\times _BY[dl]^{{\\operatorname{id}}_X\\times f}\\\\X[r]_-{\\Delta _X}&X\\times _BX}$ Note that the square ${Y[r]^-{\\gamma _f}[d]_f&X\\times _BY[d]^{{\\operatorname{id}}_X\\times f}\\\\X[r]_-{\\Delta _X}&X\\times _BX}$ is transverse cartesian.",
"If we have closed subsets $Z\\subset X$ , $W\\subset Y$ with $f(W)\\subset Z$ , and line bundle $L\\rightarrow X$ , the pentagon diagram induces the diagram in cohomology ${15pt}{H^{*,*}_W(Y,\\omega _{Y/B}\\otimes f^*L)Y[dd]_{f_*}&[l]^-{\\Delta _Y^*}H^{*,*}_{W\\times W}(Y\\times _BY, \\omega _{Y/B}\\otimes f^*L)\\\\\\ &&\\hspace{-43.0pt}H^{*,*}_{Z\\times W}(X\\times _BY, \\omega _{Y/B}\\boxtimes L)@/_55pt/[ull]_{\\gamma _f^*}[ul]_{\\ \\ (f\\times {\\operatorname{id}}_Y)^*}[dl]^{\\ \\ ({\\operatorname{id}}_X\\times f)_*}\\\\H^{*-2d, *-d}_Z(X, \\omega _{X/B}\\otimes L)&H^{*-2d, *-d}_{Z\\times Z}(X\\times _BX, \\omega _{X/B}\\otimes L)[l]^-{\\Delta ^*_X}}$ Take $\\alpha \\in H^{a,b}_Z(X, M)$ , $\\beta \\in H^{c,d}_W(Y, \\omega _{Y/B}\\otimes f^*(L\\otimes M^{-1}))$ .",
"By functoriality of $(-)^*$ and (A5) for $(-)^*$ we have $\\gamma _f^*(\\alpha \\cup _{X,Y} \\beta )=f^*(\\alpha )\\cup _Y\\beta $ and by (A2) and (A5) for $(-)_*$ we have $f_*(f^*(\\alpha )\\cup _Y\\beta )=\\Delta _X^*({\\operatorname{id}}_X\\times f)_*(\\alpha \\cup _{X,Y}\\beta )=\\alpha \\cup _X f_*(\\beta ).$ Similarly, in the presence of (A2) and (A5) for $(-)^*$ , functoriality for $(-)^*$ and $(-)_*$ and the projection formula implies (A5) for $(-)_*$ .",
"Definition 6.4 A twisted cohomology theory on ${\\mathrm {Sm}}_B$ is given by the data D1, D4, D5 above, satisfying the parts of the axioms A1, A3-A7 that only involve $H^{*,*}$ .",
"Given an $\\operatorname{SL}$ -oriented cohomology theory $(H^{*,*}, H_{*,*}, \\ldots )$ on ${\\mathrm {Sm}}_B$ , one has the underlying twisted cohomology theory $(H^{*,*}, \\ldots )$ by forgetting the proper pushforward maps.",
"Example 6.5 The primary example of an $\\operatorname{SL}$ -oriented cohomology theory on ${\\mathrm {Sm}}_B$ is the one induced by an $\\operatorname{SL}$ -oriented motivic spectrum ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ : $(X,Z,L)\\mapsto {\\mathcal {E}}^{*,*}_Z(X;L).$ One defines, for $X\\in {\\mathrm {Sm}}_B$ of dimension $d_X$ over $B$ , ${\\mathcal {E}}_{m,n}^Z(X;L):={\\mathcal {E}}^{2d_X-m, d_X-n}_Z(X;\\omega _{X/B}\\otimes L);$ we extend the definition to arbitrary $X\\in {\\mathrm {Sm}}_B$ by taking the sum over the connected components of $X$ and write this also as ${\\mathcal {E}}^{2d_X-m, d_X-n}_Z(X;\\omega _{X/B}\\otimes L)$ by considering $d_X$ as a locally constant functor on $X$ .",
"The pushforward maps for a proper morphism of relative dimension $d$ , $f:Y\\rightarrow X$ , closed subsets $W\\subset Y$ , $Z\\subset X$ with $f(W)\\subset Z$ and line bundle $L\\rightarrow X$ are given by the pushforward $f_*:{\\mathcal {E}}^{2d_Y-m,2d_Y-n}_W(Y;\\omega _{Y/B}\\otimes f^*L)\\rightarrow {\\mathcal {E}}^{2d_X-m,2d_X-n}_Z(X;\\omega _{X/B}\\otimes L).$ Comparison isomorphisms We recall the element $\\eta \\in {\\rm Hom}_{{\\operatorname{SH}}(B)}(1_B, \\mathrm {S}^{-1,-1}\\wedge 1_B)$ induced by the map of $B$ -schemes $\\eta :{\\mathbb {A}}^2\\setminus \\lbrace 0\\rbrace \\rightarrow {\\mathbb {P}}^1$ , $\\eta (a,b)=(a:b)$ .",
"As every ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ is a module for $1_B$ , we have the map $\\times \\eta :{\\mathcal {E}}\\rightarrow \\mathrm {S}^{-1,-1}\\wedge {\\mathcal {E}}$ for each $x\\in {\\operatorname{SH}}(B)$ .",
"We say that $\\eta $ acts invertibly on ${\\mathcal {E}}$ if $\\times \\eta $ is an isomorphism in ${\\operatorname{SH}}(B)$ .",
"We consider the following situation: fix an $\\operatorname{SL}$ -oriented motivic spectrum ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ .",
"This gives us the twisted cohomology theory ${\\mathcal {E}}^{*,*}$ underlying the oriented cohomology defined by ${\\mathcal {E}}$ .",
"Let $({\\mathcal {E}}^{*,*}, \\tilde{{\\mathcal {E}}}_{*,*})$ be an extension of ${\\mathcal {E}}^{*,*}$ to an oriented cohomology theory on ${\\mathrm {Sm}}_B$ , in other words, we define new pushforward maps $\\hat{f}_*:{\\mathcal {E}}^{*,*}_W(Y, \\omega _{Y/B}\\otimes f^*L)\\rightarrow {\\mathcal {E}}^{*-2d,*-d}_Z(X, \\omega _{X/B}\\otimes L)$ The main result of this section is a comparison theorem.",
"Before stating the result we recall the decomposition of ${\\operatorname{SH}}(B)[1/2]$ into plus and minus parts.",
"We have the involution $\\tau :1_B\\rightarrow 1_B$ induced by the symmetry isomorphism $\\tau :{\\mathbb {P}}^1\\wedge {\\mathbb {P}}^1\\rightarrow {\\mathbb {P}}^1\\wedge {\\mathbb {P}}^1$ .",
"In ${\\operatorname{SH}}(B)[1/2]$ , this gives us the idempotents $({\\operatorname{id}}+\\tau )/2$ , $({\\operatorname{id}}-\\tau )/2$ , and so decomposes ${\\operatorname{SH}}(B)[1/2]$ into +1 and -1 “eigenspaces” for $\\tau $ : ${\\operatorname{SH}}(B)[1/2]={\\operatorname{SH}}(B)^+\\times {\\operatorname{SH}}(B)^-$ We decompose ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)[1/2]$ as ${\\mathcal {E}}={\\mathcal {E}}_+\\oplus {\\mathcal {E}}_-$ .",
"Theorem 7.1 Suppose the pushforward maps $f_*, \\hat{f}_*:{\\mathcal {E}}^{*,*}_W(Y, \\omega _{Y/B}\\otimes f^*L)\\rightarrow {\\mathcal {E}}^{*-2d,*-d}_Z(X, \\omega _{X/B}\\otimes L)$ agree for $W$ , $Z$ , $L$ , $X=V$ a vector bundle over $Y$ and $f:Y\\rightarrow V$ the zero-section.",
"Suppose in addition that one of the following conditions holds: the $\\operatorname{SL}$ -orientation of ${\\mathcal {E}}$ extends to a $\\operatorname{GL}$ -orientation; $\\eta $ acts invertibly on ${\\mathcal {E}}$ ; 2 acts invertibly on ${\\mathcal {E}}$ and ${\\mathcal {E}}_+^{-1,0}(U)=0$ for affine $U$ in ${\\mathrm {Sm}}_B$ .",
"Then $f_*=\\hat{f}_*$ for all $X, Y, Z, W, L, f$ for which the pushforward is defined.",
"By the standard argument of deformation to the normal cone, it follows that $f_*=\\hat{f}_*$ for all $f:Y\\rightarrow X$ a closed immersion, $Z, W, L$ .",
"As every proper map in ${\\mathrm {Sm}}_B$ is projective, $f$ admits a factorization $f=p\\circ i$ , with $i:Y\\rightarrow X\\times _B{\\mathbb {P}}^N$ a closed immersion and $p:X\\times _B{\\mathbb {P}}^N\\rightarrow X$ the projection.",
"By functoriality of the pushforward maps, it suffices to check that $p_*=\\hat{p}_*$ .",
"In case (i), this follows from the uniqueness assertion in [38].",
"Indeed, the cohomology theory associated to a $\\operatorname{GL}$ -oriented motivic spectrum ${\\mathcal {E}}$ satisfies the axioms of Panin-Smirnov and the associated Thom isomorphisms give rise to an “orientation” in the sense of [38], so we may apply the results cited.",
"We note that in [38] the base-scheme is ${\\rm Spec\\,}k$ , $k$ a field, so loc.",
"cit.",
"does not immediately apply to our setting of a more general base-scheme; we say a few words about the extension of this result to our base-scheme $B$ .",
"As a proper map $f:Y\\rightarrow X$ in ${\\mathrm {Sm}}_B$ is projective, one factors $f$ as $f=p\\circ i$ , with $i:Y\\rightarrow {\\mathbb {P}}^n_X$ a closed immersion and $p:{\\mathbb {P}}^n_X\\rightarrow X$ the projection.",
"The uniqueness for a closed immersion in ${\\mathrm {Sm}}_B$ reduces to the case of the zero-section of a vector bundle by the usual method of deformation to the normal bundle, and as the pushforward by the zero-section of our two theories are the same by assumption, we have agreement in the case of a closed immersion.",
"For the projection $p$ , the proof of [38] relies on [37], where for $p$ , using the projective bundle formula, the key point is to show that both pushforwards have the same value on the unit $1_{{\\mathbb {P}}^n_X}\\in {\\mathcal {E}}^{0,0}({\\mathbb {P}}^n_X)$ .",
"The proof of this relies on the formula for the pushforward of $1_{{\\mathbb {P}}^n_X}$ under the diagonal $\\Delta _{{\\mathbb {P}}^n_X}:{\\mathbb {P}}^n_X\\rightarrow {\\mathbb {P}}^n_X\\times _X{\\mathbb {P}}^n_X$ given by [37].",
"As $\\Delta _{{\\mathbb {P}}^n_X}$ is a closed immersion, the two pushforwards under $\\Delta _{{\\mathbb {P}}^n_X}$ agree, and the proof of the formula in [37] uses only formal properties of pushforward and pullback as expressed in our axioms, plus the projective bundle formula.",
"This latter in turn relies only on properties of the Thom class of ${\\mathcal {O}}(-1)$ and localization with respect to ${\\mathbb {A}}^m_X\\subset {\\mathbb {P}}^m_X$ , and thus we may use [37] in our more general setting.",
"The argument that the pushforward of $1_{{\\mathbb {P}}^n_X}$ under $p$ can be recovered from the formula for the pushforward of $1_{{\\mathbb {P}}^n_X}$ under $\\Delta _{{\\mathbb {P}}^n_X}$ is elementary and formal, and only uses the restriction of the two theories to ${\\mathrm {Sm}\\text{-}\\mathrm {L}}_X$ , and not the fact that these restrictions come from theories over $k$ .",
"Thus, the argument used in the proof of [37] may be used to prove our result in case (i).",
"In case (ii), we use Lemma REF below.",
"Indeed, if $N$ is odd, we may apply the closed immersion $X\\times _B{\\mathbb {P}}^N\\rightarrow X\\times _B{\\mathbb {P}}^{N+1}$ as a hyperplane, so we reduce to the case $N$ even, in which case both $p_*$ and $\\hat{p}_*$ are inverse to the map $i_*$ , where $i:X\\rightarrow X\\times _B{\\mathbb {P}}^N$ is the section associated to the point $(1:0:\\ldots :0)$ of ${\\mathbb {P}}^N$ .",
"In case (iii) we may work in the category ${\\operatorname{SH}}(B)[1/2]$ .",
"We decompose ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)[1/2]$ as ${\\mathcal {E}}={\\mathcal {E}}_+\\oplus {\\mathcal {E}}_-$ and similarly decompose the pushforward maps $f_*$ and $\\hat{f}_*$ .",
"By Lemma REF , $\\eta $ acts invertibly on ${\\operatorname{SH}}(B)^-$ and the projection of $\\eta $ to ${\\operatorname{SH}}(B)^+$ is zero.",
"By Lemma REF below, the $\\operatorname{SL}$ -orientation of ${\\mathcal {E}}$ induces an $\\operatorname{SL}$ -orientation on the projection ${\\mathcal {E}}^+$ that extends to a $\\operatorname{GL}$ -orientation.",
"By (i), this implies that $f_*^+=\\hat{f}_*^+$ .",
"By (ii), $f_*^-=\\hat{f}_*^-$ , so $f_*=\\hat{f}_*$ .",
"Lemma 7.2 ([2]) Let ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ be an $\\operatorname{SL}$ -oriented motivic spectrum on which $\\eta $ acts invertibly.",
"Let $0\\in {\\mathbb {P}}^N({\\mathbb {Z}})$ be the point $(1:0\\ldots :0)$ .",
"For $X\\in {\\mathrm {Sm}}_B$ , $L\\rightarrow X$ a line bundle and $Z\\subset X$ a closed subset, the pushforward map $i_*:{\\mathcal {E}}^{*-2N,*-N}_Z(X, \\omega _{X/B}\\otimes L)\\rightarrow {\\mathcal {E}}^{*,*}_{p^{-1}(Z)}(X\\times _B{\\mathbb {P}}^N, \\omega _{{\\mathbb {P}}^N/B}\\otimes p^*L)$ is an isomorphism.",
"Using a Mayer-Vietoris sequence, we see that the statement is local on $X$ for the Zariski topology, so we may assume that $L={\\mathcal {O}}_X$ .",
"If we prove the statement for the pair $(X,X)$ and $(X\\setminus Z, X\\setminus Z)$ the local cohomology sequence gives the result for $(X,Z)$ , thus we may assume that $Z=X$ , and we reduce to showing that $i_*:{\\mathcal {E}}^{*-2N,*-N}(X, \\omega _{X/B})\\rightarrow {\\mathcal {E}}^{*,*}(X\\times _B{\\mathbb {P}}^N, \\omega _{{\\mathbb {P}}^N/B})$ is an isomorphism.",
"This is [2] in case $B={\\rm Spec\\,}k$ , $k$ a field.",
"The proof over a general base-scheme is essentially the same, we say a few words about this generalization.",
"Most of the results that are used in the proof of loc.",
"cit.",
"are are already proved in the required generality here, for example, the Thom isomorphism (REF ) of Construction REF generalizes Ananyevskiy's construction [2] from $B={\\rm Spec\\,}k$ to general $B$ .",
"The proof of [2] relies also on [2], which in our setting reduces to the fact that for $X\\in \\mathrm {Sch}_B$ , and $u\\in \\Gamma (X,{\\mathcal {O}}_X^\\times )$ a unit, the automorphism of $X\\times {\\mathbb {P}}^1$ sending $(x,(t_0: t_1))$ to $(x, (ut_0, u^{-1}t_1)$ induces the identity on $X_+\\wedge {\\mathbb {P}}^1/X$ in ${\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)$ .",
"This follows by identifying ${\\mathbb {P}}^1_X$ with ${\\mathbb {P}}({\\mathbb {A}}^2_X)$ and noting that the diagonal matrix with entries $u, u^{-1}$ is an elementary matrix in $\\operatorname{GL}_2(\\Gamma (X,{\\mathcal {O}}_X))$ .",
"Lemma 7.3 Suppose that ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ is $\\operatorname{SL}$ oriented and that ${\\mathcal {E}}^{-1,0}(U)=0$ for all affine $U$ in ${\\mathrm {Sm}}_B$ .",
"Then the induced $\\operatorname{SL}$ orientation on ${\\mathcal {E}}_+\\in {\\operatorname{SH}}(B)_+$ extends to a $\\operatorname{GL}$ orientation.",
"Let $u\\in \\Gamma (X, {\\mathcal {O}}_X^\\times )$ be a unit on some $X\\in {\\mathrm {Sm}}_B$ .",
"Then the map $\\times u:X\\times _B{\\mathbb {P}}^1\\rightarrow X\\times _B{\\mathbb {P}}^1;\\quad (x, [t_0:t_1])\\mapsto (x, [ut_0:t_1])$ induces the identity on $\\mathrm {S}^{2,1}\\wedge X/B$ in ${\\operatorname{SH}}(B)_+$ .",
"Indeed, let $[u]:X/B\\rightarrow X/B\\wedge {\\mathbb {G}}_m$ be the map induced by $u:X\\rightarrow {\\mathbb {G}}_m$ .",
"The argument given by Morel [32], that $\\times u/B={\\operatorname{id}}+\\eta [u]$ in case $B={\\rm Spec\\,}k$ , $k$ a field, is perfectly valid over a general base-scheme: this only uses the fact that for ${\\mathcal {X}}$ and ${\\mathcal {Y}}$ pointed spaces over $B$ , one has $\\Sigma ^\\infty _{\\mathrm {S}^1}{\\mathcal {X}}\\times _B{\\mathcal {Y}}\\simeq \\Sigma ^\\infty _{\\mathrm {S}^1}{\\mathcal {X}}\\oplus \\Sigma ^\\infty _{\\mathrm {S}^1}{\\mathcal {Y}}\\oplus \\Sigma ^\\infty _{\\mathrm {S}^1}({\\mathcal {X}}\\wedge {\\mathcal {Y}})$ and that the map $\\times _u:\\mathrm {S}^1\\wedge {\\mathbb {G}}_m\\wedge X_+\\rightarrow \\mathrm {S}^1\\wedge {\\mathbb {G}}_m\\wedge X_+$ is the $\\mathrm {S}^1$ -suspension of the composition $\\mathrm {S}^1\\wedge {\\mathbb {G}}_m\\wedge X_+\\xrightarrow{}\\mathrm {S}^1\\wedge ({\\mathbb {G}}_m\\times {\\mathbb {G}}_m)\\wedge X_+\\xrightarrow{}\\mathrm {S}^1\\wedge {\\mathbb {G}}_m\\wedge X_+$ where $\\mu :{\\mathbb {G}}_m\\times {\\mathbb {G}}_m\\rightarrow {\\mathbb {G}}_m$ is the multiplication.",
"As $\\eta $ goes to zero in ${\\operatorname{SH}}(B)_+$ , it follows that $\\times u/B={\\operatorname{id}}$ in ${\\operatorname{SH}}(B)_+$ .",
"Now take $g\\in \\Gamma (X, \\operatorname{GL}_n({\\mathcal {O}}_X))$ , let $u=\\operatorname{det}g$ , let $m_u\\in \\Gamma (X, \\operatorname{GL}_n({\\mathcal {O}}_X))$ be the diagonal matrix with entries $u, 1,\\ldots , 1$ and let $h=m_u^{-1}\\cdot g\\in \\Gamma (X,\\operatorname{SL}_n({\\mathcal {O}}_X))$ .",
"We have ${\\operatorname{Th}}_X({\\mathcal {O}}_X^n)=({\\mathbb {P}}^1)^{\\wedge n}\\wedge X_+.$ Since ${\\mathcal {E}}$ is $\\operatorname{SL}$ -oriented, the map ${\\operatorname{Th}}(h):{\\operatorname{Th}}_X({\\mathcal {O}}_X^n)\\rightarrow {\\operatorname{Th}}_X({\\mathcal {O}}_X^n)$ induces the identity on ${\\mathcal {E}}^{**}$ and thus ${\\operatorname{Th}}(g)^*={\\operatorname{Th}}(m_u)^*:{\\mathcal {E}}^{*,*}_+({\\operatorname{Th}}_X({\\mathcal {O}}_X^n))\\rightarrow {\\mathcal {E}}^{*,*}_+({\\operatorname{Th}}_X({\\mathcal {O}}_X^n))$ But as ${\\operatorname{Th}}(m_u)=(\\times u)\\wedge {\\operatorname{id}}$ , our previous computation shows that ${\\operatorname{Th}}(m_u)^*={\\operatorname{id}}$ .",
"Now let $V\\rightarrow X$ be a rank $r$ vector bundle on some $X\\in {\\mathrm {Sm}}_B$ , choose a trivializing affine open cover ${\\mathcal {U}}=\\lbrace U_i\\rbrace $ of $X$ and let $\\phi _i:V_{|U_i}\\rightarrow U_i\\times {\\mathbb {A}}^r$ be a local framing.",
"We have the suspension isomorphism ${\\operatorname{Th}}(V_{U_i})\\simeq {\\operatorname{Th}}(U_i\\times {\\mathbb {A}}^r)=\\Sigma _r U_{i+}$ giving the isomorphism $\\theta _i:{\\mathcal {E}}_+^{a,b}(U_i)\\rightarrow {\\mathcal {E}}^{2r+a, r+b}_{+0_{V_{|U_i}}}(V_{|U_i}).$ Since $\\operatorname{GL}_r({\\mathcal {O}}_{U_i})$ acts trivially on ${\\mathcal {E}}_+^{**}({\\operatorname{Th}}(U_i\\times {\\mathbb {A}}^r))$ , the isomorphism $\\theta _i$ is independent of the choice of framing $\\phi _i$ .",
"In addition, the assumption ${\\mathcal {E}}^{-1,0}(U_i\\cap U_j)=0$ implies ${\\mathcal {E}}^{2r-1, r}_{+0_{V_{|U_i\\cap U_j}}}(V_{|U_i\\cap U_j})=0$ for all $i,j$ .",
"By Mayer-Vietoris, the sections $\\theta _i(1_{U_i})\\in {\\mathcal {E}}^{2r, r}_{+0_{V_{|U_i}}}(V_{|U_i})$ uniquely extend to an element $\\theta _V\\in {\\mathcal {E}}^{2r, r}_{+0_V}(V)$ The independence of the $\\theta _i$ on the choice of framing and the uniqueness of the extension readily implies the functoriality of $\\theta _V$ and similarly implies the product formula $\\theta _{V\\oplus W}=p_1^*\\theta _V\\cup p_2^*\\theta _W$ .",
"By construction, $\\theta _V$ is the suspension of the unit over $U_i$ , another application of independence of the choice of framing and the uniqueness of the extension shows that this is the case over every open subset $U\\subset X$ for which $V_{|U}$ is the trivial bundle.",
"Finally, the independence and uniqueness shows that $V\\mapsto \\theta _V$ is an extension of the $\\operatorname{SL}$ orientation on ${\\mathcal {E}}_+$ induced by that of ${\\mathcal {E}}$ .",
"Lemma 7.4 For $u\\in \\Gamma (X,{\\mathcal {O}}_X^\\times )$ we have $[u]\\eta =\\eta [u]:\\Sigma ^\\infty _X̰_+\\rightarrow \\Sigma ^\\infty _X̰_+$ We use the decomposition $\\Sigma ^\\infty _X̰_+\\wedge {\\mathbb {G}}_m\\times {\\mathbb {G}}_m=\\Sigma ^\\infty _X̰_+\\wedge {\\mathbb {G}}_m\\oplus \\Sigma ^\\infty _X̰_+\\wedge {\\mathbb {G}}_m\\oplus \\Sigma ^\\infty _X̰_+\\wedge {\\mathbb {G}}_m\\wedge {\\mathbb {G}}_m$ Via this, $\\eta $ is the map $[s]\\wedge [t]\\mapsto [st]-[s]-[t]$ so $\\eta [u]$ sends $[t]$ to $[ut]-[u]-[t]$ and ${\\operatorname{id}}_{{\\mathbb {G}}_m}\\wedge \\eta [u]$ sends $[s]\\wedge [t]$ to $[s]\\wedge [ut]-[s]\\wedge [u]-[s]\\wedge [t]$ , so $[u]\\eta $ is given by $[s]\\wedge [t]\\mapsto [st]-[s]-[t]\\mapsto [u]\\wedge [st]-[u]\\wedge [s]-[u]\\wedge [t].$ We have the automorphism $\\xi $ of ${\\mathbb {G}}_m^{\\wedge 3}$ sending $[u]\\wedge [s]\\wedge [t]$ to $[s]\\wedge [t]\\wedge [u]$ .",
"We have the isomorphism in ${\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(B)$ , $\\Sigma _{S_1}^6{\\mathbb {G}}_m^{\\wedge 3}\\simeq {\\mathbb {A}}^3/{\\mathbb {A}}^3\\setminus \\lbrace 0\\rbrace $ , via which $\\Sigma _{S_1}^6\\xi $ is induced by the linear map $(u,s,t)\\mapsto (s, t, u)$ .",
"As this latter linear map has matrix in the standard basis a product of elementary matrices, $\\Sigma _{\\mathrm {S}^1}^6\\xi $ is ${\\mathbb {A}}^1$ -homotopic to the identity, so after stabilizing, ${\\operatorname{id}}_{{\\mathbb {G}}_m}\\wedge \\eta [u]$ is the map $[s]\\wedge [t]\\mapsto [s]\\wedge [t]\\wedge [u]\\mapsto [u]\\wedge [s]\\wedge [t]\\mapsto [u]\\wedge [st]-[u]\\wedge [s]-[u]\\wedge [t]=[u]\\eta ([s]\\wedge [t]).$ Lemma 7.5 The projection $\\eta _-$ of $\\eta $ to ${\\operatorname{SH}}(B)_-$ is an isomorphism and the projection $\\eta _+$ of $\\eta $ to ${\\operatorname{SH}}(B)_+$ is zero.",
"Morel proves this in [32] in the case of a field, but the proof works in general.",
"In some detail, the map $\\tau $ is the map on ${\\mathbb {A}}^2/({\\mathbb {A}}^2\\setminus \\lbrace 0\\rbrace )$ induced by the linear map $(x, y)\\mapsto (y,x)$ .",
"The matrix identity $\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}\\cdot \\begin{pmatrix}1&0\\\\-1&1\\end{pmatrix}\\cdot \\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}\\cdot \\begin{pmatrix}-1&0\\\\0&1\\end{pmatrix}$ shows that the maps $(x, y)\\rightarrow (y,x)$ and $(x,y)\\mapsto (-x, y)$ are ${\\mathbb {A}}^1$ -homotopic.",
"By the arguments in Lemma REF , this latter map induces the map $1+\\eta [-1]=1+[-1]\\eta $ in ${\\operatorname{SH}}(B)$ , giving the identity $(1+\\eta [-1])_-=(1+[-1]\\eta )_-=-{\\operatorname{id}}\\Rightarrow \\eta \\cdot (-[-1]/2)= (-[-1]/2)\\cdot \\eta ={\\operatorname{id}}_{{\\operatorname{SH}}(B)_-}$ For $\\eta _+$ , the projector to ${\\operatorname{SH}}(B)_+$ is given by the idempotent $(1/2)(\\tau +1)=(1/2)(2+\\eta [-1])$ , so $\\eta _+=(1/2)\\eta \\cdot (2+\\eta [-1])$ .",
"Since the map $\\tau :{\\mathbb {P}}^1\\wedge {\\mathbb {P}}^1\\rightarrow {\\mathbb {P}}^1\\wedge {\\mathbb {P}}^1$ is $1+\\eta [-1]$ and ${\\mathbb {P}}^1=\\mathrm {S}^1\\wedge {\\mathbb {G}}_m$ , the symmetry $\\epsilon :{\\mathbb {G}}_m\\wedge {\\mathbb {G}}_m\\rightarrow {\\mathbb {G}}_m\\wedge {\\mathbb {G}}_m$ is $-(1+\\eta [-1])$ .",
"From our formula for $\\eta ([s]\\wedge [t])$ we see that $\\eta \\epsilon =\\eta $ which gives $\\eta \\cdot (2+\\eta [-1])=0$ .",
"Applications In this section, we apply the motivic Gauß-Bonnet formula of § and the comparison results of § to various specific $\\operatorname{SL}$ -oriented cohomology theories, and thereby make computations of the motivic Euler characteristic $\\chi (X/B)$ in different contexts.",
"Motivic cohomology and cohomology of the Milnor K-theory sheaves We work over the base-scheme $B={\\rm Spec\\,}k$ , with $k$ a perfect field.",
"In ${\\operatorname{SH}}(k)$ we have the motivic cohomology spectrum ${\\operatorname{H}}{\\mathbb {Z}}$ representing Voevodsky's motivic cohomology (see e.g.",
"[28] for a construction valid in arbitrary characteristic).",
"By [49], there is a natural isomorphism ${\\operatorname{H}}{\\mathbb {Z}}^{a,b}(X)\\simeq {\\rm CH}^b(X, 2b-a)$ for $X\\in {\\mathrm {Sm}}_B$ , where ${\\rm CH}^b(X, 2b-a)$ is Bloch's higher Chow group [9].",
"${\\operatorname{H}}{\\mathbb {Z}}$ admits a localization sequence: for $i:Z\\rightarrow X$ a closed immersion of codimension $d$ in ${\\mathrm {Sm}}_k$ , there is a canonical isomorphism ${\\operatorname{H}}{\\mathbb {Z}}^{a,b}_Z(X)\\simeq {\\operatorname{H}}{\\mathbb {Z}}^{a-2d, b-d}(Z)$ See for example [10].",
"In particular, for $p:V\\rightarrow X$ a rank $r$ vector bundle over $X\\in {\\mathrm {Sm}}_k$ , we have the isomorphism ${\\operatorname{H}}{\\mathbb {Z}}^{2r, r}_{0_V}(V)\\simeq {\\operatorname{H}}{\\mathbb {Z}}^{0,0}(X)$ which gives us Thom classes $\\vartheta _V^{{\\operatorname{H}}{\\mathbb {Z}}}\\in {\\operatorname{H}}{\\mathbb {Z}}^{2r, r}_{0_V}(V)$ corresponding to the unit $1^{{\\operatorname{H}}{\\mathbb {Z}}}_X\\in {\\operatorname{H}}{\\mathbb {Z}}^{0,0}(X)$ .",
"Thus ${\\operatorname{H}}{\\mathbb {Z}}$ is a $\\operatorname{GL}$ -oriented motivic spectrum.",
"Let $X$ be a smooth projective $k$ -scheme of dimension $n$ over $k$ .",
"For a class $x\\in {\\operatorname{H}}{\\mathbb {Z}}^{2n, n}(X)$ , the isomorphism ${\\operatorname{H}}{\\mathbb {Z}}^{2n,n}(X)\\simeq {\\rm CH}^n(X,0)={\\rm CH}^n(X)$ allows one to represent $x$ as the class of a 0-cycle $\\tilde{x}=\\sum _in_i p_i$ , with the $p_i$ closed points of $X$ .",
"One has the degree $\\deg _k(p_i):=[k(p_i):k]$ and extending by linearity gives the degree $\\deg _k(\\tilde{x})$ , which one shows passes to rational equivalence to define a degree map $\\deg _k: {\\operatorname{H}}{\\mathbb {Z}}^{2n, n}(X)\\rightarrow {\\rm CH}^0({\\rm Spec\\,}k)={\\mathbb {Z}}.$ As a $\\operatorname{GL}$ -oriented theory, ${\\operatorname{H}}{\\mathbb {Z}}$ has Chern classes for vector bundles: $c_r(V)\\in {\\operatorname{H}}{\\mathbb {Z}}^{2r, r}(X)$ for $V\\rightarrow X$ a vector bundle over some $X\\in {\\mathrm {Sm}}_k$ and $r \\ge 0$ .",
"Theorem 8.1 Let $X\\in {\\mathrm {Sm}}_k$ be projective of dimension $d_X$ .",
"Then $u^{{\\operatorname{H}}{\\mathbb {Z}}}(\\chi (X/k))=\\deg _k(c_{d_X}(T_{X/k})).$ One has well-defined pushforward maps on ${\\rm CH}^*(-,*)$ for projective morphisms (see e.g.",
"[9]).",
"Via the isomorphism $ {\\operatorname{H}}{\\mathbb {Z}}^{a,b}(X)\\simeq {\\rm CH}^b(X, 2b-a)$ [49], this gives pushforward maps $\\hat{f}_*$ on ${\\operatorname{H}}{\\mathbb {Z}}^{*,*}$ for $f:Y\\rightarrow X$ a projective morphism in ${\\mathrm {Sm}}_B$ (see [9] for details), making $(X, Z)\\mapsto {\\operatorname{H}}{\\mathbb {Z}}^{*,*}_Z(X)$ a $\\operatorname{GL}$ -oriented cohomology theory on ${\\mathrm {Sm}}_k$ .",
"In addition, for $\\pi _X:X\\rightarrow {\\rm Spec\\,}k$ in ${\\mathrm {Sm}}_k$ projective of dimension $n$ , the map $\\hat{\\pi }_{X*}:{\\operatorname{H}}{\\mathbb {Z}}^{2n, n}(X)\\rightarrow {\\rm CH}^0({\\rm Spec\\,}k)={\\mathbb {Z}}$ is $\\deg _k$ , and for $i:Y\\rightarrow X$ a closed immersion, the map $\\hat{i}_*$ is given by the localization theorem, which readily implies that $\\hat{i}_*=i_*$ .",
"By our comparison theorem REF , which here really reduces to the theorem of Panin-Smirnov, it follows that $\\hat{f}_*=f_*$ for all projective $f$ .",
"Finally, one has $c_{d_X}=s^*s_*(1^{{\\operatorname{H}}{\\mathbb {Z}}}_X)=e^{{\\operatorname{H}}{\\mathbb {Z}}}(V)$ ([18]), so applying the motivic Gauß-Bonnet theorem REF gives the statement.",
"One can obtain the same result by using the cohomology of the Milnor K-theory sheaves as a bigraded cohomology theory.",
"The homotopy t-structure on ${\\operatorname{SH}}(k)$ has heart the abelian category of homotopy modules $\\operatorname{\\Pi _*}(k)$ (see [32] and [33] for details); we let ${\\operatorname{H}}_0:{\\operatorname{SH}}(k)\\rightarrow \\operatorname{\\Pi _*}(k)$ be the associated functor.",
"The fact that ${\\operatorname{H}}{\\mathbb {Z}}^{n,n}({\\rm Spec\\,}F) \\simeq \\mathrm {K}^\\mathrm {M}_n(F)$ for $F$ a field [35], [47] says that ${\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}$ is canonically isomorphic to the homotopy module $(\\mathrm {K}^\\mathrm {M}_n)_n$ , which is in fact a cycle module in the sense of Rost [44].",
"This gives us the isomorphism ${\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}^{a,b}(X)\\simeq {\\operatorname{H}}^a(X, {\\mathcal {K}}^\\mathrm {M}_b).$ The isomorphism $H^n(X, {\\mathcal {K}}^\\mathrm {M}_n)\\simeq {\\rm CH}^n(X)$ (a special case of Rost's formula for the Chow groups of a cycle module, [44]) gives us as above Thom classes $\\vartheta ^{\\mathrm {K}^\\mathrm {M}_*}(V)\\in {\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}^{2r,r}_{0_V}(V)$ , giving ${\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}$ a $\\operatorname{GL}$ -orientation.",
"As for ${\\operatorname{H}}{\\mathbb {Z}}^{*,*}$ , one has explicitly defined pushforward maps on ${\\operatorname{H}}^*(-, {\\mathcal {K}}^M_*)$ which give ${\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}^{*,*}$ the structure of a $\\operatorname{GL}$ -oriented cohomology theory on ${\\mathrm {Sm}}_k$ and for which the pushforward map for the zero-section of a vector bundle is given by the Thom isomorphism.",
"Since the pushforward on ${\\operatorname{H}}^n(X, {\\mathcal {K}}^\\mathrm {M}_n)$ agrees with the classical pushforward on ${\\rm CH}^n$ , we deduce the following using the same proof as for Theorem REF .",
"Theorem 8.2 Let $X\\in {\\mathrm {Sm}}_k$ be projective of dimension $d_X$ .",
"Then $u^{{\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}}(\\chi (X/k))=\\deg _k(c_{d_X}(T_{X/k})) \\text{ in }{\\rm CH}^0({\\rm Spec\\,}k)={\\mathbb {Z}}.$ Algebraic K-theory We now let $B$ be any regular separated base-scheme of finite Krull dimension.",
"Algebraic K-theory on ${\\mathrm {Sm}}_B$ is represented by the motivic commutative ring spectrum ${\\operatorname{KGL}}\\in {\\operatorname{SH}}(B)$ (see [48]).",
"Just as for ${\\operatorname{H}}{\\mathbb {Z}}$ , the purity theorem ${\\operatorname{KGL}}_Z^{a,b}(X)\\simeq {\\operatorname{KGL}}^{a-2c, b-c}(Z)$ for $i:Z\\rightarrow X$ a closed immersion of codimension $d$ in ${\\mathrm {Sm}}_B$ (a consequence of Quillen's localization sequence for algebraic K-theory [41]) gives Thom class $\\vartheta ^{\\operatorname{KGL}}(V)\\in {\\operatorname{KGL}}^{2r,r}_{0_V}(V)$ for $V\\rightarrow X$ a rank $r$ vector bundle over $X\\in {\\mathrm {Sm}}_B$ , and makes ${\\operatorname{KGL}}$ a $\\operatorname{GL}$ -oriented motivic spectrum.",
"Explicitly, ${\\operatorname{KGL}}$ represents Quillen K-theory on ${\\mathrm {Sm}}_B$ via ${\\operatorname{KGL}}^{a,b}\\simeq \\mathrm {K}_{2b-a}$ and the Thom class for a rank $r$ vector bundle $p:V\\rightarrow X$ is represented by the Koszul complex ${\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})$ .",
"Here $\\text{can}:p^*V^\\vee \\rightarrow {\\mathcal {O}}_V$ is the dual of the tautological section ${\\mathcal {O}}_V\\rightarrow p^*V$ and ${\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})$ is the complex whose terms are given by ${\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})^{-r}=\\Lambda ^rp^*V^\\vee $ and whose differential $\\Lambda ^rp^*V^\\vee \\rightarrow \\Lambda ^{r-1}p^*V^\\vee $ is given with respect to a local framing of $V^\\vee $ by $d(e_{i_1}\\wedge \\ldots \\wedge e_{i_r})=\\sum _{j=1}^r(-1)^{j-1}\\text{can}(e_{i_j})\\cdot e_{i_1}\\wedge \\ldots \\wedge \\widehat{e_{i_j}}\\wedge \\ldots \\wedge e_{i_r}.$ This complex is a locally free resolution of $s_*({\\mathcal {O}}_X)$ , where $s:X\\rightarrow V$ is the zero-section.",
"Thus, by the identification of ${\\operatorname{KGL}}^{2r, r}_{0_V}(V)$ with the Grothendieck group of the triangulated category of perfect complexes on $V$ with support contained in $0_V$ , ${\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})$ gives rise to a class $[{\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})]\\in {\\operatorname{KGL}}^{2r, r}_{0_V}(V)$ which maps to $1_X$ under the purity isomorphism ${\\operatorname{KGL}}^{2r, r}_{0_V}(V)\\simeq {\\operatorname{KGL}}^{0,0}(X)$ , so that we indeed have $[{\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})] = \\vartheta ^{\\operatorname{KGL}}(V)$ .",
"Just as for motivic cohomology, one has explicit pushforward maps in K-theory given by Quillen's localization and devissage theorems identifying, for $X \\in {\\mathrm {Sm}}_B$ and $Z \\subseteq X$ a closed subscheme, the K-theory with support $\\mathrm {K}^Z(X)$ with the K-theory of the abelian category of coherent sheaves $\\operatorname{Coh}_Z$ on $Z$ , denoted $\\mathrm {G}(Z)$ .",
"For a projective morphism $f:Y\\rightarrow X$ , one has the pushforward map $\\hat{f}_*:\\mathrm {G}(Y)\\rightarrow \\mathrm {G}(X)$ defined by using a suitable subcategory of $\\operatorname{Coh}_Y$ on which $f_*$ is exact.",
"On $\\mathrm {K}_0$ , this recovers the usual formula $\\hat{f}_*([{\\mathcal {F}}])=\\sum _{j=0}^{\\operatorname{dim}Y}(-1)^j[\\mathrm {R}^jf_*({\\mathcal {F}})]$ for ${\\mathcal {F}}\\in \\operatorname{Coh}_Y$ .",
"Via the isomorphisms ${\\operatorname{KGL}}_Z^{a,b}(X)\\simeq \\mathrm {G}_{2b-a}(Z)$ , this gives pushforward maps $\\hat{f}_*$ for ${\\operatorname{KGL}}^{*,*}$ , defining a $\\operatorname{GL}$ -oriented cohomology theory on ${\\mathrm {Sm}}_B$ .",
"For $s:X\\rightarrow V$ the zero-section of a vector bundle, $\\hat{s}_*$ agrees with the pushforward $s_*$ using the Thom isomorphism/localization theorem, hence by our comparison theorem (again really the theorem of Panin-Smirnov), we have $\\hat{f}_*=f_*$ for all projective $f$ .",
"Theorem 8.3 Let $\\pi _X:X\\rightarrow B$ be a smooth projective morphism with $B$ a regular separated scheme of finite Krull dimension.",
"Then $u^{\\operatorname{KGL}}(\\chi (X/B))=\\sum _{j=0}^{\\operatorname{dim}_BX}\\sum _{i=0}^{\\operatorname{dim}_BX} (-1)^{i+j}[\\mathrm {R}^j\\pi _{X*}\\Omega _{X/B}^i]\\in \\mathrm {K}_0(B)={\\operatorname{KGL}}^{0,0}(B).$ Let $p:T_{X/B}\\rightarrow X$ denote the relative tangent bundle and let $s:X\\rightarrow T_{X/B}$ denote the zero-section.",
"We have $e^{\\operatorname{KGL}}(T_{X/B})=s^*({\\operatorname{th}}(T_{X/B}))=s^*({\\operatorname{Kos}}_{T_{X/B}}(p^*T_{X/B}^\\vee , \\text{can})).$ Since $T_{X/B}^\\vee =\\Omega _{X/B}$ , and $s^*(\\text{can})$ is the zero-map, it follows that, in $\\mathrm {K}_0(X)$ , $s^*({\\operatorname{Kos}}_{T_{X/B}}(p^*T_{X/B}^\\vee , \\text{can}))=\\sum _{i=0}^{\\operatorname{dim}_BX}(-1)^i[\\Omega _{X/B}^i],$ and thus $\\pi _{X*}(e^{\\operatorname{KGL}}(T_{X/B}))=\\sum _{j=0}^{\\operatorname{dim}_BX}\\sum _{i=0}^{\\operatorname{dim}_BX} (-1)^{i+j}[\\mathrm {R}^j\\pi _{X*}\\Omega _{X/B}^i].$ We conclude by applying the motivic Gauß-Bonnet theorem.",
"Milnor-Witt cohomology and Chow-Witt groups In this case, we again work over a perfect base-field $k$ .",
"The Milnor-Witt sheaves ${\\mathcal {K}}^\\mathrm {MW}_*$ constructed by Morel ([32], [33]) give rise to an $\\operatorname{SL}$ -oriented theory as follows.",
"Morel describes an isomorphism of ${\\mathcal {K}}^\\mathrm {MW}_0$ with the sheafification ${\\mathcal {GW}}$ of the Grothendieck-Witt rings[33] defines an isomorphism ${\\operatorname{GW}}(F)\\rightarrow \\mathrm {K}^\\mathrm {MW}(F)$ , $F$ a field.",
"[33] defines ${\\mathcal {K}}^\\mathrm {MW}_*$ as an unramified sheaf and it follows from [36] that ${\\mathcal {GW}}$ is an unramified sheaf.",
"From this it is not difficult to show that the isomorphism ${\\operatorname{GW}}(F)\\rightarrow \\mathrm {K}^\\mathrm {MW}(F)$ for fields extends to an isomorphism of sheaves.",
"; the map of sheaves of abelian groups ${\\mathbb {G}}_m\\rightarrow {\\mathcal {GW}}^\\times $ sending a unit $u$ to the one-dimensional form $\\langle u\\rangle $ allows one to define, for $L\\rightarrow X$ a line bundle, a twisted version ${\\mathcal {K}}^\\mathrm {MW}_*(L):={\\mathcal {K}}^\\mathrm {MW}_*\\times _{{\\mathbb {G}}_m}L^\\times $ as a Nisnevich sheaf on $X\\in {\\mathrm {Sm}}_k$ (see [33] or [11]).",
"One may use the Rost-Schmid complex for ${\\mathcal {K}}^\\mathrm {MW}_*(L)$ [33] to compute ${\\operatorname{H}}^*_Z(X, {\\mathcal {K}}^\\mathrm {MW}_*(L))$ for $Z\\subseteq X$ a closed subset, which gives a purity theorem: for $i:Z\\rightarrow X$ a codimension $d$ closed immersion in ${\\mathrm {Sm}}_k$ and $L\\rightarrow X$ a line bundle, there is a canonical isomorphism ${\\operatorname{H}}^*_Z(X, {\\mathcal {K}}^\\mathrm {MW}_*(L))\\simeq {\\operatorname{H}}^{*-d}(Z, {\\mathcal {K}}^\\mathrm {MW}_{*-d}(i^*L\\otimes \\operatorname{det}N_i)),$ where $N_i\\rightarrow Z$ is the normal bundle of $i$ .",
"Applying this to the zero-section of a rank $r$ vector bundle $p:V\\rightarrow X$ gives the isomorphism ${\\operatorname{H}}^0(X, {\\mathcal {GW}})\\simeq {\\operatorname{H}}^r_{0_V}(V, {\\mathcal {K}}^\\mathrm {MW}_r(p^*\\operatorname{det}^{-1} V));$ in particular, given an isomorphism $\\phi :\\operatorname{det}V\\rightarrow {\\mathcal {O}}_X$ , we obtain a Thom class $\\theta _{V, \\phi }\\in {\\operatorname{H}}^r_{0_V}(V, {\\mathcal {K}}^\\mathrm {MW}_r)$ corresponding to the unit section $1_X\\in {\\operatorname{H}}^0(X, {\\mathcal {GW}})$ .",
"On the other hand, Morel's computation of the 0th graded homotopy sheaf of the sphere spectrum ([32], [33]) gives an identification ${\\operatorname{H}}_0(1_k)\\simeq ({\\mathcal {K}}^\\mathrm {MW}_n)_{n\\in {\\mathbb {Z}}}$ in $\\operatorname{\\Pi _*}(k)$ , which then gives the natural isomorphism ${\\operatorname{H}}_0(1_k)^{a+b, b}_Z(X)\\simeq {\\operatorname{H}}^a_Z(X, {\\mathcal {K}}^\\mathrm {MW}_b).$ This is moreover compatible with twisting by a line bundle $p:L\\rightarrow X$ , on the ${\\operatorname{H}}_0(1_k)$ side using the Thom space construction ${\\operatorname{H}}_0(1_k)^{*, *}_Z(X;L):={\\operatorname{H}}_0(1_k)^{*+2, *+1}_Z(L)$ and on the Milnor-Witt cohomology side using the twisted Milnor-Witt sheaves.",
"To see this, note that the “untwisted” isomorphism gives us an isomorphism ${\\operatorname{H}}_0(1_k)^{a+b+2, b+1}_{Z}({\\operatorname{Th}}(L))\\simeq {\\operatorname{H}}_0(1_k)^{a+b+2, b+1}_{0_L\\cap p^{-1}(Z)} (L)\\simeq {\\operatorname{H}}^{a+1}_{0_L\\cap p^{-1}(Z)}(X, {\\mathcal {K}}^\\mathrm {MW}_{b+1}),$ so it suffices to identify the right-hand side with ${\\operatorname{H}}^a_Z(X,{\\mathcal {K}}^\\mathrm {MW}_b(L))$ .",
"For $Y\\in {\\mathrm {Sm}}_k$ and line bundle $M\\rightarrow Y$ , the Rost-Schmid complex for ${\\mathcal {K}}^\\mathrm {MW}_m(M)$ consists in degree $a$ of sums of terms of twisted Milnor-Witt groups of the form $\\mathrm {K}^\\mathrm {MW}_{m-a}(k(y); \\Lambda ^a(\\mathfrak {m}_y/\\mathfrak {m}_y^2)^\\vee \\otimes M)$ , for $y$ a codimension $a$ point of $Y$ and $\\mathfrak {m}_y\\subset {\\mathcal {O}}_{Y,y}$ the maximal ideal.",
"To compute cohomology with supports in $W\\subseteq Y$ , one restricts to those $y\\in W$ .",
"If we now take $Y=L$ and $M$ the trivial bundle, with supports in $p^{-1}(Z)\\cap 0_L$ and $m=b+1$ , and compare with $Y=X$ , with supports in $Z$ with $m=b$ , the term for $y\\in Z$ , of codimension $a+1$ on $L$ is $\\mathrm {K}^\\mathrm {MW}_{b-a}(k(y); \\Lambda ^a(\\mathfrak {m}_y/\\mathfrak {m}_y^2)^\\vee \\otimes L)$ while the term for $y\\in Z$ , of codimension $a$ on $X$ is $\\mathrm {K}^\\mathrm {MW}_{b-a}(k(y); \\Lambda ^a(\\mathfrak {m}_y/\\mathfrak {m}_y^2)^\\vee )$ , where $\\mathfrak {m}_y$ is the maximal ideal in ${\\mathcal {O}}_{X,y}$ in both cases.",
"This gives the desired identification ${\\operatorname{H}}^{a+1}_{0_L\\cap p^{-1}(Z)}(X, {\\mathcal {K}}^\\mathrm {MW}_{b+1})\\simeq {\\operatorname{H}}^{a}_{Z}(X, {\\mathcal {K}}^\\mathrm {MW}_{b}(L)).$ The purity isomorphism (REF ) is a special case of this construction.",
"The Thom class $\\theta _{V, \\phi }\\in {\\operatorname{H}}^r_{0_V}(V, {\\mathcal {K}}^\\mathrm {MW}_r)$ gives the Thom class $\\theta _{V, \\phi }\\in {\\operatorname{H}}_0(1_k)^{2r, r}_{0_V}(V),$ making ${\\operatorname{H}}_0(1_k)$ an $\\operatorname{SL}$ -oriented theory (see e.g [26]).",
"The resulting canonical Thom class ${\\operatorname{th}}_V\\in {\\operatorname{H}}_0(1_k)^{2r, r}_{0_V}(V;\\operatorname{det}^{-1}V)={\\operatorname{H}}^r_{0_V}(V, {\\mathcal {K}}^\\mathrm {MW}_r(\\operatorname{det}^{-1} V))$ agrees with the image of $1_X\\in {\\operatorname{H}}^0(X, {\\mathcal {GW}})$ under the Rost-Schmid isomorphism (REF ).",
"Let $\\pi _X:X\\rightarrow {\\rm Spec\\,}k$ be smooth and projective over $k$ of dimension $d$ .",
"Using the Rost-Schmid complex for the twisted homotopy module one has generators for ${\\operatorname{H}}^d(X, {\\mathcal {K}}^\\mathrm {MW}_d(\\omega _{X/k}))$ as formal sums $\\tilde{x}=\\sum _i\\alpha _i\\cdot p_i$ , with $\\alpha _i\\in {\\operatorname{GW}}(k(p_i))$ and $p_i\\in X$ closed points.",
"Since $k$ is perfect, the finite extension $k(p_i)/k$ is separable and one can define $\\widetilde{\\deg }_k(\\tilde{x}):=\\sum _i{\\rm Tr}_{k(p_i)/k}\\alpha _i\\in {\\operatorname{GW}}(k)$ where ${\\rm Tr}_{k(p_i)/k}:{\\operatorname{GW}}(k(p_i))\\rightarrow {\\operatorname{GW}}(k)$ is the transfer induced by the usual trace map ${\\rm Tr}_{k(p_i)/k}:k(p_i)\\rightarrow k$ (see for example [11] ).",
"It is shown in [11] that this descends to a map $\\widetilde{\\deg }_k:{\\operatorname{H}}_0^{2d,d}(X;\\omega _{X/k})={\\operatorname{H}}^d(X, {\\mathcal {K}}^\\mathrm {MW}_d(\\omega _{X/k}))\\rightarrow {\\operatorname{H}}_0^{0,0}({\\rm Spec\\,}k)={\\operatorname{GW}}(k)$ See also [21], which identifies this map with one induced by the Scharlau trace.",
"The methods of this paper give a new proof of the result given in [26]: Theorem 8.4 Let $k$ be a perfect field of characteristic different from two.",
"For $\\pi _X:X\\rightarrow {\\rm Spec\\,}k$ smooth and projective over $k$ , we have $\\chi (X/k)=\\widetilde{\\deg }_k(e^{{\\operatorname{H}}_0(1_k)}(T_{X/k})).$ Under Morel's isomorphism ${\\rm End}_{{\\operatorname{SH}}(k)}(1_X)\\simeq {\\operatorname{GW}}(k)$ ([32], [33]) and the isomorphism ${\\operatorname{H}}^0({\\rm Spec\\,}k, {\\mathcal {K}}^\\mathrm {MW}_0)\\simeq {\\operatorname{GW}}(k)$ , the unit map $u^{{\\operatorname{H}}_0(1_k)}:1_k\\rightarrow {\\operatorname{H}}_0(1_k)$ induces the identity map on $\\pi _{0,0}$ .",
"Using this, the proof of the claim is essentially the same as the other Gauß-Bonnet theorems we have discussed, but with a bit of extra work since we are no longer in the GL-oriented case.",
"Fasel [15] has defined pushforward maps $\\hat{f}_*:{\\operatorname{H}}^a_W(X, {\\mathcal {K}}^\\mathrm {MW}_b(\\omega _{X/k}\\otimes f^*L))\\rightarrow {\\operatorname{H}}^{a-d}_Z(Y, {\\mathcal {K}}^\\mathrm {MW}_{b-d}(L))$ for each projective morphism $f:X\\rightarrow Y$ in ${\\mathrm {Sm}}_k$ of relative dimension $d$ , line bundle $L\\rightarrow Y$ , and closed subsets $Z\\subseteq Y$ , $W\\subseteq X$ with $f(W)\\subseteq Z$ .",
"In the case of the structure map $\\pi _X:X\\rightarrow {\\rm Spec\\,}k$ , the pushforward $\\tilde{\\pi }_{X*}:{\\operatorname{H}}^d(X, {\\mathcal {K}}^\\mathrm {MW}_d(\\omega _{X/k}))\\rightarrow {\\operatorname{H}}^0({\\rm Spec\\,}k, {\\mathcal {K}}^\\mathrm {MW}_0)={\\operatorname{GW}}(k)$ is the map $\\widetilde{\\deg }_k$ .",
"For $s:X\\rightarrow V$ the zero-section of a vector bundle, $\\hat{s}_*$ is the Thom isomorphism $s_*$ .",
"Thus, if we pass to the $\\eta $ -inverted theory, ${\\operatorname{H}}_0(1_X)_\\eta :={\\operatorname{H}}_0(1_k)[\\eta ^{-1}]$ , our comparison theorem REF says that $\\hat{f}_{\\eta *}=f_{\\eta *}$ for all projective morphisms $f$ in ${\\mathrm {Sm}}_k$ .",
"We have ${\\mathcal {K}}^\\mathrm {MW}_*[\\eta ^{-1}]\\simeq {\\mathcal {W}}$ , the sheaf of Witt rings, and the map ${\\mathcal {K}}^\\mathrm {MW}_0={\\mathcal {GW}}\\rightarrow {\\mathcal {K}}^\\mathrm {MW}_*[\\eta ^{-1}]\\simeq {\\mathcal {W}}$ is the canonical map $q:{\\mathcal {GW}}\\rightarrow {\\mathcal {W}}$ realizing ${\\mathcal {W}}$ as the quotient of ${\\mathcal {GW}}$ by the subgroup generated by the hyperbolic form.",
"Thus, applying our motivic Gauß-Bonnet theorem gives the identity $q(\\chi (X/k))=q(\\widetilde{\\deg }_k(e^{{\\operatorname{H}}_0(1_k)}(T_{X/k})))\\text{ in }{\\operatorname{W}}(k).$ To lift this to an equality in ${\\operatorname{GW}}(k)$ and thereby complete the proof, we use that the map $({\\operatorname{\\text{rnk}}}, q): {\\mathcal {GW}}\\rightarrow {\\mathbb {Z}}\\times {\\mathcal {W}}$ is injective, together with the fact that we can recover the rank by applying ${\\operatorname{H}}_0$ to the unit map $1_k\\rightarrow {\\operatorname{H}}{\\mathbb {Z}}$ and using Theorem REF .",
"Hermitian K-theory and Witt theory We again let our base-scheme $B$ be a regular noetherian separated base-scheme of finite Krull dimension, but now assume that 2 invertible on $B$ .",
"Our goal in this subsection is to explain how the description of the “rank” of $\\chi (X/B)$ given by Theorem REF can be refined to give a formula for $\\chi (X/k)$ itself in terms of Hodge cohomology by using hermitian K-theory.",
"By work of Panin-Walter [40], Schlichting [45], and Schlichting-Tripathi [46], hermitian K-theory ${\\operatorname{KO}}^{[*]}_*(-)$ is represented by a motivic commutative ring spectrum ${\\operatorname{BO}}\\in {\\operatorname{SH}}(B)$ (we use the notation of [3]).",
"Panin-Walter give ${\\operatorname{BO}}$ an $\\operatorname{SL}$ -orientation.",
"${\\operatorname{BO}}$ -theory also represents particular cases of Schlichting's Grothendieck-Witt groups [45], via functorial isomorphisms ${\\operatorname{BO}}^{2r, r}(X;L)\\simeq {\\operatorname{KO}}^{[r]}_0(X, L):={\\operatorname{GW}}({\\operatorname{perf}}(X), L[r], \\text{can}),$ where $L\\rightarrow X$ is a line bundle and ${\\operatorname{GW}}({\\operatorname{perf}}(X), L[r], \\text{can})$ is the Grothendieck-Witt group of $L[r]$ -valued symmetric bilinear forms on ${\\operatorname{perf}}(X)$; we recall a version of the definition here.",
"Definition 8.5 Let $L\\rightarrow X$ be a line bundle and let $r \\in {\\mathbb {Z}}$ .",
"An $L[r]$ -valued symmetric bilinear form on $C\\in {\\operatorname{perf}}(X)$ is a map $\\phi :C\\otimes ^{\\mathrm {L}}C\\rightarrow L[r]$ in ${\\operatorname{perf}}(X)$ which satisfies the following conditions.",
"$\\phi $ is non-degenerate: the induced map $C\\rightarrow {\\mathcal {RH}om}(C, L[n])$ is an isomorphism in ${\\operatorname{perf}}(X)$ .",
"$\\phi $ is symmetric: $\\phi \\circ \\tau =\\phi $ , where $\\tau :C\\otimes ^{\\mathrm {L}}C\\rightarrow C\\otimes ^{\\mathrm {L}}C$ is the commutativity isomorphism.",
"(Note that we are assuming non-degeneracy in the definition but leaving this out of the terminology for the sake of brevity.)",
"Similar to the case of algebraic K-theory discussed in §REF , for a rank $r$ vector bundle $p:V\\rightarrow X$ , the Thom class $\\theta ^{\\operatorname{BO}}_V\\in {\\operatorname{BO}}^{2r,r}(V; p^*\\operatorname{det}^{-1}V)$ is given by the Koszul complex ${\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )$ , where the symmetric bilinear form $\\phi _V:{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )\\otimes {\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )\\rightarrow p^*\\operatorname{det}^{-1}V[r]=\\Lambda ^rV^\\vee [r]$ is given by the usual exterior product $-\\wedge -: \\Lambda ^iV^\\vee \\otimes \\Lambda ^{r-i}V^\\vee \\rightarrow \\Lambda ^rV^\\vee .$ Moreover, there are isomorphisms for $i<0$ ${\\operatorname{BO}}^{2r-i, r}(X;L)\\simeq {\\operatorname{W}}^{r-i}({\\operatorname{perf}}(X), L[r], \\text{can})$ where ${\\operatorname{W}}^{r-i}({\\operatorname{perf}}(X), L[r], \\text{can})$ is Balmer's triangulated Witt group.",
"In the case $B = {\\rm Spec\\,}k$ for $k$ a field of characteristic different from two, Ananyevskiy [3] shows that this isomorphism induces an isomorphism of $\\eta $ -inverted hermitian K-theory with Witt-theory, ${\\operatorname{BO}}[\\eta ^{-1}]^{*,*}\\simeq {\\operatorname{W}}^*[\\eta ,\\eta ^{-1}],$ where one gives $\\eta $ bidegree $(-1,-1)$ and an element $\\alpha \\eta ^n$ with $\\alpha \\in {\\operatorname{W}}^m$ has bidegree $(m-n,-n)$ ; the same proof works over out general base $B$ (with assumptions as at the beginning of this subsection).",
"For $f:Y\\rightarrow X$ a proper map of relative dimension $d_f$ in ${\\mathrm {Sm}}_B$ and $L$ a line bundle on $X$ , we follow Calmès-Hornbostel [12] in defining a pushforward map $\\hat{f}_*:{\\operatorname{BO}}^{2r,r}(Y, \\omega _{Y/B}\\otimes f^*L)\\rightarrow {\\operatorname{BO}}^{2r-2d_f,r-d_f}(X, \\omega _{X/B}\\otimes L)$ by Grothendieck-Serre duality.",
"In op.",
"cit., this is worked out for the $\\eta $ -inverted theory ${\\operatorname{BO}}_\\eta $ when the base is a field; however, the same construction works for ${\\operatorname{BO}}$ over the general base-scheme $B$ and goes as follows.",
"For $r \\ge 0$ , given an $L[r]$ -valued symmetric bilinear form $\\phi :C\\otimes ^{\\mathrm {L}}C\\rightarrow \\omega _{Y/B}\\otimes f^*L[r]$ , we have the corresponding isomorphism $\\tilde{\\phi }:C\\rightarrow {\\mathcal {RH}om}(C, \\omega _{Y/B}\\otimes f^*L[r]) \\simeq {\\mathcal {RH}om}(C, \\omega _{Y/X}\\otimes f^*(\\omega _{X/B}\\otimes L[r]))$ Grothendieck-Serre duality gives the isomorphism $\\mathrm {R}f_*{\\mathcal {RH}om}(C, \\omega _{Y/X}\\otimes f^*(\\omega _{X/B}\\otimes L[r])){[r]^{\\psi }_\\sim &}{\\mathcal {RH}om}(\\mathrm {R}f_*C, \\omega _{X/B}\\otimes L[r-d_f]).$ Composing these, we obtain the isomorphism $\\psi \\circ \\tilde{\\phi }:\\mathrm {R}f_*C\\rightarrow {\\mathcal {RH}om}(\\mathrm {R}f_*C, \\omega _{X/B}\\otimes L[r-d_f]),$ corresponding to the (nondegenerate) bilinear form $\\mathrm {R}f_*(\\phi ):\\mathrm {R}f_*C\\otimes ^{\\mathrm {L}}\\mathrm {R}f_*C\\rightarrow \\omega _{X/B}\\otimes L[r-d_f],$ which one can show is symmetric.",
"We explicitly define the above pushforward map by setting $\\hat{f}_*(C, \\phi ) := (\\mathrm {R}f_*C, \\mathrm {R}f_*(\\phi ))$ .",
"Applying this in the situation that $f=\\pi _X:X\\rightarrow B$ is a smooth and proper $B$ -scheme of relative dimension $d_X$ , we may obtain the formula $\\hat{\\pi }_{X*}(e^{\\operatorname{BO}}(T_{X/B}))=(\\oplus _{i,j=0}^{\\operatorname{dim}_BX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i], {\\rm Tr}),$ where ${\\rm Tr}:(\\oplus _{i,j=0}^{\\operatorname{dim}_kX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i])\\otimes (\\oplus _{i,j=0}^{\\operatorname{dim}_kX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i])\\rightarrow {\\mathcal {O}}_B$ is the symmetric bilinear form in ${\\operatorname{perf}}(B)$ determined by the pairings $(\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)\\otimes (\\mathrm {R}^{d_X-i}\\pi _{X*}\\Omega _{X/B}^{d_X-j})\\xrightarrow{}\\mathrm {R}^{d_X}\\pi _{X*}\\Omega _{X/B}^{d_X} \\xrightarrow{}{\\mathcal {O}}_B.$ Indeed, if $s:X\\rightarrow T_{X/B}$ denotes the zero-section, we have $e^{\\operatorname{BO}}(T_{X/B})=s^*({\\operatorname{Kos}}(T_{X/B}), \\phi )=(\\oplus _{j=0}^{d_X}\\Omega ^j_{X/B}[j],s^*\\phi )$ with $s^*\\phi $ determined by the product maps $\\Omega ^j_{X/B}[j]\\otimes \\Omega ^{d_X-j}_{X/B}[d_X-j]\\rightarrow \\omega _{X/B}[d_X],$ and thus $\\hat{\\pi }_{X*}(e^{\\operatorname{BO}}(T_{X/B}))$ is $\\oplus _{i,j=0}^{\\operatorname{dim}_kX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i]$ with the symmetric bilinear form ${\\rm Tr}$ as described above.",
"Passing to the $\\eta $ -inverted theory ${\\operatorname{BO}}_\\eta $ , our comparison theorem REF gives $q\\circ \\hat{\\pi }_{X*}=q\\circ \\pi _{X*}$ as maps ${\\operatorname{BO}}_\\eta ^{2d_X, d_X}(X,\\omega _{X/B})\\rightarrow {\\operatorname{BO}}_\\eta (B)$ .",
"We check that the conditions of the comparison theorem hold just as we did for algebraic K-theory.",
"Firstly, as mentioned above, the $\\operatorname{SL}$ -orientation for ${\\operatorname{BO}}$ defined by Panin-Walter can be described as follows: the Thom class for an oriented vector bundle ($p:V\\rightarrow X$ , $\\rho :{\\mathcal {O}}_X\\xrightarrow{}\\operatorname{det}V$ ) is given by the Koszul complex ${\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )$ equipped with the symmetric bilinear form $\\phi _V$ defined by the product in the exterior algebra followed by the isomorphism $p^*\\rho ^\\vee :p^*\\operatorname{det}^{-1}V\\rightarrow {\\mathcal {O}}_V$ .",
"On the other hand, the Calmès-Hornbostel pushforward for the zero-section $s:X\\rightarrow V$ of a rank $r$ vector bundle $p:V\\rightarrow X$ with isomorphism $\\rho :{\\mathcal {O}}_X\\xrightarrow{} \\operatorname{det}V$ is as described above, sending a symmetric bilinear form $\\psi :C\\otimes ^{\\mathrm {L}}C\\rightarrow \\omega _{X/B} \\otimes s^*L[n]$ to the symmetric bilinear form $\\mathrm {R}s_*(\\psi ): \\mathrm {R}s_*C \\otimes ^{\\mathrm {L}}\\mathrm {R}s_*C\\rightarrow \\omega _{V/B} \\otimes L[n+r].$ Since $s$ is finite, $\\mathrm {R}s_*C \\simeq C$ , which in ${\\operatorname{perf}}(V)$ is canonically isomorphic to $p^*C\\otimes _{{\\mathcal {O}}_V}{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )$ .",
"We may thus view $\\mathrm {R}s_*(\\psi )$ instead as a map $[p^*C\\otimes _{{\\mathcal {O}}_V}{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )] \\otimes ^{\\mathrm {L}}[p^*C\\otimes _{{\\mathcal {O}}_V}{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )] \\rightarrow \\omega _{V/B} \\otimes L[n+r];$ tracing through its definition, one finds that this map is given by the composition $[p^*C\\otimes _{{\\mathcal {O}}_V}{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )] \\otimes ^{\\mathrm {L}}[p^*C\\otimes _{{\\mathcal {O}}_V}{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )] \\\\\\xrightarrow{}[p^*C\\otimes ^{\\mathrm {L}}p^*C]\\otimes [{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )\\otimes {\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )]\\\\\\xrightarrow{} \\omega _{X/B} \\otimes L[n] \\otimes p^*\\operatorname{det}^{-1}V[r] \\xrightarrow{} \\omega _{V/B} \\otimes L[n+r].$ As this is exactly $p^*(C, \\phi ) \\otimes {\\operatorname{th}}_{V,\\rho }$ , we see that the Calmès-Hornbostel pushforward for $s$ is the same as that defined by the Panin-Walter $\\operatorname{SL}$ -orientation on ${\\operatorname{BO}}_\\eta $ , which verifies the hypothesis in our Theorem REF .",
"Having verified this, we can prove our main result.",
"Theorem 8.6 Let $B$ be a regular noetherian separated scheme of finite Krull dimension with 2 invertible in $\\Gamma (B, {\\mathcal {O}}_B)$ .",
"Let $X$ be a smooth projective $B$ -scheme.",
"Then: We have $u^{{\\operatorname{BO}}_\\eta }(\\chi (X/B))=(\\oplus _{i,j=0}^{\\operatorname{dim}_kX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i], {\\rm Tr})$ in ${\\operatorname{BO}}_\\eta ^{0,0}(B) \\simeq {\\operatorname{W}}({\\operatorname{perf}}(B)) \\simeq {\\operatorname{W}}(B)$ .",
"Let $f:{\\operatorname{GW}}(B)\\rightarrow \\mathrm {K}_0(B)$ denote the forgetful map discarding the symmetric bilinear form.",
"Suppose that the map $(f, q):{\\operatorname{GW}}(B)\\rightarrow \\mathrm {K}_0(B)\\times W(B)$ is injective (this is the case if for example $B$ is the spectrum of a local ring).",
"Then $u^{{\\operatorname{BO}}}(\\chi (X/B))=(\\oplus _{i,j=0}^{\\operatorname{dim}_BX}{\\operatorname{H}}^i(X, \\Omega _{X/B}^j)[j-i], {\\rm Tr})$ in ${\\operatorname{GW}}({\\operatorname{perf}}(B)) \\simeq {\\operatorname{GW}}(B)$ .",
"Suppose $B$ is in ${\\mathrm {Sm}}_k$ for $k$ a perfect field.",
"Then the image $\\widetilde{\\chi (X/B)}$ of $\\chi (X/B)$ in $\\pi _{0,0}(1_B)(B) \\simeq {\\operatorname{H}}^0(B, {\\mathcal {GW}})$ is given by $\\widetilde{\\chi (X/B)}=(\\oplus _{i,j=0}^{\\operatorname{dim}_BX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i], {\\rm Tr}) \\in {\\operatorname{H}}^0(B, {\\mathcal {GW}}).$ In particular, if $B={\\rm Spec\\,}k$ , then $\\chi (X/k)=(\\oplus _{i,j=0}^{\\operatorname{dim}_BX}{\\operatorname{H}}^i(X, \\Omega _{X/k}^j)[j-i], {\\rm Tr})\\in {\\operatorname{GW}}(k).$ The first statement follows from our comparison theorem REF , as detailed above, together with the motivic Gauß-Bonnet theorem REF .",
"Statement (2) follows from (1) and our result for algebraic K-theory, Theorem REF .",
"Finally, (3) follows from (2), after we check that the unit map $u^{\\operatorname{BO}}$ induces the identity map on ${\\operatorname{GW}}(k)$ via ${\\operatorname{GW}}(k){[r]^{(\\hbox{\\tiny Morel})}_\\sim &} 1_k^{0,0}({\\rm Spec\\,}k){[r]^{u^{\\operatorname{BO}}}_\\sim &} {\\operatorname{BO}}^{0,0}({\\rm Spec\\,}k)\\simeq {\\operatorname{KO}}^{[0]}_0(k)={\\operatorname{GW}}(k),$ where the first isomorphism arises from Morel's theorem [32], [33] identifying $1_k^{0,0}({\\rm Spec\\,}k)$ with ${\\operatorname{GW}}(k)$ .",
"The one dimensional forms $\\langle \\lambda \\rangle \\in {\\operatorname{GW}}(k)$ , $\\lambda \\in k^\\times $ , generate ${\\operatorname{GW}}(k)$ , and via Morel's isomorphism $\\langle \\lambda \\rangle $ maps to the automorphism of $1_k$ induced by the automorphism $\\phi _\\lambda :{\\mathbb {P}}^1_k\\rightarrow {\\mathbb {P}}^1_k$ , $\\phi _\\lambda ((x_0:x_1))= (x_0:\\lambda \\cdot x_1)$ .",
"By [3], the image of $\\phi _\\lambda $ under the unit map $u^{\\operatorname{BO}}$ is also $\\langle \\lambda \\rangle $ , after the canonical identification ${\\operatorname{BO}}^{0,0}({\\rm Spec\\,}k)\\simeq {\\operatorname{KO}}^{[0]}_0(k) \\simeq {\\operatorname{GW}}(k)$ .",
"Corollary 8.7 Let $k$ be a perfect field of characteristic different from two.",
"Let $H \\in {\\operatorname{GW}}(k)$ denote the class of the hyperbolic form $x^2-y^2$ .",
"Let $X$ be a smooth and projective $k$ -scheme.",
"Suppose $X$ has odd dimension $2n-1$ .",
"Let $m := \\sum _{i+j<2n-1}(-1)^{i+j}\\operatorname{dim}_k{\\operatorname{H}}^i(X,\\Omega _{X/k}^j)- \\sum _{\\hbox{t}o 30pt{\\vbox {\\tiny 0\\le i< j\\\\i+j= 2n-1}}} \\operatorname{dim}_k{\\operatorname{H}}^i(X,\\Omega _{X/k}^j).$ Then $\\chi (X/k)=m\\cdot H \\in {\\operatorname{GW}}(k)$ .",
"Assume $X$ has even dimension $2n$ .",
"Let $m := \\sum _{i+j<2n}(-1)^{i+j}\\operatorname{dim}_k{\\operatorname{H}}^i(X,\\Omega _{X/k}^j)+ \\sum _{\\hbox{t}o 30pt{\\vbox {\\tiny 0\\le i< j\\\\i+j= 2n}}} \\operatorname{dim}_k{\\operatorname{H}}^i(X,\\Omega _{X/k}^j)$ and let $Q$ be the symmetric bilinear form ${\\operatorname{H}}^n(X, \\Omega ^n_{X/k})\\times {\\operatorname{H}}^n(X, \\Omega ^n_{X/k}) \\xrightarrow{}{\\operatorname{H}}^{2n}(X, \\Omega ^{2n}_{X/k})\\xrightarrow{} k.$ Then $\\chi (X/k)=m\\cdot H+Q \\in {\\operatorname{GW}}(k)$ .",
"For $V$ a finite dimensional $k$ -vector space and $n \\in {\\mathbb {Z}}$ , we have the symmetric bilinear form in ${\\operatorname{perf}}(k)$ $h_n: (V[n]\\oplus V^\\vee [-n]) \\otimes (V[n]\\oplus V^\\vee [-n]) \\rightarrow k$ whose restriction to $V[n]\\otimes V^\\vee [-n]$ is the canonical pairing of $V[n]$ with $V^\\vee [-n]\\simeq V[n]^\\vee $ , and $(-1)^n$ times this pairing on $V^\\vee [-n]\\otimes V[n]$ .",
"The corresponding class of $h_n$ in ${\\operatorname{GW}}(k)$ is $(-1)^n$ times the class of $h_0$ , as $(V[n]\\oplus V^\\vee [-n], h_n)$ is the image of the class of $V[n]$ in $\\mathrm {K}_0(k)$ under the hyperbolic map $H:\\mathrm {K}_0(-)\\rightarrow {\\operatorname{KO}}^{[0]}_0(-)$ (see e.g.",
"[50]), and $[V[n]]=(-1)^n[V[0]]$ in $\\mathrm {K}_0({\\operatorname{perf}}(k))\\simeq \\mathrm {K}_0(k)$ .",
"With this in mind, we may deduce the claim from the formula $\\chi (X/k)=(\\oplus _{i,j=0}^{\\operatorname{dim}_BX}{\\operatorname{H}}^i(X, \\Omega _{X/k}^j)[j-i], {\\rm Tr})$ of Theorem REF .",
"Indeed, in the case $\\operatorname{dim}X=2n-1$ , the symmetric bilinear form ${\\rm Tr}$ is the sum of the “hyperbolic” forms as above on ${\\operatorname{H}}^i(X,\\Omega _{X/k}^j)[j-i]\\oplus {\\operatorname{H}}^{2n-1-i}(X,\\Omega _{X/k}^{2n-1-j})[i-j]$ for $i+j<2n-1$ , or $0\\le i<j$ and $ i+j= 2n-1$ ; and the argument in the even dimensional case is the same, except that one has the remaining factor coming from the symmetric pairing on ${\\operatorname{H}}^n(X, \\Omega ^n_{X/k})$ .",
"The next result was obtained in 1974 independently by Abelson [1] and Kharlamov [24] using an argument of Milnor's relying on the Lefschetz fixed point theorem.",
"Corollary 8.8 Let $k$ be a field equipped with an embedding $\\sigma : k \\hookrightarrow {\\mathbb {R}}$ .",
"Let $X$ be a smooth projective $k$ -scheme of even dimension $2n$ .",
"Then $|\\chi ^\\mathrm {top}(X({\\mathbb {R}}))|\\le \\operatorname{dim}_k {\\operatorname{H}}^n(X, \\Omega ^n_{X/k}).$ We know that $\\chi ^\\mathrm {top}(X({\\mathbb {R}}))$ is the signature of $\\sigma _*(\\chi (X/k)) \\in {\\operatorname{GW}}({\\mathbb {R}})$ (see [26]).",
"The description of $\\chi (X/k)$ given by Corollary REF gives the desired inequality $|\\operatorname{sig}\\sigma _*(\\chi (X/k))|\\le \\operatorname{dim}_k {\\operatorname{H}}^n(X, \\Omega _{X/k}).",
"$ Remark 8.9 Let $k$ be a perfect field of characteristic different from two.",
"The formula for the Euler characteristic given in Theorem REF shows that the invariant $\\chi (X/k)$ is “motivic” in the following sense.",
"Let $X$ and $Y$ be smooth projective $k$ -schemes of respective even dimensions $2n$ and $2m$ and let $\\alpha :X\\dasharrow Y$ be a correspondence with $k$ -coefficients of degree $n$ , that is, an element $\\alpha \\in {\\rm CH}^{m+n}(X\\times Y)_k$ .",
"The correspondence $\\alpha $ induces the map of $k$ -vector spaces $\\alpha ^*:{\\operatorname{H}}^m(Y,\\Omega ^m_{Y/k})\\rightarrow {\\operatorname{H}}^n(X, \\Omega ^n_{X/k})$ .",
"Suppose that $\\alpha ^*$ is an isomorphism and is compatible with the trace pairings on ${\\operatorname{H}}^m(Y,\\Omega ^m_{Y/k})$ and ${\\operatorname{H}}^n(X, \\Omega ^n_{X/k})$ appearing in Corollary REF .",
"Then $\\chi (X/k)=\\chi (Y/k)$ in the Witt ring ${\\operatorname{W}}(k)$ .",
"For instance, supposing $k$ has characteristic zero, if the motives of $X$ and $Y$ (for homological equivalence with respect to de Rham cohomology) have a Künneth decomposition $h(X) \\simeq \\oplus _{i=0}^{2n}h^i(X)\\langle i\\rangle ,\\quad h(Y) \\simeq \\oplus _{i=0}^{2m}h^i(Y)\\langle i\\rangle $ and $\\alpha $ induces an isomorphism $\\alpha ^*:h^m(Y)\\langle m\\rangle \\rightarrow h^n(X)\\langle n\\rangle $ , compatible with the respective intersection products $h^m(Y)\\langle m\\rangle \\otimes h^m(Y)\\langle m\\rangle \\rightarrow h^{2m}(Y)\\langle 2m\\rangle \\xrightarrow{}h^0(k) \\simeq {\\mathbb {Q}},$ $h^n(X)\\langle n\\rangle \\otimes h^n(X)\\langle n\\rangle \\rightarrow h^{2n}(X)\\langle 2n\\rangle \\xrightarrow{}h^0(k) \\simeq {\\mathbb {Q}},$ then $\\chi (Y/k)=\\chi (X/k)$ in ${\\operatorname{W}}(k)$ .",
"Presumably, merely having an isomorphism of motives $h^n(X)\\langle n\\rangle \\simeq h^m(Y)\\langle m\\rangle $ would not suffice to yield $\\chi (Y/k)=\\chi (X/k)$ in ${\\operatorname{W}}(k)$ , but we do not have an example.",
"Descent for the motivic Euler characteristic Let $k$ be a perfect field of characteristic different from two.",
"With the explicit formula for $\\chi (X/k)$ given by Theorem REF , we may find $\\chi (X/k)$ for forms $X$ of some $k$ -scheme $X_0$ by the usual twisting construction; this works for all manners of descent but we confine ourselves to the case of Galois descent here.",
"Let $X_0, X$ be smooth projective $k$ -schemes of even dimension $2n$ .",
"Let $K$ be a finite Galois extension field of $k$ with Galois group $G$ .",
"Let $X_K:=X\\times _kK$ , $X_{0K}:=X_0\\times _kK$ , and suppose we have an isomorphism $\\phi :X\\times _kK\\rightarrow X_0\\times _kK$ .",
"This gives us the cocycle $\\lbrace \\psi _\\sigma \\in {\\operatorname{Aut}}_K(X_0\\times _kK)\\rbrace _{\\sigma \\in G}$ where $\\psi _\\sigma :=\\phi ^\\sigma \\circ \\phi ^{-1}$ .",
"Letting $&b_0:{\\operatorname{H}}^n(X_0,\\Omega ^n_{X_0/k})\\times {\\operatorname{H}}^n(X_0,\\Omega ^n_{X_0/k})\\rightarrow k,\\\\&b:{\\operatorname{H}}^n(X,\\Omega ^n_{X/k})\\times {\\operatorname{H}}^n(X,\\Omega ^n_{X/k})\\rightarrow k$ denote the respective symmetric bilinear forms ${\\rm Tr}(x\\cup y)$ , the isomorphism $\\phi $ induces an isometry $\\phi ^*:({\\operatorname{H}}^n(X_{0K},\\Omega ^n_{X_{0K}/K}), b_{0K})\\rightarrow ({\\operatorname{H}}^n(X_{K},\\Omega ^n_{X_{K}/K}), b_{K}),$ and the cocycle $\\lbrace \\psi _\\sigma \\rbrace _{\\sigma \\in G}$ determines a cocycle $\\lbrace (\\psi _\\sigma ^*)^{-1} \\in \\mathrm {O}(b_0)(K)\\rbrace _{\\sigma \\in G}$ .",
"Twisting by the latter cocycle allows one to recovers $b$ from $b_0$ ; explicitly, this works as follows.",
"Firstly, as usual, one recovers the $k$ -vector space ${\\operatorname{H}}^n(X,\\Omega ^n_{X/k})$ from the $K$ -vector space ${\\operatorname{H}}^n(X_{0K},\\Omega ^n_{X_{0K}/K})$ as the $G$ -invariants for the map $x\\mapsto \\psi _\\sigma ^{*-1}(x^\\sigma )$ .",
"Secondly, letting $A\\in \\operatorname{GL}({\\operatorname{H}}^n(X_{0K},\\Omega ^n_{X_{0K}/K}))$ be a change of basis matrix comparing the $k$ -forms ${\\operatorname{H}}^n(X_0,\\Omega ^n_{X_0/k})\\subset {\\operatorname{H}}^n(X_{0K},\\Omega ^n_{X_{0K}/K})$ and ${\\operatorname{H}}^n(X,\\Omega ^n_{X/k})\\subset {\\operatorname{H}}^n(X_{0K},\\Omega ^n_{X_{0K}/K})$ , we recover $b$ (up to $k$ -isometry) as $b(x,y)=b_0(Ax, Ay)=:b_0^A(x,y).$ Having performed this twist at the level of symmetric bilinear forms, we may now pass Grothendieck-Witt classes to describe the Euler characteristic of $X$ : namely, Corollary REF (2) gives $\\chi (X_0/k)=[b_0+m\\cdot H],\\quad \\chi (X/k)=[b_0^A+m\\cdot H]$ in ${\\operatorname{GW}}(k)$ .",
"Remark 8.10 In the case of a smooth projective surface $S$ with $p_g(S)=0$ , over a characteristic zero field $k$ , the twisting construction reduces to a computation involving ${\\rm CH}^1(S_{\\bar{k}})/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ as a ${\\rm Gal}(k)$ -module; here $\\bar{k}$ is the algebraic closure of $k$ and $\\sim _{\\operatorname{num}}$ is numerical equivalence.",
"Indeed, the assumption $p_g(S)=0$ implies that the cycle class map in Hodge cohomology ${\\operatorname{cyc}}^\\mathrm {Hdg}:{\\rm CH}^1(S_{\\bar{k}})\\rightarrow {\\operatorname{H}}^1(S_{\\bar{k}}, \\Omega ^1_{S/\\bar{k}})$ induces an isomorphism ${{\\rm CH}^1(S_{\\bar{k}})/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}} \\otimes _{\\mathbb {Z}}\\bar{k}\\xrightarrow{} {\\operatorname{H}}^1(S_{\\bar{k}}, \\Omega ^1_{S/\\bar{k}}) \\simeq {\\operatorname{H}}^1(S, \\Omega ^1_{S/k})\\otimes _k\\bar{k}$ and the cycle class map ${\\operatorname{cyc}}^\\mathrm {Hdg}$ transforms the intersection product on ${\\rm CH}^1(S_{\\bar{k}})/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ to the quadratic form $b_0$ on ${\\operatorname{H}}^1(S_{\\bar{k}}, \\Omega ^1_{S/\\bar{k}})$ , induced by cup product and the trace map.",
"Thus, our quadratic form $b$ on ${\\operatorname{H}}^1(S, \\Omega ^1_{S/k})$ is equivalent to the one gotten by twisting the $\\bar{k}$ -linear extension of the intersection product on ${\\rm CH}^1(S_{\\bar{k}})/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ by the natural Galois action.",
"Analogous comments hold for a “geometrically singular” variety, by which we mean a smooth projective $k$ -scheme $X$ of dimension $2n$ such that ${\\operatorname{H}}^n(X_{\\bar{k}},\\Omega ^n_{X_{\\bar{k}}/\\bar{k}})$ is spanned by cycle classes, where we replace ${\\rm CH}^1/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ with ${\\rm CH}^n/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ .",
"For example, one could take a K3 surface with Picard rank 20 over $\\bar{k}$ or a cubic fourfold $X$ with ${\\operatorname{H}}^2(X_{\\bar{k}},\\Omega ^2_{X_{\\bar{k}}/k})\\simeq \\bar{k}^{21}$ spanned by algebraic cycles.",
"Examples 8.11 As as simple example, take $S$ to be a quadric surface in ${\\mathbb {P}}^3_k$ defined by a degree two homogeneous form $q(X_0,\\ldots , X_3)$ ; we may assume that $q$ is a diagonal form, $q(X_0,\\ldots , X_3)=a_0X_0^2+\\sum _{i=1}^3a_iX_i^2\\\\=a_0(X_0+\\sqrt{-a_1/a_0}X_1)(X_0-\\sqrt{-a_1}X_1)\\\\+a_2(X_2-\\sqrt{-a_3/a_2}X_3)(X_2+\\sqrt{-a_3/a_2}X_3).$ This trivializes ${\\rm CH}^1(S)$ over $K:=k(\\sqrt{-a_0a_1}, \\sqrt{-a_2a_3})$ , namely ${\\rm CH}^1(S)={\\mathbb {Z}}\\ell _1\\oplus {\\mathbb {Z}}\\ell _2$ with $\\ell _1$ defined by $(X_0-\\sqrt{-a_1}X_1)=(X_2-\\sqrt{-a_3/a_2}X_3)=0$ and $\\ell _2$ defined by $(X_0-\\sqrt{-a_1}X_1)=(X_2+\\sqrt{-a_3/a_2}X_3)=0$ .",
"Embedding ${\\rm Gal}(K/k)\\subset {\\rm Gal}(k(\\sqrt{-a_0a_1})/k)\\times {\\rm Gal}(k(\\sqrt{-a_2a_3})/k)=\\langle \\sigma _1\\rangle \\times \\langle \\sigma _2\\rangle $ , the Galois action is given by $\\sigma _1(\\ell _1, \\ell _2)=\\sigma _2(\\ell _1, \\ell _2)= (\\ell _2, \\ell _1)$ .",
"A Galois-invariant basis is thus given by $((\\ell _1+\\ell _2), \\sqrt{a_0a_1a_2a_3}(\\ell _1-\\ell _2))$ , and the intersection form in this basis has matrix $\\begin{pmatrix}2&0\\\\0&-2a_0a_1a_2a_3\\end{pmatrix}.$ In other words, $\\chi (S/k)=\\langle 2\\rangle +\\langle -2a_0a_1a_2a_3\\rangle $ .",
"Suppose $S$ is the blowup of ${\\mathbb {P}}^2_k$ along a 0-dimensional closed subscheme $Z\\subset {\\mathbb {P}}^2_k$ , with $Z$ étale over $k$ .",
"Let $\\ell $ denote the class of a line in ${\\rm CH}^1({\\mathbb {P}}^2)$ .",
"Writing $Z_{\\bar{k}}=\\lbrace p_1,\\ldots , p_r\\rbrace $ , we have ${\\rm CH}^1(S_{\\bar{k}}) \\simeq {\\mathbb {Z}}\\cdot \\ell \\oplus (\\oplus _{i=1}^r{\\mathbb {Z}}\\cdot p_r)$ , with the evident Galois action and with intersection form the diagonal matrix $(1, -1,\\ldots , -1)$ .",
"It is then easy to show that the twisted quadratic form $\\chi (S/k)$ is $\\langle 1\\rangle -{\\rm Tr}_{Z/k}(\\langle 1\\rangle )$ .",
"These last two examples have been computed by different methods before: (1) is a special case of [26] and (2) is a special case of [26].",
"Here is a more interesting example.",
"Example 8.12 Let $\\pi :S\\rightarrow C$ be a a conic bundle over a smooth projective curve $C$ , all defined over $k$ ; we assume for simplicity that $k\\subset .",
"Let $ ZC$ be the degeneracy locus of $$: that is, $ Z$ is the reduced proper closed subscheme of $ C$ over which $$ is not smooth.",
"For each geometric point $ z$ of $ Z$, the fiber $ -1(z)$ is isomorphic to two distinct lines in $ P2$: $ -1(z)=zz'$.",
"There is a ``double section^{\\prime \\prime } $ DS$ with $ DC$ a finite degree two morphism, and with $ Dz=1=Dz'$ for all $ zZ(k)$.$ Over $\\bar{k}$ , the bundle $S$ is isomorphic to the blow-up of a ${\\mathbb {P}}^1$ -bundle $\\bar{S}_{\\bar{k}}\\rightarrow C_{\\bar{k}}$ along a finite set $Z^{\\prime }\\subset \\bar{S}_{\\bar{k}}$ with $Z^{\\prime }\\xrightarrow{} Z_{\\bar{k}}$ via $\\pi $ .",
"Suppose $Z_{\\bar{k}}=\\lbrace z_1,\\ldots z_r\\rbrace $ .",
"If we fix a closed point $c_0\\in C\\setminus Z$ of degree $d$ over $k$ , we have the following basis for ${\\rm CH}^1(S_{\\bar{k}})_{\\mathbb {Q}}/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ : $\\ell _{z_1}-\\ell _{z_1}^{\\prime },\\ldots , \\ell _{z_r}-\\ell _{z_r}^{\\prime }, D, \\pi ^{-1}(c_0).$ We have the finite degree two extension $p:\\tilde{Z}\\rightarrow Z$ , where for each $z\\in Z$ , $p^{-1}(z)$ corresponds to the pair of lines $\\ell _z, \\ell _z^{\\prime }$ .",
"Let $L:=k(\\lbrace z_1,\\ldots , z_r\\rbrace )\\supset k$ and let $G:={\\operatorname{Aut}}(L/k)$ .",
"Writing $k(\\tilde{Z})=k(Z)(\\sqrt{\\delta })$ for some $\\delta \\in {\\mathcal {O}}_Z^\\times $ , we have a basis of ${\\rm CH}^1(S_L)_{\\mathbb {Q}}/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ given by $v_1,\\ldots , v_r, D, \\pi ^{-1}(c_0),$ with $v_i:=\\sqrt{d}(\\ell _{z_i}-\\ell _{z_i}^{\\prime })$ .",
"The intersection form on $\\langle v_1,\\ldots , v_r\\rangle $ is the diagonal matrix $(-4\\delta (z_1),\\ldots , -4\\delta (z_r))$ , the subspaces $\\langle v_1,\\ldots , v_r\\rangle $ and $\\langle D, \\pi ^{-1}(c_0)\\rangle $ are perpendicular and $\\langle D, \\pi ^{-1}(c_0)\\rangle $ is hyperbolic.",
"Moreover, the automorphism group ${\\operatorname{Aut}}(L/k)$ acts on $\\langle v_1,\\ldots , v_r\\rangle $ just as it does on $\\langle z_1,\\ldots , z_r\\rangle $ .",
"From this it follows that the twisted intersection form $b$ is given by $b=H-{\\rm Tr}_{k(Z)/k}(\\langle \\delta \\rangle ),$ and hence $\\chi (S/k)=m\\cdot H-{\\rm Tr}_{k(Z)/k}(\\langle \\delta \\rangle )$ with $m=2-\\operatorname{dim}_k{\\operatorname{H}}^0(S,\\Omega ^1_{S/k})-\\operatorname{dim}_k{\\operatorname{H}}^1(S,{\\mathcal {O}}_X)=\\operatorname{dim}_{\\mathbb {Q}}{\\operatorname{H}}^0(S^{\\rm an},{\\mathbb {Q}})-\\operatorname{dim}_{\\mathbb {Q}}{\\operatorname{H}}^1(S^{\\rm an},{\\mathbb {Q}})+1,$ where $S^{\\rm an}$ is the complex manifold associated to $S_.$ As a particular example, we may take $S$ to be a cubic surface $V\\subset {\\mathbb {P}}^3_k$ with a line $\\ell $ .",
"Projection from $\\ell $ realizes $V$ as a conic bundle $\\pi :V\\rightarrow {\\mathbb {P}}^1_k$ , with degeneracy locus $Z\\subset {\\mathbb {P}}^1_k$ a reduced closed subscheme of degree 5 over $k$ .",
"The above implies that the symmetric bilinear form $b_V$ is given by $b_V=H-{\\rm Tr}_{Z/k}(\\langle \\delta \\rangle )$ and computes $\\chi (S/k) = 2H-{\\rm Tr}_{Z/k}(\\langle \\delta \\rangle )$ .",
"Remark 8.13 In [8], Bayer-Fluckiger and Serre consider the finite $k$ -scheme $W$ representing the 27 lines on a cubic surface $V$ and compute the trace form ${\\rm Tr}_{W/k}(\\langle 1\\rangle )$ in [8].",
"They identify their form $q_{6,V}$ with the trace form on ${\\operatorname{H}}^1(V,\\Omega _{V/k})$ and show that ${\\rm Tr}_{W/k}(\\langle 1\\rangle )=\\lambda ^2b_V + (\\langle -1\\rangle -\\langle 2\\rangle )b_V+ 7 -\\langle -2\\rangle .$"
],
[
"Motivic Gauß-Bonnet",
"Definition 5.1 Let $p:V\\rightarrow X$ be a rank $r$ vector bundle on some $X\\in {\\mathrm {Sm}}_B$ , and let ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ be an $\\operatorname{SL}$ -oriented motivic ring spectrum.",
"The Euler class $e^{\\mathcal {E}}(V)\\in {\\mathcal {E}}^{2r,r}(X,\\operatorname{det}^{-1}V)$ is defined as $e^{\\mathcal {E}}(V):=s^*s_*(1^{\\mathcal {E}}_X);\\quad 1^{\\mathcal {E}}_X\\in {\\mathcal {E}}^{0,0}(X)\\text{ the unit.",
"}$ Remark 5.2 By Lemma REF , $e^{\\mathcal {E}}(V):=s^*s_*(1^{\\mathcal {E}}_X)=\\bar{s}^*{\\operatorname{th}}_V$ , where $\\bar{s}:X\\rightarrow {\\operatorname{Th}}_X(V)$ is the map induced by $s$ .",
"Theorem 5.3 (Motivic Gauß-Bonnet) Let ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ be an $\\operatorname{SL}$ -oriented motivic ring spectrum, $\\pi _X:X\\rightarrow B$ a smooth and projective $B$ -scheme, let $u_{\\mathcal {E}}:1_B\\rightarrow {\\mathcal {E}}$ be the unit map.",
"Then $\\pi _{X/B*}(e^{\\mathcal {E}}(T_{X/B}))=u_{{\\mathcal {E}}*}(\\chi (X/B))\\in {\\mathcal {E}}^{0,0}(B).$ We have the canonical Thom isomorphism $\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}: {\\mathcal {E}}^{a,b}(X;\\omega _{X/B})\\rightarrow {\\mathcal {E}}^{a-2\\operatorname{dim}X, b-\\operatorname{dim}X}({\\operatorname{Th}}(-T_{X/B})).$ By Lemma REF , it suffices to show that the map $\\beta _{X/B}^*:{\\mathcal {E}}^{0,0}(X)\\rightarrow {\\mathcal {E}}^{0,0}({\\operatorname{Th}}(-T_{X/B}))$ sends $1^{\\mathcal {E}}_X$ to $\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}(e^{\\mathcal {E}}(T_{X/B}))$ ; by Remark REF , this is the same as $\\vartheta _{-T_{X/B}}(\\bar{s}^*{\\operatorname{th}}_{T_{X/B}})$ , where $\\bar{s}:X_+\\rightarrow {\\operatorname{Th}}(T_{X/B})$ is the map induced by the zero-section $s:X\\rightarrow T_{X/B}$ .",
"We use our description of $\\beta _{X/B}$ as $\\pi _{X\\#}$ applied to the composition (REF ).",
"Applying ${\\rm Hom}_{{\\operatorname{SH}}(X)}(-,\\pi _X^*{\\mathcal {E}})$ to $\\beta _{X/B}$ and using the adjunction ${\\rm Hom}_{{\\operatorname{SH}}(B)}(\\pi _{X\\#}(-), {\\mathcal {E}})\\simeq {\\rm Hom}_{{\\operatorname{SH}}(X)}(-,\\pi _X^*{\\mathcal {E}})$ , $ \\beta _{X/B}^*$ is given by the composition ${\\mathcal {E}}^{0,0}(X){[r]^a_\\sim &} {\\rm Hom}_{{\\operatorname{SH}}(X)}(1_X, \\pi _X^*{\\mathcal {E}}){[r]^b_\\sim &}{\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{-T_{X/B}}\\circ \\Sigma ^{T_{X/B}}(1_X), \\pi _X^*{\\mathcal {E}})\\\\{[r]^c_\\sim &}{\\rm Hom}_{{\\operatorname{SH}}(X)}({\\operatorname{Th}}_X(T_{X/B})), \\Sigma ^{T_{X/B}}\\pi _X^*{\\mathcal {E}})\\\\\\xrightarrow{}{\\rm Hom}_{{\\operatorname{SH}}(X)}(1_X, \\Sigma ^{T_{X/B}}\\pi _X^*{\\mathcal {E}})\\simeq {\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{-T_{X/B}}({\\operatorname{Th}}_X(T_{X/B})),\\pi _X^*{\\mathcal {E}})$ where the isomorphisms $a,b, c$ are the canonical ones.",
"The functoriality of the canonical Thom isomorphisms gives us the commutative diagram ${{\\mathcal {E}}^{0,0}(X)[r]^-{\\vartheta _{T_{X/B}}}[d]^a_\\wr &{\\mathcal {E}}^{2d_X, d_X}({\\operatorname{Th}}(T_{X/B}), \\omega _{X/B})[dd]^{\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}}\\\\{\\rm Hom}_{{\\operatorname{SH}}(X)}(1_X, \\pi _X^*{\\mathcal {E}})[d]^b_\\wr &\\\\{\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{-T_{X/B}}\\Sigma ^{T_{X/B}}(1_X),\\pi _X^*{\\mathcal {E}})[r]_-c^-\\sim &{\\rm Hom}_{{\\operatorname{SH}}(X)}({\\operatorname{Th}}_X(T_{X/B}),\\Sigma ^{T_{X/B}}\\pi _X^*{\\mathcal {E}})}$ Thus $(c\\circ b\\circ a)(1^{\\mathcal {E}}_X)= \\vartheta ^{\\mathcal {E}}_{-T_{X/B}}({\\operatorname{th}}_{T_{X/B}}).$ Applying $\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}$ as above gives us the commutative diagram ${40pt}{{\\mathcal {E}}^{2d_X, d_X}({\\operatorname{Th}}(T_{X/B}), \\omega _{X/B}) [r]^-{\\bar{s}^*}[d]_{\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}}&{\\mathcal {E}}^{2d_X, d_X}(X, \\omega _{X/B})[d]^{\\vartheta ^{\\mathcal {E}}_{-T_{X/B}}}\\\\{\\rm Hom}_{{\\operatorname{SH}}(X)}({\\operatorname{Th}}_X(T_{X/B}), \\Sigma ^{T_{X/B}}\\pi _X^*{\\mathcal {E}})[r]_-{\\bar{s}^*}[d]_\\wr &{\\rm Hom}_{{\\operatorname{SH}}(X)}(1_X, \\Sigma ^{T_{X/B}}\\pi _X^*{\\mathcal {E}})[d]^\\wr \\\\{\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{-T_{X/B}}{\\operatorname{Th}}_X(T_{X/B}), \\pi _X^*{\\mathcal {E}})[r]_-{\\Sigma ^{-T_{X/B}}(\\bar{s}^*)}&{\\rm Hom}_{{\\operatorname{SH}}(X)}(\\Sigma ^{-T_{X/B}}(1_X), \\pi _X^*{\\mathcal {E}})}$ and thus $\\beta _{X/B}^*(1^{\\mathcal {E}}_X)=\\vartheta _{-T_{X/B}}(\\bar{s}^*({\\operatorname{th}}_{T_{X/B}}))=\\vartheta _{-T_{X/B}}(e^{\\mathcal {E}}(T_{X/B})),$ as desired."
],
[
"$\\operatorname{SL}$ -oriented cohomology theories",
"Our ultimate goal is to apply the Gauß-Bonnet theorem of § when projective pushforwards are defined on a representable cohomology theory in some concrete manner, not necessarily relying on the six-functor formalism.",
"For this, we need a suitable axiomatization for such theories; we use a modification of the axioms of Panin-Smirnov [38], [39].",
"As before, our base-scheme $B$ is a noetherian, separated scheme of finite Krull dimension.",
"Definition 6.1 We let ${\\mathrm {Sm}\\text{-}\\mathrm {L}}_B$ denote the category of triples $(X, Z, L)$ with $X$ in ${\\mathrm {Sm}}_B$ , $Z\\subset X$ a closed subset and $L\\rightarrow X$ a line bundle.",
"A morphism $(f,\\tilde{f}):(X, Z, L)\\rightarrow (Y, W, M)$ is a morphism $f:X\\rightarrow Y$ with $Z\\supset f^{-1}(W)$ , together with an isomorphism of line bundles $\\tilde{f}:L\\rightarrow f^*M$ .",
"We let ${\\mathrm {Sm}\\text{-}\\mathrm {L}}^{\\text{pr}}_B$ denote the category with the same objects as ${\\mathrm {Sm}\\text{-}\\mathrm {L}}_B$ , but with morphisms $(f; \\tilde{f}):(X,Z, L)\\rightarrow (Y, W, M)$ a proper morphism $f:X\\rightarrow Y$ in ${\\mathrm {Sm}}_B$ , with $f(Z)\\subset W$ , and $\\tilde{f}:L\\rightarrow f^*M$ an isomorphism of line bundles.",
"Definition 6.2 An $\\operatorname{SL}$ -oriented cohomology theory on ${\\mathrm {Sm}}_B$ consists of the following data: A functor $H^{*,*}:{\\mathrm {Sm}\\text{-}\\mathrm {L}}_B^{\\text{\\rm op}}\\rightarrow \\mathrm {BiGr}{\\mathrm {Ab}}$ , $(X, Z, L)\\mapsto H^{*,*}_Z(X;L)$ ; we often write $f^*$ for $H^{*,*}(f,\\tilde{f})$ .",
"A functor $H_{*,*}:{\\mathrm {Sm}\\text{-}\\mathrm {L}}^{\\text{pr}}_B\\rightarrow {\\operatorname{\\rm Gr}}{\\mathrm {Ab}}$ , $(X, Z, L)\\mapsto H_{*,*}^Z(X,L)$ ; we often write $f_*$ for $H_{*,*}(f,\\tilde{f})$ .",
"Natural isomorphisms, for $X$ of dimension $d_X$ $H^{2d_X-n, d_X-m}_Z(X,\\omega _{X/B}\\otimes L)\\xrightarrow{} H_{n,m}^Z(X, L).$ An element $1\\in H^{0,0}_B(B;{\\mathcal {O}}_B)$ .",
"For $x:=(X, Z, L), y:=(Y, W, M)$ in ${\\mathrm {Sm}\\text{-}\\mathrm {L}}_B$ , a bigraded cup product map $\\cup _{x,y}:H^{*,*}_Z(X,L)\\otimes H^{*,*}_W(Y,M)\\rightarrow H^{*,*}_{Z\\times W}(X\\times _BY, p_1^*L\\otimes p_2^*M)$ For $Z\\subset W$ closed subsets of an $X\\in {\\mathrm {Sm}}_B$ a bigraded boundary map $\\delta _{X, W, Z}^{*,*}:H^{*, *}_{Z\\setminus W}(X\\setminus W;j_W^*L)\\rightarrow H^{*+1,*}_W(X, L)$ We write $H^{*,*}(X,L)$ for $H^{*,*}_X(X,L)$ and $H^{*,*}_Z(X)$ for $H^{*,*}_Z(X,{\\mathcal {O}}_X)$ ; we use the analogous notation for $H_{*,*}$ .",
"We write $\\cup $ for $\\cup _{x,y}$ and $\\delta $ for $\\delta _{X,Z,L}$ when the context makes the meaning clear.",
"For $f:Y\\rightarrow X$ a proper map of relative dimension $d$ in ${\\mathrm {Sm}}_B$ , with $Z\\subset X$ , $W\\subset Y$ closed subsets with $f(W)\\subset Z$ and $L\\rightarrow X$ a line bundle, combining D2 and D3 gives us pushforward maps $f_*:H^{*, *}_W(Y, \\omega _{Y/B}\\otimes f^*L)\\rightarrow H^{*-2d, *-d}_Z(X, \\omega _{X/B}\\otimes L)$ defined as the composition $H^{*, *}_W(Y, \\omega _{Y/B}\\otimes f^*L)\\xrightarrow{}H_{2d_Y-*, d_Y-*}^W(Y, f^*L)\\\\\\xrightarrow{}H_{2d_Y-*, d_Y-*}^Z(X, L)\\xrightarrow{}H^{*-2d, *-d}_Z(X, \\omega _{X/B}\\otimes L).$ These data are required to satisfy the following axioms: $H^{*,*}$ and $H_{*,*}$ are additive: $H^{*,*}$ transforms disjoint unions to products and $H_{*,*}$ transforms disjoint unions to coproducts; in particular, $H^{*,*}_Z(\\emptyset ,L)=0$ and $H_{*,*}^Z(\\emptyset , L)=0$ .",
"Let ${Y^{\\prime }[d]_{f^{\\prime }}[r]^-{g^{\\prime }}&Y[d]^f\\\\X^{\\prime }[r]_-g&X}$ be a cartesian diagram in $\\mathrm {Sch}_B$ , with $X, Y, X^{\\prime }, Y^{\\prime }$ in ${\\mathrm {Sm}}_B$ (sometimes called a transverse cartesian diagram in ${\\mathrm {Sm}}_B$ ) and with $f, f^{\\prime }$ proper of relative dimension $d$ .",
"This gives us the isomorphism $f^{\\prime *}\\omega _{X^{\\prime }/X}\\simeq \\omega _{Y^{\\prime }/Y}.$ Let $Z\\subset X$ be a closed subset, let $W\\subset Y$ be a closed subset with $f(W)\\subset Z$ , let $Z^{\\prime }=g^{-1}(Z)$ , $W^{\\prime }=g^{\\prime -1}(W)$ .",
"Let $L\\rightarrow X$ be a line bundle on $X$ and let $L^{\\prime }=g^{\\prime *}(L)$ .",
"Then the diagram ${H^{*, *}_{W^{\\prime }}(Y^{\\prime }, \\omega _{Y^{\\prime }/B}\\otimes \\omega _{Y^{\\prime }/Y}^{-1}\\otimes g^{\\prime *} L^{\\prime })[d]^{ f^{\\prime }_*}&[l]_-{g^{\\prime *}}H^{*, *}_W(Y, \\omega _{Y/B}\\otimes f^*L)[d]^{ f_*}\\\\H^{*-2d, *-d}_{Z^{\\prime }}(X^{\\prime }, \\omega _{X^{\\prime }/B}\\otimes \\omega _{X^{\\prime }/X}^{-1}\\otimes L^{\\prime })&[l]^-{g^*}H^{*-2d, *-d}_Z(X, \\omega _{X/B}\\otimes L)}$ commutes.",
"For $Z\\subset W$ closed subsets of an $X\\in {\\mathrm {Sm}}_B$ , let $U=X\\setminus Z$ with inclusion $j:U\\rightarrow X$ .",
"For $L\\rightarrow X$ a line bundle, this gives us the morphisms $({\\operatorname{id}}, {\\operatorname{id}}):(X, W, L)\\rightarrow (X, Z, L)$ and $(j, {\\operatorname{id}}):(U, W\\setminus Z, j^*L)\\rightarrow (X, W, L)$ .",
"Then the sequence $\\ldots \\xrightarrow{}H^{*, *}_Z(X, L)\\rightarrow H^{*,*}_W(X, L)\\\\\\xrightarrow{} H^{*,*}_{W\\setminus Z}(U, j^*L)\\xrightarrow{}H^{*+1, *}_Z(X, L)\\rightarrow \\ldots $ is exact.",
"Moreover, the maps $\\delta _{Z, W, X}$ are natural with respect to the pullback maps $g^*$ and the proper pushforward maps $f_*$ .",
"Let $i:Y\\rightarrow X$ be a closed immersion in ${\\mathrm {Sm}}_B$ , let $W\\subset Y$ be a closed subset, $L\\rightarrow X$ a line bundle.",
"Let $Z=i(W)$ , giving the morphism $(i, {\\operatorname{id}}):(Y, W, i^*L)\\rightarrow (X, Z, L)$ in ${\\mathrm {Sm}\\text{-}\\mathrm {L}}^{\\text{pr}}_B$ .",
"Then $i_*:H_{*,*}^W(Y, i^*L)\\rightarrow H_{*,*}^Z(X, L)$ is an isomorphism.",
"The cup products $\\cup $ of D4 are associative with unit 1.",
"The maps $f^*$ and $f_*$ are compatible with cup products: $(f\\times g)^*(\\alpha \\cup _{x,y}\\beta )=f^*(\\alpha )\\cup _{x,y} g^*(\\beta )$ .",
"Moreover, using the isomorphisms of D3, the cup products induce products $\\cup ^{x,y}$ on $H_{*,*}$ and one has $(f\\times g)_*(\\alpha \\cup ^{x,y}\\beta )=f_*(\\alpha )\\cup ^{x,y} g_*(\\beta )$ .",
"Finally, the boundary maps $\\delta _{Z, W, X}$ are module morphism: retaining the notation of D4, for $\\alpha \\in H^{*, *}_{Z\\setminus W}(X\\setminus W;j_W^*L)$ and $\\beta \\in H^{*, *}_{T}(Y, M)$ , we have $\\delta _{X\\times Y, Z\\times T, W\\times T}(\\alpha \\cup \\beta )=\\delta _{X, Z, W}(\\alpha )\\cup \\beta .$ Let $i:Y\\rightarrow X$ be a closed immersion in ${\\mathrm {Sm}}_B$ of codimension $c$ , $\\pi _Y:Y\\rightarrow B$ the structure map.",
"Let $1^H_Y\\in H^{0,0}(Y)$ be the element $\\pi _Y^*(1)$ .",
"Then $\\vartheta (i):=\\alpha _{X, Y}(i_*(1^H_Y))\\in H^{2c, c}_Y(X, \\operatorname{det}^{-1} N_i)$ is central, that is, for each $(U, T, M)\\in {\\mathrm {Sm}\\text{-}\\mathrm {L}}_B$ , and each $\\beta \\in H^{*,*}_T(U, M)$ , we have $\\tau ^*(\\beta \\cup \\vartheta (i))=\\vartheta (i)\\cup \\beta $ where $\\tau :X\\times _BU\\rightarrow U\\times _BX$ is the symmetry isomorphism.",
"Let $(f,{\\operatorname{id}}):(Y, W, f^*L)\\rightarrow (X, Z, L)$ be a morphism in ${\\mathrm {Sm}\\text{-}\\mathrm {L}}_B$ .",
"Suppose that the induced map $f:Y_W/B\\rightarrow X_Z/B$ is an isomorphism in ${\\operatorname{SH}}(B)$ .",
"Then $f^*:H^{*,*}_Z(X, L)\\rightarrow H^{*,*}_W(Y, f^*L)$ is an isomorphism.",
"Remark 6.3 It may seem strange that the proper pushforward maps respect products in the sense of (A5); one might rather expect a projection formula.",
"However, (A5) asks that the proper pushforward maps respect external products, not cup products, and in fact, having the pushforward and pullback maps respect products as in (A5) implies the projection formula, as one sees by considering the commutative pentagon associated to a proper morphism $f:Y\\rightarrow X$ in ${\\mathrm {Sm}}_B$ of relative dimension $d$ : ${Y[dd]_f@/^40pt/[drr]^{\\gamma _f:=(f\\times {\\operatorname{id}}_X)\\circ \\Delta _Y}[r]_-{\\Delta _Y}&Y\\times _BY[dr]_{f\\times {\\operatorname{id}}_Y}\\\\&&X\\times _BY[dl]^{{\\operatorname{id}}_X\\times f}\\\\X[r]_-{\\Delta _X}&X\\times _BX}$ Note that the square ${Y[r]^-{\\gamma _f}[d]_f&X\\times _BY[d]^{{\\operatorname{id}}_X\\times f}\\\\X[r]_-{\\Delta _X}&X\\times _BX}$ is transverse cartesian.",
"If we have closed subsets $Z\\subset X$ , $W\\subset Y$ with $f(W)\\subset Z$ , and line bundle $L\\rightarrow X$ , the pentagon diagram induces the diagram in cohomology ${15pt}{H^{*,*}_W(Y,\\omega _{Y/B}\\otimes f^*L)Y[dd]_{f_*}&[l]^-{\\Delta _Y^*}H^{*,*}_{W\\times W}(Y\\times _BY, \\omega _{Y/B}\\otimes f^*L)\\\\\\ &&\\hspace{-43.0pt}H^{*,*}_{Z\\times W}(X\\times _BY, \\omega _{Y/B}\\boxtimes L)@/_55pt/[ull]_{\\gamma _f^*}[ul]_{\\ \\ (f\\times {\\operatorname{id}}_Y)^*}[dl]^{\\ \\ ({\\operatorname{id}}_X\\times f)_*}\\\\H^{*-2d, *-d}_Z(X, \\omega _{X/B}\\otimes L)&H^{*-2d, *-d}_{Z\\times Z}(X\\times _BX, \\omega _{X/B}\\otimes L)[l]^-{\\Delta ^*_X}}$ Take $\\alpha \\in H^{a,b}_Z(X, M)$ , $\\beta \\in H^{c,d}_W(Y, \\omega _{Y/B}\\otimes f^*(L\\otimes M^{-1}))$ .",
"By functoriality of $(-)^*$ and (A5) for $(-)^*$ we have $\\gamma _f^*(\\alpha \\cup _{X,Y} \\beta )=f^*(\\alpha )\\cup _Y\\beta $ and by (A2) and (A5) for $(-)_*$ we have $f_*(f^*(\\alpha )\\cup _Y\\beta )=\\Delta _X^*({\\operatorname{id}}_X\\times f)_*(\\alpha \\cup _{X,Y}\\beta )=\\alpha \\cup _X f_*(\\beta ).$ Similarly, in the presence of (A2) and (A5) for $(-)^*$ , functoriality for $(-)^*$ and $(-)_*$ and the projection formula implies (A5) for $(-)_*$ .",
"Definition 6.4 A twisted cohomology theory on ${\\mathrm {Sm}}_B$ is given by the data D1, D4, D5 above, satisfying the parts of the axioms A1, A3-A7 that only involve $H^{*,*}$ .",
"Given an $\\operatorname{SL}$ -oriented cohomology theory $(H^{*,*}, H_{*,*}, \\ldots )$ on ${\\mathrm {Sm}}_B$ , one has the underlying twisted cohomology theory $(H^{*,*}, \\ldots )$ by forgetting the proper pushforward maps.",
"Example 6.5 The primary example of an $\\operatorname{SL}$ -oriented cohomology theory on ${\\mathrm {Sm}}_B$ is the one induced by an $\\operatorname{SL}$ -oriented motivic spectrum ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ : $(X,Z,L)\\mapsto {\\mathcal {E}}^{*,*}_Z(X;L).$ One defines, for $X\\in {\\mathrm {Sm}}_B$ of dimension $d_X$ over $B$ , ${\\mathcal {E}}_{m,n}^Z(X;L):={\\mathcal {E}}^{2d_X-m, d_X-n}_Z(X;\\omega _{X/B}\\otimes L);$ we extend the definition to arbitrary $X\\in {\\mathrm {Sm}}_B$ by taking the sum over the connected components of $X$ and write this also as ${\\mathcal {E}}^{2d_X-m, d_X-n}_Z(X;\\omega _{X/B}\\otimes L)$ by considering $d_X$ as a locally constant functor on $X$ .",
"The pushforward maps for a proper morphism of relative dimension $d$ , $f:Y\\rightarrow X$ , closed subsets $W\\subset Y$ , $Z\\subset X$ with $f(W)\\subset Z$ and line bundle $L\\rightarrow X$ are given by the pushforward $f_*:{\\mathcal {E}}^{2d_Y-m,2d_Y-n}_W(Y;\\omega _{Y/B}\\otimes f^*L)\\rightarrow {\\mathcal {E}}^{2d_X-m,2d_X-n}_Z(X;\\omega _{X/B}\\otimes L).$"
],
[
"Comparison isomorphisms",
"We recall the element $\\eta \\in {\\rm Hom}_{{\\operatorname{SH}}(B)}(1_B, \\mathrm {S}^{-1,-1}\\wedge 1_B)$ induced by the map of $B$ -schemes $\\eta :{\\mathbb {A}}^2\\setminus \\lbrace 0\\rbrace \\rightarrow {\\mathbb {P}}^1$ , $\\eta (a,b)=(a:b)$ .",
"As every ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ is a module for $1_B$ , we have the map $\\times \\eta :{\\mathcal {E}}\\rightarrow \\mathrm {S}^{-1,-1}\\wedge {\\mathcal {E}}$ for each $x\\in {\\operatorname{SH}}(B)$ .",
"We say that $\\eta $ acts invertibly on ${\\mathcal {E}}$ if $\\times \\eta $ is an isomorphism in ${\\operatorname{SH}}(B)$ .",
"We consider the following situation: fix an $\\operatorname{SL}$ -oriented motivic spectrum ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ .",
"This gives us the twisted cohomology theory ${\\mathcal {E}}^{*,*}$ underlying the oriented cohomology defined by ${\\mathcal {E}}$ .",
"Let $({\\mathcal {E}}^{*,*}, \\tilde{{\\mathcal {E}}}_{*,*})$ be an extension of ${\\mathcal {E}}^{*,*}$ to an oriented cohomology theory on ${\\mathrm {Sm}}_B$ , in other words, we define new pushforward maps $\\hat{f}_*:{\\mathcal {E}}^{*,*}_W(Y, \\omega _{Y/B}\\otimes f^*L)\\rightarrow {\\mathcal {E}}^{*-2d,*-d}_Z(X, \\omega _{X/B}\\otimes L)$ The main result of this section is a comparison theorem.",
"Before stating the result we recall the decomposition of ${\\operatorname{SH}}(B)[1/2]$ into plus and minus parts.",
"We have the involution $\\tau :1_B\\rightarrow 1_B$ induced by the symmetry isomorphism $\\tau :{\\mathbb {P}}^1\\wedge {\\mathbb {P}}^1\\rightarrow {\\mathbb {P}}^1\\wedge {\\mathbb {P}}^1$ .",
"In ${\\operatorname{SH}}(B)[1/2]$ , this gives us the idempotents $({\\operatorname{id}}+\\tau )/2$ , $({\\operatorname{id}}-\\tau )/2$ , and so decomposes ${\\operatorname{SH}}(B)[1/2]$ into +1 and -1 “eigenspaces” for $\\tau $ : ${\\operatorname{SH}}(B)[1/2]={\\operatorname{SH}}(B)^+\\times {\\operatorname{SH}}(B)^-$ We decompose ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)[1/2]$ as ${\\mathcal {E}}={\\mathcal {E}}_+\\oplus {\\mathcal {E}}_-$ .",
"Theorem 7.1 Suppose the pushforward maps $f_*, \\hat{f}_*:{\\mathcal {E}}^{*,*}_W(Y, \\omega _{Y/B}\\otimes f^*L)\\rightarrow {\\mathcal {E}}^{*-2d,*-d}_Z(X, \\omega _{X/B}\\otimes L)$ agree for $W$ , $Z$ , $L$ , $X=V$ a vector bundle over $Y$ and $f:Y\\rightarrow V$ the zero-section.",
"Suppose in addition that one of the following conditions holds: the $\\operatorname{SL}$ -orientation of ${\\mathcal {E}}$ extends to a $\\operatorname{GL}$ -orientation; $\\eta $ acts invertibly on ${\\mathcal {E}}$ ; 2 acts invertibly on ${\\mathcal {E}}$ and ${\\mathcal {E}}_+^{-1,0}(U)=0$ for affine $U$ in ${\\mathrm {Sm}}_B$ .",
"Then $f_*=\\hat{f}_*$ for all $X, Y, Z, W, L, f$ for which the pushforward is defined.",
"By the standard argument of deformation to the normal cone, it follows that $f_*=\\hat{f}_*$ for all $f:Y\\rightarrow X$ a closed immersion, $Z, W, L$ .",
"As every proper map in ${\\mathrm {Sm}}_B$ is projective, $f$ admits a factorization $f=p\\circ i$ , with $i:Y\\rightarrow X\\times _B{\\mathbb {P}}^N$ a closed immersion and $p:X\\times _B{\\mathbb {P}}^N\\rightarrow X$ the projection.",
"By functoriality of the pushforward maps, it suffices to check that $p_*=\\hat{p}_*$ .",
"In case (i), this follows from the uniqueness assertion in [38].",
"Indeed, the cohomology theory associated to a $\\operatorname{GL}$ -oriented motivic spectrum ${\\mathcal {E}}$ satisfies the axioms of Panin-Smirnov and the associated Thom isomorphisms give rise to an “orientation” in the sense of [38], so we may apply the results cited.",
"We note that in [38] the base-scheme is ${\\rm Spec\\,}k$ , $k$ a field, so loc.",
"cit.",
"does not immediately apply to our setting of a more general base-scheme; we say a few words about the extension of this result to our base-scheme $B$ .",
"As a proper map $f:Y\\rightarrow X$ in ${\\mathrm {Sm}}_B$ is projective, one factors $f$ as $f=p\\circ i$ , with $i:Y\\rightarrow {\\mathbb {P}}^n_X$ a closed immersion and $p:{\\mathbb {P}}^n_X\\rightarrow X$ the projection.",
"The uniqueness for a closed immersion in ${\\mathrm {Sm}}_B$ reduces to the case of the zero-section of a vector bundle by the usual method of deformation to the normal bundle, and as the pushforward by the zero-section of our two theories are the same by assumption, we have agreement in the case of a closed immersion.",
"For the projection $p$ , the proof of [38] relies on [37], where for $p$ , using the projective bundle formula, the key point is to show that both pushforwards have the same value on the unit $1_{{\\mathbb {P}}^n_X}\\in {\\mathcal {E}}^{0,0}({\\mathbb {P}}^n_X)$ .",
"The proof of this relies on the formula for the pushforward of $1_{{\\mathbb {P}}^n_X}$ under the diagonal $\\Delta _{{\\mathbb {P}}^n_X}:{\\mathbb {P}}^n_X\\rightarrow {\\mathbb {P}}^n_X\\times _X{\\mathbb {P}}^n_X$ given by [37].",
"As $\\Delta _{{\\mathbb {P}}^n_X}$ is a closed immersion, the two pushforwards under $\\Delta _{{\\mathbb {P}}^n_X}$ agree, and the proof of the formula in [37] uses only formal properties of pushforward and pullback as expressed in our axioms, plus the projective bundle formula.",
"This latter in turn relies only on properties of the Thom class of ${\\mathcal {O}}(-1)$ and localization with respect to ${\\mathbb {A}}^m_X\\subset {\\mathbb {P}}^m_X$ , and thus we may use [37] in our more general setting.",
"The argument that the pushforward of $1_{{\\mathbb {P}}^n_X}$ under $p$ can be recovered from the formula for the pushforward of $1_{{\\mathbb {P}}^n_X}$ under $\\Delta _{{\\mathbb {P}}^n_X}$ is elementary and formal, and only uses the restriction of the two theories to ${\\mathrm {Sm}\\text{-}\\mathrm {L}}_X$ , and not the fact that these restrictions come from theories over $k$ .",
"Thus, the argument used in the proof of [37] may be used to prove our result in case (i).",
"In case (ii), we use Lemma REF below.",
"Indeed, if $N$ is odd, we may apply the closed immersion $X\\times _B{\\mathbb {P}}^N\\rightarrow X\\times _B{\\mathbb {P}}^{N+1}$ as a hyperplane, so we reduce to the case $N$ even, in which case both $p_*$ and $\\hat{p}_*$ are inverse to the map $i_*$ , where $i:X\\rightarrow X\\times _B{\\mathbb {P}}^N$ is the section associated to the point $(1:0:\\ldots :0)$ of ${\\mathbb {P}}^N$ .",
"In case (iii) we may work in the category ${\\operatorname{SH}}(B)[1/2]$ .",
"We decompose ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)[1/2]$ as ${\\mathcal {E}}={\\mathcal {E}}_+\\oplus {\\mathcal {E}}_-$ and similarly decompose the pushforward maps $f_*$ and $\\hat{f}_*$ .",
"By Lemma REF , $\\eta $ acts invertibly on ${\\operatorname{SH}}(B)^-$ and the projection of $\\eta $ to ${\\operatorname{SH}}(B)^+$ is zero.",
"By Lemma REF below, the $\\operatorname{SL}$ -orientation of ${\\mathcal {E}}$ induces an $\\operatorname{SL}$ -orientation on the projection ${\\mathcal {E}}^+$ that extends to a $\\operatorname{GL}$ -orientation.",
"By (i), this implies that $f_*^+=\\hat{f}_*^+$ .",
"By (ii), $f_*^-=\\hat{f}_*^-$ , so $f_*=\\hat{f}_*$ .",
"Lemma 7.2 ([2]) Let ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ be an $\\operatorname{SL}$ -oriented motivic spectrum on which $\\eta $ acts invertibly.",
"Let $0\\in {\\mathbb {P}}^N({\\mathbb {Z}})$ be the point $(1:0\\ldots :0)$ .",
"For $X\\in {\\mathrm {Sm}}_B$ , $L\\rightarrow X$ a line bundle and $Z\\subset X$ a closed subset, the pushforward map $i_*:{\\mathcal {E}}^{*-2N,*-N}_Z(X, \\omega _{X/B}\\otimes L)\\rightarrow {\\mathcal {E}}^{*,*}_{p^{-1}(Z)}(X\\times _B{\\mathbb {P}}^N, \\omega _{{\\mathbb {P}}^N/B}\\otimes p^*L)$ is an isomorphism.",
"Using a Mayer-Vietoris sequence, we see that the statement is local on $X$ for the Zariski topology, so we may assume that $L={\\mathcal {O}}_X$ .",
"If we prove the statement for the pair $(X,X)$ and $(X\\setminus Z, X\\setminus Z)$ the local cohomology sequence gives the result for $(X,Z)$ , thus we may assume that $Z=X$ , and we reduce to showing that $i_*:{\\mathcal {E}}^{*-2N,*-N}(X, \\omega _{X/B})\\rightarrow {\\mathcal {E}}^{*,*}(X\\times _B{\\mathbb {P}}^N, \\omega _{{\\mathbb {P}}^N/B})$ is an isomorphism.",
"This is [2] in case $B={\\rm Spec\\,}k$ , $k$ a field.",
"The proof over a general base-scheme is essentially the same, we say a few words about this generalization.",
"Most of the results that are used in the proof of loc.",
"cit.",
"are are already proved in the required generality here, for example, the Thom isomorphism (REF ) of Construction REF generalizes Ananyevskiy's construction [2] from $B={\\rm Spec\\,}k$ to general $B$ .",
"The proof of [2] relies also on [2], which in our setting reduces to the fact that for $X\\in \\mathrm {Sch}_B$ , and $u\\in \\Gamma (X,{\\mathcal {O}}_X^\\times )$ a unit, the automorphism of $X\\times {\\mathbb {P}}^1$ sending $(x,(t_0: t_1))$ to $(x, (ut_0, u^{-1}t_1)$ induces the identity on $X_+\\wedge {\\mathbb {P}}^1/X$ in ${\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(X)$ .",
"This follows by identifying ${\\mathbb {P}}^1_X$ with ${\\mathbb {P}}({\\mathbb {A}}^2_X)$ and noting that the diagonal matrix with entries $u, u^{-1}$ is an elementary matrix in $\\operatorname{GL}_2(\\Gamma (X,{\\mathcal {O}}_X))$ .",
"Lemma 7.3 Suppose that ${\\mathcal {E}}\\in {\\operatorname{SH}}(B)$ is $\\operatorname{SL}$ oriented and that ${\\mathcal {E}}^{-1,0}(U)=0$ for all affine $U$ in ${\\mathrm {Sm}}_B$ .",
"Then the induced $\\operatorname{SL}$ orientation on ${\\mathcal {E}}_+\\in {\\operatorname{SH}}(B)_+$ extends to a $\\operatorname{GL}$ orientation.",
"Let $u\\in \\Gamma (X, {\\mathcal {O}}_X^\\times )$ be a unit on some $X\\in {\\mathrm {Sm}}_B$ .",
"Then the map $\\times u:X\\times _B{\\mathbb {P}}^1\\rightarrow X\\times _B{\\mathbb {P}}^1;\\quad (x, [t_0:t_1])\\mapsto (x, [ut_0:t_1])$ induces the identity on $\\mathrm {S}^{2,1}\\wedge X/B$ in ${\\operatorname{SH}}(B)_+$ .",
"Indeed, let $[u]:X/B\\rightarrow X/B\\wedge {\\mathbb {G}}_m$ be the map induced by $u:X\\rightarrow {\\mathbb {G}}_m$ .",
"The argument given by Morel [32], that $\\times u/B={\\operatorname{id}}+\\eta [u]$ in case $B={\\rm Spec\\,}k$ , $k$ a field, is perfectly valid over a general base-scheme: this only uses the fact that for ${\\mathcal {X}}$ and ${\\mathcal {Y}}$ pointed spaces over $B$ , one has $\\Sigma ^\\infty _{\\mathrm {S}^1}{\\mathcal {X}}\\times _B{\\mathcal {Y}}\\simeq \\Sigma ^\\infty _{\\mathrm {S}^1}{\\mathcal {X}}\\oplus \\Sigma ^\\infty _{\\mathrm {S}^1}{\\mathcal {Y}}\\oplus \\Sigma ^\\infty _{\\mathrm {S}^1}({\\mathcal {X}}\\wedge {\\mathcal {Y}})$ and that the map $\\times _u:\\mathrm {S}^1\\wedge {\\mathbb {G}}_m\\wedge X_+\\rightarrow \\mathrm {S}^1\\wedge {\\mathbb {G}}_m\\wedge X_+$ is the $\\mathrm {S}^1$ -suspension of the composition $\\mathrm {S}^1\\wedge {\\mathbb {G}}_m\\wedge X_+\\xrightarrow{}\\mathrm {S}^1\\wedge ({\\mathbb {G}}_m\\times {\\mathbb {G}}_m)\\wedge X_+\\xrightarrow{}\\mathrm {S}^1\\wedge {\\mathbb {G}}_m\\wedge X_+$ where $\\mu :{\\mathbb {G}}_m\\times {\\mathbb {G}}_m\\rightarrow {\\mathbb {G}}_m$ is the multiplication.",
"As $\\eta $ goes to zero in ${\\operatorname{SH}}(B)_+$ , it follows that $\\times u/B={\\operatorname{id}}$ in ${\\operatorname{SH}}(B)_+$ .",
"Now take $g\\in \\Gamma (X, \\operatorname{GL}_n({\\mathcal {O}}_X))$ , let $u=\\operatorname{det}g$ , let $m_u\\in \\Gamma (X, \\operatorname{GL}_n({\\mathcal {O}}_X))$ be the diagonal matrix with entries $u, 1,\\ldots , 1$ and let $h=m_u^{-1}\\cdot g\\in \\Gamma (X,\\operatorname{SL}_n({\\mathcal {O}}_X))$ .",
"We have ${\\operatorname{Th}}_X({\\mathcal {O}}_X^n)=({\\mathbb {P}}^1)^{\\wedge n}\\wedge X_+.$ Since ${\\mathcal {E}}$ is $\\operatorname{SL}$ -oriented, the map ${\\operatorname{Th}}(h):{\\operatorname{Th}}_X({\\mathcal {O}}_X^n)\\rightarrow {\\operatorname{Th}}_X({\\mathcal {O}}_X^n)$ induces the identity on ${\\mathcal {E}}^{**}$ and thus ${\\operatorname{Th}}(g)^*={\\operatorname{Th}}(m_u)^*:{\\mathcal {E}}^{*,*}_+({\\operatorname{Th}}_X({\\mathcal {O}}_X^n))\\rightarrow {\\mathcal {E}}^{*,*}_+({\\operatorname{Th}}_X({\\mathcal {O}}_X^n))$ But as ${\\operatorname{Th}}(m_u)=(\\times u)\\wedge {\\operatorname{id}}$ , our previous computation shows that ${\\operatorname{Th}}(m_u)^*={\\operatorname{id}}$ .",
"Now let $V\\rightarrow X$ be a rank $r$ vector bundle on some $X\\in {\\mathrm {Sm}}_B$ , choose a trivializing affine open cover ${\\mathcal {U}}=\\lbrace U_i\\rbrace $ of $X$ and let $\\phi _i:V_{|U_i}\\rightarrow U_i\\times {\\mathbb {A}}^r$ be a local framing.",
"We have the suspension isomorphism ${\\operatorname{Th}}(V_{U_i})\\simeq {\\operatorname{Th}}(U_i\\times {\\mathbb {A}}^r)=\\Sigma _r U_{i+}$ giving the isomorphism $\\theta _i:{\\mathcal {E}}_+^{a,b}(U_i)\\rightarrow {\\mathcal {E}}^{2r+a, r+b}_{+0_{V_{|U_i}}}(V_{|U_i}).$ Since $\\operatorname{GL}_r({\\mathcal {O}}_{U_i})$ acts trivially on ${\\mathcal {E}}_+^{**}({\\operatorname{Th}}(U_i\\times {\\mathbb {A}}^r))$ , the isomorphism $\\theta _i$ is independent of the choice of framing $\\phi _i$ .",
"In addition, the assumption ${\\mathcal {E}}^{-1,0}(U_i\\cap U_j)=0$ implies ${\\mathcal {E}}^{2r-1, r}_{+0_{V_{|U_i\\cap U_j}}}(V_{|U_i\\cap U_j})=0$ for all $i,j$ .",
"By Mayer-Vietoris, the sections $\\theta _i(1_{U_i})\\in {\\mathcal {E}}^{2r, r}_{+0_{V_{|U_i}}}(V_{|U_i})$ uniquely extend to an element $\\theta _V\\in {\\mathcal {E}}^{2r, r}_{+0_V}(V)$ The independence of the $\\theta _i$ on the choice of framing and the uniqueness of the extension readily implies the functoriality of $\\theta _V$ and similarly implies the product formula $\\theta _{V\\oplus W}=p_1^*\\theta _V\\cup p_2^*\\theta _W$ .",
"By construction, $\\theta _V$ is the suspension of the unit over $U_i$ , another application of independence of the choice of framing and the uniqueness of the extension shows that this is the case over every open subset $U\\subset X$ for which $V_{|U}$ is the trivial bundle.",
"Finally, the independence and uniqueness shows that $V\\mapsto \\theta _V$ is an extension of the $\\operatorname{SL}$ orientation on ${\\mathcal {E}}_+$ induced by that of ${\\mathcal {E}}$ .",
"Lemma 7.4 For $u\\in \\Gamma (X,{\\mathcal {O}}_X^\\times )$ we have $[u]\\eta =\\eta [u]:\\Sigma ^\\infty _X̰_+\\rightarrow \\Sigma ^\\infty _X̰_+$ We use the decomposition $\\Sigma ^\\infty _X̰_+\\wedge {\\mathbb {G}}_m\\times {\\mathbb {G}}_m=\\Sigma ^\\infty _X̰_+\\wedge {\\mathbb {G}}_m\\oplus \\Sigma ^\\infty _X̰_+\\wedge {\\mathbb {G}}_m\\oplus \\Sigma ^\\infty _X̰_+\\wedge {\\mathbb {G}}_m\\wedge {\\mathbb {G}}_m$ Via this, $\\eta $ is the map $[s]\\wedge [t]\\mapsto [st]-[s]-[t]$ so $\\eta [u]$ sends $[t]$ to $[ut]-[u]-[t]$ and ${\\operatorname{id}}_{{\\mathbb {G}}_m}\\wedge \\eta [u]$ sends $[s]\\wedge [t]$ to $[s]\\wedge [ut]-[s]\\wedge [u]-[s]\\wedge [t]$ , so $[u]\\eta $ is given by $[s]\\wedge [t]\\mapsto [st]-[s]-[t]\\mapsto [u]\\wedge [st]-[u]\\wedge [s]-[u]\\wedge [t].$ We have the automorphism $\\xi $ of ${\\mathbb {G}}_m^{\\wedge 3}$ sending $[u]\\wedge [s]\\wedge [t]$ to $[s]\\wedge [t]\\wedge [u]$ .",
"We have the isomorphism in ${\\operatorname{H}}_{\\raisebox {0.1ex}{\\scalebox {0.6}{\\hspace{0.80002pt}\\bullet }}}(B)$ , $\\Sigma _{S_1}^6{\\mathbb {G}}_m^{\\wedge 3}\\simeq {\\mathbb {A}}^3/{\\mathbb {A}}^3\\setminus \\lbrace 0\\rbrace $ , via which $\\Sigma _{S_1}^6\\xi $ is induced by the linear map $(u,s,t)\\mapsto (s, t, u)$ .",
"As this latter linear map has matrix in the standard basis a product of elementary matrices, $\\Sigma _{\\mathrm {S}^1}^6\\xi $ is ${\\mathbb {A}}^1$ -homotopic to the identity, so after stabilizing, ${\\operatorname{id}}_{{\\mathbb {G}}_m}\\wedge \\eta [u]$ is the map $[s]\\wedge [t]\\mapsto [s]\\wedge [t]\\wedge [u]\\mapsto [u]\\wedge [s]\\wedge [t]\\mapsto [u]\\wedge [st]-[u]\\wedge [s]-[u]\\wedge [t]=[u]\\eta ([s]\\wedge [t]).$ Lemma 7.5 The projection $\\eta _-$ of $\\eta $ to ${\\operatorname{SH}}(B)_-$ is an isomorphism and the projection $\\eta _+$ of $\\eta $ to ${\\operatorname{SH}}(B)_+$ is zero.",
"Morel proves this in [32] in the case of a field, but the proof works in general.",
"In some detail, the map $\\tau $ is the map on ${\\mathbb {A}}^2/({\\mathbb {A}}^2\\setminus \\lbrace 0\\rbrace )$ induced by the linear map $(x, y)\\mapsto (y,x)$ .",
"The matrix identity $\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}\\cdot \\begin{pmatrix}1&0\\\\-1&1\\end{pmatrix}\\cdot \\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}\\cdot \\begin{pmatrix}-1&0\\\\0&1\\end{pmatrix}$ shows that the maps $(x, y)\\rightarrow (y,x)$ and $(x,y)\\mapsto (-x, y)$ are ${\\mathbb {A}}^1$ -homotopic.",
"By the arguments in Lemma REF , this latter map induces the map $1+\\eta [-1]=1+[-1]\\eta $ in ${\\operatorname{SH}}(B)$ , giving the identity $(1+\\eta [-1])_-=(1+[-1]\\eta )_-=-{\\operatorname{id}}\\Rightarrow \\eta \\cdot (-[-1]/2)= (-[-1]/2)\\cdot \\eta ={\\operatorname{id}}_{{\\operatorname{SH}}(B)_-}$ For $\\eta _+$ , the projector to ${\\operatorname{SH}}(B)_+$ is given by the idempotent $(1/2)(\\tau +1)=(1/2)(2+\\eta [-1])$ , so $\\eta _+=(1/2)\\eta \\cdot (2+\\eta [-1])$ .",
"Since the map $\\tau :{\\mathbb {P}}^1\\wedge {\\mathbb {P}}^1\\rightarrow {\\mathbb {P}}^1\\wedge {\\mathbb {P}}^1$ is $1+\\eta [-1]$ and ${\\mathbb {P}}^1=\\mathrm {S}^1\\wedge {\\mathbb {G}}_m$ , the symmetry $\\epsilon :{\\mathbb {G}}_m\\wedge {\\mathbb {G}}_m\\rightarrow {\\mathbb {G}}_m\\wedge {\\mathbb {G}}_m$ is $-(1+\\eta [-1])$ .",
"From our formula for $\\eta ([s]\\wedge [t])$ we see that $\\eta \\epsilon =\\eta $ which gives $\\eta \\cdot (2+\\eta [-1])=0$ ."
],
[
"Applications",
"In this section, we apply the motivic Gauß-Bonnet formula of § and the comparison results of § to various specific $\\operatorname{SL}$ -oriented cohomology theories, and thereby make computations of the motivic Euler characteristic $\\chi (X/B)$ in different contexts."
],
[
"Motivic cohomology and cohomology of the Milnor K-theory sheaves",
"We work over the base-scheme $B={\\rm Spec\\,}k$ , with $k$ a perfect field.",
"In ${\\operatorname{SH}}(k)$ we have the motivic cohomology spectrum ${\\operatorname{H}}{\\mathbb {Z}}$ representing Voevodsky's motivic cohomology (see e.g.",
"[28] for a construction valid in arbitrary characteristic).",
"By [49], there is a natural isomorphism ${\\operatorname{H}}{\\mathbb {Z}}^{a,b}(X)\\simeq {\\rm CH}^b(X, 2b-a)$ for $X\\in {\\mathrm {Sm}}_B$ , where ${\\rm CH}^b(X, 2b-a)$ is Bloch's higher Chow group [9].",
"${\\operatorname{H}}{\\mathbb {Z}}$ admits a localization sequence: for $i:Z\\rightarrow X$ a closed immersion of codimension $d$ in ${\\mathrm {Sm}}_k$ , there is a canonical isomorphism ${\\operatorname{H}}{\\mathbb {Z}}^{a,b}_Z(X)\\simeq {\\operatorname{H}}{\\mathbb {Z}}^{a-2d, b-d}(Z)$ See for example [10].",
"In particular, for $p:V\\rightarrow X$ a rank $r$ vector bundle over $X\\in {\\mathrm {Sm}}_k$ , we have the isomorphism ${\\operatorname{H}}{\\mathbb {Z}}^{2r, r}_{0_V}(V)\\simeq {\\operatorname{H}}{\\mathbb {Z}}^{0,0}(X)$ which gives us Thom classes $\\vartheta _V^{{\\operatorname{H}}{\\mathbb {Z}}}\\in {\\operatorname{H}}{\\mathbb {Z}}^{2r, r}_{0_V}(V)$ corresponding to the unit $1^{{\\operatorname{H}}{\\mathbb {Z}}}_X\\in {\\operatorname{H}}{\\mathbb {Z}}^{0,0}(X)$ .",
"Thus ${\\operatorname{H}}{\\mathbb {Z}}$ is a $\\operatorname{GL}$ -oriented motivic spectrum.",
"Let $X$ be a smooth projective $k$ -scheme of dimension $n$ over $k$ .",
"For a class $x\\in {\\operatorname{H}}{\\mathbb {Z}}^{2n, n}(X)$ , the isomorphism ${\\operatorname{H}}{\\mathbb {Z}}^{2n,n}(X)\\simeq {\\rm CH}^n(X,0)={\\rm CH}^n(X)$ allows one to represent $x$ as the class of a 0-cycle $\\tilde{x}=\\sum _in_i p_i$ , with the $p_i$ closed points of $X$ .",
"One has the degree $\\deg _k(p_i):=[k(p_i):k]$ and extending by linearity gives the degree $\\deg _k(\\tilde{x})$ , which one shows passes to rational equivalence to define a degree map $\\deg _k: {\\operatorname{H}}{\\mathbb {Z}}^{2n, n}(X)\\rightarrow {\\rm CH}^0({\\rm Spec\\,}k)={\\mathbb {Z}}.$ As a $\\operatorname{GL}$ -oriented theory, ${\\operatorname{H}}{\\mathbb {Z}}$ has Chern classes for vector bundles: $c_r(V)\\in {\\operatorname{H}}{\\mathbb {Z}}^{2r, r}(X)$ for $V\\rightarrow X$ a vector bundle over some $X\\in {\\mathrm {Sm}}_k$ and $r \\ge 0$ .",
"Theorem 8.1 Let $X\\in {\\mathrm {Sm}}_k$ be projective of dimension $d_X$ .",
"Then $u^{{\\operatorname{H}}{\\mathbb {Z}}}(\\chi (X/k))=\\deg _k(c_{d_X}(T_{X/k})).$ One has well-defined pushforward maps on ${\\rm CH}^*(-,*)$ for projective morphisms (see e.g.",
"[9]).",
"Via the isomorphism $ {\\operatorname{H}}{\\mathbb {Z}}^{a,b}(X)\\simeq {\\rm CH}^b(X, 2b-a)$ [49], this gives pushforward maps $\\hat{f}_*$ on ${\\operatorname{H}}{\\mathbb {Z}}^{*,*}$ for $f:Y\\rightarrow X$ a projective morphism in ${\\mathrm {Sm}}_B$ (see [9] for details), making $(X, Z)\\mapsto {\\operatorname{H}}{\\mathbb {Z}}^{*,*}_Z(X)$ a $\\operatorname{GL}$ -oriented cohomology theory on ${\\mathrm {Sm}}_k$ .",
"In addition, for $\\pi _X:X\\rightarrow {\\rm Spec\\,}k$ in ${\\mathrm {Sm}}_k$ projective of dimension $n$ , the map $\\hat{\\pi }_{X*}:{\\operatorname{H}}{\\mathbb {Z}}^{2n, n}(X)\\rightarrow {\\rm CH}^0({\\rm Spec\\,}k)={\\mathbb {Z}}$ is $\\deg _k$ , and for $i:Y\\rightarrow X$ a closed immersion, the map $\\hat{i}_*$ is given by the localization theorem, which readily implies that $\\hat{i}_*=i_*$ .",
"By our comparison theorem REF , which here really reduces to the theorem of Panin-Smirnov, it follows that $\\hat{f}_*=f_*$ for all projective $f$ .",
"Finally, one has $c_{d_X}=s^*s_*(1^{{\\operatorname{H}}{\\mathbb {Z}}}_X)=e^{{\\operatorname{H}}{\\mathbb {Z}}}(V)$ ([18]), so applying the motivic Gauß-Bonnet theorem REF gives the statement.",
"One can obtain the same result by using the cohomology of the Milnor K-theory sheaves as a bigraded cohomology theory.",
"The homotopy t-structure on ${\\operatorname{SH}}(k)$ has heart the abelian category of homotopy modules $\\operatorname{\\Pi _*}(k)$ (see [32] and [33] for details); we let ${\\operatorname{H}}_0:{\\operatorname{SH}}(k)\\rightarrow \\operatorname{\\Pi _*}(k)$ be the associated functor.",
"The fact that ${\\operatorname{H}}{\\mathbb {Z}}^{n,n}({\\rm Spec\\,}F) \\simeq \\mathrm {K}^\\mathrm {M}_n(F)$ for $F$ a field [35], [47] says that ${\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}$ is canonically isomorphic to the homotopy module $(\\mathrm {K}^\\mathrm {M}_n)_n$ , which is in fact a cycle module in the sense of Rost [44].",
"This gives us the isomorphism ${\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}^{a,b}(X)\\simeq {\\operatorname{H}}^a(X, {\\mathcal {K}}^\\mathrm {M}_b).$ The isomorphism $H^n(X, {\\mathcal {K}}^\\mathrm {M}_n)\\simeq {\\rm CH}^n(X)$ (a special case of Rost's formula for the Chow groups of a cycle module, [44]) gives us as above Thom classes $\\vartheta ^{\\mathrm {K}^\\mathrm {M}_*}(V)\\in {\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}^{2r,r}_{0_V}(V)$ , giving ${\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}$ a $\\operatorname{GL}$ -orientation.",
"As for ${\\operatorname{H}}{\\mathbb {Z}}^{*,*}$ , one has explicitly defined pushforward maps on ${\\operatorname{H}}^*(-, {\\mathcal {K}}^M_*)$ which give ${\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}^{*,*}$ the structure of a $\\operatorname{GL}$ -oriented cohomology theory on ${\\mathrm {Sm}}_k$ and for which the pushforward map for the zero-section of a vector bundle is given by the Thom isomorphism.",
"Since the pushforward on ${\\operatorname{H}}^n(X, {\\mathcal {K}}^\\mathrm {M}_n)$ agrees with the classical pushforward on ${\\rm CH}^n$ , we deduce the following using the same proof as for Theorem REF .",
"Theorem 8.2 Let $X\\in {\\mathrm {Sm}}_k$ be projective of dimension $d_X$ .",
"Then $u^{{\\operatorname{H}}_0{\\operatorname{H}}{\\mathbb {Z}}}(\\chi (X/k))=\\deg _k(c_{d_X}(T_{X/k})) \\text{ in }{\\rm CH}^0({\\rm Spec\\,}k)={\\mathbb {Z}}.$"
],
[
"Algebraic K-theory",
"We now let $B$ be any regular separated base-scheme of finite Krull dimension.",
"Algebraic K-theory on ${\\mathrm {Sm}}_B$ is represented by the motivic commutative ring spectrum ${\\operatorname{KGL}}\\in {\\operatorname{SH}}(B)$ (see [48]).",
"Just as for ${\\operatorname{H}}{\\mathbb {Z}}$ , the purity theorem ${\\operatorname{KGL}}_Z^{a,b}(X)\\simeq {\\operatorname{KGL}}^{a-2c, b-c}(Z)$ for $i:Z\\rightarrow X$ a closed immersion of codimension $d$ in ${\\mathrm {Sm}}_B$ (a consequence of Quillen's localization sequence for algebraic K-theory [41]) gives Thom class $\\vartheta ^{\\operatorname{KGL}}(V)\\in {\\operatorname{KGL}}^{2r,r}_{0_V}(V)$ for $V\\rightarrow X$ a rank $r$ vector bundle over $X\\in {\\mathrm {Sm}}_B$ , and makes ${\\operatorname{KGL}}$ a $\\operatorname{GL}$ -oriented motivic spectrum.",
"Explicitly, ${\\operatorname{KGL}}$ represents Quillen K-theory on ${\\mathrm {Sm}}_B$ via ${\\operatorname{KGL}}^{a,b}\\simeq \\mathrm {K}_{2b-a}$ and the Thom class for a rank $r$ vector bundle $p:V\\rightarrow X$ is represented by the Koszul complex ${\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})$ .",
"Here $\\text{can}:p^*V^\\vee \\rightarrow {\\mathcal {O}}_V$ is the dual of the tautological section ${\\mathcal {O}}_V\\rightarrow p^*V$ and ${\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})$ is the complex whose terms are given by ${\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})^{-r}=\\Lambda ^rp^*V^\\vee $ and whose differential $\\Lambda ^rp^*V^\\vee \\rightarrow \\Lambda ^{r-1}p^*V^\\vee $ is given with respect to a local framing of $V^\\vee $ by $d(e_{i_1}\\wedge \\ldots \\wedge e_{i_r})=\\sum _{j=1}^r(-1)^{j-1}\\text{can}(e_{i_j})\\cdot e_{i_1}\\wedge \\ldots \\wedge \\widehat{e_{i_j}}\\wedge \\ldots \\wedge e_{i_r}.$ This complex is a locally free resolution of $s_*({\\mathcal {O}}_X)$ , where $s:X\\rightarrow V$ is the zero-section.",
"Thus, by the identification of ${\\operatorname{KGL}}^{2r, r}_{0_V}(V)$ with the Grothendieck group of the triangulated category of perfect complexes on $V$ with support contained in $0_V$ , ${\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})$ gives rise to a class $[{\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})]\\in {\\operatorname{KGL}}^{2r, r}_{0_V}(V)$ which maps to $1_X$ under the purity isomorphism ${\\operatorname{KGL}}^{2r, r}_{0_V}(V)\\simeq {\\operatorname{KGL}}^{0,0}(X)$ , so that we indeed have $[{\\operatorname{Kos}}_V(p^*V^\\vee , \\text{can})] = \\vartheta ^{\\operatorname{KGL}}(V)$ .",
"Just as for motivic cohomology, one has explicit pushforward maps in K-theory given by Quillen's localization and devissage theorems identifying, for $X \\in {\\mathrm {Sm}}_B$ and $Z \\subseteq X$ a closed subscheme, the K-theory with support $\\mathrm {K}^Z(X)$ with the K-theory of the abelian category of coherent sheaves $\\operatorname{Coh}_Z$ on $Z$ , denoted $\\mathrm {G}(Z)$ .",
"For a projective morphism $f:Y\\rightarrow X$ , one has the pushforward map $\\hat{f}_*:\\mathrm {G}(Y)\\rightarrow \\mathrm {G}(X)$ defined by using a suitable subcategory of $\\operatorname{Coh}_Y$ on which $f_*$ is exact.",
"On $\\mathrm {K}_0$ , this recovers the usual formula $\\hat{f}_*([{\\mathcal {F}}])=\\sum _{j=0}^{\\operatorname{dim}Y}(-1)^j[\\mathrm {R}^jf_*({\\mathcal {F}})]$ for ${\\mathcal {F}}\\in \\operatorname{Coh}_Y$ .",
"Via the isomorphisms ${\\operatorname{KGL}}_Z^{a,b}(X)\\simeq \\mathrm {G}_{2b-a}(Z)$ , this gives pushforward maps $\\hat{f}_*$ for ${\\operatorname{KGL}}^{*,*}$ , defining a $\\operatorname{GL}$ -oriented cohomology theory on ${\\mathrm {Sm}}_B$ .",
"For $s:X\\rightarrow V$ the zero-section of a vector bundle, $\\hat{s}_*$ agrees with the pushforward $s_*$ using the Thom isomorphism/localization theorem, hence by our comparison theorem (again really the theorem of Panin-Smirnov), we have $\\hat{f}_*=f_*$ for all projective $f$ .",
"Theorem 8.3 Let $\\pi _X:X\\rightarrow B$ be a smooth projective morphism with $B$ a regular separated scheme of finite Krull dimension.",
"Then $u^{\\operatorname{KGL}}(\\chi (X/B))=\\sum _{j=0}^{\\operatorname{dim}_BX}\\sum _{i=0}^{\\operatorname{dim}_BX} (-1)^{i+j}[\\mathrm {R}^j\\pi _{X*}\\Omega _{X/B}^i]\\in \\mathrm {K}_0(B)={\\operatorname{KGL}}^{0,0}(B).$ Let $p:T_{X/B}\\rightarrow X$ denote the relative tangent bundle and let $s:X\\rightarrow T_{X/B}$ denote the zero-section.",
"We have $e^{\\operatorname{KGL}}(T_{X/B})=s^*({\\operatorname{th}}(T_{X/B}))=s^*({\\operatorname{Kos}}_{T_{X/B}}(p^*T_{X/B}^\\vee , \\text{can})).$ Since $T_{X/B}^\\vee =\\Omega _{X/B}$ , and $s^*(\\text{can})$ is the zero-map, it follows that, in $\\mathrm {K}_0(X)$ , $s^*({\\operatorname{Kos}}_{T_{X/B}}(p^*T_{X/B}^\\vee , \\text{can}))=\\sum _{i=0}^{\\operatorname{dim}_BX}(-1)^i[\\Omega _{X/B}^i],$ and thus $\\pi _{X*}(e^{\\operatorname{KGL}}(T_{X/B}))=\\sum _{j=0}^{\\operatorname{dim}_BX}\\sum _{i=0}^{\\operatorname{dim}_BX} (-1)^{i+j}[\\mathrm {R}^j\\pi _{X*}\\Omega _{X/B}^i].$ We conclude by applying the motivic Gauß-Bonnet theorem."
],
[
"Milnor-Witt cohomology and Chow-Witt groups",
"In this case, we again work over a perfect base-field $k$ .",
"The Milnor-Witt sheaves ${\\mathcal {K}}^\\mathrm {MW}_*$ constructed by Morel ([32], [33]) give rise to an $\\operatorname{SL}$ -oriented theory as follows.",
"Morel describes an isomorphism of ${\\mathcal {K}}^\\mathrm {MW}_0$ with the sheafification ${\\mathcal {GW}}$ of the Grothendieck-Witt rings[33] defines an isomorphism ${\\operatorname{GW}}(F)\\rightarrow \\mathrm {K}^\\mathrm {MW}(F)$ , $F$ a field.",
"[33] defines ${\\mathcal {K}}^\\mathrm {MW}_*$ as an unramified sheaf and it follows from [36] that ${\\mathcal {GW}}$ is an unramified sheaf.",
"From this it is not difficult to show that the isomorphism ${\\operatorname{GW}}(F)\\rightarrow \\mathrm {K}^\\mathrm {MW}(F)$ for fields extends to an isomorphism of sheaves.",
"; the map of sheaves of abelian groups ${\\mathbb {G}}_m\\rightarrow {\\mathcal {GW}}^\\times $ sending a unit $u$ to the one-dimensional form $\\langle u\\rangle $ allows one to define, for $L\\rightarrow X$ a line bundle, a twisted version ${\\mathcal {K}}^\\mathrm {MW}_*(L):={\\mathcal {K}}^\\mathrm {MW}_*\\times _{{\\mathbb {G}}_m}L^\\times $ as a Nisnevich sheaf on $X\\in {\\mathrm {Sm}}_k$ (see [33] or [11]).",
"One may use the Rost-Schmid complex for ${\\mathcal {K}}^\\mathrm {MW}_*(L)$ [33] to compute ${\\operatorname{H}}^*_Z(X, {\\mathcal {K}}^\\mathrm {MW}_*(L))$ for $Z\\subseteq X$ a closed subset, which gives a purity theorem: for $i:Z\\rightarrow X$ a codimension $d$ closed immersion in ${\\mathrm {Sm}}_k$ and $L\\rightarrow X$ a line bundle, there is a canonical isomorphism ${\\operatorname{H}}^*_Z(X, {\\mathcal {K}}^\\mathrm {MW}_*(L))\\simeq {\\operatorname{H}}^{*-d}(Z, {\\mathcal {K}}^\\mathrm {MW}_{*-d}(i^*L\\otimes \\operatorname{det}N_i)),$ where $N_i\\rightarrow Z$ is the normal bundle of $i$ .",
"Applying this to the zero-section of a rank $r$ vector bundle $p:V\\rightarrow X$ gives the isomorphism ${\\operatorname{H}}^0(X, {\\mathcal {GW}})\\simeq {\\operatorname{H}}^r_{0_V}(V, {\\mathcal {K}}^\\mathrm {MW}_r(p^*\\operatorname{det}^{-1} V));$ in particular, given an isomorphism $\\phi :\\operatorname{det}V\\rightarrow {\\mathcal {O}}_X$ , we obtain a Thom class $\\theta _{V, \\phi }\\in {\\operatorname{H}}^r_{0_V}(V, {\\mathcal {K}}^\\mathrm {MW}_r)$ corresponding to the unit section $1_X\\in {\\operatorname{H}}^0(X, {\\mathcal {GW}})$ .",
"On the other hand, Morel's computation of the 0th graded homotopy sheaf of the sphere spectrum ([32], [33]) gives an identification ${\\operatorname{H}}_0(1_k)\\simeq ({\\mathcal {K}}^\\mathrm {MW}_n)_{n\\in {\\mathbb {Z}}}$ in $\\operatorname{\\Pi _*}(k)$ , which then gives the natural isomorphism ${\\operatorname{H}}_0(1_k)^{a+b, b}_Z(X)\\simeq {\\operatorname{H}}^a_Z(X, {\\mathcal {K}}^\\mathrm {MW}_b).$ This is moreover compatible with twisting by a line bundle $p:L\\rightarrow X$ , on the ${\\operatorname{H}}_0(1_k)$ side using the Thom space construction ${\\operatorname{H}}_0(1_k)^{*, *}_Z(X;L):={\\operatorname{H}}_0(1_k)^{*+2, *+1}_Z(L)$ and on the Milnor-Witt cohomology side using the twisted Milnor-Witt sheaves.",
"To see this, note that the “untwisted” isomorphism gives us an isomorphism ${\\operatorname{H}}_0(1_k)^{a+b+2, b+1}_{Z}({\\operatorname{Th}}(L))\\simeq {\\operatorname{H}}_0(1_k)^{a+b+2, b+1}_{0_L\\cap p^{-1}(Z)} (L)\\simeq {\\operatorname{H}}^{a+1}_{0_L\\cap p^{-1}(Z)}(X, {\\mathcal {K}}^\\mathrm {MW}_{b+1}),$ so it suffices to identify the right-hand side with ${\\operatorname{H}}^a_Z(X,{\\mathcal {K}}^\\mathrm {MW}_b(L))$ .",
"For $Y\\in {\\mathrm {Sm}}_k$ and line bundle $M\\rightarrow Y$ , the Rost-Schmid complex for ${\\mathcal {K}}^\\mathrm {MW}_m(M)$ consists in degree $a$ of sums of terms of twisted Milnor-Witt groups of the form $\\mathrm {K}^\\mathrm {MW}_{m-a}(k(y); \\Lambda ^a(\\mathfrak {m}_y/\\mathfrak {m}_y^2)^\\vee \\otimes M)$ , for $y$ a codimension $a$ point of $Y$ and $\\mathfrak {m}_y\\subset {\\mathcal {O}}_{Y,y}$ the maximal ideal.",
"To compute cohomology with supports in $W\\subseteq Y$ , one restricts to those $y\\in W$ .",
"If we now take $Y=L$ and $M$ the trivial bundle, with supports in $p^{-1}(Z)\\cap 0_L$ and $m=b+1$ , and compare with $Y=X$ , with supports in $Z$ with $m=b$ , the term for $y\\in Z$ , of codimension $a+1$ on $L$ is $\\mathrm {K}^\\mathrm {MW}_{b-a}(k(y); \\Lambda ^a(\\mathfrak {m}_y/\\mathfrak {m}_y^2)^\\vee \\otimes L)$ while the term for $y\\in Z$ , of codimension $a$ on $X$ is $\\mathrm {K}^\\mathrm {MW}_{b-a}(k(y); \\Lambda ^a(\\mathfrak {m}_y/\\mathfrak {m}_y^2)^\\vee )$ , where $\\mathfrak {m}_y$ is the maximal ideal in ${\\mathcal {O}}_{X,y}$ in both cases.",
"This gives the desired identification ${\\operatorname{H}}^{a+1}_{0_L\\cap p^{-1}(Z)}(X, {\\mathcal {K}}^\\mathrm {MW}_{b+1})\\simeq {\\operatorname{H}}^{a}_{Z}(X, {\\mathcal {K}}^\\mathrm {MW}_{b}(L)).$ The purity isomorphism (REF ) is a special case of this construction.",
"The Thom class $\\theta _{V, \\phi }\\in {\\operatorname{H}}^r_{0_V}(V, {\\mathcal {K}}^\\mathrm {MW}_r)$ gives the Thom class $\\theta _{V, \\phi }\\in {\\operatorname{H}}_0(1_k)^{2r, r}_{0_V}(V),$ making ${\\operatorname{H}}_0(1_k)$ an $\\operatorname{SL}$ -oriented theory (see e.g [26]).",
"The resulting canonical Thom class ${\\operatorname{th}}_V\\in {\\operatorname{H}}_0(1_k)^{2r, r}_{0_V}(V;\\operatorname{det}^{-1}V)={\\operatorname{H}}^r_{0_V}(V, {\\mathcal {K}}^\\mathrm {MW}_r(\\operatorname{det}^{-1} V))$ agrees with the image of $1_X\\in {\\operatorname{H}}^0(X, {\\mathcal {GW}})$ under the Rost-Schmid isomorphism (REF ).",
"Let $\\pi _X:X\\rightarrow {\\rm Spec\\,}k$ be smooth and projective over $k$ of dimension $d$ .",
"Using the Rost-Schmid complex for the twisted homotopy module one has generators for ${\\operatorname{H}}^d(X, {\\mathcal {K}}^\\mathrm {MW}_d(\\omega _{X/k}))$ as formal sums $\\tilde{x}=\\sum _i\\alpha _i\\cdot p_i$ , with $\\alpha _i\\in {\\operatorname{GW}}(k(p_i))$ and $p_i\\in X$ closed points.",
"Since $k$ is perfect, the finite extension $k(p_i)/k$ is separable and one can define $\\widetilde{\\deg }_k(\\tilde{x}):=\\sum _i{\\rm Tr}_{k(p_i)/k}\\alpha _i\\in {\\operatorname{GW}}(k)$ where ${\\rm Tr}_{k(p_i)/k}:{\\operatorname{GW}}(k(p_i))\\rightarrow {\\operatorname{GW}}(k)$ is the transfer induced by the usual trace map ${\\rm Tr}_{k(p_i)/k}:k(p_i)\\rightarrow k$ (see for example [11] ).",
"It is shown in [11] that this descends to a map $\\widetilde{\\deg }_k:{\\operatorname{H}}_0^{2d,d}(X;\\omega _{X/k})={\\operatorname{H}}^d(X, {\\mathcal {K}}^\\mathrm {MW}_d(\\omega _{X/k}))\\rightarrow {\\operatorname{H}}_0^{0,0}({\\rm Spec\\,}k)={\\operatorname{GW}}(k)$ See also [21], which identifies this map with one induced by the Scharlau trace.",
"The methods of this paper give a new proof of the result given in [26]: Theorem 8.4 Let $k$ be a perfect field of characteristic different from two.",
"For $\\pi _X:X\\rightarrow {\\rm Spec\\,}k$ smooth and projective over $k$ , we have $\\chi (X/k)=\\widetilde{\\deg }_k(e^{{\\operatorname{H}}_0(1_k)}(T_{X/k})).$ Under Morel's isomorphism ${\\rm End}_{{\\operatorname{SH}}(k)}(1_X)\\simeq {\\operatorname{GW}}(k)$ ([32], [33]) and the isomorphism ${\\operatorname{H}}^0({\\rm Spec\\,}k, {\\mathcal {K}}^\\mathrm {MW}_0)\\simeq {\\operatorname{GW}}(k)$ , the unit map $u^{{\\operatorname{H}}_0(1_k)}:1_k\\rightarrow {\\operatorname{H}}_0(1_k)$ induces the identity map on $\\pi _{0,0}$ .",
"Using this, the proof of the claim is essentially the same as the other Gauß-Bonnet theorems we have discussed, but with a bit of extra work since we are no longer in the GL-oriented case.",
"Fasel [15] has defined pushforward maps $\\hat{f}_*:{\\operatorname{H}}^a_W(X, {\\mathcal {K}}^\\mathrm {MW}_b(\\omega _{X/k}\\otimes f^*L))\\rightarrow {\\operatorname{H}}^{a-d}_Z(Y, {\\mathcal {K}}^\\mathrm {MW}_{b-d}(L))$ for each projective morphism $f:X\\rightarrow Y$ in ${\\mathrm {Sm}}_k$ of relative dimension $d$ , line bundle $L\\rightarrow Y$ , and closed subsets $Z\\subseteq Y$ , $W\\subseteq X$ with $f(W)\\subseteq Z$ .",
"In the case of the structure map $\\pi _X:X\\rightarrow {\\rm Spec\\,}k$ , the pushforward $\\tilde{\\pi }_{X*}:{\\operatorname{H}}^d(X, {\\mathcal {K}}^\\mathrm {MW}_d(\\omega _{X/k}))\\rightarrow {\\operatorname{H}}^0({\\rm Spec\\,}k, {\\mathcal {K}}^\\mathrm {MW}_0)={\\operatorname{GW}}(k)$ is the map $\\widetilde{\\deg }_k$ .",
"For $s:X\\rightarrow V$ the zero-section of a vector bundle, $\\hat{s}_*$ is the Thom isomorphism $s_*$ .",
"Thus, if we pass to the $\\eta $ -inverted theory, ${\\operatorname{H}}_0(1_X)_\\eta :={\\operatorname{H}}_0(1_k)[\\eta ^{-1}]$ , our comparison theorem REF says that $\\hat{f}_{\\eta *}=f_{\\eta *}$ for all projective morphisms $f$ in ${\\mathrm {Sm}}_k$ .",
"We have ${\\mathcal {K}}^\\mathrm {MW}_*[\\eta ^{-1}]\\simeq {\\mathcal {W}}$ , the sheaf of Witt rings, and the map ${\\mathcal {K}}^\\mathrm {MW}_0={\\mathcal {GW}}\\rightarrow {\\mathcal {K}}^\\mathrm {MW}_*[\\eta ^{-1}]\\simeq {\\mathcal {W}}$ is the canonical map $q:{\\mathcal {GW}}\\rightarrow {\\mathcal {W}}$ realizing ${\\mathcal {W}}$ as the quotient of ${\\mathcal {GW}}$ by the subgroup generated by the hyperbolic form.",
"Thus, applying our motivic Gauß-Bonnet theorem gives the identity $q(\\chi (X/k))=q(\\widetilde{\\deg }_k(e^{{\\operatorname{H}}_0(1_k)}(T_{X/k})))\\text{ in }{\\operatorname{W}}(k).$ To lift this to an equality in ${\\operatorname{GW}}(k)$ and thereby complete the proof, we use that the map $({\\operatorname{\\text{rnk}}}, q): {\\mathcal {GW}}\\rightarrow {\\mathbb {Z}}\\times {\\mathcal {W}}$ is injective, together with the fact that we can recover the rank by applying ${\\operatorname{H}}_0$ to the unit map $1_k\\rightarrow {\\operatorname{H}}{\\mathbb {Z}}$ and using Theorem REF ."
],
[
"Hermitian K-theory and Witt theory",
"We again let our base-scheme $B$ be a regular noetherian separated base-scheme of finite Krull dimension, but now assume that 2 invertible on $B$ .",
"Our goal in this subsection is to explain how the description of the “rank” of $\\chi (X/B)$ given by Theorem REF can be refined to give a formula for $\\chi (X/k)$ itself in terms of Hodge cohomology by using hermitian K-theory.",
"By work of Panin-Walter [40], Schlichting [45], and Schlichting-Tripathi [46], hermitian K-theory ${\\operatorname{KO}}^{[*]}_*(-)$ is represented by a motivic commutative ring spectrum ${\\operatorname{BO}}\\in {\\operatorname{SH}}(B)$ (we use the notation of [3]).",
"Panin-Walter give ${\\operatorname{BO}}$ an $\\operatorname{SL}$ -orientation.",
"${\\operatorname{BO}}$ -theory also represents particular cases of Schlichting's Grothendieck-Witt groups [45], via functorial isomorphisms ${\\operatorname{BO}}^{2r, r}(X;L)\\simeq {\\operatorname{KO}}^{[r]}_0(X, L):={\\operatorname{GW}}({\\operatorname{perf}}(X), L[r], \\text{can}),$ where $L\\rightarrow X$ is a line bundle and ${\\operatorname{GW}}({\\operatorname{perf}}(X), L[r], \\text{can})$ is the Grothendieck-Witt group of $L[r]$ -valued symmetric bilinear forms on ${\\operatorname{perf}}(X)$; we recall a version of the definition here.",
"Definition 8.5 Let $L\\rightarrow X$ be a line bundle and let $r \\in {\\mathbb {Z}}$ .",
"An $L[r]$ -valued symmetric bilinear form on $C\\in {\\operatorname{perf}}(X)$ is a map $\\phi :C\\otimes ^{\\mathrm {L}}C\\rightarrow L[r]$ in ${\\operatorname{perf}}(X)$ which satisfies the following conditions.",
"$\\phi $ is non-degenerate: the induced map $C\\rightarrow {\\mathcal {RH}om}(C, L[n])$ is an isomorphism in ${\\operatorname{perf}}(X)$ .",
"$\\phi $ is symmetric: $\\phi \\circ \\tau =\\phi $ , where $\\tau :C\\otimes ^{\\mathrm {L}}C\\rightarrow C\\otimes ^{\\mathrm {L}}C$ is the commutativity isomorphism.",
"(Note that we are assuming non-degeneracy in the definition but leaving this out of the terminology for the sake of brevity.)",
"Similar to the case of algebraic K-theory discussed in §REF , for a rank $r$ vector bundle $p:V\\rightarrow X$ , the Thom class $\\theta ^{\\operatorname{BO}}_V\\in {\\operatorname{BO}}^{2r,r}(V; p^*\\operatorname{det}^{-1}V)$ is given by the Koszul complex ${\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )$ , where the symmetric bilinear form $\\phi _V:{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )\\otimes {\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )\\rightarrow p^*\\operatorname{det}^{-1}V[r]=\\Lambda ^rV^\\vee [r]$ is given by the usual exterior product $-\\wedge -: \\Lambda ^iV^\\vee \\otimes \\Lambda ^{r-i}V^\\vee \\rightarrow \\Lambda ^rV^\\vee .$ Moreover, there are isomorphisms for $i<0$ ${\\operatorname{BO}}^{2r-i, r}(X;L)\\simeq {\\operatorname{W}}^{r-i}({\\operatorname{perf}}(X), L[r], \\text{can})$ where ${\\operatorname{W}}^{r-i}({\\operatorname{perf}}(X), L[r], \\text{can})$ is Balmer's triangulated Witt group.",
"In the case $B = {\\rm Spec\\,}k$ for $k$ a field of characteristic different from two, Ananyevskiy [3] shows that this isomorphism induces an isomorphism of $\\eta $ -inverted hermitian K-theory with Witt-theory, ${\\operatorname{BO}}[\\eta ^{-1}]^{*,*}\\simeq {\\operatorname{W}}^*[\\eta ,\\eta ^{-1}],$ where one gives $\\eta $ bidegree $(-1,-1)$ and an element $\\alpha \\eta ^n$ with $\\alpha \\in {\\operatorname{W}}^m$ has bidegree $(m-n,-n)$ ; the same proof works over out general base $B$ (with assumptions as at the beginning of this subsection).",
"For $f:Y\\rightarrow X$ a proper map of relative dimension $d_f$ in ${\\mathrm {Sm}}_B$ and $L$ a line bundle on $X$ , we follow Calmès-Hornbostel [12] in defining a pushforward map $\\hat{f}_*:{\\operatorname{BO}}^{2r,r}(Y, \\omega _{Y/B}\\otimes f^*L)\\rightarrow {\\operatorname{BO}}^{2r-2d_f,r-d_f}(X, \\omega _{X/B}\\otimes L)$ by Grothendieck-Serre duality.",
"In op.",
"cit., this is worked out for the $\\eta $ -inverted theory ${\\operatorname{BO}}_\\eta $ when the base is a field; however, the same construction works for ${\\operatorname{BO}}$ over the general base-scheme $B$ and goes as follows.",
"For $r \\ge 0$ , given an $L[r]$ -valued symmetric bilinear form $\\phi :C\\otimes ^{\\mathrm {L}}C\\rightarrow \\omega _{Y/B}\\otimes f^*L[r]$ , we have the corresponding isomorphism $\\tilde{\\phi }:C\\rightarrow {\\mathcal {RH}om}(C, \\omega _{Y/B}\\otimes f^*L[r]) \\simeq {\\mathcal {RH}om}(C, \\omega _{Y/X}\\otimes f^*(\\omega _{X/B}\\otimes L[r]))$ Grothendieck-Serre duality gives the isomorphism $\\mathrm {R}f_*{\\mathcal {RH}om}(C, \\omega _{Y/X}\\otimes f^*(\\omega _{X/B}\\otimes L[r])){[r]^{\\psi }_\\sim &}{\\mathcal {RH}om}(\\mathrm {R}f_*C, \\omega _{X/B}\\otimes L[r-d_f]).$ Composing these, we obtain the isomorphism $\\psi \\circ \\tilde{\\phi }:\\mathrm {R}f_*C\\rightarrow {\\mathcal {RH}om}(\\mathrm {R}f_*C, \\omega _{X/B}\\otimes L[r-d_f]),$ corresponding to the (nondegenerate) bilinear form $\\mathrm {R}f_*(\\phi ):\\mathrm {R}f_*C\\otimes ^{\\mathrm {L}}\\mathrm {R}f_*C\\rightarrow \\omega _{X/B}\\otimes L[r-d_f],$ which one can show is symmetric.",
"We explicitly define the above pushforward map by setting $\\hat{f}_*(C, \\phi ) := (\\mathrm {R}f_*C, \\mathrm {R}f_*(\\phi ))$ .",
"Applying this in the situation that $f=\\pi _X:X\\rightarrow B$ is a smooth and proper $B$ -scheme of relative dimension $d_X$ , we may obtain the formula $\\hat{\\pi }_{X*}(e^{\\operatorname{BO}}(T_{X/B}))=(\\oplus _{i,j=0}^{\\operatorname{dim}_BX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i], {\\rm Tr}),$ where ${\\rm Tr}:(\\oplus _{i,j=0}^{\\operatorname{dim}_kX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i])\\otimes (\\oplus _{i,j=0}^{\\operatorname{dim}_kX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i])\\rightarrow {\\mathcal {O}}_B$ is the symmetric bilinear form in ${\\operatorname{perf}}(B)$ determined by the pairings $(\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)\\otimes (\\mathrm {R}^{d_X-i}\\pi _{X*}\\Omega _{X/B}^{d_X-j})\\xrightarrow{}\\mathrm {R}^{d_X}\\pi _{X*}\\Omega _{X/B}^{d_X} \\xrightarrow{}{\\mathcal {O}}_B.$ Indeed, if $s:X\\rightarrow T_{X/B}$ denotes the zero-section, we have $e^{\\operatorname{BO}}(T_{X/B})=s^*({\\operatorname{Kos}}(T_{X/B}), \\phi )=(\\oplus _{j=0}^{d_X}\\Omega ^j_{X/B}[j],s^*\\phi )$ with $s^*\\phi $ determined by the product maps $\\Omega ^j_{X/B}[j]\\otimes \\Omega ^{d_X-j}_{X/B}[d_X-j]\\rightarrow \\omega _{X/B}[d_X],$ and thus $\\hat{\\pi }_{X*}(e^{\\operatorname{BO}}(T_{X/B}))$ is $\\oplus _{i,j=0}^{\\operatorname{dim}_kX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i]$ with the symmetric bilinear form ${\\rm Tr}$ as described above.",
"Passing to the $\\eta $ -inverted theory ${\\operatorname{BO}}_\\eta $ , our comparison theorem REF gives $q\\circ \\hat{\\pi }_{X*}=q\\circ \\pi _{X*}$ as maps ${\\operatorname{BO}}_\\eta ^{2d_X, d_X}(X,\\omega _{X/B})\\rightarrow {\\operatorname{BO}}_\\eta (B)$ .",
"We check that the conditions of the comparison theorem hold just as we did for algebraic K-theory.",
"Firstly, as mentioned above, the $\\operatorname{SL}$ -orientation for ${\\operatorname{BO}}$ defined by Panin-Walter can be described as follows: the Thom class for an oriented vector bundle ($p:V\\rightarrow X$ , $\\rho :{\\mathcal {O}}_X\\xrightarrow{}\\operatorname{det}V$ ) is given by the Koszul complex ${\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )$ equipped with the symmetric bilinear form $\\phi _V$ defined by the product in the exterior algebra followed by the isomorphism $p^*\\rho ^\\vee :p^*\\operatorname{det}^{-1}V\\rightarrow {\\mathcal {O}}_V$ .",
"On the other hand, the Calmès-Hornbostel pushforward for the zero-section $s:X\\rightarrow V$ of a rank $r$ vector bundle $p:V\\rightarrow X$ with isomorphism $\\rho :{\\mathcal {O}}_X\\xrightarrow{} \\operatorname{det}V$ is as described above, sending a symmetric bilinear form $\\psi :C\\otimes ^{\\mathrm {L}}C\\rightarrow \\omega _{X/B} \\otimes s^*L[n]$ to the symmetric bilinear form $\\mathrm {R}s_*(\\psi ): \\mathrm {R}s_*C \\otimes ^{\\mathrm {L}}\\mathrm {R}s_*C\\rightarrow \\omega _{V/B} \\otimes L[n+r].$ Since $s$ is finite, $\\mathrm {R}s_*C \\simeq C$ , which in ${\\operatorname{perf}}(V)$ is canonically isomorphic to $p^*C\\otimes _{{\\mathcal {O}}_V}{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )$ .",
"We may thus view $\\mathrm {R}s_*(\\psi )$ instead as a map $[p^*C\\otimes _{{\\mathcal {O}}_V}{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )] \\otimes ^{\\mathrm {L}}[p^*C\\otimes _{{\\mathcal {O}}_V}{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )] \\rightarrow \\omega _{V/B} \\otimes L[n+r];$ tracing through its definition, one finds that this map is given by the composition $[p^*C\\otimes _{{\\mathcal {O}}_V}{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )] \\otimes ^{\\mathrm {L}}[p^*C\\otimes _{{\\mathcal {O}}_V}{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )] \\\\\\xrightarrow{}[p^*C\\otimes ^{\\mathrm {L}}p^*C]\\otimes [{\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )\\otimes {\\operatorname{Kos}}(p^*V^\\vee , s_{\\text{can}}^\\vee )]\\\\\\xrightarrow{} \\omega _{X/B} \\otimes L[n] \\otimes p^*\\operatorname{det}^{-1}V[r] \\xrightarrow{} \\omega _{V/B} \\otimes L[n+r].$ As this is exactly $p^*(C, \\phi ) \\otimes {\\operatorname{th}}_{V,\\rho }$ , we see that the Calmès-Hornbostel pushforward for $s$ is the same as that defined by the Panin-Walter $\\operatorname{SL}$ -orientation on ${\\operatorname{BO}}_\\eta $ , which verifies the hypothesis in our Theorem REF .",
"Having verified this, we can prove our main result.",
"Theorem 8.6 Let $B$ be a regular noetherian separated scheme of finite Krull dimension with 2 invertible in $\\Gamma (B, {\\mathcal {O}}_B)$ .",
"Let $X$ be a smooth projective $B$ -scheme.",
"Then: We have $u^{{\\operatorname{BO}}_\\eta }(\\chi (X/B))=(\\oplus _{i,j=0}^{\\operatorname{dim}_kX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i], {\\rm Tr})$ in ${\\operatorname{BO}}_\\eta ^{0,0}(B) \\simeq {\\operatorname{W}}({\\operatorname{perf}}(B)) \\simeq {\\operatorname{W}}(B)$ .",
"Let $f:{\\operatorname{GW}}(B)\\rightarrow \\mathrm {K}_0(B)$ denote the forgetful map discarding the symmetric bilinear form.",
"Suppose that the map $(f, q):{\\operatorname{GW}}(B)\\rightarrow \\mathrm {K}_0(B)\\times W(B)$ is injective (this is the case if for example $B$ is the spectrum of a local ring).",
"Then $u^{{\\operatorname{BO}}}(\\chi (X/B))=(\\oplus _{i,j=0}^{\\operatorname{dim}_BX}{\\operatorname{H}}^i(X, \\Omega _{X/B}^j)[j-i], {\\rm Tr})$ in ${\\operatorname{GW}}({\\operatorname{perf}}(B)) \\simeq {\\operatorname{GW}}(B)$ .",
"Suppose $B$ is in ${\\mathrm {Sm}}_k$ for $k$ a perfect field.",
"Then the image $\\widetilde{\\chi (X/B)}$ of $\\chi (X/B)$ in $\\pi _{0,0}(1_B)(B) \\simeq {\\operatorname{H}}^0(B, {\\mathcal {GW}})$ is given by $\\widetilde{\\chi (X/B)}=(\\oplus _{i,j=0}^{\\operatorname{dim}_BX}\\mathrm {R}^i\\pi _{X*}\\Omega _{X/B}^j)[j-i], {\\rm Tr}) \\in {\\operatorname{H}}^0(B, {\\mathcal {GW}}).$ In particular, if $B={\\rm Spec\\,}k$ , then $\\chi (X/k)=(\\oplus _{i,j=0}^{\\operatorname{dim}_BX}{\\operatorname{H}}^i(X, \\Omega _{X/k}^j)[j-i], {\\rm Tr})\\in {\\operatorname{GW}}(k).$ The first statement follows from our comparison theorem REF , as detailed above, together with the motivic Gauß-Bonnet theorem REF .",
"Statement (2) follows from (1) and our result for algebraic K-theory, Theorem REF .",
"Finally, (3) follows from (2), after we check that the unit map $u^{\\operatorname{BO}}$ induces the identity map on ${\\operatorname{GW}}(k)$ via ${\\operatorname{GW}}(k){[r]^{(\\hbox{\\tiny Morel})}_\\sim &} 1_k^{0,0}({\\rm Spec\\,}k){[r]^{u^{\\operatorname{BO}}}_\\sim &} {\\operatorname{BO}}^{0,0}({\\rm Spec\\,}k)\\simeq {\\operatorname{KO}}^{[0]}_0(k)={\\operatorname{GW}}(k),$ where the first isomorphism arises from Morel's theorem [32], [33] identifying $1_k^{0,0}({\\rm Spec\\,}k)$ with ${\\operatorname{GW}}(k)$ .",
"The one dimensional forms $\\langle \\lambda \\rangle \\in {\\operatorname{GW}}(k)$ , $\\lambda \\in k^\\times $ , generate ${\\operatorname{GW}}(k)$ , and via Morel's isomorphism $\\langle \\lambda \\rangle $ maps to the automorphism of $1_k$ induced by the automorphism $\\phi _\\lambda :{\\mathbb {P}}^1_k\\rightarrow {\\mathbb {P}}^1_k$ , $\\phi _\\lambda ((x_0:x_1))= (x_0:\\lambda \\cdot x_1)$ .",
"By [3], the image of $\\phi _\\lambda $ under the unit map $u^{\\operatorname{BO}}$ is also $\\langle \\lambda \\rangle $ , after the canonical identification ${\\operatorname{BO}}^{0,0}({\\rm Spec\\,}k)\\simeq {\\operatorname{KO}}^{[0]}_0(k) \\simeq {\\operatorname{GW}}(k)$ .",
"Corollary 8.7 Let $k$ be a perfect field of characteristic different from two.",
"Let $H \\in {\\operatorname{GW}}(k)$ denote the class of the hyperbolic form $x^2-y^2$ .",
"Let $X$ be a smooth and projective $k$ -scheme.",
"Suppose $X$ has odd dimension $2n-1$ .",
"Let $m := \\sum _{i+j<2n-1}(-1)^{i+j}\\operatorname{dim}_k{\\operatorname{H}}^i(X,\\Omega _{X/k}^j)- \\sum _{\\hbox{t}o 30pt{\\vbox {\\tiny 0\\le i< j\\\\i+j= 2n-1}}} \\operatorname{dim}_k{\\operatorname{H}}^i(X,\\Omega _{X/k}^j).$ Then $\\chi (X/k)=m\\cdot H \\in {\\operatorname{GW}}(k)$ .",
"Assume $X$ has even dimension $2n$ .",
"Let $m := \\sum _{i+j<2n}(-1)^{i+j}\\operatorname{dim}_k{\\operatorname{H}}^i(X,\\Omega _{X/k}^j)+ \\sum _{\\hbox{t}o 30pt{\\vbox {\\tiny 0\\le i< j\\\\i+j= 2n}}} \\operatorname{dim}_k{\\operatorname{H}}^i(X,\\Omega _{X/k}^j)$ and let $Q$ be the symmetric bilinear form ${\\operatorname{H}}^n(X, \\Omega ^n_{X/k})\\times {\\operatorname{H}}^n(X, \\Omega ^n_{X/k}) \\xrightarrow{}{\\operatorname{H}}^{2n}(X, \\Omega ^{2n}_{X/k})\\xrightarrow{} k.$ Then $\\chi (X/k)=m\\cdot H+Q \\in {\\operatorname{GW}}(k)$ .",
"For $V$ a finite dimensional $k$ -vector space and $n \\in {\\mathbb {Z}}$ , we have the symmetric bilinear form in ${\\operatorname{perf}}(k)$ $h_n: (V[n]\\oplus V^\\vee [-n]) \\otimes (V[n]\\oplus V^\\vee [-n]) \\rightarrow k$ whose restriction to $V[n]\\otimes V^\\vee [-n]$ is the canonical pairing of $V[n]$ with $V^\\vee [-n]\\simeq V[n]^\\vee $ , and $(-1)^n$ times this pairing on $V^\\vee [-n]\\otimes V[n]$ .",
"The corresponding class of $h_n$ in ${\\operatorname{GW}}(k)$ is $(-1)^n$ times the class of $h_0$ , as $(V[n]\\oplus V^\\vee [-n], h_n)$ is the image of the class of $V[n]$ in $\\mathrm {K}_0(k)$ under the hyperbolic map $H:\\mathrm {K}_0(-)\\rightarrow {\\operatorname{KO}}^{[0]}_0(-)$ (see e.g.",
"[50]), and $[V[n]]=(-1)^n[V[0]]$ in $\\mathrm {K}_0({\\operatorname{perf}}(k))\\simeq \\mathrm {K}_0(k)$ .",
"With this in mind, we may deduce the claim from the formula $\\chi (X/k)=(\\oplus _{i,j=0}^{\\operatorname{dim}_BX}{\\operatorname{H}}^i(X, \\Omega _{X/k}^j)[j-i], {\\rm Tr})$ of Theorem REF .",
"Indeed, in the case $\\operatorname{dim}X=2n-1$ , the symmetric bilinear form ${\\rm Tr}$ is the sum of the “hyperbolic” forms as above on ${\\operatorname{H}}^i(X,\\Omega _{X/k}^j)[j-i]\\oplus {\\operatorname{H}}^{2n-1-i}(X,\\Omega _{X/k}^{2n-1-j})[i-j]$ for $i+j<2n-1$ , or $0\\le i<j$ and $ i+j= 2n-1$ ; and the argument in the even dimensional case is the same, except that one has the remaining factor coming from the symmetric pairing on ${\\operatorname{H}}^n(X, \\Omega ^n_{X/k})$ .",
"The next result was obtained in 1974 independently by Abelson [1] and Kharlamov [24] using an argument of Milnor's relying on the Lefschetz fixed point theorem.",
"Corollary 8.8 Let $k$ be a field equipped with an embedding $\\sigma : k \\hookrightarrow {\\mathbb {R}}$ .",
"Let $X$ be a smooth projective $k$ -scheme of even dimension $2n$ .",
"Then $|\\chi ^\\mathrm {top}(X({\\mathbb {R}}))|\\le \\operatorname{dim}_k {\\operatorname{H}}^n(X, \\Omega ^n_{X/k}).$ We know that $\\chi ^\\mathrm {top}(X({\\mathbb {R}}))$ is the signature of $\\sigma _*(\\chi (X/k)) \\in {\\operatorname{GW}}({\\mathbb {R}})$ (see [26]).",
"The description of $\\chi (X/k)$ given by Corollary REF gives the desired inequality $|\\operatorname{sig}\\sigma _*(\\chi (X/k))|\\le \\operatorname{dim}_k {\\operatorname{H}}^n(X, \\Omega _{X/k}).",
"$ Remark 8.9 Let $k$ be a perfect field of characteristic different from two.",
"The formula for the Euler characteristic given in Theorem REF shows that the invariant $\\chi (X/k)$ is “motivic” in the following sense.",
"Let $X$ and $Y$ be smooth projective $k$ -schemes of respective even dimensions $2n$ and $2m$ and let $\\alpha :X\\dasharrow Y$ be a correspondence with $k$ -coefficients of degree $n$ , that is, an element $\\alpha \\in {\\rm CH}^{m+n}(X\\times Y)_k$ .",
"The correspondence $\\alpha $ induces the map of $k$ -vector spaces $\\alpha ^*:{\\operatorname{H}}^m(Y,\\Omega ^m_{Y/k})\\rightarrow {\\operatorname{H}}^n(X, \\Omega ^n_{X/k})$ .",
"Suppose that $\\alpha ^*$ is an isomorphism and is compatible with the trace pairings on ${\\operatorname{H}}^m(Y,\\Omega ^m_{Y/k})$ and ${\\operatorname{H}}^n(X, \\Omega ^n_{X/k})$ appearing in Corollary REF .",
"Then $\\chi (X/k)=\\chi (Y/k)$ in the Witt ring ${\\operatorname{W}}(k)$ .",
"For instance, supposing $k$ has characteristic zero, if the motives of $X$ and $Y$ (for homological equivalence with respect to de Rham cohomology) have a Künneth decomposition $h(X) \\simeq \\oplus _{i=0}^{2n}h^i(X)\\langle i\\rangle ,\\quad h(Y) \\simeq \\oplus _{i=0}^{2m}h^i(Y)\\langle i\\rangle $ and $\\alpha $ induces an isomorphism $\\alpha ^*:h^m(Y)\\langle m\\rangle \\rightarrow h^n(X)\\langle n\\rangle $ , compatible with the respective intersection products $h^m(Y)\\langle m\\rangle \\otimes h^m(Y)\\langle m\\rangle \\rightarrow h^{2m}(Y)\\langle 2m\\rangle \\xrightarrow{}h^0(k) \\simeq {\\mathbb {Q}},$ $h^n(X)\\langle n\\rangle \\otimes h^n(X)\\langle n\\rangle \\rightarrow h^{2n}(X)\\langle 2n\\rangle \\xrightarrow{}h^0(k) \\simeq {\\mathbb {Q}},$ then $\\chi (Y/k)=\\chi (X/k)$ in ${\\operatorname{W}}(k)$ .",
"Presumably, merely having an isomorphism of motives $h^n(X)\\langle n\\rangle \\simeq h^m(Y)\\langle m\\rangle $ would not suffice to yield $\\chi (Y/k)=\\chi (X/k)$ in ${\\operatorname{W}}(k)$ , but we do not have an example."
],
[
"Descent for the motivic Euler characteristic",
"Let $k$ be a perfect field of characteristic different from two.",
"With the explicit formula for $\\chi (X/k)$ given by Theorem REF , we may find $\\chi (X/k)$ for forms $X$ of some $k$ -scheme $X_0$ by the usual twisting construction; this works for all manners of descent but we confine ourselves to the case of Galois descent here.",
"Let $X_0, X$ be smooth projective $k$ -schemes of even dimension $2n$ .",
"Let $K$ be a finite Galois extension field of $k$ with Galois group $G$ .",
"Let $X_K:=X\\times _kK$ , $X_{0K}:=X_0\\times _kK$ , and suppose we have an isomorphism $\\phi :X\\times _kK\\rightarrow X_0\\times _kK$ .",
"This gives us the cocycle $\\lbrace \\psi _\\sigma \\in {\\operatorname{Aut}}_K(X_0\\times _kK)\\rbrace _{\\sigma \\in G}$ where $\\psi _\\sigma :=\\phi ^\\sigma \\circ \\phi ^{-1}$ .",
"Letting $&b_0:{\\operatorname{H}}^n(X_0,\\Omega ^n_{X_0/k})\\times {\\operatorname{H}}^n(X_0,\\Omega ^n_{X_0/k})\\rightarrow k,\\\\&b:{\\operatorname{H}}^n(X,\\Omega ^n_{X/k})\\times {\\operatorname{H}}^n(X,\\Omega ^n_{X/k})\\rightarrow k$ denote the respective symmetric bilinear forms ${\\rm Tr}(x\\cup y)$ , the isomorphism $\\phi $ induces an isometry $\\phi ^*:({\\operatorname{H}}^n(X_{0K},\\Omega ^n_{X_{0K}/K}), b_{0K})\\rightarrow ({\\operatorname{H}}^n(X_{K},\\Omega ^n_{X_{K}/K}), b_{K}),$ and the cocycle $\\lbrace \\psi _\\sigma \\rbrace _{\\sigma \\in G}$ determines a cocycle $\\lbrace (\\psi _\\sigma ^*)^{-1} \\in \\mathrm {O}(b_0)(K)\\rbrace _{\\sigma \\in G}$ .",
"Twisting by the latter cocycle allows one to recovers $b$ from $b_0$ ; explicitly, this works as follows.",
"Firstly, as usual, one recovers the $k$ -vector space ${\\operatorname{H}}^n(X,\\Omega ^n_{X/k})$ from the $K$ -vector space ${\\operatorname{H}}^n(X_{0K},\\Omega ^n_{X_{0K}/K})$ as the $G$ -invariants for the map $x\\mapsto \\psi _\\sigma ^{*-1}(x^\\sigma )$ .",
"Secondly, letting $A\\in \\operatorname{GL}({\\operatorname{H}}^n(X_{0K},\\Omega ^n_{X_{0K}/K}))$ be a change of basis matrix comparing the $k$ -forms ${\\operatorname{H}}^n(X_0,\\Omega ^n_{X_0/k})\\subset {\\operatorname{H}}^n(X_{0K},\\Omega ^n_{X_{0K}/K})$ and ${\\operatorname{H}}^n(X,\\Omega ^n_{X/k})\\subset {\\operatorname{H}}^n(X_{0K},\\Omega ^n_{X_{0K}/K})$ , we recover $b$ (up to $k$ -isometry) as $b(x,y)=b_0(Ax, Ay)=:b_0^A(x,y).$ Having performed this twist at the level of symmetric bilinear forms, we may now pass Grothendieck-Witt classes to describe the Euler characteristic of $X$ : namely, Corollary REF (2) gives $\\chi (X_0/k)=[b_0+m\\cdot H],\\quad \\chi (X/k)=[b_0^A+m\\cdot H]$ in ${\\operatorname{GW}}(k)$ .",
"Remark 8.10 In the case of a smooth projective surface $S$ with $p_g(S)=0$ , over a characteristic zero field $k$ , the twisting construction reduces to a computation involving ${\\rm CH}^1(S_{\\bar{k}})/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ as a ${\\rm Gal}(k)$ -module; here $\\bar{k}$ is the algebraic closure of $k$ and $\\sim _{\\operatorname{num}}$ is numerical equivalence.",
"Indeed, the assumption $p_g(S)=0$ implies that the cycle class map in Hodge cohomology ${\\operatorname{cyc}}^\\mathrm {Hdg}:{\\rm CH}^1(S_{\\bar{k}})\\rightarrow {\\operatorname{H}}^1(S_{\\bar{k}}, \\Omega ^1_{S/\\bar{k}})$ induces an isomorphism ${{\\rm CH}^1(S_{\\bar{k}})/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}} \\otimes _{\\mathbb {Z}}\\bar{k}\\xrightarrow{} {\\operatorname{H}}^1(S_{\\bar{k}}, \\Omega ^1_{S/\\bar{k}}) \\simeq {\\operatorname{H}}^1(S, \\Omega ^1_{S/k})\\otimes _k\\bar{k}$ and the cycle class map ${\\operatorname{cyc}}^\\mathrm {Hdg}$ transforms the intersection product on ${\\rm CH}^1(S_{\\bar{k}})/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ to the quadratic form $b_0$ on ${\\operatorname{H}}^1(S_{\\bar{k}}, \\Omega ^1_{S/\\bar{k}})$ , induced by cup product and the trace map.",
"Thus, our quadratic form $b$ on ${\\operatorname{H}}^1(S, \\Omega ^1_{S/k})$ is equivalent to the one gotten by twisting the $\\bar{k}$ -linear extension of the intersection product on ${\\rm CH}^1(S_{\\bar{k}})/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ by the natural Galois action.",
"Analogous comments hold for a “geometrically singular” variety, by which we mean a smooth projective $k$ -scheme $X$ of dimension $2n$ such that ${\\operatorname{H}}^n(X_{\\bar{k}},\\Omega ^n_{X_{\\bar{k}}/\\bar{k}})$ is spanned by cycle classes, where we replace ${\\rm CH}^1/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ with ${\\rm CH}^n/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ .",
"For example, one could take a K3 surface with Picard rank 20 over $\\bar{k}$ or a cubic fourfold $X$ with ${\\operatorname{H}}^2(X_{\\bar{k}},\\Omega ^2_{X_{\\bar{k}}/k})\\simeq \\bar{k}^{21}$ spanned by algebraic cycles.",
"Examples 8.11 As as simple example, take $S$ to be a quadric surface in ${\\mathbb {P}}^3_k$ defined by a degree two homogeneous form $q(X_0,\\ldots , X_3)$ ; we may assume that $q$ is a diagonal form, $q(X_0,\\ldots , X_3)=a_0X_0^2+\\sum _{i=1}^3a_iX_i^2\\\\=a_0(X_0+\\sqrt{-a_1/a_0}X_1)(X_0-\\sqrt{-a_1}X_1)\\\\+a_2(X_2-\\sqrt{-a_3/a_2}X_3)(X_2+\\sqrt{-a_3/a_2}X_3).$ This trivializes ${\\rm CH}^1(S)$ over $K:=k(\\sqrt{-a_0a_1}, \\sqrt{-a_2a_3})$ , namely ${\\rm CH}^1(S)={\\mathbb {Z}}\\ell _1\\oplus {\\mathbb {Z}}\\ell _2$ with $\\ell _1$ defined by $(X_0-\\sqrt{-a_1}X_1)=(X_2-\\sqrt{-a_3/a_2}X_3)=0$ and $\\ell _2$ defined by $(X_0-\\sqrt{-a_1}X_1)=(X_2+\\sqrt{-a_3/a_2}X_3)=0$ .",
"Embedding ${\\rm Gal}(K/k)\\subset {\\rm Gal}(k(\\sqrt{-a_0a_1})/k)\\times {\\rm Gal}(k(\\sqrt{-a_2a_3})/k)=\\langle \\sigma _1\\rangle \\times \\langle \\sigma _2\\rangle $ , the Galois action is given by $\\sigma _1(\\ell _1, \\ell _2)=\\sigma _2(\\ell _1, \\ell _2)= (\\ell _2, \\ell _1)$ .",
"A Galois-invariant basis is thus given by $((\\ell _1+\\ell _2), \\sqrt{a_0a_1a_2a_3}(\\ell _1-\\ell _2))$ , and the intersection form in this basis has matrix $\\begin{pmatrix}2&0\\\\0&-2a_0a_1a_2a_3\\end{pmatrix}.$ In other words, $\\chi (S/k)=\\langle 2\\rangle +\\langle -2a_0a_1a_2a_3\\rangle $ .",
"Suppose $S$ is the blowup of ${\\mathbb {P}}^2_k$ along a 0-dimensional closed subscheme $Z\\subset {\\mathbb {P}}^2_k$ , with $Z$ étale over $k$ .",
"Let $\\ell $ denote the class of a line in ${\\rm CH}^1({\\mathbb {P}}^2)$ .",
"Writing $Z_{\\bar{k}}=\\lbrace p_1,\\ldots , p_r\\rbrace $ , we have ${\\rm CH}^1(S_{\\bar{k}}) \\simeq {\\mathbb {Z}}\\cdot \\ell \\oplus (\\oplus _{i=1}^r{\\mathbb {Z}}\\cdot p_r)$ , with the evident Galois action and with intersection form the diagonal matrix $(1, -1,\\ldots , -1)$ .",
"It is then easy to show that the twisted quadratic form $\\chi (S/k)$ is $\\langle 1\\rangle -{\\rm Tr}_{Z/k}(\\langle 1\\rangle )$ .",
"These last two examples have been computed by different methods before: (1) is a special case of [26] and (2) is a special case of [26].",
"Here is a more interesting example.",
"Example 8.12 Let $\\pi :S\\rightarrow C$ be a a conic bundle over a smooth projective curve $C$ , all defined over $k$ ; we assume for simplicity that $k\\subset .",
"Let $ ZC$ be the degeneracy locus of $$: that is, $ Z$ is the reduced proper closed subscheme of $ C$ over which $$ is not smooth.",
"For each geometric point $ z$ of $ Z$, the fiber $ -1(z)$ is isomorphic to two distinct lines in $ P2$: $ -1(z)=zz'$.",
"There is a ``double section^{\\prime \\prime } $ DS$ with $ DC$ a finite degree two morphism, and with $ Dz=1=Dz'$ for all $ zZ(k)$.$ Over $\\bar{k}$ , the bundle $S$ is isomorphic to the blow-up of a ${\\mathbb {P}}^1$ -bundle $\\bar{S}_{\\bar{k}}\\rightarrow C_{\\bar{k}}$ along a finite set $Z^{\\prime }\\subset \\bar{S}_{\\bar{k}}$ with $Z^{\\prime }\\xrightarrow{} Z_{\\bar{k}}$ via $\\pi $ .",
"Suppose $Z_{\\bar{k}}=\\lbrace z_1,\\ldots z_r\\rbrace $ .",
"If we fix a closed point $c_0\\in C\\setminus Z$ of degree $d$ over $k$ , we have the following basis for ${\\rm CH}^1(S_{\\bar{k}})_{\\mathbb {Q}}/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ : $\\ell _{z_1}-\\ell _{z_1}^{\\prime },\\ldots , \\ell _{z_r}-\\ell _{z_r}^{\\prime }, D, \\pi ^{-1}(c_0).$ We have the finite degree two extension $p:\\tilde{Z}\\rightarrow Z$ , where for each $z\\in Z$ , $p^{-1}(z)$ corresponds to the pair of lines $\\ell _z, \\ell _z^{\\prime }$ .",
"Let $L:=k(\\lbrace z_1,\\ldots , z_r\\rbrace )\\supset k$ and let $G:={\\operatorname{Aut}}(L/k)$ .",
"Writing $k(\\tilde{Z})=k(Z)(\\sqrt{\\delta })$ for some $\\delta \\in {\\mathcal {O}}_Z^\\times $ , we have a basis of ${\\rm CH}^1(S_L)_{\\mathbb {Q}}/\\hspace{-4.0pt}\\sim _{\\operatorname{num}}$ given by $v_1,\\ldots , v_r, D, \\pi ^{-1}(c_0),$ with $v_i:=\\sqrt{d}(\\ell _{z_i}-\\ell _{z_i}^{\\prime })$ .",
"The intersection form on $\\langle v_1,\\ldots , v_r\\rangle $ is the diagonal matrix $(-4\\delta (z_1),\\ldots , -4\\delta (z_r))$ , the subspaces $\\langle v_1,\\ldots , v_r\\rangle $ and $\\langle D, \\pi ^{-1}(c_0)\\rangle $ are perpendicular and $\\langle D, \\pi ^{-1}(c_0)\\rangle $ is hyperbolic.",
"Moreover, the automorphism group ${\\operatorname{Aut}}(L/k)$ acts on $\\langle v_1,\\ldots , v_r\\rangle $ just as it does on $\\langle z_1,\\ldots , z_r\\rangle $ .",
"From this it follows that the twisted intersection form $b$ is given by $b=H-{\\rm Tr}_{k(Z)/k}(\\langle \\delta \\rangle ),$ and hence $\\chi (S/k)=m\\cdot H-{\\rm Tr}_{k(Z)/k}(\\langle \\delta \\rangle )$ with $m=2-\\operatorname{dim}_k{\\operatorname{H}}^0(S,\\Omega ^1_{S/k})-\\operatorname{dim}_k{\\operatorname{H}}^1(S,{\\mathcal {O}}_X)=\\operatorname{dim}_{\\mathbb {Q}}{\\operatorname{H}}^0(S^{\\rm an},{\\mathbb {Q}})-\\operatorname{dim}_{\\mathbb {Q}}{\\operatorname{H}}^1(S^{\\rm an},{\\mathbb {Q}})+1,$ where $S^{\\rm an}$ is the complex manifold associated to $S_.$ As a particular example, we may take $S$ to be a cubic surface $V\\subset {\\mathbb {P}}^3_k$ with a line $\\ell $ .",
"Projection from $\\ell $ realizes $V$ as a conic bundle $\\pi :V\\rightarrow {\\mathbb {P}}^1_k$ , with degeneracy locus $Z\\subset {\\mathbb {P}}^1_k$ a reduced closed subscheme of degree 5 over $k$ .",
"The above implies that the symmetric bilinear form $b_V$ is given by $b_V=H-{\\rm Tr}_{Z/k}(\\langle \\delta \\rangle )$ and computes $\\chi (S/k) = 2H-{\\rm Tr}_{Z/k}(\\langle \\delta \\rangle )$ .",
"Remark 8.13 In [8], Bayer-Fluckiger and Serre consider the finite $k$ -scheme $W$ representing the 27 lines on a cubic surface $V$ and compute the trace form ${\\rm Tr}_{W/k}(\\langle 1\\rangle )$ in [8].",
"They identify their form $q_{6,V}$ with the trace form on ${\\operatorname{H}}^1(V,\\Omega _{V/k})$ and show that ${\\rm Tr}_{W/k}(\\langle 1\\rangle )=\\lambda ^2b_V + (\\langle -1\\rangle -\\langle 2\\rangle )b_V+ 7 -\\langle -2\\rangle .$"
]
] | 1808.08385 |
[
[
"Analyzing Learned Representations of a Deep ASR Performance Prediction\n Model"
],
[
"Abstract This paper addresses a relatively new task: prediction of ASR performance on unseen broadcast programs.",
"In a previous paper, we presented an ASR performance prediction system using CNNs that encode both text (ASR transcript) and speech, in order to predict word error rate.",
"This work is dedicated to the analysis of speech signal embeddings and text embeddings learnt by the CNN while training our prediction model.",
"We try to better understand which information is captured by the deep model and its relation with different conditioning factors.",
"It is shown that hidden layers convey a clear signal about speech style, accent and broadcast type.",
"We then try to leverage these 3 types of information at training time through multi-task learning.",
"Our experiments show that this allows to train slightly more efficient ASR performance prediction systems that - in addition - simultaneously tag the analyzed utterances according to their speech style, accent and broadcast program origin."
],
[
"Introduction",
"Predicting automatic speech recognition (ASR) performance on unseen speech recordings is an important Grail of speech research.",
"In a previous paper [5], we presented a framework for modeling and evaluating ASR performance prediction on unseen broadcast programs.",
"CNNs were very efficient encoding both text (ASR transcript) and speech to predict ASR word error rate (WER).",
"However, while achieving state-of-the-art performance prediction results, our CNN approach is more difficult to understand compared to conventional approaches based on engineered features such as TransRaterhttps://github.com/hlt-mt/TranscRater for instance.",
"This lack of interpretability of the representations learned by deep neural networks is a general problem in AI.",
"Recent papers started to address this issue and analyzed hidden representations learned during training of different natural language processing models [12], [16], [1], [14], [2], [15].",
"Contribution.",
"This work is dedicated to the analysis of speech signal embeddings and text embeddings learnt by the CNN during training of our ASR performance prediction model.",
"Our goal is to better understand which information is captured by the deep model and its relation with conditioning factors such as speech style, accent or broadcast program type.",
"For this, we use a data set presented in [5] which contains a large amount of speech utterances taken from various collections of French broadcast programs.",
"Following a methodology similar to [1], our deep performance prediction model is used to generate utterance level features that are given to a shallow classifier trained to solve secondary classification tasks.",
"It is shown that hidden layers convey a clear signal about speech style, accent and show.",
"We then try to leverage these 3 types of information at training time through multi-task learning.",
"Our experiments show that this allows to train slightly more efficient ASR performance prediction systems that - in addition - simultaneously tag the analyzed utterances according to their speech style, accent and broadcast program origin.",
"Outline.",
"The paper is organized as follows.",
"In section , we present a brief overview of related works and present our ASR performance prediction system in section .",
"Then, we detail our methodology to evaluate learned representations in section .",
"Our multi-task learning experiments for ASR performance prediction are presented in section .",
"Finally, section concludes this work."
],
[
"Related works",
"Several works tried to understand learned representations for NLP tasks such as Automatic Speech Recognition (ASR) and Neural Machine Translation (NMT).",
"[14] and [2] tried to better understand the hidden representations of NMT models which were given to a shallow classifier in order to predict syntactic labels [14], part-of-speech labels or semantic ones [2].",
"It was shown that lower layers are better at POS tagging, while higher layers are better at learning semantics.",
"[12] and [1] analyzed the feature representations from a deep ASR model using t-SNE visualization [11] and tried to understand which layers better capture the phonemic information by training a shallow phone classifier.",
"Also relevant is the work of [15] who proposed an in-depth investigation on three kinds of speaker embeddings learned for a speaker recognition task, i.e.",
"i-vector, d-vector and RNN/LSTM based sequence-vector (s-vector).",
"Classification tasks were designed to facilitate better understanding of the encoded speaker representations.",
"Multi-task learning was also proposed to integrate different speaker embeddings and improve speaker verification performance."
],
[
" ASR performance prediction system",
"In [5], we proposed a new approach using convolution neural networks (CNNs) to predict ASR performance from a collection of heterogeneous broadcast programs (both radio and TV).",
"We particularly focused on the combination of text (ASR transcription) and signal (raw speech) inputs which both proved useful for CNN prediction.",
"We also observed that our system remarkably predicts WER distribution on a collection of speech recordings.",
"To obtain speech transcripts (ASR outputs) for the prediction model, we built our own French ASR system based on the KALDI toolkit [13].",
"A hybrid HMM-DNN system was trained using 100 hours of broadcast news from Quaerohttp://www.quaero.org, ETAPE [7], ESTER 1 & ESTER 2 [6] and REPERE [8] collections.",
"ASR performance was evaluated on the held out corpora presented in table REF (used to train and evaluate ASR prediction) and its averaged value was 22.29% on the TRAIN set, 22.35% on the DEV set and 31.20% on the TEST set (which contains more challenging broadcast programs).",
"Figure REF shows our network architecture.",
"The network input can be either a pure text input, a pure signal input (raw signal) or a dual (text+speech) input.",
"To avoid memory issues, signals are downsampled to 8khz and models are trained on six-second speech turns (shorter speech turns are padded with zeros).",
"For text input, the architecture is inspired from [9] (green in Figure REF ): the input is a matrix of dimensions 296x100 (296 is the longest ASR hypothesis length in our corpus ; 100 is the dimension of pre-trained word embeddings on a large held out text corpus of 3.3G words).",
"For speech input, we use the best architecture (m18) proposed in [4] (colored in red in Figure REF ) of dimensions 48000 x 1 (48000 samples correspond to 6s of speech).",
"For WER prediction, our best approach (called CNN$_{Softmax}$ ) used $softmax$ probabilities and an external fixed WER$_{Vector}$ which corresponds to a discretization of the WER output space (see [5] for more details).",
"The best performance obtained is 19.24% MAEMean Absolute Error (MAE) is a common metric to evaluate WER prediction ; it computes the absolute deviation between the true and predicted WERs, averaged over the number of utterances in the test set.",
"using text+speech input.",
"Our ASR prediction system is built using both Keras [3] and Tensorflowhttps://www.tensorflow.org.",
"Figure: Architecture of our CNN with text (green) and signal (red) inputs for WER predictionIn the next section, we analyze the representations learnt in the higher layers (3 blocks colored in yellow and dotted in Figure REF ) for pure text (TXT), pure speech (RAW-SIG) and both (TXT+RAW-SIG)."
],
[
"Methodology",
"In this section, we attempt to understand what our best ASR performance prediction system [5] learned.",
"We analyze the text and speech representations obtained by our architecture.",
"Alike [1], the joint text+speech model is used to generate utterance level features (hidden representations of speech turns colored in yellow in Figure REF ) that are given to a shallow classifier trained to solve secondary classification tasks such as: STYLE: classify the utterances between (spontaneous and non spontaneous) styles (see table REF ), ACCENT: classify the utterances between native and non native speech (see also table REF , we used the speaker annotations provided with our datasets in order to label our utterances in native/non native speech), SHOW: classify the utterances in different broadcast programs (as described in table REF , each utterance of our corpus is labeled with a broadcast program name).",
"As a more visual analysis, we also plot an example of hidden representations projected to a 2-D space using t-distributed Stochastic Neighbor Embedding (t-SNE) [11].https://lvdmaaten.github.io/tsne/code/tsne_python.zip"
],
[
"Shallow classifiers",
"We built three shallow classifiers (SHOW, STYLE, ACCENT) with a similar architecture.",
"The classifier is a feed-forward neural network with one hidden layer (size of the hidden layer is set to 128) followed by dropout (rate of 0.5) and a ReLU non-linearity.",
"Finally, a $softmax$ layer is used for mapping onto the label set size.",
"We chose this simple formulation as we are interested in evaluating the quality of the representations learned by our ASR prediction model, rather than optimizing the secondary classification tasks.",
"The network input size depends on which layer to analyze (see figure REF ).",
"Training is performed using $Adam$ [10] (using default parameters) over shuffled mini-batches in order to minimize the cross-entropy loss.",
"The models are trained for 30 epochs with a batch size of 16 speech utterances.",
"After training, we keep the model with the best performance on DEV set and report its performance on the TEST set.",
"The classifier outputs are evaluated in terms of accuracy."
],
[
"Data",
"A data set from [5] was employed in our experiments, divided into three subsets: training (TRAIN), development (DEV) and test (TEST).",
"Speech utterances come from various French broadcast collections gathered during projects or shared tasks: Quaero, ETAPE, ESTER 1 & ESTER 2 and REPERE.",
"The TEST set contains unseen broadcast programs that are different from those present in TRAIN and DEV [5].",
"Table: NO_CAPTIONTable: NO_CAPTIONTables REF and REF show the whole data set in terms of speech turns available for each classification task.",
"We clearly see that the data is unbalanced for the three categories (STYLE, ACCENT, SHOW).",
"Since we are interested in evaluating the discriminative power of our learned representations for these 3 tasks, we extracted a balanced version of our TRAIN/DEV/TEST sets by filtering among over-represented labels (final number of kept utterances corresponds to bold numbers in table REF and REF ).",
"Table REF shows the distribution of our final balanced TRAIN/DEV/TEST sets as well as the number of categories for each task.For the SHOW classification task, the FRANCE3-DEBATE shows were finally removed since they represent a too small amount of speech turns.",
"Table: NO_CAPTION"
],
[
"Results",
"For each classification task, we build a shallow classifier using the hidden representations of TXT, RAW-SIG and TXT+RAW-SIG blocks as input.",
"The experimental results are presented in table REF for both DEV and TEST sets separated by two vertical bars ($||$ ).",
"Classification performance is all above a random baseline accuracy ($>$ 50% for STYLE and ACCENT and $>$ 20% for SHOW).",
"This shows that training a deep WER prediction system gives representation layers that contain a meaningful amount of information about speech style, speech accent and broadcast program label.",
"Predicting utterance style (spontaneous/non spontaneous) is slightly easier than predicting accent (native/non native) especially from text input.",
"One explanation might be that speech utterances are short ($<6s$ ) while accent identification needs probably longer sequences.",
"We also observe that using both text and speech improves the learned representations for the STYLE task while it is less clear for the ACCENT task (for which improvement seen on DEV is not confirmed on TEST).",
"Finally, text input is significantly better than speech input whereas we could have expected better performance from speech for the SHOW task (speech signals convey information about the audio characteristics of a broadcast program).",
"It means that text input contains correlated information with broadcast-program type, speech style and speaker's accent.",
"In case of SHOW task, our performance prediction system is able to capture information (vocabulary, topic, syntax, etc.)",
"about a specific broadcast program type, based on textual features and to differ it from others (radio programs, TV debate programs, phone calls, broadcast news programs, etc.).",
"Likewise, the textual information captured is very different between spontaneous/non-spontaneous speech styles and native/non-native speaker's accents.",
"Among the representations analyzed, the outputs of the CNNs (A1,B1) lead to the best classification results, in line with previous findings about convolutions as feature extractors.",
"Performance then drops using the higher (fully connected) layers that do not generate better representations for detecting style, accent or show.",
"Table: NO_CAPTIONFigure: Visualization of utterance representations from C2 layer for different speech styles (S spontaneous - NS non spontaneous) - (a) utt.",
"length is 4s≤\\le D<<5s and (b) 5s≤\\le D<<6sWe visualize an example of utterance representations from C2(TXT+RAW-SIG) layer in figure REF using the t-SNE.",
"For a fixed utterance duration 4s$\\le $ D$<$ 5s (716 speech turns) and 5s$\\le $ D$<$ 6s (489 speech turns), non spontaneous utterances are plotted in blue while spontaneous ones are in pink.",
"The C2 layer produces clusters which shows that spontaneous utterances are in the upper-left part of the 2D space.",
"This suggests that C2 hidden representation captures a weak signal about speaking style.",
"Finally, figure REF is the confusion matrix produced using C2(TXT+RAW-SIG) layer.",
"The classifiers very well predicted TELSONNE category (Accuracy of 82%), which contains many phone calls from the radio listeners.",
"This show is rather different from the 4 other shows in DEV (broadcast debates and news).",
"Figure: Confusion matrix for SHOW classification using C2(TXT+RAW-SIG) layer as input, evaluated on DEVTable: NO_CAPTION"
],
[
"Multi-task learning",
"We have seen in the previous section that, while training an ASR performance prediction system, hidden layers convey a clear signal about speech style, accent and show.",
"This suggests that these 3 types of information might be useful to structure the deep ASR performance prediction models.",
"In this section, we investigate the effect of knowledge of these labels (style, accent, show) at training time on prediction systems qualities.",
"For this, we perform multi-task learning providing the additional information about broadcast type, speech style and speaker's accent during training.",
"The architecture of the multi-task model is similar to the single-task WER prediction model of Figure REF but we add additional outputs: a $softmax$ function is added for each new classification task after the last fully connected layer (C2).",
"The output dimension depends on the task: 6 for SHOW and 2 for STYLE and ACCENT tasks.",
"We use the full (unbalanced) data set described in tables REF and REF .",
"Training of the multitask model uses $Adadelta$ update rule and all parameters are initialized from scratch (8.70M).",
"Models are performed for 50 epochs with batch size of 32.",
"MAE is used as the loss function for WER prediction task while cross-entropy loss is used for the classification tasks.",
"In the composite (multitask) loss, we assign a weight of 1 for MAE loss (main task) and a smaller weight of 0.3 (tuned using a grid search on DEV dataset) for cross-entropy (secondary classification task) loss(es).",
"After training, we take the model that lead to the best MAE on DEV set and report its performance on TEST.",
"We build several models that simultaneously address 1, 2, 3 and 4 tasks.",
"The models are evaluated with a specific metric for each task: MAE & KendallCorrelation between true ASR values and predicted ASR values for WER prediction task and Accuracy for classification tasks.",
"Table REF summarizes the experimental results on DEV and TEST sets, separated by two vertical bars ($||$ ).",
"We considered the mono-task model described in [5] (and summarized in section 3) as a baseline system.",
"We recall that we evaluated the SHOW classification task only on the DEV set (TEST broadcast programs are new and were unseen in the TRAIN).",
"First of all, we notice that performance of classification tasks in muti-task scenarios are very good: we are able to train efficient ASR performance prediction systems that simultaneously tag the analyzed utterances according to their speech style, accent and broadcast program origin.",
"Such multi-task systems might be useful diagnostic tools to analyze and predict ASR on large speech collections.",
"Moreover, our best multi-task systems display a better performance (MAE, Kendall) than the baseline system, which means that the implicit information given about style, accent and broadcast program type can be helpful to structure the system's predictions.",
"For example, in 2-task case, the best model is obtained on WER+SHOW tasks with a difference of +0.41%, +2.25% for MAE and Kendall respectively (on DEV) compared to the baseline on WER prediction task.",
"However, it is also important to mention that the impact of multi-task learning on the main task (ASR performance prediction) is limited: only slight improvements on the test set are observed for MAE and Kendall metrics.",
"Anyway, the systems trained seem complementary since their combination (averaging, over all multi-task systems, predicted WERs at utterance level) leads to significant performance improvement (MAE and Kendall)."
],
[
"Conclusion",
"This paper presented an analysis of learned representations of our deep ASR performance prediction system.",
"Experiments show that hidden layers convey a clear signal about speech style, accent, and broadcast type.",
"We also proposed a multi-task learning approach to simultaneously predict WER and classify utterances according to style, accent and broadcast program origin."
]
] | 1808.08573 |
[
[
"Semi-Autoregressive Neural Machine Translation"
],
[
"Abstract Existing approaches to neural machine translation are typically autoregressive models.",
"While these models attain state-of-the-art translation quality, they are suffering from low parallelizability and thus slow at decoding long sequences.",
"In this paper, we propose a novel model for fast sequence generation --- the semi-autoregressive Transformer (SAT).",
"The SAT keeps the autoregressive property in global but relieves in local and thus is able to produce multiple successive words in parallel at each time step.",
"Experiments conducted on English-German and Chinese-English translation tasks show that the SAT achieves a good balance between translation quality and decoding speed.",
"On WMT'14 English-German translation, the SAT achieves 5.58$\\times$ speedup while maintains 88\\% translation quality, significantly better than the previous non-autoregressive methods.",
"When produces two words at each time step, the SAT is almost lossless (only 1\\% degeneration in BLEU score)."
],
[
"Introduction",
"Neural networks have been successfully applied to a variety of tasks, including machine translation.",
"The encoder-decoder architecture is the central idea of neural machine translation (NMT).",
"The encoder first encodes a source-side sentence ${\\bf x}= x_1\\dots x_m$ into hidden states and then the decoder generates the target-side sentence ${\\bf y} = y_1\\dots y_n$ from the hidden states according to an autoregressive model $p(y_t|y_1\\dots y_{t-1},{\\bf x})$ Recurrent neural networks (RNNs) are inherently good at processing sequential data.",
"sutskever2014sequence,cho2014learning successfully applied RNNs to machine translation.",
"bahdanau2014neural introduced attention mechanism into the encoder-decoder architecture and greatly improved NMT.",
"GNMT [25] further improved NMT by a bunch of tricks including residual connection and reinforcement learning.",
"Figure: The different levels of autoregressive properties.",
"Lines with arrow indicate dependencies.",
"We mark the longest dependency path with bold red lines.",
"The length of the longest dependency path decreases as we relieve the autoregressive property.",
"An extreme case is non-autoregressive, where there is no dependency at all.The sequential property of RNNs leads to its wide application in language processing.",
"However, the property also hinders its parallelizability thus RNNs are slow to execute on modern hardware optimized for parallel execution.",
"As a result, a number of more parallelizable sequence models were proposed such as ConvS2S [7] and the Transformer [24].",
"These models avoid the dependencies between different positions in each layer thus can be trained much faster than RNN based models.",
"When inference, however, these models are still slow because of the autoregressive property.",
"A recent work [8] proposed a non-autoregressive NMT model that generates all target-side words in parallel.",
"While the parallelizability is greatly improved, the translation quality encounter much decrease.",
"In this paper, we propose the semi-autoregressive Transformer (SAT) for faster sequence generation.",
"Unlike gu2017non, the SAT is semi-autoregressive, which means it keeps the autoregressive property in global but relieves in local.",
"As the result, the SAT can produce multiple successive words in parallel at each time step.",
"Figure REF gives an illustration of the different levels of autoregressive properties.",
"Experiments conducted on English-German and Chinese-English translation show that compared with non-autoregressive methods, the SAT achieves a better balance between translation quality and decoding speed.",
"On WMT'14 English-German translation, the proposed SAT is 5.58$\\times $ faster than the Transformer while maintaining 88% of translation quality.",
"Besides, when produces two words at each time step, the SAT is almost lossless.",
"It is worth noting that although we apply the SAT to machine translation, it is not designed specifically for translation as gu2017non,lee2018deterministic.",
"The SAT can also be applied to any other sequence generation task, such as summary generation and image caption generation."
],
[
"Related Work",
"Almost all state-of-the-art NMT models are autoregressive [22], [3], [25], [7], [24], meaning that the model generates words one by one and is not friendly to modern hardware optimized for parallel execution.",
"A recent work [8] attempts to accelerate generation by introducing a non-autoregressive model.",
"Based on the Transformer [24], they made lots of modifications.",
"The most significant modification is that they avoid feeding the previously generated target words to the decoder, but instead feeding the source words, to predict the next target word.",
"They also introduced a set of latent variables to model the fertilities of source words to tackle the multimodality problem in translation.",
"lee2018deterministic proposed another non-autoregressive sequence model based on iterative refinement.",
"The model can be viewed as both a latent variable model and a conditional denoising autoencoder.",
"They also proposed a learning algorithm that is hybrid of lower-bound maximization and reconstruction error minimization.",
"The most relevant to our proposed semi-autoregressive model is [12].",
"They first autoencode the target sequence into a shorter sequence of discrete latent variables, which at inference time is generated autoregressively, and finally decode the output sequence from this shorter latent sequence in parallel.",
"What we have in common with their idea is that we have not entirely abandoned autoregressive, but rather shortened the autoregressive path.",
"A related study on realistic speech synthesis is the parallel WaveNet [18].",
"The paper introduced probability density distillation, a new method for training a parallel feed-forward network from a trained WaveNet [23] with no significant difference in quality.",
"There are also some work share a somehow simillar idea with our work: character-level NMT [6], [15] and chunk-based NMT [26], [11].",
"Unlike the SAT, these models are not able to produce multiple tokens (characters or words) each time step.",
"oda2017neural proposed a bit-level decoder, where a word is represented by a binary code and each bit of the code can be predicted in parallel."
],
[
"The Transformer",
"Since our proposed model is built upon the Transformer [24], we will briefly introduce the Transformer.",
"The Transformer uses an encoder-decoder architecture.",
"We describe the encoder and decoder below."
],
[
"The Encoder",
"From the source tokens, learned embeddings of dimension $d_{model}$ are generated which are then modified by an additive positional encoding.",
"The positional encoding is necessary since the network does not leverage the order of the sequence by recurrence or convolution.",
"The authors use additive encoding which is defined as: $PE(pos, 2i) &= sin(pos/10000^{2i/d_{model}})\\\\PE(pos, 2i+1) &= cos(pos/10000^{2i/d_{model}})$ where $pos$ is the position of a word in the sentence and $i$ is the dimension.",
"The authors chose this function because they hypothesized it would allow the model to learn to attend by relative positions easily.",
"The encoded word embeddings are then used as input to the encoder which consists of $N$ blocks each containing two layers: (1) a multi-head attention layer, and (2) a position-wise feed-forward layer.",
"Multi-head attention builds upon scaled dot-product attention, which operates on a query Q, key K and value V: $Attention(Q,K,V) = softmax(\\frac{QK^T}{\\sqrt{d_k}})V$ where $d_k$ is the dimension of the key.",
"The authors scale the dot product by $1/\\sqrt{d_k}$ to avoid the inputs to softmax function growing too large in magnitude.",
"Multi-head attention computes $h$ different queries, keys and values with $h$ linear projections, computes scaled dot-product attention for each query, key and value, concatenates the results, and projects the concatenation with another linear projection: $&H_i = Attention(QW_i^Q, KW_i^K, VW_i^V) \\\\&MultiHead(Q,K,V) = Concat(H_1,\\dots H_h)$ in which $W_i^Q, W_i^K \\in \\mathbb {R}^{d_{model}\\times d_k}$ and $W_i^V \\in \\mathbb {R}^{d_{model}\\times d_v}$ .",
"The attention mechanism in the encoder performs attention over itself ($Q=K=V$ ), so it is also called self-attention.",
"The second component in each encoder block is a position-wise feed-forward layer defined as: $FFN(x) = max(0, xW_1+b_1)W_2+b_2$ where $W_1\\in \\mathbb {R}^{d_{model}\\times d_{ff}}$ , $W_2\\in \\mathbb {R}^{d_{ff}\\times d_{model}}$ , $b_1\\in \\mathbb {R}^{d_{ff}}$ , $b_2\\in \\mathbb {R}^{d_{model}}$ .",
"For more stable and faster convergence, residual connection [9] is applied to each layer, followed by layer normalization [2].",
"For regularization, dropout [21] are applied before residual connections.",
"Figure: The architecture of the Transformer, also of the SAT, where the red dashed boxes point out the different parts of these two models."
],
[
"The Decoder",
"The decoder is similar with the encoder and is also composed by $N$ blocks.",
"In addition to the two layers in each encoder block, the decoder inserts a third layer, which performs multi-head attention over the output of the encoder.",
"It is worth noting that, different from the encoder, the self-attention layer in the decoder must be masked with a causal mask, which is a lower triangular matrix, to ensure that the prediction for position $i$ can depend only on the known outputs at positions less than $i$ during training."
],
[
"The Semi-Autoregressive Transformer",
"We propose a novel NMT model—the Semi-Autoregressive Transformer (SAT)—that can produce multiple successive words in parallel.",
"As shown in Figure REF , the architecture of the SAT is almost the same as the Transformer, except some modifications in the decoder."
],
[
"Group-Level Chain Rule",
"Standard NMT models usually factorize the joint probability of a word sequence $y_1\\dots y_n$ according to the word-level chain rule $p(y_1\\dots y_n|{\\bf x}) = \\prod _{t=1}^{n} p(y_t|y_1\\dots y_{t-1},{\\bf x})$ resulting in decoding each word depending on all previous decoding results, thus hindering the parallelizability.",
"In the SAT, we extend the standard word-level chain rule to the group-level chain rule.",
"We first divide the word sequence $y_1\\dots y_n$ into consecutive groups $&G_1,G_2,\\dots ,G_{[(n-1)/K]+1} =\\\\&y_1\\dots y_K, y_{K+1}\\dots y_{2K}, \\dots , y_{[(n-1)/K]\\times K+1}\\dots y_{n}$ where $[\\cdot ]$ denotes floor operation, $K$ is the group size, and also the indicator of parallelizability.",
"The larger the $K$ , the higher the parallelizability.",
"Except for the last group, all groups must contain $K$ words.",
"Then comes the group-level chain rule $p(y_1\\dots y_n|{\\bf x}) = \\prod _{t=1}^{[(n-1)/K]+1} p(G_t|G_1\\dots G_{t-1},{\\bf x})$ This group-level chain rule avoids the dependencies between consecutive words if they are in the same group.",
"With group-level chain rule, the model no longer produce words one by one as the Transformer, but rather group by group.",
"In next subsections, we will show how to implement the model in detail."
],
[
"Long-Distance Prediction",
"In autoregressive models, to predict $y_t$ , the model should be fed with the previous word $y_{t-1}$ .",
"We refer it as short-distance prediction.",
"In the SAT, however, we feed $y_{t-K}$ to predict $y_t$ , to which we refer as long-distance prediction.",
"At the beginning of decoding, we feed the model with $K$ special symbols ${<}\\text{s}{>}$ to predict $y_1\\dots y_K$ in parallel.",
"Then $y_1\\dots y_K$ are fed to the model to predict $y_{K+1}\\dots y_{2K}$ in parallel.",
"This process will continue until a terminator ${<}{/}\\text{s}{>}$ is generated.",
"Figure REF gives illustrations for both short and long-distance prediction.",
"Figure: Short-distance prediction (top) and long-distance prediction (bottom)."
],
[
"Relaxed Causal Mask",
"In the Transformer decoder, the causal mask is a lower triangular matrix, which strictly prevents earlier decoding steps from peeping information from later steps.",
"We denote it as strict causal mask.",
"However, in the SAT decoder, strict causal mask is not a good choice.",
"As described in the previous subsection, in long-distance prediction, the model predicts $y_{K+1}$ by feeding with $y_1$ .",
"With strict causal mask, the model can only access to $y_1$ when predict $y_{K+1}$ , which is not reasonable since $y_1\\dots y_K$ are already produced.",
"It is better to allow the model to access to $y_1\\dots y_K$ rather than only $y_1$ when predict $y_{K+1}$ .",
"Therefore, we use a coarse-grained lower triangular matrix as the causal mask that allows peeping later information in the same group.",
"We refer to it as relaxed causal mask.",
"Given the target length $n$ and the group size $K$ , relaxed causal mask $M \\in \\mathbb {R}^{n\\times n}$ and its elements are defined below: $ M[i][j]={\\left\\lbrace \\begin{array}{ll}1 & \\text{if $j < ([(i-1)/K]+1)\\times K$} \\\\0 & \\text{other}\\end{array}\\right.",
"}$ For a more intuitive understanding, Figure REF gives a comparison between strict and relaxed causal mask.",
"Figure: Strict causal mask (left) and relaxed causal mask (right) when the target length n=6n=6 and the group size K=2K=2.We mark their differences in bold."
],
[
"The SAT",
"Using group-level chain rule instead of word-level chain rule, long-distance prediction instead of short-distance prediction, and relaxed causal mask instead of strict causal mask, we successfully extended the Transformer to the SAT.",
"The Transformer can be viewed as a special case of the SAT, when the group size $K$ = 1.",
"The non-autoregressive Transformer (NAT) described in gu2017non can also be viewed as a special case of the SAT, when the group size $K$ is not less than maximum target length.",
"Table REF gives the theoretical complexity and acceleration of the model.",
"We list two search strategies separately: beam search and greedy search.",
"Beam search is the most prevailing search strategy.",
"However, it requires the decoder states to be updated once every word is generated, thus hinders the decoding parallelizability.",
"When decode with greedy search, there is no such concern, therefore the parallelizability of the SAT can be maximized.",
"Table: Theoretical complexity and acceleration of the SAT.",
"aa denotes the time consumed on the decoder network (calculating a distribution over the target vocabulary) each time step and bb denotes the time consumed on search (searching for top scores, expanding nodes and pruning).",
"In practice, aa is usually much larger than bb since the network is deep."
],
[
"Experiments",
"We evaluate the proposed SAT on English-German and Chinese-English translation tasks."
],
[
"Experimental Settings",
"Datasets For English-German translation, we choose the corpora provided by WMT 2014 [4].",
"We use the newstest2013 dataset for development, and the newstest2014 dataset for test.",
"For Chinese-English translation, the corpora we use is extracted from LDCThe corpora include LDC2002E18, LDC2003E14, LDC2004T08 and LDC2005T0.. We chose the NIST02 dataset for development, and the NIST03, NIST04 and NIST05 datasets for test.",
"For English and German, we tokenized and segmented them into subword symbols using byte-pair encoding (BPE) [20] to restrict the vocabulary size.",
"As for Chinese, we segmented sentences into characters.",
"For English-German translation, we use a shared source and target vocabulary.",
"Table REF summaries the two corpora.",
"Table: Summary of the two corpora.Table: Results on English-German translation.",
"Latency is calculated on a single NVIDIA TITAN Xp without batching.",
"For comparison, we also list results reported by gu2017non,kaiser2018fast,lee2018deterministic.",
"Note that gu2017non,lee2018deterministic used PyTorch as their platform, but we and kaiser2018fast used TensorFlow.",
"Even on the same platform, implementation and hardware may not exactly be the same.",
"Therefore, it is not fair to directly compare BLEU and latency.",
"A fairer way is to compare performance degradation and speedup, which are calculated based on their own baseline.Baseline We use the base Transformer model described in vaswani2017attention as the baseline, where $d_{model}=512 \\text{ and } N=6$ .",
"In addition, for comparison, we also prepared a lighter Transformer model, in which two encoder/decoder blocks are used ($N$ = 2), and other hyper-parameters remain the same.",
"Hyperparameters Unless otherwise specified, all hyperparameters are inherited from the base Transformer model.",
"We try three different settings of the group size $K$ : $K$ = 2, $K$ = 4, and $K$ = 6.",
"For English-German translation, we share the same weight matrix between the source and target embedding layers and the pre-softmax linear layer.",
"For Chinese-English translation, we only share weights of the target embedding layer and the pre-softmax linear layer.",
"Search Strategies We use two search strategies: beam search and greedy search.",
"As mentioned in Section REF , these two strategies lead to different parallelizability.",
"When beam size is set to 1, greedy search is used, otherwise, beam search is used.",
"Knowledge Distillation Knowledge distillation [10], [13] describes a class of methods for training a smaller student network to perform better by learning from a larger teacher network.",
"For NMT, kim2016sequence proposed a sequence-level knowledge distillation method.",
"In this work, we apply this method to train the SAT using a pre-trained autoregressive Transformer network.",
"This method consists of three steps: (1) train an autoregressive Transformer network (the teacher), (2) run beam search over the training set with this model and (3) train the SAT (the student) on this new created corpus.",
"Initialization Since the SAT and the Transformer have only slight differences in their architecture (see Figure REF ), in order to accelerate convergence, we use a pre-trained Transformer model to initialize some parameters in the SAT.",
"These parameters include all parameters in the encoder, source and target word embeddings, and pre-softmax weights.",
"Other parameters are initialized randomly.",
"In addition to accelerating convergence, we find this method also slightly improves the translation quality.",
"Training Same as vaswani2017attention, we train the SAT by minimize cross-entropy with label smoothing.",
"The optimizer we use is Adam [14] with $\\beta _1=0.9$ , $\\beta _2=0.98$ and $\\varepsilon =10^{-9}$ .",
"We change the learning rate during training using the learning rate funtion described in vaswani2017attention.",
"All models are trained for 10K steps on 8 NVIDIA TITAN Xp with each minibatch consisting of about 30k tokens.",
"For evaluation, we average last five checkpoints saved with an interval of 1000 training steps.",
"Evaluation Metrics We evaluate the translation quality of the model using BLEU score [19].",
"Implementation We implement the proposed SAT with TensorFlow [1].",
"The code and resources needed for reproducing the results are released at https://github.com/chqiwang/sa-nmt."
],
[
"Results on English-German",
"Table REF summaries results of English-German translation.",
"According to the results, the translation quality of the SAT gradually decreases as $K$ increases, which is consistent with intuition.",
"When $K$ = 2, the SAT decodes 1.51$\\times $ faster than the Transformer and is almost lossless in translation quality (only drops 0.21 BLEU score).",
"With $K$ = 6, the SAT can achieve 2.98$\\times $ speedup while the performance degeneration is only 8%.",
"When using greedy search, the acceleration becomes much more significant.",
"When $K$ = 6, the decoding speed of the SAT can reach about $5.58\\times $ of the Transformer while maintaining 88% of translation quality.",
"Comparing with gu2017non,kaiser2018fast,lee2018deterministic, the SAT achieves a better balance between translation quality and decoding speed.",
"Compared to the lighter Transformer ($N$ = 2), with $K$ = 4, the SAT achieves a higher speedup with significantly better translation quality.",
"In a real production environment, it is often not to decode sentences one by one, but batch by batch.",
"To investigate whether the SAT can accelerate decoding when decoding in batches, we test the decoding latency under different batch size settings.",
"As shown in Table REF , the SAT significantly accelerates decoding even with a large batch size.",
"Table: Time needed to decode one sentence under various batch size settings.",
"A single NVIDIA TIAN Xp is used in this test.It is also good to know if the SAT can still accelerate decoding on CPU device that does not support parallel execution as well as GPU.",
"Results in Table REF show that even on CPU device, the SAT can still accelerate decoding significantly.",
"Table: Time needed to decode one sentence on CPU device.",
"Sentences are decoded one by one without batching.",
"KK=1 denotes the Transformer.Table: Results on Chinese-English translation.",
"Latency is calculated on NIST02."
],
[
"Results on Chinese-English",
"Table REF summaries results on Chinese-English translation.",
"With $K$ = 2, the SAT decodes 1.69$\\times $ while maintaining 97% of the translation quality.",
"In an extreme setting where $K$ = 6 and beam size = 1, the SAT can achieve 6.41$\\times $ speedup while maintaining 83% of the translation quality."
],
[
"Analysis",
"Effects of Knowledge Distillation As shown in Figure REF , sequence-level knowledge distillation is very effective for training the SAT.",
"For larger $K$ , the effect is more significant.",
"This phenomenon is echoing with observations by gu2017non,oord2017parallel,lee2018deterministic.",
"In addition, we tried word-level knowledge distillation [13] but only a slight improvement was observed.",
"Figure: Performance of the SAT with and without sequence-level knowledge distillation.Position-Wise Cross-Entropy In Figure REF , we plot position-wise cross-entropy for various models.",
"To compare with the baseline model, the results in the figure are from models trained on the original corpora, i.e., without knowledge distillation.",
"As shown in the figure, position-wise cross-entropy has an apparent periodicity with a period of $K$ .",
"For positions in the same group, the position-wise cross-entropy increase monotonously, which indicates that the long-distance dependencies are always more difficult to model than short ones.",
"It suggests the key to further improve the SAT is to improve the ability of modeling long-distance dependencies.",
"Figure: Position-wise cross-entropy for various models on English-German translation.Table: Three sample Chinese-English translations by the SAT and the Transformer.",
"We mark repeated words or phrases by red font and underline.Case Study Table REF lists three sample Chinese-English translations from the development set.",
"As shown in the table, even when produces $K$ = 6 words at each time step, the model can still generate fluent sentences.",
"As reported by gu2017non, instances of repeated words or phrases are most prevalent in their non-autoregressive model.",
"In the SAT, this is also the case.",
"This suggests that we may be able to improve the translation quality of the SAT by reducing the similarity of the output distribution of adjacent positions."
],
[
"Conclusion",
"In this work, we have introduced a novel model for faster sequence generation based on the Transformer [24], which we refer to as the semi-autoregressive Transformer (SAT).",
"Combining the original Transformer with group-level chain rule, long-distance prediction and relaxed causal mask, the SAT can produce multiple consecutive words at each time step, thus speedup decoding significantly.",
"We conducted experiments on English-German and Chinese-English translation.",
"Compared with previously proposed non-autoregressive models [8], [16], [12], the SAT achieves a better balance between translation quality and decoding speed.",
"On WMT'14 English-German translation, the SAT achieves 5.58$\\times $ speedup while maintaining 88% translation quality, significantly better than previous methods.",
"When produces two words at each time step, the SAT is almost lossless (only 1% degeneration in BLEU score).",
"In the future, we plan to investigate better methods for training the SAT to further shrink the performance gap between the SAT and the Transformer.",
"Specifically, we believe that the following two directions are worth study.",
"First, use object function beyond maximum likelihood to improve the modeling of long-distance dependencies.",
"Second, explore new method for knowledge distillation.",
"We also plan to extend the SAT to allow the use of different group sizes $K$ at different positions, instead of using a fixed value."
],
[
"Acknowledgments",
"We would like to thank the anonymous reviewers for their valuable comments.",
"We also thank Wenfu Wang, Hao Wang for helpful discussion and Linhao Dong, Jinghao Niu for their help in paper writting."
]
] | 1808.08583 |
[
[
"NNLO PDFs for the LHC"
],
[
"Abstract We consider some trends, achievements and a series of remaining problems in the precision determination of parton distribution functions.",
"For the description of the scaling violations of the deep-inelastic scattering data, forming the key ingredients to all PDF fits, a solid theoretical framework is of importance.",
"It is provided by the fixed flavor number scheme in describing the heavy-quark contributions which is found in good agreement with the present experimental data in a very wide range of momentum transfers.",
"In this framework also a consistent determination of the heavy-quark masses is possible at high precision.",
"The emerging Drell-Yan data measured at hadron colliders start to play a crucial role in disentangling the quark species, particularly at small and large values of $x$.",
"These new inputs demonstrate a good overall consistency with the earlier constraints on the PDFs coming from fixed-target experiments.",
"No dramatic change is observed in the PDFs in case of a consistent account of the higher-order QCD corrections and when leaving enough flexibility in the PDF shape parameterization.}"
],
[
"DESY 18-150, DO-TH 18/19 NNLO Parton Distributions for the LHC NNLO PDFs for the LHC Sergey AlekhinThis work was supported in part by Bundesministerium für Bildung und Forschung (contract 05H15GUCC1).",
"II.",
"Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany; Institute for High Energy Physics,142281 Protvino, Russia E-mail: sergey.alekhin@desy.de Johannes Blümlein Deutsches Elektronensynchrotron DESY, Platanenallee 6, D–15738 Zeuthen, Germany E-mail: Johannes.Bluemlein@desy.de Sven-Olaf Moch II.",
"Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany E-mail: sven-olaf.moch@desy.de We consider some trends, achievements and a series of remaining problems in the precision determination of parton distribution functions.",
"For the description of the scaling violations of the deep-inelastic scattering data, forming the key ingredients to all PDF fits, a solid theoretical framework is of importance.",
"It is provided by the fixed flavor number scheme in describing the heavy-quark contributions which is found in good agreement with the present experimental data in a very wide range of momentum transfers.",
"In this framework also a consistent determination of the heavy-quark masses is possible at high precision.",
"The emerging Drell-Yan data measured at hadron colliders start to play a crucial role in disentangling the quark species, particularly at small and large values of $x$ .",
"These new inputs demonstrate a good overall consistency with the earlier constraints on the PDFs coming from fixed-target experiments.",
"No dramatic change is observed in the PDFs in case of a consistent account of the higher-order QCD corrections and when leaving enough flexibility in the PDF shape parameterization.",
"Loops and Legs in Quantum Field Theory (LL2018) 29 April 2018 - 04 May 2018 St. Goar, Germany After a long period of phenomenological studies, contemporary particle physics has reached the level of percent accuracy for the parton distribution functions (PDFs).",
"However, some important features still need further clarification [1].",
"This concerns in particular the asymptotic behavior for small and large values of Bjorken $x$ .",
"The first issue is in turn related to the theoretical foundations for the description of small-$x$ deep-inelastic scattering (DIS) processes, including the heavy-quark contributions to the structure functions due to charm and bottom.",
"The latter provide very essential constraints on the PDFs in the small-$x$ region.",
"The heavy quark contribution to deep-inelastic scattering (DIS) is commonly considered within two competitive factorization schemes, one with a fixed number of flavors (FFN) and another variable number of flavors (VFN).",
"A detailed comparison of these two approaches was performed in Ref.",
"[1] and the FFN scheme was found to provide a better description of the existing HERA data on DIS charm production.",
"Indeed, the superiority of the FFN scheme versus a VFN scheme within the kinematic region covered by HERA had been observed already very early on, cf. Ref. [2].",
"Figure: The exact result for the 3-loop pure single OME a Qq, ps (3)0 a^{(3)\\,0}_{Qq,\\,\\rm ps} and the comparison to previous approximations of Ref.",
"based on the limited set of Mellin moments from Ref.",
".The FFN scheme turns out to provide a more consistent setting for the heavy quark masses than in the case of the VFN scheme See also Ref.",
"[3] for an updated comparison with the use of recent HERA data on the heavy-quark production.",
"Present theoretical calculations in the FFN scheme include the next-to-next-to-leading order (NNLO) Wilson coefficients [4], which are modeled using the available asymptotics in different kinematic regimes between heavy-quark production at threshold and the high-energy limit.",
"The asymptotic expressions of these two regimes are matched using the factorized form of the massive Wilson coefficients expressed in terms of massless coefficient functions and the massive operator matrix elements (OMEs), which are valid at momentum transfers $Q^2 \\gg m_h^2 $ , where $m_h$ denotes heavy quark mass [6], [5], [8], [7].",
"At NNLO one needs for this purpose the 3-loop OMEs, which are known exactly in part [10], [6], [5], [12], [11], [7] and are available in main terms in form of an approximation [4] based on the fixed number of Mellin moments, calculated in Ref. [9].",
"Such approximations are commonly less accurate at small $x$ , however their uncertainty can be validated using exact results, e.g., the recently calculated pure-singlet OME [5].",
"It turns out, that the exact pure-singlet term is well within the uncertainties quoted for the approximate form obtained earlier from the first five non-vanishing Mellin moments [9], see Fig.",
"REF .",
"Moreover, the exact pure-singlet term can be employed to derive the gluon OME using the Casimir-scaling approximation.",
"The expressions for the NNLO massive Wilson coefficients in the FFN scheme comprise all these ingredients The 3-loop massive OMEs obtained in this way can be also used to compute NNLO PDFs in the VFN scheme, see e.g.",
"Ref. [6]..",
"An important improvement in this formalism concerns definition of the heavy-quark mass.",
"While the perturbative calculations are usually based on the pole mass-scheme, one rather turns to the $\\overline{\\mathrm {MS}}\\, $ running-mass for reasons of perturbative stability [13].",
"Good agreement with the existing data is achieved by using this framework See Refs.",
"[12], [5], [11] for the scheme transformation to the $\\overline{\\rm MS}$ scheme up to 3-loops..",
"Figure: The 1σ1\\sigma band for the NNLO quark iso-spin asymmetry(d ¯-u ¯)/(d ¯+u ¯)(\\bar{d}-\\bar{u})/(\\bar{d}+\\bar{u}) in the 3-flavor scheme at the scaleof μ=3 GeV \\mu =3~{\\rm GeV} as a function of Bjorken xx obtained in variants of the ABMP16 PDFfit with the data on production of WW-bosons (left-titled hash),ZZ-bosons (right-titled hash), and both WW- and ZZ-boson (shaded area)excluded form the fit.The value for the $c$ -quark $\\overline{\\mathrm {MS}}\\, $ mass obtained in the recent ABMP16 fit [14] $m_c(m_c)=1.252\\pm 0.018 {\\rm (exp.",
")}\\pm 0.010 {\\rm (th.",
")}$ is in a very good agreement with other precision determinations, e.g.",
"based on the $e^+e^-$ data [15].",
"The inclusive DIS data have a limited potential to disentangle the distributions of the quark species, particularly at small $x$ .",
"This is due to the fact that the HERA data consist only of proton data.",
"Meanwhile, however, the Drell-Yan (DY) data from the LHC are of sufficient quality to determine the different flavor distributions very well up to energies of 13 TeV.",
"These data probe the PDFs in a wide range of $x$ , down to $x \\sim 10^{-4}$ and provide a variety of constraints on the quark distributions due to the production of both, $W^\\pm $ - and $Z$ -bosons.",
"The impact of this input on the PDF determination is demonstrated for instance in ABMP16 fit [14], which includes a wide collection of the $W^\\pm $ - and $Z$ -production data from the ATLAS, CMS and LHCb experiments at the LHC and from the DØ experiment at Tevatron.",
"By discarding these data sets in a test variant of the ABMP16 fit we find an essential deterioration in the determination of the quark distributions, leading to a greatly expanded uncertainty in the iso-spin asymmetry $(\\bar{d}-\\bar{u})/(\\bar{d}+\\bar{u})$ at small $x$ .",
"In the absence Figure: The pulls for the ATLAS data on thepp→W ± +X→l ± ν+Xpp \\rightarrow W^\\pm +X \\rightarrow l^\\pm \\nu + X production (a) and (b)and pp→Z+X→l + l - +Xpp \\rightarrow Z+X \\rightarrow l^+l^- + X (c): central region, (d): forward region) at s=7\\sqrt{s} = 7 TeVcollected at luminosity of 35 pb -1 ^{-1} (2011) (blue squares)and 4.6 fb -1 ^{-1} (2016) (red circles)with cuts on the lepton's transverse momentum P T l >20 GeV P_T^l>20~{\\rm GeV}as a function of the lepton pseudo-rapidity η\\eta versus NNLO predictionsobtained using FEWZ (version 3.1) , and the ABMP16 PDFs.of DY data this piece is essentially unconstrained, see Fig.",
"REF .",
"Therefore, in earlier PDF parameterizations, it was commonly set to zero for $x\\rightarrow 0$ .",
"The collider DY data prefer a sizable negative value at $x\\sim 10^{-4}$ and a symmetric non-strange sea is observed at $x\\lesssim 10^{-5}$ only [16].",
"In general, the available DY data a very consistent.",
"However, with rising experimental accuracy some tension between different experiments or even within one experiment may emerge.",
"In particular, this concerns the recent ATLAS data on $W^\\pm $ - and $Z$ -production at a center-of-mass (c.m.s.)",
"energy of 7 TeV [17].",
"This sample is in good agreement with the earlier data obtained by the same collaboration from the low-luminosity run [18].",
"It is in part related to the $W^\\pm $ -production, see Fig.",
"REF .",
"Meanwhile, the $Z$ -production cross sections at central rapidity moved somewhat higher than the earlier ones.",
"The tension is at the level of 1-2$\\sigma $ .",
"It makes it difficult to describe the recent ATLAS data with the PDFs tuned to the previous release.",
"Moreover, the epWZ16 PDFs extracted by ATLAS from data of Ref.",
"[17], in combination with the inclusive DIS sample from HERA, demonstrate some unusual features, namely the strange sea is greatly enhanced if compared to strange suppression factors of $\\sim 0.5$ as commonly obtained in the PDF fits.",
"To the most extent such an enhancement can be explained by a particular PDF shape employed in the analysis of Ref. [17].",
"This shape had been suggested for the HERAPDF fit based on the HERA data only long ago.",
"Therefore it contains many constraints due to the limited potential of inclusive DIS in disentangling quark distributions.",
"By applying these constraints the non-strange sea distributions are artificially suppressed and this suppression is compensated in the ATLAS analysis by the strangeness enhancement, which finally leads to an abnormal strange sea suppression factor [21].",
"If instead a flexible enough PDF shape is used, the strangeness preferred by the ATLAS data is in a reasonable agreement with the earlier determinations, although some tension at $x\\sim 0.01$ still persists, see Fig.",
"REF .",
"This tension is evidently related to the impact of the upward shift in the central $Z$ -production observed for the recent ATLAS measurements, see Fig.",
"REF .",
"However, it is worth noting that the ATLAS data for forward-rapidity demonstrate a different trend, although being statistically less significant.",
"Besides, the CMS data on $Z$ -production are also somewhat lower than the ATLAS results, see Ref.",
"[21] for details.",
"Therefore this tension still deserves further clarification.",
"Another problematic aspect of the DY data analysis concerns the accuracy of the tools, which are needed for the computation of the cross sections with account of realistic experimental cuts on the lepton transverse momentum.",
"The fully exclusive NNLO codes FEWZ [19], [20] and DYNNLO [22], [23], which accomplish these computations, are not in perfect agreement in the relevant kinematical region, see Fig.",
"REF .",
"Figure: The same as in Fig.",
"for the forward ZZ-productions data and the NNLO predictionsobtained with FEWZ (version 3.1) , (red circles) and DYNNLO (version 1.4) , (blue squares).In general, the predictions by DYNNLO are lower than the ones by FEWZ by $\\sim 1\\%$ .",
"However, at the edges of the distributions this difference rises to 10%.",
"The discrepancies between DYNNLO and FEWZ were partially understood as being due to the numerical integration accuracy [24] and due to effects of experimental cuts on the lepton transverse momentum in higher-order QCD computations [25], but at the moment the theoretical accuracy is limiting the related studies In the ATLAS analysis [17] the DYNNLO calculations are used for nominal results and the difference between DYNNLO and FEWZ is taken as a theoretical uncertainty..",
"The DY collider data also help to constrain the large-$x$ region of the quark distributions, in particular for the ratio $d/u$ .",
"In this context the DØ measurement of the $W$ charge asymmetry [26] provides the statistically most significant constraint.",
"Since $W$ -boson production is not measured directly, the $W$ -asymmetry is derived in the DØ analysis from the measurement of the electrons stemming from the $W$ decays.",
"This is possible in a unique way at leading order (LO) only, while account of the higher-order corrections requires additional modeling.",
"This, in particular, causes sensitivity to the $W$ -asymmetry obtained by the choice of the PDFs used.",
"It leads to a certain tension between the $W$ -asymmetry data and the original $e$ -asymmetry ones, if the PDFs are varied.",
"In particular, the predictions of the $W$ -asymmetry for the DØ kinematics obtained with the ABMP15 PDFs based on the DØ data on the $e$ -asymmetry [27], are in substantial disagreement with the DØ data on $W$ -asymmetry, see Fig.",
"REF .",
"Figure: The pulls of the DØ data on the WWcharge asymmetry versus the predictions obtainedwith FEWZ (version 3.1) at NNLO in QCD and the ABMP15PDFs constrained by the DØ ee-asymmetry data as a function of the WW-boson rapidity η W \\eta _W.The shaded area displays the PDF uncertainties in the predictions.The potential of the DØ measurements on the large-$x$ asymptotics of the $d/u$ ratio was checked in the recent CJ15 PDF fit [28].",
"An advantage of this analysis is a flexible PDF shape, which allows for a non-vanishing value of $(d/u)\\vert _{x=1}$ .",
"The CJ15 analysis combines both the $W$ -asymmetry and the $e$ -asymmetry DØ data.",
"The large-$x$ $d/u$ ratio is mainly driven by the $W$ -asymmetry data due to its statistical significance.",
"The impact of these data is quite sensitive on the theoretical accuracy of the analysis.",
"The $d/u$ ratio obtained with the LO description leads to higher values than the one obtained accounting for the next-to-leading order (NLO) corrections, see Fig.",
"REF .",
"Furthermore, the uncertainties in the $d/u$ -ratio do substantially rise in the NLO fit.",
"This is evidently due to the smearing of the predictions by the gluon-initiated contribution and the propagation of the uncertainty of the gluon-distribution into the ratio of $d/u$ extracted from the fit.",
"The theory framework of the CJ15 fit is based on a $K$ -factor approximation of the $W$ -production cross section, with the NLO predictions represented as a product of the LO approximation and the pre-computed ratio of the NLO and LO cross sections.",
"In case of the $\\bar{p}p$ initial state such an approach reproduces the initial LO predictions.",
"Therefore the CJ15 result on $d/u$ should be biased upwards due to the missing NLO corrections, see Ref.",
"[29] for details.",
"The value of $d/u$ preferred by the DØ data on the $e$ -asymmetry [27] is substantially lower than the $W$ -asymmetry results and even extends to negative values at $x\\rightarrow 1$ , although with large uncertainties, see Fig.",
"REF .",
"Comparing it with the NLO determination based on the $W$ -asymmetry, we conclude that there is no strong evidence in favor of a non-vanishing $(d/u)\\vert _{x=1}$ from the analysis of the DØ data.",
"Moreover, the $e$ -asymmetry data, preferring a smaller value of $d/u$ , are less model-dependent than the $W$ -asymmetry.",
"The interpretation of the DØ data in the PDF fit turns out to be essential for the related phenomenology of electroweak single-top production since the latter reaction is to a great extent driven by the quark-initiated subprocesses.",
"Therefore a trend observed for the $d/u$ ratio in the variants of the PDF fit with a different treatment of the DØ experimental input is reflected in the ratio of the top and anti-top production cross sections $R_{t/\\bar{t}}$ computed with respective PDFs, see Fig.",
"REF .",
"For the fit based on the $e$ -asymmetry data the value of $R_{t/\\bar{t}}$ is larger by $\\sim 2 \\sigma $ than for the one obtained from the LO fit using the $W$ -asymmetry data.",
"This is comparable to the spread in the predictions of different PDFs, which can be explained in part by the selection of the DY collider data and their treatment.",
"In summary, we have considered some current trends, achievements and problems in the precision determination of PDFs.",
"For the DIS data a solid theoretical framework is available with the FFN scheme used for description of the heavy-quark contribution.",
"It provides good agreement with existing experimental data in a wide range of momentum transfers and implies a consistent setting of the heavy-quark masses, which are basic parameters of the Standard Model.",
"The emerging DY data collected at the hadron colliders start to play a crucial role in disentangling quark species, particularly at small and large values of $x$ .",
"These new inputs demonstrate a good overall consistency with the earlier constraints on PDFs coming from the fixed-target experiments.",
"No dramatic change in the PDFs is caused in case of consistent account of the higher-order QCD corrections and using PDF shapes which are flexible enough for fitting the experimental data.",
"Figure: Left: The same as in Fig.",
"for the ratio d/ud/u obtained using the CJ15 PDFshape andwith addition of the DØ data onWW- and ee-asymmetry, described within various approximations(vertical hash: WW-asymmetry at LO,left-titled hash: the same at NLO,right-tilted hash: ee-asymmetry at NLOin comparison with the nominal CJ15 PDFs(shaded area).",
"Right: The ratio of single top to anti-top production crosssection in pppp collisions at c.m.s.",
"energy 7 TeV computed with the PDFsobtained in these variants of the fit in comparison with the ATLASdata and the predictions ofABMP16 , CT14 ,MMHT14 ,NNPDF3.0 andNNPDF3.1 PDFs."
]
] | 1808.08404 |
[
[
"Convolutional Neural Networks for Aerial Vehicle Detection and\n Recognition"
],
[
"Abstract This paper investigates the problem of aerial vehicle recognition using a text-guided deep convolutional neural network classifier.",
"The network receives an aerial image and a desired class, and makes a yes or no output by matching the image and the textual description of the desired class.",
"We train and test our model on a synthetic aerial dataset and our desired classes consist of the combination of the class types and colors of the vehicles.",
"This strategy helps when considering more classes in testing than in training."
],
[
"Introduction",
"Aerial imagery, captured by drones or Unmanned Aerial Vehicles (UAVs), is a great tool for surveillance because of its wide field of view and the ability of drones to access places that would otherwise be difficult to visit.",
"Aerial imagery has also other applications like border security, search and rescue tasks, and image and video understanding.",
"Also, it can be used in human-human, human-vehicle, and vehicle-vehicle interaction understanding.",
"The wide field of view advantages of aerial imagery, however, result in objects of interest occupying small number of pixels in each image.",
"In comparison to the regular view or street view images, aerial images have less information and details about vehicles as well as other objects in the image.",
"Therefore, it is common that a vehicle in an aerial view is missed because of the background or other objects.",
"On the other hand, false positive predictions are also highly probable.",
"The other issue that makes the resolution challenge harder is the limitation in the computational resources.",
"Although it is possible to take a high resolution image, processing a large image will result in a huge unavoidable computational costs, specially if we are interested in implementing an online aerial vehicle detection system.",
"Figure: (a) An aerial image.",
"(b) Some exemplar objects of interest (vehicles).The application of aerial vehicle detection and recognition can be more specific if the goal of the system is not just limited to detect vehicles but to find specific vehicles.",
"For example, a detection system can concentrate on searching for a specific car with a specific color, type, and other descriptions (e.g., yellow taxi, large green truck).",
"In this scenario, the detection system can be used in the applications like finding a suspicious vehicle or target vehicle among several other vehicles, objects, and backgrounds.",
"In this specific goal, in addition to the resolution challenge, there is an open-ended challenge.",
"Providing a comprehensive dataset that covers all the probable objects variations and classes (e.g., vehicle types, colors, shapes, and other variations) is impossible.",
"Therefore, the detection system has to respond to unseen targets.",
"In other words, during the testing phase it sees sample variations that it has not seen in its training phase.",
"This challenge can be called as unseen challenge or open-ended challenge.",
"Visual Question Answering (VQA) systems take an image and an open-ended textual question about the given image, and try to provide an answer to the question in a textual format [1].",
"The core idea behind the VQA task is to answer to unseen questions that could be considered here as classes.",
"Antol et al.",
"by using LSTM and MLP structure achieves acceptable results on their large dataset consisting about 0.25M images, 0.76M questions, and 10M answershttp://www.visualqa.org/.",
"In order to alleviate the challenge of objects occupying small number of pixels, we split the problem into two sub-problems [2].",
"We first assume that a deep detector like Single Shot Multibox Detector (SSD) [3] extracts objects or areas of interest, and second, we use a deep convolutional network to recognize which of the extracted objects of interest are also the vehicles we wish to detect.",
"In this paper, we propose a framework that can handle the problem of open-ended classification or prediction.",
"In a classical image classification system an image is processed and an output label is produced.",
"However, in this paper, we use another novel architecture in which it receives an image and a desired textual description of the class, represented by a code vector, and makes a yes or no decision about the correctness of the input label.",
"In other words, it decides if the input image has the desired textual description of the class label or not (see Figure 2).",
"Figure: (a) A classical classifier that receives an image and predicts a label code for the image class.",
"(b) The proposed architecture that can consider classes that have not been seen during the training.This paper is organized as follows.",
"In Section II, we review the literature of deep object detectors and visual question answering.",
"Then in Section III, we investigate our proposed framework.",
"In Section IV, we explain the dataset, experiments, and results.",
"Finally, conclusion and future works are explained in the Section V."
],
[
"Related Works",
"The combination of choosing a good hand-engineered image feature descriptors like histogram of gradients (HOG) [4] and scale-invariant feature transform (SIFT) [5], and choosing a classifier like support vector machine (SVM) [6] or multilayer perceptron (MLP) have been the main focus of research papers in the area of image classification for several years.",
"However, in recent years deep convolutional neural networks, which have outperformed other methods on different datasets like VOC [7] and Imagenet [8], are attracting interest in the field of image classification.",
"Deep convolutional neural networks like LeNet [9] and AlexNet [10] have confirmed their effectiveness as classifiers that only receive raw images without any type of image feature descriptors.",
"R-CNN [11] can be considered as the first considerable and well-known deep structure for the application of object detection.",
"This method takes an input image, then a classical regions of interest generator like selective search [12] creates about 2000 regions proposals, and a deep convolutional neural network (CNN) extracts visual features, and finally an SVM classifier determines if there is any specific object in these proposals or not.",
"However, doing all these steps separately makes R-CNN slow and inaccurate.",
"They also achieved a good detection result since a high-capacity convolutional neural networks was applied to bottom-up region proposals, and because they used a pre-trained model for their initial points.",
"Fast R-CNN [13] is the next top method in the object detection literature.",
"In contrast to R-CNN, fast R-CNN has its own classifier layer.",
"It takes an image and multiple of regions of interest proposals, then a CNN creates the feature maps for the proposals and a softmax layer and a regressor layer to find the objects in the image.",
"The other important advantage for Fast R-CNN is that it does not need a disk storage for feature caching.",
"The next step, which led to real-time object detection with region proposal networks, was Faster R-CNN [14].",
"Faster R-CNN eliminates the need for extra regions of interest proposal generator.",
"In other words, generating proposals is also done using a convolutional framework.",
"Their method employs the anchor concept that helps to better catch different objects with different sizes and aspect ratios.",
"It must be noted that Faster R-CNN employs Fast R-CNN for some parts of its algorithm.",
"YOLO [15] might be considered as the first work in which the object detection problem is considered as a regression problem, while the previous works focused on classifiers to perform detection.",
"A single CNN predicts bounding boxes as well as the class probabilities from only the input images, without any need for the proposals.",
"This is an end-to-end method for detecting objects in the images.",
"While YOLO is well-known for its extreme speed, its accuracy is not as good as the other top methods.",
"Single Shot Multibox detector or SSD [3] is another end-to-end single shot object detector that uses a deep learning architecture which in addition to the speed, it has an outstanding accuracy.",
"SSD is a convolutional network that predicts a large number of bounding boxes and scores, which theoretically helps to detect 8732 objects in just one image.",
"The output of SSD, like the previous method, is passed through a non-maximum suppression operator to reduce the overlapping results.",
"SSD is made of a base network, which has a structure like VGG [16].",
"This base network creates a preliminary representation for the next steps which is done using the extra feature layers.",
"These auxiliary layers decrease in size progressively and are responsible for detecting small to large objects.",
"In other words, smaller objects are detected in the earlier layers while the larger ones are detected in the last layers.",
"SSD uses the idea of default boxes, a concept similar to anchors, that helps the network to learn specific filters for specific scales and aspect ratios.",
"These are responsible for objects at different scales and aspect ratios.",
"Figure: Proposed method for the task of aerial vehicle detection.",
"Detected objects of interest are described using the features extracted by the convolutional layers.",
"Desired classes are represented using bag of words and then fully connected layers to build a common latent space in which the yes or no decision is made on top of this sub-space.Equation (1) is the SSD's cost function.",
"SSD has an objective function consisting of two parts, one for the localization cost that codes the location of the bounding boxes and the objects, and one for the confidence cost that determines the degree of certainty for the presence of an object in the specific bounding box or location.",
"$l(x,c,l,g)=1/N(L_{conf}(x,c)+{L_{loc}(x,l,g)}).$ The localization cost is a smooth L1 loss and codes the error between the bounding boxes from the ground-truth and the predicted one.",
"In contrast, the confidence cost is a softmax loss that can be considered as a classification cost too.",
"Note that $x$ is the parameter that determines the presence of an object in the SSD's default boxes.",
"During the training, this information can be calculated using the ground-truth information and the default boxes positions.",
"$c$ is for the class parameters, and $l$ and $g$ are the parameters for the predicted locations and ground-truth locations, respectively."
],
[
"Proposed Method",
"We propose a framework in which, first, an SSD [3], which has shown promising performances in the aerial image object detection literature [2] and [17], generates a number of objects of interest proposals for an input aerial image.",
"These proposals might contain vehicle, background, or other objects.",
"In other words, we would have a set of object patches or object chips per original image.",
"In contrast to the classical structures of the classifiers that receive an input image and predict the labels at the output, we propose an architecture that receives an image as well as the textual description of the desired class as the inputs.",
"This architecture predicts a yes and no decision that shows if the image has the desired class label or not (see Figure 2).",
"One of the main reasons for changing the structure is that in this new structure, the classifier is not limited to pretrained or predefined class labels.",
"In other worlds, class labels are also used like open-ended images.",
"We use a VGG-16 architecture [16] with only one fully connected layer to extract visual descriptors.",
"This convolutional structure consists of five convolutional layers with the following details.",
"The first two layers consist of two similar layers with the depth of 64 and 128, respectively.",
"The next three layers have three similar layers with the depth of 256, 512, and again 512, respectively.",
"Just after each layer, there is a max pooling structure to reduce the spatial size and increase the generalization.",
"A fully connected layer is used just after the fifth layer (see Figure 3) and its values are fed into the next step which is a fully-connected layer for fusion of visual features extracted by the VGG structure and the textual features that is described below.",
"Meanwhile, the textual description of the desired classes are coded using the bag of words representation, and then a fully connected layer transforms this information into the next space.",
"The textual description of the desired classes in our experiments consist of the color and the types of the vehicles, but they can be more complicated with more details about the vehicles.",
"As Figure 3 shows, the visual features, which are extracted by the VGG network and the textual features, which are extracted by the fully-connected layers, are fused and form a visual-textual sub-space.",
"This is done by using a fully-connected fusion layer.",
"Finally, a two-class softmax classifier is placed on top of the last layer and trained to predict if the image has the desired class or not (Figure 3 shows the details about the proposed method).",
"It is worth noting that all the weights of the network, including the visual feature extractor, the textual feature extractor and the softmax classifier are optimized together.",
"This fact will force each component of our algorithm to influence the optimization of other components."
],
[
"Experiments",
"In order to evaluate our proposed framework, we use our dataset that contains real aerial images and synthetic cars and trucks which are placed on the streets (see Figure 1 and Figure 4).",
"Vehicles can have seven different colors: black, white, gray, yellow, green, blue, and red.",
"The two types of vehicles in conjunction with these seven colors describe a 14-class recognition problem.",
"The synthetic aerial dataset contains about 5000 vehicles with information about the vehicle types and colors.",
"See Figure 4 for some examples.",
"More details about the dataset are provided in the following subsection.",
"Figure: Some samples from our synthetic aerial dataset."
],
[
"Dataset",
"Whilst a limited number of datasets are available [18], capturing Wide Area Motion Imagery can be an expensive and difficult process.",
"In addition to the obvious problems of obtaining specialist (often classified military) camera equipment, there is also the need to organize aircraft, pilots and permission to fly.",
"There are also broader legal implications of performing surveillance of a real city or community - a person could be identified by a house number he/she visits.",
"Critically, there is no definitive form of ground truth (e.g., vehicle types & positions) for comparative evaluation of methods - nor a means of placing a camera in a particular position, orientation and time, which is a useful capability for developing new algorithms.",
"A brief overview of the image and ground truth generation method is described here (see Figure 5), a more detailed discussion is presented in [19].",
"Firstly, the ground truth data is generated using a MATLAB simulation that controls an instance of SUMO traffic simulator (Simulation of Urban MObility) [20].",
"The MATLAB simulation can seamlessly transfer entities such as vehicles and people, between SUMO and itself.",
"SUMO is used to navigate vehicles as per a basic set of rules of the road, and also navigates pedestrians along sidewalks and crosswalks.",
"The MATLAB simulation handles the high level goals of the people such as when and where to visit (e.g., shop or workplace) the mode of transport to be used.",
"It also manages other bespoke \"`micro-simulations\"' such as walking from the sidewalk to the building doorway, and people using bus stops.",
"Figure: (left) ARGUS field of view illustrates the large area simulated (approx 6km x 6km).",
"(right) 6 examples of various areas at the same point in time.Similarly, images are then generated from the ground truth at each time step using a MATLAB controlled instance of the X-Plane flight simulator [21].",
"The MATLAB image generator can position the flight simulator's viewpoint, and spawn 3D vehicles of the correct type and color, finally triggering image captures and converting to the MATLAB matrix format (for saving or further processing).",
"A configuration based on the DARPA/BAE Systems ARGUS-IS imager [22] is used that contains an array of 368 5-Megapixel subcameras.",
"This generates an image of 1.8-Gigapixels (or 5-Gigabytes of uncompressed RGB imagery per frame) capturing a circular area of approximately 6km diameter (at a 6km altitude).",
"Equations describing the configuration of the subcamera array can be found in [19].",
"The final output per frame (each timestep) is the ground truth data saved in an XML format, 368 PNG subcamera images each with an accompanying metadata file for the subcamera position and orientation.",
"The example applications within [19] detail a tiled video playback tool, and an analysis tool for interpreting the ground truth data directly without the imagery (e.g., tracing paths taken by vehicles)."
],
[
"Results",
"In order to test our proposed method, we implemented two different experimental setups.",
"First, we trained the proposed method on 75 percent of the dataset (all the 14 classes), which contains visual and textual information about the vehicle types and colors.",
"Then, the remaining samples were used for the testing.",
"Figure 6 shows the accuracy, true positive, and true negative percentages for the testing set.",
"The three results indicate the promising power of the proposed method for recognizing the vehicles and their types and colors for the synthetic aerial images.",
"To check the ability of the proposed method on unseen classes (open-ended setup), we repeated the experiment in a way that we only trained on 13 classes and set one class aside for testing.",
"This experimental setup is repeated 14 times for all the 14 classes.",
"Figure 7 shows the accuracy of the system for the unseen classes.",
"Based on this experiment, we can see that the proposed method is capable of recognizing unseen vehicles but belonging to similar classes.",
"Figure: The performance of the proposed method on the synthetic aerial dataset.In order to understand the underlying process in the system, we tried to visualize the textual information of the desired class labels in the hidden layer.",
"Just for this experiment, we forced the fully connected layers for the textual feature to have a two dimensional representation.",
"Figure 8 shows this two dimensional sub-space.",
"It is clear that the two different vehicle types, trucks and cars, have a similar feature pattern and lie in a similar manifold.",
"The colors of the vehicles in these two vehicle types are also in the same order.",
"Therefore, it might be true that if the system is not trained on yellow cars, for example, it can respond to the unseen yellow car samples during the testing phase by considering the similar manifolds of the trucks and cars."
],
[
"Conclusion",
"In this paper, we investigated the problem of aerial vehicle detection.",
"We assumed that a deep detector with a fast and accurate performance like single shot multi-box detector or SSD generates a number of objects of interest for each aerial image, which can be called as image proposals or image patches.",
"In the next step, we used a VGG-16 structure framework to extract visual information for the generated image proposals.",
"On the other hand, the bag of words representation and fully-connected layers are used to make a textual feature representation for the desired classes.",
"The visual and textual information are fused and make a common latent sub-space, which is called visual-textual sub-space.",
"Based on this sub-space a softmax classifier is trained to generate yes or no outputs that correspond to the cases when the input image patch has the desired class label or not.",
"It is important to note that all the weights of the second step including the convolutional layers, the fully connected layers, and the softmax classifier, are optimized simultaneously.",
"In other words, visual and textual features are trained together.",
"We tested our system on a synthetic aerial dataset that contains information about the vehicle types and vehicle colors.",
"Results on this dataset, showed that in addition to the promising performance when recognizing seen or trained classes, our framework can recognize unseen classes as well, and this is the advantage of the open-ended framework.",
"For the future works, collecting or synthesizing more complicated datasets that have both visual and rich textual information would result in the further improvements in the field.",
"Figure: The accuracy of the proposed method in the unseen experiment.Figure: 2D visualization of the desired classes.",
"The line separates the cars and trucks (vehicle types), and the colors correspond to the vehicle colors."
]
] | 1808.08560 |
[
[
"What is an answer? - remarks, results and problems on PIO formulas in\n combinatorial enumeration, part I"
],
[
"Abstract For enumerative problems, i.e.",
"computable functions f from N to Z, we define the notion of an effective (or closed) formula.",
"It is an algorithm computing f(n) in the number of steps that is polynomial in the combined size of the input n and the output f(n), both written in binary notation.",
"We discuss many examples of enumerative problems for which such closed formulas are, or are not, known.",
"These problems include (i) linear recurrence sequences and holonomic sequences, (ii) integer partitions, (iii) pattern-avoiding permutations, (iv) triangle-free graphs and (v) regular graphs.",
"In part I we discuss problems (i) and (ii) and defer (iii)--(v) to part II.",
"Besides other results, we prove here that every linear recurrence sequence of integers has an effective formula in our sense."
],
[
"Introduction",
"We define what is an effective (closed, explicit) formula (solution, algorithm) for a problem in enumerative combinatorics or number theory.",
"An enumerative problem or a counting function is a computable function $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ (in part II also a computable function $f:\\;\\lbrace 0,1\\rbrace ^*\\rightarrow \\mathbb {Z}$ ) given usually via an algorithm computing it.",
"The algorithm, often inefficient, usually follows straightforwardly from the statement of the problem.",
"Our notation: $|X|$ and $\\#X$ denote cardinality of a set $X$ , $\\mathbb {N}=\\lbrace 1,2,\\dots \\rbrace $ , $\\mathbb {N}_0=\\lbrace 0,1,\\dots \\rbrace $ , $\\mathbb {Z}$ is the ring of integers, $\\mathbb {Q}$ and $\\mathbb {C}$ are, respectively, the fields of rational and complex numbers and $\\lbrace 0,1\\rbrace ^*$ is the set of finite binary words.",
"The asymptotic symbols $o(\\cdot )$ , $O(\\cdot )$ , $\\Omega (\\cdot )$ , $\\Theta (\\cdot )$ , $\\cdot \\ll \\cdot $ , $\\cdot \\sim \\cdot $ and $\\mathrm {poly}(\\cdot )$ have their usual meaning ($\\ll $ is synonymous to $O$ and $\\mathrm {poly}(x)=O((1+|x|)^d)$ for some $d\\in \\mathbb {N}$ ).",
"We will discuss many enumerative problems and effective formulas.",
"Here are some examples of enumerative problems $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ .",
"The Catalan numbers $f(n)=c_n=\\frac{1}{n}\\binom{2n-2}{n-1}$ which count planted trees with $n$ vertices, and many other structures.",
"Let $f(n)=c_n$ for even $c_n$ and $f(n)=1$ for odd $c_n$ .",
"A linear recurrence sequence $f(n+k)=\\sum _{i=0}^{k-1}a_if(n+i)$ , given by initial values $f(1),\\dots ,f(k)\\in \\mathbb {Z}$ and recurrence coefficients $a_0,\\dots ,a_{k-1}\\in \\mathbb {Z}$ .",
"We may be interested in $f(n)=\\sum _{\\lambda \\in P(n)}\\left(\\Vert \\lambda \\Vert ^{\\Vert \\lambda \\Vert ^{\\Vert \\lambda \\Vert }}-\\lfloor \\log \\Vert \\lambda \\Vert \\rfloor \\right)$ where $\\lambda $ runs through all partitions of $n$ and $\\Vert \\lambda \\Vert $ is the number of parts.",
"Or in the surplus $f(n)$ of the partitions of $n$ into an even number of distinct parts from $\\lbrace 1_1,1_2,2_1,2_2,3_1,\\dots \\rbrace $ (two-sorted natural numbers) over those with an odd number of distinct parts, that is, $f(n)$ is the coefficient of $q^n$ in the expansion of $\\prod _{k=1}^{\\infty }(1-q^k)^2$ .",
"Another function $f(n)$ counts 1324-avoiding permutations $a_1a_2\\dots a_n$ of $[n]=\\lbrace 1,2,\\dots ,n\\rbrace $ ; the avoidance means that no four indices $1\\le i_1<i_2<i_3<i_4\\le n$ exist with $a_{i_1}<a_{i_3}<a_{i_2}<a_{i_4}$ .",
"Or $f(n)$ may be the number of labeled triangle-free graphs on the vertex set $[n]$ .",
"In the binary words setting, if $\\lambda =0^{m_0}1^{m_1}\\dots (n-1)^{m_{n-1}}$ , $m_i\\in \\mathbb {N}_0$ , is a multiset with $m_0+m_1+\\dots +m_{n-1}=n$ , then $f(\\lambda )\\in \\mathbb {N}_0$ counts the labeled simple graphs $G$ on the vertex set $[n]$ such that for $i=0,1,\\dots ,n-1$ exactly $m_i$ vertices of $G$ have degree $i$ .",
"We encode the input $\\lambda $ as an element of $\\lbrace 0,1\\rbrace ^*$ in an appropriate way which we will discuss later in part II.",
"This shows the variety of problems in enumeration one can consider and investigate.",
"We say something on each of them from the perspective of Definition REF .",
"Here we consider Examples 1–5 and defer the remaining ones to part II.",
"We put forward our definition of an effective formula for an enumerative problem.",
"The acronym PIO stands for polynomial in input and output.",
"Definition 1.1 (PIO formula) .",
"For a counting function $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ , a PIO formula is an algorithm, called a PIO algorithm, that for some constants $c,d\\in \\mathbb {N}$ for every input $n\\in \\mathbb {N}$ computes the output $f(n)\\in \\mathbb {Z}$ in at most $c\\cdot m(n)^d=O(m(n)^d)=\\mathrm {poly}(m(n))$ steps, where $m(n)=m_f(n):=\\log (1+n)+\\log (2+|f(n)|)$ measures the combined complexity of the input and the output.",
"Similarly for counting function $f:\\;X\\rightarrow \\mathbb {Z}$ defined on a subset $X\\subset \\mathbb {N}$ .",
"We think this is the precise and definitive notion of a “closed formula”, and the yardstick one should use, possibly with some ramifications or weakenings, to determine if a solution to an enumerative problem is effective.",
"We call counting functions possessing PIO formulas shortly PIO functions.",
"Definition REF repeats (more explicitly) the proposal made already in M. Klazar [88] in 2010, the innovation is that meanwhile we learned that the relevant complexity class PIO exists for a long time in the literature.",
"In fact, after submitting this text for publication we found out that J. Shallit mentioned briefly Definition REF as a “very good formula” in [133] in 2016.",
"We comment on the definition.",
"The steps mean steps of the formal specification of an algorithm as a multitape Turing machine.",
"We are primarily interested in the bit complexity but sometimes consider also the algebraic complexity, the number of required arithmetic operations.",
"We do not consider the space complexity which concerns memory requirements.",
"The shifts $1+n$ and $2+|f(n)|$ serve for removing arguments 0 and 1, inconvenient for logarithms.",
"The two natural logarithms come from the decadic or binary (but not unary!)",
"encoding of numbers: $\\log (2+|f(n)|)=\\Theta (r)$ where $r$ is the number of bits in the binary code for $f(n)\\in \\mathbb {Z}$ .",
"The rationale behind the definition is that any algorithm solving a nontrivial enumerative problem $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ needs minimum roughly $m(n)$ steps just for reading the input $n$ and printing the output $f(n)$ , and an effective algorithm takes only polynomially many steps in this minimum.",
"We include the output in the complexity of the problem because in enumeration typically the output is much larger than the input, and thus considering only the input complexity (and ignoring the time it takes to print the output) is bound to lead to confusion.",
"But we will see, and a moment of reflection reveals it, that not large outputs but on the contrary the unexpectedly small ones pose difficulty — for them one has much less time for effective computation.",
"We reflect such “cancellative” problems by selecting $\\mathbb {Z}$ , and not $\\mathbb {N}$ or $\\mathbb {N}_0$ , as the codomain of counting functions.",
"In Example 1, $m(n)=\\Theta (n)$ because $\\log (2+c_n)=\\Theta (n)$ , and in Example 2, $m(n)=\\Theta (n)$ for even $c_n$ and $m(n)=\\Theta (\\log (1+n))$ when $c_n$ is odd.",
"We state the definition of the complexity class PIO ([149], [65], [153]) which we above specialized to counting functions; $|u|$ denotes the length $n$ of a binary word $u=a_1a_2\\dots a_n\\in \\lbrace 0,1\\rbrace ^*$ .",
"Definition 1.2 (complexity class PIO) .",
"A function $f:\\;\\lbrace 0,1\\rbrace ^*\\rightarrow \\lbrace 0,1\\rbrace ^*$ is in the complexity class PIO if there is an algorithm that for every input $u\\in \\lbrace 0,1\\rbrace ^*$ computes the output $f(u)$ in time polynomial in $\\max (|u|,|f(u)|)$ .",
"PIO belongs to standard complexity classes, see the “complexity zoo” [153], but it appears not to be widely known.",
"It was introduced implicitly by M. Yanakakis [149] and later explicitly and independently by Y. Gurevich and S. Shelah [65].",
"Researchers in database theory measure by it complexity of algorithms, see for example S. Cohen, B. Kimelfeld and Y. Sagiv [46], S. Cohen and Y. Sagiv [47], Y. Kanza and Y. Sagiv [85] or M. Vardi [143].",
"We are not aware of any mention of the class PIO in enumerative combinatorics where we think it has a natural place.",
"The title alludes to the pioneering work [146] of the late H. S. Wilf who was the first to ponder in the light of computational complexity the question what it precisely means to give an effective, or a nontrivial, solution — an answer — to a problem in enumeration.",
"The first of two of his definitions says that a nontrivial solution (he actually uses the term effective solution) is an algorithm that computes $f(n)$ for a counting function $f:\\;\\mathbb {N}\\rightarrow \\mathbb {N}_0$ in the number of steps that is $o(\\mathrm {List}(n))$ where “$\\mathrm {List}(n)=$ the complexity of producing all of the members of the set $S_n$ [where $f(n)=|S_n|$ ], one at a time, by the speediest known method, and counting them.” ([146]).",
"As it depends on the complexity of the current “speediest known method”, it depends on time and progress of knowlege, and so it is not really a mathematical definition but more a heuristic to measure effectivity of algorithms.",
"In part II we give another specification of this notion.",
"But we have to add here that this is not a bug but a feature of the first definition: “We will see that a corollary of this attitude is that our decision as to what constitutes an answer may be time-dependent: as faster algorithms for listing the objects become available, a proposed formula for counting the objects will have to be comparably faster to evaluate.” ([146]).",
"The second definition of H. S. Wilf says that for superpolynomially growing $f(n)$ (“a problem in the class $\\nu \\pi $ ”) an effective solution (he actually uses the term that a problem is $p$ -solved) is an algorithm that computes $f(n)$ in $\\mathrm {poly}(n)$ steps ([146]).",
"We also have to mention that he does not restrict only to bit complexity but allows also other measures of complexity, “such as multiplication or division of numbers in a certain size range, or bit operations, or function evaluations, etc.” ([146]).",
"Shortcomings of H. S. Wilf's second definition, rectified in Definition REF , are that it does not take into account complexity of the output and restricts only to functions $f$ with superpolynomial growth.",
"(In the second definition the function $f(n)$ is not bounded from above, but it appears that tacitly it is of at most broadly exponential growth, $\\log (1+f(n))\\ll n^d$ .)",
"Thus Example 2 falls outside his framework, which is kind of unsatisfactory, and so we sought better definition.",
"H. S. Wilf illustrates his two definitions with the function (we quote from [146]) $f(n)&=&\\sum _{\\lambda \\in P(n)}\\frac{2^{g(\\lambda )}}{1^{m_1}\\cdot m_1!\\cdot 2^{m_2}\\cdot m_2!\\cdot \\ldots \\cdot n^{m_n}\\cdot m_n!",
"}\\ \\mbox{ where}\\\\g(\\lambda )&=&\\frac{1}{2}\\bigg (\\sum _{i,j=1}^n\\mathrm {gcd}(i,j)m_im_j-\\sum _{k\\ge 1}m_{2k-1}\\bigg )$ and $\\lambda =1^{m_1}2^{m_2}\\dots n^{m_n}$ runs through the partitions of $n$ — $f(n)$ counts the unlabeled (i.e., nonisomorphic) graphs on the vertex set $[n]$ (G. Pólya [118]).",
"Since $f(n)\\sim 2^{n(n-1)/2}/n!$ , we have $\\mathrm {List}(n)=\\Omega (2^{(1+o(1))n^2/2})$ , but the formula based on the displayed sum over $P(n)$ computes $f(n)$ in $O(p(n)n^d)=O(\\exp (c\\sqrt{n}))$ steps (where $c>0$ is a constant and $p(n)=|P(n)|$ is the number of partitions of $n$ ), which is asymptotically much smaller.",
"Thus we have a nontrivial solution.",
"An effective solution is in question because no algorithm is known that would compute $f(n)$ in $O(n^d)$ steps.",
"H. S. Wilf asks if such algorithm exists, but so far his question remains unanswered.",
"H. S. Wilf's article [146] is discussed by D. Zeilberger [151] and it gave rise to the notion of a Wilfian formula (also called a polynomial enumeration scheme), which is an algorithm working in time polynomial in $n$ , for enumerative problems $n\\mapsto f(n)\\in \\mathbb {N}_0$ of the type $\\Omega (n^c)=\\log (2+f(n))=O(n^d)$ (for some real constants $0<c<d$ ).",
"It appears in the works on enumeration of Latin squares by D. S. Stones [140] or permutations with forbidden patterns by V. Vatter [144], B. Nakamura and D. Zeilberger [109], B. Nakamura [108] and others.",
"Recently [146] was invoked by V. S. Miller [104] for counting squares in $F_2^{n\\times n}$ by a nontrivial formula making exponentially many steps (an effective solution is not known) or by M. Kauers and D. Zeilberger [86].",
"Perhaps our proposal in Definition REF has a certain reinventing-the-wheel quality because nowadays, unlike in the times of, say, L. Comtet [48] or J. Riordan [126] when computational complexity did not exist, it is a common knowledge that an effective solution to a problem means, in the first approximation, a polynomial time algorithm (see, for example, B. Edixhoven and J.-M. Couveignes [51] or V. Becher, P. A. Heiber and T. A. Slaman [14]).",
"For this common knowledge we are indebted to A. Cobham [45] and J. Edmonds [52] in 1965.",
"But, then, it seems not to be a common knowledge that one should consistently include the complexity of the output in the complexity of an enumerative problem.",
"Also, one can still read in contemporary literature on enumerative combinatorics statements to the effect that there is no precise definition or determination of a closed formula or answer to an enumerative problem.",
"For example, M. Aigner [3] writes that “There is no straightforward answer as to what “determining” a counting function means.” or F. Ardila [10] concludes that “So what is a good answer to an enumerative problem?",
"[emphasize in original] Not surprisingly, there is no definitive answer to this question.” On the other hand, the text of P. J. Cameron [33], [34] contains an interesting discussion of the complexity matters which does reflect the output complexity of enumerative problems, but does not result in concrete definition of a closed formula.",
"We believe that Definition REF gives the definitive and more or less straightforward answer.",
"Recently (I wrote the main bulk of this article in autumn 2016 and add this in March 2018, and now see that it is even August) the excellent survey [112] of I. Pak appeared that also tackles the question what is an effective formula in enumeration but it changes nothing on our above discussion.",
"Content and main results.",
"In the following two sections we discuss, from the perspective of Definition REF , in their order the eight examples given at the beginning.",
"At least, this we initially intended but as the text started get too long, we decided to split it in two parts.",
"Here we consider Examples 1–5 and defer Examples 6–8 to part II.",
"The length of our text is caused only by the great variety of enumerative problems offering themselves for investigation.",
"Section 2 deals with Examples 1–3.",
"After establishing in Propositions REF and REF PIO formulas for Examples 1 and 2, where Example 2 is chosen to illustrate peculiarity of this notion, we prove in Theorem REF that every integral linear recurrence sequence has a PIO formula (but with a non-effective complexity bound).",
"This seems so far not to be reflected in the literature.",
"Propositions REF and REF –REF gather tools for proving Theorem REF .",
"We present some problems and results on holonomic sequences which generalize linear recurrence sequences.",
"For example, in Problem REF we ask if every holonomic sequence $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ is a PIO function.",
"In Section 3 we consider enumerative problems inspired by Examples 4 and 5.",
"Propositions REF and REF revisit efficient evaluation of the partition function $p(n)$ .",
"It still holds from us some secrets, for example, it is not known how to compute efficiently the parity of $p(n)$ (Problem REF ).",
"In Proposition REF we show that counting functions like Example 4 are PIO functions and in Proposition REF we prove it for partitions with distinct parts.",
"Hence Corollary REF : compositions of $n$ with distinct parts are counted by a PIO function.",
"Proposition REF is a general result implying that, for example, the partitions of $n$ whose multiplicities of parts divide $n$ are counted by a PIO function.",
"If parts are required to divide $n$ , PIO formula seems not to be known.",
"Another corollary is Corollary REF : if $g:\\;\\mathbb {N}\\rightarrow \\mathbb {N}$ strictly increases, grows only polynomially and is polynomial-time computable then the partitions of $n$ with parts in $\\lbrace g(1),\\,g(2),\\,\\dots \\rbrace $ are counted by a PIO function.",
"Yet another Corollary REF shows that the number of partitions of $n$ into distinct squares is a PIO function.",
"Corollary REF gives PIO formulas for counting functions of partitions with prescribed multiplicities.",
"Problem REF , inspired by H. S. Wilf [147], asks if we can effectively count partitions of $n$ with distinct multiplicities of parts.",
"The well known theorem of E. T. Bell [15] (Proposition REF ) says that partitions of $n$ that take parts from a fixed finite set $A\\subset \\mathbb {N}$ are counted by a quasipolynomial in $n$ .",
"In Corollaries REF and REF we point out that the argument proving Proposition REF gives with almost no change more general results.",
"Corollary REF returns to Example 4: if the function $g:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ has a finite support then $f(n)=\\sum _{\\lambda \\in P(n)}g(\\Vert \\lambda \\Vert )$ is a PIO function because it is a quasipolynomial in $n$ .",
"We mention further quasipolynomial enumerative results on partitions, Proposition REF due to D. Zeilberger [152] and Proposition REF due to G. E. Andrews, M. Beck and N. Robbins [8].",
"Rather general quasipolynomial enumerative result was obtained by T. Bogart, J. Goodricks and K. Woods [19] (Theorem REF ).",
"Problem REF asks if one can effectively count partitions of $n$ into powers of a fixed integer $m\\ge 2$ and Theorem REF quotes a recent positive resolution of this problem by I. Pak and D. Yeliussizov [113], [114].",
"Proposition REF points out that the two basic cancellative counting problems on partitions, $n\\mapsto \\sum _{\\lambda \\in Q(n)}(-1)^{\\Vert \\lambda \\Vert }$ and $n\\mapsto \\sum _{\\lambda \\in P(n)}(-1)^{\\Vert \\lambda \\Vert }$ (where $P(n)$ are all partitions of $n$ , and $Q(n)$ are those with distinct parts), are both PIO functions.",
"The former follows from the pentagonal identity of L. Euler, and the latter from J. W. L. Glaisher's identity [62].",
"In the former case we have almost complete cancellation but in the latter case only little cancellation.",
"Problem REF asks when such cancellation for $(-1)^{\\Vert \\lambda \\Vert }$ -count of partitions occurs.",
"We look at this problem for partitions into squares and for partitions into parts from $l$ -sorted $\\mathbb {N}$ (Example 5 is $l=2$ ), including the case $l=24$ that gives the Ramanujan tau function — the last Problem REF concerns computation of coefficients in powers of R. Dedekind's $\\eta $ -function.",
"When we below state and prove results on PIO functions for enumerative problems, we are not content with just saying that a PIO algorithm for the problem exists — it would be ironic to refer in such a way to efficient algorithms — but we always indicate if and how the PIO algorithm can be constructed from the given data.",
"See, for example, Theorem REF or Corollary REF ."
],
[
"The numbers of Catalan and Fibonacci",
"We start with Example 1.",
"A planted tree is a finite tree with a distinguished vertex, called a root, and with every set of children of a vertex linearly ordered.",
"Its size is the number of vertices.",
"(For a long time I used to call this kind of trees rooted plane trees, which also some literature uses, but F. Bergeron, G. Labelle and P. Leroux [16] showed me that this terminology is imprecise: embedding in the plane gives to the children of the root only cyclic, not linear, ordering.)",
"Proposition 2.1 .",
"Let $c_n$ be the $n$ -th Catalan number, the number of (unlabeled) planted trees with size $n$ .",
"Then $n\\mapsto c_n$ is a PIO function and the PIO algorithm is given by the recurrence displayed below.",
"Proof.",
"We only need to know the recursive structure of planted trees.",
"There is just one planted tree with size 1, and for $n\\ge 2$ every planted tree $T$ of size $n$ bijectively decomposes in an ordered pair $(U,V)$ of planted trees with sizes adding to $n$ ; $U$ is the subtree of $T$ rooted in the first child of $T$ 's root and $V$ is the rest of $T$ .",
"Thus the combinatorial recurrence $c_1=1$ and, for $n\\ge 2$ , $c_n=\\sum _{k=1}^{n-1}c_kc_{n-k}\\;.$ It implies that $c_n\\ge 2c_{n-1}$ for $n\\ge 3$ and by induction $c_n\\gg 2^n$ .",
"On the other hand, by induction $c_n\\le (n-1)!\\le n^n$ for every $n\\ge 1$ and $\\log (2+c_n)\\ll n^2$ (such crude but easy to obtain bound suffices for our purposes).",
"Hence $n\\ll m(n)\\ll n^2$ in this enumerative problem and we need to compute $c_n$ in $O(n^d)$ steps.",
"We do it on a Turing machine with six tapes $T_1,\\dots ,T_6$ .",
"Recall that elementary school algorithms multiply two $O(m)$ -bit numbers in $O(m^2)$ steps and add them in $O(m)$ steps (see comments below).",
"Tape $T_1$ stores, in this order, the binary codes for $c_1,c_2,\\dots ,c_{n-1}$ .",
"For $k=1,2,\\dots ,n-1$ we do the following.",
"We find $c_k$ and $c_{n-k}$ on $T_1$ and write them on the respective tapes $T_2$ and $T_3$ .",
"This costs $O(\\log (1+c_1)+\\dots +\\log (1+c_{n-1}))=O(n^3)$ steps, say.",
"We compute in $O(n^4)$ steps the product $c_kc_{n-k}=:s$ and write it on $T_4$ .",
"Tape $T_5$ stores $\\sum _{i=1}^{k-1}c_ic_{n-i}=:t$ .",
"In $O(n^2)$ steps we compute the sum $s+t$ and store it on $T_6$ .",
"We conclude by rewritting in $O(n^2)$ steps the content of $T_5$ with that of $T_6$ .",
"After the step $k=n-1$ , tape $T_5$ contains $t=\\sum _{i=1}^{n-1}c_ic_{n-i}=c_n$ and we copy this in $O(n^2)$ steps on $T_1$ .",
"The computation of $c_n$ from $c_1,c_2,\\dots ,c_{n-1}$ takes $O(n\\cdot n^4)=O(n^5)$ steps.",
"The recurrence, implemented by the six-tape Turing machine, computes $c_n$ from beginning in $\\sum _{k=2}^nO(k^5)=O(n^6)=O(m(n)^6)$ steps, and $n\\mapsto c_n$ is a PIO function.",
"$\\Box $ We give some comments.",
"If the Turing machine has only one tape, and the two $O(m)$ -bit numbers to be added are stored on it one after another, it is impossible to add them in $O(m)$ steps as the reading head has to move back and forth between them, and one needs $\\Theta (m^2)$ steps for addition.",
"This can be proven similarly as F. C. Hennie [71] proved the $\\Theta (m^2)$ lower bound on recognition of $m$ -bit palindromes.",
"Therefore we use multitape Turing machines.",
"The cost of adding or multiplying two numbers is not only the cost of the operation but in practice includes also the cost of recalling both operands from the memory, and therefore we analyzed above the computation of $c_n$ in more details.",
"But these technicalities cause at worst only polynomial slowdown and are not important for our main concern that is a purely qualitative alternative: there is a polynomial time PIO algorithm for the enumerative problem considered or its existence is not known.",
"The Catalan numbers $(c_n)_{n\\ge 1}=(1,\\,1,\\,2,\\,5,\\,14,\\,42,\\,132,\\,429,\\,1430,\\,4862,\\,\\dots )$ form sequence A000108 in the database OEIS (The Online Encyclopedia of Integer Sequences) [154].",
"Using stronger bound $\\log (2+c_n)=\\Theta (n)$ and faster integer multiplication (consult D. Harvey, J. van der Hoeven and G. Lecerf [70] and D. Harvey and J. van der Hoeven [69] for the state of art and history of multiplication algorithms) we can evaluate $c_n$ in, say, $O(n^3\\log ^d(1+n))$ steps.",
"It is not a surprise that Catalan numbers can be effectively computed, but it depends on what “effectively” precisely means.",
"The combinatorial recurrence computes $c_n$ in $\\mathrm {poly}(n)$ arithmetic operations, and to get $\\mathrm {poly}(n)$ steps we need that all numbers involved in the computation have $O(n^d)$ digits for a fixed $d$ .",
"This is ensured by (i) the bound $c_n\\le n^n$ and (ii) the non-negativity of coefficients in the recurrence which entails that the result $c_n$ upper bounds every number arising in evaluating the recurrence.",
"But we also need that each $c_n$ has $\\Omega (n^c)$ digits for a fixed real $c>0$ , which is ensured by the bound $c_n\\gg 2^n$ , so that $\\mathrm {poly}(n)$ steps means $\\mathrm {poly}(m(n))$ steps and we really have an effective algorithm.",
"If, say, for infinitely many $n$ we had the bound $c_n=O(1)$ then $\\mathrm {poly}(n)$ steps would cease to mean an effective algorithm in the sense of Definition REF and we would have to try more, as in Example 2.",
"To establish qualitatively a PIO formula for $c_n$ , the combinatorial recurrence suffices and one does not need generating functions or “advanced” formulas for $c_n$ like ($n\\in \\mathbb {N}$ ) $c_n=\\frac{1}{n}\\binom{2n-2}{n-1}=(-1)^{n+1}\\frac{4^n}{2}\\binom{\\frac{1}{2}}{n}\\ \\mbox{ or }\\ c_{n+1}=\\frac{4n-2}{n+1}c_n\\;.$ The last recurrence gives a more efficient PIO formula than the combinatorial recurrence.",
"The Catalan numbers have asymptotics $c_n\\sim cn^{-3/2}4^n$ with a constant $c>0$ .",
"For asymptotic methods in enumeration consult P. Flajolet and R. Sedgewick [58], and R. Pemantle and M. C. Wilson [117] for the multivariate case.",
"We move to Example 2.",
"As we will see shortly, the case $f(n)=1$ occurs for infinitely many $n$ .",
"Then the reader probably realizes that even though the two functions $f_c$ and $f_o$ , defined as $f_c(n):=c_n$ and $f_o(n):=n$ for even $n$ and $f_o(n):=1$ for odd $n$ , are PIO functions and compose to the counting function $f(n)=f_o(f_c(n))$ of Example 2, composition of their PIO algorithms is not a PIO algorithm for $f(n)$ .",
"It is an algorithm that always does $\\Omega (n^d)$ steps, but we need an algorithm that for $n$ with odd $c_n$ makes only $O(\\log ^d(1+n))$ steps.",
"Naturally, we need to determine effectively which numbers $c_n$ are odd.",
"Proposition 2.2 .",
"Let $c_n$ be the $n$ -th Catalan number.",
"Then the function $f:\\;\\mathbb {N}\\rightarrow \\mathbb {N}$ , $f(n)=c_n$ for even $c_n$ and $f(n)=1$ for odd $c_n$ , is a PIO function.",
"The PIO algorithm is described below.",
"Proof.",
"The combinatorial recurrence for $c_n$ shows that $c_n$ is odd iff $n=2^m$ for an $m\\in \\mathbb {N}_0$ : $c_1=1$ is odd, for odd $n>1$ the number $c_n=2(c_1c_{n-1}+\\dots +c_{(n-1)/2}c_{(n+1)/2})$ is even and, similarly, for even $n$ the number $c_n=c_{n/2}^2+2(c_1c_{n-1}+\\dots +c_{(n-2)/2}c_{(n+2)/2})$ has the same parity as $c_{n/2}$ .",
"Thus we effectively compute $f(n)$ as follows.",
"For given $n\\in \\mathbb {N}$ we first in $O(\\log ^2(1+n))$ steps determine if $n$ is a power of 2.",
"If it is so, we output 1 in $O(1)$ steps.",
"Else we output, using the PIO formula for $n\\mapsto c_n$ , in $O(n^6)$ or so steps the value $c_n$ .",
"This gives a formula for $f(n)$ that for even $c_n$ makes $O(n^6)$ steps and for odd $c_n$ only $O(\\log ^2(1+n))$ steps, which is $O(m(n)^6)$ steps for every $n\\in \\mathbb {N}$ .",
"$\\Box $ Alternatively, we get a PIO algorithm for the $f(n)$ of Proposition REF or, more generally, we determine effectively if a fixed $m\\in \\mathbb {N}$ divides $c_n$ (or, more generally, a hypergeometric term), by means of A.-M. Legendre's formula $k=\\nu _p(n!",
")=\\sum _{j\\ge 1}\\lfloor n/p^j\\rfloor $ for the largest $k\\in \\mathbb {N}_0$ for which $p^k$ divides $1\\cdot 2\\cdot \\ldots \\cdot n$ (or by means of a generalization of the formula to products of numbers in arithmetic progressions).",
"At the close of the section we mention other effective formulas for modular reductions of $c_n$ and similar numbers.",
"Example 2 illustrates the fact that composition of two PIO algorithms need not be a PIO algorithm.",
"In fact, as one expects, composition of two PIO functions need not be a PIO function.",
"An example of such functions is easily constructed by taking a computable function not in PIO, e.g.",
"a function $f:\\;\\lbrace 0,1\\rbrace ^*\\rightarrow \\lbrace 0,1\\rbrace $ in $\\mathrm {EXP}\\backslash \\mathrm {P}$ , see Ch. H.",
"Papadimitriou [115].",
"Y. Gurevich and S. Shelah [65] elaborate such example.",
"Example 3 concerns the ubiquitous linear recurrence sequences.",
"Best known of them are the Fibonacci numbers $f_n$ , given by $f_0=0,f_1=1$ , and $f_{n+2}=f_{n+1}+f_n$ for every $n\\in \\mathbb {N}_0$ .",
"The sequence $(f_n)=(f_n)_{n\\ge 1}=(1,\\,1,\\,2,\\,3,\\,5,\\,8,\\,13,\\,21,\\,34,\\,55,\\,89,\\,144,\\,\\dots )$ is sequence [154].",
"Since $2f_{n-2}\\le f_n\\le 2f_{n-1}$ for $n\\ge 3$ , we have exponential bounds $(\\sqrt{2})^n\\ll f_n\\ll 2^n$ , $n\\in \\mathbb {N}$ .",
"The precise asymptotics is $f_n\\sim c\\phi ^n$ where $c>0$ is a constant and $\\phi =1.61803\\dots $ satisfies $\\phi ^2-\\phi -1=0$ .",
"In general, a linear recurrence sequence in $\\mathbb {Z}$, abbreviated as a LRS, is a function $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ determined by $2k$ integers $a_0,\\dots ,a_{k-1},f(1),\\dots ,f(k)$ with $k\\in \\mathbb {N}_0$ and $a_0\\ne 0$ , and the recurrence relation $f(n+k)=a_{k-1}f(n+k-1)+a_{k-2}f(n+k-2)+\\dots +a_0f(n),\\ n\\in \\mathbb {N}\\;.$ (Later we point out that allowing $a_i$ outside $\\mathbb {Z}$ does not give new LRS.)",
"Note that we require $a_0\\ne 0$ and the recurrence to hold from the beginning.",
"By reverting the recurrence every LRS $f$ extends naturally to $f:\\;\\mathbb {Z}\\rightarrow \\mathbb {Q}$ , for example $(1,2,4,8,\\dots )$ extends to $f(n)=2^{n-1}$ , $n\\in \\mathbb {Z}$ .",
"Thus $(0,1,1,1,\\dots )$ is not a LRS, as can be seen be reverting the purported recurrence, but it is true that this sequence differs from a LRS in just one term.",
"If $k=0$ or if $f(1)=f(2)=\\dots =f(k)=0$ , we get the zero sequence that has $f(n)=0$ for every $n\\in \\mathbb {N}$ .",
"We say that $f$ (more precisely, the recurrence) has order $k$.",
"Like the Fibonacci numbers, every LRS has an easy exponential upper bound $|f(n)|\\le c^n$ for every $n\\in \\mathbb {N}$ , with the constant $c=k\\max _i|a_i|\\max _{i\\le k}|f(i)|$ .",
"Lower bound is a different story, see Proposition REF .",
"The defining recurrence computes every LRS $f(n)$ in $\\Theta (n)$ arithmetic operations.",
"We recall, on the Fibonacci numbers $f_n$ , the well known and beautiful formula (algorithm) based on binary powering that computes $f(n)$ in only $\\mathrm {poly}(\\log n)$ arithmetic operations.",
"By E. Bach and J. Shallit [11] or D. E. Knuth [91], it appears first in J. C. P. Miller and D. J. Spencer Brown [103].",
"For $f_n$ the formula reads: if $n=\\sum _{i=0}^kb_i2^i$ with $b_i\\in \\lbrace 0,1\\rbrace $ is the binary expansion of $n\\in \\mathbb {N}_0$ (where $0=02^0$ ) then $f_n=(0,1)\\cdot \\prod _{i=0}^k\\underbrace{\\left(\\dots \\left(\\left(\\left(\\begin{array}{ll}1&1\\\\1&0\\end{array}\\right)^{b_i}\\right)^2\\right)^2\\dots \\right)^2}_{i\\ \\mathrm {squarings}}\\cdot \\left(\\begin{array}{l}1\\\\0\\end{array}\\right)\\;.$ Indeed, if $M$ is the stated $2\\times 2$ matrix and $F_n$ is the column $(f_{n+1},f_n)^T$ then, since the first row of $M$ records the recurrence for $f_n$ and matrix multiplication is associative, we have $MF_n=F_{n+1}$ and $F_n=M^nF_0$ .",
"The power $M^n$ is then computed by repeated squaring from the $b_i$ s. Since $k=O(\\log (n+1))$ , the formula computes $f_n$ in only $O(\\log (n+1))$ multiplications of two integral $2\\times 2$ matrices, so in only $O(\\log (n+1))$ arithmetic operations with integers.",
"This easily extends to general LRS.",
"For more information on repeated squaring and computing terms in linear recurrence sequences see J. von zur Gathen and J. Gerhard [61].",
"Recent contribution to the literature on computing linear recurrence sequences is S. G. Hyun, S. Melczer and C. St-Pierre [77].",
"For qualitative bit complexity such cleverness seems superfluous, the $n$ -th term $f(n)$ of a LRS is an $O(n)$ digit number (which cannot be printed in fewer steps) and already the defining recurrence computes $f(n)$ in $\\mathrm {poly}(n)$ steps, which might be viewed as an efficient computation.",
"But not from the point of view of Definition REF .",
"Since $f(n)$ may have as few as $O(1)$ digits, to get a PIO formula we need to locate effectively these small values (cf.",
"Example 2) and compute them in $\\mathrm {poly}(\\log n)$ steps.",
"This is not automatic by the above displayed $\\mathrm {poly}(\\log n)$ arithmetic operations formula because only $\\mathrm {poly}(\\log n)$ digit numbers may be used, and for general LRS the displayed formula contains negative numbers (the result need not upper bound intermediate values, as in $1=2^n-(2^n-1)$ ).",
"It can be done, albeit on the verge of non-effectivity, and we prove the following theorem.",
"Theorem 2.3 .",
"Every linear recurrence sequence $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ in $\\mathbb {Z}$ of order $k\\in \\mathbb {N}_0$ is a PIO function.",
"In the proof we indicate an algorithm ${\\cal A}$ with inputs $(a,b,n)\\in \\mathbb {Z}^k\\times \\mathbb {Z}^k\\times \\mathbb {N}$ , where $k\\in \\mathbb {N}_0$ , and outputs in $\\mathbb {Z}$ such that for every fixed tuple $(a,b)=(a_0,\\dots ,a_{k-1},f(1),\\dots ,f(k))$ with $a_0\\ne 0$ , ${\\cal A}$ is a PIO algorithm computing the LRS $f(n)$ determined by $(a,b)$ .",
"We will see that the implicit constant in the $O(m(n)^d)$ complexity bound in these PIO algorithms is currently non-effective — at the present state of knowledge we cannot provide for it any specific value.",
"The theorem follows by combining standard results from the theory of linear recurrence sequences, see the monograph [54] of G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, but we did not find it mentioned in [54] or anywhere else.",
"We review tools for the proof of Theorem REF .",
"For background on linear recurrence sequences see [54], W. M. Schmidt [131] or R. P. Stanley [138].",
"By $\\overline{\\mathbb {Q}}\\subset \\mathbb {C}$ we denote the field of algebraic numbers, consisting of all roots of monic polynomials from $\\mathbb {Q}[x]$ .",
"The subring of algebraic integers is formed by all roots of monic polynomials from $\\mathbb {Z}[x]$ .",
"A power sum is an expression $s(x)=\\sum _{i=1}^lp_i(x)\\alpha _i^x$ where $l\\in \\mathbb {N}_0$ , $\\alpha _i\\in \\overline{\\mathbb {Q}}$ are distinct and nonzero numbers, and $p_i\\in \\overline{\\mathbb {Q}}[x]$ are nonzero polynomials.",
"The numbers $\\alpha _i$ are the roots of the power sum.",
"A sequence $f:\\;\\mathbb {N}\\rightarrow \\overline{\\mathbb {Q}}$ is represented by a power sum $s(x)$ if $f(n)=s(n)$ for every $n\\in \\mathbb {N}$ .",
"The empty power sum with $l=0$ represents the zero sequence.",
"We consider the more general linear recurrence sequences in $\\mathbb {Q}$, shortly LRS in $\\mathbb {Q}$ — they are defined as LRS, only their values and coefficients of defining recurrences lie in the field $\\mathbb {Q}$ instead of its subring $\\mathbb {Z}$ .",
"If $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Q}$ is a LRS in $\\mathbb {Q}$ given by a recurrence $f(n+k)=\\sum _{i=0}^{k-1}a_if(n+i)$ , $a_i\\in \\mathbb {Q}$ and $a_0\\ne 0$ , the recurrence polynomial $p(x)$ is $p(x)=x^k-a_{k-1}x^{k-1}-a_{k-2}x^{k-2}-\\dots -a_0\\in \\mathbb {Q}[x]\\;.$ If the recurrence for $f$ has minimum order, we denote $p(x)$ by $p_f(x)$ and call it the characteristic polynomial of $f$; in a moment we show that there is always a unique $p_f(x)$ .",
"For example, the characteristic polynomial of the zero sequence is the constant polynomial $p_{\\equiv 0}(x)=1$ .",
"For a sequence $f:\\;\\mathbb {N}\\rightarrow \\mathbb {C}$ and a polynomial $p(x)=\\sum _{i=0}^ka_ix^i\\in \\mathbb {C}[x]$ we let $pf:\\;\\mathbb {N}\\rightarrow \\mathbb {C}$ denote the sequence given by $pf(n)=\\sum _{i=0}^ka_if(n+i)$ .",
"We say that $p$ annihilates $f$ if $pf$ is the zero sequence.",
"The set of rational polynomials annihilating $f$ is denoted by $V(f)\\subset \\mathbb {Q}[x]$ .",
"Clearly, $V(f)$ consists exactly of all rational recurrence polynomials for $f$ (with $a_0=0$ and non-unit leading coefficient allowed).",
"The following results are well known but we prove them here for reader's convenience and as an workout for the author.",
"Proposition 2.4 .",
"Power sums and linear recurrence sequences have the following properties.",
"Every sequence $f:\\;\\mathbb {N}\\rightarrow \\overline{\\mathbb {Q}}$ has at most one power sum representation.",
"If $f:\\;\\mathbb {N}\\rightarrow \\mathbb {C}$ is a sequence then $V(f)$ is an ideal in the ring $\\mathbb {Q}[x]$ .",
"If $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Q}$ is a LRS in $\\mathbb {Q}$ then $V(f)=\\langle p_f(x)\\rangle $ for a unique monic polynomial $p_f\\in \\mathbb {Q}[x]$ with $p_f(0)\\ne 0$ .",
"This unique generator $p_f$ of $V(f)$ is called the characteristic polynomial of $f$ and gives the unique minimum order rational recurrence for $f$ .",
"A sequence $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Q}$ is a LRS in $\\mathbb {Q}$ if and only if it is represented by a power sum $s(x)$ .",
"If it is the case, the roots $\\alpha _i$ of $s(x)$ are exactly the roots of $p_f(x)$ .",
"Proof.",
"1.",
"It suffices to show that no nonempty power sum represents the zero sequence.",
"For a power sum $s(x)$ as above we define $\\deg s(x)=\\sum _{i=1}^l(\\deg p_i(x)+1)\\in \\mathbb {N}_0$ .",
"Clearly, only the empty power sum has degree 0.",
"For any nonempty $s(x)$ we define the new power sum $\\Delta s(x)=s(x+1)-\\alpha _1s(x)$ .",
"Note that, crucially, $\\deg \\Delta s(x)=\\deg s(x)-1$ .",
"Also, if $s(x)$ represents the zero sequence then so does $\\Delta s(x)$ , and no $s(x)$ with $\\deg s(x)=1$ represents the zero sequence.",
"(In fact, if $\\deg s(x)=1$ then $s(n)\\ne 0$ for every $n\\in \\mathbb {N}$ .)",
"The last three facts together imply that no nonempty power sum represents the zero sequence.",
"2 and 3.",
"It is easy to see that for any $p,q\\in \\mathbb {C}[x]$ and any sequence $f:\\;\\mathbb {N}\\rightarrow \\mathbb {C}$ we have $(p+q)f=pf+qf$ and $(pq)f=p(qf)$ .",
"Thus $V(f)$ is an ideal in $\\mathbb {Q}[x]$ .",
"Every ideal in $\\mathbb {Q}[x]$ is principal, is generated by a single element, because the ring $\\mathbb {Q}[x]$ is Euclidean.",
"Requiring the generator monic makes it unique because the units of $\\mathbb {Q}[x]$ are exactly the nonzero constants.",
"Finally, $p_f(0)\\ne 0$ because $f$ being a LRS in $\\mathbb {Q}$ implies that $V(f)$ contains a $p$ with $p(0)\\ne 0$ .",
"4.",
"Suppose that $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Q}$ is a LRS in $\\mathbb {Q}$ with minimum order $k$ and characteristic polynomial $p_f(x)$ .",
"Thus in the ring of formal power series $\\mathbb {Q}[[x]]$ we have the equality $\\sum _{n\\ge 0}f(n)x^n=\\frac{q(x)}{q_f(x)}$ where $q_f(x)=x^kp_f(1/x)$ , $q\\in \\mathbb {Q}[x]$ has degree $<k$ and $q(x)$ and $q_f(x)$ are coprime (by the minimality of $k$ ).",
"The value $f(0)$ is computed from $f(1)$ , $f(2),\\dots ,f(k)$ by the reverted recurrence; now it would be more convenient if $f$ had domain $\\mathbb {N}_0$ or even $\\mathbb {Z}$ but counting functions have domain $\\mathbb {N}$ .",
"After decomposing the rational function $q(x)/q_f(x)$ in partial fractions, expanding them in $\\overline{\\mathbb {Q}}[[x]]$ in generalized geometric series, and comparing coefficients of $x^n$ , we get a representation of $f(n)$ by a power sum $s(x)$ .",
"The roots of $s(x)$ are exactly the roots of $p_f(x)$ because of the coprimality of $q(x)$ and $q_f(x)$ .",
"Let $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Q}$ be represented by a power sum $s(x)=\\sum _{i=1}^lp_i(x)\\alpha _i^x$ : $f(n)=s(n)$ for every $n\\in \\mathbb {N}$ .",
"We may assume that $f$ is not the zero sequence and so $s(x)$ is nonempty.",
"We show that $f$ is a LRS in $\\mathbb {Q}$ .",
"The argument is of interest because of three invocations of Lemma REF below and because it uses negative $n\\in \\mathbb {Z}$ .",
"If $d=\\max _i\\deg p_i(x)$ then $s(x)$ is a $\\overline{\\mathbb {Q}}$ -linear combination of the $t=(d+1)l$ expressions $x^j\\alpha _i^x$ for $0\\le j\\le d$ and $1\\le i\\le l$ .",
"So is every shift $s(x+r)$ for $r\\in \\mathbb {N}$ , as can be seen by expanding $(x+r)^j=\\sum _{b=0}^j\\binom{j}{b}r^{j-b}x^b$ and $\\alpha _i^{x+r}=\\alpha _i^r\\alpha _i^x$ .",
"By Lemma REF there exist coefficients $\\beta _0,\\beta _1,\\dots ,\\beta _t\\in \\overline{\\mathbb {Q}}$ , not all zero, such that $\\beta _0s(x)+\\beta _1s(x+1)+\\dots +\\beta _ts(x+t)=0$ identically.",
"But we need coefficients not only in $\\overline{\\mathbb {Q}}$ but in $\\mathbb {Q}$ .",
"We set $V=\\lbrace (f(n),f(n+1),\\dots ,f(n+t))\\;|\\;n\\in \\mathbb {N}\\rbrace \\subset \\mathbb {Q}^{t+1}\\subset \\overline{\\mathbb {Q}}^{t+1}$ and select a maximum subset $B\\subset V$ of linearly independent (over $\\overline{\\mathbb {Q}}$ ) vectors.",
"By Lemma REF , $|B|\\le t+1$ .",
"Every vector $z\\in V$ is a $\\overline{\\mathbb {Q}}$ -linear combination of the vectors in $B$ .",
"If $|B|=t+1$ , the matrix whose rows are the linearly independent vectors $z\\in B$ is a square matrix and thus has linearly independent columns.",
"But the system $z\\cdot (x_0,x_1,\\dots ,x_t)=0,\\ z\\in B\\;,$ has a nontrivial solution $x_i=\\beta _i\\in \\overline{\\mathbb {Q}}$ which means that the columns are linearly dependent.",
"Hence $|B|\\le t$ .",
"But $B\\subset \\mathbb {Q}^{1+t}$ and thus by Lemma REF this system has a nontrivial solution $x_i=\\gamma _i\\in \\mathbb {Q}$ , with not all $\\gamma _i$ zero.",
"So $z\\cdot (\\gamma _0,\\gamma _1,\\dots ,\\gamma _t)=0\\ \\mbox{ and }\\ \\gamma _0s(n)+\\gamma _1s(n+1)+\\dots +\\gamma _ts(n+t)=0$ for every $z\\in V$ and every $n\\in \\mathbb {N}$ .",
"By part 1, then $\\gamma _0s(x)+\\gamma _1s(x+1)+\\dots +\\gamma _ts(x+t)=0$ identically (the left side is the empty power sum) and the last displayed equality thus holds for every $n\\in \\mathbb {Z}$ .",
"Let $u\\in \\mathbb {N}_0$ and $v\\in \\mathbb {N}_0$ be the respective minimum and maximum index $r$ with $\\gamma _r\\ne 0$ , and let $w=v-u\\in \\mathbb {N}_0$ .",
"Then $&&\\gamma _vf(n+w)+\\gamma _{v-1}f(n+w-1)+\\dots +\\gamma _uf(n)\\\\&&=\\gamma _ts(n-u+t)+\\gamma _{t-1}s(n-u+t-1)+\\dots +\\gamma _0s(n-u)=0$ for every $n\\in \\mathbb {N}$ (the arguments of $s(\\cdot )$ may be negative).",
"After dividing by $\\gamma _v$ and rearranging we see that $f$ is a LRS in $\\mathbb {Q}$ of order $w$ .",
"The roots of $p_f(x)$ and of $s(x)$ coincide by the previously proved opposite implication and by uniqueness of $s(x)$ proved in part 1.",
"$\\Box $ Note that nonzero power sums may vanish for infinitely many $n\\in \\mathbb {N}$ , for example $s(x)=1^x+(-1)(-1)^x$ on $2\\mathbb {N}$ , and the first part is therefore more subtle result than for polynomials.",
"Recalling the zeros of $\\sin (\\pi x)$ we have $\\exp (\\pi ix)-\\exp (-\\pi ix)=\\exp (\\pi i)^x+(-1)\\exp (-\\pi i)^x=0\\ \\mbox{ for every }\\ x\\in \\mathbb {Z}$ but this, of course, is not a counterexample to the first part (why?).",
"Also, $f_n=\\frac{1}{\\sqrt{5}}\\left(\\frac{1+\\sqrt{5}}{2}\\right)^n-\\frac{1}{\\sqrt{5}}\\left(\\frac{1-\\sqrt{5}}{2}\\right)^n$ is the familiar power sum representation of the Fibonacci numbers.",
"The next lemma, used several times in the previous proof, is a well known result from linear algebra.",
"Its proof is left to the interested reader as an exercise.",
"Lemma 2.5 .",
"Let $m,n\\in \\mathbb {N}$ with $m<n$ .",
"Every linear homogeneous system $a_{j,1}x_1+a_{j,2}x_2+\\dots +a_{j,n}x_n=0_K,\\ j=1,2,\\dots ,m\\;,$ with $m$ equations, $n$ unknowns $x_i$ , and coefficients $a_{j,i}$ in a field $K$ has a nontrivial solution $x_i\\in K$ with not all $x_i=0_K$ .",
"Recall that a root of unity is a number $\\alpha \\in \\mathbb {C}$ such that $\\alpha ^k=1$ for some $k\\in \\mathbb {N}$ , i.e.",
"$\\alpha $ is a root of $x^k-1$ .",
"The minimum such $k$ is the order of $\\alpha $ .",
"We say that a power sum $s(x)$ is degenerate if some root $\\alpha _i$ or some ratio $\\alpha _i/\\alpha _j$ of two roots is a root of unity different from 1, else $s(x)$ is non-degenerate.",
"So we allow 1 as a root in a non-degenerate power sum, and empty power sum is non-degenerate.",
"For any sequence $f:\\;\\mathbb {N}\\rightarrow X$ and numbers $m\\in \\mathbb {N}$ and $j\\in [m]$ , the $m$ -section $f_{j,m}:\\;\\mathbb {N}\\rightarrow X$ of $f$ is the subsequence of values of $f$ on the residue class $j$ mod $m$ : $f_{j,m}(n)=f(j+m(n-1)),\\ n\\in \\mathbb {N}\\;.$ If $f:\\;\\mathbb {N}\\rightarrow \\overline{\\mathbb {Q}}$ is represented by a power sum $s(x)=\\sum _{i=1}^lp_i(x)\\alpha _i^x$ , then the $m$ -section $f_{j,m}$ is represented by the power sum $s_{j,m}(x)=\\sum _{i=1}^l\\alpha _i^{j-m}p_i(j-m+mx)(\\alpha _i^m)^x=\\sum _{i=1}^rq_i(x)\\beta _i^x$ where $r\\le l$ and $\\lbrace \\beta _i\\;|\\;i=1,\\dots ,r\\rbrace \\subset \\lbrace \\alpha _i^m\\;|\\;i=1,\\dots ,l\\rbrace $ — we collect like terms in the middle expression so that the numbers $\\beta _i$ are distinct and the polynomials $q_i(x)$ nonzero.",
"For example, the degenerate power sum $s(x)=2^x+(-2)^x$ has 2-sections $s_{1,2}(x)=0$ (the empty power sum with $r=0$ ) and $s_{2,2}(x)=2\\cdot 4^x$ .",
"It is not hard to prove that for any $m\\in \\mathbb {N}$ , $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Q}$ is a LRS in $\\mathbb {Q}$ if and only if every $m$ -section $f_{j,m}$ is a LRS in $\\mathbb {Q}$ .",
"Proposition 2.6 .",
"The following holds for roots of unity and power sums.",
"If $p\\in \\mathbb {Z}[x]$ is a monic polynomial with $p(0)\\ne 0$ and every root of $p$ has modulus at most 1, then every root of $p$ is a root of unity.",
"If $s(x)=\\sum _{i=1}^lp_i(x)\\alpha _i^x$ is a power sum such that every $\\alpha _i$ is an algebraic integer, $|\\alpha _i|\\le 1$ for every $i$ , and $s(n)\\in \\mathbb {Q}$ for every $n\\in \\mathbb {N}$ , then every $\\alpha _i$ is a root of unity.",
"Proof.",
"1.",
"This is called Kronecker's theorem.",
"See U. Zannier [150] or E. Bombieri and W. Gubler [21] or V. V. Prasolov [123].",
"2.",
"By the assumption and part 4 of Proposition REF , the sequence $f(n)=s(n)$ , $n\\in \\mathbb {N}$ , is a LRS in $\\mathbb {Q}$ .",
"By parts 3 and 4 of Proposition REF , the numbers $\\alpha _i$ are exactly the roots of the characteristic polynomial $p_f(x)\\in \\mathbb {Q}[x]$ .",
"Since all $\\alpha _i$ are algebraic integers, so are the coefficients of $p_f(x)$ (by expressing them in terms of the $\\alpha _i$ s).",
"But this implies that $p_f(x)\\in \\mathbb {Z}[x]$ .",
"Using part 1 of the present proposition, we get that all $\\alpha _i$ are roots of unity.",
"$\\Box $ On the Internet or even in paper literature one can encounter the erroneous claim that if $\\alpha \\in \\overline{\\mathbb {Q}}$ with $|\\alpha |=1$ then $\\alpha $ is a root of unity.",
"The number $\\frac{4+3i}{5}$ is a counterexample.",
"Part 1 of Proposition REF shows when arguments of this sort are correct.",
"Concerning exponential lower bounds on growth of linear recurrence sequences, there is the next deep result.",
"Proposition 2.7 .",
"If $f:\\;\\mathbb {N}\\rightarrow \\overline{\\mathbb {Q}}$ is represented by a non-degenerate power sum whose roots have maximum modulus $\\beta >1$ , then for every $\\varepsilon >0$ there is an $n_0\\in \\mathbb {N}$ such that $|f(n)|>\\beta ^{(1-\\varepsilon )n}\\mbox{ for every } n>n_0\\;.$ Proof.",
"This is [54] where the proof is omitted.",
"At [54] the result is attributed to J.-H. Evertse [55] and independently A. van der Poorten and H. P. Schlickewei [122].",
"J.-H. Evertse [55] attributes it to A. van der Poorten [120].",
"See also A. van der Poorten [121].",
"$\\Box $ We deduce the following growth dichotomy for LRS that effectively separates small and large values.",
"Proposition 2.8 .",
"Suppose $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ is a LRS of order $k\\in \\mathbb {N}_0$ , represented by a power sum $s(x)$ .",
"We let $m\\in \\mathbb {N}$ be the least common multiple of the orders of the roots of unity among the roots $\\alpha _i$ of $s(x)$ and their ratios $\\alpha _i/\\alpha _j$ , and let $J\\subset [m]$ be the set of $j\\in \\mathbb {N}$ for which the power sum $s_{j,m}(x)$ is empty or has the single root 1.",
"Then there exist a real constant $c>1$ and an $n_0\\in \\mathbb {N}$ such that for every $j\\in [m]$ the following holds.",
"If $j\\in J$ then $f_{j,m}(n)$ is a rational polynomial in $n\\in \\mathbb {N}$ with degree less than $k$ .",
"If $j\\notin J$ then $|f_{j,m}(n)|>c^n$ for every $n>n_0$ .",
"Proof.",
"By parts 3 and 4 of Proposition REF , all roots $\\alpha _i$ of $s(x)$ are algebraic integers.",
"Take a $j\\in [m]$ and consider the power sum $s_{j,m}(x)=\\sum _{i=1}^rq_i(x)\\beta _i^x$ representing $f_{j,m}(n)$ .",
"The $\\beta _i$ s are $m$ -th powers of $\\alpha _i$ s and are algebraic integers too.",
"Also, $s_{j,m}(x)$ is non-degenerate.",
"Suppose that $|\\beta _i|\\le 1$ for every $i=1,2,\\dots ,r$ .",
"By part 2 of Proposition REF all $\\beta _i$ are roots of unity.",
"But then non-degeneracy of $s_{j,m}(x)$ implies that either $r=0$ , $s_{j,m}(x)$ is empty and $f_{j,m}$ is the zero sequence, or $r=1$ , $\\beta _1=1$ and $f_{j,m}(n)=q_1(n)\\in \\mathbb {Z}$ for every $n\\in \\mathbb {N}$ .",
"It follows that $q_1\\in \\mathbb {Q}[x]$ and $\\deg q_1\\le \\max _i\\deg p_i<k$ ($p_i$ are the polynomials in $s(x)$ ).",
"Thus we get the first case with $j\\in J$ .",
"If $|\\beta _i|>1$ for some $i$ , Proposition REF gives the second case with $j\\notin J$ .",
"$\\Box $ Unfortunately, currently no proof of Proposition REF is known giving an explicit upper bound on the threshold $n_0$ , only its existence is proven.",
"Therefore also the $n_0$ of Proposition REF is non-effective (we cannot compute it).",
"Effective versions of much weaker inequalities are not known.",
"Already T. Skolem [135] proved that if $f(n)$ is a nonzero LRS represented by a non-degenerate power sum then $|f(n)|\\ge 1$ for every $n>n_0$ , that is, $f(n)=0$ has only finitely many solutions $n\\in \\mathbb {N}$ .",
"To obtain an effective version of this result with an explicit upper bound on $n_0$ , that is, on the sizes of solutions, is a famous open problem, mentioned for example in T. Tao [141] or in B. Poonen [119].",
"Before we turn to the proof of Theorem REF we state a corollary of Proposition REF .",
"Recall that a sequence $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ is a quasi-polynomial if for some $m\\in \\mathbb {N}$ polynomials $q_1,\\dots ,q_m\\in \\mathbb {Q}[x]$ we have $f_{j,m}(n)=q_j(n)$ for every $j\\in [m]$ and $n\\in \\mathbb {N}$ .",
"Corollary 2.9 .",
"If a LRS $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ has subexponential growth, $\\limsup _{n\\rightarrow \\infty } |f(n)|^{1/n}\\le 1\\;,$ then $f(n)$ is a quasi-polynomial.",
"We remark that one can prove Proposition REF and Corollary REF in a conceptually simpler (but probably not much shorter) way without Kronecker's theorem, using incomensurability of the frequencies of the roots of non-degenerate power sums.",
"Proof of Theorem REF .",
"Let $k\\in \\mathbb {N}_0$ and $2k$ integers $a_0,\\dots ,a_{k-1}$ , $f(1),\\dots $ , $f(k)$ , $a_0\\ne 0$ , be given.",
"We describe a PIO algorithm for the LRS $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ defined by $f(n+k)=\\sum _{i=0}^{k-1}a_if(n+i)\\;.$ We take the recurrence polynomial $p(x)=x^k-a_{k-1}x^{k-1}-\\dots -a_0$ of $f$ , decompose the generating function $\\sum _{n\\ge 0}f(n)x^n=\\frac{r(x)}{q(x)}\\in \\mathbb {Q}(x),\\ q(x)=x^kp(1/x)\\ \\mbox{ and }\\ \\deg r(x)<k\\;,$ into partial fractions and as in the proof of the first implication in part 4 of Proposition REF determine from them the power sum $s(x)$ representing $f(n)$ .",
"From $s(x)$ we determine the number $m$ and set $J\\subset [m]$ as defined in Proposition REF .",
"For each $j\\in J$ we find the polynomial $q_j\\in \\mathbb {Q}[x]$ such that $\\deg q_j<k$ and $q_j(n)=f_{j,m}(n)$ for $n=1,2,\\dots ,k$ .",
"This precomputation can be done algorithmicly.",
"Now for an input $n\\in \\mathbb {N}$ we compute the residue $j\\in [m]$ of $n$ modulo $m$ .",
"If $j\\in J$ , we output $f(n)=q_j((n+m-j)/m)$ .",
"If $j\\notin J$ , we compute $f(n)$ by the defining recurrence.",
"Correctness of the algorithm follows from Proposition REF .",
"We bound its time complexity in terms of $m(n)$ .",
"The precomputation takes $O(1)$ steps and determining $j$ takes $\\mathrm {poly}(\\log n)$ steps.",
"If $j\\in J$ , computing $q_j((n+m-j)/m)=f(n)$ takes $\\mathrm {poly}(\\log n)$ steps because we do $O(1)$ arithmetic operations with $O(\\log (1+n))$ digit numbers.",
"If $j\\notin J$ , computing $f(n)$ by the defining recurrence takes $\\mathrm {poly}(n)$ steps because $f(n)$ is an $O(n)$ digit number for every $n\\in \\mathbb {N}$ .",
"As for $m(n)$ , if $j\\in J$ then $f(n)$ is an $O(\\log (1+n))$ digit number ($f_{j,m}(n)$ grows only polynomially) and $m(n)=\\Theta (\\log (1+n))$ .",
"If $j\\notin J$ then $f(n)$ is an $\\Omega (n)$ digit number (by case 2 of Proposition REF ) and $m(n)=\\Theta (n)$ .",
"No matter if $j\\in J$ or not, for every $n\\in \\mathbb {N}$ the algorithm does $\\mathrm {poly}(m(n))$ steps and is a PIO algorithm.",
"$\\Box $ Since the constant in the $\\Omega (n)$ lower bound at the end of the proof is non-effective, the complexity bound $\\mathrm {poly}(m(n))=O(m(n)^d)$ involves a non-effective constant as well.",
"Before we turn to holonomic sequences, we discuss the effect of domain extension for recurrence coefficients of a LRS.",
"In the definition we required them to lie in $\\mathbb {Z}$ .",
"Could one get more integer-valued sequences if the coefficients lie in a larger domain than $\\mathbb {Z}$ ?",
"The answer is no.",
"We already proved in the proof of the second implication in part 4 of Proposition REF that if $K\\subset L$ is an extension of fields and $f:\\;\\mathbb {N}\\rightarrow K$ is a LRS in $L$ then $f$ is in fact a LRS in $K$ .",
"This folklore result on linear recurence sequences is mentioned for example in M. Stoll [139].",
"Recurrence coefficients outside $\\mathbb {Q}$ thus give nothing new.",
"One can also prove that if $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ is a LRS in $\\mathbb {Q}$ (recurrence coefficients lie in $\\mathbb {Q}$ ), then $f$ is in fact a LRS (another recurrence exists with coefficients in $\\mathbb {Z}$ ).",
"See R. P. Stanley [138] for the proof by generating functions and references for this result, known as the Fatou lemma.",
"One could also try to extend Theorem REF to linear recurrence sequences in $\\mathbb {Q}$ .",
"For this one extends the codomain of counting functions from $\\mathbb {Z}$ to $\\mathbb {Q}$ and in the definition of $m(n)$ (Definition REF ) replaces $|f(n)|$ with $\\max (|a|,|b|)$ where $f(n)=\\frac{a}{b}\\in \\mathbb {Q}$ , $\\mathrm {gcd}(a,b)=1$ .",
"We hope to return to this question later.",
"We conclude the section with some results and problems on computing terms in holonomic sequences.",
"These generalize LRS and are also quite common in enumerative combinatorics and number theory.",
"For simplicity we restrict to integer-valued sequences.",
"A sequence $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ is holonomic (synonymous terms in use are $P$ -recursive and polynomially recursive) if for some $k$ rational functions $a_0,\\dots ,a_{k-1}\\in \\mathbb {Z}(x)$ , $k\\in \\mathbb {N}_0$ and $a_0(x)\\ne 0$ , we have $f(n+k)=a_{k-1}(n)f(n+k-1)+a_{k-2}(n)f(n+k-2)+\\dots +a_0(n)f(n)$ for every $n>n_0$ .",
"Now the recurrence cannot hold in general from the beginning because of possible zeros of the denominators in the $a_i(x)$ .",
"Examples of such sequences are $f(n)=n!$ or the Catalan numbers $f(n)=c_n$ (see the “advanced” recurrence for $c_n$ ).",
"Unfortunately, holonomic sequences lack some analog of the power sum representation; for a form of the matrix exponential representation (used in the matrix formula for the Fibonacci numbers) see Ch.",
"Reutenauer [124].",
"We propose the following problem.",
"Problem 2.10 .",
"Is it true that every holonomic sequence $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ is a PIO function?",
"A. Bostan, P. Gaudry and E. Schost [26] give an algorithm computing the $n$ -th term of a holonomic sequence in $O(n^{1/2}\\log ^d(1+n))$ arithmetic operations.",
"Since Example 2 and Proposition REF deal with an effective computation of the function $n\\mapsto c_n\\ \\mathrm {mod}\\ 2$ , we mention a problem and some results on effective computation of modular reductions of holonomic sequences.",
"For a sequence $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ and $m\\in \\mathbb {N}$ , the modular reduction $n\\mapsto f(n)\\ \\mathrm {mod}\\ m$ has values in the fixed set of residues $[m]$ , and so a PIO formula for it means a computation in $\\mathrm {poly}(\\log n)$ steps.",
"Trivially, modular reduction of every LRS is eventually periodic and has therefore a PIO formula.",
"As we saw in the proof of Proposition REF , $c_n$ modulo 2 is not eventually periodic.",
"We propose the following problem.",
"Problem 2.11 .",
"Is it true that for every $m\\in \\mathbb {N}$ and every holonomic sequence $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ its modular reduction $n\\mapsto (f(n)\\ \\mathrm {mod}\\ m)\\in [m]$ is a PIO function, that is, can be computed in $O(\\log ^d(1+n))$ steps?",
"The answer is affirmative for algebraic $f$, that is, if the generating series $f(x)=\\sum _{n\\ge 1}f(n)x^n$ satisfies a polynomial equation, $P(x,f(x))=0$ for a nonzero polynomial $P\\in \\mathbb {Z}[x,y]$ (it is not hard to show that every algebraic sequence is holonomic).",
"This applies to the Catalan numbers as $c(x)=\\sum _{n\\ge 1}c_nx^n$ satisfies $c(x)^2-c(x)+x=0$ .",
"See A. Bostan, X. Caruso, G. Christol and P. Dumas [24] for fast algorithm computing modular reduction of algebraic $f$ .",
"In fact, the answer is affirmative for an even larger subclass of holonomic sequences, namely for rational diagonals.",
"These are sequences $f:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ representable for $n\\in \\mathbb {N}$ as $f(n)=a_{n,n,\\dots ,n}\\ \\mbox{ where }\\ \\sum _{n_1,\\dots ,n_k\\ge 1}a_{n_1,\\dots ,n_k}x_1^{n_1}\\dots x_k^{n_k}=\\frac{P(x_1,\\dots ,x_k)}{Q(x_1,\\dots ,x_k)}$ for some polynomials $P,Q\\in \\mathbb {Z}[x_1,\\dots ,x_k]$ , $Q\\ne 0$ (one can show that every algebraic sequence is a rational diagonal, and that every rational diagonal is holonomic).",
"See [24], A. Bostan, G. Christol and P. Dumas [25] and E. Rowland and R. Yassawi [130] (and some of the references therein) for more information on these two results.",
"At the conclusion of [129] E. Rowland mentions that a conjecture of G. Christol [38] implies affirmative answer to Problem REF for any at most exponentially growing holonomic sequence.",
"See C. Krattenthaler and T. W. Müller [96] (and other works of the authors cited therein) for another approach to computation of modular reductions of algebraic sequences."
],
[
"Integer partitions",
"Let us discuss PIO formulas for enumerative problems related to and motivated by the initial Examples 4 and 5.",
"A partition $\\lambda $ of a number $n\\in \\mathbb {N}_0$ is a multiset of natural numbers summing to $n$ .",
"We write partitions in two formats, $\\lambda =1^{m_1}2^{m_2}\\dots n^{m_n},\\ m_i\\in \\mathbb {N}_0,\\ \\mbox{ and }\\ \\lambda =(\\lambda _1\\ge \\lambda _2\\ge \\dots \\ge \\lambda _k),\\ \\lambda _i\\in \\mathbb {N},\\, k\\in \\mathbb {N}_0\\;.$ So $|\\lambda |:=n=\\sum _im_ii=\\sum _i\\lambda _i$ .",
"The numbers $1,2,\\dots ,n$ and $\\lambda _1,\\lambda _2,\\dots ,\\lambda _k$ are the parts of $\\lambda $ and the $m_i$ are their multiplicities.",
"We denote the number of parts in $\\lambda $ by $\\Vert \\lambda \\Vert $ , so $\\Vert \\lambda \\Vert =m_1+m_2+\\dots +m_n=k$ .",
"The set of all partitions of $n$ is $P(n)$ , their number is $p(n)=|P(n)|$ , and $P:=\\bigcup _{n\\ge 0}P(n)$ .",
"For the empty partition $\\emptyset =()$ we have $|()|=\\Vert ()\\Vert =0$ , $P(0)=\\lbrace ()\\rbrace $ and $p(0)=1$ .",
"Similar quantities are $Q(n)$ , $q(n)$ , and $Q$ , defined for partitions with distinct parts (all $m_i\\le 1$ , i.e.",
"$\\lambda _i>\\lambda _{i+1}$ ).",
"$P_k(n)$ are the partitions of $n$ with $k$ parts, and similarly $Q_k(n)$ are those with $k$ distinct parts.",
"The sequence $p:\\;\\mathbb {N}\\rightarrow \\mathbb {N}$ is [154] and begins $(p(n))_{n\\ge 1}=(1,\\,2,\\,3,\\,5,\\,7,\\,11,\\,15,\\,22,\\,30,\\,42,\\,56,\\,77,\\,101,\\,135,\\,176,\\,\\dots )\\;;$ $(q(n))_{n\\ge 1}=(1,1,2,2,3,4,5,\\dots )$ is [154].",
"We give two proofs for the well known fact that $p(n)$ can be efficiently computed and is a PIO function in our parlance, and so is $q(n)$ .",
"What would be a non-efficient computation?",
"For example, the “cave man formula” in [151]: $p(n)=\\sum _{\\lambda \\in P(n)}1$ .",
"By the multiplicity format, $q(n)\\le p(n)\\le (n+1)^n$ .",
"To get a lower bound, for a given $n\\in \\mathbb {N}$ , $n\\ge 4$ , consider the maximum $m\\in \\mathbb {N}$ with $1+2+\\dots +m=\\binom{m+1}{2}\\le \\frac{n}{2}-1$ .",
"Then $m=\\Theta (n^{1/2})$ and $n=\\sum _{i\\in X}i+\\left(n-\\sum _{i\\in X}i\\right),\\ X\\subset [m]\\;,$ are $2^m$ different partitions in $Q(n)$ .",
"So $p(n)\\ge q(n)\\ge 2^m\\gg \\exp (\\Omega (n^{1/2}))$ for $n\\in \\mathbb {N}$ .",
"Thus for the partition function $p(n)$ we have $n^{1/2}\\ll m(n)\\ll n^2$ and need to compute $p(n)$ in $\\mathrm {poly}(n)$ steps.",
"Asymptotically, $p(n)\\sim \\frac{\\exp (\\pi \\sqrt{2n/3})}{4\\cdot 3^{1/2}\\cdot n}\\ \\mbox{ and }\\ q(n)\\sim \\frac{\\exp (\\pi \\sqrt{n/3})}{4\\cdot 3^{1/4}\\cdot n^{3/4}}\\ \\mbox{ as $n\\rightarrow \\infty $}$ (G. H. Hardy and S. Ramanujan [67], G. Meinardus [102], G. E. Andrews [6], V. Kotěšovec [94]).",
"Thus, more precisely, $p(n)$ and $q(n)$ have $\\Theta (n^{1/2})$ digits and $m(n)=\\Theta (n^{1/2})$ .",
"For the sake of brevity we treat the PIO algorithms and their complexity in Propositions REF and REF below more schematically and do not discuss their implementation by multitape Turing machines as we did in Proposition REF .",
"These omitted details could be easily filled in, and it is easy to see that the deduced polynomiality of algorithms holds true.",
"Proposition 3.1 .",
"For $k,n\\in \\mathbb {N}$ with $1\\le k\\le n$ let $p_k(n)=|P_k(n)|$ be the number of partitions of $n$ with $k$ parts, and let $p_k(n)=0$ and $P_k(n)=\\emptyset $ if $k>n$ .",
"Then for every $n\\ge 1$ we have $p_n(n)=p_1(n)=1$ , and for every $n\\ge 2$ and every $k$ with $1<k<n$ we have $p_k(n)=p_k(n-k)+p_{k-1}(n-1)\\;.$ Consequently, $p(n)=p_1(n)+p_2(n)+\\dots +p_n(n)$ is a PIO function.",
"Proof.",
"The values $p_n(n)=p_1(n)=1$ are trivial.",
"The displayed recurrence mirrors the set partition $P_k(n)=A\\cup B$ where $A$ are the partitions of $n$ with all $k$ parts at least 2 and $B$ are the remaining partitions with at least one part 1.",
"Decreasing each part in every $\\lambda \\in A$ by 1 gives the bijection $A\\rightarrow P_k(n-k)$ and removing one part 1 from every $\\lambda \\in B$ gives the bijection $B\\rightarrow P_{k-1}(n-1)$ , whence the recurrence.",
"For given input $n\\in \\mathbb {N}$ we use the recurrence and the initial and border values and generate the array of $O(n^2)$ numbers $(p_k(m)\\;|\\;1\\le k\\le m\\le n)$ in $O(n^2)$ additions.",
"Another $n-1$ additions produce $p(n)$ .",
"Every number involved in the computation has $O(n^2)$ digits (in fact, $O(n^{1/2})$ digits), and therefore the algorithm makes $O(n^4)$ steps (in fact, $O(n^{5/2})$ steps), which is $O(m(n)^8)$ steps (in fact, $O(m(n)^5)$ steps) as $m(n)\\gg n^{1/2}$ .",
"Therefore the stated recurrence schema is a PIO formula for $p(n)$ .",
"$\\Box $ The interested reader will find in M. Bodirsky, C. Gröpl and M. Kang [17] a recurrence schema, in its priciple similar to the previous one but much more involved in details, that computes in polynomial time the number of labeled planar graphs on the vertex set $[n]$ ; computation of this number for $n=50$ in one hour is reported.",
"The second proof shows that L. Euler's generating function formula $\\sum _{n\\ge 0}p(n)q^n=\\prod _{k\\ge 1}\\frac{1}{1-q^k}=\\prod _{k\\ge 1}(1+q^k+q^{2k}+\\dots )$ also gives a PIO formula for $p(n)$ .",
"Let $[x^n]a(x):=a_n$ if $a(x)=\\sum _{n\\ge 0}a_nx^n$ .",
"Proposition 3.2 .",
"Let $m,n\\in \\mathbb {N}$ .",
"The product $ab$ of two polynomials $a,b\\in \\mathbb {Z}[x]$ such that $\\deg a,\\deg b\\le n$ and each coefficient in them has at most $m$ digits can be computed in the obvious way in $O(m^2n^3)$ steps.",
"Thus $p(n)=[q^n]\\prod _{k=1}^n(1+q^k+q^{2k}+\\dots +q^{\\lfloor n/k\\rfloor k})$ is a PIO function.",
"Proof.",
"Each of the $1+\\deg a+\\deg b=O(n)$ sums $[x^k]ab=\\sum _{i+j=k}[x^i]a\\cdot [x^j]b$ , $k\\le \\deg a+\\deg b$ , has $O(n)$ summands, multiplication in each summand takes $O(m^2)$ steps, and each addition costs $O(m+n)$ steps (we add two numbers with $\\ll m+\\log (1+n)\\ll m+n$ digits).",
"The list of coefficients of $ab$ is thus computed in $\\ll n(m^2n+(m+n)n)=O(m^2n^3)$ steps.",
"The value $p(n)$ is a coefficient in the product of $n$ polynomials with degrees at most $n$ and coefficients 0 and 1.",
"We apply the lemma about the product of two polynomials $n-1$ times and each time we multiply two polynomials with degrees at most $n^2$ and with coefficients of size $\\le p(n^2)$ that have $\\ll n^4$ digits.",
"Thus we compute the product of the $n$ polynomials in $O(n(n^4)^2(n^2)^3)=O(n^{15})=O(m(n)^{30})$ steps (recall that $m(n)\\gg n^{1/2}$ ) and see that $p(n)$ is a PIO function.",
"$\\Box $ Often less elementary recurrences are invoked to efficiently compute $p(n)$ : $p(n)=\\sum _{i\\ge 1}(-1)^{i+1}(p(n-a_i)+p(n-b_i))\\ \\mbox{ or }\\ p(n)=\\frac{1}{n}\\sum _{i=1}^{n}\\sigma (i)p(n-i)$ where $n=1,2,\\dots $ , $a_i=\\frac{i(3i-1)}{2}$ and $b_i=\\frac{i(3i+1)}{2}$ are so called (generalized) pentagonal numbers, $p(0)=1$ , $p(n):=0$ for $n<0$ , and $\\sigma (n):=\\sum _{d\\,|\\,n}d$ is the sum of divisors function (G. E. Andrews [6]).",
"For more recurrences for $p(n)$ see Y. Choliy, L. W. Kolitsch and A. V. Sills [37].",
"The pentagonal recurrence yields an algorithm computing the list of values $(p(m)\\;|\\;1\\le m\\le n)$ in $O(n^2)$ steps (D. E. Knuth [92]) while the algorithm of Proposition REF makes $O(n^{5/2})$ steps.",
"N. Calkin, J. Davis, K. James, E. Perez and C. Swannack [32] give an algorithm producing this list in $O(n^{3/2}\\log ^2n)$ steps, which is close to optimum complexity (because it takes $\\Omega (n^{3/2})$ steps just to print it).",
"The exponent 30 in the proof of Proposition REF is hilarious but the reader understands that we do not optimize bounds and instead focus on simplicity of arguments.",
"We can decrease it by computing the product $a(x)b(x)$ more quickly but in a less elementary way in $O((mn)^{1+o(1)})$ steps by [61].",
"A clever implementation of the Hardy–Ramanujan–Rademacher analytic formula for $p(n)$ ([6]) by F. Johansson [80] computes $p(n)$ in $O(n^{1/2}\\log ^{4+o(1)}n)=O(m(n)^{1+o(1)})$ steps, again in close to optimum complexity.",
"F. Johansson reports computing $p(10^6)$ by his algorithm in milliseconds and $p(10^{19})$ in less than 100 hours; see [81] for his computation of $p(10^{20})$ .",
"Proposition REF represents $p(n)$ as a coefficient in a polynomial from $\\mathbb {Z}[x]$ .",
"J. H. Bruiner and K. Ono [30] recently found another (more complicated) representation in this spirit, which has $(1-24n)p(n)$ as the next to leading coefficient in a monic polynomial from $\\mathbb {Q}[x]$ , see also J. H. Bruiner, K. Ono and A. V. Sutherland [31].",
"Computation-wise for $p(n)$ it lags far behind the H–R–R formula ([31]).",
"There is an extensive literature on modular properties of $p(n)$ (for both meanings of “modular”), see K. Ono [110], [111] and the references therein.",
"N. Calkin et al.",
"[32] can generate the list $(p(k)\\ \\mathrm {mod}\\ m\\;|\\;1\\le k\\le n)$ for special prime moduli $m$ depending on $n$ in $O(n^{1+o(1)})$ steps.",
"But unlike for the Catalan numbers and algebraic sequences, so far we do not know an efficient way to determine the parity of individual numbers $p(n)$ .",
"Problem 3.3 .",
"Is the parity of $p(n)$ , the function $n\\mapsto (p(n)\\ \\mathrm {mod}\\ 2)\\in [2]$ , a PIO function?",
"That is, can one compute it in $O(\\log ^d(1+n))$ steps (bit operations)?",
"By [81], “With current technology, the most efficient way to determine $p(n)$ modulo a small integer is to compute the full value $p(n)$ and then reduce it.” (cf.",
"the discussion of Example 2).",
"Cannot we do better?",
"The parity of $p(n)$ was investigated by T. R. Parkin and D. Shanks [116] already in 1967.",
"At the end of their article they ask if it can be computed in $O(n)$ steps.",
"We generalize Proposition REF by replacing 1 in $p(n)=\\sum _{\\lambda \\in P(n)}1$ with a positive PIO function of the number of parts.",
"This includes Example 4.",
"Proposition 3.4 .",
"If $g:\\;\\mathbb {N}\\rightarrow \\mathbb {N}$ is a PIO function then $f:\\;\\mathbb {N}\\rightarrow \\mathbb {N}$ , $f(n)=\\sum _{\\lambda \\in P(n)}g(\\Vert \\lambda \\Vert )\\;,$ is a PIO function too.",
"Construction of the PIO algorithm for $f$ from that for $g$ is described in the proof.",
"Proof.",
"We set $G(n)=\\max (g(1),g(2),\\dots ,g(n))$ .",
"Clearly, $f(n)=\\sum _{k=1}^ng(k)p_k(n)\\;,$ and so $f(n)\\ge p(n)+G(n)-1\\;.$ Thus the combined input and output complexity of $f(n)$ satisfies $m(n)=\\log (1+n)+\\log (2+f(n))\\gg \\log (p(n)+G(n))\\gg n^{1/2}+\\log (2+G(n))\\;.$ By the assumption on $g$ we compute the list $(g(k)\\;|\\;1\\le k\\le n)$ in $\\ll n(\\log (1+n)+\\log (2+G(n)))^d$ steps, for a fixed $d\\in \\mathbb {N}$ .",
"By Proposition REF we compute the list $(p_k(n)\\;|\\;1\\le k\\le n)$ in $O(n^{5/2})$ steps.",
"The product $g(k)p_k(n)$ is computed by elementary school multiplication in $\\ll \\log ^2p(n)+\\log ^2(2+G(n))\\ll (n^{1/2}+\\log (2+G(n)))^2$ steps, and this also bounds the cost of each addition in the sum.",
"The displayed sum therefore computes $f(n)$ in $&&\\ll n(\\log (1+n)+\\log (2+G(n)))^d+n^{5/2}+n(n^{1/2}+\\log (2+G(n)))^2\\\\&&\\ll (n^{1/2}+\\log (2+G(n)))^{d+4}\\ll m(n)^{d+4}$ steps.",
"$\\Box $ Functions covered by the proposition include $f(n)=p(n)$ for $g(n)=1$ and the contrived counting function $f(n)$ of Example 4.",
"For $g(n)=n$ we get the total number of parts in all partitions of $n$ , so $f(n)=\\sum _{\\lambda \\in P(n)}\\Vert \\lambda \\Vert =\\sum _{i=1}^n\\tau (i)p(n-i)$ is a PIO function.",
"Here $\\tau (i)$ denotes the number of divisors of $i$ .",
"One can deduce the last sum (which itself is a PIO formula for $f(n)$ , given one for $p(n)$ , no matter that we cannot compute effectively $i\\mapsto \\tau (i)$ ) by differentiating the generating function $\\sum _{\\lambda \\in P}y^{\\Vert \\lambda \\Vert }x^{|\\lambda |}=\\frac{1}{(1-yx)(1-yx^2)\\dots }$ by $y$ and then setting $y=1$ .",
"For partitions with distinct parts we have similar results.",
"Proposition 3.5 .",
"If $g:\\;\\mathbb {N}\\rightarrow \\mathbb {N}$ is a PIO function then $f:\\;\\mathbb {N}\\rightarrow \\mathbb {N}$ , $f(n)=\\sum _{\\lambda \\in Q(n)}g(\\Vert \\lambda \\Vert )\\;,$ is a PIO function too.",
"Construction of the PIO algorithm for $f$ from that for $g$ is described in the proof.",
"Proof.",
"Now $f(n)=\\sum _{k=1}^ng(k)q_k(n)$ where $q_k(n)=|Q_k(n)|$ is the number of partitions of $n$ with $k$ distinct parts.",
"For $q_k(n)$ we have the recurrence schema $q_1(n)=q_n(n)=1$ for every $n\\ge 1$ , $q_k(n)=0$ for $k>n$ , and $q_k(n)=q_k(n-k)+q_{k-1}(n-k)$ for $n\\ge 2$ and $1<k<n$ .",
"Compared to Proposition REF , this differs in the last summand: now the set of $\\lambda \\in Q_k(n)$ with one part 1 bijectively corresponds to $Q_{k-1}(n-k)$ , delete the 1 and decrease each of the remaining $k-1$ parts by 1.",
"We only replace $p(n)$ with $q(n)$ and $p_k(n)$ with $q_k(n)$ and argue as in the proof of Proposition REF .",
"$\\Box $ Now for $g(n)=n$ we get that the total number of parts in all $\\lambda \\in Q(n)$ , $f(n)=\\sum _{\\lambda \\in Q(n)}\\Vert \\lambda \\Vert =\\sum _{i=1}^n\\tau ^{\\pm }(i)q(n-i)\\;,$ is a PIO function.",
"Here $\\tau ^{\\pm }(i):=\\sum _{d\\,|\\,i}(-1)^{d+1}$ is the surplus of the odd divisors of $i$ over the even ones.",
"The last sum (again by itself a PIO formula for $f(n)$ , given one for $q(n)$ ) follows by the same manipulation with the generating function $\\sum _{\\lambda \\in Q}y^{\\Vert \\lambda \\Vert }x^{|\\lambda |}=(1+yx)(1+yx^2)\\dots $ as before.",
"We remark that ${\\textstyle (\\sum _{\\lambda \\in P(n)}\\Vert \\lambda \\Vert )_{n\\ge 1}}&=&(1,\\,3,\\,6,\\,12,\\,20,\\,35,\\,54,\\,86,\\,128,\\,192,\\,\\dots )\\mbox{ and }\\\\{\\textstyle (\\sum _{\\lambda \\in Q(n)}\\Vert \\lambda \\Vert )_{n\\ge 1}}&=&(1,\\,3,\\,3,\\,5,\\,8,\\,10,\\,13,\\,18,\\,25,\\,30,\\,\\dots )$ are respective sequences [154] and [154].",
"The first one was investigated by “Miss S. M. Luthra, University of Delhi” [99] (see p. 485 for the formula with $\\tau (n)$ ), and the second by A. Knopfmacher and N. Robbins [89] (they deduce the formula with $\\tau ^{\\pm }(n)$ ).",
"Recall that a composition $c$ of $n\\in \\mathbb {N}$ is an “ordered partition” of $n$ , that is, a tuple $c=(c_1,c_2,\\dots ,c_k)\\in \\mathbb {N}^k$ with $c_1+c_2+\\dots +c_k=n$ .",
"It is well known and easy to show that there are $2^{n-1}$ compositions of $n$ .",
"What is the number $f_{cdp}(n)$ of compositions of $n$ with distinct parts?",
"Corollary 3.6 .",
"The number $f_{cdp}(n)$ of compositions of $n$ with no part repeated is a PIO function.",
"The PIO algorithm for $f_{cdp}$ is described in the proof.",
"Proof.",
"The mapping $(c_1,c_2,\\dots ,c_k)\\mapsto (c_{i_1}>c_{i_2}>\\dots >c_{i_k})$ sending a composition of $n$ with distinct parts to its decreasing reordering is a $k!$ -to-1 mapping from the set of compositions of $n$ with $k$ distinct parts onto $Q_k(n)$ .",
"Thus $f_{cdp}(n)=\\sum _{k=1}^nk!q_k(n)$ and the result is an instance of Proposition REF for $g(n)=n!$ (clearly, $n\\mapsto n!$ is a PIO function, also see P. B. Borwein [23]).",
"$\\Box $ The sequence $(f_{cdp}(n))_{n\\ge 1}=(1,1,3,3,5,11,13,19,27,\\dots )$ is [154].",
"B. Richmond and A. Knopfmacher [125] note that $f_{cdp}(n)=\\exp ((1+o(1))(2n)^{1/2}\\log n)$ and obtain a more precise asymptotics.",
"See the book of S. Heubach and T. Mansour [72] for many more enumeration problems for compositions and words, especially with forbidden patterns.",
"We pose the following problem.",
"Problem 3.7 .",
"Give general sufficient conditions on functions $g:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ ensuring that $f(n)=\\sum _{\\lambda \\in P(n)}g(\\Vert \\lambda \\Vert )\\ \\mbox{ and }\\ f(n)=\\sum _{\\lambda \\in Q(n)}g(\\Vert \\lambda \\Vert )$ are PIO functions.",
"Propositions REF and REF say that it suffices when $g$ is a positive PIO function, but it would be more interesting to have general sufficient conditions allowing negative values of $g$ .",
"Corollary REF and Proposition REF are motivated by this problem too.",
"We generalize Proposition REF .",
"Many enumerative problems on partition, but of course not all, fit in the general schema of counting partitions with prescribed parts and multiplicities: for every triple $(n,i,j)\\in \\mathbb {N}^2\\times \\mathbb {N}_0$ we say if $i^j$ , part $i$ with multiplicity $j$ , may or may not appear in the counted partitions of $n$ .",
"If $m(n)\\gg n^c$ with $c>0$ for the counting problem, the simple algorithm of Proposition REF gives a PIO formula.",
"We spell it out explicitly.",
"Proposition 3.8 .",
"Suppose that $X\\subset \\mathbb {N}$ is a set, $g(n,i,j)\\in \\lbrace 0,1\\rbrace $ is a function defined for $n,i\\in \\mathbb {N}$ and $j\\in \\mathbb {N}_0$ with $i,j\\le n$ and computable in $\\mathrm {poly}(n)$ steps, and that the function $f:\\;\\mathbb {N}\\rightarrow \\mathbb {N}_0$ , defined by $f(n)&=&|\\lbrace \\lambda =1^{j_1}2^{j_2}\\dots n^{j_n}\\in P(n)\\;|\\;\\mbox{$g(n,i,j_i)=1$ for every $i\\in [n]$}\\rbrace |\\\\&=&[q^n]\\prod _{i=1}^n\\sum _{j=0}^ng(n,i,j)q^{ij}\\;,$ grows for $n\\in X$ as $f(n)\\gg \\exp (n^c)$ with a constant $c>0$ .",
"Then the restriction $f:\\;X\\rightarrow \\mathbb {N}_0$ is a PIO function.",
"The proof shows that the algorithm for $g$ constructively gives the PIO algorithm for $f$ , provided that the function $f$ gets as inputs only elements of $X$ .",
"Proof.",
"In the formula for $f(n)$ we have a product of $n$ polynomials with degrees at most $n^2$ each and with the coefficients 0 and 1 that can be computed in $\\mathrm {poly}(n)$ steps.",
"We argue as in the proof of Proposition REF and deduce that $f(n)$ can be computed in $\\mathrm {poly}(n)$ steps, which means $\\mathrm {poly}(m(n))$ steps for $n\\in X$ by the assumption on growth of $f$ .",
"$\\Box $ This applies to partitions of $n$ into distinct parts, odd parts, squares, etc.",
"but first we illustrate the proposition with two problems where the condition defining counted partitions $\\lambda \\in P(n)$ depends on $n$ — then our reflex to write a formula for $\\sum _{n\\ge 0}f(n)q^n$ , often an infinite product of simple factors, fails us as it cannot be done.",
"They are the functions $f_m,f_p:\\mathbb {N}\\rightarrow \\mathbb {N}$ where $f_m(n)$ (resp.",
"$f_p(n)$ ) counts partitions of $n$ such that every nonzero multiplicity (resp.",
"every part with nonzero multiplicity) divides $n$ .",
"Then $(f_m(n))_{n\\ge 1}&=&(1,\\,2,\\,3,\\,5,\\,4,\\,10,\\,6,\\,17,\\,14,\\,26,\\,13,\\,66,\\,19,\\,63,\\,60,\\,\\dots )\\mbox{ and}\\\\(f_p(n))_{n\\ge 1}&=&(1,\\,2,\\,2,\\,4,\\,2,\\,8,\\,2,\\,10,\\,5,\\,11,\\,2,\\,45,\\,2,\\,14,\\,14,\\,\\dots )$ are respective sequences [154] and [154].",
"By Proposition REF , applied with $X=\\mathbb {N}$ and $g(n,i,j)=1$ if $j$ divides $n$ and $g(n,i,j)=0$ else, the function $f_m(n)$ is a PIO function (with PIO algorithm given in Proposition REF ) because $g(n,i,j)$ is computable even in $\\mathrm {poly}(\\log n)$ steps and $f_m(n)\\ge q(n)\\gg \\exp (\\Omega (n^{1/2}))$ (by the above lower bound on $q(n)$ ).",
"The second function $f_p(n)$ was investigated by D. Bowman, P. Erdös and A. Odlyzko [27] who proved that $(1+o(1))\\bigg (\\frac{\\tau (n)}{2}-1\\bigg )\\log n<\\log f_p(n)<(1+o(1))\\frac{\\tau (n)}{2}\\log n\\;,$ in fact with stronger bounds in place of the $o(1)$ terms.",
"A PIO formula for $f_p(n)$ is apparently not known (in contrast to $f_m(n)$ , the growth condition is not satisfied and small values occur).",
"After M. Agrawal, N. Kayal and N. Saxena [2] we however know a PIO function $f:\\;\\mathbb {N}\\rightarrow \\lbrace 0,2\\rbrace $ with the property that $f(n)=2\\iff f_p(n)=2$ for every $n\\in \\mathbb {N}$ — we can efficiently compute $f_p(n)$ for infinitely many $n$ , namely the prime numbers.",
"We turn to the more standard situation when $g(n,i,j)$ does not depend on $n$ .",
"Then we easily obtain Corollary REF below.",
"For its proof we need the next lemma which is also used in the proof of Corollary REF .",
"Lemma 3.9 .",
"Let $a_1,a_2,\\dots ,a_k\\in \\mathbb {N}$ be distinct numbers such that if $d\\in \\mathbb {N}$ divides each $a_i$ then $d=1$ .",
"There is an $n_0$ , specified in the proof, such that for every $n\\in \\mathbb {N}$ with $n>n_0$ the equation $n=a_1x_1+\\dots +a_kx_k$ has a solution $x_1,\\dots ,x_k\\in \\mathbb {N}_0$ .",
"The same holds even if the $k$ numbers $x_i$ are required be distinct.",
"Proof.",
"1.",
"The ideal $\\langle a_1,\\dots ,a_k\\rangle $ in the ring $\\mathbb {Z}$ shows that $1=a_1b_1+\\dots +a_kb_k$ for some $b_i\\in \\mathbb {Z}$ .",
"It is not hard to see that one may take all $b_i$ with $|b_i|\\le A^{k-1}$ if $|a_i|\\le A$ for every $i$ .",
"We set $c=a_1\\max _i|b_i|$ and $d=\\sum _ia_ic$ .",
"It follows that $n_0=d-1$ works because any $n\\ge d$ is expressed by the nonnegative solution $x_1=c+l+jb_1$ , $x_i=c+jb_i$ if $i>1$ , for appropriate $l\\in \\mathbb {N}_0$ and $j\\in \\lbrace 0,1,\\dots ,a_1-1\\rbrace $ .",
"2.",
"Now we set $c=2a_1\\max _i|b_i|$ and $d=\\sum _ia_ic(k+1-i)$ .",
"Then $n_0=d-1$ works, because any $n\\ge d$ is expressed by the nonnegative solution with distinct coordinates $x_1=ck+l+jb_1$ , $x_i=c(k+1-i)+jb_i$ if $i>1$ , again for appropriate $l\\in \\mathbb {N}_0$ and $j\\in \\lbrace 0,1,\\dots ,a_1-1\\rbrace $ .",
"$\\Box $ The first part of the lemma is well known and is the simplest version of the Frobenius problem, see the book [4] of J. L. R. Alfonsín for more information.",
"We look at restricted partitions with parts in a prescribed set $A\\subset \\mathbb {N}$ ; let $P_A(n)\\subset P(n)$ be their set and $p_A(n):=|P_A(n)|$ .",
"We show that we can count them efficiently if the elements of $A$ can be efficiently recognized and $A$ is not too sparse.",
"To be precise, in general we probably only “can” count them efficiently because the proof relies on quantities $d\\in \\mathbb {N}$ and $B\\subset \\mathbb {N}$ that in general appear not to be computable.",
"Corollary 3.10 .",
"Suppose that the function $g=g(n):\\;\\mathbb {N}\\rightarrow \\mathbb {N}$ increases, is computable in $\\mathrm {poly}(n)$ steps and grows only polynomially, $g(n)<(1+n)^d$ for every $n\\in \\mathbb {N}$ and a constant $d\\in \\mathbb {N}$ , and define $f(n)=p_A(n)$ for $A=\\lbrace g(1),g(2),\\dots \\rbrace $ , $\\sum _{n\\ge 0}f(n)q^n=\\prod _{i\\ge 1}\\bigg (1+\\sum _{j=1}^{\\infty }q^{g(i)j}\\bigg )=\\prod _{i\\ge 1}\\frac{1}{1-q^{g(i)}}\\;.$ Then $f(n)$ is a PIO function.",
"We show how to compute the PIO algorithm for $f(n)$ from the algorithm for $g(n)$ if we are given the number $d=\\mathrm {gcd}(A)=\\mathrm {gcd}(g(1),g(2),\\dots )\\in \\mathbb {N}$ and a finite set $B\\subset A$ with $\\mathrm {gcd}(B)=d$ .",
"Proof.",
"Let $g(n)$ , $A$ , $d$ and $B$ be as stated (it is easy to see from prime factorizations that such finite subset $B$ exists) and let $n_0=\\max (\\lbrace 0\\rbrace \\cup \\lbrace n\\in d\\mathbb {N}\\;|\\;p_B(n)=0\\rbrace )\\in \\mathbb {N}_0\\;.$ Then $n_0<\\infty $ by part 1 of Lemma REF applied to the numbers $\\frac{1}{d}B$ , and in fact we can compute $n_0$ from the given $B$ .",
"So $f(n)=0$ if $n\\notin d\\mathbb {N}$ and for $n\\in \\mathbb {N}$ we can decide in $\\mathrm {poly}(\\log n)$ steps if $n\\in d\\mathbb {N}$ .",
"Subsets $S\\subset \\lbrace g(1),g(2),\\dots ,g(m)\\rbrace \\backslash B$ , where $m\\in \\mathbb {N}$ is maximum with $g(1)+g(2)+\\dots +g(m)\\le n-n_0-d\\;,$ prove that for $n\\in d\\mathbb {N}$ with $n\\ge n_0+d$ one has $f(n)\\gg \\exp (\\Omega (n^{1/(d+1)}))$ (for each $S$ we complete the sum of its elements by an appropriate partition with parts in $B$ to a partition of $n$ ).",
"We compute $f(n)$ effectively as follows.",
"For the input $n\\in \\mathbb {N}$ we check in $\\mathrm {poly}(\\log n)$ steps if $n\\notin d\\mathbb {N}$ and if $n\\le n_0$ .",
"In the former case we output $f(n)=0$ and in the latter case we compute $f(n)$ by brute force.",
"If neither case occurs, we have $n\\in d\\mathbb {N}$ and $n>n_0$ and compute $f(n)$ by Proposition REF , applied with $X=d\\mathbb {N}\\backslash [n_0]$ and $g(n,i,j)$ defined as $g(n,i,j)=0$ if $j\\ge 1$ and $i\\notin A$ , and $g(n,i,j)=1$ else (it is clear that the assumptions are satisfied, we can check if $i\\notin A$ in $\\mathrm {poly}(i)$ steps).",
"It follows that this is a PIO algorithm for $f(n)$ .",
"Also, we have constructed it explicitly from the algorithm for $g(n)$ and the knowledge of $d$ and $B$ .",
"$\\Box $ In general the quantities $d$ and $B$ probably are not computable from the algorithm for the function $g(n)$ .",
"Hence, probably, the PIO algorithm for $f(n)$ cannot be computed given only the algorithm for $g(n)$ .",
"For example, we may take $g(n)=n^2$ (so $d=1$ and $B=\\lbrace 1\\rbrace $ ) and compute the number $f_{sq}(n)$ of partitions of $n$ into squares, $\\sum _{n\\ge 0}f_{sq}(n)q^n=\\prod _{k\\ge 1}(1-q^{k^2})^{-1}$ .",
"By Corollary REF we get a PIO function $(f_{sq}(n))_{n\\ge 1}=(1,\\,1,\\,1,\\,2,\\,2,\\,2,\\,2,\\,3,\\,4,\\,4,\\,4,\\,5,\\,6,\\,6,\\,6,\\,8,\\,\\dots )\\;,$ [154].",
"The function $f_{sq}(n)$ was investigated by J. Bohman, C.-E. Fröberg and H. Riesel [20].",
"As we showed in the proof, $f_{sq}(n)\\gg \\exp (\\Omega (n^{1/3}))$ .",
"More generally, already in 1934 E. M. Wright found in [148] the asymptotics for the number $p_{S_k}(n)$ of partitions of $n$ into $k$ -th powers $S_k=\\lbrace n^k\\;|\\;n\\in \\mathbb {N}\\rbrace $ ($k\\in \\mathbb {N}$ ): $p_{S_k}(n)\\sim \\frac{\\Delta }{(2\\pi )^{(k+1)/2}}\\cdot \\frac{k^{1/2}}{(k+1)^{3/2}}\\cdot n^{\\frac{1}{k+1}-\\frac{3}{2}}\\cdot \\exp (\\Delta n^{1/(k+1)})$ where $\\Delta =(k+1)\\cdot \\big ((1/k)\\cdot \\Gamma (1+1/k)\\cdot \\zeta (1+1/k)\\big )^{1-1/(k+1)}\\;.$ More recently the asymptotics for $p_{S_k}(n)$ with $k=2$ (i.e., $f_{sq}(n)$ ) was treated by R. C. Vaughan [145], for general $k$ by A. Gafni [60], and for $p_A(n)$ with $A$ formed by values of an integral polynomial by A. Dunn and N. Robles [49].",
"Why not partition $n$ into distinct squares, $f_{dsq}(n):=[q^n]\\prod _{k\\ge 1}(1+q^{k^2})$ ?",
"The initially somewhat dull sequence $(f_{dsq}(n))_{n\\ge 1}=(1,\\,0,\\,0,\\,1,\\,1,\\,0,\\,0,\\,0,\\,1,\\,1,\\,0,\\,0,\\,1,\\,1,\\,0,\\,1,\\,1,\\,0,\\,\\dots )\\;,$ [154], eventually takes off (the first $n$ with $f(n)\\ge 2$ is $n=25$ ) and R. Sprague [137] proved in 1948 that $n=128$ is the last number with $f_{dsq}(n)=0$ .",
"See M. D. Hirschhorn [74] for “an almost complete proof” that $f_{dsq}(n)\\sim c_2n^{-5/6}\\exp (c_1n^{1/3})$ where $c_1=3c_3^{2/3}$ , $c_2=c_3^{1/3}/\\sqrt{6\\pi }$ , and $c_3=\\sqrt{\\pi }(2-\\sqrt{2})\\zeta (3/2)/8$ .",
"We have Corollary 3.11 .",
"The function $f_{dsq}(n):\\;\\mathbb {N}\\rightarrow \\mathbb {N}_0$ counting partitions of $n$ into distinct squares, which is the same as counting partitions of $n$ such that each part $i\\in \\mathbb {N}$ has multiplicity either 0 or $i$ , is a PIO function.",
"The PIO algorithm is described in the proof.",
"Proof.",
"This follows from Proposition REF , applied with $X=\\lbrace 129,130,\\dots \\rbrace $ and $g(n,i,j)=1$ iff $(i=k^2\\,\\&\\,j=1)$ or $j=0$ , if we show that $f_{dsq}(n)\\gg \\exp (n^c)$ on $X$ for some $c>0$ .",
"To obtain such lower bound for $f_{dsq}(n)$ we begin with a lemma: for every $j\\in [4]$ and $n\\ge 4$ we have $|\\lbrace A\\subset [n]\\;|\\;|A|\\equiv j\\;(\\mathrm {mod}\\;4)\\rbrace |\\gg 2^n$ because any $B\\subset [n-3]$ can be enlarged by adding one of $n-2,n-1,n$ to have cardinality $j$ modulo 4.",
"Now for given $n\\in \\mathbb {N}$ with $n\\equiv j\\;(\\mathrm {mod}\\;4)$ , $j\\in [4]$ and $n>1000$ , consider the maximum $m\\in \\mathbb {N}$ with $1^2+3^2+5^2+\\dots +(2m-1)^2\\le n-4\\cdot 129$ .",
"Clearly, $m=\\Theta (n^{1/3})$ .",
"For every subset $C\\subset [m]$ with $|C|\\equiv j\\;(\\mathrm {mod}\\;4)$ the sum $S_C=\\sum _{i\\in C}(2i-1)^2$ is also $j$ modulo 4.",
"By R. Sprague's theorem mentioned above and our selection of $m$ we may partition $\\frac{n-S_C}{4}\\in \\mathbb {N}$ into distinct squares, say $\\frac{n-S_C}{4}=x_1^2+\\dots +x_k^2$ .",
"But then $n=S_C+(2x_1)^2+\\dots +(2x_k)^2$ is a partition of $n$ into distinct squares, and distinct subsets $C$ yield distinct such partitions.",
"Using the lemma we get that $f_{dsq}(n)\\gg 2^m=\\exp (\\Omega (n^{1/3}))$ for $n\\in X$ .",
"$\\Box $ As M. D. Hirschhorn [74] himself admits, also in [75], his proof of the asymptotics for $f_{dsq}(n)$ is not complete.",
"Problem 3.12 .",
"Derive rigorously asymptotics for $f_{dsq}(n)$ .",
"Could not theta functions tell us something about $f_{sq}(n)$ or $f_{dsq}(n)$ ?",
"It transpires that $f_{dsq}(n)$ (with other partition counting functions) comes up in quantum statistical physis.",
"M. N. Tran, M. V. N. Murty and R. K. Bhaduri [142] and, recently, M. V. N. Murthy, M. Brack, R. K. Bhaduri and J. Bartel [106] investigate the asymptotics of $f_{dsq}(n)$ .",
"The main term is derived, with some further terms in the asymptotic expansion, but it is not clear to us whether these arguments are rigorous, and so we leave Problem REF as it is.",
"We partitioned $n$ into squares with arbitrary multiplicities, why not partition $n$ into arbitrary parts but with square multiplicities, $f_{sm}(n):=[q^n]\\prod _{i\\ge 1}(1+\\sum _{j\\ge 1}q^{ij^2})$ ?",
"In general setting this leads to the next counterpart to Corollary REF .",
"Now we have no restriction on the growth of the set of allowed multiplicities, it may very well be finite.",
"Corollary 3.13 .",
"Suppose that $1\\le g_1<g_2<\\dots $ is a finite or infinite nonempty increasing sequence of natural numbers such that $n\\mapsto g_n$ is computable in $\\mathrm {poly}(n)$ steps and define $f(n)=f_A(n)\\in \\mathbb {N}_0$ to be the number of partitions $\\lambda \\in P(n)$ with all nonzero multiplicities in $A=\\lbrace g_1,g_2,\\dots \\rbrace $ , $\\sum _{n\\ge 0}f(n)q^n=\\prod _{i\\ge 1}\\bigg (1+\\sum _{j\\ge 1}q^{ig_j}\\bigg )\\;.$ Then $f(n)$ is a PIO function.",
"We show how to compute the PIO algorithm for $f(n)$ from the algorithm for $g_n$ if we are given the number $d=\\mathrm {gcd}(A)=\\mathrm {gcd}(g(1),g(2),\\dots )\\in \\mathbb {N}$ and a finite set $B\\subset A$ with $\\mathrm {gcd}(B)=d$ .",
"Proof.",
"Let $g_n$ , $A$ , $d$ and $B$ be as stated and let $n_0=\\max (\\lbrace 0\\rbrace \\cup \\lbrace n\\in d\\mathbb {N}\\;|\\;f_B(n)=0\\rbrace )\\in \\mathbb {N}_0\\;.$ Then $n_0<\\infty $ by the second part of Lemma REF and we can compute $n_0$ from $B$ .",
"So $f(n)=0$ if $n\\notin d\\mathbb {N}$ , and for any $n\\in \\mathbb {N}$ the membership of $n$ in $d\\mathbb {N}$ is decidable in $\\mathrm {poly}(\\log n)$ steps.",
"We obtain a lower bound for $f(n)$ with large $n\\in d\\mathbb {N}$ as in the proof of Corollary REF .",
"Namely, we may assume that $\\frac{g_1}{d}$ is odd (there is certainly a $g_k$ with odd $\\frac{g_k}{d}$ ) and take arbitrary $n\\in d\\mathbb {N}$ with $n>2n_0+1$ and $\\frac{n}{d}\\equiv r$ modulo 2, $r\\in [2]$ .",
"Let $m\\in \\mathbb {N}$ be maximum with $(1+3+5+\\dots +(2m-1))g_1\\le n-2(n_0+1)$ .",
"Then $m=\\Theta (n^{1/2})$ and there are $\\gg 2^m$ subsets $C\\subset [m]$ such that $|C|\\frac{g_1}{d}\\equiv r$ modulo 2.",
"For each $C$ we have $(n-g_1\\sum _{i\\in C}(2i-1))/2\\in d\\mathbb {N}\\backslash [n_0]$ and this number equals $\\sum _{g\\in D}i_gg$ for some $D\\subset B$ and $|D|$ distinct numbers $i_g\\in \\mathbb {N}$ .",
"But then $n=\\sum _{i\\in C}(2i-1)g_1+\\sum _{g\\in D}(2i_g)g$ is a partition of $n$ into parts $\\lbrace 2i-1\\;|\\;i\\in C\\rbrace $ , each with multiplicity $g_1$ , and parts $\\lbrace 2i_g\\;|\\;g\\in D\\rbrace $ , with respective multiplicities $\\lbrace g\\;|\\;g\\in D\\rbrace $ .",
"All multiplicities are in $A$ and distinct subsets $C$ give distinct such partitions.",
"Thus for $n\\in d\\mathbb {N}$ with $n>n_0$ one has $f(n)\\gg 2^m=\\exp (\\Omega (n^{1/2}))$ .",
"We compute $f(n)$ effectively as follows.",
"For the input $n\\in \\mathbb {N}$ we check in $\\mathrm {poly}(\\log n)$ steps if $n\\notin d\\mathbb {N}$ and if $n\\le n_0$ .",
"In the former case we output $f(n)=0$ and in the latter case we compute $f(n)$ by brute force.",
"If neither case occurs, we have $n\\in d\\mathbb {N}$ and $n>n_0$ and compute $f(n)$ by Proposition REF , applied with $X=d\\mathbb {N}\\backslash [n_0]$ and $g(n,i,j)$ defined for $j\\ge 1$ and $j\\notin A$ as $g(n,i,j)=0$ and $g(n,i,j)=1$ else (clearly the assumption is satisfied, this $g(n,i,j)$ is computable in $\\mathrm {poly}(n)$ steps).",
"We have for $f(n)$ a PIO algorithm which we have constructed explicitly from the algorithm for $g_n$ and the knowledge of $d$ and $B$ .",
"$\\Box $ Similarly to Corollary REF , the PIO algorithm for $f(n)$ probably cannot be computed given only the algorithm for $g_n$ .",
"The sequence $(f_{sm}(n))_{n\\ge 1}=(1,\\,1,\\,2,\\,3,\\,3,\\,5,\\,6,\\,8,\\,12,\\,12,\\,17,\\,23,\\,27,\\,32,\\,41,\\,52,\\,\\dots )$ corresponding to $g_n=n^2$ ($d=1$ and $B=\\lbrace 1\\rbrace $ ) was for a long time we were preparing this article absent in OEIS, but checking it once more in August 2018 we found out that S. Manyama had added it as [154] in May 2018, thank you.",
"An interesting counting problem for partitions outside the framework of Proposition REF is the number $f_{dm}(n)$ of partitions of $n$ into parts with distinct nonzero multiplicities, so $f_{dm}(n)=[q^n]^*\\prod _{i=1}^n\\bigg (1+\\sum _{j=1}^nq^{ij}\\bigg )$ where the “star extraction” of the coefficient means that in the product we only accept monomials $q^{i_1j_1}q^{i_2j_2}\\dots q^{i_kj_k}$ , $1\\le i_1<\\dots <i_k\\le n$ , with $|\\lbrace j_1,\\dots ,j_k\\rbrace |=k$ .",
"The sequence $(f_{dm}(n))_{n\\ge 1}=(1,\\,2,\\,2,\\,4,\\,5,\\,7,\\,10,\\,13,\\,15,\\,21,\\,28,\\,31,\\,45,\\,55,\\,62,\\,\\dots )$ is [154].",
"The problem to investigate $f_{dm}(n)$ was posed by H. S. Wilf [147] in 2010.",
"To get a lower bound on $f_{dm}(n)$ , for given $n\\in \\mathbb {N}$ consider the maximum $m\\in \\mathbb {N}$ such that $1m+2(m-1)+\\dots +m1=(m+1)\\sum _{i=1}^mi-\\sum _{i=1}^mi^2\\le n$ .",
"Then $m=\\Theta (n^{1/3})$ and the subsets of $\\lbrace 2^{m-1},3^{m-2},\\dots ,m^1\\rbrace $ , where the exponents are used in the multiplicities sense, show that $f_{dm}(n)\\ge 2^{m-1}\\gg \\exp (\\Omega (n^{1/3}))$ (we can always add at least $m$ 1s to get a distinct multiplicities partition of $n$ ).",
"Thus $f_{dm}(n)$ still has a broadly exponential growth but it is not clear how to compute it efficiently.",
"Problem 3.14 .",
"Is the number $f_{dm}(n)$ of distinct-multiplicity partitions of $n$ a PIO function?",
"That is, can we compute it in $O(n^d)$ steps for a constant $d\\in \\mathbb {N}$ ?",
"D. Zeilberger [152] says: “I conjecture that the fastest algorithm takes exponential time, but I have no idea how to prove that claim.” See [154] for the first 500 or so terms of $(f_{dm}(n))_{n\\ge 1}$ .",
"J. A.",
"Fill, S. Janson and M. D. Ward [56] proved that $f_{dm}(n)=\\exp ((1+o(1))\\frac{1}{3}(6n)^{1/3}\\log n)$ and D. Kane and R. C. Rhoades [84] obtained an even more precise asymptotics.",
"So far we mostly considered partition counting functions $f(n)$ of broadly exponential growth, satisfying $\\log (2+f(n))\\gg n^c$ for a constant $c>0$ .",
"We turn to broadly polynomial growth when $\\log (2+f(n))\\ll \\log ^d(1+n)$ for a constant $d\\in \\mathbb {N}$ .",
"In Corollary REF we effectively computed $p_A(n)$ for infinite and not too sparse sets $A\\subset \\mathbb {N}$ .",
"At the opposite extreme lie finite sets $A\\subset \\mathbb {N}$ , for them $p_A(n)\\ll n^{|A|}$ .",
"By the classical result of E. T. Bell [15] (going back to J. Sylvester in 1857, as E. T. Bell himself acknowledges in [15]), then $p_A(n)$ can also be effectively computed.",
"Proposition 3.15 (E. T. Bell, 1943) .",
"For any finite set $A\\subset \\mathbb {N}$ , the number $p_A(n)$ of partitions $\\lambda =(\\lambda _1\\ge \\dots \\ge \\lambda _k)\\in P(n)$ with all $\\lambda _i\\in A$ is a rational quasipolynomial in $n$ .",
"Hence for every finite $A$ , $p_A(n)$ is a PIO function.",
"Below we extend Proposition REF and indicate how to obtain the PIO algorithm.",
"Recall from the previous section that a quasipolynomial $f:\\;\\mathbb {Z}\\rightarrow \\mathbb {C}$ is determined by a modulus $m\\in \\mathbb {N}$ and $m$ polynomials $p_i\\in \\mathbb {C}[x]$ , $i\\in [m]$ , such that $f(n)=p_i(n)$ if $n\\equiv i$ modulo $m$ .",
"If $d\\ge \\deg p_i(x)$ for every $i\\in [m]$ , we say that the quasipolynomial $f$ has class $(m,d)$ .",
"Replacing in the definition $\\mathbb {C}$ with $\\mathbb {Q}$ , we get rational quasipolynomials.",
"Equivalently, $f$ is a quasipolynomial if and only if $f(n)=a_k(n)n^k+\\dots +a_1(n)n+a_0(n)$ for some periodic functions $a_i:\\;\\mathbb {Z}\\rightarrow \\mathbb {C},\\mathbb {Q}$ .",
"If $f(n)=p_i(n)$ , $n\\equiv i$ modulo $m$ , only holds for every $n\\ge N$ for some $N$ , we speak of eventual quasipolynomials.",
"Particular cases are (eventually) periodic sequences $f:\\mathbb {N}_0\\rightarrow \\mathbb {C},\\mathbb {Q}$ which are the constant (eventual) quasipolynomials, each polynomial $p_i(x)$ is (eventually) constant.",
"There are many treatments of E. T. Bell's result in the literature.",
"We mention only R. P. Stanley [138], Ø. J. Rødseth and J. A.",
"Sellers [128], M. Cimpoeaş and F. Nicolae [41], [42], the inductive proof of R. Jakimczuk [78] or the recent S. Robins and Ch.",
"Vignat [127].",
"The last reference gives a very nice, simple and short proof of Proposition REF by generating functions, which moreover presents a PIO formula for $p_A(n)$ , $A=\\lbrace a_1,a_2,\\dots ,a_k\\rbrace $ , explicitly: $p_A(n)=\\sum _{j\\in J,\\,a_1j_1+\\dots +a_kj_k\\equiv n\\;(\\mathrm {mod}\\;D)}\\binom{\\frac{1}{D}(n-a_1j_1-\\dots -a_kj_k)+k-1}{k-1}$ where $D$ is any common multiple of $a_1,\\dots ,a_k$ and $J=[0,\\frac{D}{a_1}-1]\\times \\dots \\times [0,\\frac{D}{a_k}-1]$ .",
"We give yet another proof via a closure property.",
"We prove that the family of rational quasipolynomials is closed under convolution.",
"The (Cauchy) convolution of functions $f,g:\\;\\mathbb {N}_0\\rightarrow \\mathbb {C}$ is the function $f*g:\\;\\mathbb {N}_0\\rightarrow \\mathbb {C}$ , $(f*g)(n)=\\sum _{i=0}^nf(i)g(n-i)=\\sum _{i+j=n}f(i)g(j)\\;.$ Convolution gives coefficients in products of power series, if $a,b\\in \\mathbb {C}[[x]]$ then $[x^k]ab=([x^n]a)*([x^n]b)(k)$ .",
"If we can compute a rational quasipolynomial $f(n)$ and know its class $(m,d)$ , we effectively have a PIO algorithm for $f(n)$ .",
"Namely, we compute the first (in fact, any) $d+1$ values $f(n)$ in each congruence class $n\\equiv i$ modulo $m$ and find by Lagrange interpolation the $m$ polynomials $p_i\\in \\mathbb {Q}[x]$ fitting them.",
"The $p_i(x)$ then constitute the PIO algorithm for $f(n)$ .",
"Thus it is useful to know how classes of quasipolynomials transform under convolution (and linear combination).",
"For other closure properties of sequences under the operation of convolution see S. A. Abramov, M. Petkovšek and H. Zakrajšek [1].",
"Proposition 3.16 .",
"Let $f,g:\\;\\mathbb {N}_0\\rightarrow \\mathbb {Q}$ , $\\alpha ,\\beta \\in \\mathbb {Q}$ and $N,N^{\\prime }\\in \\mathbb {N}_0$ .",
"If $f$ and $g$ are rational quasipolynomials then so is $f*g$ .",
"If $f$ and $g$ have classes, respectively, $(m,d)$ and $(m^{\\prime },d^{\\prime })$ then $f*g$ has class $(M,d+d^{\\prime }+1)$ where $M$ is any common multiple of $m$ and $m^{\\prime }$ .",
"If $f$ and $g$ are eventual rational quasipolynomials then so is $\\alpha f+\\beta g$ .",
"If $f$ and $g$ have classes, respectively, $(m,d)$ for $n\\ge N$ and $(m^{\\prime },d^{\\prime })$ for $n\\ge N^{\\prime }$ then $\\alpha f+\\beta g$ has class $(M,\\max (d,d^{\\prime }))$ for $n\\ge \\max (N,N^{\\prime })$ where $M$ is any common multiple of $m$ and $m^{\\prime }$ .",
"If $f$ and $g$ are eventual rational quasipolynomials then so is $f*g$ .",
"If $f$ and $g$ have classes, respectively, $(m,d)$ for $n\\ge N$ and $(m^{\\prime },d^{\\prime })$ for $n\\ge N^{\\prime }$ then $f*g$ has class $(M,d+d^{\\prime }+1)$ for $n\\ge N+N^{\\prime }$ where $M$ is any common multiple of $m$ and $m^{\\prime }$ .",
"Proof.",
"1.",
"By [138] we have $\\sum _{n\\ge 0}f(n)q^n=\\frac{a(q)}{(1-q^m)^{d+1}}\\ \\mbox{ and }\\ \\sum _{n\\ge 0}g(n)q^n=\\frac{b(q)}{(1-q^{m^{\\prime }})^{d^{\\prime }+1}}$ for some polynomials $a,b\\in \\mathbb {Q}[q]$ with degree smaller than that of the denominator.",
"Using the identity $1-q^{ke}=(1-q^e)(1+q^e+q^{2e}+\\dots +q^{(k-1)e}),\\ e,k\\in \\mathbb {N}\\;,$ which is the main trick in [127], we can write for any common multiple $M$ of $m$ and $m^{\\prime }$ the product in the same form $\\sum _{n\\ge 0}f(n)q^n\\cdot \\sum _{n\\ge 0}g(n)q^n=\\frac{a(q)}{(1-q^m)^{d+1}}\\cdot \\frac{b(q)}{(1-q^{m^{\\prime }})^{d^{\\prime }+1}}=\\frac{c(q)}{(1-q^M)^{d+d^{\\prime }+2}}$ where $c\\in \\mathbb {Q}[q]$ with $\\deg c<M(d+d^{\\prime }+2)$ .",
"By expanding the denominator in generalized geometric series we get the result.",
"2.",
"This is immediate to see.",
"3.",
"Let $\\sum _{n\\ge 0}f(n)q^n=a(q)+b(q)$ and $\\sum _{n\\ge 0}g(n)q^n=c(q)+d(q)$ where $a,c\\in \\mathbb {Q}[[q]]$ have sequences of coefficients that are rational quasipolynomials with the stated classes for $n\\ge 0$ and $b,d\\in \\mathbb {Q}[q]$ are polynomials with degrees $\\deg b<N$ and $\\deg d<N^{\\prime }$ .",
"Let $M$ be any common multiple of $m$ and $m^{\\prime }$ .",
"Then $(f*g)(n)$ is the sequence of coefficients in $(a+b)(c+d)=ac+ad+bc+bd\\;.$ By parts 1 and 2, the sequences of coefficients in the last four summands are eventual rational quasipolynomials with classes, respectively, $(M,d+d^{\\prime }+1)$ for $n\\ge 0$ , $(m,d)$ for $n\\ge N^{\\prime }$ (the sequence of coefficients in $ad$ is a linear combination of $\\deg d+1$ shifts, by numbers $<N^{\\prime }$ , of that in $a$ ), $(m^{\\prime },d^{\\prime })$ for $n\\ge N$ , and $(1,0)$ for $n\\ge N+N^{\\prime }-1$ .",
"The result for $f*g$ follows by applying part 2 thrice.",
"$\\Box $ We generalize Proposition REF as follows.",
"Corollary 3.17 .",
"Let $k,m\\in \\mathbb {N}$ and $g:\\;\\mathbb {N}\\times \\mathbb {N}_0\\rightarrow \\lbrace 0,1\\rbrace $ be a function such that $g(i,0)=1$ for $i>k$ , $g(i,j)=0$ for $i>k$ and $j>0$ , and for $i\\le k$ each of the $k$ 0-1 sequences $(g(i,j))_{j\\ge 0}$ is $m$ -periodic.",
"Then the function $f:\\;\\mathbb {N}\\rightarrow \\mathbb {N}_0$ , $f(n)&=&\\#\\lbrace \\lambda =1^{j_1}2^{j_2}\\dots n^{j_n}\\in P(n)\\;|\\;\\mbox{$g(i,j_i)=1$ for every $i=1,2,\\dots ,n$}\\rbrace \\\\&=&[q^n]\\prod _{i=1}^k\\sum _{j=0}^{\\infty }g(i,j)q^{ij}\\;,$ is a rational quasipolynomial with class $(m\\cdot k!, k-1)$ .",
"The PIO algorithm for $f(n)$ follows constructively from $k,m$ and $g$ .",
"We state the “eventual” version.",
"Let $k,m$ and $g$ be as before, with the modification that each sequence $(g(i,j))_{j\\ge 0}$ , $i\\le k$ , is $m$ -periodic only for $j\\ge N$ , for some given $N\\in \\mathbb {N}_0$ .",
"Then the function $f(n)$ , defined as above, is an eventual rational quasipolynomial of class $(m\\cdot k!, k-1)$ for $n\\ge \\binom{k+1}{2}N$ .",
"The PIO algorithm for $f(n)$ follows constructively from $k,m,N$ and $g$ .",
"Proof.",
"For each $i\\le k$ we set $G_i(q)=\\sum _{j=0}^{\\infty }g(i,j)q^{ij}$ .",
"So $G_i\\in \\lbrace 0,1\\rbrace [[q]]$ has $im$ -periodic sequence of coefficients (i.e., with class $(im,0)$ ).",
"By $k-1$ applications of part 1 of Proposition REF , $f(n)=[q^n]\\prod _{i=1}^kG_i(q)$ is a rational quasipolynomial with class $(m\\cdot k!, k-1)$ .",
"In the “eventual” version, $G_i\\in \\lbrace 0,1\\rbrace [[q]]$ has $im$ -periodic sequence of coefficients for $n\\ge iN$ .",
"The result follows by $k-1$ applications of part 3 of Proposition REF .",
"$\\Box $ For finite $A\\subset \\mathbb {N}$ , Proposition REF is the instance with $k=\\max (A)$ and $g(i,j)=1$ if and only if $i\\in A$ or $j=0$ .",
"More generally Corollary REF implies, for example, eventual quasipolynomiality of the numbers of partitions of $n$ with parts in $A=\\lbrace 3,4,27\\rbrace $ and such that 3 appears an even number of times, except that multiplicity 2018 is not allowed, and the multiplicity of 27 equals 2 or 7 modulo 11.",
"Not much changes in the proof, using again Proposition REF , of the next generalization, and we leave it as an exercise.",
"Corollary 3.18 .",
"Let $k,m,d\\in \\mathbb {N}$ and $g:\\;\\mathbb {N}\\times \\mathbb {N}_0\\rightarrow \\mathbb {Z}$ be a function such that $g(i,0)=1$ for $i>k$ , $g(i,j)=0$ for $i>k$ and $j>0$ , and for every $i\\le k$ the function $j\\mapsto g(i,j)$ is a rational quasipolynomial of class $(m,d)$ .",
"Then ($\\lambda =1^{j_1}2^{j_2}\\dots n^{j_n}$ ) $f(n):=\\sum _{\\lambda \\in P(n)}\\prod _{i=1}^ng(i,j_i)\\in \\mathbb {Z}$ is a rational quasipolynomial with class $(m\\cdot k!,k(d+1)-1)$ .",
"The PIO algorithm for $f(n)$ follows constructively from $k,m,d$ and $g$ .",
"If each function $g(i,j)$ , $i\\le k$ , is an eventual rational quasipolynomial of class $(m,d)$ for $j\\ge N$ , for some given $N\\in \\mathbb {N}_0$ , then $f(n)$ is an eventual rational quasipolynomial with class $(m\\cdot k!,k(d+1)-1)$ for $n\\ge \\binom{k+1}{2}N$ .",
"The PIO algorithm for $f(n)$ follows constructively from $k,m,d,N$ and $g$ .",
"For example, the weighted number of partitions $\\lambda \\in P_A(n)$ , $A=\\lbrace 3,4,27\\rbrace $ , with the weight of $\\lambda $ equal to $(-1)^mm^5+3m$ where $m$ is the multiplicity of 4, is a rational quasipolynomial (we leave determination of its class as an exercise for the reader).",
"Support of a function is the set of arguments where it attains nonzero values.",
"In Propositions REF and REF we made our life easy by positivity of the function $g(n)$ .",
"Another easy case occurs when $g(n)$ is almost always zero.",
"Corollary 3.19 .",
"If $g:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ has finite support $S\\subset \\mathbb {N}$ with $s=\\max (S)$ then both functions $f_1,f_2:\\;\\mathbb {N}\\rightarrow \\mathbb {Z}$ , $f_1(n)=\\sum _{\\lambda \\in P(n)}g(\\Vert \\lambda \\Vert )\\ \\mbox{ and }\\ f_2(n)=\\sum _{\\lambda \\in Q(n)}g(\\Vert \\lambda \\Vert )\\;,$ are rational quasipolynomials with class $(s!,s-1)$ .",
"The PIO algorithm for $f_i(n)$ follows constructively from $s$ and $g$ .",
"(Recall that $Q(n)$ are the partitions of $n$ with no part repeated.)",
"Proof.",
"Using conjugation of partitions, which is the involution $(\\lambda _1\\ge \\lambda _2\\ge \\dots \\ge \\lambda _k)\\leftrightarrow 1^{\\lambda _1-\\lambda _2}2^{\\lambda _2-\\lambda _3}\\dots (k-1)^{\\lambda _{k-1}-\\lambda _k}k^{\\lambda _k}\\;,$ we get the well known identity $p_k(n)=p_{[k]}(n)-p_{[k-1]}(n)$ — the number of partitions of $n$ with $k$ parts equals the number of those partitions of $n$ with parts in $[k]$ that use part $k$ .",
"Thus $f_1(n)=\\sum _{\\lambda \\in P(n)}g(\\Vert \\lambda \\Vert )=\\sum _{k\\in S}g(k)(p_{[k]}(n)-p_{[k-1]}(n))\\;.$ By part 1 of Proposition REF , $p_{[k]}(n)$ is a rational quasipolynomial of class $(k!,k-1)$ .",
"The result follows by part 2 of Proposition REF .",
"As for $f_2(n)$ , we have $f_2(n)=\\sum _{k\\in S}g(k)q_k(n)$ .",
"Conjugation of partitions yields the identity $q_k(n)=p_{[k]}(n)-\\bigg |\\bigcup _{i=1}^kP_{[k]\\backslash \\lbrace i\\rbrace }(n)\\bigg |$ — the number of partitions of $n$ into $k$ distinct parts equals the number of those partitions of $n$ with parts in $[k]$ that use each part $1,2,\\dots ,k$ .",
"Applying the principle of inclusion and exclusion, we express $f_2(n)$ as an integral linear combination of the functions $p_A(n)$ with $A\\subset [s]$ .",
"The result follows by part 2 of Proposition REF .",
"$\\Box $ In the literature one can find several interesting enumerative results on partitions involving quasipolynomials.",
"We state the results of D. Zeilberger [152] and of G. E. Andrews, M. Beck and N. Robbins [8], for other quasipolynomial results see A. D. Christopher and M. D. Christober [39] and V. Jelínek and M. Klazar [79].",
"Proposition 3.20 (D. Zeilberger, 2012) .",
"For every finite set $A\\subset \\mathbb {N}$ the number $f(n)$ of the partitions $\\lambda \\in P_A(n)$ that have distinct nonzero multiplicities is a quasipolynomial in $n$ (and so a PIO function).",
"Proposition 3.21 (G. E. Andrews, M. Beck and N. Robbins, 2015) .",
"Let $t\\in \\mathbb {N}_0$ with $t\\ge 2$ .",
"The number $f(n)$ of $\\lambda =(\\lambda _1\\ge \\dots \\ge \\lambda _k)\\in P(n)$ with $\\lambda _1-\\lambda _k=t$ is a quasipolynomial in $n$ (and so a PIO function).",
"For $t=0$ and 1 the reader can check that, respectively, $f(n)=\\tau (n)$ and $f(n)=n-\\tau (n)$ where $\\tau (n)$ is the number of divisors of $n$ (and not the Ramanujan function which we will discuss too).",
"Sadly, despite their simplicity, we do not know if these are PIO formulas because we do not know how to efficiently factorize numbers.",
"In fact, [8] contains a more general result for prescribed differences between parts.",
"Quite general result in enumeration and logic involving quasipolynomials was achieved by T. Bogart, J. Goodrick and K. Woods [19].",
"In the statement we extend in the obvious way the notion of eventual quasipolynomial to function defined on an eventually periodic subset of $\\mathbb {N}_0$ .",
"Theorem 3.22 (T. Bogart, J. Goodrick and K. Woods, 2017) .",
"Let $d\\in \\mathbb {N}$ and $t\\mapsto X_t\\subset \\mathbb {Z}^d,\\ t\\in \\mathbb {N}\\;,$ be a sequence of sets $X_t$ that is defined by a formula in 1-parametric Pressburger arithmetic.",
"Then the set $Y\\subset \\mathbb {N}$ of $t\\in \\mathbb {N}$ for which $|X_t|<\\infty $ is eventually periodic and $f:\\;Y\\rightarrow \\mathbb {N}_0$ , $f(t)=|X_t|$ , is an eventual quasipolynomial (and so a PIO function).",
"The way of definition of the sequence $t\\mapsto X_t$ is that the membership $(x_1,x_2,\\dots ,x_d)\\in X_t$ is defined, for some $k\\in \\mathbb {N}$ , by a formula built by logical connectives and quantification of integer variables from atomic inequalities of the form $a_1y_1+a_2y_2+\\dots +a_ky_k\\le b\\ \\mbox{ where }\\ a_i,b\\in \\mathbb {Z}[t]$ (here enters the single parameter $t$ in the problem) and the $y_i$ are integer variables including the $x_i$ .",
"Consult [19] for details and examples and, of course, for the proof.",
"T. Bogart, J. Goodrick, D. Nguyen and K. Woods prove in [18] that for more than one parameter, polynomial-time computability disappears (assuming $\\mathrm {P}\\ne \\mathrm {NP}$ ).",
"We state Propositions REF and REF and Theorem REF in their original form and so do not indicate how to get PIO algorithms from the given data, but with some effort such extensions probably could be obtained from the proofs.",
"In Corollary REF , for $i\\le k$ each 0-1 sequence $(g(i,j))_{j\\ge 0}$ recording allowed multiplicities of part $i$ follows a linear (periodic or eventually periodic) pattern.",
"What happens for, say, quadratic patterns?",
"What is the number $f_{x^2+2y^2}(n)$ of partitions $\\lambda =1^{x^2}2^{y^2}\\in P(n)$ , $x,y\\in \\mathbb {N}_0$ , that is, partitions of $n$ into parts 1 and 2 with square multiplicities?",
"A nice formula exists: $f_{x^2+2y^2}(n)=\\frac{\\tau _{1,8}(n)+\\tau _{3,8}(n)-\\tau _{5,8}(n)-\\tau _{7,8}(n)+\\delta }{2}$ where $\\tau _{i,m}(n)$ counts divisors of $n$ that are $i$ modulo $m$ , $\\delta =1$ if $n$ is a square or twice a square and $\\delta =0$ else.",
"This goes back to P. Dirichlet in 1840, see M. D. Hirschhorn [73] for a proof.",
"From the reason we already mentioned we do not know if it is a PIO formula.",
"We do not know how to count efficiently solutions of equations like $n=x^2+2y^2$ or $n=x^2+y^2$ ($x,y\\in \\mathbb {N}_0$ or, more classically, $x,y\\in \\mathbb {Z}$ ).",
"But if only one of the patterns is quadratic, we can again count efficiently.",
"For example, we can count efficiently solutions of $n=x+2y^2$ : $f_{x+2y^2}(n):=\\#\\lbrace \\lambda =1^{j_1}2^{j_2^2}\\in P(n)\\;|\\;j_i\\in \\mathbb {N}_0\\rbrace =\\lfloor \\sqrt{n/2}\\rfloor +1$ is a PIO function (but not a quasipolynomial).",
"We compute it in $\\mathrm {poly}(\\log n)$ steps as follows.",
"To compute integral square root $n\\mapsto \\lfloor \\sqrt{n}\\rfloor $ in $\\mathrm {poly}(\\log n)$ steps, initialize $m:=0$ , add to $m$ in $m:=m+2^r$ the largest power of two such that $m^2\\le n$ , and repeat.",
"When $m$ cannot be increased by adding a power of two, $m=\\lfloor \\sqrt{n}\\rfloor $ .",
"See [61] for faster algorithms.",
"We hope to treat generalizations of $f_{x+2y^2}(n)$ elsewhere.",
"In the setup of Corollary REF , we get for $f(n)$ broadly polynomial growth if $g(n)$ grows at least exponentially.",
"The classical example is for $m\\in \\mathbb {N}$ with $m\\ge 2$ the counting function $f_{mp}(n)=f_{mp}(n,m):=p_{\\lbrace 1,m,m^2,m^3,\\dots \\rbrace }(n)$ counting the partitions of $n$ in powers of $m$ , so called $m$ -ary partitions.",
"For $m=2$ we get the binary partitions.",
"Binary partitions with distinct parts are easy to count as $\\prod _{k=0}^{\\infty }(1+q^{2^k})=\\sum _{n=0}^{\\infty }q^n$ (partition theorist's joke).",
"But it appears not easy to count effectively general binary partitions or $m$ -ary partitions.",
"“Effectively” here means, of course, in $\\mathrm {poly}(\\log n)$ steps: $m(n)=\\log (1+n)+\\log (2+f_{mp}(n))=\\Theta (\\log ^2(1+n))$ because K. Mahler [100] proved that $f_{mp}(n)=\\exp ((1+o(1))(1/2\\log m)\\log ^2n)$ .",
"More precise asymptotic relations were derived by N. G. de Bruijn [29], C.-E. Fröberg [59] and others.",
"Problem 3.23 .",
"Let $m\\in \\mathbb {N}$ with $m\\ge 2$ .",
"Is the function ${\\textstyle f_{mp}(n)=\\#\\lbrace (x_i)_{i\\ge 0}\\subset \\mathbb {N}_0\\;|\\;\\sum _{i\\ge 0}x_im^i=n\\rbrace }$ counting $m$ -ary partitions of $n$ a PIO function?",
"That is, can we compute it in $O(\\log ^d(1+n))$ steps (bit operations) for a fixed $d\\in \\mathbb {N}$ ?",
"An interesting algorithm of V. P. Bakoev [12] suggests that the answer might be positive.",
"Proposition 3.24 (V. P. Bakoev, 2004) .",
"Let $m\\in \\mathbb {N}$ with $m\\ge 2$ be given.",
"There is an algorithm computing $f_{mp}(m^n)$ for every $n\\in \\mathbb {N}$ in $O(n^3)$ arithmetic operations.",
"From the literature on $m$ -ary partitions we further mention T. Edgar [50], M. D. Hirschhorn and J. A.",
"Sellers [76] and D. Krenn and S. Wagner [97] (which deals mostly with $m$ -ary compositions).",
"It is easy to see that the number $f_{bp}(n):=f_{mp}(n,2)$ of binary partitions of $n$ follows the recurrence $f_{bp}(0)=1$ and $f_{bp}(n)=f_{bp}(n-1)+f_{bp}(n/2)$ for $n\\ge 1$ (where $f_{bp}(n/2)=0$ if $n/2\\notin \\mathbb {N}_0$ ).",
"The reduction $f_{bp}^{\\prime }(n):=f_{bp}(2n)$ follows the recurrence $f_{bp}^{\\prime }(0)=1$ and ($n\\ge 1$ ) $f_{bp}^{\\prime }(n)=f_{bp}^{\\prime }(n-1)+f_{bp}^{\\prime }(\\lfloor n/2\\rfloor )$ and forms the sequence [154], $(f_{bp}(2n))_{n\\ge 0}=(f_{bp}^{\\prime }(n))_{n\\ge 0}=(1,\\,2,\\,4,\\,6,\\,10,\\,14,\\,20,\\,26,\\,36,\\,46,\\,60,\\,74,\\,\\dots )\\;,$ investigated by D. E. Knuth [90] fifty years ago.",
"Recently, I. Pak [112] has announced positive resolution of Problem REF in I. Pak and D. Yeliussizov [113], [114]: (we quote verbatim from [112]) Theorem 3.25 (I. Pak and D. Yeliussizov) .",
"Let ${\\cal A}=\\lbrace a_1,a_2,\\dots \\rbrace $ , and suppose $a_k/a_{k-1}$ is an integer $\\ge 2$ , for all $k>1$ .",
"Suppose also that membership $x\\in {\\cal A}$ can be decided in $\\mathrm {poly}(\\log x)$ time.",
"Then $\\lbrace p_{{\\cal A}}(n)\\rbrace $ can be computed in time $\\mathrm {poly}(\\log n)$ .",
"In conclusion of Section 3 and of our article we turn to cancellative problems related to the initial Example 5.",
"Sums of integers with large absolute values still may be small, even 0.",
"In enumeration it means that a formula, effective for nonnegative summands, may no longer be effective (in the sense of Definition REF ) for integral summands, if cancellations occur.",
"In the next proposition we give both an example and a non-example of such cancellation.",
"The former is a classics but the latter may be not so well known.",
"Proposition 3.26 .",
"Both functions $q^{\\pm }(n):=\\sum _{\\lambda \\in Q(n)}(-1)^{\\Vert \\lambda \\Vert }\\ \\mbox{ and }\\ p^{\\pm }(n):=\\sum _{\\lambda \\in P(n)}(-1)^{\\Vert \\lambda \\Vert }$ are PIO functions.",
"Concretely, $\\sum _{n=0}^{\\infty }q^{\\pm }(n)q^n=\\prod _{k=1}^{\\infty }(1-q^k)=1+\\sum _{n=1}^{\\infty }(-1)^n(q^{n(3n-1)/2}+q^{n(3n+1)/2})$ and $\\sum _{n=0}^{\\infty }p^{\\pm }(n)q^n=\\prod _{k=1}^{\\infty }\\frac{1}{1+q^k}=\\prod _{k=1}^{\\infty }(1+(-q)^{2k-1})=1+\\sum _{n=1}^{\\infty }(-1)^nq_o(n)q^n$ where $q_o(n):=\\#\\lbrace \\lambda \\in Q(n)\\;|\\;\\lambda _i\\equiv 1\\ (\\mathrm {mod}\\ 2)\\rbrace $ .",
"Proof.",
"The first identity is the famous pentagonal identity of L. Euler [53] (or [6], [68]).",
"Replacing $q(n)$ with $q^{\\pm }(n)$ leads to almost complete cancellation to values just 0 and $\\pm 1$ , and $m(n)=\\Theta (\\log (1+n))$ for $q^{\\pm }(n)$ .",
"The algorithms of Propositions REF (recurrence schema) and REF (coefficient extraction from a generating polynomial) still work but do $\\mathrm {poly}(n)$ steps and are not effective for computing $n\\mapsto q^{\\pm }(n)$ .",
"For a PIO formula more efficient algorithm is needed.",
"Fortunately, the pentagonal identity provides it.",
"We easily determine in $\\mathrm {poly}(\\log n)$ steps the existence of a solution $i\\in \\mathbb {N}$ to the equation $n=\\frac{i(3i\\pm 1)}{2}$ and its parity, simply by computing the integral square root as discussed above in connection with $f_{x+2y^2}(n)$ .",
"The second identity, more precisely the middle equality, follows at once from $\\frac{1}{1+q^k}=\\frac{1-q^k}{1-q^{2k}}$ .",
"Now the replacement of $p(n)$ with $p^{\\pm }(n)$ leads to almost no cancellation because $|p^{\\pm }(n)|=q_o(n)=[q^n]\\prod _{k=1}^{\\infty }(1+q^{2k-1})\\sim \\frac{\\exp (\\pi \\sqrt{n/6})}{2^{3/2}\\cdot 6^{1/4}\\cdot n^{3/4}}$ ([154]; V. Kotěšovec [94]; G. Meinardus [102]) remains of broadly exponential growth.",
"Thus for $p^{\\pm }(n)$ we have $m(n)\\gg n^{1/2}$ and both algorithms for $p(n)$ remain effective for $p^{\\pm }(n)$ .",
"Hence, more easily than for $q^{\\pm }(n)$ , $p^{\\pm }(n)$ is a PIO function.",
"$\\Box $ We had derived the second identity and then we learned in A. Ciolan [43] that it is in fact due to J. W. L. Glaisher [62].",
"The above examples lead us to the following question.",
"Problem 3.27 .",
"Find general sufficient conditions on the functions $a=a_i:\\;\\mathbb {N}\\rightarrow \\mathbb {N}_0\\ \\mbox{ and }\\ b=b_{i,j,k}:\\;\\mathbb {N}\\times \\mathbb {N}_0\\times \\mathbb {N}\\rightarrow \\lbrace 0,1\\rbrace $ ensuring that $f(n)=[q^n]\\prod _{i=1}^{\\infty }\\prod _{k=1}^{a_i}\\sum _{j=0}^{\\infty }b_{i,j,k}(-1)^jq^{ij}$ is a PIO function.",
"Find asymptotics of $f(n)$ .",
"Thus $f(n)$ is the $(-1)^{\\Vert \\lambda \\Vert }$ -count of the partitions $\\lambda $ of $n$ into parts $i\\in \\mathbb {N}$ coming in $a_i$ sorts $(i,1),(i,2),\\dots ,(i,a_i)$ such that the part $(i,k)^j$ may appear if and only if $b_{i,j,k}=1$ .",
"For $a_i=1$ , $b_{i,0,1}=b_{i,1,1}=1$ , and $b_{i,j,k}=0$ else we get $q^{\\pm }(n)$ , and for $a_i=1$ , $b_{i,j,1}=1$ , and $b_{i,j,k}=0$ else we get $p^{\\pm }(n)$ .",
"For $a_i=2$ , $b_{i,0,1}=b_{i,1,1}=b_{i,0,2}=b_{i,1,2}=1$ , and $b_{i,j,k}=0$ else we get the function $f(n)$ of Example 5.",
"More generally, for $l\\in \\mathbb {N}$ the counting functions $\\sum _{n=0}^{\\infty }q^{\\pm ,l}(n)q^n:=\\prod _{n=1}^{\\infty }(1-q^n)^l$ correspond to $a_i=l$ , $b_{i,0,1}=b_{i,1,1}=b_{i,0,2}=b_{i,1,2}=\\dots =b_{i,0,l}=b_{i,1,l}=1$ , and $b_{i,j,k}=0$ else; $q^{\\pm }(n)=q^{\\pm ,1}(n)$ and Example 5 is $q^{\\pm ,2}(n)$ .",
"A related open problem, due to D. Newman, is mentioned in G. E. Andrews and D. Newman [9]: In $\\sum _{n\\ge 0}p(n)q^n=\\prod _{k=1}^{\\infty }(1+p^k+p^{2k}+\\dots )\\;,$ can one change some signs in the last product so that on the left side the $p(n)$ turn to coefficients 0 and $\\pm 1$ ?",
"Another example of non-cancellation in Problem REF is the result of A. Ciolan [43]: if $S_2:=\\lbrace 1,4,9,16,\\dots \\rbrace $ , $B:=\\Gamma (3/2)\\zeta (3/2)/2\\sqrt{2}$ and $t_n:=\\sum _{\\lambda \\in P(n),\\,\\lambda _i\\in S_2}(-1)^{\\Vert \\lambda \\Vert }=[q^n]\\prod _{k\\ge 1}\\frac{1}{1+q^{k^2}}$ then $t_n\\sim (-1)^n\\frac{\\exp \\left(3(B/2)^{2/3}n^{1/3}\\right)}{(3\\pi )^{1/2}(2n)^{5/6}/B^{1/3}}\\;.$ In [154], for example $(t_n)_{n=32}^{49}=(1,\\,-2,\\,3,\\,-4,\\,3,\\,-2,\\,1,\\,0,\\,1,\\,-2,\\,3,\\,-4,\\,3,\\,-2,\\,1,\\,0,\\,0,\\,-2)\\;,$ V. Kotěšovec gave this asymptotic formula as well, without proof.",
"If $s_n:=[q^n]\\prod _{k\\ge 1}(1-q^{k^2})$ is the corresponding number for distinct squares, with the help of MAPLE we get the values $(s_n)_{n=0}^{15}=(1,\\,-1,\\,0,\\,0,\\,-1,\\,1,\\,0,\\,0,\\,0,\\,-1,\\,1,\\,0,\\,0,\\,1,\\,-1,\\,1,\\,0)$ or $(s_n)_{n=2990}^{3000}=(111,\\,-112,\\,61,\\,46,\\,-114,\\,116,\\,-21,\\,11,\\,-30,\\,-17,\\,37)$ and $\\max _{n\\le 3000}|s_n|=319$ .",
"It is [154].",
"Problem 3.28 .",
"Is $(s_n)$ unbounded?",
"We finish with the generalization of Example 5 to the numbers $q^{\\pm ,l}(n)$ , $l\\in \\mathbb {N}$ .",
"These result from an almost complete cancellation because by the pentagonal identity, $\\left|q^{\\pm ,l}(n)\\right|\\le [q^n]\\bigg (\\sum _{n=0}^{\\infty }\\left|q^{\\pm ,1}(n)\\right|q^n\\bigg )^l\\le [q^n](1-q)^{-l}=\\binom{n+l-1}{l-1}\\;.$ So $q^{\\pm ,l}(n)=O(n^{l-1})$ , $m(n)=\\Theta _l(\\log (1+n))$ for this counting problem and effective computation of $q^{\\pm ,l}(n)$ means computation in $\\mathrm {poly}(\\log n)$ steps.",
"Besides the pentagonal identity for $l=1$ , another nice identity occurs for $l=3$ : $\\prod _{n\\ge 1}(1-q^n)^3=\\sum _{n\\ge 0}(-1)^{n}(2n+1)q^{n(n+1)/2}\\;,$ due to C. Jacobi (G. H. Hardy and E. M. Wright [68]).",
"Thus also $q^{\\pm ,3}(n)$ is a PIO function.",
"For $l=2$ , Example 5, we get $(q^{\\pm ,2}(n))_{n\\ge 0}=(1,\\,-2,\\,-1,\\,2,\\,1,\\,2,\\,-2,\\,0,\\,-2,\\,-2,\\,1,\\,0,\\,0,\\,2,\\,3,\\,-2,\\,2,\\,0,\\,\\dots )$ or $(q^{\\pm ,2}(n))_{n=58}^{75}=(0,\\,-2,\\,0,\\,-2,\\,0,\\,-2,\\,2,\\,0,\\,-4,\\,0,\\,0,\\,-2,\\,-1,\\,2,\\,0,\\,2,\\,0,\\,0)\\;,$ [154], not showing any clear pattern.",
"J. W. L. Glaisher [63] writes: “I had no hope that these coefficients would follow any simple law, as in the Eulerian or Jacobian series; for, if such a law existed, it could not fail to have been discovered long ago by observation.” Let us see how we advanced in 130 years.",
"In August 2018 the “links” section of the entry [154] (author N. J. A.",
"Sloane) lists these references: table of first 10000 values by S. Manyama, G. E. Andrews [7], M. Boylan [28], S. Finch [57], J. W. L. Glaisher [63], J. T. Joichi [82], V. G. Kač and D. H. Peterson [83], M. Koike [93], V. Kotěšovec [95], Y. Martin [101], T. Silverman [134], index to 74 sequences in [101] by M. Somos, M. Somos [136], and article Ramanujan Theta Functions in E. Weisstein [156].",
"J. W. L. Glaisher [63] did discover and prove a kind of simple pattern for this sequence, which we state in the elegant form given in [57] (the other references seem not relevant for computation of $q^{\\pm ,2}(n)$ ): $q^{\\pm ,2}(n)=G(12n+1)$ where $G:\\mathbb {N}\\rightarrow \\mathbb {Z}$ is a multiplicative function, which means that $G(ab)=G(a)G(b)$ whenever $a,b\\in \\mathbb {Z}$ are coprime numbers, defined on prime powers $p^r$ , $r\\in \\mathbb {N}$ , by $G(p^r)=\\left\\lbrace \\begin{array}{lll}1&\\dots &p\\equiv 7,\\,11\\ (\\mathrm {mod}\\ 12),\\,r\\equiv 0\\ (\\mathrm {mod}\\ 2),\\\\(-1)^{r/2}&\\dots &p\\equiv 5\\ (\\mathrm {mod}\\ 12),\\,r\\equiv 0\\ (\\mathrm {mod}\\ 2),\\\\r+1&\\dots &p\\equiv 1\\ (\\mathrm {mod}\\ 12),\\,(-3)^{(p-1)/4}\\equiv 1\\ (\\mathrm {mod}\\ p),\\\\(-1)^r(r+1)&\\dots &p\\equiv 1\\ (\\mathrm {mod}\\ 12),\\,(-3)^{(p-1)/4}\\equiv -1\\ (\\mathrm {mod}\\ p),\\\\0&\\dots &\\mbox{otherwise}\\;.\\end{array}\\right.$ This is a PIO formula for $G(n)=G(p^r)$ if $n$ is a known prime power.",
"But in general we do not know if $q^{\\pm ,2}(n)$ is a PIO function because we do not know how to effectively factorize numbers.",
"For example, $q^{\\pm ,2}(58)=G(697)=G(17)G(41)=0\\times \\dots =0$ and $q^{\\pm ,2}(59)=G(709)=(-1)^1(1+1)=-2$ because 709 is a prime that is 1 modulo 12 and $(-3)^{177}=-3^{2^7}3^{2^5}3^{2^4}3\\equiv -1$ modulo 709 as $3^{16}\\equiv 495$ , $3^{32}\\equiv 420$ and $3^{128}\\equiv 29$ .",
"For repeated parts, $\\sum _{n\\ge 0}p^{\\pm ,2}(n)q^n:=\\prod _{n\\ge 1}\\frac{1}{(1+q^n)^2}=\\prod _{n\\ge 1}(1+(-q)^{2n-1})^2\\;,$ we get $(p^{\\pm ,2}(n))_{n\\ge 0}=(1,\\,-2,\\,1,\\,-2,\\,4,\\,-4,\\,5,\\,-6,\\,9,\\,-12,\\,13,\\,-16,\\,21,\\,-26,\\,\\dots )\\;,$ [154], and see, like before, that $p^{\\pm ,2}(n)$ is also $(-1)^n$ times the number of partitions of $n$ into 2-sorted distinct odd numbers and that we have almost no cancellation.",
"For $l=24$ a shift of $q^{\\pm ,24}(n)$ gives the Ramanujan tau function $\\tau (n)$, $\\sum _{n\\ge 0}\\tau (n)q^n:=q\\prod _{n\\ge 1}(1-q^n)^{24}\\;.$ So $(\\tau (n))_{n\\ge 1}=(1,\\,-24,\\,252,\\,-1472,\\,4830,\\,-6048,\\,-16744,\\,84480,\\,-113643,\\,\\dots )\\;,$ [154].",
"Combinatorially, $\\tau (n)$ is the $(-1)^{\\Vert \\lambda \\Vert }$ -count of partitions $\\lambda $ of $n-1$ into parts in 24-sorted $\\mathbb {N}$ (not that this would really help for deriving properties of $\\tau (n)$ ).",
"Is $\\tau (n)$ a PIO function, can we compute it effectively, in $\\mathrm {poly}(\\log n)$ steps (as we noted above, $\\tau (n)=O(n^{23})$ )?",
"The Wikipedia article [155] on tau function is silent about this fundamental aspect but the simple and unsatisfactory answer is again that we do not know.",
"If we could effectively factorize numbers, we could besides decoding secret messages also compute $\\tau (n)$ effectively: (i) $\\tau (mn)=\\tau (m)\\tau (n)$ if $m$ and $n$ are coprime, (ii) $\\tau (p^{k+2})=\\tau (p)\\tau (p^{k+1})-p^{11}\\tau (p^k)$ for every prime number $p$ and every $k\\in \\mathbb {N}_0$ and (iii) $\\tau (p)$ can be computed in $\\mathrm {poly}(\\log p)$ steps for every prime $p$ .",
"The first two properties, conjectured by S. Ramanujan, were proved by L. J. Mordell [105] and the whole book [51], edited and mostly written by B. Edixhoven and J.-M. Couveignes, is devoted to exposition of an algorithm proving (iii).",
"Problem 3.29 .",
"Are, in the current state of knowledge, the known PIO functions $q^{\\pm ,l}(n)$ only those for $l=1$ and 3?",
"For which $l\\in \\mathbb {N}$ can one compute $q^{\\pm ,l}(n)$ in $\\mathrm {poly}(\\log n)$ steps with the help of an oracle that can factorize integers efficiently?",
"From the extensive literature on the numbers $q^{\\pm ,l}(n)$ we further mention only H. H. Chan, S. Cooper and P. Ch.",
"Toh [35], [36] (check the former for $l=26$ ), E. Clader, Y. Kemper and M. Wage [44] and J.-P. Serre [132].",
"Acknowledgments.",
"The OEIS database [154] was very helpful.",
"I thank I. Pak for valuable comments and references.",
"Martin Klazar Department of Applied Mathematics Charles University, Faculty of Mathematics and Physics Malostranské náměstí 25 11800 Praha Czechia klazar@kam.mff.cuni.cz"
]
] | 1808.08449 |
[
[
"Effect of double pulse irradiation on the morphology of a picosecond\n laser produced chromium plasma"
],
[
"Abstract We describe the measurements to control the morphology and hence the characteristics of a picosecond laser produced chromium plasma plume upon double-pulse (DP) irradiation compared to its single-pulse (SP) counterpart.",
"DP schemes are realized by employing two geometries wherein the inter-pulse delay ($\\tau_p$) in the collinear geometry and the spatial separation ($\\Delta x$) are the control parameters for schemes DP$_1$ and DP$_2$ respectively.",
"The aspect ratio (plume length/plume width) decreases upon increasing parameters such as pressure, delay between pulses and the energy of the second pulse in DP1 scheme.",
"Interestingly, the expansion conditions of the plume which occurs at higher pressures for SP scheme could be recreated in DP1 scheme for a lower pressure $\\sim$ 10$^{-6}$ Torr.",
"This could be potentially applied for immediate applications such as high harmonic generation and quality thin film production."
],
[
"Introduction",
"Laser produced plasmas (LPP) have wide employability in a variety of applications [1] such as high-order harmonic generation (HHG)[2], [3], attosecond pulse generation[4], [5], [6], EUV generation [7], [8], wake field acceleration [9], [10], material processing [11], pulsed laser deposition (PLD) [12], nanoparticle and nanocluster generation [13], [14] etc.",
"Despite the availability of different theories and expansion models [15], [16], [17], [18], [19], the transient nature [20] of the plasma plume makes it difficult to predict the expansion dynamics and plume composition completely.",
"Therefore, detailed experimental investigation of the plasma plumes by employing commonly used diagnostic techniques such as optical emission spectroscopy (OES) [22], [21], [19], optical time of flight (OTOF) [20], Langmuir probe [23], Thomson scattering [24], interferometry and shadowgraphy [25], [26] can be used to characterize plasmas for the applications mentioned above.",
"In addition to this, plume imaging using an intensified charge coupled device (ICCD) [19], [27] could help to unravel the morphology and expansion of the expanding plasma plume with a better temporal resolution to about a few nanoseconds.",
"Attaining specific plasma parameters such as number density and plasma temperature [28], [29] are crucial for most of the applications.",
"These parameters depend on various factors [30] such as the laser wavelength [31], pulse duration [32], fluence [33], spot size of irradiation [34], ambient gas [35] as well as the material properties [36].",
"In addition to this, irradiation schemes used for the generation of plasma also plays a major role in controlling the plume dynamics and its emission characteristics [29] to a great extent.",
"Double-pulse (DP) irradiation schemes in both orthogonal and collinear geometry has been investigated in the past to find its effect on line emission properties [37], [38].",
"However, these studies were largely carried out for nanosecond (ns) and femtosecond (fs) laser pulses [39], [40], [41], [42], [43] with implications to LIBS and material processing applications.",
"DP schemes have provided a better laser-plasma coupling depending on the nature of the preformed plasma for optimized conditions.",
"An increase in the line emission intensity from various species [44], [45], modest increase in the plasma temperature as well as modification of the plume morphology were reported when the DP scheme was employed with an optimal delay of $\\approx $ 500-1000 ps between pulses for fs irradiations [46].",
"Theoretical calculations predicted an increase in the plasma temperature when the inter-pulse delay between the two pulses is $\\le $ 200 ps [47].",
"Other independent investigations using DP schemes with different wavelengths[48], [49] indicated a drastic increase in the emission intensity and plasma temperatures (2-300 times enhancement) for an inter-pulse delay of a few microseconds to tens of microseconds (1 $\\mu $ s in [48] and 25$\\mu $ s in [49]).",
"The current experiment aims at the investigation of picosecond laser produced chromium plasma using fast imaging techniques.",
"Uncompressed ps pulse from the laser amplifier is widely used to generate plasmas in LPP based HHG experiments [50] while the compressed fs pulse drives harmonics.",
"Attosecond pulses of $\\sim $ 300 as has been previously reported [50] and it is possible to have phase-matched harmonic generation from Cr due to the co-existence of multiply charged species along with neutrals.",
"Hence Cr is an important candidate for future HHG experiments and investigation of ps laser produced Cr plasma is necessary to proceed further in this direction.",
"While experiments in the past focused on the effect of double pulse on ns and fs LPPs, inadequate information is available on the effect of DP using picosecond (ps) laser pulses.",
"Current experiment not only investigate the characteristic expansion to bridge the gap between DP using ns pulses and fs pulses, but also reveal the use of different irradiation schemes for different applications namely, pulsed laser deposition and high harmonic generation.",
"Collinear DP scheme (DP$_1$ ) is experimentally investigated by varying important parameters such as the inter-pulse delay between the first and second pulse($\\tau _p$ ), energies in the first ($E_1$ ) and second ($E_2$ ) pulses and the ambient/background pressure ($P_{bg}$ ) to investigate the distinction in the morphology of plasma plumes.",
"In addition to this, two plasmas are generated closer to each other and the modification in the plasma plume structure is investigated by varying the separation between the two plasmas ($\\Delta x$ , represented as DP$_2$ ).",
"The variation in velocity of fast and slow species are studied in detail along with the studies of modification in the aspect ratio (which is defined as the ratio of plume length/plume width) of the plume with respect to the variation in the above mentioned parameters."
],
[
"Experimental",
"Plasma is generated by focusing a $\\sim $ 60 ps laser pulses of maximum energy ($E_i$ ) 550 uJ from a multi-pass amplifier (Odin II, Quantronix, operated at 1kHz) to a spot size of $\\sim $ 80 $\\mu $ m using a 500 mm plano-convex lens onto a pure Cr (ACI Alloys Inc, USA) target in nitrogen ambient.",
"A fast, synchronized mechanical shutter positioned along the beam path is used to control the number of irradiations and the target is translated 200 $\\mu $ m in single-pulse (SP) and DP$_1$ schemes and $\\ge $ 300 $\\mu $ m for DP$_2$ after each irradiations to avoid ablation from the pit formed by previous irradiation.",
"A 1024$\\times $ 1024, Gen II ICCD (Pi:MAX 1024f, Princeton Instruments), with a temporal resolution of $\\sim $ 2 ns is used to measure the plume dynamics for different gate delays ($t_d$ ) with gate width ($t_w$ ) fixed at 10% of the $t_d$ .",
"Double-pulse scheme is implemented by using a combination of half wave plate and a polarizing cube beam splitter as shown in figure REF a.",
"The path labeled as A in figure REF a is the fixed arm and the path labeled as B is the delay arm, which allows $\\tau _p$ to be varied from 0 - 1000 ps.",
"Measurements are repeated to investigate the effect of $\\tau _p$ , energy in first ($E_1$ )and second pulses ($E_2$ ) and $P_{bg}$ for DP$_1$ and the distance between two spots ($\\Delta x$ ) in DP$_2$ and are then compared with its SP counterpart.",
"The irradiation schemes, SP, DP$_1$ and DP$_2$ used in the experiments are shown in figure REF b (i), REF b (ii) and REF b (iii) respectively.",
"A detailed description of the results on optimizing the LPP are discussed below, which has importance while employing LPP as the source medium for HHG process."
],
[
"Results and Discussion",
"In DP, $E_i$ is divided into $E_1$ and $E_2$ (such that $E_1$ + $E_2$ = $E_i$ ) to ablate the Cr target from a same spot for $DP_1$ and from different spots for $DP_2$ .",
"This approach not only benefited to investigate the influence of $\\tau _p$ , but also find out the effect of different energy ratios between pulses on same spot so as to optimize the characteristics of the plume.",
"Interestingly, this experiment explore the possibilities to use the pre-plasma (i.e, the plasma generated by the first pulse) as a dynamic background to confine the main plasma (i.e, the plasma generated by the second pulse) such that the main plasma expands similar to the case in SP scheme at high ambient pressures [29].",
"On the other hand for DP$_2$ , ablation from closely spaced spots on the target surface is measured for $\\tau _p$ = 0 at $P_{bg}$ = 10$^{-6}$ Torr.",
"Two plasmas generated in DP$_2$ interacts with each other on expansion leading to a completely distinct complex plume dynamics.",
"These characteristic expansion dynamics due to the different irradiation schemes, on varying the specified parameters are discussed in detail in the following sections and with their relevances."
],
[
"Plume expansion in SP scheme with irradiation energy $\\simeq $ 550 $\\mu $ J are recorded for various $t_d$ s for orthogonal polarizations to investigate the effect of H and V polarization on the plume dynamics at a $P_{bg} \\simeq $ 10$^{-6}$ Torr; the results of which are given in the figure REF .",
"The plume morphology is found to be independent of polarization (figure REF (a)).",
"Plume length ($l_{LPP}$ ) and plume width ($w_{LPP}$ ) are calculated using the 1/e$^2$ times the maximum intensity in the measured plume image cross sections.",
"These are important parameters when analyzing the plume morphology.",
"${l_{LPP}}/{w_{LPP}}$ , the aspect ratio (AR) of the plume defines the overall plasma structure and it is investigated in detail to understand plume morphology.",
"Figure REF (b) shows the variation of AR with respect to $t_d$ and is $\\ge $ 2 illustrating that the plume expands two times along the length (normal to the target) than its width (lateral expansion) displaying a cylindrical geometry to the plume expansion.",
"Further, the fast and slow species are found to expand with $V_f$ (velocity of fast species) $\\approx $ 42 km/s and $V_s$ (velocity of slow species) $\\approx $ 10 km/s and are shown in the table given in figure REF (c).",
"The plume dynamics, AR of the plume, $V_f$ and $V_s$ in the plume remain unchanged upon changing the polarization.",
"Evidently, the plume expansion dynamics is independent of the polarization of irradiation and choosing any arbitrary linear polarization doesn't influence the results; particularly when employing DP scheme with orthogonal polarizations.",
"Previous experiments on double-pulse scheme using fs laser pulses reported an optimum $\\tau _P$ $\\approx $ 500 ps -1000 ps [46] to reach a larger plasma temperature and better emission yields.",
"Therefore, experiments are performed initially to investigate the important differences between SP and DP$_1$ schemes in nitrogen ambient at $P_{bg}$ = 10$^{-6}$ Torr.",
"SP scheme is realized with energies $E_i/2$ and $E_i$ , whereas DP$_1$ scheme is realized with energies $E_1$ = $E_2$ = $E_i/2$ (i.e, equal energies in both paths) with $\\tau _p$ = 0 ps ($DP_{10}$ ) and 1000 ps ($DP_{11}$ ).",
"These measurements describe that the nature of plume expansion does not change for all irradiation schemes except for $DP_{11}$ as shown in Figure REF (a).",
"Figure: Plume dynamics at P bg P_{bg} = 10 -6 ^{-6} Torr for SP and DP 1 _1 schemes for a maximum irradiation energy of 550 μ\\mu J.",
"(a) 2 cm ×\\times 1.5 cm images of the plasma plume acquired using an ICCD for t d t_d as mentioned at the left side of the figure and t w t_w = 10% of the t d t_d.",
"E i /2E_i/2, E i E_i, DP 10 DP_{10} and DP 11 DP_{11} refers to SP at half of the maximum energy, SP at the maximum energy, DP 1 _1 with τ p \\tau _p = 0 ps and τ p \\tau _p = 1000 ps respectively.",
"(b) Variation in AR with t d t_d for the different irradiation schemes presented in (a).",
"The plume is more spherical for DP 11 DP_{11} when compared to all other schemes.",
"(c) Variation in the average V f V_f and V s V_s in the plume as a function of the different irradiation schemes.While plumes in SP with $E_i/2$ and $E_i$ and $DP_{10}$ are found to have similar expansion features resembling a cylindrical plasma [32] as in the case of a fs LPP plume, the $DP_{11}$ show a prominent lateral expansion resembling a ns LPP plume.",
"To explain this further, it can be assumed that the energy of irradiation on the target by the second pulse may be $\\le $ $E_i/2$ as it passes through a preformed spatio-temporally expanded plasma (for a $\\tau _P$ = 1000 ps), which might absorb/scatter a fraction of the energy depending on the cross sections of the process.",
"This in turn let the plasma to expand laterally and can be studied by comparing the variation of AR (figure REF (b)) with respect to other irradiation schemes.",
"While AR approaches to 1 for $DP_{11}$ illustrating a spherical morphology (which could in turn improve the plume homogeneity), the AR is $\\ge $ 1.5 with a maximum of 3 at $t_d$ = 90 ns displays a larger axial expansion for all other cases.",
"$V_f$ and $V_s$ (as shown in figure REF (c)) are calculated from the emission intensity profiles obtained from the imaging data.",
"While the velocities of the slow species does not vary much in $E_i/2$ , $E_i$ and $DP_{10}$ , it decreases considerably for $DP_{11}$ .",
"Plume in $E_i/2$ reaches shorter distance when compared to $E_i$ and $DP_{10}$ as expected since plume expansion depend on irradiation energy [51] for a given LPP.",
"$V_f$ is relatively larger and is $\\approx $ 32 km/s, 42 km/s, 41 km/s and 25 km/s respectively for $E_i/2$ , $E_i$ , $DP_{10}$ and $DP_{11}$ .",
"The slow species move with $V_s$ $\\approx $ 10 km/s for $E_i/2$ , $E_i$ and $DP_{10}$ , whereas it is $\\approx $ 5 km/s for $DP_{11}$ and therefore the effect of $\\tau _p$ on the plume dynamics for DP$_1$ needs further investigation; which will be discussed in the following sections.",
"Figure: Variation in aspect ratio of the plasma plume as a function of gate delay for (a) different inter-pulse delays, τ p \\tau _p and (b) for varying energy in the first pulse, E 1 E_1.",
"It can be seen that the plume is more spherical in double pulse scheme with an inter-pulse delay ≥\\ge 400 ps and for cases where energy in the first pulse is less than or equal to the energy in the second pulse.",
"Variation in the average velocity of the fast and slow components of the plume for DP 1 _1 scheme with (c) different inter-pulse delays and (d) for different energy in the first pulse."
],
[
"Effect of $\\tau _p$ on plume hydrodynamics is investigated to find the morphological changes in the plume as well as the changes in the velocity of fast and slow species.",
"Measurements are carried out for various $t_d$ 's with $\\tau _p$ varying from 0 ps to 1000 ps for $E_1$ = $E_2$ = $E_i/2$ and the results are given in figure REF (a).",
"The plume morphology changes from a cylindrical expansion for $\\tau _p$ = 0 ps to a spherical expansion for $\\tau _p$ = 1000 ps.",
"Plume is found to exhibit a relatively low lateral expansion with AR $\\approx $ 2 (for $t_d$ = 30 ns) and $\\approx $ 3 (for $t_d$ = 90 ns) for $\\tau _p$ = 0 ps changes to a plume with relatively large lateral expansion with AR $\\approx $ 1 (for $t_d$ = 30 ns) and $\\approx $ 1.5 (for $t_d$ = 90 ns) for $\\tau _p$ = 1000 ps; indicating a factor of two change in the plasma length when $\\tau _p$ is varied from 0 ps to 1000ps.",
"Interestingly, lateral expansion of the plume is evident after $\\tau _p$ = 400 ps, depicting an enhancement in the interaction of the generated second plasma produced with the pre-formed plasma, thereby increasing the possibilities for confinement.",
"It is clear from the figure REF (a) that the the plume length increases and reaches a maximum quickly within 100 ns after ablation and a gradual decrease for longer $t_d$ s. Velocities of fast and slow components in the plume were calculated from the above mentioned measurements and is given in figure REF (c).",
"$V_f$ is found to decrease from $\\approx $ 40 km/s to $\\approx $ 25 km/s and $V_s$ decreases from $\\approx $ 10 km/s to $\\approx $ 5 km/s respectively on increasing $\\tau _p$ from 0 ps to 1000 ps.",
"This complements the argument on plume confinement and it could be inferred from these measurements that a longer $\\tau _p$ can be chosen to have greater lateral expansion whenever required.",
"Further, measurements are carried out to investigate the effect of energy $E_1$ in the $DP_1$ .",
"The energies used for irradiation was varied in such a way that the total energy of irradiation; i.e, $E_1$ + $E_2$ = $E_i$ = 550 $\\mu $ J.",
"Since the previous measurements concluded that longer $\\tau _p$ yields a better lateral expansion, the following set of measurements are carried out with $\\tau _p$ = 1000 ps.",
"These inter-delayed pulses generate two plasmas one after another depending on the number density ($n_c$ ) of the plasma formed by $E_1$ irradiation.",
"The plume shape in these measurements are found to have considerable variation when $E_1$ is increased from 100 $\\mu $ J to 550 $\\mu $ J and the results of the variation of AR with respect to $t_d$ for different $E_1$ is given in figure REF (b).",
"The plume has the better confinement leading to a larger lateral expansion of the plume and hence a spherical morphology for $E_1$ = 200 $\\mu $ J; whereas a transition from spherical to cylindrical geometry is observed when $E_1 \\ge $ 300 $\\mu $ J.",
"In this case, the plasma plume generated by the first pulse expands spatio-temporally and the plasma produced by the second pulse experiences a dynamic plasma surrounding.",
"Though the generation of the second plasma and its expansion into the pre-generated plasma depend considerably on the number density of the preformed plasma, we could record the generation of a second plasma in all our experimental conditions.",
"For eg.",
"when $E_1$ = 100 $\\mu $ J, a better ablation by the second pulse happened because of the modified heated target surface and a poor shielding of the second pulse to reach the target surface by the the pre-formed plasma due to the insufficient number density.",
"For $E_1\\le E_2$ , plasma formed by the second pulse finds a highly dynamic and transient ambient created by the plume formed from the first pulse.",
"Complex process that involves the interaction of two plasmas may therefore result in a plume with larger radial expansion when compared to the SP counter part having similar total energy of irradiation.",
"On the other hand, when $E_1 \\ge $ 300 $\\mu $ J, a fraction of $E_2$ may be absorbed by the plume via inverse bremsstrahlung processes and thereby results in generating a plume with less lateral expansion when compared to the DP$_1$ case wherein $E_1 \\le E_2$ .",
"AR is closer to 1 for plasmas generated when $E_1 \\le $ 300 $\\mu $ J and it increases with an increase in $E_1$ .",
"Further, $V_f$ and $V_s$ were found to get reduced in DP$_1$ scheme when compared to SP with the minimum velocity of both species occurring at $E_1$ = 200 $\\mu $ J; which is an indication of collisional interactions/confinement happening in the plasma plume."
],
[
"Effect of ambient pressure",
"Ambient pressure plays an important role in the dynamics of the plasma plume [35] and hence plasma can devised for various applications by optimizing $P_{bg}$ .",
"Therefore, the experiments were repeated at different pressures for SP and $DP_1$ schemes.",
"While SP scheme was carried out using the maximum irradiation energy available ($E_{i}$ = 550 $\\mu $ J), $DP_1$ scheme was carried out with $E_1$ = $E_2$ = $E_i/2$ and $\\tau _p$ = 1000 ps for ambient pressures ranging from 10$^{-6}$ Torr to 10$^{1}$ Torr.",
"Figure: Dynamics of the plasma plume for SP and DP irradiation schemes with respect to variation in the ambient pressure are shown here.",
"1.5 cm ×\\times 2 cm images of the plasma plume acquired using an ICCD for t d t_d as mentioned in the figure legends and t w t_w = 10% of the t d t_d for (a) SP scheme and (b) DP scheme.",
"Graph showing the variation in aspect ratio with respect to t d t_d for (c) SP scheme and (d) DP 1 _1 scheme.",
"(e) Variation in the average velocity of the fast and slow components of the plume in SP and DP 1 _1 schemes.Figure REF (a) and REF (b) represents the plasma plume expansion for SP and $DP_1$ schemes.",
"Different expansion features are observed for SP and DP$_1$ scheme while $P_{bg}$ is varied.",
"In SP, while plume expands adiabatically reaching farther distances with cylindrical plume shape (i.e.",
"with lesser lateral expansion) at lower pressures from 10$^{-6}$ Torr to 10$^{-2}$ Torr, a sudden change to the plume structure is recorded at 10$^{-1}$ Torr with a noticeable lateral expansion.",
"This pressure regime is considered to be really interesting and rich with many complex process has been reported in several observations [21].",
"For higher pressures, plume expansion is significantly resisted by the surrounding atmosphere leading to an enhanced lateral expansion imparting a better spherical shape for pressures 10$^0$ Torr and 10$^1$ Torr.",
"The plume length is found to have smallest values among the current measurements, indicating a plume confinement at higher pressures for SP schemes.",
"AR of the plume (see figure REF (c)) compliments this claim since the value approaches 1 when $P_{bg}$ reaches 10$^0$ Torr from 10$^{-6}$ Torr.",
"Interestingly at 10$^1$ Torr, plume confinement is so dominant that the AR drops to values $<$ 1 indicating a larger plume width.",
"While slow species in the plume is present at all pressures, the fast species are visible only at pressures $\\le $ 10$^{-2}$ Torr.",
"$V_f$ decreases with the increase of pressure, whereas for the slow species, $V_s$ remains unchanged up to 10$^{-1}$ Torr and then decreases on a further increase in pressure.",
"$DP_1$ experiments are repeated for pressures from 10$^{-6}$ Torr to 10$^2$ Torr displays an entirely different structure, but a similar trend for all $t_d$ 's upon comparison with SP; even though the values of AR are lower.",
"Figure REF (b) shows the plume expansion dynamics and figure REF (d) presents the variation in AR with respect to $t_d$ for DP$_1$ scheme.",
"Effect of second pulse causes the plasma to expand more laterally than the axial expansion.",
"Plume width is slightly more for DP at pressures from 10$^{-6}$ Torr to 10$^{-2}$ Torr and hence the shape of the plume found to be more spherical (with reduced AR values), when compared to the axially confined plume in SP case.",
"Fast and slow components moves with reduced velocities (see figure REF (e)) whereas AR is reduced when compared to the SP scheme.",
"This confirms that the DP1 scheme not only confines the plume length, but also causes a lateral expansion, thereby creating a plasma that exhibit a structure similar to a ns LPP.",
"At 10$^{-1}$ Torr pressure, it could be seen that the plume structure is more spherical at all $t_d$ 's and the plume size is still bigger and brighter for $t_d \\ge $ 300 ns.",
"Though a bigger and confined plasma plume at $t_d \\ge $ 300 ns is observed for SP case, the plume is much brighter and has a better spherical shape in $DP_1$ scheme.",
"The plume is confined such that the plume width is $\\ge $ the plume length for 10$^{0}$ Torr and 10$^{1}$ Torr pressures, leading to an AR $< $ 1.",
"However, at later stages of expansion, it could be seen that the AR approaches 1, leading to a spherical plume at these $t_d$ 's.",
"Variation of the AR and velocities of fast and slow species in the $DP_1$ scheme are given in figure REF (e).",
"Figure: Dynamics of the plasma plume for double pulse scheme (DP 2 _2) as shown in figure b (iii) at 10 -6 ^{-6} Torr for E irr E_{irr} = 560 μ\\mu J.",
"(a) 1.5 cm ×\\times 2 cm images of the plasma plume acquired using an ICCD for a time delay (t d t_d) as mentioned in the figure legends and integration time (t w t_w) is 10% of the t d t_d.",
"The distance between the laser irradiation spots (Δx\\Delta x) is given on the top of each column.",
"(b) Variation in aspect ratio with respect to t d t_d for different Δx\\Delta x.",
"(c) Variation in the average velocity of the fast and slow components of the plume with respect to Δx\\Delta x."
],
[
"Double-pulse at different spatial points",
"Apart from the DP$_1$ scheme wherein parameters like $\\tau _p$ , variation in $E_1$ and $E_2$ and $P_{bg}$ were optimized to tailor the plume morphology, another double pulse scheme, $DP_2$ was employed to study the variation in the plume morphology.",
"In this scheme, plasmas are generated on the target using two identical pulses of energies $E_1$ = $E_2$ = $E_i/2$ with $\\tau _p$ = 0 ps for a $P_{bg}$ = 10$^{-6}$ Torr, such that both pulses reach the target surface at the same time and simultaneously creates two closely separated plasmas with a separation = $\\Delta x$ .",
"Figure REF (a) shows the expansion features of the plasma plumes when $\\Delta x$ is varied from 0 $\\mu $ m to 600 $\\mu $ m. The plasma is found to be highly directional and cylindrical in shape for all cases and directionality increases on increasing $\\Delta x$ .",
"From careful detailed analysis it is found that the plume width and plume length increases at constant rate causing the AR to remain constant soon after it reaches a maximum value and it persists for longer $t_d$ s. It is also found that when $\\Delta x$ is increased from 0 $\\mu $ m to 600 $\\mu $ m, the maximum value of the AR decreases as shown in Figure REF (b).",
"While the $V_f$ shows an intermediate value between SP and $DP_1$ schemes, $V_s$ displays larger velocities in comparison with the other two schemes.",
"More specifically, $V_f$ decreases when $\\Delta x$ is increased from 0 $\\mu $ m to 200 $\\mu $ m and thereafter remains almost constant for all other $\\Delta x$ 's.",
"$V_s$ , however decreases by increasing $\\Delta x$ and it is found to have longer life time in the plume, thus forming major component of the plume in all cases.",
"To understand the process further, the plume interaction and formation of the directed plume in $DP_2$ case, the expansion of plasma plume when $\\Delta x$ = 600 $\\mu $ m is analyzed in detail and is given in figure REF .",
"Figure: Time-gated images of the plasma plume evolution captured for a t d t_d as mentioned in each picture and t w t_w = 10% of t d t_d.",
"The images are captured for E 1 E_1 = E 2 E_2 = E i /2E_i/2, τ p \\tau _p = 0 ps, P bg P_{bg} = 10 -6 ^{-6} and Δx\\Delta x = 600 μ\\mu m. Dimension of image is 1 cm ×\\times 0.8 cm.When the two pulses hit the target surface, two distinct plumes are formed as in the figure REF , (with legend 30 ns) and they propagate rather independently for a given time and space.",
"On expansion these two plasmas can interact along their edges as in figure REF ,for 50 ns and 70 ns, leading to the creation of a stagnation layer between the two plasmas which initiate the process of collisional interaction between them[52].",
"As a result of the interaction between fast and slow components at different times, the interaction becomes more complex leading plume to move together, thereby having a longer persistence of slow components.",
"Due to the interaction along the edges of the plasma for $t_d \\ge $ 50 ns, the AR has a larger value and these interactions imparts a highly directional and cylindrical nature for the plume expansion.",
"These kind of plasmas could be suitably used for the production of quality thin films as plume dynamics displays a better cylindrical expanding plume shape, that attains a constant plume length after $\\approx $ 200 ns with better homogeneity in its cross section perpendicular to the axial expansion direction."
],
[
"Conclusion",
"Plasmas generated by laser pulses of 60 ps duration at 800 nm, are studied under different irradiation schemes.",
"Single-pulse (SP) and double-pulse (DP) schemes are used to understand their effects on the plasma plume morphology, which in-turn would help devising them for applications as in high-harmonic generation which is considered here.",
"Double-pulse scheme has been carried out in two different ways: back to back ( or collinear) irradiation of the pulses on the same spatial point ($DP_1$ ) and irradiation of the two pulses at slightly different spatial points ($DP_2$ ).",
"Plume morphology in all these cases are different and are found to differ significantly by varying the inter-pulse delay ($\\tau _p$ ), energy in the first and second pulse ($E_1$ and $E_2$ ) and background pressure ($P_{bg}$ ) for DP$_1$ and with respect to the spatial separation ($\\Delta x$ ) for DP$_2$ .",
"DP$_1$ helps in obtaining a more spherical plasma, with aspect ratio close to 1 when compared to SP and DP$_2$ scheme that generates a cylindrical plasma where aspect ratio is greater than 1 for $\\tau _p \\ge $ 400 ps.",
"Also, a spherical plume with aspect ratio greater than 1 is observed When the energy in the first path is less than or equal to the energy in the second path.",
"For SP schemes wherein the background pressures are varied, plume confinement is observed on an increase in the background gas with an aspect ratio closer to 1 for higher pressures.",
"Furthermore, the effect of double pulse with an increase in background yields to a more spherical expansion when compared to their respective SP counterparts.",
"Therefore, the plume expansion conditions as in the SP case with higher $P_{bg}$ can be reproduced in $DP_1$ scheme at lower pressures by wisely choosing $\\tau _p$ , $E_1$ and $E_2$ .",
"This would help in improving the temperature and number densities of the plasmas at relatively lower pressures and also ensure a more homogeneous plasma that would support phase matching/ quasi-phase matching conditions for efficient high-order harmonic generation from plasmas.",
"Whereas plume morphology is entirely different in DP$_2$ when compared to that of SP and DP$_1$ .",
"A more directed and cylindrical plume with aspect ratio $>$ 1 is always observed, which could be suitably utilized for producing quality thin films by laser ablation/plasma-assisted thin film deposition."
],
[
"Acknowledgements",
"This project is funded by the Australian Research Council Linkage project grant No.",
"LP140100813.",
"Kavya H. Rao was supported through an “Australian Government Research Training Program Scholarship\" and N. Smijesh was supported by the Griffith University Postdoctoral Fellowship Scheme."
]
] | 1808.08678 |
[
[
"Big bounce and closed universe from spin and torsion"
],
[
"Abstract We analyze the dynamics of a homogeneous and isotropic universe in the Einstein$--$Cartan theory of gravity.",
"The coupling between the spin and torsion prevents gravitational singularities and replaces the Big Bang with a nonsingular big bounce, at which the universe transitions from contraction to expansion.",
"We show that a closed universe exists only when the product of the scale factor and temperature is higher than a particular threshold, contrary to a flat universe and an open universe, which are not restricted.",
"During inflation, this product must increase to another threshold, so that the universe can reach dark-energy acceleration."
],
[
"Cosmology in Einstein–Cartan (EC) gravity",
"The simplest mechanism generating a nonsingular big bounce and inflation, involving only one unknown parameter and no hypothetical fields, arises in the EC theory of gravity [1].",
"EC is the simplest and most natural theory of gravity with torsion, with the Lagrangian density for the gravitational field proportional to the Ricci scalar, as in general relativity.",
"The conservation law for the total (orbital plus spin) angular momentum of fermions in curved spacetime, consistent with the Dirac equation, requires that the antisymmetric part of the affine connection, the torsion tensor [2], is not constrained to zero [3], [4].",
"Instead, torsion is determined by the field equations obtained from varying the action with respect to the torsion tensor [5], [6], [7], [8], [9], [10], [11], [12], [13], [14].",
"In EC, the spin of fermions is the source of torsion.",
"The multipole expansion [15] of the conservation law for the spin tensor in EC gives a spin tensor that describes fermionic matter as a spin fluid (ideal fluid with spin) [16].",
"Once the torsion is integrated out, EC reduces to general relativity with an effective spin fluid as a matter source [10], [11], [12], [13], [14].",
"The effective energy density and pressure of a spin fluid are given by $\\tilde{\\epsilon }=\\epsilon -\\alpha n_\\textrm {f}^2,\\quad \\tilde{p}=p-\\alpha n_\\textrm {f}^2,$ where $\\epsilon $ and $p$ are the thermodynamic energy density and pressure, $n_\\textrm {f}$ is the number density of fermions, and $\\alpha =\\kappa (\\hbar c)^2/32$ [17].",
"The negative corrections from the spin-torsion coupling in (REF ) generate gravitational repulsion, which prevents the formation of gravitational singularities and replaces the Big Bang with a nonsingular bounce, at which the universe transitions from contraction to expansion [18], [19], [20].",
"These corrections lead to a violation of the strong energy condition by the spin fluid when $\\epsilon +3p-4\\alpha n_\\textrm {f}^2$ drops below 0, thus evading the singularity theorems [17].",
"Accordingly, this violation could be thought of as the cause of the bounce.",
"The dynamics of the EC universe filled with a spin fluid (REF ) has been studied in [21], [22], [23], [24], [25], with a parity-violating extension in [26], and with torsion coupled to the spinor field in [27].",
"The expansion of the closed universe with torsion and quantum particle production shortly after a bounce is almost exponential for a finite period of time, explaining inflation [1].",
"Depending on the particle production rate, the universe may undergo several bounces until it produces enough matter to reach a size where the cosmological constant starts cosmic acceleration.",
"This expansion also predicts the cosmic microwave background radiation parameters that are consistent with the Planck 2015 observations [28], [29], as was shown in [30].",
"The avoidance of singularities can also occur in cosmological models based on Riemann–Cartan geometries without spin density: Poincaré gauge theories with quadratic terms in curvature and torsion [31], [32], scalar-tensor theories with torsion [33], [34], and higher-dimensional geometries with torsion [35].",
"Therefore, it seems to be a generic feature of the Riemann–Cartan spacetime, rather than a particular feature of EC.",
"We consider EC because it has other interesting consequences.",
"The spin-torsion coupling modifies the Dirac equation, adding a term that is cubic in spinor fields [36].",
"As a result, fermions must be spatially extended [37], [38], which could eliminate infinities arising in Feynman diagrams involving fermion loops.",
"In the presence of torsion, the four-momentum operator components do not commute and thus the integration in the momentum space in Feynman diagrams must be replaced with the summation over the discrete momentum eigenvalues.",
"The resulting sums are finite: torsion naturally regularizes ultraviolet-divergent integrals in quantum electrodynamics [39], [40].",
"Torsion may also explain the matter-antimatter asymmetry and dark matter [41], and the cosmological constant [42].",
"The analysis in [1] considered a closed, homogeneous, and isotropic universe in EC.",
"However, the calculations of the maximum temperature and the minimum scale factor at a bounce neglected the factor $k=1$ in the Friedmann equations (which is justified during and after inflation but not at a bounce before inflation), de facto considering a flat universe.",
"In this article, we refine these calculations by taking $k$ into account and analyzing the expansion of the universe for all three cases: $k=1$ (closed universe), $k=0$ (flat universe), and $k=-1$ (open universe).",
"We discover that a closed universe exists only when the product of the scale factor and temperature is higher than a particular threshold, whereas open and flat universes are not restricted by such a condition.",
"Accordingly, a closed universe forms in a region of space within a trapped null surface [1] when this threshold is reached."
],
[
"Dynamics of scale factor and temperature",
"If we assume that the universe is homogeneous and isotropic, then it is described by the Friedmann–Lemaître–Robertson–Walker metric in the isotropic spherical coordinates [43]: $ds^2=c^2 dt^2-\\frac{a^2(t)}{(1+kr^2/4)^2}(dr^2+r^2 d\\theta ^2+r^2\\sin ^2\\theta \\,d\\phi ^2),$ where $a(t)$ is the scale factor as a function of the cosmic time $t$ .",
"The Einstein field equations for this metric become the Friedmann equations [43]: $\\frac{\\dot{a}^2}{c^2}+k=\\frac{1}{3}\\kappa \\epsilon a^2$ and $\\frac{\\dot{a}^2+2a\\ddot{a}}{c^2}+k=-\\kappa pa^2,$ where a dot denotes the derivative with respect to $t$ and $\\kappa =8\\pi G/c^4$ .",
"Multiplying the first Friedmann equation by $a$ and differentiating over time, and subtracting from it the second Friedmann equation multiplied by $\\dot{a}$ gives an equation that has the form of the first law of thermodynamics for an adiabatic universe: $\\frac{d}{dt}(\\epsilon a^3)+p\\frac{d}{dt}(a^3)=0.$ For EC, the Friedmann equations have the same form but the energy density and pressure are replaced by $\\tilde{\\epsilon }$ and $\\tilde{p}$ [21], [22], [24], [25]: $\\frac{\\dot{a}^2}{c^2}+k=\\frac{1}{3}\\kappa (\\epsilon -\\alpha n_\\textrm {f}^2)a^2$ and $\\frac{d}{dt}[(\\epsilon -\\alpha n_\\textrm {f}^2)a^3]+(p-\\alpha n_\\textrm {f}^2)\\frac{d}{dt}(a^3)=0.$ The spin fluid in the early universe is formed by an ultrarelativistic matter in kinetic equilibrium, for which $\\epsilon =h_\\star T^4$ , $p=\\epsilon /3$ , and $n_\\textrm {f}=h_{n\\textrm {f}}T^3$ , where $T$ is the temperature of the universe, $h_\\star =(\\pi ^2/30)(g_\\textrm {b}+(7/8)g_\\textrm {f})k_\\textrm {B}^4/(\\hbar c)^3$ , and $h_{n\\textrm {f}}=(\\zeta (3)/\\pi ^2)(3/4)g_\\textrm {f}k_\\textrm {B}^3/(\\hbar c)^3$ [44].",
"The quantities $g_\\textrm {b}$ and $g_\\textrm {f}$ are the numbers of spin states for all elementary bosons and fermions, respectively.",
"For standard-model particles, $g_\\textrm {b}=29$ and $g_\\textrm {f}=90$ .",
"In the presence of spin and torsion, the first Friedmann equation is therefore [1], [45] $\\frac{{\\dot{a}}^2}{c^2}+k=\\frac{1}{3}\\kappa (h_\\star T^4-\\alpha h_{n\\textrm {f}}^2 T^6)a^2.$ The first law of thermodynamics (REF ) gives [1] $\\Bigl (\\frac{\\dot{a}}{a}+\\frac{\\dot{T}}{T}\\Bigr )\\Bigl (1-\\frac{3\\alpha h_{n\\textrm {f}}^2}{2h_\\star }T^2\\Bigr )=0,$ which yields $\\frac{\\dot{a}}{a}+\\frac{\\dot{T}}{T}=0.$ Remarkably, the last equation is the same as that for the relativistic universe without spin and torsion."
],
[
"Analysis of solutions for a closed universe",
"Let us consider a closed relativistic universe.",
"We define nondimensional quantities: $& & x=\\frac{T}{T_\\textrm {cr}}, \\\\& & y=\\frac{a}{a_\\textrm {cr}}, \\\\& & \\tau =\\frac{ct}{a_\\textrm {cr}},$ where $T_\\textrm {cr}=\\Bigl (\\frac{2h_\\star }{3\\alpha h_{n\\textrm {f}}^2}\\Bigr )^{1/2}=9.410\\times 10^{31}\\,\\textrm {K}$ and $a_\\textrm {cr}=\\frac{9\\hbar c}{8\\sqrt{2}}\\Bigl (\\frac{\\alpha h_{n\\textrm {f}}^4}{h_\\star ^3}\\Bigr )^{1/2}=3.701\\times 10^{-36}\\,\\textrm {m}.$ Henceforth, we will use the dot to denote the derivative with respect to the new time coordinate $\\tau $ .",
"Equation (REF ) can be written as $\\dot{y}^2+1=(3x^4-2x^6)y^2.$ Equation (REF ) can be integrated to $xy=C,$ where $C$ is a positive constant (because the scale factor and temperature are greater than 0).",
"Substitution of this relation into Equation (REF ) gives $\\dot{y}^2+1=\\frac{3C^4}{y^2} - \\frac{2C^6}{y^4}.$ Since the left-hand side of this equation is positive, $y$ cannot reach zero because the right-hand side of this equation would have to become negative.",
"Consequently, a cosmological singularity is never produced for any value of $C$ .",
"The big bounce and big crunch of a closed universe are turning points (there is no expansion or contraction at these points), so therefore they are determined by a condition $\\dot{y}= 0.$ Using this relation, Equation (REF ) can be resolved for a quadratic, in terms of $y$ and $C$ .",
"The resulting quadratic equation for $y^2$ is $y^4-3y^2 C^4+2C^6 =0.$ The solutions of this equation are: $y^2_\\pm = \\frac{3C^4 \\pm \\sqrt{9C^8-8C^6}}{2}.$ At the big bounce, $y=y_-$ , and at the big crunch, $y=y_+$ .",
"These turning points of a closed universe exist if the expression under the square root in Equation (REF ) is positive or zero.",
"In order for that expression to remain positive or zero, $C$ must be greater than or equal to $\\sqrt{8/9}$ .",
"Consequently, an inequality $C \\ge \\sqrt{8/9},$ equivalent to $aT \\ge \\sqrt{8/9}a_\\textrm {cr}T_\\textrm {cr},$ is a necessary condition for creating a closed universe in a region of space with the local values of $a$ and $T$ .",
"If $C = \\sqrt{8/9}$ , then the turning points coincide, $y_-=y_+$ , and the universe is stationary (no expansion or contraction) with the constant value of the scale factor of $y=\\sqrt{32/27}$ .",
"Since the values of $y$ and $C$ are now both known, the corresponding constant value of $x$ can be found using Equation (REF ) to be $\\sqrt{3}/2$ .",
"Such a stationary universe has a constant scale factor $a=\\sqrt{32/27}a_\\textrm {cr}$ and temperature $T=(\\sqrt{3}/2)T_\\textrm {cr}$ .",
"If $C$ is greater than $\\sqrt{8/9}$ , then the universe has two turning points and both the big bounce and big crunch occur.",
"The value of $\\dot{y}^2$ is nonnegative in the range from $y=y_-$ to $y=y_+$ , so therefore the universe oscillates between $y=y_-$ and $y=y_+$ .",
"If $C>\\sqrt{8/9}$ , then Equation (REF ) can be rearranged to give $y^2_\\pm = 3C^4\\Bigl [\\frac{1\\pm \\sqrt{1-\\frac{8}{9C^2}}}{2}\\Bigr ].$ In the limit $C\\gg 1$ , using the formula $(1-x)^n \\approx 1+nx$ for $|x|\\ll 1$ gives $y^2 = 3C^4\\Bigl [\\frac{1\\pm (1-\\frac{4}{9C^2})}{2}\\Bigr ].$ Accordingly, $y_-\\approx \\sqrt{2/3}C$ and $y_+\\approx \\sqrt{3}C^2$ .",
"The absolute minimum value for $y$ among all possible values of $C$ can be determined from a condition $dy^2_-/dC=0$ , giving $C = 1.$ For this value of $C$ , the nondimensionalized minimum scale factor and the corresponding maximum temperature are $x = 1, \\quad y_\\textrm {min} = 1.$ Accordingly, the constant $a_\\textrm {cr}$ is equal to the least possible value of the scale factor of a closed universe in EC.",
"A closed universe with spin and torsion is nonsingular ($y \\ge 1$ and thus $y>0$ ).",
"The squared values of $y_\\textrm {min}$ , $y_\\textrm {max}$ , $x_\\textrm {min}$ , and $x_\\textrm {max}$ for different values of $C$ are shown in Table REF .",
"The greatest possible value of the temperature in a closed universe in EC is $\\sqrt{3/2}T_\\textrm {cr}$ .",
"When $C$ is much greater than 1, the universe can expand by a factor of $9C^2/2$ and its temperature can decrease by the same factor, reaching the value at which the transition from the radiation (relativistic) domination to the matter (nonrelativistic) domination occurs.",
"Table: The minima and maxima of the nondimensionalized temperature xx and scale factor yy for different values of the integration constant CC in a closed universe.",
"The domain of CC is [8/9,∞)[\\sqrt{8/9},\\infty ).Equation (REF ) can be solved analytically.",
"We substitute $y=[E-F\\cos (2\\theta )]^{1/2},\\quad \\dot{y}=[E-F\\cos (2\\theta )]^{-1/2}F\\sin (2\\theta )\\dot{\\theta },$ where $E$ and $F$ are positive constants such that $E>F$ , obtaining $[F^2\\sin ^2(2\\theta )\\dot{\\theta }^2][E-F\\cos (2\\theta )]+[E-F\\cos (2\\theta )]^2-3C^4[E-F\\cos (2\\theta )]+2C^6=0.$ Putting $2E=3C^4$ and $E^2-F^2=2C^6$ reduces this equation into $\\dot{\\theta }^2[E-F\\cos (2\\theta )]=1,$ which integrates to $\\tau =\\int _0^\\theta [E-F\\cos (2\\theta )]^{1/2}d\\theta =(E-F)^{1/2}\\int _0^\\theta [1+\\xi ^2\\sin ^2\\theta ]^{1/2}d\\theta ,$ where $\\xi ^2=2F/(E-F)$ , and $\\theta =0$ at $t=0$ .",
"The integral in this equation is the elliptic integral of the second kind with an imaginary $\\xi $ .",
"Therefore, the time dependence $y(\\tau )$ of the scale factor is given in the parametric form by the first equation in (REF ) and Equation (REF ) with $E=3C^4/2,\\quad F=\\sqrt{9C^8/4-2C^6}.$ The parameter $\\theta $ runs from 0 (bounce) through $\\pi /2$ (crunch) to $\\pi $ (next bounce) for one cycle.",
"The minimum and maximum scale factors are given by $y_\\pm =\\sqrt{E\\pm F}$ , in agreement with (REF ).",
"The value of $F$ is real, giving the condition (REF ).",
"Figures REF , REF , and REF show the nondimensionalized scale factor $y$ as a function of the nondimensionalized time $\\tau $ near a nonsingular, smooth bounce with $C=1$ , $C=10$ , and $C=100$ , respectively.",
"Figure REF shows the nondimensionalized scale factor $y$ as a function of the nondimensionalized time $\\tau $ for one cycle with $C=1,10,100$ using a logarithmic scale.",
"Figure: Nondimensionalized scale factor yy as a function of the nondimensionalized time τ\\tau near a bounce with C=1C=1.The time τ=0\\tau =0 is set at the bounce.Figure: Nondimensionalized scale factor yy as a function of the nondimensionalized time τ\\tau near a bounce with C=10C=10.The time τ=0\\tau =0 is set at the bounce.Figure: Nondimensionalized scale factor yy as a function of the nondimensionalized time τ\\tau near a bounce with C=100C=100.The time τ=0\\tau =0 is set at the bounce.Figure: Nondimensionalized scale factor yy as a function of the nondimensionalized time τ\\tau for one cycle with C=1,10,100C=1,10,100 using a logarithmic scale.The results of this section show that a necessary condition for the creation of an expanding universe in a given volume of space is $C>\\sqrt{8/9}$ for all points in that volume.",
"For all allowed values of $C$ , a closed universe in nonsingular."
],
[
"Analysis of solutions for a flat and an open universe",
"For a flat relativistic universe, Equation (REF ) becomes $\\dot{y}^2=\\frac{3C^4}{y^2} - \\frac{2C^6}{y^4}.$ The corresponding quadratic equation for $y^2$ at the turning points is $3y^2-2C^2 =0.$ The physical solution of this equation is $y = \\sqrt{2/3}C,$ which is the minimum value of the nondimensionalized scale factor.",
"This value is equal to the nondimensionalized scale factor at a bounce for the case of a closed universe in the limit $C\\gg 1$ .",
"Consequently, a flat universe in EC is also nonsingular, because $C$ is positive.",
"Since a flat universe has only one turning point (at a bounce), it expands to infinity.",
"Contrary to a closed universe, there are no further restrictions on the value of $C$ .",
"For an open relativistic universe, Equation (REF ) becomes $\\dot{y}^2-1=\\frac{3C^4}{y^2} - \\frac{2C^6}{y^4}.$ The corresponding quadratic equation for $y^2$ at the turning points is $y^4+3y^2 C^4-2C^6 =0.$ The physical solution of this equation is $y^2 = \\frac{-3C^4+\\sqrt{9C^8+8C^6}}{2},$ with $y>0$ .",
"This solution is the minimum value of the nondimensionalized scale factor.",
"As for a flat universe, an open universe is also nonsingular and has only one turning point (at a bounce), so it expands to infinity.",
"There are no further restrictions on the value of $C$ ."
],
[
"Analysis with cosmological constant",
"A flat and an open universe expand to infinity.",
"Without further considerations, a closed expanding universe has two turning points (provided that $C>\\sqrt{8/9}$ ) and therefore it reaches the maximum value of the scale factor, after which it contracts.",
"To avoid the big crunch and the subsequent contraction, another term in the first Friedmann equation is needed that can cause the acceleration of a late universe.",
"The simplest term is given by a cosmological constant, which enters the first Friedmann equation (for relativistic matter) according to $\\dot{y}^2 + 1 = \\frac{3C^4}{y^2}- \\frac{2C^6}{y^4} + \\lambda y^2,$ where $\\lambda >0$ is the nondimensionalized cosmological constant: $\\lambda =\\frac{1}{3}\\Lambda a_\\textrm {cr}^2 = 5.0\\times 10^{-124}.$ This small value results from the small cosmological constant $\\Lambda =1.1\\times 10^{-52}\\,\\textrm {m}^{-2}$ .",
"For a late universe, where the values of $y$ are large, the $y^{-4}$ component of this equation can be ignored, reducing this equation to $\\dot{y}^2 + 1 = \\frac{3C^4}{y^2} + \\lambda y^2.$ Putting $\\dot{y} = 0$ to find the turning points, the equation can then be rewritten as the quadratic given below: $\\lambda z^2 - z + 3C^4 = 0,$ where $z=y^2$ .",
"Solving the quadratic shows that in order for no turning point to exist (in a late universe), $\\lambda $ must be greater than $1/(12C^4)$ .",
"This condition can also be written as $C > (12\\lambda )^{-1/4}.$ A closed universe expands to infinity if the cosmological constant is sufficiently high.",
"Such expansion is asymptotically exponential: $\\dot{y}^2\\approx \\lambda y^2$ in a late universe gives $y\\sim \\exp (\\sqrt{\\lambda }\\tau )$ .",
"According to the results of Section and the condition (REF ), a closed universe forms in a given region of space when the local value of $C$ is equal to $\\sqrt{8/9}$ .",
"When $C>\\sqrt{8/9}$ , the universe expands.",
"To expand to infinity, the universe must have a mechanism to increase the value of $C$ from $\\sqrt{8/9}$ to $(12\\lambda )^{-1/4}$ .",
"If $(12\\lambda )^{-1/4}<\\sqrt{8/9}$ , then the universe expands to infinity regardless.",
"A natural and physical mechanism for the growth of $C$ is provided by quantum particle-pair production in strong gravitational fields [46], [47], [48], [49], [50], [51], [52].",
"This mechanism also generates a brief period of exponential expansion of a very early universe [1], thus naturally deriving inflation [53], [54], [55].",
"This analysis would be valid for a relativistic universe.",
"However, a relativistic universe transitions from the radiation-dominated era to the matter-dominated era and becomes nonrelativistic.",
"Equation (REF ) becomes $\\dot{y}^2 + 1 = \\frac{B}{y} + \\lambda y^2,$ where $B$ is a positive constant.",
"This transition occurs when $\\frac{3C^4}{y^2}=\\frac{B}{y}$ and at temperature $T_\\textrm {eq}=8.8\\times 10^3$ K. Using (REF ) and (REF ), we obtain $B=3C^3 x_\\textrm {eq}^3,$ where $x_\\textrm {eq}=\\frac{T_\\textrm {eq}}{T_\\textrm {cr}}=9.4\\times 10^{-29}.$ A nonrelativistic universe expands to infinity if the cosmological constant is sufficiently high.",
"The turning points of Equation (REF ) are given by a cubic equation $y^3-\\frac{y}{\\lambda }+\\frac{B}{\\lambda }=0.$ This equation has no real positive solutions if [9] $B>\\frac{2}{3\\sqrt{3\\lambda }}.$ Consequently, (REF ) gives the following condition for the absence of turning points in a late universe: $C>\\Bigl (\\frac{2}{9\\sqrt{3\\lambda }}\\Bigr )^{1/3}\\frac{1}{x_\\textrm {eq}}=1.9\\times 10^{48}.$ If this condition is satisfied, the universe expands to infinity.",
"Numerical analysis of generic gravitational collapse in general relativity shows that each spatial point in the interior of a black hole locally evolves toward the singularity as an independent, spatially homogeneous, closed universe [56], [57].",
"In EC, this evolution does not reach a singularity, but instead undergoes a nonsingular bounce after which such a universe expands [1].",
"Accordingly, if our Universe is closed, its contraction before the big bounce could correspond to gravitational collapse of matter inside a newly formed black hole existing in another universe [58], [59], [60], [61], [62], [63], [64], [65], [66], [67].",
"In this scenario, the formation of our Universe corresponds to the moment when the quantity $C$ , representing the product of the scale factor and temperature, begins to satisfy the inequality (REF ) in a given volume of space in the black hole.",
"During inflation, this quantity increases because of quantum particle-pair production in strong gravitational fields, and must reach the threshold (REF ) so that the Universe could start the observed current acceleration.",
"If this threshold is not reached, the closed universe contracts to another bounce and starts another cycle of expansion.",
"The last bounce before reaching the threshold can be regarded as the Big Bang of the Universe.",
"This work was funded by the University Research Scholar program at the University of New Haven."
]
] | 1808.08327 |
[
[
"Strong and Weak Optimizations in Classical and Quantum Models of\n Stochastic Processes"
],
[
"Abstract Among the predictive hidden Markov models that describe a given stochastic process, the {\\epsilon}-machine is strongly minimal in that it minimizes every R\\'enyi-based memory measure.",
"Quantum models can be smaller still.",
"In contrast with the {\\epsilon}-machine's unique role in the classical setting, however, among the class of processes described by pure-state hidden quantum Markov models, there are those for which there does not exist any strongly minimal model.",
"Quantum memory optimization then depends on which memory measure best matches a given problem circumstance."
],
[
"Introduction",
"When studying classical stochastic processes, we often seek models and representations of the underlying system that allow us to simulate and predict future dynamics.",
"If the process is memoryful, then models that generate it or predict its future actions must also have memory.",
"Memory, however, comes at some resource cost; both in a practical sense—consider, for instance, the substantial resources required to generate predictions of weather and climate [1], [2]—and in a theoretical sense—seen in analyzing thermodynamic systems such as information engines [3].",
"It is therefore beneficial to seek out a process' minimally resource-intensive implementations.",
"Predicting and simulating classical processes, and monitoring the memory required, led to a generalization of statistical mechanics called computational mechanics [4], [5], [6], [7].",
"To date computational mechanics focused on discrete stochastic processes.",
"These are probability measures $\\mathbb {P}\\left(\\dots x_{-1} x_0 x_1 \\dots \\right)$ over strings of symbols taking values in a finite alphabet $\\mathcal {A}$ .",
"The minimal information processing required to predict the sequence is represented by a type of hidden Markov model called the .",
"The statistical complexity $$ —the memory rate for to simultaneously generate many copies of a process—is a key quantity and a proposed invariant for measuring the process' structural complexity.",
"When simulating classical processes, quantum systems can be constructed that have smaller memory requirements than the [8], [9].",
"The $q$ -machine is a particular implementation of quantum simulation that has shown advantage in memory rate over a wide range of processes; often the advantage is unbounded [10], [11], [12], [13].",
"For quantum models, the minimal memory rate $C_q$ has been determined in cases such as the Ising model [11] and the Perturbed Coin Process [14], where the $q$ -machine attains the minimum rate.",
"And so, though a given $q$ -machine's $C_q$ can be readily calculated [15], in many cases the absolute minimum $C_q$ is not known.",
"Properly accounting for memory requires an appropriate formalism for resources themselves.",
"The field of resource theory has recently emerged in quantum information theory as a toolkit for addressing resource consumption in the contexts of entanglement, thermodynamics, and numerous other quantum and classical resources [16].",
"Its fundamental challenge is to determine when one system, or resource, can be converted to another using a predetermined set of free operations.",
"Resource theory is closely allied with two other areas of mathematics, namely majorization and lattice theory.",
"Figure REF depicts their relationships.",
"Figure: Triumvirate of resource theory, majorization, and lattice theory.On the one hand, majorization is a preorder relation $\\succsim $ on positive vectors (typically probability distributions) computed by evaluating a set of inequalities [17].",
"If the majorization relations hold between two vectors, then one can be converted to the other using a certain class of operations.",
"Majorization is used in some resource theories to numerically test for convertibility between two resources [18], [19], [20].",
"Lattice theory, on the other hand, concerns partially ordered sets and their suprema and infima, if they exist [21].",
"Functions that quantify the practical uses of a resource are monotonic with respect to the partial orders induced by convertibility and majorization.",
"Optimization of practical measures of memory is then related to the problem of finding the extrema of the lattice.",
"Majorization and resource convertibility are both relations that generate lattice-like structures on the set of systems.",
"Here, we examine the memory costs of classical and quantum models of stochastic processes via majorization.",
"Using lattice-theoretic intuition, we then define the concept of strong optimization, which occurs when a particular model simultaneously optimizes all measures of memory via its extremal position in the lattice.",
"We show that among classical predictive models, the is strongly minimal.",
"Following this, we show that the is strongly maximal to a subset of quantum models but that no strongly minimal quantum model exists in some circumstances.",
"These results constitute initial steps to a resource theory of memoryful information processing."
],
[
"Majorization and optimization",
"The majorization of positive vectors provides a qualitative description of how concentrated the quantity of a vector is over its components.",
"For ease of comparison, consider vectors $\\mathbf {p}=(p_i)$ , $i\\in \\lbrace 1,\\dots ,n\\rbrace $ , whose components all sum to some constant value, which we take to be unity: i=1n pi = 1 , and are nonnegative: $p_i \\ge 0$ .",
"For our purposes, we interpret these vectors as probability distributions.",
"Our introduction to majorization here follows Ref.",
"[17].",
"The historical definition of majorization is also the most intuitive, starting with the concept of a transfer operation.",
"[Transfer operation] A transfer operation $\\mathbf {T}$ on a vector $\\mathbf {p}=(p_i)$ selects two indices $i,j\\in \\lbrace 1,\\dots ,n\\rbrace $ , such that $p_i>p_j$ , and transforms the components in the following way: (Tp)i = pi- (Tp)j = pj+ , where $0 < \\epsilon < p_i-p_j$ , while leaving all other components equal; $(Tp)_k = p_k$ for $k\\ne i,j$ .",
"Intuitively, these operations reduce concentration, since they act to equalize the disparity between two components, in such a way as to not create greater disparity in the opposite direction.",
"This is the principle of transfers.",
"Suppose now that we have two vectors $\\mathbf {p}=(p_i)$ and $\\mathbf {q}=(q_i)$ and that there exists a sequence of transfer operations $\\mathbf {T}_1,\\dots ,\\mathbf {T}_m$ such that $\\mathbf {T}_m \\circ \\cdots \\circ \\mathbf {T}_1 \\mathbf {p} = \\mathbf {q}$ .",
"We will say that $\\mathbf {p}$ majorizes $\\mathbf {q}$ ; denoted $\\mathbf {p}\\succsim \\mathbf {q}$ .",
"The relation $\\succsim $ defines a preorder on the set of distributions, as it is reflexive and transitive but not necessarily antisymmetric.",
"There are, in fact, a number of equivalent criteria for majorization.",
"We list three relevant to our development in the following composite theorem.",
"[Majorization Criteria] Given two vectors $\\mathbf {p}=(p_i)$ and $\\mathbf {q}=(q_i)$ with the same total sum, let their orderings be given by the permuted vectors $\\mathbf {p}^{\\downarrow } = (p^{\\downarrow }_i)$ and $\\mathbf {q}^{\\downarrow } = (q^{\\downarrow }_i)$ such that $p^{\\downarrow }_1 > p^{\\downarrow }_2 > \\dots > p^{\\downarrow }_n$ and the same for $\\mathbf {q}^{\\downarrow }$ .",
"Then the following statements are equivalent: Hardy-Littlewood-Pólya: For every $1\\le k\\le n$ , i=1k pi i=1k qi ; Principle of transfers: $\\mathbf {p}$ can be transformed to $\\mathbf {q}$ via a sequence of transfer operations; Schur-Horn: There exists a unitary matrix $\\mathbf {U}=(U_{ij})$ such that $\\mathbf {q} = \\mathbf {Dp}$ , where $\\mathbf {D}=\\left(\\left|U_{ij}\\right|^2\\right)$ , a uni-stochastic matrix.",
"The Hardly-Littlewood-Pólya criterion provides a visual representation of majorization in the form of the Lorenz curve.",
"For a distribution $\\mathbf {p}=(p_i)$ , the Lorenz curve is simply the function $\\beta _\\mathbf {p}(k) = \\sum _{i=1}^k p_i^{\\downarrow }$ .",
"See Fig.",
"REF .",
"We can see that $\\mathbf {p} \\succsim \\mathbf {q}$ so long as the area under $\\beta _\\mathbf {q}$ is completely contained in the area under $\\beta _\\mathbf {p}$ .",
"Figure: 𝐩\\mathbf {p} and 𝐪\\mathbf {q} arecomparable and the first majorizes the second: 𝐩≿𝐪\\mathbf {p} \\succsim \\mathbf {q}.",
"(Here we chose 𝐩=(34,18,18,0,0)\\mathbf {p} =({3}{4},{1}{8},{1}{8},0,0) and 𝐪=(25,15,15,110,110)\\mathbf {q} =({2}{5},{1}{5},{1}{5},{1}{10},{1}{10}).Tick marks indicate kinks in the Lorenz curve.",
")The Lorenz curve can be understood via a social analogy, by examining rhetoric of the form “The top $x$ % of the population owns $y$ % of the wealth”.",
"Let $y$ be a function of $x$ in this statement, and we have the Lorenz curve of a wealth distribution.",
"(Majorization, in fact, has its origins in the study of income inequality.)",
"If neither $\\mathbf {p}$ nor $\\mathbf {q}$ majorizes the other, they are incomparable.",
"(See Fig.",
"REF .)",
"As noted, majorization is a preorder, since there may exist distinct $\\mathbf {p}$ and $\\mathbf {q}$ such that $\\mathbf {p}\\succsim \\mathbf {q}$ and $\\mathbf {q} \\succsim \\mathbf {p}$ .",
"This defines an equivalence relation $\\sim $ between distributions.",
"Every preorder can be converted into a partial order by considering equivalence classes $[\\mathbf {p}]_\\sim $ .",
"Figure: 𝐩\\mathbf {p} and 𝐪\\mathbf {q} are incomparable.",
"(Here we chose 𝐩=(35,110,110,110,110)\\mathbf {p} = ({3}{5},{1}{10},{1}{10},{1}{10},{1}{10}) and 𝐪=(13,13,13,0,0)\\mathbf {q} = ({1}{3},{1}{3},{1}{3},0,0).",
")If majorization, in fact, captures important physical properties of the distributions, we should expect that these properties may be quantified.",
"The class of monotones that quantify the preorder of majorization are called Schur-convex and Schur-concave functions.",
"[Schur-convex (-concave) functions] A function $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}$ is called Schur-convex (-concave) if $\\mathbf {p}\\succsim \\mathbf {q}$ implies $f(\\mathbf {p}) \\ge f(\\mathbf {q})$ ($f(\\mathbf {p}) \\le f(\\mathbf {q})$ ).",
"An important class of Schur-concave functions consists of the Rényi entropies: H(p) = 11-2(i=1n pi) .",
"In particular, the three limits H(p) = 1H(p) = -i=1n pi2 pi , H0(p) = 0H(p) = 2 |{1i n: pi > 0}| , and H(p) = H(p) = -2 1i npi , —Shannon entropy, topological entropy, and min-entropy, respectively—describe important practical features of a distribution.",
"In order, they describe the asymptotic rate at which the outcomes can be accurately conveyed, the single-shot resource requirements for the same task, and the probability of error in guessing the outcome if no information is conveyed at all (or, alternatively, the single-shot rate at which randomness can be extracted from the distribution) [22], [23].",
"As such, they play a significant role in communication and memory storage.",
"The example of two incomparable distributions $\\mathbf {p}$ and $\\mathbf {q}$ can be analyzed in terms of the Rényi entropies if we plot $H_\\alpha \\left(\\mathbf {p}\\right)$ and $H_\\alpha \\left(\\mathbf {q}\\right)$ as a function of $\\alpha $ , as in Fig.",
"REF .",
"Figure: Rényi entropies of the two incomparable distributions𝐩\\mathbf {p} and 𝐪\\mathbf {q} from Fig..The central question we explore in the following is applying majorization to determine when it is possible to simultaneously optimize all entropy monotones or, alternatively, to determine if each monotone has a unique solution.",
"This leads to defining strong maxima and strong minima.",
"[Strong maximum (minimum)] Let $S$ be a set of probability distributions.",
"If a distribution $\\mathbf {p}\\in S$ satisfies $\\mathbf {p}\\precsim \\mathbf {q}$ ($\\mathbf {p}\\succsim \\mathbf {q}$ ), for all $\\mathbf {q}\\in S$ , then $\\mathbf {p}$ is a strong maximum (minimum) of the set $S$ .",
"The extrema names derive from the fact that the strong maximum maximizes the Rényi entropies and the strong minimum minimizes them.",
"One can extend the definitions to the case where $\\mathbf {p} \\notin S$ , but is the least-upper-bound such that any other $\\mathbf {p}^{\\prime }$ satisfying $\\mathbf {p}^{\\prime }\\precsim \\mathbf {q}$ must obey $\\mathbf {p}^{\\prime }\\precsim \\mathbf {p}$ .",
"This case would be called a strong supremum (or in the other direction a strong infimum).",
"However, these constructions may not be unique as $\\succsim $ is a preorder and not a partial order.",
"However, if we sort by equivalence class, then the strongly maximal (minimal) class is unique if it exists.",
"In lattice-theoretic terms, the strong maximum is essentially the lattice-theoretic notion of a meet and the strong minimum is a join [21].",
"One example of strong minimization is found in quantum mechanics.",
"Let $\\rho $ be a state and $X$ be a maximal diagonalizing measurement.",
"For a given measurement $Y$ , let $\\left.\\rho \\right|_Y$ be the corresponding probability distribution that comes from measuring $\\rho $ with $Y$ .",
"Then $\\left.\\rho \\right|_X \\succsim \\left.\\rho \\right|_Y$ for all maximal projective measurements $Y$ .",
"(This follows from the unitary matrices that transform from the basis of $X$ to that of $Y$ , and the Schur-Horn lemma.)",
"Another, recent example is found in Ref.",
"[24], where the set $B_{\\epsilon }\\left(\\mathbf {p}\\right)$ of all distributions $\\epsilon $ -close to $\\mathbf {p}$ under the total variation distance $\\delta $ is considered: B(p) = {q:(p,q)} .",
"This set has a strong minimum, called the steepest distribution $\\overline{\\mathbf {p}}{}_\\epsilon $ , and a strong maximum, called the flattest distribution $\\underline{\\mathbf {p}}{}_\\epsilon $ .",
"When a strong minimum or maximum does not exist, we refer to the individual extrema of the various monotones as weak extrema.",
"We close with a technical note on how to compare distributions over different numbers of events.",
"There are generally two standards for such comparisons that depend on application.",
"In the resource theory of informational nonequilibrium [20], one compares distributions over different numbers of events by “squashing” their Lorenz curves so that the $x$ -axis ranges from 0 to 1.",
"Under this comparison, the distribution $\\mathbf {p}_3 = (1,0,0)$ has more informational nonequilibrium than $\\mathbf {p}_2=(1,0)$ .",
"In the following, however, we adopt the standard of simply extending the smaller distribution by adding events of zero probability.",
"In this, $\\mathbf {p}_3$ and $\\mathbf {p}_2$ are considered equivalent.",
"This choice is driven by our interest in the Rényi entropic costs and not in the overall nonequilibrium.",
"(The latter is more naturally measured by Rényi negentropies $\\bar{H}_\\alpha \\left(\\mathbf {p}\\right) = \\log n -{H}_\\alpha \\left(\\mathbf {p}\\right)$ , where $n$ is the number of events.)"
],
[
"Strong minimality of the ",
"The general task we set ourselves is simulating classical processes.",
"[Bi-infinite process] A bi-infinite process over an alphabet $\\mathcal {A}$ is a probability measure $\\mathbb {P}(\\overleftrightarrow{x})$ over the set of all bi-infinite strings $\\overleftrightarrow{x}= \\overleftarrow{x}_t \\overrightarrow{x}_t \\in \\mathcal {A}^{\\infty }$ , where the past $\\overleftarrow{x}_t = \\dots x_{-1+t}x_t$ and the future $\\overrightarrow{x}_t = x_t x_{t+1} \\dots $ are constructed by concatenating elements of $\\mathcal {A}$ .",
"Though defined over bi-infinite strings, the measure gives probabilities for seeing finite-length words $w=x_1\\dots x_\\ell $ , defined as $\\mathbb {P}(w) =\\mathbb {P}\\left(\\lbrace \\overleftrightarrow{x} = \\overleftarrow{x}_t w\\overrightarrow{x}_{t+\\ell } : t \\in \\mathbb {N}\\rbrace \\right)$ .",
"This can be taken as an alternate definition of the process measure.",
"Here, we focus on finite predictive models.",
"[Finite predictive model] A finite predictive model is a triplet $\\mathbf {M}=(\\mathcal {R},\\mathcal {A},\\lbrace \\mathbf {T}^{(x)}:x\\in \\mathcal {A}\\rbrace )$ of hidden states $\\mathcal {R}$ , an output alphabet $\\mathcal {A}$ , and nonnegative transition matrices $\\mathbf {T}^{(x)} = \\left(T^{(x)}_{\\rho \\rho ^{\\prime }} \\right)$ with $x\\in \\mathcal {A}$ and $\\rho ,\\rho ^{\\prime }\\in \\mathcal {R}$ , satisfying the properties: Irreducibility: $\\mathbf {T} = \\sum _{x\\in \\mathcal {A}} \\mathbf {T}^{(x)}$ is stochastic and irreducible.",
"Unifilarity: $T^{(x)}_{\\rho \\rho ^{\\prime }} = \\mathbb {P}\\left(x|\\rho \\right)\\delta _{\\rho ^{\\prime },f(\\rho ,x)}$ for some conditional probability $\\mathbb {P}\\left(x|\\rho \\right)$ and deterministic function $f$ .",
"A finite predictive model is a type of hidden Markov model [25], whose dynamic is to transition between states at each timestep while emitting a symbol with probabilities determined by the transition matrices $\\mathbf {T}^{(x)}$ .",
"Unifilarity ensures that, given the model state $\\sigma \\in \\mathcal {R}$ and symbol $x \\in \\mathcal {A}$ , the next state $\\sigma ^\\prime \\in \\mathcal {R}$ is unique.",
"Given a finite predictive model $\\mathbf {M}$ , the state transition matrix $\\mathbf {T}$ has a single left-eigenstate ${\\pi }$ of eigenvalue 1, by the Perron-Frobenius theorem, satisfying ${\\pi }^\\top \\mathbf {T} ={\\pi }^\\top $ .",
"We call this state distribution the stationary state.",
"Using it, we define the process $\\mathbb {P}_\\mathbf {M}$ generated by $\\mathbf {M}$ as $\\mathbb {P}_\\mathbf {M}(w) = {\\pi }^\\top \\mathbf {T}^{(x_1)}\\cdots \\mathbf {T}^{(x_\\ell )} {1}$ , where $w=x_1\\dots x_\\ell $ and 1 is the vector with all 1's for its components.",
"$\\mathbb {P}_\\mathbf {M}$ describes a stationary process.",
"If we let ${\\delta }_\\rho $ represent the state-distribution that assigns the state $\\rho \\in \\mathcal {R}$ probability 1, then $\\mathbb {P}_{\\mathbf {M},\\rho }(w) = {\\delta }_{\\rho }^\\top \\mathbf {T}^{(x_1)}\\cdots \\mathbf {T}^{(x_\\ell )} {1}$ is the probability of seeing word $w$ after starting in state $\\rho $ .",
"Given a model with stationary distribution ${\\pi }$ , we define the model's Rényi memory as $H_\\alpha \\left(\\mathbf {M}\\right) =H_\\alpha \\left({\\pi }\\right)$ .",
"This includes the topological memory $H_0\\left(\\mathbf {M}\\right)$ , the statistical memory $H\\left(\\mathbf {M}\\right)=H_1\\left(\\mathbf {M}\\right)$ , and the min-memory $H_\\infty \\left(\\mathbf {M}\\right)$ .",
"Given a process $\\mathbf {P}$ , we define the Rényi complexity as the minimal memory $^{(\\alpha )} =\\min _{\\mathbf {M}}H_\\alpha \\left(\\mathbf {M}\\right)$ over all models that generate $\\mathbf {P}$ [4].",
"These include the topological complexity $C^{(0)}_\\mu $ , the statistical complexity $=^{(1)}$ , and the min-complexity $C^{(\\infty )}_\\mu $ .",
"Among the class of finite predictive models, a particularly distinguished member is the [4]: [Generator ] A generator is a finite predictive model $\\mathbf {M}=(\\mathcal {S},\\mathcal {A},\\lbrace \\mathbf {T}^{(x)}:x\\in \\mathcal {A}\\rbrace )$ such that for each pair of distinct states $\\rho , \\rho ^{\\prime } \\in \\mathcal {S}$ , there exists a word $w$ such that $\\mathbb {P}_{\\mathbf {M},\\rho }(w)\\ne \\mathbb {P}_{\\mathbf {M},\\rho ^{\\prime }}(w)$ .",
"In other words, a generator must be irreducible, unifilar, and its states must be probabilistically distinct, so that no pair of distinct states predict the same future.",
"An important result of computational mechanics is that the generator is unique with respect to the process it generates [26].",
"This is a combined consequence of the equivalence of the generator definition with another, called the history , which is provably unique [6].",
"That is, given an $\\mathbf {M}$ , there is no other that generates $\\mathbb {P}_{\\mathbf {M}}$ .",
"A further important result is that the minimizes both the statistical complexity $$ and the topological complexity $^{(0)}$ .",
"To fix intuitions, consider now several examples of models and their processes.",
"First, consider the Biased Coin Process, a memoryless process in which, at each time step, a coin is flipped with probability $p$ of generating a 1 and probability $1-p$ of generating a 0.",
"Figure REF displays three models for it.",
"Model (a) is the process' , and models (b) and (c) are each 2-state alternative finite predictive models.",
"Notice that in both models (b) and (c), the two states generate equivalent futures.",
"Figure: (a) for a coin flipped with bias pp.",
"(b) Alternate representationwith pp to be in state BB and 1-p1-p to be in state CC.",
"(c) Alternaterepresentation with biases pp to stay in current state and 1-p1-p to switchstates.Continuing, Fig.",
"REF displays two alternative models of the Even-Odd Process.",
"This process produces sequences formed by concatenating strings of an odd number of 1s to strings of an even number of 0s.",
"We see in (a) the process' .",
"In (b), we see an alternative finite predictive model, and notice that its states $E$ and $F$ predict the same futures, and so are not probabilistically distinct.",
"We notice that they both play the role of state $C$ in the , in terms of the futures they predict.",
"Figure: (a) for Even-Odd Process.",
"(b) Refinement of the Even-Odd Process, where the 's state CC has been split into states EE and FF.We can compare these examples using Lorenz curves of the state distributions, as shown in Fig.",
"REF .",
"Here, recall, we adopted the convention of comparing two distributions over a different number of states by extending the smaller system to include zero-probability states.",
"We notice that the state distribution always majorizes the state distribution of the alternative machines.",
"Figure: (a) Lorenz curves for Fig.",
"(a)'s and Fig.",
"(b)'s alternative predictor ofthe Biased Coin Process.",
"(b) Same comparison for the Even-Odd ProcessFig.",
"(a) andalternative predictor Fig.",
"(b).The key to formalizing this observation is the following lemma.",
"[State Merging] Let $\\mathbf {M}=(\\mathcal {R},\\mathcal {A},\\lbrace \\mathbf {T}^{(x)}:x\\in \\mathcal {A}\\rbrace )$ be a finite predictive model that is not an .",
"Then the machine created by merging its probabilistically equivalent states is the of the process $\\mathbb {P}_\\mathbf {M}$ generated by $\\mathbf {M}$ .",
"Let $\\sim $ be the equivalence relation $\\rho \\sim \\rho ^{\\prime }$ if $\\mathbb {P}_{\\mathbf {M},\\rho }(w) =\\mathbb {P}_{\\mathbf {M},\\rho ^{\\prime }}(w)$ for all $w$ .",
"Let $\\mathcal {S}$ consist of the set of equivalence classes $[\\rho ]_{\\sim }$ generated by this relation.",
"For a given class $\\sigma \\in \\mathcal {S}$ , consider the transition probabilities associated with each $\\rho \\in \\sigma $ .",
"For each $x\\in \\mathcal {A}$ such that $\\mathbb {P}\\left(x|\\rho \\right) > 0$ , there is a outcome state $\\rho _x =f(x,\\rho )$ .",
"Comparing with another state in the same class $\\rho ^{\\prime }\\in \\sigma $ , we have the set of outcome states $\\rho _x^{\\prime } = f(x,\\rho ^{\\prime })$ .",
"For the future predictions of both states $\\rho $ and $\\rho ^{\\prime }$ to be equivalent, they must also be equivalent after seeing the symbol $x$ .",
"That is, $\\mathbb {P}_{\\mathbf {M},\\rho }(w) =\\mathbb {P}_{\\mathbf {M},\\rho ^{\\prime }}(w)$ for all $w$ also implies $\\mathbb {P}_{\\mathbf {M},\\rho }(xw)=\\mathbb {P}_{\\mathbf {M},\\rho ^{\\prime }}(xw)$ for all $w$ .",
"But $\\mathbb {P}_{\\mathbf {M},\\rho }(xw) = \\mathbb {P}_{\\mathbf {M},\\rho _x}(w)$ , and so we have $\\rho _x\\sim \\rho _x^{\\prime }$ for all $x\\in \\mathcal {A}$ .",
"The upshot of these considerations is that we can define a consistent and unifilar transition dynamic $\\lbrace \\widetilde{\\mathbf {T}}{}^{(x)}:x\\in \\mathcal {A}\\rbrace $ on $\\mathcal {S}$ given by the matrices $\\widetilde{T}{}^{(x)}_{\\sigma \\sigma ^{\\prime }} =\\widetilde{T}{}^{(x)}_{\\rho \\rho ^{\\prime }}$ for any $\\rho \\in \\sigma $ and $\\rho ^{\\prime }\\in \\sigma ^{\\prime }$ .",
"It inherits unifilarity from the original model $\\mathbf {M}$ as well as irreducibility.",
"It has probabilistically distinct states because we have already merged all of the probabilistically equivalent states.",
"Therefore, the resulting machine $\\mathbf {M}_{\\mathcal {S}} =(\\mathcal {S},\\mathcal {A},\\lbrace \\widetilde{\\mathbf {T}}{}^{(x)}:x\\in \\mathcal {A}\\rbrace )$ is the of the process $\\mathbb {P}_\\mathbf {M}$ generated by $\\mathbf {M}$ .",
"The state-merging procedure here is an adaptation of the Hopcroft algorithm for minimization of deterministic (nonprobabilistic) finite automata, which is itself an implementation of the Nerode equivalence relation, [27].",
"It has been applied previously to analyze synchronization in [28].",
"Using Lemma , we can prove the main result of this section: [Strong Minimality of ] Let $\\mathbf {M}_{\\mathcal {S}} =(\\mathcal {S},\\mathcal {A},\\lbrace \\widetilde{\\mathbf {T}}{}^{(x)}:x\\in \\mathcal {A}\\rbrace )$ be the of process $\\mathbb {P}$ and $\\mathbf {M}_{\\mathcal {R}}=(\\mathcal {R},\\mathcal {A},\\lbrace \\mathbf {T}^{(x)}:x\\in \\mathcal {A}\\rbrace )$ be any other finite generating machine.",
"Let the stationary distributions be ${\\pi }_\\mathcal {S}=\\left(\\pi _{\\mathcal {S},\\sigma }\\right)$ and ${\\pi }_\\mathcal {R}=\\left(\\pi _{\\mathcal {R},\\rho }\\right)$ , respectively.",
"Then ${\\pi }_\\mathcal {S}\\succsim {\\pi }_\\mathcal {R}$ .",
"By Lemma , the states of the $\\mathbf {M}_{\\mathcal {S}}$ are formed by merging equivalence classes $\\sigma =[\\rho ]$ on the finite predictive model $\\mathbf {M}_{\\mathcal {R}}$ .",
"Since the machines are otherwise equivalent, the stationary probability $\\pi _{\\mathcal {S},\\sigma }$ is simply the sum of the stationary probabilities for each $\\rho \\subseteq \\sigma $ , given by $\\pi _{\\mathcal {R},\\rho }$ .",
"That is: S, = R R, .",
"One can then construct ${\\pi }_\\mathcal {R}$ from ${\\pi }_\\mathcal {S}$ by a series of transfer operations in which probability is shifted out of the state $\\sigma $ into new states $\\rho $ .",
"Since the two states are related by a series of transfer operations, ${\\pi }_\\mathcal {S}\\succsim {\\pi }_\\mathcal {R}$ .",
"It immediately follows from this that not only does the minimize the statistical complexity $$ and the topological complexity $C^{(0)}_\\mu $ , but it also minimizes every other Rényi complexity $C^{(\\alpha )}_\\mu $ as well.",
"The uniqueness of the is extremely important in formulating this result.",
"This property of follows from the understanding of predictive models as partitions of the past and of the as corresponding to the coarsest graining of these predictive partitions [6].",
"Other paradigms for modeling will not necessarily have this underlying structure and so may not have strongly minimal solutions.",
"In the following, we see this is, in fact, the case for pure-state quantum machines."
],
[
"Strong quantum advantage",
"A pure-state quantum model can be generalized from the classical case by replacing the classical states $\\sigma $ with quantum-mechanical pure states $\\left|\\eta _\\sigma \\right>$ and the symbol-labeled transition matrices $\\mathbf {T}^{(x)}$ with symbol-labeled Kraus operators $K^{(x)}$ .",
"[Pure-state quantum model] A pure-state quantum model is a quintuplet $\\mathbf {M}=(\\mathcal {H},\\mathcal {A},\\mathcal {S},\\Sigma =\\lbrace \\left|\\eta _\\sigma \\right>:\\sigma \\in \\mathcal {S}\\rbrace ,\\lbrace K^{(x)}:x\\in \\mathcal {A}\\rbrace )$ of a Hilbert space $\\mathcal {H}$ , an output alphabet $\\mathcal {A}$ , pure states $\\left|\\eta _\\sigma \\right>$ corresponding to some set of state labels $\\mathcal {S}$ , and nonnegative Kraus operators $K^{(x)}$ with $x\\in \\mathcal {A}$ satisfying the properties: Completeness: The Kraus operators satisfy $\\sum _x K^{(x)\\dagger }K^{(x)}=I$ .",
"Unifilarity: $K^{(x)}\\left|\\eta _\\sigma \\right> \\propto \\left|\\eta _{f(\\sigma ,x)}\\right>$ for some deterministic function $f(\\sigma ,x)$ .",
"This is a particular kind of hidden quantum Markov model [29] in which we assume the dynamics can be described by the evolution of pure states.",
"This is practically analogous to the assumption of unifilarity in the classical predictive setting.",
"It is not necessarily the case that the states $\\lbrace \\left|\\eta _\\sigma \\right>\\rbrace $ form an orthonormal basis; rather, nonorthonormality is the intended advantage [8], [9].",
"Overlap between the states allows for a smaller von Neumann entropy for the stationary state of the process.",
"We formalize this shortly.",
"It is assumed that the Kraus operators have a unique stationary state $\\rho _\\pi $ .",
"One way to compute it is to note that taking $\\mathbb {P}(x|\\sigma )= \\left<\\eta _\\sigma \\right|K^{(x)\\dagger }K^{(x)}\\left|\\eta _\\sigma \\right>$ and the function $\\sigma \\mapsto f(\\sigma ,x)$ determines a finite predictive model as defined above.",
"The model's stationary state ${\\pi }=(\\pi _\\sigma )$ is related to the stationary state of the quantum model via: = |><| .",
"The process generated by a pure-state quantum model has the word distribution, for words $w=x_1 \\dots x_\\ell $ : P(w) = Tr[K(x)K(x1)K(x1)K(xL)] .",
"The eigenvalues $\\lbrace \\lambda _i\\rbrace $ of the stationary state $\\rho _\\pi $ form a distribution ${\\lambda } = \\left(\\lambda _i\\right)$ .",
"The Rényi entropies of these distributions form the von Neumann-Rényi entropies of the states: S() = H() .",
"We noted previously that for a given state these are strongly minimal over the entropies of all projective, maximal measurements on the state.",
"Given a model $\\mathbf {M}$ with stationary state $\\rho _\\pi $ , we may simply write $S_\\alpha \\left(\\mathbf {M}\\right) = S_\\alpha \\left(\\rho _\\pi \\right)$ as the Rényi memory of the model.",
"Important limits, as before, are the topological memory $S_0\\left(\\mathbf {M}\\right)$ , the statistical memory $S\\left(\\mathbf {M}\\right) = S_1\\left(\\mathbf {M}\\right)$ , and the min-memory $S_\\infty \\left(\\mathbf {M}\\right)$ , which represent physical limitations on memory storage for the generator.",
"To properly compare pure-state quantum models and classical predictive models, we define the classical equivalent model of a pure-state quantum model.",
"[Classical equivalent model] Let $\\mathbf {M}=(\\mathcal {H},\\mathcal {A},\\mathcal {S},\\Sigma =\\lbrace \\left|\\eta _\\sigma \\right>:\\sigma \\in \\mathcal {S}\\rbrace ,\\lbrace K^{(x)}:x\\in \\mathcal {A}\\rbrace )$ be a pure-state quantum model, with probabilities and deterministic function $\\mathbb {P}(x|\\sigma ) =\\left<\\eta _\\sigma \\right|K^{(x)\\dagger }K^{(x)}\\left|\\eta _\\sigma \\right>$ and $\\sigma \\mapsto f(\\sigma ,x)$ , respectively.",
"Its classical equivalent $\\mathbf {M}_\\mathrm {Cl.",
"}$ is the classical finite predictive model with state set $\\mathcal {S}$ , alphabet $\\mathcal {A}$ and symbol-based transition matrices $\\mathbf {T}^{(x)}$ generated by the state-to-symbol probabilities $\\mathbb {P}(x|\\sigma )$ and deterministic function $f(\\sigma ,x)$ .",
"We now prove that a finite classical predictive model strongly maximizes all pure-state quantum models of which it is the classical equivalent.",
"[Strong quantum advantage] Let $\\mathbf {M}=(\\mathcal {H},\\mathcal {A},\\mathcal {S},\\Sigma =\\lbrace \\left|\\eta _\\sigma \\right>:\\sigma \\in \\mathcal {S}\\rbrace ,\\lbrace K^{(x)}:x\\in \\mathcal {A}\\rbrace )$ be a pure-state quantum model with stationary state $\\rho _\\pi $ , and let $\\mathbf {M}_\\mathrm {Cl.",
"}$ be the classical equivalent model with stationary state ${\\pi }=(\\pi _\\sigma )$ (with $\\sigma =1,\\dots ,n$ ).",
"Let $d=\\dim \\mathcal {H}$ and $n=\\left|\\mathcal {S}\\right|$ .",
"(We have $n\\ge d$ : if not, then we can take a smaller Hilbert space that spans the states.)",
"Let ${\\lambda }=(\\lambda _i)$ be an $n$ -dimensional vector where the first $d$ components are the eigenvalues of $\\rho _\\pi $ and the remaining elements are 0.",
"Then ${\\lambda }\\succsim {\\pi }$ .",
"We know that: = S |><| = S |><| , where $\\left|\\phi _\\sigma \\right> = \\sqrt{\\pi _\\sigma }\\left|\\eta _\\sigma \\right>$ .",
"However, we can also write $\\rho _\\pi $ in the eigenbasis: = i=1d i |i><i| = i=1d |i><i| , where $\\left|\\psi _i\\right> = \\sqrt{\\lambda _i}\\left|i\\right>$ .",
"Then the two sets of vectors can be related via: |> = i=1d Ui |i> , where $U_{\\sigma i}$ is a $n\\times d$ matrix comprised of $d$ rows of orthonormal $n$ -dimensional vectors [30].",
"Now, we have: = <|> = i=1d |Ui|2 i .",
"Note that $U_{\\sigma i}$ is not square, but since we have taken $\\lambda _i = 0$ for $i> d$ , we can simply extend $U_{\\sigma i}$ into a square unitary matrix by filling out the bottom $n-d$ rows with more orthonormal vectors.",
"This leaves the equation unchanged.",
"We can then write: = i=1n |Ui|2 i .",
"Then by Theorem 1, ${\\lambda }\\succsim {\\pi }$ .",
"$\\square $ $S_\\alpha (\\mathbf {M}) \\le H_\\alpha \\left(\\mathbf {M}_{\\mathrm {Cl.",
"}}\\right)$ for all $\\alpha \\ge 0$ .",
"$S_\\alpha (\\rho _\\pi ) \\le H_\\alpha ({\\pi })$ for all $\\alpha \\ge 0$ follows from the definitions of the von Neumann-Rényi entropies and the Schur-concavity of $H_\\alpha $ .",
"$\\square $ Many alternative pure-state quantum models may describe the same process.",
"The “first mark”, so to speak, for quantum models is the $q$ -machine [9], [15], which directly embeds the dynamics of the into a quantum system while already leveraging the memory advantage due to state overlap.",
"[$q$ -Machine] Given an $\\lbrace \\mathcal {S},\\mathcal {A},\\lbrace \\mathbf {T}^{(x)}:x\\in \\mathcal {A}\\rbrace \\rbrace $ , where ${T}^{(x)}_{\\sigma \\sigma ^{\\prime }}=\\mathbb {P}(x|\\sigma ) \\delta _{\\sigma ^{\\prime },f(\\sigma ,x)}$ for some deterministic function $f(\\sigma ,x)$ , construct the corresponding $q$ -machine in the following way: The states $\\left|\\eta _\\sigma \\right>$ are built to satisfy the recursive relation: <|'> = xAP(x|)P(x|')<f(,x)|f(',x)> .",
"$\\mathcal {H}$ is the space spanned by the states $\\left|\\eta _\\sigma \\right>$ .",
"The Kraus operators $K^{(x)}$ are determined by the relations: K(x)|> = P(x|)|f(,x)> .",
"One can check that this satisfies the completeness relations and has the correct probability dynamics for the process generated by the .",
"That the $q$ -machine offers statistical memory advantage with respect to the was previously shown in [31] and with respect to topological memory in [14].",
"Theorem and Corollary imply these as well as advantage with respect to other Rényi measures of memory."
],
[
"Weak quantum minimality",
"An open problem is to determine the minimal quantum pure-state representation of a given classical process.",
"This problem is solved in some specific instances such as the Ising model [11] and the Perturbed Coin Process [14].",
"In these cases it is known to be the $q$ -machine.",
"We denote the smallest value of the Rényi entropy of the stationary state as $C_{q}^{(\\alpha )} = \\min _{\\mathbf {M}} S_\\alpha \\left(\\mathbf {M}\\right)$ , called the quantum Rényi complexities, including the limits, the quantum topological complexity $C_{q}^{(0)}$ , the quantum min-complexity $C_{q}^{(\\infty )}$ , and the quantum statistical complexity $C_{q}=C_{q}^{(1)}$ .",
"If a strongly minimal quantum pure-state model exists, these complexities are all attained by the same pure-state model.",
"One of our primary results in this section is that for some processes, this does not occur.",
"We start by examining two examples.",
"The first, the MBW Process introduced in Ref.",
"[29], demonstrates a machine whose $q$ -machine is not minimal in the von Neumann complexity.",
"Consider the process generated by the 4-state MBW machine shown in Fig.",
"REF .",
"Figure: The 4-state MBW Process as a Markov chain (which is the ).This process' HMM is simply a Markov chain, and its representation in Fig.",
"REF is its .",
"Denote this classical representation by $\\mathbf {M}_4$ .",
"If we take $\\lbrace \\left|A\\right>, \\left|B\\right>,\\left|C\\right>, \\left|D\\right> \\rbrace $ as an orthonormal basis of a Hilbert space, we can construct the $q$ -machine with the states: |A> = 12|A> + 12(|C>+|D>) , |B> = 12|B> + 12(|C>+|D>) , |C> = 12|C> + 12(|A>+|B>) , and |D> = 12|D> + 12(|A>+|B>) .",
"Since it is a Markov chain, we can write the Kraus operators as $K_x =\\left|\\eta _x\\right>\\left<\\epsilon _x\\right|$ , where $\\left<\\epsilon _x|\\eta _{x^{\\prime }}\\right> \\propto \\sqrt{\\mathbb {P}(x|x^{\\prime })}$ .",
"This is a special case of the construction used in Ref.",
"[13].",
"For $q$ -machines of Markov chains, then, the dual basis is just $\\left<\\epsilon _x\\right| =\\left<x\\right|$ .",
"We denote the $q$ -machine model of the 4-state MBW Process as $\\mathbf {Q}_4$ .",
"Let's examine the majorization between $\\mathbf {Q}_4$ and the Markov model via the Lorenz curves of ${\\lambda }$ , the eigenvalues of $\\rho _\\pi $ , and the stationary state of the Markov chain.",
"See Fig.",
"REF .",
"Figure: Lorenz curves for the 4-state MBW 𝐌 4 \\mathbf {M}_4and the associated qq-machine 𝐐 4 \\mathbf {Q}_4.It turns out that there is a smaller quantum model embedded in two dimensions, with states: |A'> = |0> , |B'> = |1> , |C'> = 12(|0>+|1>) , and |D'> = 12(|0>-|1>) .",
"In this case, $\\left<\\epsilon _x^{\\prime }\\right|=\\frac{1}{\\sqrt{2}}\\left<\\eta _x^{\\prime }\\right|$ derives the $q$ -machine.",
"This gives the proper transition probabilities for the 4-state MBW model.",
"This dimensionally smaller model we denote $\\mathbf {D}_4$ .",
"Figure REF compares the Lorenz curve of its stationary eigenvalues ${\\lambda }^{\\prime }$ to those of $\\mathbf {Q}_4$ .",
"One sees that it does not majorize the $q$ -machine, but it does have a lower statistical memory: $S(\\mathbf {D}_4) = 1.0$ and $S(\\mathbf {Q}_4) \\approx 1.2$ bit.",
"(On the other hand, the $q$ -machine has a smaller min-memory, with $S_\\infty (\\mathbf {D}_4) = 1.0$ and $S_\\infty (\\mathbf {Q}_4) \\approx 0.46$ .)",
"Figure: Lorenz curves for the 4-state MBW qq-machine𝐐 4 \\mathbf {Q}_4 and a dimensionally smaller model𝐃 4 \\mathbf {D}_4.Now consider something in the opposite direction.",
"Consider the 3-state MBW model, denoted $\\mathbf {M}_3$ and displayed in Fig.",
"REF .",
"This is a generalization of the previous example to three states instead of four.",
"We will compute the corresponding $q$ -machine $\\mathbf {Q}_3$ and show that there also exists a dimensionally smaller representation $\\mathbf {D}_3$ .",
"In this case, however, $\\mathbf {D}_3$ is not smaller in its statistical memory.",
"The $q$ -machine $\\mathbf {Q}_3$ of this Markov chain is given by the states: |A> = 23|A> + 16(|B>+|C>) , |B> = 23|B> + 16(|A>+|C>) , and |C> = 23|C> + 16(|A>+|B>) , and Kraus operators defined similarly to before.",
"We can examine the majorization between the $q$ -machine and the Markov model by plotting the Lorenz curves of ${\\lambda }$ , the eigenvalues of $\\rho _\\pi $ , and the stationary state of the Markov chain, shown in Fig.",
"REF .",
"Figure: 3-state MBW Process as a Markov chain (which is the process' ).Figure: Lorenz curves for the 3-state MBW 𝐌 3 \\mathbf {M}_3 and the associatedqq-machine 𝐐 3 \\mathbf {Q}_3.The lower-dimensional model $\\mathbf {D}_3$ is given by the states: |A> = |0> , |B> = 12|0>+32|1> , and |C> = 12|0>-32|1> , with $\\left<\\epsilon _x^{\\prime }\\right|=\\sqrt{\\frac{2}{3}}\\left<\\eta _x^{\\prime }\\right|$ .",
"This gives the proper transition probabilities for the 3-state MBW model.",
"Figure REF compares the Lorenz curve of its stationary eigenvalues ${\\lambda }^{\\prime }$ to that of $\\mathbf {Q}_3$ .",
"We see that it does not majorize $\\mathbf {Q}_3$ .",
"And, this time, this is directly manifested by the fact that the smaller-dimension model has a larger entropy: $S(\\mathbf {D}_3) = 1.0$ and $S(\\mathbf {Q}_3) \\approx 0.61$ bit.",
"Figure: Lorenz curves for the 3-state MBW qq-machine,𝐐 3 \\mathbf {Q}_3 and adimensionally smaller model 𝐃 3 \\mathbf {D}_3.After seeing the 's strong minimality with respect to other classical models and its strong maximality with respect to quantum models, it is certainly tempting to conjecture that a strongly minimal quantum model exists.",
"However, the examples we just explored cast serious doubt.",
"None of the examples covered above are strong minima.",
"One way to prove that no strong minimum exists for, say, the 3-state MBW process requires showing that there does not exist any other quantum model in 2 dimensions that generates the process.",
"This would imply that no other model can majorize $\\mathbf {D}_3$ .",
"And, since this model is not strongly minimal, no strongly minimal solution can exist.",
"Appendix proves exactly this—thus, demonstrating a counterexample to the strong minimality of quantum models.",
"[Weak Minimality of $\\mathbf {D}_3$ ] The quantum model $\\mathbf {D}_3$ weakly minimizes topological complexity for all quantum generators of the 3-state MBW Process; consequently, the 3-state MBW Process has no strongly minimal model.",
"Figure: Proposed majorization saddle structure of model-space: The (labeledϵ\\epsilon ) is located at a saddle-point with respect to majorization,where classical deviations (state-splitting) move up the lattice andquantum deviations (utilizing state overlap) move down the lattice."
],
[
"Concluding remarks",
"Majorizing states provides a means to compare a process' alternative models in both the classical and quantum regimes.",
"Majorization implies the simultaneous minimization of a large host of functions.",
"As a result we showed that: The majorizes all classical predictive models of the same process, and so simultaneously minimizes many different measures of memory cost.",
"The $q$ -machine, and indeed any quantum realization of the , always majorizes the , and so simultaneously improves on all the measures of memory cost.",
"For at least one process, there does not exist any quantum pure-state model that majorizes all quantum pure-state models of that process.",
"Thus, while an $\\epsilon $ -machine may be improved upon by different possible quantum models, there is not a unique one quantum model that is unambiguously the “best” choice.",
"Imagining the as an invariant “saddle-point” in the majorization structure of model-space, Fig.",
"REF depicts the implied geometry.",
"That is, we see that despite its nonminimality among all models, the still occupies a topologically important position in model-space—one that is invariant to one's choice of memory measure.",
"However, no similar model plays the topologically minimal role for quantum pure-state models.",
"The quantum statistical complexity $C_q$ has been offered up as an alternative quantum measure of structural complexity—a rival of the statistical complexity $$ [32].",
"One implication of our results here is that the nature of this quantum minimum $C_q$ is fundamentally different than that of $$ .",
"This observation should help further explorations into techniques required to compute $C_q$ and the physical circumstances in which it is most relevant.",
"That the physical meaning of $C_q$ involves generating an asymptotically large number of realizations of a process may imply that it cannot be accurately computed by only considering machines that generate a single realization.",
"This is in contrast to $$ which, being strongly minimized, must be attainable in the single-shot regime along with measures like $^{(0)}$ and $^{(\\infty )}$ .",
"In this way, the quantum realm again appears ambiguous.",
"Ambiguity in structural complexity has been previously observed in the sense that there exist pairs of processes, $A$ and $B$ , such that $(A)>(B)$ but $C_q(A)<C_q(B)$ [33].",
"The classical and quantum paradigms for modeling can disagree on simplicity—there is no universal Ockham's Razor.",
"How this result relates to strong versus weak optimization deserves further investigation.",
"The methods and results here should also be extended to analyze classical generative models which, in many ways, bear resemblances in their functionality to the quantum models [34], [35], [36].",
"These drop the requirement of unifilarity, similar to how the quantum models relax the notion of orthogonality.",
"Important questions to pursue in this vein are whether generative models are strongly maximized by the and whether they have their own strong minimum or, like the quantum models, only weak minima in different contexts.",
"To close, we only explored finite-state, discrete-time processes.",
"Processes with infinite memory [37] and continuous generation [38], [39] are also common in nature.",
"Applying our results to understand these requires further mathematical development."
],
[
"Acknowledgments",
"The authors thank Fabio Anza, John Mahoney, Cina Aghamohammadi, and Ryan James for helpful discussions.",
"As a faculty member, JPC thanks the Santa Fe Institute and the Telluride Science Research Center for their hospitality during visits.",
"This material is based upon work supported by, or in part by, John Templeton Foundation grant 52095, Foundational Questions Institute grant FQXi-RFP-1609, the U.S. Army Research Laboratory and the U. S. Army Research Office under contract W911NF-13-1-0390 and grant W911NF-18-1-0028, and via Intel Corporation support of CSC as an Intel Parallel Computing Center."
],
[
"Appendix: Weak Minimality of $\\mathbf {D}_3$",
"Here, we prove that $\\mathbf {D}_3$ is the unique 2D representation of the 3-state MBW process.",
"We show this by considering the entire class of 2D models and applying the completeness constraint.",
"We note that a pure-state quantum model of the 3-state MBW process must have three states $\\left|\\eta _A\\right>$ , $\\left|\\eta _B\\right>$ , and $\\left|\\eta _C\\right>$ , along with three dual states $\\left<\\epsilon _A\\right|$ , $\\left<\\epsilon _B\\right|$ , and $\\left<\\epsilon _C\\right|$ such that: <A|A> = eiAA23 , <A|B> = eiAB16 , and <A|C> = eiAC16 , <B|A> = eiBA16 , <B|B> = eiBB23 , and <B|C> = eiBC16 , and: <C|A> = eiCA16 , <C|B> = eiCB16 , <C|C> = eiCC23 .",
"We list the available geometric symmetries that leave the final stationary state unchanged: Phase transformation on each state, $\\left|\\eta _x\\right>\\mapsto e^{i\\phi _x}\\left|\\eta _x\\right>$ ; Phase transformation on each dual state, $\\left|\\epsilon _x\\right>\\mapsto e^{i\\phi _x}\\left|\\epsilon _x\\right>$ ; and Unitary transformation $\\left|\\eta _x\\right> \\mapsto U\\left|\\eta _x\\right>$ and $\\left<\\epsilon _x\\right| \\mapsto \\left<\\epsilon _x\\right|U^\\dagger $ .",
"From these symmetries we can fix gauge in the following ways: Set $\\left<0|\\eta _x\\right>$ to be real and positive for all $x$ .",
"Set $\\phi _{AA}=\\phi _{BB}=\\phi _{CC}=0$ .",
"Set $\\left<0|\\eta _A\\right>=0$ and set $\\left<1|\\eta _B\\right>$ to be real and positive.",
"These gauge fixings allow us to write: |A> = |0> , |B> = B |0> +B|1> , and |C> = C |0> +eiC|1> , for $\\alpha _B,\\alpha _C\\ge 0$ , $\\beta _B=\\sqrt{1-\\alpha _B^2}$ and $\\beta _C=\\sqrt{1-\\alpha _C^2}$ and a phase $\\theta $ .",
"That these states are embedded in a 2D Hilbert space means there must exist some linear consistency conditions.",
"For some triple of numbers $\\mathbf {c}=(c_A, c_B, c_C)$ we can write: cA|A> +cB|B> +cC|C> = 0 .",
"Up to a constant, we use our parameters to choose: (cA, cB, cC) = (eiBCB-C, -eiCB, 1) .",
"Consistency requires that this relationship between vectors is preserved by the Kraus operator dynamic.",
"Consider the matrix $\\mathbf {A}=(A_{xy}) = \\left(\\left<\\epsilon _x|\\eta _y\\right>\\right)$ .",
"The vector $\\mathbf {c}$ must be a null vector of $\\mathbf {A}$ ; i.e.",
"$\\sum _y A_{xy}c_y=0$ .",
"This first requires that $A_{xy}$ be degenerate.",
"One way to enforce this to check that the characteristic polynomial $\\det (\\mathbf {A}-\\lambda \\mathbf {I}_3)$ has an overall factor of $\\lambda $ .",
"For simplicity, we compute the characteristic polynomial of $A\\sqrt{6}$ : (6A-I3) = (2-)3 + (ei(AB+BC+CA)+ei(BA+CB+AC)) - (2-)(ei(AB+BA)+ei(AC+CA)+ei(BC+CB)) .",
"To have an overall factor of $\\lambda $ , we need: 0 = 8 + (ei(AB+BC+CA)+ei(BA+CB+AC)) -2(ei(AB+BA)+ei(AC+CA)+ei(BC+CB)) .",
"Typically, there will be several ways to choose phases to cancel out vectors, but in this case since the sum of the magnitudes of the complex terms is 8, the only way to cancel is at the extreme point where $\\phi _{AB}=-\\phi _{BA}=\\phi _1$ , $\\phi _{BC}=-\\phi _{CB}=\\phi _2$ , and $\\phi _{CA}=-\\phi _{AC}=\\phi _3$ and: 1+2+3= .",
"To recapitulate the results so far, $\\mathbf {A}$ has the form: A = 16 (ccc 2 ei1 -ei(1+2) e-i1 2 ei2 -e-i(1+2) e-i2 2 ) .",
"We now need to enforce that $\\sum _y A_{xy}c_y=0$ .",
"We have the three equations: 2cA+ei1cB -ei(1+2)cC = 0 , 2cB+e-i1cA +ei2cC = 0 , and 2cC+e-i2cB -e-i(1+2)cA = 0 .",
"It can be checked that these are solved by cA = ei(1+2)cC and cB = - ei2cC .",
"Taking our formulation of the $\\mathbf {c}$ vector, we immediately have $\\beta _B=\\beta _C=\\beta $ (implying $\\alpha _B=\\alpha _C=\\alpha $ ), $\\phi _2 =\\theta $ , and: e-i3 = (1-ei) =-2i()ei/2 =()ei(-)/2 .",
"This means: = 12 |(2)| and 3 = -+sgn()2 , where we take $-\\pi \\le \\theta \\le \\pi $ and $\\mathrm {sgn}(\\theta )$ is the sign of $\\theta $ .",
"Note, however, that for $-\\frac{\\pi }{3}< \\theta < \\frac{\\pi }{3}$ , we have $|\\csc (\\theta )|> 1$ , so these values are unphysical.",
"Thus, we see that all parameters in our possible states $\\left|\\eta _x\\right>$ , as well as all the possible transition phases, are dependent on the single parameter $\\theta $ .",
"To construct the dual basis, we start with the new forms of the states: |A> = |0>, |B> = |0> + |1> , and |C> = |0> + ei|1> .",
"We note directly that we must have: <A|0> = 23, <B|0> = 16e-i1 , and <C|0> = 16ei3 , from how the dual states contract with $\\left|\\eta _A\\right>$ .",
"These can be used with the contractions with $\\left|\\eta _B\\right>$ to get: <A|1> = 123(12ei1-) , <B|1> = 123(1-12e-i1) , and <C|1> = 1223(e-i2-ei3) .",
"It is quickly checked that these coefficients are consistent with the action on on $\\left|\\eta _C\\right>$ by making liberal use of $e^{-i\\phi _3} =\\alpha (1-e^{i\\theta })$ .",
"Recall that with the correct dual states, the Kraus operators take the form: KA = |A><A| , KB = |B><B| , and KC = |C><C| .",
"Completeness requires: |A><A| + |B><B| + |C><C| = I .",
"Define the vectors $u_x = \\left<\\epsilon _x | 0\\right>$ and $v_x =\\left<\\epsilon _x | 1\\right>$ .",
"One can check that the above relationship implies $\\sum _x u_x^\\ast u_x = \\sum _x v_x^\\ast v_x = 1$ and $\\sum _x u_x v^\\ast _x = 0$ .",
"However, for our model, it is straightforward (though a bit tedious) to check that: x uxux = 23+16+16 = 1 and x vxvx = 12(1+2-1) .",
"Using the definitions of $\\alpha $ , $\\beta $ , and $\\phi _1$ , the second equation can be simplified to: x vxvx = 2+224-22 .",
"This is unity only when $\\csc ^2\\frac{\\theta }{2}=1$ , which requires that $\\theta = \\pi $ .",
"This is, indeed, the model $\\mathbf {D}_3$ that we have already seen.",
"This establishes that the only two-dimensional pure-state quantum model which reproduces the 3-state MBW process is the one with a nonminimal statistical memory $S(\\rho _\\pi )$ .",
"This means there cannot exist a quantum representation of the 3-state MBW process that majorizes all other representations of the same.",
"For, if it existed, it must be a two-dimensional model and also minimize $S(\\rho _\\pi )$ ."
]
] | 1808.08639 |
[
[
"Identifying Domain Adjacent Instances for Semantic Parsers"
],
[
"Abstract When the semantics of a sentence are not representable in a semantic parser's output schema, parsing will inevitably fail.",
"Detection of these instances is commonly treated as an out-of-domain classification problem.",
"However, there is also a more subtle scenario in which the test data is drawn from the same domain.",
"In addition to formalizing this problem of domain-adjacency, we present a comparison of various baselines that could be used to solve it.",
"We also propose a new simple sentence representation that emphasizes words which are unexpected.",
"This approach improves the performance of a downstream semantic parser run on in-domain and domain-adjacent instances."
],
[
"Introduction",
"Semantic parsers map text to logical forms, which can then be used by downstream components to fulfill an action.",
"Consider, for example, a system for booking air travel, in which a user provides natural language input, and a downstream subsystem is able to make or cancel flight reservations.",
"Users of the system typically have a general understanding of its purpose, so the input will revolve around the correct topic of air travel.",
"However, they are unlikely to know the limits of the system's functionality, and may provide inputs for which the expected action is beyond its capabilities, such as asking to change seats on a flight reservation.",
"Because the logical schema is designed with fulfillment in mind, no logical form can capture the semantics of these sentences, making it impossible for the parser to generate a correct parse.",
"Any output the parser generates will cause unintended actions to be executed downstream.",
"For example, asking to change seats might be misparsed and executed as changing flights.",
"Instead, the parser should identify that this input is beyond its scope so the condition can be handled.In the final page of the paper, we suggest a few immediate downstream system behaviors when a domain-adjacent instance is identified, but others have investigated the related problem of teaching the system new behavior [4].",
"In this paper, we formalize this pervasive problem, which we call domain-adjacent instance identification.",
"Figure: In this example, an air travel semantic parser is trained on data containing in-domain predicates.",
"A test instance cannot be parsed correctly if it contains any domain-adjacent or out-of-domain predicates.While this task is similar to that of identifying out-of-domain input instances (e.g., banking with respect to air travel), it is much more subtle — the instances come from roughly the same domain as the parser's training examples, and thus use very similar language.",
"Domain adjacency is a property with respect to the parser's output schema, independent of the data used to train it.",
"In this paper, we formalize this task, and propose a simple approach for representing sentences in which words are weighted by how likely they are to differentiate between in-domain and domain-adjacent instances.",
"Note that while this approach can also be applied to out-of-domain instances, in this paper we are interested in its performance on domain-adjacent instances.",
"We describe an evaluation framework for this new task and, finally, evaluate our proposed method against a set of baselines, comparing performance on the domain-adjacent classification problem and a downstream semantic parsing task."
],
[
"Problem Setting",
"A semantic parser can be seen as a function $\\varphi $ that maps sentences $x$ in a natural language $\\mathcal {L}$ to logical forms $y \\in \\mathcal {Y}$ .",
"Assuming the existence of an oracle parser $\\hat{\\varphi }$ , the problem we propose in this paper is that of determining, for a given test instance $x$ , whether it belongs to the domain $\\Phi $ of $\\hat{\\varphi }$ , i.e., if its semantics can be encoded in the schema $\\mathcal {Y}$ .",
"In real-world usage, the input sentences $x$ will be generated by a human user, who associates the capabilities of the parser to a particular topic (e.g., air travel).",
"Thus most of the $x \\in \\mathcal {L} \\setminus \\Phi $ will share topic with the $\\hat{x} \\in \\Phi $ .",
"Because of the similarity between these $x$ and $\\hat{x}$ , we call this task identification of domain-adjacent instances."
],
[
"Approach",
"Our goal is to identify input instances whose semantics are not representable in the parser's output schema, and we assume only an in-domain dataset is available at training time.",
"Our approach is based on determining similarity to these training instances.",
"We split the task in two parts: 1) encode the sentences to a compact representation that preserves the needed information, and 2) given these representations, identify which sentences are so dissimilar that they are unlikely to be parseable with any schema that covers the training set."
],
[
"Sentence Representation",
"Among recent work in distributional semantics, averaging the word vectors to represent a sentence [18], [1] has proven to be a simple and robust approach.",
"However, we have an intuition that words which are unexpected in their context given the training data may be a strong signal that an instance is domain-adjacent.",
"To incorporate this signal, we propose a weighted average, in which the weight corresponds to how unexpected the word is in its context.",
"For example, given in-domain predicates from Figure REF , in the domain-adjacent sentence “Upgrade my flight to SFO with my miles”, upgrade should receive a much higher weight than flight or SFO.",
"Our weighting scheme is as follows: We use the cosine distance between the expected ($\\bar{v}_i$ ) and the actual ($\\hat{v}_i$ ) domain-specific word embedding at a given position ($i$ ) in a sentence to compute its weight: $w_i = 1 - cos(\\bar{v}_i, \\hat{v}_i)$ .",
"The expected word embedding is computed using the context embeddings, $\\bar{v}_i = \\sum _{j=i-c, j\\ne i}^{i+c} \\hat{v}_j$ , where $\\hat{v}_j$ is a domain-specific word embedding, in a window of size $c$ around position $i$ .",
"Intuitively, $w_i$ represents how surprising the word is in the context.",
"Since our training set is too small to directly learn domain-specific embeddings, we learn a mapping from general pre-trained embeddings.",
"We train a continuous bag-of-words model [13] in which we pass pre-trained embeddings ($v_i$ ) instead of 1-hot vectors, as input to the embedding layer.",
"The layer thus learns a mapping from pre-trained to domain-specific embeddings ($\\hat{v}_i$ ).",
"We use this mapping to compute new embeddings for words that are missing from the training set.",
"Only words that do not have pre-trained embeddings are ignored.",
"Finally, for a sentence with $n$ words, we take the weighted average of the pre-trained embeddings of the words in the sentence, using the weights from above: $S = \\left( \\sum _{i=1}^{n} w_i v_i \\right) / \\left( \\sum _{i=1}^{n} w_i \\right)$ .",
"This approach assigns high weight to words that differ significantly from what is expected based on the training data.",
"By combining these weights with the pre-trained word embeddings, we allow the model to incorporate external information, improving generalization beyond the training set."
],
[
"Domain-Adjacent Model",
"A number of techniques can be applied to predict whether a sentence is domain-adjacent from its continuous representation.",
"Of the methods we tried, we found k-nearest neighbors [2] to perform best: to classify a sentence, we calculate the average cosine distance between its embedding and its $k$ nearest neighbors in the training data, and label it domain-adjacent if this value is greater than some threshold.",
"This simpler model relies more heavily on the external information brought in by pre-trained word embeddings, while more complex models seem to overfit to the training data."
],
[
"Evaluation",
"In this section, we introduce an evaluation framework for this new task.",
"We consider training and test sets from a single domain, with only the latter containing domain-adjacent instances.",
"Test instances are classified individually, and we measure performance on in-domain/domain-adjacent classification and semantic parsing."
],
[
"Dataset and Semantic Parser",
"We simulate this setting by adapting the Overnight dataset [17].",
"This dataset is composed of queries drawn from eight domains, each having a set of seven to eighteen distinct semantic predicates.",
"Queries consist of a crowd-sourced textual sentence and its corresponding logical form containing one or more of these domain-specific semantic predicates.",
"For each domain, we select a set of predicates to exclude from the logical schema (see Table REF ), and remove all instances containing these predicates from the training set (since they are now domain-adjacent).",
"We then train a domain-adjacent model and semantic parser on the remaining training data and attempt to identify the domain-adjacent examples in the test data.",
"We use the train/test splits from [17].",
"In all experiments, we use the SEMPRE parser [5]."
],
[
"Baselines",
"Because this is a novel task, and results are not comparable to previous work, we report results from a variety of baseline systems.",
"The first baseline, Confidence, identifies instances as domain-adjacent if the semantic parser's confidence in its predictions is below some threshold.",
"The remaining baselines follow the two-part approach from Section .",
"Autoencoder is inspired by [15]'s work on identifying out-of-domain examples.",
"For the sentence representation, this method uses a bi-LSTM with self-attention, trained to predict the semantic predicates, and concatenates the final hidden state from each direction as the sentence representation.",
"An autoencoder is used as the domain-adjacent classifier.",
"The remaining methods use the nearest neighbor model discussed in Section REF .",
"For sentence representations, we include baselines drawn from different neural approaches.",
"In CBOW, we simply average the pre-trained word embeddings in the sentence.",
"In CNN, we train a two-layer CNN with a final softmax layer to predict the semantic predicates for a sentence.",
"We concatenate the mean pooling of each layer as the sentence representation.",
"In LSTM, we use the same sentence representation as in Autoencoder, with the nearest neighbor domain-adjacent model.",
"Finally, Surprise is the approach presented in Section REF ."
],
[
"Direct Evaluation",
"We first directly evaluate the identification of domain-adjacent instances: Table REF reports the area under a receiver operating characteristic curve (AUC) for the considered models [6].",
"Surprise generally performs the best on this evaluation; and, in general, the simpler models tend to perform better, suggesting that more complex approaches tune too much to the training data.",
"Qualitatively, for domains where the Surprise model performs better, it places higher weight on words we would consider important for distinguishing domain-adjacent sentences.",
"For example in “show me recipes with longer preparation times than rice pudding” from Recipes, “longer” and “preparation” have the highest weights.",
"In Social, there are two in-domain predicates (employmentStartDate and educationEndDate) which use very similar wording to those that are domain-adjacent, making it difficult to isolate surprising words.",
"The weights in this domain seem to instead emphasize unusual wordings such as “soonest” in “employees with the soonest finish date”."
],
[
"Ablation Analysis",
"In order to determine the contribution of each one of the components of Surprise, we performed an ablation analysis comparing the following modifications of the method: CBOW, as described above, using an unweighted average of pre-trained embeddings; Frequency, using a weighted average of pre-trained embeddings, with weights based on inverse document frequency in the training set; PreTrained, using the surprise schema but with weights determined using pre-trained embeddings; and the full Suprise as presented above.",
"Each approach adds one component (weighting, surprise-based weights, and domain-specific embeddings) with respect to the previous one.",
"The results of the experiment are shown in Table REF .",
"We can see that Frequency performs slightly worse than CBOW and PreTrained performs even worse than that.",
"We can conclude that the combination of the weighting schema and the tuned vectors is what makes Suprise effective."
],
[
"Downstream Task Evaluation",
"We next evaluate how including the domain-adjacent predictions affects the performance of a semantic parser.",
"In a real setting, when the semantic parser is presented with domain-adjacent input that is beyond its scope, the correct behavior is to label it as such so that it can be handled properly by downstream components.",
"To simulate this behavior, we set the gold parse for domain-adjacent instances to be an empty parse, and automatically assign an empty parse to any instance that is identified as domain-adjacent.",
"We report accuracy of the semantic parser with 20% domain-adjacent test data.",
"We include two additional models: NoFilter, in which nothing is labeled domain-adjacent, and Oracle, in which all the domain-adjacent instances are correctly identified.",
"For each baseline requiring a threshold, we set it such that 3% of the instances in the dev set would be marked as domain-adjacent (intuitively, this represents the error-tolerance of the system).",
"Table REF shows the results for this experiment.",
"In general, the relative performance is similar to that in the direct evaluation (e.g.",
"Surprise tends to do well on most domains, but performs poorly on Basketball and Social in particular).",
"However, in this evaluation, misclassifying an instance as domain-adjacent if the semantic parser would have accurately parsed it is worse than misclassifying the instance if the semantic parser could not have accurately parsed it.",
"For example, in Social we can thus infer that Surprise is marking some instances as domain-adjacent that would otherwise be accurately parsed as the performance there is actually worse than for NoFilter."
],
[
"Related Work",
"Domain-adjacency identification is a new task, but relatively little effort has been devoted to even the related task of identifying out-of-domain instances (i.e., from completely separate domains) for semantic parsers.",
"[7] approached the problem by clustering sentences based on shared subgraphs in their general semantic parses; [15] classify sentences with autoencoder reconstruction error.",
"Prior distributional semantics work to create compact sentential representations generated specific embeddings for downstream tasks [9], [11], [16].",
"Recently, work has focused on domain-independent embeddings, learned without downstream task supervision.",
"[12], [8], and [10] learn representations by predicting the surrounding sentences.",
"[18] use paraphrases assupervision.",
"[14] represent sentences by the low-rank subspace spanned by the embeddings of the words in them; [3] use a weighted average of word embeddings, with their projection onto the first principal component across all sentences in the corpus removed.",
"Another relatively sparse area of related work is handling the domain-adjacent instances once they have been identified.",
"The simplest thing to do is to return a generic error.",
"For user-facing applications, one such message might state that the system can't handle that specific query.",
"[4] approach this problem by having the user break down the domain-adjacent instance into a sequence of simpler textual instructions and then attempting to map those to known logical forms."
],
[
"Conclusion",
"Identifying domain-adjacent instances is a practical issue that can improve downstream semantic parsing precision, and thus provide a smoother and more reliable user experience.",
"In this paper, we formalize this task, and introduce a novel sentence embedding approach which outperforms baselines.",
"Future work includes exploring alternative ways of incorporating information outside of the given training set and experimenting with various combinations of semantic parsers and upstream domain-adjacency models.",
"Another area of future research is how the underlying system should recover when domain-adjacent instances are detected."
]
] | 1808.08626 |
[
[
"Antenna Array Based Positional Modulation with a Two-Ray Multi-Path\n Model"
],
[
"Abstract Traditional directional modulation (DM) designs are based on the assumption that there is no multi-path effect between transmitters and receivers.",
"One problem with these designs is that the resultant systems will be vulnerable to eavesdroppers which are aligned with or very close to the desired directions, as the received modulation pattern at these positions is similar to the given one.",
"To solve the problem, a two-ray multi-path model is studied for positional modulation and the coefficients design problem for a given array geometry and a location-optimised antenna array is solved, where the multi-path effect is exploited to generate a given modulation pattern at desired positions, with scrambled values at positions around them."
],
[
"Introduction",
"Directional modulation (DM), as a security technique to keep known constellation mappings in a desired direction or directions, while scrambling them for the remaining ones, was introduced in [1] by combining the direct radiation beam and reflected beams in the far-field.",
"In [2], a reconfigurable array was designed by switching elements for each symbol to make their constellation points not scrambled in desired directions, but distorted in other directions.",
"A method named dual beam DM was introduced in [3], where the I and Q signals are transmitted by different antennas.",
"In [4], [5], phased arrays were employed to show that DM can be implemented by phase shifting the transmitted antenna signals properly.",
"Multi-carrier based phased antenna array design for directional modulation was studied in [6], followed by a combination of DM and polarisation design in [7].",
"The bit error rate (BER) performance of a system based on a two-antenna array was studied using the DM technique for eight phase shift keying modulation in [8].",
"A more systematic pattern synthesis approach was presented in [9], followed by a time modulation technique for DM to form a four-dimensional (4-D) antenna array in [10].",
"However, eavesdroppers aligned with or very close to the desired direction/directions will be a problem for secure signal transmission, as their received modulation patterns are similar to the given one.",
"To make sure that a given modulation pattern can only be received at certain desired positions, one solution is adopting a multi-path model, where signals via both line of sight (LOS) and reflected paths are combined at the receiver side [11], [12], [13], [14], [15].",
"In this work, the typical two-ray multi-path model is further studied based on an antenna array and a closed-form solution is provided.",
"Such a two-ray model is more realistic in the millimetre wave band given the more directional propagation model in this frequency band.",
"Furthermore, the antenna location optimisation problem is investigated in the context of positional modulation and a compressive-sensing based design is proposed.",
"The remaining part of this paper is structured as follows.",
"A review of the two-ray model is given in Sec.",
".",
"Positional modulation design based on a given array geometry and an array with optimised antenna locations are presented in Sec.",
".",
"Design examples are provided in Sec.",
", followed by conclusions in Sec.",
"."
],
[
"Review of Two-Path Model",
"An $N$ -element omni-directional linear antenna array for transmit beamforming [11] is shown in Fig.",
"REF , where the spacing between the zeroth and the $n$ -th antennas is represented by $d_n$ for $n = 1,\\ldots , N-1$ , with the transmission angle $\\theta \\in [-90^{\\circ },90^{\\circ }]$ .",
"The weight coefficient of each antenna is denoted by $w_{n}$ , for $n=0, \\ldots , N-1$ .",
"The desired position is represented by $L$ with a distance $D_1$ to the transmit array and a vertical distance $h$ to the broadside direction where $h$ is positive for $L$ above the broadside direction and negative for the opposite.",
"The projection of $D_1$ onto the broadside direction is represented by $D_2$ .",
"The positions of eavesdroppers $E$ are shown on the circumference of the circle, with the radius $\\bar{r}$ and angle $\\eta \\in [0^\\circ ,360^\\circ )$ to the circle centre $L$ .",
"For eavesdroppers in the direction $\\eta $ , we have the corresponding $\\hat{h}$ and $\\hat{l}$ , representing the vertical height and horizontal length relative to the centre point L, with $\\bar{r} = \\sqrt{\\hat{h}^2+\\hat{l}^2}$ , and the distance to transmitters is represented by $D_3$ .",
"To produce the required reflected path, a reflecting surface with a distance $H$ above and perpendicular to the antenna array is created to form the two-ray model.",
"The reflected distances $R_1$ and $R_2$ represent the path length before and after reflection, and the transmission angle for the reflected path is determined by $\\zeta \\in (0^{\\circ },90^{\\circ }]$ .",
"For signals transmitted to the desired location $L$ , as shown in Fig.",
"REF , we have $\\begin{split}D_2 &= \\sqrt{D_1^2-h^2},\\quad \\theta = \\tan ^{-1}(h/D_2),\\\\\\zeta &= \\tan ^{-1}((2H-h)/D_2).\\end{split}$ For signals transmitted to the eavesdroppers, $\\begin{split}\\hat{h}(\\eta ) &= \\bar{r}\\sin \\eta ,\\quad \\hat{l}(\\eta ) = \\bar{r}\\cos \\eta ,\\\\D_3 &= \\sqrt{(D_2+\\hat{l})^2+(h+\\hat{h})^2}.\\end{split}$ The corresponding $\\theta (\\eta )$ and $\\zeta (\\eta )$ for the LOS and reflected paths can be formulated as $\\begin{split}\\theta (\\eta ) &= \\tan ^{-1}((h+\\hat{h})/(D_2+\\hat{l})),\\\\\\zeta (\\eta ) &= \\tan ^{-1}((2H-\\hat{h}-h)/(D_2+\\hat{l})).\\end{split}$ Then, for the reflected path, $R_1(\\zeta )$ and $R_2(\\zeta )$ are given by $R_1(\\zeta ) = H/\\sin \\zeta ,\\quad R_2(\\zeta ) = (H-h-\\hat{h})/\\sin \\zeta .$ The steering vector for the LOS path and the reflected path in two-ray model are, respectively, given by $\\begin{split}\\textbf {s}(\\omega ,\\theta ) &= [1, e^{j\\omega d_1\\sin \\theta /c}, \\ldots , e^{j\\omega d_{N-1}\\sin \\theta /c}]^{T},\\\\\\hat{\\textbf {s}}(\\omega ,\\zeta ) &= [1, e^{j\\omega d_1\\sin \\zeta /c}, \\ldots , e^{j\\omega d_{N-1}\\sin \\zeta /c}]^{T}.\\end{split}$ Moreover, phase shift and power attenuation caused by these multiple paths need to be considered [11].",
"When $\\hat{h}$ and $\\hat{l}$ are both zero-valued, as shown in (REF ), $D_3 = D_1$ .",
"Therefore, we can consider the length $D_1$ as a special case of the length $D_3$ .",
"Then the phase shifts for LOS paths is given by $\\psi (\\theta ) = 2\\pi \\times rem(D_3(\\theta ),\\lambda ),$ where $rem(A,\\lambda )$ represents the remainder of $A$ divided by $\\lambda $ .",
"The phase shift for the reflected path is determined by $R_1(\\zeta )+R_2(\\zeta )$ and given by $\\phi (\\zeta ) = \\pi +2\\pi \\times rem(R_1(\\zeta )+R_2(\\zeta ),\\lambda ),$ where $\\pi $ is caused by the reflecting surface.",
"The attenuation ratio for a LOS is given by [11] $\\nu (\\theta ) = D/D_3(\\theta ).$ Here $D$ is assumed to be the distance where the received signal has unity power.",
"Similarly, the attenuation ratio for the signal received via the reflected path is given by $\\xi (\\zeta ) = D/(R_1(\\zeta )+R_2(\\zeta )).$ Then, in the two-ray model, the beam response of the array, represented by $p(\\theta ,\\zeta )$ , is a combination of signals through the LOS path and the reflected path, $\\begin{split}&p(\\theta ,\\zeta )=\\\\&\\nu (\\theta )e^{j\\psi (\\theta )}(\\textbf {w}^{H}\\textbf {s}(\\omega ,\\theta ))+\\xi (\\zeta )e^{j\\phi (\\zeta )}(\\textbf {w}^{H}\\hat{\\textbf {s}}(\\omega ,\\zeta )),\\end{split}$ with the weight vector $\\textbf {w} = [w_{0}, w_{1}, \\ldots , w_{N-1}]^{T}$ ."
],
[
"Positional modulation design for a given array geometry",
"The objective of positional modulation design is to find a set of weight coefficients creating signals with a given modulation pattern to desired locations, while the modulations of the signals received around them are distorted.",
"For $M$ -ary signaling, such as multiple phase shift keying (MPSK), there are $M$ sets of desired array responses $p_m(\\theta ,\\zeta )$ , with a corresponding weight vector $\\textbf {w}_{m}=[w_{0,m}, \\ldots , w_{N-1,m}]^T$ , $m=0, \\ldots , M-1$ .",
"Assuming in total $R$ locations in the design ($r$ desired locations and $R-r$ eavesdropper locations), we can have the corresponding transmission angles $\\theta _k$ for LOS and $\\zeta _k$ for the reflected path to the $k$ -th position, $k = 0,\\ldots ,R-1$ .",
"Then an $N\\times r$ matrix $\\textbf {S}_{L}$ is constructed as the set of steering vectors for the LOS path to desired receivers, and similarly we have $\\textbf {S}_{E} = [\\textbf {s}(\\omega ,\\theta _0),\\textbf {s}(\\omega ,\\theta _1), \\ldots , \\textbf {s}(\\omega ,\\theta _{R-r-1})]$ (an $N\\times (R-r)$ matrix) for steering vectors to eavesdroppers.",
"The corresponding steering vectors for the reflected path to desired receivers and eavesdroppers are given by $\\hat{\\textbf {S}}_{L}$ and $\\hat{\\textbf {S}}_{E}$ , respectively.",
"$\\textbf {p}_{m,L}$ ($1\\times r$ vector) and $\\textbf {p}_{m,E}$ ($1\\times (R-r)$ vector) are required responses for the desired locations and the eavesdroppers for the $m$ -th constellation point.",
"Moreover, the phase shifts for the LOS and reflected paths to both eavesdroppers and desired receivers, and their corresponding attenuation ratios are given by $\\begin{split}\\mathbf {\\psi }_{E} & = [\\psi (\\theta _0), \\psi (\\theta _1), \\ldots , \\psi (\\theta _{R-r-1})],\\\\\\mathbf {\\psi }_{L} & = [\\psi (\\theta _{R-r}), \\psi (\\theta _{R-r+1}), \\ldots , \\psi (\\theta _{R-1})],\\\\\\mathbf {\\phi }_{E} & = [\\phi (\\zeta _0), \\phi (\\zeta _1), \\ldots , \\phi (\\zeta _{R-r-1})],\\\\\\mathbf {\\phi }_{L} & = [\\phi (\\zeta _{R-r}), \\phi (\\zeta _{R-r+1}), \\ldots , \\phi (\\zeta _{R-1})],\\\\\\mathbf {\\nu }_{E} & = [\\nu (\\theta _0), \\nu (\\theta _1), \\ldots , \\nu (\\theta _{R-r-1})],\\\\\\mathbf {\\nu }_{L} & = [\\nu (\\theta _{R-r}), \\nu (\\theta _{R-r+1}), \\ldots , \\nu (\\theta _{R-1})],\\\\\\mathbf {\\xi }_{E} & = [\\xi (\\zeta _0), \\xi (\\zeta _1), \\ldots , \\xi (\\zeta _{R-r-1})],\\\\\\mathbf {\\xi }_{L} & = [\\xi (\\zeta _{R-r}), \\xi (\\zeta _{R-r+1}), \\ldots , \\xi (\\zeta _{R-1})].\\end{split}$ Then, for the $m$ -th constellation point, the coefficients can be formulated as $\\begin{split}&\\underset{\\textbf {w}_m}{\\text{min}}||\\textbf {p}_{m,E}-(\\mathbf {\\nu }_{E}\\cdot e^{j\\mathbf {\\psi }_{E}}\\cdot (\\textbf {w}_{m}^{H}\\textbf {S}_{E})+\\mathbf {\\xi }_{E}\\cdot e^{j\\mathbf {\\phi }_{E}}\\cdot (\\textbf {w}_m^{H}\\hat{\\textbf {S}}_{E}))||_{2}\\\\&\\text{subject to}\\\\&\\mathbf {\\nu }_{L}\\cdot e^{j\\mathbf {\\psi }_{L}}\\cdot (\\textbf {w}_{m}^{H}\\textbf {S}_{L})+\\mathbf {\\xi }_{L}\\cdot e^{j\\mathbf {\\phi }_{L}}\\cdot (\\textbf {w}_m^{H}\\hat{\\textbf {S}}_{L}) = \\textbf {p}_{m,L},\\end{split}$ where $\\cdot $ is the dot product.",
"Its solution can be solved by the method of Lagrange multipliers, and the optimum value for the weight vector $\\textbf {w}_{m}$ is given by $\\begin{split}\\textbf {w}_{m} = &\\textbf {K}_5^{-1}(\\hat{\\textbf {S}}_{E}\\textbf {K}_2\\textbf {p}^H_{m,E}-\\textbf {S}_{E}\\textbf {K}_1\\textbf {p}^H_{m,E}\\\\&-\\textbf {K}_6^H\\textbf {S}_{L}\\textbf {K}_3-\\textbf {K}_6^H\\hat{\\textbf {S}}_{L}\\textbf {K}_4)\\end{split}$ where $\\begin{split}\\textbf {K}_1 &= diag(\\mathbf {\\nu }_{E}diag(e^{j\\mathbf {\\psi }_{E}})),\\quad \\textbf {K}_2 = diag(\\mathbf {\\xi }_{E}diag(e^{j\\mathbf {\\phi }_{E}})),\\\\\\textbf {K}_3 &= diag(\\mathbf {\\nu }_{L}diag(e^{j\\mathbf {\\psi }_{L}})),\\quad \\textbf {K}_4 = diag(\\mathbf {\\xi }_{L}diag(e^{j\\mathbf {\\phi }_{L}})),\\\\\\textbf {K}_5 &= \\textbf {S}_{E}\\textbf {K}_1\\textbf {K}_1^H\\textbf {S}^H_{E}+\\textbf {S}_{E}\\textbf {K}_1\\textbf {K}_2^H\\textbf {S}^H_{E}\\\\&+\\hat{\\textbf {S}}_{E}\\textbf {K}_2\\textbf {K}^H_1\\textbf {S}^H_{E}+\\hat{\\textbf {S}}_{E}\\textbf {K}_2\\textbf {K}^H_2\\hat{\\textbf {S}}^H_{E},\\\\\\textbf {K}_6 &= (\\textbf {p}_{m,E}\\textbf {K}^H_2\\hat{\\textbf {S}}^H_{E}\\textbf {K}_5^{-H}\\textbf {S}_{L}\\textbf {K}_3-\\textbf {p}_{m,E}\\textbf {K}^H_1\\textbf {S}^H_{E}\\textbf {K}_5^{-H}\\textbf {S}_{L}\\textbf {K}_3\\\\&-\\textbf {p}_{m,E}\\textbf {K}^H_2\\hat{\\textbf {S}}^H_{E}\\textbf {K}_5^{-H}\\hat{\\textbf {S}}_{L}\\textbf {K}_4-\\textbf {p}_{m,E}\\textbf {K}^H_1\\textbf {S}^H_{E}\\textbf {K}_5^{-H}\\hat{\\textbf {S}}_{L}\\textbf {K}_4\\\\&-\\textbf {p}_{m,L})\\\\&\\times (\\textbf {K}_3^H\\textbf {S}^H_{L}\\textbf {K}_5^{-H}\\textbf {S}_{L}\\textbf {K}_3+\\textbf {K}_4^H\\hat{\\textbf {S}}^H_{L}\\textbf {K}_5^{-H}\\textbf {S}_{L}\\textbf {K}_3\\\\&+\\textbf {K}_3^H\\textbf {S}^H_{L}\\textbf {K}_5^{-H}\\hat{\\textbf {S}}_{L}\\textbf {K}_4+\\textbf {K}_4^H\\hat{\\textbf {S}}^H_{L}\\textbf {K}_5^{-H}\\hat{\\textbf {S}}_{L}\\textbf {K}_4)^{-1}.\\end{split}$"
],
[
"Positional modulation design for an optimised locations array",
"Equation (REF ) is for designing the positional modulation coefficients for a given set of antenna locations.",
"In practice, we may opt to find optimised locations to construct an array for an improved performance, which can be considered as a sparse antenna array design problem [16], [17].",
"Many methods have been proposed for the design of a general sparse antenna array, including the genetic algorithm [18], [19], [20], simulated annealing [21], and compressive sensing (CS) [22], [23], [24], [25], and in this section, CS-based methods is studied.",
"For CS-based sparse array design for positional modulation, a given aperture is densely sampled with a large number ($N$ ) of potential antennas, as shown in Fig REF , and the values of $d_{n}$ , for $n=1, 2, \\ldots , N-1$ , are selected to give a uniform grid.",
"Through selecting the minimum number of non-zero valued weight coefficients, where the corresponding antennas are kept, and the rest of the antennas with zero-valued coefficients are removed, to generate a response close to the desired one, sparseness of the design is acquired [5], [6].",
"Then for the $m$ -th constellation point, the cost function is $\\underset{\\textbf {w}_m}{\\text{min}}||\\textbf {w}_{m}||_1$ and the constraints are $||\\textbf {p}_{m,E}-(\\mathbf {\\nu }_{E}\\cdot e^{j\\mathbf {\\psi }_{E}}\\cdot (\\textbf {w}_{m}^{H}\\textbf {S}_{E})+\\mathbf {\\xi }_{E}\\cdot e^{j\\mathbf {\\phi }_{E}}\\cdot (\\textbf {w}_m^{H}\\hat{\\textbf {S}}_{E}))||_{2}\\le \\alpha $ and $\\mathbf {\\nu }_{L}\\cdot e^{j\\mathbf {\\psi }_{L}}\\cdot (\\textbf {w}_{m}^{H}\\textbf {S}_{L})+\\mathbf {\\xi }_{L}\\cdot e^{j\\mathbf {\\phi }_{L}}\\cdot (\\textbf {w}_m^{H}\\hat{\\textbf {S}}_{L}) = \\textbf {p}_{m,L}$ , where $||\\cdot ||_1$ is the $l_1$ norm, used as an approximation to the $l_0$ norm and $\\alpha $ is the allowed difference between the desired and designed responses.",
"As each antenna element corresponds to $M$ weight coefficients and these $M$ coefficients correspond to $M$ symbols, to remove the $n$ -th antenna, we need all coefficients in the following vector $\\tilde{\\textbf {w}}_{n}$ to be zero-valued or $||\\tilde{\\textbf {w}}_{n}||_2 = 0$ [5], [6], $\\tilde{\\textbf {w}}_{n}=[w_{n,0}, \\ldots , w_{n,M-1}],$ where $w_{n,m}$ represents the coefficients on the $n$ -th antenna for the $m$ -th symbol.",
"Then, to calculate the minimum number of antenna elements, we gather all $||\\tilde{\\textbf {w}}_{n}||_2$ for $n = 0,\\ldots , N-1$ to form a new vector $\\hat{\\textbf {w}}$ , $\\hat{\\textbf {w}} = [||\\tilde{\\textbf {w}}_{0}||_{2}, ||\\tilde{\\textbf {w}}_{1}||_2, \\ldots , ||\\tilde{\\textbf {w}}_{N-1}||_2]^{T}.$ Moreover, we need to impose positional modulation constraints including $\\begin{split}\\textbf {W} &= [\\textbf {w}_{0}, \\textbf {w}_{1}, \\ldots , \\textbf {w}_{M-1}],\\textbf {P}_{E} = [\\textbf {p}_{0,E}, \\textbf {p}_{1,E}, \\ldots , \\textbf {p}_{M-1,E}]^T,\\\\\\textbf {P}_{L} &= [\\textbf {p}_{0,L}, \\textbf {p}_{1,L}, \\ldots , \\textbf {p}_{M-1,L}]^T,\\\\\\tilde{\\mathbf {\\nu }}_{E} &= \\mathbf {\\nu }_{E}\\otimes \\text{ones}(M,1),\\;\\tilde{\\mathbf {\\nu }}_{L} = \\mathbf {\\nu }_{L}\\otimes \\text{ones}(M,1),\\\\\\tilde{\\mathbf {\\xi }}_{E} &= \\mathbf {\\xi }_{E}\\otimes \\text{ones}(M,1),\\;\\tilde{\\mathbf {\\xi }}_{L} = \\mathbf {\\xi }_{L}\\otimes \\text{ones}(M,1),\\\\\\tilde{\\mathbf {\\psi }}_{E} &= \\mathbf {\\psi }_{E}\\otimes \\text{ones}(M,1),\\;\\tilde{\\mathbf {\\psi }}_{L} = \\mathbf {\\psi }_{L}\\otimes \\text{ones}(M,1),\\\\\\tilde{\\mathbf {\\phi }}_{E} &= \\mathbf {\\phi }_{E}\\otimes \\text{ones}(M,1),\\;\\tilde{\\mathbf {\\phi }}_{L} = \\mathbf {\\phi }_{L}\\otimes \\text{ones}(M,1),\\end{split}$ where $\\otimes $ stands for the Kronecker product, and $\\text{ones}(M,1)$ is an $M\\times 1$ matrix of ones.",
"Then the group sparsity based sparse array design for DM [5], [6] can be formulated as $\\begin{split}&\\underset{\\textbf {W}}{\\text{min}}||\\hat{\\textbf {w}}||_{1}\\\\&\\text{subject to}\\\\&||\\textbf {P}_{E}-(\\tilde{\\mathbf {\\nu }}_{E}\\cdot e^{j\\tilde{\\mathbf {\\psi }}_{E}}\\cdot (\\textbf {W}^{H}\\textbf {S}_{E})+\\tilde{\\mathbf {\\xi }}_{E}\\cdot e^{j\\tilde{\\mathbf {\\phi }}_{E}}\\cdot (\\textbf {W}^{H}\\hat{\\textbf {S}}_{E}))||_{2}\\le \\alpha \\\\&\\tilde{\\mathbf {\\nu }}_{L}\\cdot e^{j\\tilde{\\mathbf {\\psi }}_{L}}\\cdot (\\textbf {W}^{H}\\textbf {S}_{L})+\\tilde{\\mathbf {\\xi }}_{L}\\cdot e^{j\\tilde{\\mathbf {\\phi }}_{L}}\\cdot (\\textbf {W}^{H}\\hat{\\textbf {S}}_{L}) = \\textbf {P}_{L}.\\end{split}$ As the reweighted $l_1$ norm minimisation has a closer approximation to the $l_0$ norm [26], [27], [28], we can further modify (REF ) into the reweighted form in a similar way as in [5], where at the $u$ -th iteration, $\\begin{split}&\\underset{\\textbf {W}}{\\text{min}}\\sum \\limits _{n=0}^{N-1} \\delta _{n}^u||\\tilde{\\textbf {w}}_{n}^u||_2\\\\&\\text{subject to}\\quad ||\\textbf {P}_{E}-(\\tilde{\\mathbf {\\nu }}_{E}\\cdot e^{j\\tilde{\\mathbf {\\psi }}_{E}}\\cdot ((\\textbf {W}^{u})^H\\textbf {S}_{E})\\\\&+\\tilde{\\mathbf {\\xi }}_{E}\\cdot e^{j\\tilde{\\mathbf {\\phi }}_{E}}\\cdot ((\\textbf {W}^{u})^H\\hat{\\textbf {S}}_{E}))||_{2}\\le \\alpha \\\\&\\tilde{\\mathbf {\\nu }}_{L}\\cdot e^{j\\tilde{\\mathbf {\\psi }}_{L}}\\cdot ((\\textbf {W}^{u})^H\\textbf {S}_{L})+\\tilde{\\mathbf {\\xi }}_{L}\\cdot e^{j\\tilde{\\mathbf {\\phi }}_{L}}\\cdot ((\\textbf {W}^{u})^H\\hat{\\textbf {S}}_{L}) = \\textbf {P}_{L}.\\end{split}$ Here the superscript $u$ indicates the $u$ -th iteration, and $\\delta _{n}$ is the reweighting term for the $n$ -th row of coefficients, given by $\\delta _{n}^u=(||\\tilde{\\textbf {w}}_{n}^{u-1}||_2+\\gamma )^{-1}$ .",
"($\\gamma >0$ is required to provide numerical stability and the iteration process is described as in [5].)",
"The problem in (REF ) and (REF ) can be solved by cvx [29], [30]."
],
[
"Design examples",
"In this section, we provide several representative design examples to show the performance of the proposed formulations in the two-ray model.",
"Without loss of generality, we assume there is one desired location at the circle centre with $\\theta = 0^{\\circ }$ , and $H = 500\\lambda $ , $D_1 = D = 1000\\lambda $ .",
"Eavesdroppers are located at the circumference of the circle with $\\bar{r} = 8.4\\lambda $ and $\\eta \\in [0^{\\circ }, 360^{\\circ })$ , sampled every $1^{\\circ }$ .",
"With the radius $\\bar{r}$ and the angle $\\eta $ based on (REF ), it can be seen that all eavesdroppers are in the directions of $\\theta \\in (-0.5^\\circ , 0.5^\\circ )$ , i.e.",
"aligned with or very close to the desired user.",
"The desired response is a value of one magnitude (the gain is 0dB) with $90^\\circ $ phase shift at the desired location (QPSK), i.e.",
"symbols `00', `01', `11', `10' correspond to $45^{\\circ }$ , $135^{\\circ }$ , $-135^{\\circ }$ and $-45^{\\circ }$ , respectively, and a value of $0.1$ (magnitude) with random phase shifts at eavesdroppers.",
"Moreover the bit error rate (BER) result is also presented.",
"Here the signal to noise ratio (SNR) is set at 12 dB at the desired location, and we assume the additive white Gaussian noise (AWGN) level is at the same level for all eavesdroppers.",
"The number of antenna elements for the ULA design is $N = 30$ , while for the sparse array design, the maximum aperture of the array is set to $20\\lambda $ with 401 equally spaced potential antennas.",
"To make a fair comparison, we use the value of error norm between desired and designed array responses calculated from the ULA design (REF ) as the threshold $\\alpha $ for the sparse array design.",
"$\\gamma = 0.001$ used in the reweighted $l_1$ norm minimisation (REF ) indicates that antennas associated with a weight value smaller than $0.001$ will be removed.",
"The resultant beam and phase patterns for the eavesdroppers based on the ULA design (REF ) are shown in Figs.",
"REF and REF , where the beam response level at all locations of the eavesdroppers ($\\eta \\in [0^\\circ , 360^\\circ $ )) is lower than 0dB which is the beam response for the desired locations.",
"The phase of signal at these eavesdroppers are random while the desired phase for these four symbols should be QPSK modulation, as mentioned before.",
"The beam and phase patterns for the sparse array design in (REF ) are not shown as they have similar characteristics to ULA's beam and phase responses.",
"As shown in Table REF , with a fewer number of antennas, the sparse array design results provide a better match to the desired responses based on the error norm of array responses.",
"Considering the imperfect knowledge of the geometry, e.g.",
"the locations of eavesdroppers are not exactly the same as the locations we thought.",
"Here we assume eavesdroppers are distributed on the circumferences of the circles with $\\bar{r} = 8\\lambda $ and $\\bar{r} = 8.8\\lambda $ , while the set of weight coefficients are designed for $\\bar{r} = 8.4\\lambda $ .",
"Fig.",
"REF shows the BERs based on the ULA design (REF ) in the multi-path model, where BERs at these eavesdroppers in these cases are still much higher than the rate in the desired location ($10^{-5}$ ).",
"While in LOS model, as shown in Fig.",
"REF , BERs based on $\\bar{r} = 8.4\\lambda $ at some positions of the eavesdroppers are close to $10^{-3}$ , lower than the counterpart ($10^{-1}$ ) in the multi-path model, indicated by dash line in Fig.",
"REF , demonstrating the effectiveness of the multi-path scheme.",
"Moreover, for eavesdroppers close to the desired direction and also integer wavelengths away from the desired location, e.g.",
"$\\bar{r} = 8\\lambda $ , $\\eta = 0^\\circ $ and $\\eta = 180^\\circ $ , the BERs reach $10^{-5}$ , same as in desired locations, much lower than the BERs at these positions in the multi-path model, further demonstrating the effectiveness of the proposed positional modulation designs.",
"The BERs for the sparse array design (REF ) are not shown as they have similar features to the ULA designs.",
"Table: Summary of the design results.Figure: Resultant beam and phase patterns based on the ULA design () for eavesdroppers.Figure: BERs patterns for the eavesdroppers and desired receiver based on ULA designs (a) in multi-path model () and (b) in LOS model."
],
[
"Conclusions",
"In this paper, a two-ray transmission model has been studied for positional modulation, where signals via LOS and reflected paths are combined at the receiver side.",
"With the positional modulation technique, signals with a given modulation pattern can only be received at desired locations, but scrambled for positions around them.",
"By the proposed designs, the multi-path effect is exploited to overcome the drawback of traditional DM design when eavesdroppers are aligned with or very close to the desired users.",
"Examples for a given array geometry and an optimised sparse array have been provided to verify the effectiveness of the proposed designs.",
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]
] | 1808.08383 |
[
[
"The Complexity of $(\\Delta + 1)$Coloring inCongested Clique, Massively\n Parallel Computation,and Centralized Local Computation"
],
[
"Abstract We present new randomized algorithms that improve the complexity of the classic $(\\Delta+1)$-coloring problem, and its generalization $(\\Delta+1)$-list-coloring, in three well-studied models of distributed, parallel, and centralized computation: Distributed Congested Clique: We present an $O(1)$-round randomized algorithm for $(\\Delta+1)$-list coloring in the congested clique model of distributed computing.",
"This settles the asymptotic complexity of this problem.",
"It moreover improves upon the $O(\\log^\\ast \\Delta)$-round randomized algorithms of Parter and Su [DISC'18] and $O((\\log\\log \\Delta)\\cdot \\log^\\ast \\Delta)$-round randomized algorithm of Parter [ICALP'18].",
"Massively Parallel Computation: We present a $(\\Delta+1)$-list coloring algorithm with round complexity $O(\\sqrt{\\log\\log n})$ in the Massively Parallel Computation (MPC) model with strongly sublinear memory per machine.",
"This algorithm uses a memory of $O(n^{\\alpha})$ per machine, for any desirable constant $\\alpha>0$, and a total memory of $\\widetilde{O}(m)$, where $m$ is the size of the graph.",
"Notably, this is the first coloring algorithm with sublogarithmic round complexity, in the sublinear memory regime of MPC.",
"For the quasilinear memory regime of MPC, an $O(1)$-round algorithm was given very recently by Assadi et al.",
"[SODA'19].",
"Centralized Local Computation: We show that $(\\Delta+1)$-list coloring can be solved with $\\Delta^{O(1)} \\cdot O(\\log n)$ query complexity, in the centralized local computation model.",
"The previous state-of-the-art for $(\\Delta+1)$-list coloring in the centralized local computation model are based on simulation of known LOCAL algorithms."
],
[
"Introduction, Related Work, and Our Results",
"In this paper, we present improved randomized algorithms for vertex coloring in three models of distributed, parallel, and centralized computation: the congested clique model of distributed computing, the massively parallel computation model, and the centralized local computation model.",
"We next overview these results in three different subsections, while putting them in the context of the state of the art.",
"The next section provides a technical overview of the known algorithmic tools as well as the novel ingredients that lead to our results."
],
[
"$(\\Delta + 1)$ -coloring and {{formula:81b4eb4a-c22c-4499-a661-3016c3d6dbb2}} -list Coloring.",
"Our focus is on the standard $\\Delta +1$ vertex coloring problem, where $\\Delta $ denotes the maximum degree in the graph.",
"All our results work for the generalization of the problem to $(\\Delta + 1)$ -list coloring problem, defined as follows: each vertex $v$ in the graph $G=(V, E)$ is initially equipped with a set of colors $\\Psi (v)$ such that $|\\Psi (v)| = \\Delta + 1$ .",
"The goal is to find a proper vertex coloring where each vertex $v\\in V$ is assigned a color in $\\Psi (v)$ such that no two adjacent vertices are colored the same."
],
[
"Models of Distributed Computation.",
"There are three major models for distributed graph algorithms, namely $\\mathsf {LOCAL}$ , $\\mathsf {CONGEST}$ , and $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ .",
"In the $\\mathsf {LOCAL}$ model [46], [53], the input graph $G=(V,E)$ is identical to the communication network and each $v\\in V$ hosts a processor that initially knows $\\deg (v)$ , a unique $\\Theta (\\log n)$ -bit $\\operatorname{ID}(v)$ , and global graph parameters $n = |V|$ and $\\Delta =\\max _{v\\in V} \\deg (v)$ .",
"Each processor is allowed unbounded computation and has access to a stream of private random bits.",
"Time is partitioned into synchronized rounds of communication, in which each processor sends one unbounded message to each neighbor.",
"At the end of the algorithm, each $v$ declares its output label, e.g., its own color.",
"The $\\mathsf {CONGEST}$ model [53] is a variant of $\\mathsf {LOCAL}$ where there is an $O(\\log n)$ -bit message size constraint.",
"The $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ model, introduced in [49], is a variant of $\\mathsf {CONGEST}$ that allows all-to-all communication: Each vertex initially knows its adjacent edges of the input graph $G=(V, E)$ .",
"In each round, each vertex is allowed to transmit $n-1$ many $O(\\log n)$ -bit messages, one addressed to each other vertex.",
"In this paper, our new distributed result is an improvement for coloring in the $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ model.",
"It is worth noting that the $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ model has been receiving extensive attention recently, see e.g., [58], [24], [13], [45], [23], [51], [39], [37], [18], [38], [15], [29], [21], [31], [32], [30], [57], [52], [14]."
],
[
"State of the Art for Coloring in $\\mathsf {LOCAL}$ and {{formula:168dcb15-95c5-426d-9db1-1f59da213b84}} .",
"Most prior works on distributed coloring focus on the $\\mathsf {LOCAL}$ model.",
"The current state-of-the-art randomized upper bound for the $(\\Delta +1)$ -list coloring problem is $O(\\log ^\\ast \\Delta ) + O(\\mathsf {Det}_{\\scriptscriptstyle d}({\\operatorname{poly}}\\log n)) = O(\\mathsf {Det}_{\\scriptscriptstyle d}({\\operatorname{poly}}\\log n))$ of [20] (which builds upon the techniques of [40]), where $\\mathsf {Det}_{\\scriptscriptstyle d}(n^{\\prime }) = 2^{O(\\sqrt{\\log \\log n^{\\prime }})}$ is the deterministic complexity of $(\\deg +1)$ -list coloring on $n^{\\prime }$ -vertex graphs [56].",
"In the $(\\deg +1)$ -list coloring problem, each $v$ has a palette of size $\\deg (v)+1$ .",
"This algorithm follows the graph shattering framework [10], [31].",
"The pre-shattering phase takes $O(\\log ^\\ast \\Delta )$ rounds.",
"After that, the remaining uncolored vertices form connected components of size $O({\\operatorname{poly}}\\log n)$ .",
"The post-shattering phase then applies a $(\\deg +1)$ -list coloring deterministic algorithm to color all these vertices."
],
[
"State of the Art for Coloring in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ .",
"Hegeman and Pemmaraju [37] gave algorithms for $O(\\Delta )$ -coloring in the $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ model, which run in $O(1)$ rounds if $\\Delta \\ge \\Theta (\\log ^4 n )$ and in $O(\\log \\log n)$ rounds otherwise.",
"It is worth noting that $O(\\Delta )$ coloring is a significantly more relaxed problem in comparison to $\\Delta +1$ coloring.",
"For instance, we have long known a very simple $O(\\Delta )$ -coloring algorithm in $\\mathsf {LOCAL}$ -model algorithm with round complexity $2^{O(\\sqrt{\\log \\log n})}$ [10], but only recently such a round complexity was achieved for $\\Delta +1$ coloring [20], [40].",
"Our focus is on the much more stringent $\\Delta +1$ coloring problem.",
"For this problem, the $\\mathsf {LOCAL}$ model algorithms of [20], [40] need messages of $O(\\Delta ^2 \\log n)$ bits, and thus do not extend to $\\mathsf {CONGEST}$ or $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ .",
"For $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ model, the main challenge is when $\\Delta > \\sqrt{n}$ , as otherwise, one can simulate the algorithm of [20] by leveraging the all-to-all communication in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ which means each vertex in each round is capable of communicating $O(n \\log n)$ bits of information.",
"Parter [52] designed the first sublogarithmic-time $(\\Delta +1)$ coloring algorithm for $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ , which runs in $O(\\log \\log \\Delta \\log ^\\ast \\Delta )$ rounds.",
"The algorithm of [52] is able to reduce the maximum degree to $O(\\sqrt{n})$ in $O(\\log \\log \\Delta )$ iterations, and each iteration invokes the algorithm of [20] on instances of maximum degree $O(\\sqrt{n})$ .",
"Once the maximum degree is $O(\\sqrt{n})$ , the algorithm of [20] can be implemented in $O(\\log ^\\ast \\Delta )$ rounds in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ .",
"Subsequent to [52], the upper bound was improved to $O(\\log ^\\ast \\Delta )$ in [57].",
"Parter and Su [57] observed that the algorithm of [52] only takes $O(1)$ iterations if we only need to reduce the degree to $n^{1/2 + \\epsilon }$ , for some constant $\\epsilon > 0$ , and they achieved this by modifying the internal details of [20] to reduce the required message size to $O(\\Delta ^{8/5} \\log n)$ ."
],
[
"Our Result.",
"For the $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ model, we present a new algorithm for $(\\Delta +1)$ -list coloring in the randomized congested clique model running in $O(1)$ rounds.",
"This improving on the previous best known $O(\\log ^\\ast \\Delta )$ -round algorithm of Parter and Su [57] and settles the asymptotic complexity of the problem.",
"Theorem 1.1 There is an $O(1)$ -round algorithm that solves the $(\\Delta +1)$ -list coloring problem in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ , with success probability $1 - 1/{\\operatorname{poly}}(n)$ .",
"The proof is presented in two parts: If $\\Delta \\ge \\log ^{4.1} n$ , the algorithm of thm:largedegree solves the $(\\Delta +1)$ -list coloring problem in $O(1)$ rounds; otherwise, the algorithm of thm:smalldegree solves the problem in $O(1)$ rounds."
],
[
"Model.",
"The Massively Parallel Computation (MPC) model was introduced by Karloff et al.",
"[44], as a theoretical abstraction for practical large-scale parallel processing settings such as MapReduce [22], Hadoop [62], Spark [63], and Dryad [41], and it has been receiving increasing more attention over the past few years [44], [33], [48], [16], [4], [17], [37], [3], [60], [42], [19], [5], [1], [30], [35], [11], [7], [9], [6].",
"In the $\\mathsf {MPC}$ model, the system consists of a number of machines, each with $S$ bits of memory, which can communicate with each other in synchronous rounds through a complete communication network.",
"Per round, each machine can send or receive at most $S$ bits in total.",
"Moreover, it can perform some ${\\operatorname{poly}}(S)$ computation, given the information that it has.",
"In the case of graph problems, we assume that the graph $G$ is partitioned among the machines using a simple and globally known hash function such that each machine holds at most $S$ bits, and moreover, for each vertex or potential edge of the graph, the hash function determines which machines hold that vertex or edge.",
"Thus, the number of machines is $\\Omega (m/S)$ and ideally not too much higher, where $m$ denotes the number of edges.",
"At the end, each machine should know the output of the vertices that it holds, e.g., their color."
],
[
"State of the Art for Coloring.",
"The $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ algorithms discussed above can be used to obtain $\\mathsf {MPC}$ algorithms with the same asymptotic round complexity if machines have memory of $S=\\Omega (n\\log n)$ bits.",
"In particular, the work of Parter and Su [57] leads to an $O(\\log ^* \\Delta )$ -round $\\mathsf {MPC}$ algorithm for machines with $S=\\Omega (n\\log n)$ bits.",
"However, this $\\mathsf {MPC}$ algorithm would have two drawbacks: (A) it uses $\\Omega (n^2 \\log n)$ global memory, and thus would require $(n^2 \\log n)/S$ machines, which may be significantly larger than $\\tilde{O}(m)/S$ .",
"This is basically because the algorithm makes each vertex of the graph learn some $\\widetilde{\\Theta }(n)$ bits of information.",
"(B) It is limited to machines with $S=\\Omega (n\\log n)$ memory, and it does not extend to the machines with strongly sublinear memory, which is gaining more attention recently due to the increase in the size of graphs.",
"We note that for the regime of machines with super-linear memory, very recently, Assadi, Chen, and Khanna [2] gave an $O(1)$ -round algorithm which uses only $O(n\\log ^3 n)$ global memory.Here “global memory” refers to the memory used for communication.",
"Of course we still need $\\tilde{O}(m)$ memory to store the graph.",
"However, this algorithm also relies heavily on $S=\\Omega (n\\log ^3 n)$ memory per machine and cannot be run with weaker machines that have strongly sublinear memory."
],
[
"Our Result.",
"We provide the first sublogarithmic-time algorithm for $(\\Delta +1)$ coloring and $(\\Delta +1)$ -list coloring in the $\\mathsf {MPC}$ model with strongly sublinear memory per machine: Theorem 1.2 There is an $\\mathsf {MPC}$ algorithm that, in $O(\\log ^*\\Delta +\\sqrt{\\log \\log n}) = O(\\sqrt{\\log \\log n})$ rounds, w.h.p.",
"computes a $(\\Delta +1)$ list-coloring of an $n$ -vertex graph with $m$ edges and maximum degree $\\Delta $ and that uses $O(n^{\\alpha })$ memory per machine, for an arbitrary constant $\\alpha >0$ , as well as a total memory of $\\widetilde{O}(m)$ .",
"The proof is presented in subsec:MPC."
],
[
"Model.",
"This Local Computation Algorithms (LCA) model is a centralized model of computation that was introduced in [59]; an algorithm in this model is usually called an $\\mathsf {LCA}$ .",
"In this model, there is a graph $G = (V, E)$ where the algorithm is allowed to make the following queries: Degree Query: Given $\\operatorname{ID}(v)$ , the oracle returns $\\deg (v)$ .",
"Neighbor Query: Given $\\operatorname{ID}(v)$ and an index $i \\in [1,\\Delta ]$ , if $\\deg (v) \\le i$ , the oracle returns $\\operatorname{ID}(u)$ , where $u$ is the $i$ th neighbor of $v$ ; otherwise, the oracle returns $\\bot $ .",
"It is sometimes convenient to assume that there is a query that returns the list of all neighbors of $v$ .",
"This query can be implemented using one degree query and $\\deg (v)$ neighbor queries.",
"For randomized algorithms, we assume that there is an oracle that given $\\operatorname{ID}(v)$ returns an infinite-length random sequence associated with the vertex $v$ .",
"Similarly, for problems with input labels (e.g., the color lists in the list coloring problem), the input label of a vertex $v$ can be accessed given $\\operatorname{ID}(v)$ .",
"Given a distributed problem $\\mathcal {P}$ , an $\\mathsf {LCA}$ $\\mathcal {A}$ accomplishes the following.",
"Given $\\operatorname{ID}(v)$ , the algorithm $\\mathcal {A}$ returns $\\mathcal {A}(v) =$ the output of $v$ , after making a small number of queries.",
"It is required that the output of $\\mathcal {A}$ at different vertices are consistent with one legal solution of $\\mathcal {P}$ .",
"The complexity measure for an $\\mathsf {LCA}$ is the number of queries.",
"It is well-known [55] that any $\\tau $ -round $\\mathsf {LOCAL}$ algorithm $\\mathcal {A}$ can be transformed into an $\\mathsf {LCA}$ $\\mathcal {A}^{\\prime }$ with query complexity $\\Delta ^{\\tau }$ .",
"The $\\mathsf {LCA}$ $\\mathcal {A}^{\\prime }$ simply simulates the $\\mathsf {LOCAL}$ algorithm $\\mathcal {A}$ by querying all radius-$\\tau $ neighborhood of the given vertex $v$ .",
"See [47] for a recent survey about the state-of-the-art in the centralized local model."
],
[
"State of the Art $\\mathsf {LCA}$ for Coloring",
"The previous state-of-the-art for $(\\Delta +1)$ -list coloring in the centralized local computation model are based on simulation of known $\\mathsf {LOCAL}$ algorithms.",
"The deterministic $O(\\sqrt{\\Delta } {\\operatorname{poly}}\\log \\Delta + \\log ^\\ast n)$ -round $\\mathsf {LOCAL}$ algorithm of [28], [8]Precisely, the complexity is $O(\\sqrt{\\Delta } \\log ^{2.5} \\Delta + \\log ^\\ast n)$ in [28], and this has been later improved to $O(\\sqrt{\\Delta \\log \\Delta } \\log ^\\ast \\Delta + \\log ^\\ast n)$ in [8].",
"can be implemented in the centralized local computation model with query complexity $\\Delta ^{O(\\sqrt{\\Delta } {\\operatorname{poly}}\\log \\Delta )} \\cdot O(\\log ^\\ast n)$ ; the randomized $O(\\log ^\\ast \\Delta ) +2^{O(\\sqrt{\\log \\log n})}$ -round $\\mathsf {LOCAL}$ algorithm of [20] can be implemented in the centralized local computation model with query complexity $\\Delta ^{O(\\log ^\\ast \\Delta )} \\cdot O(\\log n)$ ."
],
[
"Our Result.",
"We show that $(\\Delta +1)$ -list coloring can be solved with $\\Delta ^{O(1)} \\cdot O(\\log n)$ query complexity.",
"Note that $\\Delta ^{O(1)} \\cdot O(\\log n)$ matches a “natural barrier” for randomized algorithms based on the graph shattering framework, as each connected component in the post-shattering phase has this size $\\Delta ^{O(1)} \\cdot O(\\log n)$ .",
"Theorem 1.3 There is an centralized local computation algorithm that solves the $(\\Delta +1)$ -list coloring problem with query complexity $\\Delta ^{O(1)} \\cdot O(\\log n)$ , with success probability $1 - 1/{\\operatorname{poly}}(n)$ .",
"The proof is presented in sect-implement-color-bidding."
],
[
"Technical Overview: Tools and New Ingredients",
"In this section, we first review some of the known technical tools that we will use in our algorithms, and then we overview the two new technical ingredients that lead to our improved results (in combination with the known tools)."
],
[
"Notes and Notations.",
"When talking about randomized algorithms, we require the algorithm to succeed with high probability (w.h.p.",
"), i.e., to have success probability at least $1 - 1/{\\operatorname{poly}}(n)$ .",
"For each vertex $v$ , we write $N(v)$ to denote the set of neighbors of $v$ .",
"If there is an edge orientation, $N^{\\operatorname{out}}(v)$ refers to the set of out-neighbors of $v$ .",
"We write $N^k(v) = \\lbrace u \\in V \\ | \\ \\operatorname{dist}(u,v) \\le k\\rbrace $ .",
"We use subscript to indicate the graph $G$ under consideration, e.g., $N_G(v)$ or $N^{\\operatorname{out}}_G(v)$ .",
"In the course of our algorithms, we slightly abuse the notation to also use $\\Psi (v)$ to denote the set of available colors of $v$ .",
"i.e., the subset of $\\Psi (v)$ that excludes the colors already taken by its neighbors in $N(v)$ .",
"The number of excess colors at a vertex is the number of available colors minus the number of uncolored neighbors.",
"Moreover, we make an assumption that each color can be represented using $O(\\log n)$ bits.",
"This is without loss of generality (in all of the models under consideration in our paper), since otherwise we can hash the colors down to this magnitude, as we allow a failure probability of $1/{\\operatorname{poly}}(n)$ for randomized algorithms."
],
[
"Lenzen's Routing.",
"The routing algorithm of Lenzen [45] for $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ allows us to deliver all messages in $O(1)$ rounds, as long as each vertex $v$ is the source and the destination of at most $O(n)$ messages.",
"This is a very useful (and frequently used) communication primitive for designing $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ algorithms.",
"Lemma 2.1 (Lenzen's Routing) Consider a graph $G=(V,E)$ and a set of point-to-point routing requests, each given by the $\\operatorname{ID}$ s of the corresponding source-destination pair.",
"As long as each vertex $v$ is the source and the destination of at most $O(n)$ messages, namely $O(n \\log n)$ bits of information, we can deliver all messages in $O(1)$ rounds in the $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ model."
],
[
"The Shattering Framework.",
"Our algorithm follows the graph shattering framework [10], which first performs some randomized process (known as pre-shattering) to solve “most” of the problem, and then performs some clean-up steps (known as post-shattering) to solve the remaining part of the problem.",
"Typically, the remaining graph is simpler in the sense of having small components and having a small number of edges.",
"Roughly speaking, at each step of the algorithm, we specify an invariant that all vertices must satisfy in order to continue to participate.",
"Those bad vertices that violate the invariant are removed from consideration, and postponed to the post-shattering phase.",
"We argue that the bad vertices form connected components of size $\\Delta ^{O(1)} \\cdot O(\\log n)$ with probability $1 - 1/{\\operatorname{poly}}(n)$ ; we use this in designing $\\mathsf {LCA}$ .",
"Also, the total number of edges induced by the bad vertices is $O(n)$ .",
"Therefore, using Lenzen's routing, in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ we can gather all information about the bad vertices to one distinguished vertex $v^\\star $ , and then $v^\\star $ can color them locally.",
"More precisely, we have the following lemma [10], [27]; see Sect:proof-shattering-lemma for the proof.",
"Lemma 2.2 (The Shattering Lemma) Let $c \\ge 1$ .",
"Consider a randomized procedure that generates a subset of vertices $B \\subseteq V$ .",
"Suppose that for each $v \\in V$ , we have $\\operatorname{Pr}[v \\in B] \\le \\Delta ^{-3c}$ , and this holds even if the random bits not in ${N}^{c}(v)$ are determined adversarially.",
"Then, the following is true.",
"With probability $1 - n^{- \\Omega (c^{\\prime })}$ , each connected component in the graph induced by $B$ has size at most $(c^{\\prime }/c) \\Delta ^{2c} \\log _{\\Delta } n$ .",
"With probability $1 - O(\\Delta ^c) \\cdot \\exp (-\\Omega (n \\Delta ^{-c}))$ , the number of edges induced by $B$ is $O(n)$ ."
],
[
"Round Compression in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ and {{formula:61e90aa5-a0cc-456f-adf0-5f0c01284324}} by Information Gathering.",
"Suppose we are given a $\\tau $ -round $\\mathsf {LOCAL}$ algorithm $\\mathcal {A}$ on a graph of maximum degree $\\Delta $ .",
"A direct simulation of $\\mathcal {A}$ on $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ costs also $\\tau $ rounds.",
"However, if each vertex $v$ already knows all information in its radius-$\\tau $ neighborhood, then $v$ can locally compute its output in zero rounds.",
"In general, this amount of information can be as high as $\\Theta (n^2)$ , since there could be $\\Theta (n^2)$ edges in the radius-$\\tau $ neighborhood of $v$ .",
"For the case of $\\Delta ^{\\tau } = O(n)$ , it is possible to achieve an exponential speed-up in the round complexity in the $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ , compared to that of $\\mathsf {LOCAL}$ .",
"In particular, in this case, each vertex $v$ can learn its radius-$\\tau $ neighborhood in just $O(\\log \\tau )$ rounds in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ .",
"Roughly speaking, after $k$ rounds, we are able to simulate the product graph $G^{2^k}$ , which is the graph where any two vertices with distance at most $2^{k}$ in graph $G$ are adjacent.",
"This method is known as graph exponentiation [50], and it has been applied before in the design of algorithms in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ and $\\mathsf {MPC}$ models, see e.g., [32], [34], [57], [52], [6]."
],
[
"Round Compression via Opportunistic Information Gathering.",
"Our goal is to achieve the $O(1)$ round complexity in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ , so an exponential speed-up compared to the $\\mathsf {LOCAL}$ model will not be enough.",
"Consider the following “opportunisitc” way of simulating a $\\mathsf {LOCAL}$ algorithm $\\mathcal {A}$ in the $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ model.",
"Each vertex $u$ sends its local information (which has $O(\\Delta \\log n)$ bits) to each vertex $v \\in V$ with some fixed probability $p = O(1/\\Delta )$ , independently, and it hopes that there exists a vertex $v \\in V$ that gathers all the required information to calculate the outcome of $\\mathcal {A}$ at $u$ .",
"To ensure that for each $u$ , there exists such a vertex $v$ w.h.p., it suffices that $p^{\\Delta ^{\\tau }} \\gg \\frac{\\log n}{n}$ .",
"We note that a somewhat similar idea was key to the $O(1)$ -round MST algorithm of [43] for $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ .",
"lem:speedup, presented below, summarizes the criteria for this method to work; see sect:speed-up-alg for the proof of the lemma.",
"Denote $\\ell _{\\operatorname{in}}$ as the number of bits needed to represent the random bits and the input for executing $\\mathcal {A}$ at a vertex.",
"Denote $\\ell _{\\operatorname{out}}$ as the number of bits needed to represent the output of $\\mathcal {A}$ at a vertex.",
"We assume that each vertex $v$ initially knows a set $N_{\\ast }(v) \\subseteq N(v)$ such that throughout the algorithm $\\mathcal {A}$ , each vertex $v$ only receives information from vertices in $N_{\\ast }(v)$ .",
"We write $\\Delta _{\\ast }= \\max _{v \\in v} |N_{\\ast }(v)|$ .",
"Note that it is possible that $u \\in N_{\\ast }(v)$ but $v \\notin N_{\\ast }(u)$ .",
"In this case, during the execution of $\\mathcal {A}$ , all messages sent via the edge $\\lbrace u, v\\rbrace $ are from $u$ to $v$ .",
"Denote $N_{\\ast }^k(v)$ as the set of all vertices $u$ such that there is a path $(v = w_0, w_1, \\ldots , w_{x-1} = u)$ such that $x \\le k$ and $w_i \\in N_{\\ast }(w_{i-1})$ for each $i \\in [1, x-1]$ .",
"Intuitively, if $\\mathcal {A}$ takes $\\tau $ rounds, then all information needed for vertex $v \\in V$ to calculate its output is the IDs and the inputs of all vertices in $N_{\\ast }^{\\tau }(v)$ .",
"Lemma 2.3 (Opportunistic Speed-up) Let $\\mathcal {A}$ be a $\\tau $ -round $\\mathsf {LOCAL}$ algorithm on $G = (V, E)$ .",
"There is an $O(1)$ -round simulation of $\\mathcal {A}$ in in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ , given that (i) $\\Delta _{\\ast }^{\\tau } \\log (\\Delta _{\\ast }+ \\ell _{\\operatorname{in}}/\\log n) = O(\\log n)$ , (ii) $\\ell _{\\operatorname{in}}= O(n)$ , and (iii) $\\ell _{\\operatorname{out}}= O(\\log n)$ ."
],
[
"Our New Technical Ingredients, In a Nutshell",
"The results in our paper are based on the following two novel technical ingredients, which are used in combination with the known tools mentioned above: (i) a new graph partitioning algorithm for coloring and (ii) a sparsification of the CLP coloring algorithm [20].",
"We note that the first ingredient suffices for our $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ result for graphs with maximum degree at least ${\\operatorname{poly}}(\\log n)$ , and also for our $\\mathsf {MPC}$ result.",
"This ingredient is presented in sect-high-deg-decomp.",
"The second ingredient, which is also more involved technically, is used for extending our $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ result to graphs with smaller maximum degree, as well as for our $\\mathsf {LCA}$ result.",
"This ingredient is presented in sec:sparseCLP.",
"Here, we provide a brief overview of these ingredients and how they get used in our results."
],
[
"Ingredient 1 — Graph Partitioning for Coloring.",
"We provide a simple random partitioning that significantly simplifies and extends the one in [52], [57].",
"The main change will be that, besides partitioning the vertices randomly, we also partition the colors randomly.",
"In particular, this new procedure partitions the vertices and colors in a way that allows us to easily apply CLP in a black box manner.",
"Concretely, our partitioning breaks the graph as well as the respective palettes randomly into many subgraphs $B_1, \\ldots , B_k$ of maximum degree $O(\\sqrt{n})$ and size $O(\\sqrt{n})$ , while ensuring that each vertex in these subgraphs receives a random part of its palette with size close to the maximum degree of the subgraph.",
"The palettes for each part are disjoint, which allows us to color all parts in parallel.",
"There will be one left-over subgraph $L$ , with maximum degree $\\tilde{O}(\\Delta ^{3/4})$ , as well as sufficiently large remaining palettes for each vertex in this left-over subgraph.",
"Application in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ : Since each subgraph has $O(n)$ edges, all of $B_1, \\ldots , B_k$ can be colored, in parallel, in $O(1)$ rounds, using Lenzen's routing (lem:routing).",
"The left-over part $L$ is handled by recursion.",
"We show that when $\\Delta > \\log ^{4.1} n$ , we are done after $O(1)$ levels of recursion.",
"Application in Low-memory MPC: We perform recursive calls on not only on $L$ but also on $B_1, \\ldots , B_k$ .",
"After $O(1)$ levels of recursion, the maximum degree can be made $O(n^{\\beta })$ , for any given constant $\\beta > 0$ , which enables us to run the CLP algorithm on a low memory $\\mathsf {MPC}$ .",
"We note that the previous partitioning approach [52], [57] is unable to reduce the maximum degree to below $\\sqrt{n}$ ; this is a significant limitation that our partitioning overcomes."
],
[
"Ingredient 2 — Sparsification of the CLP Algorithm.",
"In general, to calculate the output of a vertex $v$ in a $\\tau $ -round $\\mathsf {LOCAL}$ algorithm $\\mathcal {A}$ , the output may depend on all of the $\\tau $ -hop neighborhood of $v$ and we may need to query $\\Delta ^{\\tau }$ vertices.",
"To efficiently simulate $\\mathcal {A}$ in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ or to transform $\\mathcal {A}$ to an $\\mathsf {LCA}$ , a strategy is to “sparsify” the algorithm $\\mathcal {A}$ so that the number of vertices a vertex has to explore to decide its output is sufficiently small.",
"This notion of sparsification is a key idea behind some recent algorithms [32], [34].",
"In the present paper, a key technical ingredient is providing such a sparsification for the $(\\Delta +1)$ coloring algorithm of CLP [20].",
"The pre-shattering phase of the CLP algorithm [20] consists of three parts: (i) initial coloring, (ii) dense coloring, and (iii) color bidding.",
"Parts (i) and (ii) take $O(1)$ rounds;In the preliminary versions (arXiv:1711.01361v1 and STOC'18) of [20], dense coloring takes $O(\\log ^\\ast \\Delta )$ time.",
"This time complexity has been later improved to $O(1)$ in a revised full version of [20] (arXiv:1711.01361v2).",
"part (iii) takes $\\tau = O(\\log ^\\ast \\Delta )$ rounds.",
"In this paper, we sparsify the color bidding part of the CLP algorithm.",
"We let each vertex $v$ sample $O({\\operatorname{poly}}\\log \\Delta )$ colors from its palette at the beginning of this procedure, and we show that with probability $1 - 1/{\\operatorname{poly}}(\\Delta )$ , these colors are enough for $v$ to correctly execute the algorithm.",
"Based on the sampled colors, we can do an $O(1)$ -round pre-processing step to let each vertex $v$ identify a subset of neighbors $N_{\\ast }(v) \\subseteq N(v)$ of size $\\Delta _{\\ast }= O({\\operatorname{poly}}\\log \\Delta )$ neighbors $N_{\\ast }(v) \\subseteq N(v)$ , and $v$ only needs to receive messages from neighbors in $N_{\\ast }(v)$ in the subsequent steps of the algorithm.",
"Application in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ : For the case $\\Delta = O({\\operatorname{poly}}\\log n)$ , the parameters $\\tau = O(\\log ^\\ast \\Delta )$ and $\\Delta _{\\ast }= O({\\operatorname{poly}}\\log \\Delta ) = O({\\operatorname{poly}}(\\log \\log n))$ satisfy the condition for applying the opportunistic speedup lemma (lem:speedup), and so the pre-shattering phase of the CLP algorithm can be simulated in $O(1)$ rounds in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ .",
"Application in Centralized Local Computation: With sparsification, the pre-shattering phase of the CLP algorithm can be transformed into an $\\mathsf {LCA}$ with $\\Delta ^{O(1)} \\cdot \\Delta _{\\ast }^\\tau = \\Delta ^{O(1)}$ queries.",
"The recent work [2] on $(\\Delta +1)$ -coloring in $\\mathsf {MPC}$ is also based on some form of palette sparsification, as follows.",
"They showed that if each vertex samples $O(\\log n)$ colors uniformly at random, then w.h.p., the graph still admits a proper coloring using the sampled colors.",
"Since we only need to consider the edges $\\lbrace u,v\\rbrace $ where $u$ and $v$ share a sampled color, this effectively reduces the degree to $O(\\log ^2 n)$ .",
"For an $\\mathsf {MPC}$ algorithm with $\\tilde{O}(n)$ memory per processor, the entire sparsified graph can be sent to one processor, and a coloring can be computed there, using any coloring algorithm, local or not.",
"This sparsification is not applicable for our setting.",
"In particular, in our sparsified CLP algorithm, we need to ensure that the coloring can be computed by a $\\mathsf {LOCAL}$ algorithm with a small locality volume; this is because the final coloring is constructed distributedly via the opportunistic speedup lemma (lem:speedup)."
],
[
"Coloring of High-degree Graphs via Graph Partitioning ",
"In this section, we describe our graph partitioning algorithm, which is the first new technical ingredient in our results.",
"As mentioned in subsec:ingredients, this ingredient on its own leads to our $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ result for graphs with $\\Delta =\\Omega ({\\operatorname{poly}}(\\log n))$ and also our $\\mathsf {MPC}$ result, as we will explain in subsec:CongestedClique and subsec:MPC, respectively.",
"The algorithm will be applied recursively, but it is required that the failure probability is at most $1 - 1/{\\operatorname{poly}}(n)$ in all recursive calls, where $n$ is the number of vertices in the original graph.",
"Thus, in this section, $n$ does not refer to the number of vertices in the current subgraph $G = (V,E)$ under consideration."
],
[
"The Graph Partitioning Algorithm.",
"The graph partitioning is parameterized by two constants $\\gamma $ and $\\lambda $ satisfying $\\gamma \\ge 2$ and $\\lambda = \\frac{1}{2} + \\frac{2}{3\\gamma +2}$ .",
"Consider a graph $G = (V, E)$ with maximum degree $\\Delta $ .",
"Note that $G$ is a subgraph of the $n$ -vertex original graph, and so $n \\ge |V|$ .",
"Each vertex $v \\in V$ has a palette $\\Psi (v)$ of size $|\\Psi (v)| \\ge \\max \\lbrace \\deg _G(v), \\Delta ^{\\prime }\\rbrace + 1$ , where $\\Delta ^{\\prime } = \\Delta - \\Delta ^{\\lambda }$ .",
"Denote $G[S]$ as the subgraph induced by the vertices $S \\subseteq V$ .",
"For each vertex $v \\in V$ , denote $\\deg _S(v)$ as $|N(v) \\cap S|$ .",
"The algorithm is as follows, where we set $k = \\sqrt{\\Delta }$ .",
"Vertex Set: The partition $V = B_1 \\cup \\dots \\cup B_{k} \\cup L$ is defined by the following procedure.",
"Including each $v \\in V$ to the set $L$ with probability $q = \\Theta \\left( \\sqrt{\\frac{\\log n }{\\Delta ^{1/4}}}\\right)$ .",
"Each remaining vertex joins one of $B_1 , \\ldots , B_{k}$ uniformly at random.",
"Note that $\\operatorname{Pr}[v \\in B_i] = p(1-q)$ , where $p = 1/k = 1/\\sqrt{\\Delta }$ .",
"Palette: Denote $C = \\bigcup _{v \\in V} \\Psi (v)$ as the set of all colors.",
"The partition $C = C_1 \\cup \\dots \\cup C_{k}$ is defined by having each color $c \\in C$ joins one of $C_1 , \\ldots , C_{k}$ uniformly at random.",
"Note that $\\operatorname{Pr}[c \\in C_i] = p$ .",
"We require that with probability $1 - 1/{\\operatorname{poly}}(n)$ , the output of the partitioning algorithm satisfies the following properties, assuming that $\\Delta = \\omega (\\log ^{\\gamma } n)$ .",
"i) Size of Each Part: It is required that $|E(G[B_i])| = O(|V|)$ , for each $i \\in [k]$ .",
"Also, it is required that $|L| = O(q |V|) = O(\\frac{\\sqrt{\\log n}}{\\Delta ^{1/4}}) \\cdot |V|$ .",
"ii) Available Colors in $B_i$ : For each $i\\in \\lbrace 1, \\ldots ,k\\rbrace $ and $v \\in B_i$ , the number of available colors in $v$ in the subgraph $B_i$ is $g_i(v) := |\\Psi (v) \\cap C_i|$ .",
"It is required that $g_i(v) \\ge \\max \\lbrace \\deg _{B_i}(v), \\Delta _i - \\Delta _i^{\\lambda }\\rbrace +1$ , where $\\Delta _i := \\max _{v \\in B_i}\\deg _{B_i}(v)$ .",
"iii) Available Colors in $L$ : For each $v \\in L$ , define $g_L(v) := |\\Psi (v)| - (\\deg _G(v) - \\deg _L(v))$ .",
"It is required that $g_L(v) \\ge \\max \\lbrace \\deg _L(v), \\Delta _L - \\Delta _L^{\\lambda }\\rbrace +1$ for each $v \\in L$ , where $\\Delta _L := \\max _{v \\in L}\\deg _{L}(v)$ .",
"Note that $g_L(v)$ represents a lower bound on the number of available color in $v$ after all of $B_1, \\ldots , B_k$ have been colored.",
"iv) Remaining Degrees: The maximum degrees of $B_i$ and $L$ are $\\deg _{B_i}(v)\\le \\Delta _i = O(\\sqrt{\\Delta })$ and $\\deg _{L}(v) \\le \\Delta _L = O(q\\Delta ) = O(\\frac{\\sqrt{\\log n}}{\\Delta ^{1/4}}) \\cdot \\Delta $ .",
"For each vertex individually, we have $\\deg _{B_i}(v)\\le \\max \\lbrace O(\\log n), O(1/\\sqrt{\\Delta }) \\cdot \\deg (v)\\rbrace $ and $\\deg _{L}(v)\\le \\max \\lbrace O(\\log n), O(q) \\cdot \\deg (v)\\rbrace $ .",
"Intuitively, we will use this graph partitioning in the following way.",
"First compute the decomposition of the vertex set and the palette, and then color each $B_i$ using colors in $C_i$ .",
"Since $|E(G[B_i])| = O(|V|) = O(n)$ , in the $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ model we are able to send the entire graph $G[B_i]$ to a single distinguished vertex $v_i^\\star $ , and then $v_i^\\star $ can compute a proper coloring of $G[B_i]$ locally.",
"This procedure can be done in parallel for all $i$ .",
"If $|E(G[L])| = O(n)$ , then similarly we can let a vertex to compute a proper coloring of $G[L]$ ; otherwise we apply the graph partitioning recursively on $G[L]$ , with the same parameter $n$ .",
"Lemma 3.1 Suppose $|\\Psi (v)| \\ge \\max \\lbrace \\deg _G(v), \\Delta ^{\\prime }\\rbrace +1$ with $\\Delta ^{\\prime } = \\Delta - \\Delta ^{\\lambda }$ , and $|V| > \\Delta = \\omega (\\log ^{\\gamma }n)$ , where $\\gamma $ and $\\lambda $ are two constants satisfying $\\gamma \\ge 2$ and $\\lambda = \\frac{1}{2} + \\frac{2}{3\\gamma +2}$ .",
"The two partitions $V = B_1 \\cup \\dots \\cup B_{k} \\cup L$ and $C = \\bigcup _{v \\in V} \\Psi (v) = C_1 \\cup \\dots \\cup C_{k}$ satisfy the required properties, with probability $1 - 1/{\\operatorname{poly}}(n)$ .",
"We prove that the properties i), ii), iii), and iv) hold with high probability.",
"Note that for some of the bounds, it is straightforward to observe that they hold in expectation.",
"i) Size of Each Part: We first show that $|E(G[B_i])| = O(|V|)$ , for each $i \\in [k]$ , with probability $1 - 1/{\\operatorname{poly}}(n)$ .",
"To have $|E(G[B_i])| = O(|V|)$ , it suffices to have $\\deg _{B_i}(v) = O(p \\Delta )$ for each $v$ , and $|B_i| = O(p |V|)$ , since $p = 1/ \\sqrt{\\Delta }$ .",
"Recall that we already have $\\operatorname{E}[\\deg _{B_i}(v)] \\le (1-q)p \\Delta < p \\Delta $ and $\\operatorname{E}[|B_i|] =(1-q)p|V| < p |V|$ , so we only need to show that these parameters concentrate at their expected values with high probability.",
"This can be established by a Chernoff bound, as follows.",
"Note that we have $\\epsilon _{1} < 1$ and $\\epsilon _{2} < 1$ .",
"In particular, the inequality $\\epsilon _{1} < 1$ holds because of the assumption $\\Delta = \\omega (\\log ^{\\gamma }n) \\ge \\omega (\\log ^{2}n)$ .",
"$\\operatorname{Pr}[ \\deg _{B_i}(v) \\le (1+\\epsilon _{1}) (1-q)p\\Delta ] &= 1 - \\exp (-\\Omega (\\epsilon _{1}^2 (1-q)p\\Delta )) = 1 - O(1/{\\operatorname{poly}}(n)), \\\\\\text{where } \\epsilon _{1} &=\\Theta \\left(\\sqrt{\\frac{\\log n}{(1-q) p \\Delta }}\\right) = \\Theta \\left(\\sqrt{\\frac{\\log n}{p \\Delta }}\\right).\\\\\\\\\\operatorname{Pr}[ |B_i| \\le (1+\\epsilon _{2}) (1-q)p|V|] &= 1 - \\exp (-\\Omega (\\epsilon _{2}^2 (1-q)p|V|)) = 1 - O(1/{\\operatorname{poly}}(n)), \\\\\\text{where } \\epsilon _{2} &=\\Theta \\left(\\sqrt{\\frac{\\log n}{(1-q) p |V|}}\\right) = \\Theta \\left(\\sqrt{\\frac{\\log n}{p |V|}}\\right).$ Next, we show the analogous results for $L$ , i.e., with probability $1 - 1/{\\operatorname{poly}}(n)$ , both $|L|/|V|$ and $\\Delta _L/\\Delta $ are $O(q) = O\\left(\\frac{\\sqrt{\\log n}}{\\Delta ^{1/4}}\\right)$ , where $\\Delta _L = \\max _{v \\in L}\\deg _{L}(v)$ .",
"Similarly, we already have $\\operatorname{E}[\\deg _{L}(v)] \\le q \\Delta $ and $\\operatorname{E}[|L|] = q|V|$ , and remember that $q = O(\\frac{\\sqrt{\\log n}}{\\Delta ^{1/4}})$ , so we only need to show that these parameters concentrate at their expected values with high probability, by a Chernoff bound.",
"$\\operatorname{Pr}[ \\deg _{L}(v) \\le (1+\\epsilon _{3}) q\\Delta ] &= 1 - \\exp (-\\Omega (\\epsilon _{3}^2 q\\Delta )) =1 - O(1/{\\operatorname{poly}}(n)), \\\\\\text{where } \\epsilon _{3} &=\\Theta \\left(\\sqrt{\\frac{\\log n}{q \\Delta }}\\right).\\\\\\\\\\operatorname{Pr}[ |L| \\le (1+\\epsilon _{4}) q|V|] &=1 - \\exp (-\\Omega (\\epsilon _{4}^2 q|V|)) =1 - O(1/{\\operatorname{poly}}(n)), \\\\\\text{where } \\epsilon _{4} &=\\Theta \\left(\\sqrt{\\frac{\\log n}{q |V|}}\\right).$ Similarly, we have $\\epsilon _{3} < 1$ and $\\epsilon _{4} < 1$ .",
"In particular, $\\epsilon _{3} < 1$ because $\\Delta = \\omega (\\log ^{\\gamma }n) \\ge \\omega (\\log ^2 n)$ .",
"ii) Available Colors in $B_i$ : Now we analyze the number of available color for each set $B_i$ .",
"Recall that for each $v \\in B_i$ , the number of available colors in $v$ in the subgraph $B_i$ is $g_i(v) := |\\Psi (v) \\cap C_i|$ .",
"We need to prove the following holds with probability $1 - 1/{\\operatorname{poly}}(n)$ : (i) $|\\Psi (v) \\cap C_i| \\ge \\deg _{B_i}(v) + 1$ , and (ii) $|\\Psi (v) \\cap C_i| \\ge \\Delta _i - \\Delta _i^{\\lambda } +1$ , where $\\Delta _i := \\max _{v \\in B_i}\\deg _{B_i}(v)$ .",
"We will show that with probability $1 - 1/{\\operatorname{poly}}(n)$ , we have $|\\Psi (v) \\cap C_i| \\ge \\Delta _i + 1$ for each $B_i$ and each $v \\in B_i$ , and this implies the above (i) and (ii).",
"Recall that $\\Delta ^{\\prime } = \\Delta \\left(1 - \\Delta ^{-(1-\\lambda )}\\right)$ , $q = \\Theta \\left(\\frac{\\sqrt{\\log n}}{\\Delta ^{1/4}}\\right) \\gg \\Delta ^{-(1-\\lambda )}$ ,The assumptions $\\gamma \\ge 2$ and $\\lambda = \\frac{1}{2} + \\frac{2}{3\\gamma +2}$ imply that $\\lambda \\in (1/2,3/4]$ , and so $\\Delta ^{-(1-\\lambda )} \\le \\Delta ^{-1/4} \\ll q$ .",
"and $\\epsilon _{1} = \\Theta \\left(\\frac{\\sqrt{\\log n}}{\\Delta ^{1/4}}\\right)$ .",
"By selecting $q \\ge 3\\epsilon _{1} = \\Theta \\left(\\frac{\\sqrt{\\log n}}{\\Delta ^{1/4}}\\right)$ , we have $(1-\\epsilon _{1})p\\Delta ^{\\prime } = (1-\\epsilon _{1})\\left(1-\\Delta ^{-(1-\\lambda )}\\right)p\\Delta \\ge (1+\\epsilon _{1})(1-q)p\\Delta + 1.$ We already know that $\\Delta _i \\le (1+\\epsilon _{1})(1-q)p\\Delta $ with probability $1 - 1/{\\operatorname{poly}}(n)$ .",
"In order to have $|\\Psi (v) \\cap C_i| \\ge \\Delta _i + 1$ , we only need to show that $|\\Psi (v) \\cap C_i| \\le (1-\\epsilon _{1})p\\Delta ^{\\prime }$ with probability $1 - 1/{\\operatorname{poly}}(n)$ .",
"For the expected value, we know that $\\operatorname{E}[|\\Psi (v) \\cap C_i|] = p |\\Psi (v)| \\ge p \\Delta ^{\\prime }$ .",
"By a Chernoff bound, we have $\\operatorname{Pr}[ |\\Psi (v) \\cap C_i|\\le (1-\\epsilon _{1})p\\Delta ^{\\prime }] = 1 - \\exp (-\\Omega (\\epsilon _{1}^2 p \\Delta ^{\\prime })) = 1 - O(1/{\\operatorname{poly}}(n)).$ iii) Available Colors in $L$ : Next, we consider the number of available colors in $L$ .",
"We show that with probability $1 - 1/{\\operatorname{poly}}(n)$ , for each $v \\in L$ , we have $g_L(v) \\ge \\max \\lbrace \\deg _L(v), \\Delta _L - \\Delta _L^{\\lambda }\\rbrace +1$ , where $g_L(v) = |\\Psi (v)| - (\\deg _G(v) - \\deg _L(v))$ .",
"It is straightforward to see that $g_L(v) \\ge \\deg _L(v)+1$ , since $g_L(v) = (|\\Psi (v)| - \\deg _G(v)) + \\deg _L(v) \\ge 1 + \\deg _L(v)$ .",
"Thus, we only need to show that $g_L(v) \\ge \\Delta _L - \\Delta _L^{\\lambda } +1$ .",
"In this proof, without loss of generality we assume $\\deg _G(v) = |\\Psi (v)| - 1 \\ge \\Delta ^{\\prime }$ .If this is not the case, we can increase the degree of $v$ in a vacuous way by adding dummy neighbors to it.",
"For instance, we can add a clique of size $\\Delta $ next to $v$ (to be simulated by $v$ ), remove a large enough matching from this clique and instead connect the endpoints to $v$ .",
"Since $\\operatorname{E}[\\deg _L(v)] = q \\deg _G(v) \\ge q \\Delta ^{\\prime }$ , by a Chernoff bound, we have $\\operatorname{Pr}[ \\deg _{L}(v) \\ge (1-\\epsilon _{3}) q \\Delta ^{\\prime }] &= 1 - \\exp (-\\Omega (\\epsilon _{3}^2 q \\Delta ^{\\prime })) =1 - O(1/{\\operatorname{poly}}(n))$ Remember that $\\epsilon _{3} =\\Theta \\left(\\sqrt{\\frac{\\log n}{q \\Delta }}\\right) = \\Theta \\left(\\sqrt{\\frac{\\log n}{q \\Delta ^{\\prime }}}\\right)$ , and we already know that $\\epsilon _{3} < 1$ .",
"Using this concentration bound, the following calculation holds with probability $1 - 1/{\\operatorname{poly}}(n)$ .",
"$g_L(v)&\\ge (1 - \\epsilon _{3})q\\Delta ^{\\prime }\\\\&\\ge q \\Delta ^{\\prime } - O\\left(\\sqrt{q \\Delta ^{\\prime } \\log n}\\right)\\\\&\\ge q \\Delta - q \\Delta ^{\\lambda } - O\\left(\\sqrt{q \\Delta \\log n}\\right).$ Combining this with $\\Delta _L \\le (1+\\epsilon _{3}) q\\Delta = q\\Delta + O(\\sqrt{q \\Delta \\log n})$ , we obtain $g_L(v) \\ge \\Delta _L - q\\Delta ^{\\lambda } - O(\\sqrt{q \\Delta \\log n})$ .",
"Note that $q\\Delta ^{\\lambda } + O(\\sqrt{q \\Delta \\log n}) = o\\left( (q\\Delta )^{\\lambda }\\right) = o\\left(\\Delta _L^{\\lambda }\\right)$ ,The bound $\\sqrt{q \\Delta \\log n} \\ll (q\\Delta )^{\\lambda }$ can be derived from the assumptions $\\lambda = \\frac{1}{2} + \\frac{2}{3\\gamma +2}$ and $\\Delta = \\omega (\\log ^{\\gamma }n)$ , as follows: $q\\Delta = \\Theta (\\Delta ^{\\frac{3}{4}} \\log ^{\\frac{1}{2}} n) = \\omega (\\log ^{\\frac{3}{4} \\gamma +\\frac{1}{2}}n) \\Rightarrow \\sqrt{q \\Delta \\log n} = (q \\Delta )^{\\frac{1}{2}} \\log ^{1/2} n \\ll (q \\Delta )^{\\frac{1}{2}} (q \\Delta )^{\\frac{1}{2}\\left(\\frac{3}{4} \\gamma +\\frac{1}{2}\\right)^{-1}} = (q \\Delta )^{\\lambda }$ .",
"and so we finally obtain $g_L(v) \\ge \\Delta _L - \\Delta _L^{\\lambda } + 1$ .",
"iv) Remaining Degrees: The degree upper bounds of $\\Delta _i$ and $\\Delta _L$ follow immediately from the concentration bounds on $\\deg _{B_i}(v)$ and $\\deg _{L}(v)$ calculated in the proof of i).",
"The bounds $\\deg _{B_i}(v)\\le \\max \\lbrace O(\\log n), O(1/\\sqrt{\\Delta }) \\cdot \\deg (v)\\rbrace $ and $\\deg _{L}(v)\\le \\max \\lbrace O(\\log n), O(q) \\cdot \\deg (v)\\rbrace $ can be derived by a straightforward application of Chernoff bound."
],
[
"Congested Clique Algorithm for High-Degree Graphs",
"In this section, we show that the $(\\Delta + 1)$ -list coloring problem can be solved in $O(1)$ rounds in the $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ model when the degrees are assumed to be sufficiently high.",
"The formal statement is captured in thm:largedegree.",
"First, we show that the partitioning algorithm can indeed be implemented in the $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ model.",
"Then, we show how to color the parts resulting from the graph partitioning efficiently.",
"The proof of thm:largedegree is completed by showing that only $O(1)$ recursive applications of the partitioning are required."
],
[
"Implementation of the Graph Partitioning.",
"The partitions can be computed in $O(1)$ rounds on $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ .",
"Partitioning the vertex set $V$ is straightforward, as every vertex can make the decision independently and locally, whereas it is not obvious how to partition $C$ to make all vertices agree on the same partition.",
"Note that we can assume $|C| \\le (\\Delta +1)|V|$ ; if $|C|$ is greater than $(\\Delta +1)|V|$ initially, then we can let each vertex decrease its palette size to $\\Delta +1$ by removing some colors in its palette, and we will have $|C| \\le (\\Delta +1)|V|$ after removing these colors.",
"A straightforward way of partitioning $C$ is to generate $\\Theta (|C|\\log n)$ random bits at a vertex $v$ locally, and then $v$ broadcasts this information to all other vertices.",
"Note that it takes $O(\\log k) = O(\\log |V|) = O(\\log n)$ bits to encode which part of $C_1 \\cup \\cdots \\cup C_k$ each $c \\in C$ is in.",
"A direct implementation of the approach cannot be done in $O(1)$ rounds, due to the message size constraint of $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ , as each vertex can send at most $\\Theta (n \\log n)$ bits in each round.",
"To solve this issue, observe that it is not necessary to use total independent random bits for each $c \\in C$ , and $\\Theta (\\log n)$ -wise independence suffices.",
"More precisely, suppose $X$ is the summation of $n$ $K$ -wise independent 0-1 random variables with mean $p$ , and so $\\mu = \\operatorname{E}[X] = np$ .",
"A Chernoff bound with $K$ -wise Independence [61] guarantees that $\\operatorname{Pr}[X \\ge (1+q)\\mu ] \\le \\exp \\left(-\\min \\lbrace K, q^2 \\mu \\rbrace \\right).$ In order to guarantee a failure probability of $1/ {\\operatorname{poly}}(n)$ in all applications of Chernoff bound in lem:partition, it suffices that $K = \\Theta (\\log n)$ .",
"Therefore, to compute the decomposition $C = C_1 \\cup \\cdots \\cup C_k$ with $K$ -wise independent random bits, we only need $O(K \\cdot \\log ( |C| \\log k)) = O(\\log ^2 n)$ total independent random bits.",
"Broadcasting $O(\\log ^2 n)$ bits of information to all vertices can be done in $O(1)$ rounds via Lenzen's routing (lem:routing)."
],
[
"The Algorithm of $(\\Delta + 1)$ -list coloring on High-degree Graphs.",
"We next present our $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ -model coloring algorithm for high-degree graphs, using the partitioning explained above.",
"Theorem 3.2 Suppose $\\Delta = \\Omega (\\log ^{4+\\epsilon }n)$ for some constant $\\epsilon > 0$ .",
"There is an algorithm that solves $(\\Delta +1)$ -list coloring in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ in $O(1)$ rounds.",
"[Proof] We show that a constant-depth recursive applications of lem:partition suffices to give an $O(1)$ -round $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ $(\\Delta + 1)$ -list coloring algorithm for graphs with $\\Delta = \\Omega (\\log ^{4+\\epsilon }n)$ , for any constant $\\epsilon > 0$ .",
"Consider the graph $G = (V, E)$ .",
"First, we apply the graph partitioning algorithm of lem:partition to partition vertices $V$ into subsets $B_1, \\ldots , B_{k}, L$ with parameter $n = |V|$ , and $k = \\sqrt{\\Delta }$ .",
"After that, let arbitrary $k = \\sqrt{\\Delta }$ vertices to be responsible for coloring each $G[B_i]$ .",
"Each of these $k$ vertices, in parallel, gathers all information of $G[B_i]$ from vertices $B_i$ , and then computes a proper coloring of $G[B_i]$ , where each vertex $v \\in B_i$ uses only the palette $\\Psi (v) \\cap C_i$ .",
"The existence of such a proper coloring is guaranteed by Property (ii).",
"Using this approach, we can color all vertices in $V \\setminus L$ in $O(1)$ rounds using Lenzen's routing.",
"Note that Property (i) guarantees that $|E(G[B_i])| = O(n)$ .",
"Finally, each vertex $v \\in L$ removes the colors that have been taken by its neighbors in $V\\backslash L$ from its palette $\\Psi (v)$ .",
"In view of Property (iii), after this operation, the number of available colors for each $v \\in L$ is at least $g_L(v) \\ge \\max \\lbrace \\deg _L(v), \\Delta _L - \\Delta _L^{\\lambda }\\rbrace +1$ .",
"Now the subgraph $G[L]$ satisfies all conditions required to apply lem:partition, so long as $\\Delta _L = \\omega (\\log ^{\\gamma } n)$ .",
"We will see that this condition is always met in our application.",
"We then recursively apply the algorithm of the lemma on the subgraph induced by vertices $L$ with the same parameter $n$ .",
"The recursion stops once we reach a point that $|E(G[L])| = O(n)$ , and so we can apply Lenzen's routing to let one vertex $v$ gather all information of $G[L]$ and compute its proper coloring.",
"Now we analyze the number of iterations needed to reach a point that $|E(G[L])| = O(n)$ .",
"Here we use $\\gamma = 2$ and $\\lambda = 3/4$ .We choose $\\gamma = 2$ (the smallest possible) to minimize the degree requirement in thm:largedegree.",
"Define $V_1 = V$ and $\\Delta _1 = \\Delta $ as the vertex set and the maximum degree for the first iteration.",
"Let $V = B_1 \\cup \\dots \\cup B_{k} \\cup L$ be the outcome of the first iteration, and define $V_2 = L$ and $\\Delta _2 = \\Delta _L$ .",
"Similarly, for $i > 2$ , we define $V_i$ and $\\Delta _i$ based on the set $L$ in the outcome of the graph partitioning algorithm for the $(i-1)$ th iteration.",
"We have the following formulas.",
"$\\Delta _1 &= \\Delta \\\\\\Delta _i &= \\Delta _{i-1} \\cdot O\\left(\\frac{\\sqrt{\\log n}}{\\Delta _{i-1}^{1/4}}\\right) & \\text{ by Property iv) }\\\\|V_1| &= n\\\\|V_i| &= |V_{i-1}| \\cdot O\\left(\\frac{\\sqrt{\\log n}}{\\Delta _{i-1}^{1/4}}\\right) & \\text{ by Property i) }\\multicolumn{2}{l}{\\text{Let $\\alpha > 0$ be chosen such that $\\Delta = \\Delta _1 = (\\log n)^{2+\\alpha }$, and assume $\\alpha = \\Omega (1)$ and $i = O(1)$.",
"We can calculate the value of $\\Delta _i$ and $|V_i|$ as follows.",
"}}\\\\\\Delta _i &= O\\left((\\log n)^{2 + \\alpha \\cdot (\\lambda )^{i-1}}\\right)\\\\|V_i| &= n \\cdot O\\left((\\log n)^{\\alpha \\left( (\\lambda )^{i-1} - 1\\right)}\\right)$ Thus, given that $\\alpha = \\Omega (1)$ and $i = O(1)$ , the condition of $\\Delta _i = \\omega (\\log ^{\\gamma } n) = \\omega (\\log ^2 n)$ for applying lem:partition must be met.",
"Next, we analyze the number of iterations it takes to make $\\Delta _i |V_i|$ sufficiently small.",
"In the $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ model, if $\\Delta _i |V_i| = O(n)$ , then we are able to compute a proper coloring of $V_i$ in $O(1)$ rounds by information gathering.",
"Let us write $\\Delta = \\log ^{2 + \\alpha }n$ , where $\\alpha = 2 + \\beta $ .",
"The lemma statement implies that $\\beta = \\Omega (1)$ .",
"Note that the condition for $\\Delta _i |V_i| = O(n)$ can be re-written as $2 \\alpha \\left(1 - (\\lambda )^{i-1}\\right) \\ge 2+\\alpha .$ Combining this with $\\alpha = 2 + \\beta $ , a simple calculation shows that this condition is met when $i \\ge \\log \\left(\\frac{8(\\beta +2)}{3 \\beta }\\right) / \\log \\left(4/3\\right).$ Since $\\beta = \\Omega (1)$ , we have $\\log \\left(\\frac{8(\\beta +2)}{3 \\beta }\\right) / \\log \\left(4/3\\right) = O(1)$ , and so our algorithm takes only $O(1)$ iterations.",
"In particular, when $\\beta \\ge 10.8$ , i.e., $\\Delta = \\Omega (\\log ^{12.8}n)$ , we have $\\Delta _4 |V_4| = O(n)$ , and so 3 iterations suffice.",
"Since each iteration can be implemented in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ in $O(1)$ rounds, overall we get an algorithm with round complexity $O(1)$ .",
"Remark 3.3 Similar to the proof of thm:largedegree, the graph partitioning algorithm also leads to an $O(1)$ -round $\\mathsf {MPC}$ coloring algorithm with $S = \\widetilde{O}(n)$ memory per processor and $\\widetilde{O}(m)$ total memory.",
"This gives an simple alternate proof (with a slightly worse memory size) of the main result of [2] that $(\\Delta +1)$ -coloring can be solved with $S = \\widetilde{O}(n)$ memory per processor."
],
[
"Massively Parallel Computation with Strongly Sublinear Memory",
"We now show how to apply lem:partition as well as the CLP algorithm of [20], as summarized in the following lemma, to prove thm:MPCCol.",
"Lemma 3.4 ([20], [52]) Let $G$ be an $n$ -vertex graph with $m$ edges and maximum degree $\\Delta $ .",
"Suppose any vertex $v$ has a palette $|\\Psi (v)|$ that satisfies $|\\Psi (v)| \\ge \\max \\left\\lbrace \\deg _G(v)+1,\\Delta -\\Delta ^{3/5}\\right\\rbrace $ .",
"Then the list-coloring problem can be solved w.h.p.",
"in $O(\\sqrt{\\log \\log n})$ rounds of low-memory MPC with local memory $O(n^{\\alpha })$ for an arbitrary constant $\\alpha \\in (0,1)$ and total memory $\\widetilde{O}\\left( \\sum _v \\deg _G(v)^2 \\right)$ if $\\Delta ^2 = O \\left( n^{\\alpha } \\right)$ .",
"The proof of lemma:CLP almost immediately follows from [20], [52]; there are only few changes that have to be made in order to turn their $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ algorithm into a low-memory $\\mathsf {MPC}$ algorithm.",
"The details are deferred to app:LowMemMPC.",
"[Proof of thm:MPCCol] We present a recursive algorithm based on the randomized partitioning algorithm of lem:partition.",
"If $\\Delta = {\\operatorname{poly}}(\\log n)$ then the conditions of lemma:CLP are satisfied trivially; we can solve the problem in $O(\\log ^*\\Delta +\\sqrt{\\log \\log n}) = O(\\sqrt{\\log \\log n})$ rounds of low-memory MPC with total memory $\\widetilde{O} ( n \\cdot \\Delta ^2 ) = \\widetilde{O}(m)$ .",
"Otherwise, we execute the following algorithm."
],
[
"Randomized Partitioning:",
"Let $G$ be the graph that we want to color.",
"We apply the randomized partitioning algorithm of lem:partition to $G$ , which gives us sets $B_1, \\cdots , B_k$ and $L$ , as well as color sets $C_1, \\cdots , C_k$ .",
"The goal is now to first color $B_1,\\ldots ,B_k$ with colors from $C_1, \\ldots ,C_k$ , respectively.",
"Since the colors in the sets $C_i$ are disjoint, this gives a proper coloring of $B:=\\bigcup _{i=1}^k B_i$ .",
"Then, for every vertex in $L$ , we remove all colors already used by neighbors in $B$ from the palettes, leaving us with a list-coloring problem of the graph induced by $L$ with maximum degree $\\Delta _L$ .",
"In the following, we first describe how to color each set $B_i$ with colors in $C_i$ , and then how to solve the remaining list-coloring problem in $L$ .",
"For the parameters in lem:partition, we use $\\gamma = 6$ and $\\lambda = 3/5$ .The choice $\\lambda = 3/5$ is to ensure that the number of available colors for each vertex in each subgraph meets the palette size constraint specified in lemma:CLP."
],
[
"List-Coloring Problem in $B_i$ :",
"If the maximum degree $\\Delta _{i}$ in $B_i$ satisfies $\\Delta _i^2=O(n^{\\alpha })$ , then, by lem:partition ii), $B_i$ satisfies the conditions of lemma:CLP We thus can apply the algorithm of lemma:CLP to $B_i$ .",
"Otherwise, we recurse on $B_i$ .",
"Note that this is possible since, by lem:partition ii) applied to $G$ , $B_i$ satisfies the conditions of lem:partition."
],
[
"List-Coloring Problem in $L$ :",
"If the maximum degree $\\Delta _{L}$ in $L$ satisfies $\\Delta _L^2=O(n^{\\alpha })$ , then, by lem:partition iii) applied to $G$ , $L$ satisfies the conditions of lemma:CLP.",
"We thus can apply the algorithm of lemma:CLP to $L$ .",
"Otherwise, we recurse on $L$ .",
"Note that this is possible since by lem:partition iii), $L$ satisfies the conditions of lem:partition."
],
[
"Number of Iterations:",
"Since the maximum degree in $L$ reduces by a polynomial factor in every step, after at most $O(1/\\alpha )$ steps, the resulting graph has maximum degree at most $O(n^{\\alpha /2})$ , where we satisfy the conditions of lemma:CLP, and hence do not recurse further.",
"Note that when recursing on sets $B_i$ , the degree drop is even larger, and hence the same reasoning applies to bound the number of iterations."
],
[
"Memory Requirements:",
"It is obvious that the recursive partitioning of the input graph $G$ does not incur any overhead in the memory, neither local nor global.",
"Now, let $\\mathcal {H}$ be the set of all graphs $H$ on which we apply the algorithm of lemma:CLP.",
"As we only apply this algorithm when the maximum degree $\\Delta _H$ of $H$ is $O(n^{\\alpha /2})$ or ${\\operatorname{poly}}( \\log n)$ , we clearly have $\\Delta _H^2=O(n^{\\alpha })$ , so the algorithm lemma:CLP is guaranteed to run with local memory $O(n^{\\alpha })$ .",
"It remains to show how to guarantee the total memory requirement of $\\widetilde{O}(m)$ , where $m$ is the number of edges in the input graph $G$ , as promised in thm:MPCCol.",
"First, observe that due to the specifications of lemma:CLP, we can write the total memory requirement as $\\sum _{H\\in \\mathcal {H}}\\sum _{v\\in H} (\\deg _H(v))^2$ .",
"First, assume that the graph $G$ has been partitioned at least three times to get to $H$ .",
"By lem:partition iv), the degree of any vertex $v$ in $H$ is either $\\tilde{O}(1)$ or at most $\\deg _G(v) \\cdot \\tilde{O}\\left(\\Delta ^{-\\frac{1}{4}}\\right) \\cdot \\tilde{O}\\left(\\Delta ^{-\\frac{1}{4} \\cdot \\frac{3}{4}}\\right) \\cdot \\tilde{O}\\left(\\Delta ^{-\\frac{1}{4} \\cdot (\\frac{3}{4})^2}\\right) = \\deg _G(v) \\cdot \\tilde{O}\\left(\\Delta ^{-37/64}\\right) < \\tilde{O}\\left(\\sqrt{\\deg _G(v)}\\right).$ Note that in the above calculation we assume $v$ always goes to the left-over part $L$ in all three iterations.",
"If $v$ goes to $B_i$ , then the degree shrinks faster.",
"Remember that we set $q = \\tilde{O}(\\Delta ^{-1/4})$ .",
"Hence, we require a total memory of $\\widetilde{O}\\left(\\sum _{H\\in \\mathcal {H}} \\sum _{v\\in H} (\\deg _H (v))^2 \\right) = \\widetilde{O}\\left( \\sum _{H\\in \\mathcal {H}} \\sum _{v\\in H} \\deg _G(v) \\right) = \\widetilde{O} \\left( \\sum _{v \\in G} \\deg _G (v) \\right) = \\widetilde{O}(m) \\ .$ Note that the algorithm can be easily adapted to always perform at least three partitioning steps if $\\Delta _H$ is bounded from below by a sufficiently large ${\\operatorname{poly}}(\\log n)$ , because then the conditions of lem:partition are satisfied.",
"On the other hand, if $\\Delta _H = {\\operatorname{poly}}(\\log n)$ , it is follows immediately that $\\widetilde{O}\\left( \\sum _v (\\deg _H(v))^2 \\right) = {\\operatorname{poly}}(\\log n)=\\widetilde{O}(1)$ .",
"Put together, we have $ \\sum _{H\\in \\mathcal {H}}\\sum _{v\\in H} (\\deg _H(v))^2 = \\widetilde{O}(m)$ .",
"FnFunctionend ProcProcedureend Forsimfordo simultaneously Forfordo IfElseIfElseifthenelse ifelse generatecolorGenerateColor samplecolorsSampleColors sparsifiedcolorbiddingSparsifiedColorBidding sparsifiedcoloringSparsifiedColoring"
],
[
"Distributed Coloring with Palette Sparsification ",
"In this section, we present our sparsification for the $\\mathsf {LOCAL}$ -model coloring algorithm of CLP [20], which is the second novel technical ingredient in our results.",
"As a consequence, this sparsification gives us (i) an $\\mathsf {LCA}$ solving $(\\Delta +1)$ list coloring with query complexity $\\Delta ^{O(1)} \\cdot O(\\log n)$ and (ii) an $O(1)$ -round $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ algorithm solving $(\\Delta +1)$ list coloring for the case $\\Delta = O({\\operatorname{poly}}\\log n)$ , using the speedup lemma (lem:speedup)."
],
[
"The Chang-Li-Pettie Coloring Algorithm.",
"We will not sparsify the entire algorithm of [20].",
"The algorithm of [20] is based on the graph shattering framework.",
"Each vertex successfully colors itself with probability $1 - 1/{\\operatorname{poly}}(\\Delta )$ during the pre-shattering phase of [20], and so by the shattering lemma (lem:shatter), the remaining uncolored vertices $V_{{\\sf {Bad}}}$ form connected components of size $\\Delta ^{O(1)} O({\\operatorname{poly}}\\log n)$ .In the analysis of [20], this can also be made $O({\\operatorname{poly}}\\log n)$ , regardless of $\\Delta $ .",
"The post-shattering phase then applies a deterministic $(\\deg +1)$ -list coloring algorithm to color them.",
"lem:shatter guarantees that the number of edges within $V_{\\sf {Bad}}$ is $O(n)$ , and so they can be colored in $O(1)$ rounds in the $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ model.",
"Similarly, dealing with $V_{\\sf {Bad}}$ only adds an $\\Delta ^{O(1)} \\cdot O(\\log n)$ -factor overhead for $\\mathsf {LCA}$ .",
"Thus, we only need to focus on the pre-shattering phase, which consists of the following three steps.",
"Initial Coloring Step: This step is an $O(1)$ -round procedure that generates excess colors at vertices that are locally sparse.",
"Dense Coloring Step: This step is an $O(1)$ -round procedure that colors most of the locally dense vertices.",
"Color Bidding Step: This step is an $O(\\log ^\\ast \\Delta )$ -round procedure that colors most of the remaining uncolored vertices, using the property that these vertices have large number of excess colors.",
"For our $\\mathsf {LCA}$ and $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ algorithms, the plan is to run the initial coloring step and the dense coloring step by a direct simulation, which costs $O(1)$ rounds.",
"Then, we will give a sparsified version of the color bidding step where each vertex $v$ only need to receive the information from $O({\\operatorname{poly}}\\log \\Delta )$ of its neighbors to decide its output."
],
[
"A Black Box Coloring Algorithm.",
"In view of the above, we will use part of the algorithm of [20] as a black box.",
"The specification of this black box is as follows.",
"Consider an instance of the $(\\Delta +1)$ -list coloring on the graph $G=(V,E)$ .",
"The black box algorithm colors a subset of $V$ such that the remaining uncolored vertices are partitioned into three subsets $V_{\\sf {Good}}$ , $V_{{\\sf {Bad}}}$ , and $R$ meeting the following conditions.",
"Good Vertices: The edges within $V_{\\sf {Good}}$ are oriented as a DAG, and each vertex $v \\in V_{\\sf {Good}}$ is associated with a parameter $p_v \\le |\\Psi (v)| - \\deg (v)$ satisfying the conditions $p^\\star = \\min _{v \\in V} p_v \\ge \\Delta / \\log \\Delta $ and $\\sum _{u \\in N^{\\operatorname{out}}(v)} 1/ p_u \\le 1/C$ , where $C > 0$ can be any specified constant.Here $\\Psi (v)$ is the set of available colors at $v$ , i.e., the colors in the palette of $v$ that have not been taken by $v$ 's neighbors.",
"Here $\\deg (v)$ refers to the number of uncolored neighbors of $v$ in $V_{\\sf {Good}}$ .",
"We use $\\operatorname{outdeg}(v)$ to refer to the number of out-neighbors of $v$ .",
"Intuitively, $p_v \\le |\\Psi (v)| - \\deg (v)$ is a lower bound on the number of excess colors at $v$ .",
"Recall that $N^{\\operatorname{out}}(v)$ refers to the set of out-neighbors of $v$ .",
"Bad Vertices: The probability that a vertex $v \\in V$ joins $V_{\\sf {Bad}}$ is $1 - 1/{\\operatorname{poly}}(\\Delta )$ .",
"In particular, in view of lem:shatter, with probability $1 - 1/ {\\operatorname{poly}}(n)$ , they form connected components of size $\\Delta ^{O(1)} \\cdot O( \\log n)$ , and the number of edges within the bad vertices is $O(n)$ .",
"Remaining Vertices: The subgraph induced by $R$ has a constant maximum degree.",
"lem:CLP-summary follows from [20], after some minor modifications.",
"For the sake of completeness we show the details of how we obtain lem:CLP-summary from the results in [20] in Appendix .",
"Note that for the case of $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ , as long as $\\Delta = O(\\sqrt{n})$ , lem:CLP-summary can be implemented in $O(1)$ rounds.",
"Lemma 4.1 ([20]) Consider an instance of the $(\\Delta +1)$ -list coloring on the graph $G=(V,E)$ .",
"There is an $O(1)$ -round $\\mathsf {LOCAL}$ algorithm that colors a subset of vertices such that the remaining uncolored vertices are partitioned into three subsets $V_{\\sf {Good}}$ , $V_{{\\sf {Bad}}}$ , and $R$ meeting the above conditions, and the algorithm uses $O(\\Delta ^2 \\log n)$ -bit messages."
],
[
"A Sparsified Color Bidding Algorithm ",
"In view of lem:CLP-summary, we focus on the subgraph induced by $V_{\\sf {Good}}$ , and denote it as $G_0=(V_0,E_0)$ .",
"The graph $G_0$ is a directed acyclic graph.",
"The set of available colors for $v$ is denoted as $\\Psi _0(v)$ .",
"Our goal is to give a proper coloring of $G_0$ .",
"An important property of $G_0$ is that each vertex $v \\in V$ is associated with a parameter $p_v \\le |\\Psi _0(v)| - \\deg _{G_0}(v)$ such that $\\sum _{u \\in N^{\\operatorname{out}}(v)} 1/ p_u \\le 1/C_0$ , where $C_0$ can be any specified large constant.",
"Intuitively, $p_v$ gives the lower bound of the number of excess colors at vertex $v$ .",
"It is guaranteed that $p^\\star = \\min _{v\\in V_0} p_v \\ge \\Delta / \\log \\Delta $ .",
"Parameters $C_0$ and $p^\\star $ are initially known to all vertices in $V_0$ ."
],
[
"Review of the Color Bidding Algorithm.",
"The above conditions might look a bit strange, but it allows us to find a proper coloring in $O(\\log ^\\ast \\Delta )$ rounds in the $\\mathsf {LOCAL}$ model by applying $O(\\log ^\\ast \\Delta )$ iterations of the procedure ColorBidding [20], as follows.",
"Each color $c \\in \\Psi (v)$ is added to $S_v$ with probability $\\frac{C}{2 |\\Psi (v)|}$ independently.",
"If there exists a color $c^\\star \\in S_v$ that is not selected by any vertex in $N^{\\operatorname{out}}(v)$ , $v$ colors itself $c^\\star $ .",
"We give a very high-level explanation about how this works.",
"For the first iteration we use $C = C_0$ .",
"Intuitively, for each color $c \\in S_v$ , the probability that $c$ is selected by an out-neighbor of $v$ is $\\sum _{u \\in N^{\\operatorname{out}}(v)} C/ (2|\\Psi (u)|) \\le \\sum _{u \\in N^{\\operatorname{out}}(v)} C/ (2p_u) \\le 1/2.$ In the calculation we use the inequality $\\sum _{u \\in N^{\\operatorname{out}}(v)} 1/ p_u \\le 1/C_0$ that is guaranteed by lem:CLP-summary.",
"The probability that $v$ fails to color itself is roughly $1/2^{|S_v|}$ , which is exponentially small in $C_0$ , as in expectation $|S_v| = C_0/2$ .",
"Thus, for the next iteration we may use a parameter $C$ that is exponentially small in $C_0$ , and so after $O(\\log ^\\ast \\Delta )$ iterations, we are done."
],
[
"Parameters.",
"Let $\\beta > 0$ be a constant to be determined.",
"Let $p^\\star \\in [\\Delta / \\log \\Delta , \\Delta ]$ be the parameter specified in the conditions for lem:CLP-summary.",
"The $C$ -parameters used in the algorithms $C_0, \\ldots , C_{k-1}$ are defined as follows.",
"For the base case, $C_0$ is the parameter $C$ specified in the conditions for lem:CLP-summary.",
"Given that $C_{i}$ has been defined, we set $C_{i+1} =2\\left\\lceil \\left( \\min \\left\\lbrace \\frac{1}{2} \\exp (C_i/6) C_i, \\ \\log ^{\\beta } p^\\star \\right\\rbrace \\right) / 2\\right\\rceil - 2.$ In other words, $C_{i+1}$ is the result of rounding $ \\min \\left\\lbrace \\frac{1}{2} \\exp (C_i/6) C_i, \\ \\log ^{\\beta } p^\\star \\right\\rbrace $ down to the nearest even number.",
"The number of iterations $k$ is chosen as the smallest index such that $C_{k-1} = 2\\left\\lceil \\log ^{\\beta } p^\\star / 2\\right\\rceil - 2$ .",
"It is clear that $k = O(\\log ^\\ast \\Delta )$ , as $p^\\star \\le \\Delta +1$ .",
"We will use this sequence $C_0, \\ldots , C_{k-1}$ in our sparsified color bidding algorithm.",
"This sequence is slightly different than the one used in [20].",
"The last number in the sequence used in [20] is set to be $\\sqrt{p^\\star }$ , but here we set it to be $O({\\operatorname{poly}}\\log p^\\star )$ .",
"Having a larger $C$ -parameter leads to a smaller failure probability, but it comes at a cost that we have to sample more colors, and this means that each vertex needs to communicate with more neighbors to check for conflict."
],
[
"Overview of the Proof.",
"We first review the analysis of the multiple iterations of ColorBidding in [20], and then we discuss how we sparsify this algorithm.",
"The proof in [20] maintains an invariant $\\mathcal {I}_i(v)$ for each vertex $v$ that is uncolored at the beginning of each iteration $i$ , as follows.In this section, $G$ refers the current graph under consideration, i.e., it excludes all vertices that have been colored or removed in previous iterations.",
"We use $G_0$ to refer to the original graph.",
"$&\\mathcal {I}_i(v): \\sum _{u \\in N^{\\operatorname{out}}_G(v)} 1/ p_u \\le 1/C_i.$ We will use the same $p_u$ because the number of excess colors of a vertex never decreases.",
"By lem:CLP-summary, this invariant is met for $i=0$ .",
"The vertices $u$ not satisfying the invariant $\\mathcal {I}_{i}(v)$ are considered bad, and are removed from consideration.",
"The analysis of [20] shows that Suppose all vertices $u$ in $G$ at the beginning of the $i$ th iteration satisfy the $\\mathcal {I}_{i}(u)$ .",
"Then at end of this iteration, for each vertex $u$ , with probability $1 - 1/{\\operatorname{poly}}(\\Delta )$ , either $u$ has been successfully colored, or $\\mathcal {I}_{i+1}(u)$ is satisfied.",
"For the last iteration, Given that all vertices $u$ in $G$ satisfy $\\mathcal {I}_{k-1}(u)$ , then $v$ is successfully colored at iteration $k$ with probability $1 - 1/{\\operatorname{poly}}(\\Delta )$ .",
"By the shattering lemma (lem:shatter), all vertices that remain uncolored at the end of the algorithm induce a subgraph with $O(n)$ edges.",
"In particular, in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ we are able to color them in $O(1)$ additional rounds.",
"To sparsify the algorithm, our strategy is to let each vertex sample the colors needed in all iterations at the beginning of the algorithm.",
"It is straightforward to see that each vertex only needs to use $O({\\operatorname{poly}}\\log \\Delta )$ colors throughout the algorithm, with probability $1 - 1/{\\operatorname{poly}}(\\Delta )$ .",
"After sampling the colors, if $u$ finds that $v \\in N^{\\operatorname{out}}(u)$ do not share any sampled color, then there is no need for $u$ to communicate with $v$ .",
"This effectively reduces the maximum degree to $\\Delta ^{\\prime } = O({\\operatorname{poly}}\\log \\Delta )$ .",
"If $\\Delta = O({\\operatorname{poly}}\\log n)$ , then $\\Delta ^{\\prime } = O({\\operatorname{poly}}(\\log \\log n))$ , which is enough to apply the opportunistic speedup lemma (lem:speedup).",
"There is one issue needed to be overcome.",
"That is, verifying whether $\\mathcal {I}_i(u)$ is met has to be done on the original graph $G$ , as we have to go over all vertices $v \\in N^{\\operatorname{out}}_G(u)$ , regardless of whether $u$ and $v$ have shared sampled colors.",
"One way to deal with this issue is to simply not remove the vertices $u$ violating $\\mathcal {I}_i(u)$ , but if we do it this way, then when we calculate the failure probability of a vertex $v$ , we have to apply a union bound over all vertices $u$ within radius $\\tau = O(\\log ^\\ast \\Delta )$ to $v$ that $u$ does not violate the invariant for each iteration.",
"Due to this union bound, we can only upper bound the size of the connected components of bad vertices by $\\Delta ^{O(\\log ^\\ast \\Delta )} \\cdot O(\\log n)$ , so this does not lead to an improved $\\mathsf {LCA}$ .We remark that this is only an issue for $\\mathsf {LCA}$ , and this is not an issue for application in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ .",
"In the shattering lemma (lem:shatter), for the parameters $\\Delta = O({\\operatorname{poly}}\\log n)$ and $c = O(\\log ^\\ast \\Delta )$ , we can still bound the number of edges within the bad vertices $B$ by $O(n)$ .",
"To resolve this issue, we observe that the invariant $\\mathcal {I}_i(u)$ might be too strong for our purpose, since intuitively if $v \\in N^{\\operatorname{out}}(u)$ does not share any sampled colors with $u$ , then $v$ should not be able to affect $u$ throughout the algorithm.",
"In this paper, we will consider an alternate invariant $\\mathcal {I}_i^{\\prime }(u)$ that can be checked in the sparsified graph.",
"More precisely, in each iteration, each vertex $v$ will do a two-stage sampling to obtain two color sets $S_v \\subseteq T_v \\subseteq \\Psi (v)$ .",
"The set $S_v$ has size $C/2$ , and the set $T_v$ has size $\\log ^{\\beta } \\Delta $ , where $\\beta >0$ is a constant to be determined.",
"The alternate invariant $\\mathcal {I}_i^{\\prime }(v)$ is defined as $\\mathcal {I}_i^{\\prime }(v): \\left|T_v \\setminus \\bigcup _{u \\in N^{\\operatorname{out}}_G(v)}S_u \\right| \\ge |T_v|/3.$ This invariant $\\mathcal {I}_i^{\\prime }(v)$ can be checked by having $v$ communicating only with its neighbors that share a sampled color with $v$ .",
"Intuitively, if $\\mathcal {I}_i(v)$ holds, then $\\mathcal {I}_i^{\\prime }(v)$ holds with probability $1 - 1/{\\operatorname{poly}}(\\Delta )$ .",
"It is also straightforward to see that $\\mathcal {I}_i^{\\prime }(v)$ implies that $v$ has a high probability of successfully coloring itself in this iteration, as $S_v$ is a size-$(C_i/2)$ uniformly random subset of $T_v$ .",
"In subsequent discussion, we say that $v$ is rich if $\\mathcal {I}_i^{\\prime }(v)$ is met.",
"Other than not satisfying $\\mathcal {I}_i^{\\prime }(v)$ , there are two other bad events that we need to consider: (Informally) $v$ has too many neighbors that share a sampled color with $v$ ; in this case, we say that $v$ is overloaded.",
"This is a bad event since the goal of the palette sparsification is to reduce the number of neighbors that $v$ needs to receive information from.",
"Most of the sampled colors of $v$ reserved for iteration $i$ have already be taken by the neighbors of $v$ during the previous iterations $1, \\ldots , i-1$ , so $v$ does not have enough colors to correctly run the algorithm for the $i$ th iteration; in this case, we say that $v$ is lazy."
],
[
"The Sparsified Color Bidding Algorithm.",
"We are now in a position to describe the sparsified version of ColorBidding.",
"For the sake of clarity we use the following notations to describe the palette of a vertex $u$ .",
"Recall that $\\Psi _0(u)$ refers to the palette of $u$ initially in the original graph $G_0$ .",
"At the beginning of an iteration, we write $\\Psi ^+(u)$ to denote the set of available colors at $u$ , and write $\\Psi ^-(u)$ to denote the set of colors already taken by vertices in $N_{G_0}(u)$ .",
"Note that $\\Psi ^+(u) = \\Psi _0(u) \\setminus \\Psi ^-(u)$ .",
"The function describe how we sample the colors $S_u$ and $T_u$ in an iteration.",
"Intuitively, we use $k_1 = C/2$ and $k_2 = \\log ^\\beta p^\\star $ as the target set sizes.",
"The set ${R}$ represents a length-$K$ sequence of colors that $u$ pre-sampled for the $i$ th iteration, where $K = \\log ^{3+\\beta } p^\\star $ , and ${R}(j)$ represents the $j$ th color of ${R}$ .",
"We will later see that ${R}$ is generated in such a way that each ${R}(j)$ is a uniformly random color chosen from $\\Psi _0(u)$ , where $\\Psi _0(u)$ is the set of available colors of $v$ initially in $G_0$ .",
"The set $S^{-}$ represents the set $\\Psi ^-(u)$ which consists of the colors already taken by the vertices in $N_{G_0}(u)$ before iteration $i$ .",
"[H] $k_1$ , $k_2$ , $S^-$ , ${R}$ $T \\leftarrow \\varnothing $ $j \\leftarrow 1$ $k_2 \\log ^{3} p^\\star $ $c \\leftarrow {{R}}(j)$ $c \\notin S^-$ $T \\leftarrow T \\cup \\lbrace c\\rbrace $ $|T| = k_1$ $T_1 \\leftarrow T$ $|T| = k_2$ $(T_1, T)$ $(\\varnothing , \\varnothing )$ The procedure is the sparsified version of ColorBidding.",
"In this procedure, it is straightforward to verify that the outcome $S_v \\leftarrow T_1$ and $T_v \\leftarrow T$ of $C/2$ , $\\log ^{\\beta } p^\\star $ , $\\Psi ^-(v), {{R}}_v$ satisfies either one of the following: $S_v = \\varnothing $ and $T_v = \\varnothing $ .",
"This happens when most of the pre-sampled colors for this iteration have been taken by the neighboring vertices.",
"We will later show that this occurs with probability $1/{\\operatorname{poly}}(\\Delta )$ .",
"Given that each ${{R}}_v(j)$ is a uniformly random color of $\\Psi _0(v)$ , we have: (i) $S_v$ is a size-$(C/2)$ uniformly random subset of $T_v$ , and (ii) $T_v$ is a size-$\\left(\\log ^{\\beta }p^\\star \\right)$ uniformly random subset of $\\Psi _0(v)\\setminus \\Psi ^-(v)$ .",
"That is, these two sets $S_v$ and $T_v$ are sampled uniformly randomly from the set of available colors of $v$ , i.e., $\\Psi _0(v) \\setminus \\Psi ^-(v)$ .",
"The condition for $v$ to be overloaded is defined in the procedure .",
"Intuitively, $v$ is said to be overloaded at iteration $i$ if the colors in ${R}_v^{(i)}$ have appeared in $\\bigcup _{0 \\le i^{\\prime } \\le i} {R}_u^{(i^{\\prime })}$ , for too many neighbors $u \\in N_{G_0}(v)$ ; this is undesirable as we want the degree of the sparsified graph to be small.",
"[H] $G$ , $C$ , $\\Psi ^-$ , $\\lbrace {R}_v\\rbrace _{v \\in V_0}$ each vertex $v \\in V(G)$ 1.",
"$(S_v, T_v) \\leftarrow $ $C/2$ , $\\log ^{\\beta } p^\\star $ , $\\Psi ^-(v), {{R}}_v$ .",
"If $v$ is overloaded, reset $(S_v, T_v) \\leftarrow (\\varnothing , \\varnothing )$ .",
"We call $v$ lazy if $S_v = \\varnothing $ .",
"2.",
"$v$ collects information about $S_u$ from all neighbors $u \\in N^{\\operatorname{out}}_G(v)$ .",
"3.",
"If $\\left|T_v \\setminus \\bigcup _{u \\in N^{\\operatorname{out}}_G(v)}S_u \\right| \\ge |T_v|/3$ , i.e., at most $2/3$ of colors $v$ sampled in $T_v$ are selected in $S_u$ of some neighbors $u \\in N^{\\operatorname{out}}_G(v)$ , then we call $v$ rich.",
"If (i) $v$ is not rich or (ii) $v$ is lazy, then $v$ marks itself Bad and it skips the next step.",
"4.",
"If there is a color $c \\in S_v$ that is not in $\\bigcup _{u \\in N^{\\operatorname{out}}_G(v)}S_u$ , we call $v$ lucky with color $c$ .",
"If $v$ is lucky with $c$ , $v$ colors itself $c$ .",
"Tie is broken arbitrarily.",
"The procedure represents the entire coloring algorithm, which consists of $k = O(\\log ^\\ast \\Delta )$ iterations of .",
"The notation $G[U]$ refers to the subgraph induced by $U$ .",
"Note that the set $U$ does not include the vertices that are marked Bad, i.e., once a vertex $v$ marked itself Bad, it stops attempting to color itself; but a Bad vertex might still need to provide information to other vertices in subsequent iterations.",
"[H] $G \\leftarrow G_0$ $K \\leftarrow \\log ^{3+ \\beta } \\Delta $ (*[h]Obviously $k = O(\\log ^* p^\\star - \\log ^* C_0) = O(\\log ^* \\Delta )$ .",
")$i \\leftarrow 0$ $k-1$ 1.",
"$G \\leftarrow G[U]$ , where $U$ consists of the yet uncolored vertices in $G$ that are not Bad.",
"2.",
"Each vertex $v \\in V(G_0)$ generates a color sequence ${{R}}^{(i)}_v(1), \\ldots , {{R}}^{(i)}_v(K)$ by the following rule: for each $j = 1,\\ldots ,K$ , ${{R}}^{(i)}_v(j)$ is a color in $\\Psi _0(v)$ , chosen uniformly at random, independently.",
"3.",
"Each vertex $v \\in V(G)$ gathers the information about $\\lbrace {{R}}^{(i^{\\prime })}_u\\rbrace _{\\begin{array}{c}0 \\le i^{\\prime } \\le i\\end{array}}$ from each neighbor $u \\in N_{G_0}(v)$ .",
"4.",
"If there exist three indices $i^{\\prime } \\in [0, i]$ , $j \\in [1, K]$ , and $j^{\\prime } \\in [1, K]$ such that ${{R}}^{(i)}_v(j) = {{R}}^{(i^{\\prime })}_u(j^{\\prime })$ , we say $u \\in N_{G_0}(v)$ is a significant neighbor of $v \\in V(G)$ .",
"If $v$ has more than $K^2 \\log \\Delta $ significant neighbors, we call $v$ overloaded.",
"5.",
"Each vertex $v \\in V(G)$ gathers the information about the colors that have been taken by the vertices in $N_{G_0}(v)$ .",
"Let $\\Psi ^-(v)$ be the set of these colors.",
"6.",
"Call $G$ , $C_i$ , $\\Psi ^-$ , $\\lbrace {R}_v^{(i)}\\rbrace _{v \\in V_0}$ .",
"It is straightforward to see that can be implemented in such a way that after an $O(1)$ -round pre-processing step, each vertex $v$ is able to identify $O({\\operatorname{poly}}\\log \\Delta )$ neighbors such that $v$ only need to receive information from these vertices during .",
"In the pre-processing step, we let each vertex $v$ sample the color sequences ${R}_v^{(i)}$ for each $0 \\le i \\le k-1$ , and let each vertex $v$ learn the set of colors sampled by $N_{G_0}(v)$ .",
"Based on this information, before the first iteration begins, $v$ is able to identify at most $K^2 \\log \\Delta = O({\\operatorname{poly}}\\log \\Delta )$ neighbors of $v$ for each iteration $i$ such that $v$ is sure that $v$ does not need to receive information from all other neighbors during the $i$ th iteration.",
"See sect-implement-color-bidding for details.",
"For the rest of sect-color-bidding-main, we focus on the analysis of .",
"For each iteration $i$ , recall that $\\Psi ^+(v) = \\Psi _0(v) \\setminus \\Psi ^-(v)$ is the set of available colors at $v$ at the beginning of this iteration.",
"For a vertex $v \\in V_0$ , and its neighbor $u \\in N_{G_0}(v)$ , we say that $u$ is a $c$ -significant neighbor of $v$ in iteration $i$ if $c = {{R}}_u^{(i^{\\prime })}(j)$ for some $i^{\\prime } \\in [1, i]$ and $j \\in [1,k]$ .",
"Consider the beginning of the $i$ th iteration of the for-loop in .",
"In the graph $G=(V,E) \\leftarrow G[U]$ under consideration in this iteration, we say that a vertex $v \\in V$ is $(C, D)$ -honest if the following two conditions are met.",
"(i) $\\sum _{u \\in N^{\\operatorname{out}}_G(v)} 1/p_u \\le 1/C$ .",
"(ii) For each color $c \\in \\Psi _0(v)$ , $v$ has at most $D$ $c$ -significant neighbors $u \\in N_{G_0}(v)$ in the previous iteration.",
"Clearly all vertices are $(C_0,0)$ -honest in $G = G_0$ at the beginning of iteration $i = 0$ .",
"lem:shrinkii shows that $(C, D)$ -honest vertices are well-behaved.",
"Lemma 4.2 Consider the $i$ th iteration of in .",
"Let $U$ be the set of yet uncolored vertices after this iteration.",
"Suppose a vertex $v$ is $(C, D)$ -honest, with $C \\le \\log ^{\\beta } p^\\star $ and $D \\le 2K \\cdot k = O(K \\log ^\\ast \\Delta )$ , at the beginning of this iteration, then The following holds.",
"$\\operatorname{Pr}[v \\text{\\normalfont \\ does not successfully color itself}] \\le \\exp (-C/6) + \\exp (-\\Omega (\\log ^{\\beta } \\Delta ))$ .",
"$\\operatorname{Pr}[v \\mathrm {\\ marks\\ itself\\ {\\bf Bad}}] \\le \\exp (-\\Omega (\\log ^{\\beta } \\Delta ))$ .",
"$\\operatorname{Pr}[\\mathrm {at \\ the \\ beginning \\ of \\ the \\ next \\ iteration,} \\ v \\in U \\ \\mathrm { or } \\ v \\ \\mathrm { is \\ not} \\ (C^{\\prime }, D^{\\prime })\\text{-}\\mathrm {honest}] \\\\ \\le \\exp (-\\Omega (\\log ^{\\beta } \\Delta ))$ , where $C^{\\prime } = \\min \\left\\lbrace \\frac{1}{2} \\exp (C/6) C, \\ \\log ^{\\beta } p^\\star \\right\\rbrace $ and $D^{\\prime } = D + 2K$ .",
"The probability calculation only relies on the distribution of random bits generated in ${N}_{G_0}^2(v)$ in this iteration, i.e., $\\lbrace {{R}}^{(i)}_u\\rbrace _{u \\in {N}_{G_0}^2(v)}$ .",
"In particular, the result holds even if random bits generated outside ${N}_{G_0}^2(v)$ are determined adversarially.",
"Note that lem:shrinkii only relies on the assumption that the vertex $v$ under consideration is $(C,D)$ -honest, and the lemma works even many of neighboring of $v$ are not $(C,D)$ -honest.",
"This is in contrast to most of the analysis of graph shattering algorithms where the analysis relies on the assumption that all vertices at the beginning of each iteration to satisfy certain invariants.",
"Based on lem:shrinkii, we show that colors a vertex with a sufficiently high probability that enables us to apply the shattering lemma.",
"Lemma 4.3 The algorithm gives a partial coloring of $G_0$ such that the probability that a vertex $v$ does not successfully color itself with a color in $\\Psi _0(v)$ is $O(k) \\cdot \\exp \\left(-\\Omega (\\log ^{\\beta } \\Delta )\\right) \\ll 1 /{\\operatorname{poly}}(\\Delta ),$ and this holds even if the random bits generated outside ${N}^2_{G_0}(v)$ are determined adversarially.",
"We consider the sequence $D_0 = 0$ and $D_{i+1} = D_i + 2K$ .",
"Suppose the algorithm does not color a vertex $v$ , then $v$ must falls into one of the following cases.",
"There is an index $i \\in [0, k-2]$ such that $v$ is $(C_i, D_i)$ -honest at the beginning of iteration $i$ , but $v$ is not $(C_{i+1}, D_{i+1})$ -honest at the beginning of iteration $i+1$ .",
"By lem:shrinkii (iii), this occurs with probability at most $(k-1) \\cdot \\exp \\left(-\\Omega (\\log ^{\\beta } \\Delta )\\right)$ .",
"There is an index $i \\in [0, k-1]$ such that $v$ is $(C_i, D_i)$ -honest at the beginning of iteration $i$ , but $v$ marks itself Bad in iteration $i$ .",
"By lem:shrinkii (ii), this occurs with probability at most $k \\cdot \\exp \\left(-\\Omega (\\log ^{\\beta } \\Delta )\\right)$ .",
"For the last iteration $i = k-1$ , the vertex $v$ is $(C_{k-1}.",
"D_{k-1})$ -honest at the beginning of iteration $k-1$ , but $v$ does not successfully colors itself by a color in its palette.",
"in iteration $k-1$ .",
"By lem:shrinkii (iii), this occurs with probability at most $ \\exp \\left(-\\Omega (\\log ^{\\beta } \\Delta )\\right)$ .",
"Note that our analysis only relies on the distribution of random bits generated in ${N}_{G_0}^2(v)$ , which is guaranteed by lem:shrinkii.",
"That is, even if the adversary is able to decide the random bits of vertices outside of ${N}_{G_0}^2(v)$ throughout the algorithm , the probability that $v$ does not successfully color itself is still at most $O(k) \\cdot \\exp \\left(-\\Omega (\\log ^{\\beta } \\Delta )\\right)$ ."
],
[
"Analysis for the Sparsified Color Bidding Algorithm ",
"In this section, we prove lem:shrinkii.",
"We focus on the $i$ th iteration of the algorithm, where the vertex $v$ is $(C,D)$ -honest, and there is no guarantee about the $(C,D)$ -honesty of all other vertices.",
"For this vertex $v$ , we write $E_v^{\\mathrm {overload}}$ , $E_v^{\\mathrm {lazy}}$ , $E_v^{\\mathrm {rich}}$ , and $E_v^{\\mathrm {lucky}}$ to denote the event that $v$ is overloaded, lazy, rich, and lucky.",
"Note that a lucky vertex must be rich and not lazy, and an overloaded vertex must be lazy.",
"In this proof we frequently use this inequality $\\Delta + 1 \\ge |\\Psi _0(v)| \\ge |\\Psi ^+(v)| \\ge p_v \\ge p^\\star = \\Omega (\\Delta / \\log \\Delta )$ .",
"Our analysis only considers the random bits generated by vertices within ${N}_{G_0}^2(v)$ in this iteration.",
"Claim 4.4 The probability that $v$ has more than $D^{\\prime }=D+2K$ $c$ -significant neighbors $u \\in N_{G_0}(v)$ for some color $c \\in \\Psi _0(v)$ in this iteration $i$ is at most $\\exp (-\\Omega (\\log ^{3+\\beta } \\Delta ))$ , and this implies that $\\operatorname{Pr}[E_v^{\\mathrm {overload}}] \\le \\exp (-\\Omega (\\log ^{3+\\beta } \\Delta ))$ .",
"Since $v$ is $(C, D)$ -honest, our plan is to show that for each color $c \\in \\Psi _0(v)$ , the number of new $c$ -significant neighbor $u \\in N_{G_0}(v)$ brought by the color sequences in the $i$ th iteration ${{R}}_u^{({i})}$ , is at most $2K$ with probability $1 - \\exp (-\\Omega (\\log ^{3+\\beta } \\Delta ))$ .",
"Write $N_{G_0}(v) = \\lbrace u_1, \\ldots , u_s\\rbrace $ , and let $X_r = 1 \\lbrace \\text{color $c$ appears in ${{R}}_{u_r}^{{(i)}}$} \\rbrace $ , $Y = \\sum _{1 \\le r \\le s} X_j$ .",
"Then $Y$ is an upper bound on the number of new $c$ -significant neighbors.",
"Since $v$ is $(C, D)$ -honest, the total number of $c$ -significant neighbors is at most $D + Y$ .",
"To prove this claim, it suffices to bound the probability of $Y > 2K$ .",
"Note that $X_1, \\ldots , X_s$ are independent, and $\\operatorname{E}[Y] = \\sum _{1 \\le r\\le s}\\operatorname{E}[X_r] \\le \\sum _{1 \\le r \\le s} \\left(1 - \\left(1 - \\frac{1}{|\\Psi _0(u_r)|}\\right)^K\\right) \\le s \\cdot \\frac{K}{\\Delta +1}\\le \\frac{K \\Delta }{\\Delta +1} < K.$ By a Chernoff bound, we have $\\operatorname{Pr}[Y \\ge 2 K] \\le \\exp (-\\Omega (K)) = \\exp (-\\Omega (\\log ^{3+\\beta } \\Delta ))$ .",
"By a union bound over all $c \\in \\Psi _0(v)$ we are done.",
"Given that $v$ has no more than $D^{\\prime }$ $c$ -significant neighbors in this iteration for every color $c \\in \\Psi _0(v)$ , we infer that $v$ has at most $|{{R}}_u^{({i})}| D^{\\prime } \\le K \\cdot D^{\\prime } = K \\cdot (D + 2K) \\ll K^2 \\log \\Delta $ significant neighbors, which implies that $v$ is not overloaded.",
"Hence we also have $\\operatorname{Pr}[E_v^{\\mathrm {overload}}] \\le \\exp (-\\Omega (\\log ^{3+\\beta } \\Delta ))$ .",
"Claim 4.5 $\\operatorname{Pr}[E_v^{\\mathrm {lazy}}] \\le \\exp (-\\Omega (\\log ^{2+\\beta } \\Delta )).$ Remember that $v$ is lazy if either (i) $v$ is overloaded, or (ii) gives $(\\varnothing , \\varnothing )$ .",
"In view of claim-11111, we only need to show that with probability at most $\\exp (-\\Omega (\\log ^{2+\\beta } \\Delta ))$ , gives $(\\varnothing , \\varnothing )$ .",
"Consider the sampling process in , and suppose that we are in the middle of the process, and $T$ is the current set of colors we have obtained.",
"Suppose $|T| = r$ currently, i.e., we have selected $r$ colors from $\\Psi ^+(v)$ .",
"The probability that the next color ${{R}}_{j}$ we consider is different from these $r$ colors in $T$ is at least $\\frac{|\\Psi ^+(v)| - r}{|\\Psi _0(v)|} \\ge \\frac{|\\Psi ^+(v)| - \\log ^{\\beta } p^\\star }{|\\Psi _0(v)|} \\ge \\frac{\\Omega (\\Delta / \\log \\Delta )}{\\Delta +1}= \\Omega (1/\\log \\Delta ).$ Remember that $|\\Psi ^+(v)| \\ge p_v = \\Omega (\\Delta / \\log \\Delta )$ and $\\log ^{\\beta } p^\\star = O(\\log ^{\\beta }\\Delta )$ .",
"Also remember that gives $(\\varnothing , \\varnothing )$ if after we go over all $k_2 \\log ^3 p^\\star = \\log ^{3+\\beta } p^\\star $ elements in the sequence ${{R}}$ , the size of $T$ is still less than $k_2 = \\log ^{\\beta } p^\\star $ .",
"The probability that this event occurs is at most $\\operatorname{Pr}[\\operatorname{Binomial}(n^{\\prime },p^{\\prime }) < t^{\\prime }] < \\operatorname{Pr}[\\operatorname{Binomial}(n^{\\prime },p^{\\prime }) < n^{\\prime }p^{\\prime }/2],$ where $n^{\\prime } = \\log ^{3+\\beta } p^\\star = \\Theta (\\log ^{3+\\beta }\\Delta )$ , $p^{\\prime } = \\Omega (1/\\log \\Delta )$ , and $t^{\\prime } = \\log ^{\\beta } p^\\star = \\Theta (\\log ^{\\beta }\\Delta ) \\ll n^{\\prime }p^{\\prime }$ .",
"By a Chernoff bound, this event occurs with probability at most $\\exp (-\\Omega (n^{\\prime }p^{\\prime })) = \\exp (-\\Omega (\\log ^{2+\\beta } \\Delta ))$ .",
"Claim 4.6 $\\operatorname{Pr}[\\overline{E_v^{\\mathrm {rich}}}] \\le \\exp (-\\Omega (\\log ^{\\beta } \\Delta ))$ .",
"Recall that $v$ is rich if $\\left|T_v \\setminus \\bigcup _{u \\in N^{\\operatorname{out}}_G(v)}S_u \\right| \\ge |T_v|/3$ .",
"If $v$ is lazy, then $T_v = \\varnothing $ , so $v$ is automatically rich.",
"Thus, in subsequent discussion we assume $v$ is not lazy.",
"We write $N^{\\operatorname{out}}_G(v) = \\lbrace u_1, \\ldots , u_s\\rbrace $ and let $X_r = |T_v \\cap S_{u_r}|$ .",
"To prove the lemma, it suffices to show that for $Y = \\sum _{r = 1}^s X_r$ , we have $\\operatorname{Pr}[Y \\ge \\frac{2}{3} |T_v|] \\le \\exp (-\\Omega (\\log ^{\\beta } p^{\\star })))$ .",
"We consider the random variable $X_r = |T_v \\cap S_{u_r}|$ .",
"For notational simplicity, we write $u = u_r$ .",
"If $u$ is lazy, then $X_r = 0$ .",
"Suppose $u$ is not lazy.",
"The set $S_u \\subseteq \\Psi ^+(u)$ is the result of randomly choosing distinct $C/2$ colors $c_1, \\ldots , c_{C/2}$ from $\\Psi ^+(u)$ , one by one.",
"For each $j \\in [1, C/2]$ , define $Z_{u, j}$ as the indicator random variable that $c_j \\in T_v$ .",
"Then $X_r = \\sum _{j=1}^{C/2} Z_{u,j}$ .",
"We have the following observation.",
"In the process, when we pick the $j$ th color, regardless of the already chosen colors $c_1, \\ldots , c_{j-1}$ , the probability that the color picked is in $T_j$ is at most $\\frac{|T_v|}{|\\Psi ^+(u)|-(j-1)} \\le \\frac{|T_v|}{|\\Psi ^+(u)|-(C/2)}$ .",
"Thus, we have $ \\operatorname{E}[Z_{u,j}] \\le \\frac{|T_v|}{|\\Psi ^+(u)|-(C/2)}\\le \\frac{|T_v|}{p_u-(C/2)}\\le \\frac{1.1|T_v|}{p_u},$ since $C/2 \\le (\\log ^{\\beta } p^\\star )/2 = O({\\operatorname{poly}}\\log \\Delta )$ and $p_u = \\Omega (\\Delta /\\log \\Delta )$ .",
"Therefore, in order to bound $Y = X_1 + \\cdots + X_s$ from above, we can assume w.l.o.g.",
"each $X_r$ is the sum of $C/2$ i.i.d.",
"random variables, and each of them is a bernoulli random variable with $p = \\frac{1.1|T_v|}{p_u}$ , and so $Y$ is the summation of $s \\cdot (C/2)$ independent 0-1 random variables.",
"Since $v$ is $(C, D)$ -honest, we have $\\sum _{u \\in N^{\\operatorname{out}}_G(v)} 1/ p_u \\le 1/C$ .",
"The expected value of $Y$ can be upper bounded as follows.",
"$\\operatorname{E}[Y]&\\le \\frac{C}{2} \\sum _{v \\in N^{\\operatorname{out}}_G(u)} \\frac{1.1 |T_v|}{p_v} \\le \\frac{1.1}{2} |T_v|.$ By a Chernoff bound, we obtain $\\operatorname{Pr}\\left[\\overline{E_v^{\\mathrm {rich}}}\\right] \\le \\operatorname{Pr}\\left[Y \\ge \\left(\\frac{1}{1.1} \\cdot \\frac{4}{3}\\right) \\left( \\frac{1.1}{2} |T_v| \\right)\\right] \\le \\exp \\left(-\\Omega \\left( \\frac{1.1}{2} |T_v|\\right)\\right) \\le \\exp \\left(-\\Omega \\left(\\log ^{\\beta } p^{\\star }\\right)\\right).$ Using the above three claims, we now prove the three conditions specified in lem:shrinkii."
],
[
"Proof of i).",
"Conditioning on $E_v^{\\mathrm {rich}}\\cap \\overline{E_v^{\\mathrm {lazy}}}$ , $v$ is lucky with some color unless it fails to select any of $|T_v|/3$ specific colors from $T_v \\subseteq \\Psi ^+(v)$ .",
"Remember that $E_v^{\\mathrm {rich}}$ implies that $\\left|T_v \\setminus \\bigcup _{u \\in N^{\\operatorname{out}}_G(v)}S_u \\right| \\ge |T_v|/3$ , and if any one of them is in $S_v$ , then $v$ successfully colors itself.",
"Also remember that $S_v$ is a size-$(C/2)$ subset of $T_v$ chosen uniformly at random.",
"Thus, $\\operatorname{Pr}[\\overline{E_v^{\\mathrm {lucky}}} \\;|\\; E_v^{\\mathrm {rich}}\\cap \\overline{E_v^{\\mathrm {lazy}}}]\\le \\frac{\\binom{\\frac{2}{3}|T_v|}{C/2}}{\\binom{|T_v|}{C/2}} \\le \\left( \\frac{2}{3} \\right)^{C/2} \\le \\exp (-C/6).$ By claim-22222 and claim-33333, we have: $\\operatorname{Pr}[\\overline{E_v^{\\mathrm {lucky}}}] &\\le \\operatorname{Pr}[\\overline{E_v^{\\mathrm {lucky}}} \\;|\\; E_v^{\\mathrm {rich}}\\cap \\overline{E_v^{\\mathrm {lazy}}}] + \\operatorname{Pr}[E_v^{\\mathrm {lazy}}] + \\operatorname{Pr}[\\overline{E_v^{\\mathrm {rich}}}] \\\\&\\le \\exp (-C/6) + \\exp (-\\Omega (\\log ^{\\beta } p^\\star )).$"
],
[
"Proof of ii).",
"This also follows from By claim-22222 and claim-33333.",
"$\\operatorname{Pr}[v \\mathrm {\\ marks\\ itself\\ {\\bf Bad}}] &= \\operatorname{Pr}[\\overline{E_v^{\\mathrm {rich}}} \\cup E_v^{\\mathrm {lazy}}] \\\\&\\le \\operatorname{Pr}[\\overline{E_v^{\\mathrm {rich}}}] + \\operatorname{Pr}[E_v^{\\mathrm {lazy}}] \\le \\exp (-\\Omega (\\log ^{\\beta } p^\\star )).$"
],
[
"Proof of iii).",
"Define $Y$ as the summation of $1/p_u$ over all vertices $u \\in N^{\\operatorname{out}}_{G}(v)$ such that $u \\notin U$ in the next iteration.",
"We prove that the probabilities of (a) $Y \\le 1/C^{\\prime }$ and (b) $v$ has more than $D^{\\prime }$ $c$ -significant neighbors $u \\in N_{G_0}(v)$ in this iteration are both at most $\\exp (-\\Omega (\\log ^{\\beta } \\Delta ))$ .",
"For (b), it follows from claim-11111.",
"For the rest of the proof, we deal with (a).",
"Write $N^{\\operatorname{out}}_{G}(v) = \\lbrace u_1, \\ldots , u_s\\rbrace $ .",
"Consider the event $E_r^\\ast = E_{u_r}^{\\mathrm {lucky}} \\cup \\overline{E_{u_r}^{\\mathrm {rich}}} \\cup E_{u_r}^{\\mathrm {lazy}}$ that $u_r$ does not join $U$ in the next iteration, i.e., $u_r$ successfully colors itself or marks itself Bad.",
"For each $r \\in [1, s]$ , define the random variable $Z_r$ as follows.",
"Let $Z_r = 0$ if the event $ E_{u_r}^{\\mathrm {lucky}} \\cup \\overline{E_{u_r}^{\\mathrm {rich}}} \\cup E_{u_r}^{\\mathrm {lazy}}$ occurs, and $Z_r = 1/{p_{u_r}}$ otherwise.",
"Clearly we have $Y = \\sum _{r=1}^s Z_r$ .",
"Note that $\\operatorname{E}[Y] \\le \\exp (-C/6) \\cdot (1/C)$ , because $\\operatorname{Pr}[\\overline{E_r^\\ast }] = \\operatorname{Pr}[\\overline{E_{u_r}^{\\mathrm {lucky}}} \\cap E_{u_r}^{\\mathrm {rich}}\\cap \\overline{E_{u_r}^{\\mathrm {lazy}}}] \\le \\operatorname{Pr}[\\overline{E_{u_r}^{\\mathrm {lucky}}} \\; | \\; E_{u_r}^{\\mathrm {rich}}\\cap \\overline{E_{u_r}^{\\mathrm {lazy}}}] \\le \\exp (-C/6),$ as calculated above in the proof of Condition (i).",
"Since $v$ is $(C,D)$ -honest, we have $\\sum _{u \\in N^{\\operatorname{out}}_G(v)} 1/ p_u \\le 1/C$ .",
"Combining these two inequalities, we obtain that $\\operatorname{E}[Y] \\le \\exp (-C/6) \\cdot (1/C)$ .",
"Recall that $C^{\\prime } = \\min \\left\\lbrace \\frac{1}{2} \\exp (C/6) C, \\ \\log ^{\\beta } p^\\star \\right\\rbrace $ , and so $\\operatorname{E}[Y] \\le 1/(2C^{\\prime })$ .",
"Next, we prove the desired concentration bound on $Y$ .",
"Each variable $Z_r$ is within the range $[a_r, b_r]$ , where $a_r = 0$ and $b_r = 1/{p_{u_r}}$ .",
"We have $\\sum _{r=1}^s (b_r - a_r)^2 \\le \\sum _{u \\in N^{\\operatorname{out}}_G(v)} 1/ p_u^2\\le \\sum _{u \\in N^{\\operatorname{out}}_G(v)} 1/ (p_u \\cdot p^\\star )\\le 1/ (C p^\\star ).$ Recall $\\operatorname{E}[Y] \\le 1/(2C^{\\prime })$ .",
"By Hoeffding's inequality, we obtain $\\operatorname{Pr}[Y \\ge 1/C^{\\prime }]&\\le \\exp \\left(\\frac{-2/(2C^{\\prime })^2}{\\sum _{r=1}^s (b_r - a_r)^2}\\right).$ By assumptions specified in the lemma, $(1/C^{\\prime })^2 = \\Omega (1 / \\log ^{2\\beta } p^\\star )$ and $1 / \\sum _{r=1}^s (b_r - a_r)^2 = \\Omega (C p^\\star ) = \\Omega ( p^\\star / \\log ^{\\beta } p^\\star )$ .",
"Thus, $ \\operatorname{Pr}[Y \\ge 1/C^{\\prime }] \\le \\exp (-\\Omega (p^\\star \\log ^{-3\\beta } p^\\star )) \\le \\exp (-\\Omega (\\sqrt{p^\\star })) \\ll \\exp (-\\Omega (\\log ^{\\beta } p^\\star )).$ There is a subtle issue regarding the applicability of Hoeffding's inequality.",
"The variables $\\lbrace X_1, \\ldots , X_k\\rbrace $ are not independent, but we argue that we are still able to apply Hoeffding's inequality.",
"Assume that $N^{\\operatorname{out}}(v) = (u_1, \\ldots , u_s)$ is sorted in reverse topological order, and so for each $1 \\le a \\le s$ , we have $N^{\\operatorname{out}}(u_a)\\cap \\lbrace u_{a}, \\ldots , u_s\\rbrace = \\varnothing $ .",
"We reveal the random bits in the following manner.",
"First of all, we reveal the set $T_u$ for all vertices $u$ .",
"Now the event regarding whether a vertex is rich or is lazy has been determined.",
"Then, for $r = 1$ to $s$ , we reveal the set $\\lbrace S_u \\ | \\ u = u_r \\text{ or } u \\in N^{\\operatorname{out}}(u_r)\\rbrace $ .",
"This information is enough for us to decide the outcome of $Z_r$ .",
"Note that in this process, conditioning on arbitrary outcome of $Z_1, \\ldots , Z_{r-1}$ and all random bits revealed prior to revealing the set $S_{u_r}$ , The probability that $\\overline{E_r^\\ast }$ occurs is still at most $\\exp (-C/6)$ ."
],
[
"Implementation for the Sparsified Color Bidding Algorithm ",
"In this section, we present an implementation of in the $\\mathsf {LOCAL}$ model such that after an $O(1)$ -round pre-processing step, each vertex $v$ is able to identify a $O({\\operatorname{poly}}\\log \\Delta )$ -size subset $N_{\\ast }(v) \\subseteq N_{G_0}(v)$ of neighboring vertices such that $v$ only needs to receive information from these vertices during ."
],
[
"Fixing All Random Bits.",
"Instead of having each vertex $v$ generate the color sequence ${{R}}_v^{(i)}$ at iteration $i$ , we determined all of $\\lbrace {{R}}_v^{(i)}\\rbrace _{\\begin{array}{c} 0 \\le i \\le k-1\\end{array}}$ in the pre-processing step.",
"After fixing these sequences, we can regard as a deterministic $\\mathsf {LOCAL}$ algorithm, where $\\lbrace {{R}}_v^{(i)}\\rbrace _{\\begin{array}{c} v \\in V_0,\\ 0 \\le i \\le k-1\\end{array}}$ can be seen as the input for the algorithm.",
"To gather this information, we need to use messages of $k \\cdot K \\cdot O(\\log n) = O({\\operatorname{poly}}\\log \\Delta ) \\cdot O(\\log n)$ bits, where $k =O(\\log ^\\ast \\Delta )$ is the number of iterations, and $K$ is the length of the color sequence ${{R}}_v^{(i)}$ for an iteration."
],
[
"Determining the Set $N_{\\ast }(v)$ .",
"We show how to let each vertex $v$ determine a $O({\\operatorname{poly}}\\log \\Delta )$ -size set $N_{\\ast }(v) \\subseteq N_{G_0}(v)$ based on the following information $\\lbrace {{R}}_u^{(i)}\\rbrace _{\\begin{array}{c} u \\in {N}_{G_0}(v),\\ 0 \\le i \\le k-1\\end{array}}.$ such that $v$ only needs to receive messages from $N_{\\ast }(v)$ during the execution of .",
"We make the following two observations.",
"In order for $v$ to execute at iteration $i$ correctly, $v$ does not need to receive information from $u \\in N_{G_0}(v)$ if all colors in $\\lbrace {{R}}_u^{(i^{\\prime })}\\rbrace _{\\begin{array}{c} u \\in {N}_{G_0}(v),\\ 0 \\le i^{\\prime } \\le i\\end{array}}$ do not overlap with the colors in ${{R}}_u^{(i)}$ .",
"In other words, $v$ only needs information from its significant neighbors.",
"If $v$ is overloaded at iteration $i$ , then $v$ knows that it is lazy in this iteration, and so the outcome of at iteration $i$ is that $v$ sets $S_v = T_v = \\varnothing $ , and $v$ marks itself Bad.",
"The above two observations follow straightforwardly from the description of .",
"Therefore, we can define the set $N_{\\ast }(v)$ as follows.",
"Add $u \\in N_{G_0}(v)$ to $N_{\\ast }(v)$ if there exists an index $i \\in [0, k-1]$ such that (i) $u$ is a significant neighbor of $v$ at iteration $i$ , and (ii) $v$ is not overloaded at iteration $i$ .",
"By the definition of overloaded vertices, we know that if $v$ is not overloaded at iteration $i$ , then $v$ has at most $K^2 \\log \\Delta = O({\\operatorname{poly}}\\log \\Delta )$ significant neighbors for iteration $i$ .",
"Thus, $|N_{\\ast }(v)| = O({\\operatorname{poly}}\\log \\Delta )$ .",
"Note that the set $N_{\\ast }(v)$ can be locally calculated at $v$ during the pre-processing step."
],
[
"Summary.",
"Algorithm can be implemented in $\\mathsf {LOCAL}$ in the following way.",
"Pre-Processing Step.",
"This step is randomized, and it takes one round and uses messages of $O({\\operatorname{poly}}\\log \\Delta ) \\cdot O(\\log n)$ bits.",
"After this step, each vertex has calculated a set $N_{\\ast }(v)$ with $|N_{\\ast }(v)| = O({\\operatorname{poly}}\\log \\Delta )$ .",
"Main Steps.",
"This is a deterministic $O(\\log ^\\ast \\Delta )$ -round procedure.",
"During the procedure, each vertex $v$ only receives messages from $N_{\\ast }(v)$ .",
"The output of each vertex is a color (or a special symbol $\\bot $ indicating that $v$ is uncolored), which can be represented by $\\ell _{\\operatorname{out}}= O(\\log n)$ buts.",
"The input of each vertex consists of its color sequences for all iterations, which can be represented in $\\ell _{\\operatorname{in}}= O({\\operatorname{poly}}\\log \\Delta ) \\cdot O(\\log n)$ bits.",
"Using the above implementation of , we show that there is an $\\mathsf {LCA}$ that solves $(\\Delta +1)$ -list coloring with $\\Delta ^{O(1)} \\cdot O(\\log n)$ queries.",
"[Proof of thm:lca-main] Consider the following algorithm for solving $(\\Delta +1)$ -list coloring.",
"Run the $O(1)$ -round algorithm of lem:CLP-summary.",
"After that, each vertex $v$ has four possible status: (i) $v$ has been colored, (ii) $v$ is in $V_{\\sf {Good}}$ , (iii) $v$ is in $V_{{\\sf {Bad}}}$ , or $v$ is in $R$ .",
"This can be done with $\\Delta ^{O(1)}$ queries.",
"The set $R$ induces a subgraph with constant maximum degree.",
"The $\\mathsf {LCA}$ for $(\\deg +1)$ -list coloring in [28] implies that each $v \\in R$ only needs $O(\\log ^\\ast n)$ queries of vertices in $R$ to compute its color.",
"By lem:CLP-summary, each connected component in $V_{{\\sf {Bad}}}$ has size $\\Delta ^{O(1)} \\cdot O(\\log n)$ .",
"We let each vertex $v \\in V_{{\\sf {Bad}}}$ learns the component $S$ it belongs to, and apply a deterministic algorithm to color $S$ .",
"All vertices in $V_{\\sf {Good}}$ run the algorithm .",
"This adds an $(\\Delta \\cdot \\Delta _{\\ast }^k)$ -factor in the query complexity, where $\\Delta _{\\ast }= \\max _{v \\in V_0} |N_{\\ast }(v)| = O({\\operatorname{poly}}\\log \\Delta )$ , and $k = O(\\log ^\\ast \\Delta )$ is the number of iterations of .",
"Note that in , when we query a vertex $v$ , the set $N_{\\ast }(v)$ can be calculated from the random bits in $N_{G_0}(v) \\cup \\lbrace v\\rbrace $ .",
"By lem:spcolor and lem:shatter, the vertices left uncolored after induces connected components of size $\\Delta ^{O(1)} \\cdot O(\\log n)$ .",
"Similarly, we let each uncolored vertex $v$ learns the component $S$ it belongs to, and apply a deterministic algorithm to color $S$ .",
"By the standard procedure for converting an $\\mathsf {LOCAL}$ algorithm to an $\\mathsf {LCA}$ , it is straightforward to implement the above algorithm as an $\\mathsf {LCA}$ with query complexity $\\Delta ^{O(1)} \\cdot \\Delta _{\\ast }^k \\cdot \\left( \\Delta ^{O(1)} \\cdot O(\\log n) \\right)= \\Delta ^{O(1)} \\cdot O(\\log n).",
"\\Box $ Next, we show that by applying with the speedup lemma (lem:speedup), we can solve $(\\Delta +1)$ -list coloring in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ in $O(1)$ rounds when $\\Delta = O({\\operatorname{poly}}\\log n)$ .",
"Theorem 4.7 Suppose $\\Delta = O({\\operatorname{poly}}\\log n)$ .",
"There is an algorithm that solves $(\\Delta +1)$ -list coloring in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ in $O(1)$ rounds.",
"Recall from lem:routing that one round in $\\mathsf {LOCAL}$ with messages of at most $O(n \\log n)$ bits can be simulated in $O(1)$ rounds in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ .",
"The first step of the algorithm is to run the black box algorithm for lem:CLP-summary, which takes $O(1)$ rounds in $\\mathsf {LOCAL}$ with messages of size $O(\\Delta ^2 \\log n) \\ll O(n \\log n)$ .",
"The set $R$ trivially induces a subgraph with $O(n)$ edges.",
"By lem:shatter, $V_{{\\sf {Bad}}}$ induces a subgraph with $O(n)$ edges.",
"We use lem:routing to color them in $O(1)$ rounds.",
"Now we focus on $V_{\\sf {Good}}$ .",
"We execute the algorithm using the above implementation.",
"The pre-processing step takes $O(1)$ rounds in $\\mathsf {LOCAL}$ with messages of size $O({\\operatorname{poly}}\\log \\Delta ) \\cdot O(\\log n) \\ll O(n \\log n)$ .",
"For the main steps, we apply the speedup lemma (lem:speedup) with $\\tau = O(\\log ^\\ast \\Delta )$ , $\\ell _{\\operatorname{out}}= O(\\log n)$ , $\\ell _{\\operatorname{in}}= O({\\operatorname{poly}}\\log \\Delta ) \\cdot O(\\log n)$ , and $\\Delta _{\\ast }= O({\\operatorname{poly}}\\log \\Delta )$ .",
"Since we assume $\\Delta = O({\\operatorname{poly}}\\log n)$ , this satisfies the criterion for lem:speedup, and so this procedure can be executed on $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ in $O(1)$ rounds.",
"The algorithm does not color all vertices in $V_{\\sf {Good}}$ .",
"However, by lem:spcolor and lem:shatter, we know that these uncolored vertices induces a subgraph with $O(n)$ edges.",
"We use lem:routing to color them in $O(1)$ rounds."
],
[
"Probabilistic Tools",
"In this section we review some probabilistic tools used in this paper."
],
[
"Chernoff Bound.",
"Let $X$ be the summation of $n$ independent 0-1 random variables with mean $p$ .",
"Multiplicative Chernoff bounds give the following tail bound of $X$ with mean $\\mu = np$ .",
"$\\operatorname{Pr}[X \\ge (1+\\delta )\\mu ] \\le {\\left\\lbrace \\begin{array}{ll}\\exp (\\frac{- \\delta ^2 \\mu }{3}) & \\text{if } \\delta \\in [0,1]\\\\\\exp (\\frac{- \\delta \\mu }{3}) & \\text{if } \\delta > 1.\\end{array}\\right.",
"}$ Note that these bounds hold even when $X$ is the summation of $n$ negatively correlated 0-1 random variables [26], [25] with mean $p$ , i.e., total independent is not required.",
"These bounds also hold when $\\mu > np$ is an overestimate of $\\operatorname{E}[X]$ ."
],
[
"Chernoff Bound with $k$ -wise Independence.",
"Suppose $X$ is the summation of $n$ $k$ -wise independent 0-1 random variables with mean $p$ .",
"We have $\\mu \\ge \\operatorname{E}[X] = np$ and the following tail bound [61].",
"$\\operatorname{Pr}[X \\ge (1+\\delta )\\mu ] \\le \\exp \\left(-\\min \\lbrace k, \\delta ^2 \\mu \\rbrace \\right).$ In particular, when $k = \\Omega (\\delta ^2 \\mu )$ , we obtain the same asymptotic tail bound as that of Chernoff bound with total independence."
],
[
"Chernoff Bound with Bounded Independence.",
"Suppose $X$ is the summation of $n$ independent 0-1 random variables with bounded dependency $d$ , and let $\\mu \\ge \\operatorname{E}[X]$ , where $X = \\sum _{i=1}^n X_i$ .",
"Then we have [54]: $\\operatorname{Pr}[X \\ge (1+\\delta )\\mu ] \\le O(d) \\cdot \\exp (-\\Omega (\\delta ^2 \\mu / d)).$"
],
[
"Hoeffding's Inequality.",
"Consider the scenario where $X = \\sum _{i=1}^n X_i$ , and each $X_i$ is an independent random variable bounded by the interval $[a_i, b_i]$ .",
"Let $\\mu \\ge \\operatorname{E}[X]$ .",
"Then we have the following concentration bound [36].",
"$\\operatorname{Pr}[X \\ge (1+\\delta )\\mu ] \\le \\exp \\left(\\frac{-2(\\delta \\mu )^2}{\\sum _{i=1}^n (b_i - a_i)^2}\\right).$"
],
[
"Proof of the Shattering Lemma ",
"In this section, we prove the shattering lemma (lem:shatter).",
"Let $c \\ge 1$ .",
"Consider a randomized procedure that generates a subset of vertices $B \\subseteq V$ .",
"Suppose that for each $v \\in V$ , we have $\\operatorname{Pr}[v \\in B] \\le \\Delta ^{-3c}$ , and this holds even if the random bits not in ${N}^{c}(v)$ are determined adversarially.",
"Then, the following is true.",
"With probability at least $1 - n^{- \\Omega (c^{\\prime })}$ , each connected component in the graph induced by $B$ has size at most $(c^{\\prime }/c) \\Delta ^{2c} \\log _{\\Delta } n$ .",
"With probability $1 - O(\\Delta ^c) \\cdot \\exp (-\\Omega (n \\Delta ^{-c}))$ , the number of edges induced by $B$ is $O(n)$ .",
"Statement (1) is well-known; see e.g., [10], [27].",
"Here we provide a proof for Statement (2).",
"For each edge $e = \\lbrace u,v\\rbrace $ , write $X_e$ to be the indicator random variable such that $X_e = 1$ if $u \\in B$ and $v \\in B$ .",
"Let $X = \\sum _{e \\in E} X_e$ .",
"It is clear that $\\operatorname{Pr}[X_e] \\le 2 \\Delta ^{-3c}$ , and so $\\mu = \\operatorname{E}[X] \\le n \\Delta ^{1 - 3c} \\ll n$ .",
"By a Chernoff bound with bounded dependence $d = 2 \\Delta ^{c}$ the probability that $X > n$ is $O(\\Delta ^c) \\cdot \\exp (-\\Omega (n \\Delta ^{-c}))$ ."
],
[
"Fast Simulation of $\\mathsf {LOCAL}$ Algorithms in {{formula:a8462dfd-0545-46f0-9c4e-e9189221382f}}",
"In this section, we prove lem:speedup.",
"Let $\\mathcal {A}$ be a $\\tau $ -round $\\mathsf {LOCAL}$ algorithm on $G = (V, E)$ .",
"We show that there is an $O(1)$ -round simulation of $\\mathcal {A}$ in in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ , given that (i) $\\Delta _{\\ast }^{\\tau } \\log (\\Delta _{\\ast }+ \\ell _{\\operatorname{in}}/\\log n) = O(\\log n)$ , (ii) $\\ell _{\\operatorname{in}}= O(n)$ , and (iii) $\\ell _{\\operatorname{out}}= O(\\log n)$ .",
"Assume $\\mathcal {A}$ is in the following canonical form.",
"Each vertex first generates certain amount of local random bits, and then collects all information in its $\\tau $ -neighborhood.",
"The information includes not only the graph topology, but also IDs, inputs, and the random bits of these vertices.",
"After gathering this information, each vertex locally computes its output based on the information it gathered.",
"Consider the following procedure in $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ for simulating $\\mathcal {A}$ .",
"In the first phase, for each ordered vertex pair $(u,v)$ , with probability $p$ to be determined, $u$ sends all its local information to $v$ .",
"The local information can be encoded in $\\Theta (\\Delta _{\\ast }\\log n + \\ell _{\\operatorname{in}})$ bits.",
"This includes the local input of $u$ , the local random bits needed for $u$ to run $\\mathcal {A}$ , and the list of IDs in $N_{\\ast }(u) \\cup \\lbrace u\\rbrace $ .",
"In the second phase, for each ordered vertex pair $(u,v)$ , if $v$ has gathered all the required information to calculate the output of $\\mathcal {A}$ at $u$ , then $v$ sends to $u$ the output of $\\mathcal {A}$ at $u$ .",
"At first sight, the procedure seems to take $\\omega (1)$ rounds because of the $O(\\log n)$ -bit message size constraint of $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ .",
"However, if we set $p = \\Theta \\left( \\frac{1}{\\Delta _{\\ast }+ \\ell _{\\operatorname{in}}/\\log n}\\right)$ , the expected number of $O(\\log n)$ -bit messages sent from or received by a vertex is $n p \\cdot \\Theta (\\Delta _{\\ast }+ \\ell _{\\operatorname{in}}/ \\log n) = O(n)$ .",
"More precisely, let $X_u$ be the number vertices $v \\in V$ to which $u$ sends its local information in the first phase; similarly, let $Y_v$ be the number of vertices $u \\in V$ sending their local information to a $v$ .",
"We have $\\operatorname{E}[X_u] = np$ , for each $u \\in V$ , and $\\operatorname{E}[Y_v] = np$ , for each $v \\in V$ .",
"By a Chernoff bound, so long as $np = \\Omega (\\log n)$ , with probability $1 - \\exp (-\\Omega (np)) = 1/ {\\operatorname{poly}}(n)$ , we have $X_u = O(np)$ , for each $u \\in V$ , and $Y_v = O(np)$ , for each $v \\in V$ .",
"That is, the number of $O(\\log n)$ -bit messages sent from or received by a vertex is at most $n p \\cdot \\Theta (\\Delta _{\\ast }+ \\ell _{\\operatorname{in}}/ \\log n) = O(n)$ , w.h.p.",
"We verify that $np = \\Omega (\\log n)$ .",
"Condition (i) implicitly requires $\\Delta _{\\ast }= O(\\log n)$ , and Condition (ii) requires $\\ell _{\\operatorname{in}}= O(n)$ .",
"Therefore, $np = \\Theta \\left(\\frac{n}{\\Delta + \\ell _{\\operatorname{in}}/\\log n}\\right) = \\Omega (\\log n)$ .",
"Thus, we can route all messages in $O(1)$ rounds using Lenzen's routing (Lemma REF ), and so the first phase can be done in $O(1)$ rounds.",
"Condition (iii) guarantees that $\\ell _{\\operatorname{out}}= O(\\log n)$ , and so the messages in the second phase can be sent directly in $O(1)$ rounds.",
"What remains to do is to show that for each $u \\in V$ , w.h.p., there is a vertex $v \\in V$ that receives messages from all vertices in $N_{\\ast }^{\\tau }(u)$ during the first phase, and so $v$ is able to calculate the output of $u$ locally.",
"Denote $E_{u,v}$ as the event that $v \\in V$ that receives messages from all vertices in $N_{\\ast }^{\\tau }(u)$ during the first phase, and denote $E_{u}$ as the event that at least one of $\\lbrace E_{u,v} \\ | \\ v \\in V \\rbrace $ occurs.",
"We have $\\operatorname{Pr}[ E_{u,v} ] \\ge p^{\\Delta _{\\ast }^{\\tau }}$ , since $|N_{\\ast }^{\\tau }(u)| \\le \\Delta _{\\ast }^{\\tau }$ .",
"Thus, $\\operatorname{Pr}[E_u] \\ge 1 - ( 1 - p^{\\Delta _{\\ast }^{\\tau }} )^n$ .",
"Condition (i) guarantees that $\\Delta ^{\\tau } \\log (\\Delta _{\\ast }+ \\ell _{\\operatorname{in}}/\\log n) = O(\\log n)$ .",
"By setting $p = \\epsilon / (\\Delta _{\\ast }+ \\ell _{\\operatorname{in}}/\\log n)$ for some sufficiently small constant $\\epsilon $ , we haves $\\Delta _{\\ast }^{\\tau } \\log p \\ge -\\frac{1}{2} \\log n$ , and it implies $p^{\\Delta _{\\ast }^{\\tau }} \\ge 1/\\sqrt{n}$ .",
"Therefore, $\\operatorname{Pr}[E_u] \\ge 1 - ( 1 - p^{\\Delta _{\\ast }^{\\tau }} )^n = 1 - \\exp (-\\Omega (\\sqrt{n}))$ .",
"Thus, the simulation gives the correct output for all vertices w.h.p.",
"We remark that the purpose of the condition $\\ell _{\\operatorname{out}}= O(\\log n)$ is only to allow the messages in the second phase to be sent directly in $O(1)$ rounds.",
"With a more careful analysis and using Lenzen's routing , the condition $\\ell _{\\operatorname{out}}= O(\\log n)$ can be relaxed to $\\ell _{\\operatorname{out}}= O(n)$ , though in our application we only need $\\ell _{\\operatorname{out}}= O(\\log n)$ ."
],
[
"Proof of lem:CLP-summary ",
"In this section, we briefly review the algorithm of [20] and show how to obtain lem:CLP-summary from [20].",
"The algorithm uses a sparsity sequence defined by $\\epsilon _1 = \\Delta ^{-1/10}$ , $\\epsilon _i = \\sqrt{\\epsilon _{i-1}}$ for $i>1$ , and $\\ell = \\Theta (\\log \\log \\Delta )$ is the largest index such that $\\frac{1}{\\epsilon _{\\ell }} \\ge K$ for some sufficiently large constant $K$ .",
"The algorithm first do an $O(1)$ -round procedure (initial coloring step) to color a fraction of the vertex set $V$ , and denote $V^\\star $ as the set of remaining uncolored vertices.",
"The set $V^\\star $ is decomposed into $\\ell +1$ subsets $(V_1,\\ldots , V_{\\ell }, V_{\\mathsf {sp}})$ according to local sparsity.",
"The algorithm then applies another $O(1)$ -round procedure (dense coloring step) to color a fraction of vertices in $V_1 \\cup \\cdots \\cup V_{\\ell }$ .",
"The remaining uncolored vertices in $V^\\star $ after the above procedure (initial coloring step and dense coloring step) are partitioned into three subsets: $U$ , $R$ , and $V_{\\mathsf {bad}}$ .The algorithm in [20] for coloring layer-1 large blocks has two alternatives.",
"Here we always use the one that puts the remaining uncolored vertices in one of $R$ or $V_{\\mathsf {bad}}$ , where each vertex is added to $V_{\\mathsf {bad}}$ with probability $\\Delta ^{-\\Omega (c)}$ .",
"The set $R$ induces a constant-degree graph.",
"The set $V_{\\mathsf {bad}}$ satisfies the property that each vertex is added to $V_{\\mathsf {bad}}$ with probability $\\Delta ^{-\\Omega (c)}$ , where $c$ can be any given constant, independent on the runtime.",
"The vertices in $U$ satisfy the following properties.",
"Excess Colors: We have $V_1 \\cap U = \\varnothing $ .",
"Each $v \\in V_i \\cap U$ , with $i > 1$ , has $\\Omega (\\epsilon _{i-1}^2 \\Delta )$ excess colors.",
"Each $v \\in V_{\\mathsf {sp}}\\cap U$ has $\\Omega (\\epsilon _\\ell ^2 \\Delta ) = \\Omega (\\Delta )$ excess colors.",
"The number of excess colors at a vertex $v$ is defined by the number of available colors of $v$ minus the number of uncolored neighbors of $v$ .",
"Number of Neighbors: For each $v \\in U$ , and for each $i \\in [2, \\ell ]$ , the number of uncolored neighbors of $v$ in $V_i\\cap U$ is $O(\\epsilon _i^5 \\Delta ) = O(\\epsilon _{i-1}^{2.5} \\Delta )$ .",
"The number of uncolored neighbors of $v$ in $V_{\\mathsf {sp}} \\cap U$ is of course at most $\\Delta = O(\\epsilon _{\\ell }^{2.5} \\Delta )$ , since $\\epsilon _{\\ell }$ is a constant.",
"At this moment, the two sets $V_{\\mathsf {bad}}$ and $R$ satisfy the required condition specified in lem:CLP-summary.",
"In what follows, we focus on $U$ ."
],
[
"Orientation.",
"We orient the graph induced by the uncolored vertices in $U$ as follows.",
"For any edge $\\lbrace u,v\\rbrace $ , we orient it as $(u,v)$ if one of the following is true: (i) $u \\in V_{\\mathsf {sp}}$ but $v \\notin V_{\\mathsf {sp}}$ , (ii) $u \\in V_i$ and $v \\in V_j$ with $i > j$ , (iii) $u$ and $v$ are within the same part in the partition $V^\\star = V_1 \\cup \\ldots V_{\\ell } \\cup V_{\\mathsf {sp}}$ and $\\operatorname{ID}(v) < \\operatorname{ID}(u)$ .",
"This results in a directed acyclic graph.",
"We write $N^{\\operatorname{out}}(v)$ to denote the set of out-neighbors of $v$ in this graph."
],
[
"Lower Bound of Excess Colors.",
"In view of the above, there exist universal constants $\\eta > 0$ and $C > 0$ such that the following is true.",
"For each $i \\in [2,\\ell ]$ and each uncolored vertex $v \\in V_i \\setminus V_{\\mathsf {bad}}$ , we set $p_v = \\eta \\epsilon _{i-1}^2 \\Delta $ .",
"For each $v \\in V_{\\mathsf {sp}} \\setminus V_{\\mathsf {bad}}$ , we set $p_v = \\eta \\epsilon _{\\ell }^2 \\Delta $ .",
"By selecting a sufficiently small $\\eta $ , the number $p_v$ is always a lower bound on the number of excess colors at $v$ ."
],
[
"The Number of Excess Colors is Large.",
"Recall that to color the graph quickly we need the number of excess colors to be sufficiently large with respect to out-degree.",
"If $v \\in V_i \\cap U$ with $i \\ge 2$ , it satisfies $|N^{\\operatorname{out}}(v)| = \\sum _{j=2}^{i} O(\\epsilon _{j-1}^{2.5} \\Delta ) = O(\\epsilon _{i-1}^{2.5} \\Delta )$ .",
"In this case, $p_v / |N^{\\operatorname{out}}(v)| = \\Omega (\\epsilon _{i-1}^{-0.5})$ .",
"If $v \\in V_{\\mathsf {sp}} \\cap U$ , then of course $|N^{\\operatorname{out}}(v)| \\le \\Delta = O(\\epsilon _{\\ell }^2 \\Delta )$ , since $\\epsilon _{\\ell }$ is a constant.",
"In this case, $p_v / |N^{\\operatorname{out}}(v)| = \\Omega (\\epsilon _{\\ell }^{-0.5})$ .",
"However, due to the high variation on the palette size in our setting, $p_v / |N^{\\operatorname{out}}(v)|$ is not a good measurement for the gap between the number of excess colors and out-degree at $v$ .",
"The inverse of the expression $\\sum _{u \\in N^{\\operatorname{out}}(v)} 1 / p_u$ turns out to be a better measurement, as it takes into account the number of excess colors in each out-neighbor.",
"There is a constant $C > 0$ such that for each uncolored vertex $v \\in V^\\star \\setminus (V_{\\mathsf {bad}} \\cup R)$ , we have $\\sum _{u \\in N^{\\operatorname{out}}(v)} 1 / p_u \\le 1/C$ .",
"The calculation is as follows.",
"$&\\text{If $v \\in V_i \\cap U$ \\ $(i>1)$,} \\ & & \\text{then }\\sum _{u \\in N^{\\operatorname{out}}(v)} 1 / p_u=\\sum _{j=2}^{i} O\\mathopen {}\\left( \\frac{\\epsilon _{j-1}^{2.5} \\Delta }{\\epsilon _{j-1}^2 \\Delta } \\right)\\mathclose {} = \\sum _{j=2}^{i} O({\\epsilon _{j-1}^{0.5}}) = O(\\epsilon _{i-1}^{0.5}) < 1/C.\\\\&\\text{If $v \\in V_{\\mathsf {sp}} \\cap U$, } \\ & & \\text{then }\\sum _{u \\in N^{\\operatorname{out}}(v)} 1 / p_u=\\sum _{j=2}^{\\ell +1} O\\mathopen {}\\left( \\frac{\\epsilon _{j-1}^{2.5} \\Delta }{\\epsilon _{j-1}^2 \\Delta } \\right)\\mathclose {} = \\sum _{j=2}^{\\ell +1} O({\\epsilon _{j-1}^{0.5}}) = O(\\epsilon _{\\ell }^{0.5}) < 1/C.$ For a specific example, if $v$ is an uncolored vertex in $V_2 \\setminus V_{\\mathsf {bad}}$ , then $p_v = \\eta \\epsilon _1^2 \\Delta = \\eta \\Delta ^{0.8}$ is the lower bound on the number of excess colors at $v$ , and $v$ has out-degree $|N^{\\operatorname{out}}(v)| = O(\\epsilon _1^{2.5} \\Delta ) = O(\\Delta ^{0.75})$ , and we have $\\sum _{u \\in N^{\\operatorname{out}}(v)} 1 / p_u = O(\\epsilon _1^{0.5}) = O(\\Delta ^{-0.05}) < 1/C$ .",
"Intuitively, this means that the gap between the number of excess colors and the out-degree at $v$ is $\\Omega (\\Delta ^{0.05})$ ."
],
[
"Summary.",
"Currently the graph induced by $U$ satisfies the following conditions.",
"Each vertex $v$ is associated with a parameter $p_v = \\eta \\epsilon _j^2 \\Delta $ (for some $j \\in [1,\\ell ]$ ) such that the number of excess colors at $v$ is at least $p_v = \\Omega (\\epsilon _j^2 \\Delta )$ , but the number of out-neighbors of $v$ is at most $O(\\epsilon _j^{2.5} \\Delta )$ .",
"In particular, we always have $\\sum _{u \\in N^{\\operatorname{out}}(v)} 1/p_u = O(\\epsilon _{j}^{0.5}) < 1/C$ , where $C > 0$ is a universal constant.",
"The current $p_v$ -values for vertices in $U$ almost satisfy the required condition for $V_{\\sf {Good}}$ specified in lem:CLP-summary.",
"Lower Bound of $p^\\star $ .",
"Define $p^\\star $ as the minimum $p_v$ -value among all uncolored vertices $v \\in V^\\star $ .",
"Currently we only have $p^\\star \\ge \\eta \\epsilon _1^2 \\Delta = \\eta \\Delta ^{0.8}$ , but in lem:CLP-summary it is required that $p^\\star \\ge \\Delta / \\log \\Delta $ .",
"Lower Bound of $C$ .",
"Currently we have $\\sum _{u \\in N^{\\operatorname{out}}(v)} 1/ p_u \\le 1/C$ for some universal constant $C$ , but in lem:CLP-summary it is required that $C > 0$ can be any given constant.",
"For the rest of the section, we show that there is an $O(1)$ -round that is able to improve the lower bound of $p^\\star $ to $p^\\star \\ge \\Delta / \\log \\Delta $ and increase the parameter $C$ to any specified constant.",
"The procedure will colors a fraction of vertices in $U$ and puts some vertices in $U$ to the set $V_{\\mathsf {bad}}$ .",
"We first consider improving the lower bound of $p^\\star $ .",
"This is done by letting all vertices whose $p_v$ -value are too small (i.e., less than $\\Delta / \\log \\Delta $ ) to jointly run lem:color-remain-copy.",
"For these vertices, we have $\\sum _{u \\in N^{\\operatorname{out}}(v)} 1/ p_u \\le O\\left(\\log ^{-1/4} \\Delta \\right)$ ,Each vertex $v$ is associated with a parameter $p_v = \\eta \\epsilon _j^2 \\Delta $ , and we have $\\sum _{u \\in N^{\\operatorname{out}}(v)} 1/p_u = O(\\epsilon _{j}^{0.5}) = O(p_v^{1/4})$ .",
"and so we can use $C = \\Omega \\left( \\log ^{1/4} \\Delta \\right)$ in lem:color-remain-copy.",
"The algorithm of lem:color-remain-copy takes only $O(1)$ rounds.",
"All participating vertices that still remain uncolored join $V_{\\mathsf {bad}}$ .",
"Lemma D.1 ([20]) Consider a directed acyclic graph, where vertex $v$ is associated with a parameter $p_v \\le |\\Psi (v)| - \\deg (v)$ We write $p^\\star = \\min _{v\\in V} p_v$ .",
"Suppose that there is a number $C = \\Omega (1)$ such that all vertices $v$ satisfy $\\sum _{u \\in N^{\\operatorname{out}}(v)} 1/ p_u \\le 1/C$ .",
"Let $d^\\star $ be the maximum out-degree of the graph.",
"There is an $O(\\log ^\\ast (p^\\star ) - \\log ^\\ast (C))$ -time algorithm achieving the following.",
"Each vertex $v$ remains uncolored with probability at most $\\exp (-\\Omega (\\sqrt{p^\\star })) + d^\\star \\exp (-\\Omega (p^\\star ))$ .",
"This is true even if the random bits generated outside a constant radius around $v$ are determined adversarially.",
"Now the lower bound on $p^\\star $ is met.",
"We show how to increase the $C$ -value to any given constant we like in $O(1)$ rounds.",
"We apply lem:shrink-copy using the current $p^\\star $ and $C$ .",
"After that, we can set the new $C$ -value to be $C^{\\prime } = C \\cdot \\exp (C/6) / (1+\\lambda )$ , after putting each vertex $v$ not meeting the following condition to $V_{\\mathsf {bad}}$ : $\\text{Sum of $1/p_u$ over all remaining uncolored vertices $u$ in $N^{\\operatorname{out}}(v)$ is at most $1/C^{\\prime } = \\frac{1+\\lambda }{ \\exp (C/6) C }$.",
"}$ If $\\lambda $ is chosen as a small enough constant, we have $C^{\\prime } > C$ .",
"After a constant number of iterations, we can increase the $C$ -value to any constant we like.",
"Now, all conditions in lem:CLP-summary are met for the three sets $R$ , $V_{\\mathsf {bad}}$ , and $V_{\\sf {Good}} \\leftarrow U$ .",
"Lemma D.2 ([20]) There is an one-round algorithm meeting the following conditions.",
"Let $v$ be any vertex.",
"Let $d$ be the summation of $1/p_u$ over all vertices $u$ in $N^{\\operatorname{out}}(v)$ that remain uncolored after the algorithm.",
"Then the following holds.",
"$\\operatorname{Pr}\\left[d \\ge \\frac{1+\\lambda }{ \\exp (C/6) C }\\right] &\\le \\exp \\left(-2 \\lambda ^2 p^\\star \\exp (-C/3) / C \\right) +d^\\star \\exp (-\\Omega (p^\\star )).$"
],
[
"The CLP Algorithm in the Low-Memory $\\mathsf {MPC}$ Model",
"In this section, we show which changes have to be made to [20] to get a low-memory $\\mathsf {MPC}$ algorithm, thus proving lemma:CLP.",
"There are two main issues in the low-memory $\\mathsf {MPC}$ model that we need to take care of.",
"First, the total memory of the system is limited to $\\tilde{\\Theta }(m + n)$ , where $m$ and $n$ are the number of edges and vertices in the input graph, respectively.",
"Second, the local memory per machine is restricted to $O(n^{\\alpha })$ , for an arbitrary constant $\\alpha >0$ .",
"These two restrictions force us to be careful about the amount of information sent between the machines.",
"In particular, no vertex can receive messages from more than $O(n^{\\alpha })$ other vertices in one round (as opposed to the $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ , where a vertex can receive up to $O(n)$ messages per round).",
"The key feature of our partitioning algorithm is that we can reduce the coloring problem to several instances of coloring graphs with maximum degree $\\Delta = O(n^{\\alpha / 2})$ .",
"Given this assumption, we can implement the CLP algorithm in the low-memory $\\mathsf {MPC}$ model almost line by line as done by Parter [52] for the $\\mathsf {CONGESTED}\\text{-}\\mathsf {CLIQUE}$ .",
"Therefore, here we simply point out the differences in the algorithm and refer the reader to the paper by Parter for further technical details."
],
[
"Dense Vertices.",
"Put briefly, a vertex is $\\gamma $ -dense, if a $(1 - \\gamma )$ -fraction of the edges incident on it belong to at least $(1 - \\gamma ) \\cdot \\Delta $ triangles.",
"An $\\gamma $ -almost clique is a connected component of $\\gamma $ -dense vertices that have at most $\\gamma \\cdot \\Delta $ vertices outside the component.",
"Each such component has a weak diameter of at most 2.",
"These components can be computed in 2 rounds by each vertex learning its 2-hop neighborhood.",
"This process is performed $O(\\log \\log \\Delta )$ times in parallel which incurs a factor of $O(\\log \\log \\Delta )$ in the memory requirements, which is negligible.",
"Furthermore, the algorithm requires running a coloring algorithm within the dense components.",
"Since the component size is at most $\\Delta \\ll \\Delta ^2$ , we can choose one vertex in the component as a leader and the leader vertex can locally simulate the coloring algorithm without breaking the local memory restriction."
],
[
"Memory Bounds.",
"Once the 2-hop neighborhoods of nodes have been learned, no more memory overhead is required.",
"Since we have $\\Delta \\ll n^{\\alpha /2}$ , learning the 2-hop neighborhoods does not violate the local memory restriction of $O(n^{\\alpha })$ .",
"For the total memory bound, storing the 2-hop neighborhoods requires at most $\\widetilde{O}(\\sum _v (\\deg _G (v))^2)$ memory."
],
[
"Post-Shattering and Clean-up.",
"Another step that we cannot use as a black box is a subroutine that colors a graph that consists of connected components of $O({\\operatorname{poly}}\\log n)$ size.",
"Regardless of the component sizes being small, all vertices over all components might not fit the memory of a single machine.",
"Hence, similarly to the CLP algorithm in the $\\mathsf {LOCAL}$ model, we use the best deterministic list coloring algorithm to color the components.",
"For general graphs, currently the best runtime in the $\\mathsf {LOCAL}$ model is obtained by applying the algorithm by Panconesi and Srinivasan [56] with runtime of $ 2^{O(\\sqrt{\\log n^{\\prime }})} $ , where $n^{\\prime } = O({\\operatorname{poly}}\\log n)$ is the maximum size of the small components.",
"We can improve this bound exponentially in the $\\mathsf {MPC}$ model by using the known graph exponentiation technique [45], [12] and obtain a runtime of $O(\\sqrt{\\log \\log n})$ .",
"The graph exponentiation technique works as follows.",
"Suppose that every vertex knows all the vertices and the topology of its $2^{i - 1}$ -hop neighborhood in round $i - 1$ for some integer $i \\ge 0$ .",
"Then, in round $i$ , every vertex can communicate the topology of its $2^{i - 1}$ -hop neighborhood to all the vertices in its $2^{i - 1}$ -hop neighborhood.",
"This way, every vertex learns its $2^{i}$ -hop neighborhood in round $i$ and hence, every vertex can simulate any $2^i$ -round $\\mathsf {LOCAL}$ algorithm in $i$ rounds.",
"We observe that, in the components of $O({\\operatorname{poly}}\\log n)$ size, the $2^{i}$ -hop neighborhood of any vertex for any $i$ fits into the memory of a single machine since the number of vertices in the neighborhood is clearly bounded by $O({\\operatorname{poly}}\\log n)$ .",
"The same observation yields that the total memory of $\\tilde{O}(m)$ suffices."
]
] | 1808.08419 |
[
[
"Inequalities of Riesz-Sobolev type for compact connected Abelian groups"
],
[
"Abstract A version of the Riesz-Sobolev convolution inequality is formulated and proved for arbitrary compact connected Abelian groups.",
"Maximizers are characterized and a quantitative stability theorem is proved, under natural hypotheses.",
"A corresponding stability theorem for sets whose sumset has nearly minimal measure is also proved, sharpening recent results of other authors.",
"For the special case of the group $\\mathbb{R}/\\mathbb{Z}$, a continuous deformation of sets is developed, under which an appropriately scaled Riesz-Sobolev functional is shown to be nondecreasing."
],
[
"Introduction",
"Let $G$ be a compact connected Abelian topological group, equipped with Haar measure $\\mu $ .",
"Throughout this paper, the measure $\\mu $ is assumed to be complete.",
"We say that $\\mu $ is normalized to mean that $\\mu (G)=1$ .",
"By a measurable subset of $G$ we will always mean a $\\mu $ –measurable subset.",
"$\\mu _*$ denotes the associated inner measure.",
"Let ${\\mathbb {T}}={\\mathbb {R}}/{\\mathbb {Z}}$ , equipped with Lebesgue measure $m$ , with $m({\\mathbb {T}})=1$ .",
"Our first result is a Riesz-Sobolev–type inequality for $G$ , of which the following is one of several formulations.",
"Assuming $\\mu $ to be normalized, to any measurable set $A\\subset G$ is associated the set ${A^\\star }\\subset {\\mathbb {T}}$ , which is defined to be the closed interval centered at 0 satisfying $m({A^\\star }) = \\mu (A)$ .",
"Convolution on $G$ is defined by $f*g(x) = \\int _G f(x-y)g(y)\\,d\\mu (y)$ .",
"${\\mathbf {1}}_A$ denotes the indicator function of $A$ .",
"Theorem 1.1 Let $G$ be a compact connected Abelian topological group, equipped with normalized Haar measure.",
"For any measurable subsets $A,B,C\\subset G$ , $ \\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu \\le \\int _{{C^\\star }} {\\mathbf {1}}_{{A^\\star }}*{\\mathbf {1}}_{{B^\\star }}\\,dm.$ Kneser's inequality [18] $ \\mu _*(A+B) \\ge \\min (\\mu (A)+\\mu (B),\\mu (G))$ may be viewed as a limiting case of (REF ).",
"A mildly stronger formulation is(REF ) follows from (REF ) for $G={\\mathbb {T}}^d$ by a simple argument involving points of density, since $A+B=A+_0B$ if every point of each of $A,B$ is a point of density.",
"For general groups $G$ , (REF ) follows from the special case of ${\\mathbb {T}}^d$ by approximating by elements of the algebra generated by Bohr sets.",
"Alternatively, a stronger form of (REF ) is proved in [22].",
"$ \\mu (A+_0 B) \\ge \\min (\\mu (A)+\\mu (B),\\mu (G))$ where $A+_0 B$ is the open set $ A+_0 B:=\\lbrace x: {\\mathbf {1}}_A*{\\mathbf {1}}_B(x)>0\\rbrace .$ Indeed, $\\mu _*(A+B)\\ge \\mu _*(A+_0 B) = \\mu (A+_0 B)$ .",
"Our main theme is the quantitative characterization of triples $(A,B,C)$ that maximize, or nearly maximize, the functional $\\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu $ among all sets of specified Haar measures.",
"However, the inequality (REF ) seems to have attracted little attention in the setting of compact groups, so some of its aspects relevant to this characterization are also developed here.",
"In the parameter range of primary interest, (REF ) can be restated with an alternative expression for the right-hand side.",
"Theorem 1.2 For any compact connected Abelian topological group $G$ and any measurable subsets $A,B,C\\subset G$ satisfying $\\left\\lbrace {\\begin{array}{c}|\\mu (A)-\\mu (B)|\\le \\mu (C)\\le \\mu (A)+\\mu (B),\\\\\\mu (A)+\\mu (B)+\\mu (C)\\le 2,\\end{array}} \\right.",
"$ one has $ \\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu \\le \\tfrac{1}{2} (ab + bc + ca) - \\tfrac{1}{4} (a^2+b^2+c^2)= ab-\\tfrac{1}{4}(a+b-c)^2$ where $(a,b,c) = (\\mu (A),\\mu (B),\\mu (C))$ .",
"The conclusion (REF ) can also be stated $ \\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu \\le \\mu (A)\\mu (B)-\\tfrac{1}{4} (\\mu (A)+\\mu (B)-\\mu (C))^2$ where $\\tau $ is defined by $\\mu (C) = \\mu (A)+\\mu (B)-2\\tau $ .",
"Both hypotheses (REF ) are invariant under permutations of $(A,B,C)$ .",
"Likewise, the modified form $\\int _{-C} {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu $ , where $-C=\\lbrace -x: x\\in C\\rbrace $ , is invariant under permutations of $(A,B,C)$ .",
"Equality holds in (REF ), under the indicated hypotheses on $(\\mu (A),\\mu (B),\\mu (C))$ , when $G={\\mathbb {T}}$ and $(A,B,C) = ({A^\\star },{B^\\star },{C^\\star })$ .",
"Thus (REF ) is a direct restatement of (REF ) in this parameter regime.",
"If the hypothesis (REF ) is violated, then (REF ) is easily verified directly, using the trivial upper bound $\\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu \\le \\min (\\mu (A)\\mu (B),\\mu (B)\\mu (C),\\mu (C)\\mu (A))$ which follows from $\\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu \\le \\int _G {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu = \\mu (A)\\mu (B)$ and permutation invariance.",
"In this paper we will focus primarily on the regime in which the hypotheses (REF ) hold.",
"The formulation (REF ) is analogous to the Riesz-Sobolev inequality for ${\\mathbb {R}}^d$ , but now the symmetrization ${A^\\star }$ is a subset of ${\\mathbb {T}}$ , rather than of $G$ .",
"As is the case for ${\\mathbb {R}}^d$ , the inequality for indicator functions implies the generalization $ \\langle f*g,h\\rangle _G\\le \\langle f^\\star *g^\\star ,h^\\star \\rangle _{\\mathbb {T}}$ for arbitrary nonnegative measurable functions defined on $G$ , with the pairing $\\langle \\varphi ,\\psi \\rangle _G = \\int _G \\varphi \\psi \\,d\\mu $ of real-valued functions, and with the natural extension of the definition of symmetrization $f^\\star $ from indicator functions to general nonnegative functions.",
"Thus if ${\\mathbb {T}}$ is identified with $(-\\tfrac{1}{2},\\tfrac{1}{2}]$ by identifying each equivalence class in ${\\mathbb {R}}/{\\mathbb {Z}}$ by its unique representative in this domain, then $f^\\star $ is even, is nonincreasing on $[0,\\tfrac{1}{2}]$ , and is equimeasurable with $f$ .",
"Theorem REF has a stronger conclusion than (REF ).",
"For $G={\\mathbb {T}}$ , Theorem REF was proved by Baernstein [3], and was stated by Luttinger [20].",
"For general compact connected Abelian groups, inequality (REF ) should be regarded as an equivalent formulation of an inequality of Tao [22].",
"The deduction of (REF ) as a consequence of the formulation in [22] is carried out in §§ and below.",
"The formulation of our inverse theorems requires several definitions.",
"Definition 1.1 Two measurable sets $A,A^{\\prime }\\subset G$ are equivalent if $\\mu (A\\,\\Delta \\,A^{\\prime })=0$ .",
"Likewise, two ordered triples ${\\mathbf {E}}=(E_1,E_2,E_3)$ and ${\\mathbf {E}}^{\\prime }=(E^{\\prime }_1,E^{\\prime }_2,E^{\\prime }_3)$ are equivalent if $E_j$ is equivalent to $E^{\\prime }_j$ for each $j\\in \\lbrace 1,2,3\\rbrace $ .",
"Definition 1.2 For $x\\in {\\mathbb {T}}= {\\mathbb {R}}/{\\mathbb {Z}}$ , $ \\Vert x \\Vert _{\\mathbb {T}}= |y|$ where $y\\in [-\\tfrac{1}{2},\\tfrac{1}{2}]$ is congruent to $x$ modulo 1.",
"Definition 1.3 A rank one Bohr set ${\\mathcal {B}}\\subset G$ is a set of the form ${\\mathcal {B}}= {\\mathcal {B}}(\\phi ,\\rho ,c) = \\lbrace x\\in G: \\Vert \\phi (x)-c \\Vert _{\\mathbb {T}}\\le \\rho \\rbrace ,$ where $\\phi : G\\rightarrow {\\mathbb {T}}$ is a continuous homomorphism, $c\\in {\\mathbb {T}}$ , and $\\rho \\in [0,\\tfrac{1}{2}]$ .",
"By a homomorphism $\\phi :G\\rightarrow {\\mathbb {T}}$ , we will always mean a continuous homomorphism.",
"Definition 1.4 Two rank one Bohr subsets ${\\mathcal {B}}_1,{\\mathcal {B}}_2$ of $G$ are parallel if they can be represented as ${\\mathcal {B}}_j = {\\mathcal {B}}(\\phi _j,c_j,\\rho _j)$ with $\\phi _1=\\phi _2$ .",
"An ordered triple $({\\mathcal {B}}_1,{\\mathcal {B}}_2,{\\mathcal {B}}_3)$ of rank one Bohr subsets of $G$ is parallel if these three sets are pairwise parallel.",
"An ordered triple $({\\mathcal {B}}_1,{\\mathcal {B}}_2,{\\mathcal {B}}_3)$ of Bohr sets ${\\mathcal {B}}_j={\\mathcal {B}}(\\phi _j,c_j,\\rho _j)$ is compatibly centered if $c_3=c_1+c_2$ .",
"Definition 1.5 Let $(E_1,E_2,E_3)$ be an ordered triple of measurable subsets of $G$ .",
"$(E_1,E_2,E_3)$ is admissible if $0<\\mu (E_i)<1$ for each $i\\in \\lbrace 1,2,3\\rbrace $ , $\\mu (E_1)+\\mu (E_2)+\\mu (E_3)<2$ , and $\\mu (E_k)\\le \\mu (E_i)+\\mu (E_j)$ for each permutation $(i,j,k)$ of $(1,2,3)$ .",
"$(E_1,E_2,E_3)$ is strictly admissible if it is admissible and $\\mu (E_k)< \\mu (E_i)+\\mu (E_j)$ for every permutation $(i,j,k)$ of $(1,2,3)$ .",
"For any $\\eta >0$ , $(E_1,E_2,E_3)$ is $\\eta $ –strictly admissible if it is admissible and $\\mu (E_k)\\le \\mu (E_i)+\\mu (E_j) - \\eta \\max (\\mu (E_1),\\mu (E_2),\\mu (E_3))$ for every permutation $(i,j,k)$ of $(1,2,3)$ .",
"Admissibility is a property only of the measures $\\mu (E_j)$ , and we will sometimes say instead that $(\\mu (E_1),\\mu (E_2),\\mu (E_3))$ is admissible.",
"The condition that $\\mu (E_k)\\le \\mu (E_i)+\\mu (E_j)$ for every permutation $(i,j,k)$ of $(1,2,3)$ can be equivalently formulated as the condition $|\\mu (E_i)-\\mu (E_j)| \\le \\mu (E_k)\\le \\mu (E_i)+\\mu (E_j)$ for any single permutation.",
"Corresponding equivalences hold for strict admissibility and $\\eta $ –strict admissibility.",
"Simple consequences of $\\eta $ –strict admissibility are $ \\mu (E_i) &\\ge |\\mu (E_j)-\\mu (E_k)| + \\eta \\max (\\mu (E_1),\\mu (E_2),\\mu (E_3)),\\\\ \\mu (E_i)&\\ge \\eta \\mu (E_j) $ for every permutation $(i,j,k)$ of $(1,2,3)$ .",
"Definition 1.6 The ordered triple $(A,B,C)$ of measurable subsets of $G$ is $\\eta $ –bounded if it satisfies $\\mu (A) + \\mu (B) + \\mu (C)\\le (2-\\eta )\\mu (G),\\\\\\min (\\mu (A),\\mu (B),\\mu (C))\\ge \\eta \\mu (G).$ If $(A,B,C)$ is $\\eta $ –strictly admissible and satisfies (REF ) then $\\max (\\mu (A),\\mu (B),\\mu (C))\\le \\tfrac{2-\\eta }{2+\\eta }\\le 1-\\tfrac{\\eta }{2}.$ Indeed, suppose that $\\mu (C)$ is largest.",
"Since $\\mu (A) + \\mu (B) \\ge (1+\\eta )\\mu (C)$ , $(2+\\eta ) \\mu (C) \\le \\mu (A)+\\mu (B)+\\mu (C)\\le 2-\\eta $ .",
"$\\Box $ Theorem 1.3 (Uniqueness of maximizers up to symmetries) Let $G$ be a compact connected Abelian topological group equipped with Haar measure $\\mu $ satisfying $\\mu (G)=1$ .",
"Let $(A,B,C)$ be an admissible triple of measurable subsets of $G$ .",
"Then $\\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu = \\int _{C^*} {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star }\\,dm$ if and only if $(A,B,C)$ is equivalent to a compatibly centered parallel ordered triple of rank one Bohr sets.",
"As recognized by Burchard [6], if $\\mu (C) > \\mu (A)+\\mu (B)$ then no characterization of cases of equality is possible without the admissibility hypothesis, beyond the trivial necessary and sufficient condition that ${\\mathbf {1}}_A*{\\mathbf {1}}_B$ should vanish $\\mu $ –almost everywhere on the complement of $C$ .",
"Our main stability theorem is the following.",
"Theorem 1.4 (Stability) For each $\\eta >0$ there exist $\\delta _0>0$ and ${\\mathbf {C}}<\\infty $ with the following property.",
"Let $G$ be a compact connected Abelian topological group equipped with Haar measure $\\mu $ satisfying $\\mu (G)=1$ .",
"Let $(A,B,C)$ be an $\\eta $ –strictly admissible and $\\eta $ -bounded ordered triple of measurable subsets of $G$ .",
"Let $0\\le \\delta \\le \\delta _0$ .",
"If $\\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu \\ge \\int _{C^\\star }{\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star }\\,dm-\\delta $ then there exists a compatibly centered parallel ordered triple $({\\mathcal {B}}_A,{\\mathcal {B}}_B,{\\mathcal {B}}_C)$ of rank one Bohr sets satisfying $\\mu (A\\,\\Delta \\,{\\mathcal {B}}_A) \\le {\\mathbf {C}}\\delta ^{1/2}$ and likewise for $(B,{\\mathcal {B}}_B)$ and $(C,{\\mathcal {B}}_C)$ .",
"The next two theorems generalize Theorems REF and REF from indicator functions of sets to more general functions.",
"Theorem REF will be used in our proof of Theorem REF .",
"Theorem 1.5 Let $G$ be a compact connected Abelian topological group equipped with Haar measure $\\mu $ satisfying $\\mu (G)=1$ .",
"For any measurable functions $f,g,h:G\\rightarrow [0,1]$ , $ \\langle f*g,h\\rangle _G\\le \\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star },{\\mathbf {1}}_{C^\\star }\\rangle _{\\mathbb {T}}$ where ${A^\\star },{B^\\star },{C^\\star }\\subset {\\mathbb {T}}$ are intervals centered at 0 satisfying $\\big (m({A^\\star }),m({B^\\star }),m({C^\\star })\\big )=\\big ({\\textstyle \\int _G f\\,d\\mu ,\\,\\int _G g\\,d\\mu ,\\, \\int _G h\\,d\\mu }\\big ).$ Inequality (REF ) is known for ${\\mathbb {R}}^d$ .",
"A corresponding stability theorem extends Theorem REF from indicator functions of sets to more general functions.",
"Theorem 1.6 For each $\\eta >0$ there exists ${\\mathbf {C}}<\\infty $ with the following property.",
"Let $G$ be a compact connected Abelian topological group equipped with Haar measure $\\mu $ satisfying $\\mu (G)=1$ .",
"Let $f,g,h:G\\rightarrow [0,1]$ be measurable.",
"Let $({A^\\star },{B^\\star },{C^\\star })\\subset {\\mathbb {T}}$ be intervals centered at 0 with Lebesgue measures $(\\int f\\,d\\mu ,\\int g\\,d\\mu ,\\int h\\,d\\mu )$ .",
"Let ${\\mathcal {D}}=\\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star },{\\mathbf {1}}_{C^\\star }\\rangle _{\\mathbb {T}}-\\langle f*g,h\\rangle _G.$ Suppose that $({A^\\star },{B^\\star },{C^\\star })$ is $\\eta $ –strictly admissible and $\\eta $ –bounded.",
"If ${\\mathcal {D}}$ is sufficiently small as a function of $\\eta $ alone then there exists a compatibly centered parallel triple $({\\mathcal {B}}_f,{\\mathcal {B}}_g,{\\mathcal {B}}_h)$ of rank one Bohr subsets of $G$ satisfying $ \\Vert f-{\\mathbf {1}}_{{\\mathcal {B}}_f} \\Vert _{L^1(G,\\mu )}\\le {\\mathbf {C}}{\\mathcal {D}}^{1/2}$ and likewise for $(g,{\\mathbf {1}}_{{\\mathcal {B}}_g})$ and $(h, {\\mathbf {1}}_{{\\mathcal {B}}_h})$ .",
"Underlying our analysis are analogous results concerning Kneser's inequality (REF ).",
"Continue to consider any compact connected Abelian topological group $G$ , equipped with Haar measure $\\mu $ .",
"If $A,B\\subset G$ are measurable sets that satisfy $\\mu (A)+\\mu (B)<\\mu (G)$ and $\\min (\\mu (A),\\mu (B))>0$ , then according to a theorem of Kneser [18], $\\mu _*(A+B)=\\mu (A)+\\mu (B)$ if and only if there exists a pair of parallel rank one Bohr sets satisfying $A\\subset {\\mathcal {B}}_A$ , $B\\subset {\\mathcal {B}}_B$ , and $\\mu ({\\mathcal {B}}_A\\setminus A) = \\mu ({\\mathcal {B}}_B\\setminus B)=0$ .",
"For compact Abelian groups that are not necessarily connected, matters are more complicated; see for instance [16].",
"Moreover, Tao [22] and Griesmer [17] have proved associated stability, or quantitative uniqueness, theorems.",
"Most relevant to our considerations is this result from [17]: For every $\\varepsilon ,\\eta >0$ there exists $\\delta >0$ such that if $A,B\\subset G$ are measurable sets satisfying the auxiliary hypotheses $\\mu (A)\\ge \\eta \\mu (G)$ , $\\mu (B)\\ge \\eta \\mu (G)$ , $\\mu (A)+\\mu (B)\\le (1-\\eta )\\mu (G)$ and the main hypothesis $\\mu _*(A+B)\\le \\mu (A)+\\mu (B)+\\delta \\mu (G)$ , then there exists a pair of parallel rank one Bohr sets $({\\mathcal {B}}_A,{\\mathcal {B}}_B)$ satisfying $A\\subset {\\mathcal {B}}_A$ , $B\\subset {\\mathcal {B}}_B$ , and $\\mu ({\\mathcal {B}}_A\\setminus A) + \\mu ({\\mathcal {B}}_B\\setminus B)<\\varepsilon \\mu (G).$ This is the result required for our analysis on general compact connected groups.",
"We also prove a more quantitative version, Theorem REF below.",
"Both [22] and [17] extend this result by weakening the hypothesis to one which involves an upper bound only on the Haar measure of $\\lbrace x\\in A+B: {\\mathbf {1}}_A*{\\mathbf {1}}_B(x)\\ge \\rho \\rbrace $ for sufficiently small $\\rho >0$ .",
"Our development does not rely on this extension.",
"A more quantitative stability theorem for sumsets is the following.",
"Theorem 1.7 For each $\\eta ,\\eta ^{\\prime }>0$ there exist $\\delta _0>0$ and ${\\mathbf {C}}<\\infty $ with the following property.",
"Let $G$ be any compact connected Abelian topological group equipped with normalized Haar measure $\\mu $ .",
"Let $A,B\\subset G$ be a pair of measurable sets satisfying $\\min (\\mu (A),\\mu (B)) \\ge \\eta ^{\\prime }$ and $\\mu (A)+\\mu (B)\\le 1-\\eta $ .",
"If $\\mu (A+_0 B)\\le \\mu (A)+\\mu (B)+\\delta \\min (\\mu (A),\\mu (B))$ and $\\delta \\le \\delta _0$ , then there exists a pair of parallel rank one Bohr sets $({\\mathcal {B}}_A,{\\mathcal {B}}_B)$ such that $A\\subset {\\mathcal {B}}_A$ , $B\\subset {\\mathcal {B}}_B$ , and $\\mu ({\\mathcal {B}}_A\\setminus A) + \\mu ({\\mathcal {B}}_B\\setminus B) \\le {\\mathbf {C}}\\delta \\min (\\mu (A),\\mu (B)).$ Candela and de Roton [8] have proved a theorem of this type for the special case $G={\\mathbb {T}}$ in which the relationship between $m({\\mathcal {B}}_A\\setminus A)$ and $m_*(A+B)-m(A)-m(B)$ is made quite precise, for an interesting range of parameters.",
"We believe that their theorem extends to arbitrary compact connected Abelian groups, with the same relationship between parameters.",
"Organization of the paper.",
"We begin by reviewing in § an inequality of Tao [22], stating several equivalent reformulations and establishing a refinement.",
"This refinement is used in § to prove the Riesz-Sobolev–type inequality of Theorem REF .",
"The defect ${\\mathcal {D}}(A,B,C) = \\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star },{\\mathbf {1}}_{C^\\star }\\rangle - \\langle {\\mathbf {1}}_A*{\\mathbf {1}}_B,{\\mathbf {1}}_C\\rangle $ and a related defect ${\\mathcal {D}}^{\\prime }(A,B,\\tau )$ , in terms of which much of our analysis is naturally phrased, are introduced in §.",
"In § we discuss two key principles, submodularity and complementation.",
"At the heart of our analysis of stability for the Riesz-Sobolev-type inequality (REF ) is a connection, developed in [10] for $G={\\mathbb {R}}$ , between (near) equality in the Riesz-Sobolev equality and (near) equality in the sumset inequality for certain associated sets.",
"This connection only applies directly in the case in which two of the three sets $A,B,C$ have equal measures.",
"§ reviews this connection and adapts it to general connected compact Abelian groups.",
"§ begins a reduction of the general case to the special case of two sets of equal measures.",
"This reduction does not proceed in the same way as the corresponding reduction in [10] for the Euclidean case.",
"§ establishes the conclusion of Theorem REF in its quantitative form, for the perturbative regime in which $(A,B,C)$ is assumed to be within a certain threshold distance of a compatibly centered parallel triple of rank one Bohr sets.",
"§ digresses to establish quantitative stability for Kneser's inequality in the perturbative regime in which $A,B$ are assumed to be moderately close to a pair of parallel rank one Bohr sets.",
"These perturbative results are elements of our more general analysis of stability in the absence of any perturbative hypotheses.",
"Theorem REF and Theorem REF concern relaxed variants of the Riesz-Sobolev-type inequality and its companion inverse stability theorem.",
"They are proved in § and §, respectively.",
"§ and § analyze the special case in which the defect ${\\mathcal {D}}(A,B,C)$ is small and one of the three sets is well approximated by a rank one Bohr set.",
"§ treats the sub-subcase in which $G={\\mathbb {T}}$ and $C$ is an interval.",
"In §, we reduce matters from general groups $G$ to ${\\mathbb {T}}$ .",
"The situation that arises on ${\\mathbb {T}}$ in this way belongs to the more general framework of Theorems REF and REF .",
"That framework comes into play at this juncture.",
"The proof of Theorem REF is completed in §.",
"Another thread is taken up in § and §, which are concerned with the important group $G={\\mathbb {T}}$ .",
"This thread is founded on the monotonicity of a normalized version of the functional $\\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,dm$ under a certain continuous deformation of $A,B,C$ .",
"This deformation is developed in §.",
"As an application, in § we establish Theorem REF , a refinement for $G={\\mathbb {T}}$ of Theorem REF which in an appropriate sense eliminates the dependence of the conclusion on a lower bound for $\\min (m(A),m(B),m(C))$ .",
"One could alternatively bypass the analysis in § of the situation in which $G={\\mathbb {T}}$ and $C$ is an interval, by invoking the theory for ${\\mathbb {T}}$ established in §.",
"Notation 1.7 For the sake of economy, we will often refer to (REF ) as the Riesz-Sobolev inequality, or the Riesz-Sobolev inequality for $G$ .",
"Throughout the remainder of the paper, $G$ denotes a compact connected Abelian topological group equipped with a complete Haar measure $\\mu $ that is normalized in the sense that $\\mu (G)=1$ .",
"This is a hypothesis of all lemmas and propositions, though it is not included in their statements.",
"It is implicitly asserted that all constants in upper and lower bounds in theorems, propositions, lemmas, and inequalities are independent of $G$ , except when the special case $G={\\mathbb {T}}$ is explicitly indicated.",
"$m$ denotes Lebesgue measure for ${\\mathbb {T}}= {\\mathbb {R}}/{\\mathbb {Z}}$ .",
"$m(E)$ is alternatively denoted by $|E|$ in some parts of the discussion.",
"$C$ with no subscript is used to denote a subset of $G$ , rather than a constant.",
"$c$ , $c^{\\prime }$ , and ${\\mathbf {C}}$ denote unspecified positive finite constants, whose values may change freely from one occurrence to the next.",
"$C_n$ denotes a constant that is fixed for a relatively short portion of the discussion.",
"It will be convenient in the analysis of the functional $\\langle {\\mathbf {1}}_{E_1}*{\\mathbf {1}}_{E_2},{\\mathbf {1}}_{E_3}\\rangle $ to be able to freely interchange the sets $E_j$ .",
"For that purpose, we work with a more symmetric variant.",
"For measurable functions $f_j:G\\rightarrow [0,\\infty )$ , ${\\mathcal {T}}_G(f_1,f_2,f_3) = \\iint _{x+y+z=0} f_1(x)f_2(y)f_3(z)\\,d\\lambda (x,y,z)$ where $\\lambda $ is the measure on $\\lbrace (x,y,z)\\in G^3: x+y+z=0\\rbrace $ defined by pulling back the measure $\\mu \\times \\mu $ on $G\\times G$ via the mapping $(x,y,z)\\mapsto (x,y)$ .",
"This definition of $\\lambda $ is invariant with respect to permutation of the three coordinates.",
"Equivalently, $ {\\mathcal {T}}_G(\\mathbf {f})= {\\mathcal {T}}_G(f_1,f_2,f_3) = \\iint _{G^2} f_1(x)f_2(y)f_3(-x-y)\\,d\\mu (x)\\,d\\mu (y).",
"$ For a three-tuple ${\\mathbf {E}}= (E_j: j\\in \\lbrace 1,2,3\\rbrace )$ of sets, we write ${\\mathcal {T}}_G({\\mathbf {E}}) = {\\mathcal {T}}_G(\\mathbf {f})$ with $f_j={\\mathbf {1}}_{E_j}$ .",
"We sometimes work simultaneously on a general group $G$ and on ${\\mathbb {T}}$ , and write ${\\mathcal {T}}_G$ and/or ${\\mathcal {T}}_{\\mathbb {T}}$ to distinguish between the functionals associated to the two groups.",
"Defining $\\overline{{\\mathcal {D}}}(A,B,C):={\\mathcal {T}}_{\\mathbb {T}}(A^\\star ,B^\\star ,C^\\star )-{\\mathcal {T}}_G(A,B,C),$ one has $\\overline{{\\mathcal {D}}}(A,B,C)={\\mathcal {D}}(A,B,-C)$ .",
"The authors are grateful to Rupert Frank, who kindly called their attention to a slip in the proof of Theorem REF in an earlier draft.",
"A version of Theorem REF is proved by Frank and Lieb in the Euclidean setting in arbitrary dimension [13], in the special case where two of the functions are identical and the third is the indicator function of a ball."
],
[
"Refinement of a related inequality",
"In this section we review an inequality of Tao [22], discuss multiple equivalent reformulations, and formulate and prove a sharper inequality, from which the Riesz-Sobolev inequality (REF ) for $G$ will subsequently be derived.",
"The inequality of [22] states that for any compact connected Abelian group $G$ with normalized Haar measure $\\mu $ , for any measurable $A,B\\subset G$ , $\\int _G \\min ({\\mathbf {1}}_A*{\\mathbf {1}}_B,\\tau )\\,d\\mu \\ge \\tau \\min (\\mu (A)+\\mu (B)-\\tau ,1)\\ \\forall \\, 0\\le \\tau \\le \\max (\\mu (A),\\mu (B)).$ The inequality is trivial in the range $\\min (\\mu (A),\\mu (B))\\le \\tau \\le \\max (\\mu (A),\\mu (B))$ , in the sense that for arbitrary $A,B$ equality holds when $\\tau $ is equal to the minimum or maximum, while for $\\tau $ in the open interval $\\big (\\min (\\mu (A),\\mu (B)),\\,\\max (\\mu (A),\\mu (B))\\big )$ , the left-hand side is equal to $\\mu (A)\\cdot \\mu (B)$ and (REF ) holds with strict inequality.",
"(REF ) also holds with equality whenever $\\mu (A)+\\mu (B)\\ge 1+\\tau $ , for in that case, ${\\mathbf {1}}_A*{\\mathbf {1}}_B(x) = \\mu (A \\cap (x-B)) \\ge \\mu (A)+\\mu (B)-1\\ge \\tau $ for every $x\\in G$ , so both the left– and right–hand sides are equal to $\\tau $ .",
"(REF ) never holds when $\\tau > \\max (\\mu (A),\\mu (B))$ .",
"If $G={\\mathbb {T}}$ and $A,B\\subset {\\mathbb {T}}$ are intervals centered at 0 then equality holds in (REF ) whenever $\\tau \\le \\min (\\mu (A),\\mu (B))$ .",
"Therefore this inequality can be equivalently restated as $\\int _G \\min ({\\mathbf {1}}_A*{\\mathbf {1}}_B,\\tau )\\,d\\mu \\ge \\int _{\\mathbb {T}}\\min ({\\mathbf {1}}_{{A^\\star }}*{\\mathbf {1}}_{{B^\\star }},\\tau )\\,dm\\ \\forall \\, 0\\le \\tau \\le \\min (\\mu (A),\\mu (B)).$ By virtue of the identities $ \\int _G {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu = \\mu (A)\\cdot \\mu (B) $ and $\\max (f,g)+\\min (f,g) = f+g$ , (REF ) can be equivalently reformulated as $\\int _G \\max ({\\mathbf {1}}_A*{\\mathbf {1}}_B-\\tau ,0)\\,d\\mu \\le (\\mu (A)-\\tau )(\\mu (B)-\\tau )\\\\\\ \\forall \\, \\tau \\in [\\mu (A)+\\mu (B)-1,\\, \\min (\\mu (A),\\mu (B))]$ with $\\int _G \\max ({\\mathbf {1}}_A*{\\mathbf {1}}_B-\\tau ,0)\\,d\\mu = \\mu (A)\\mu (B)-\\tau $ for all $\\tau \\in [0,\\mu (A)+\\mu (B)-1]$ .",
"Likewise, (REF ) can be reformulated as $\\int _G \\max ({\\mathbf {1}}_A*{\\mathbf {1}}_B-\\tau ,0)\\,d\\mu \\le \\int _{\\mathbb {T}}\\max ({\\mathbf {1}}_{{A^\\star }}*{\\mathbf {1}}_{{B^\\star }}-\\tau ,0)\\,dm\\\\\\forall \\,0\\le \\tau \\le \\min (\\mu (A),\\mu (B)).$ These four inequalities are equivalent in the sense that any one of them follows from any other one by simple manipulations augmented by the above discussion of the cases in which $\\min (\\mu (A),\\mu (B))\\le \\tau \\le \\max (\\mu (A),\\mu (B))$ or $\\tau \\le \\mu (A)+\\mu (B)-1$ .",
"The inequalities (REF ) through (REF ) can be reformulated in terms of superlevel sets and associated distribution functions.",
"The following notation (REF ) will be used throughout the paper.",
"Definition 2.1 For measurable sets $A,B\\subset G$ and for $t\\ge 0$ , the associated superlevel set is $ S_{A,B}(t) = \\lbrace x\\in G: {\\mathbf {1}}_A*{\\mathbf {1}}_B(x)>t\\rbrace .$ Superlevel sets appear in fundamental formulae for the functionals of interest here: $&\\int _G {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu = \\mu (A)\\cdot \\mu (B) = \\int _0^\\infty \\mu (S_{A,B}(t))\\,dt,\\\\&\\int _{S_{A,B}(\\tau )}{\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu = \\tau \\mu (S_{A,B}(\\tau )) +\\int _\\tau ^\\infty \\mu (S_{A,B}(t))\\,dt,\\\\&\\int _G \\max ({\\mathbf {1}}_A*{\\mathbf {1}}_B- \\tau ,0)\\,d\\mu = \\int _\\tau ^\\infty \\mu (S_{A,B}(t))\\,dt.$ Thus (REF ) can be written $ \\int _\\tau ^\\infty \\mu (S_{A,B}(t))\\,dt\\le (\\mu (A)-\\tau )(\\mu (B)-\\tau )\\\\\\ \\forall \\, \\tau \\in [\\mu (A)+\\mu (B)-1,\\, \\min (\\mu (A),\\mu (B))].$ The next result sharpens (REF ) and will be the basis of our proof of Theorem REF .",
"Theorem 2.1 Let $G$ be a compact connected Abelian topological group, equipped with normalized Haar measure $\\mu $ .",
"Suppose that $ 0\\le \\tau \\le \\min (\\mu (A),\\mu (B))$ and that $ \\mu (A)+\\mu (B) + \\mu (S_{A,B}(\\tau ))\\le 2.$ Let $\\sigma = \\tfrac{1}{2} (\\mu (A)+\\mu (B)-\\mu (S_{A,B}(\\tau )))$ .",
"Then $ \\int _\\tau ^\\infty \\mu (S_{A,B}(t))\\,dt\\le (\\mu (A)-\\tau )(\\mu (B)-\\tau )-h,$ where $h=\\left\\lbrace \\begin{alignedat}{2}&(\\sigma -\\tau )^2&&\\text{if $\\sigma \\le \\min (\\mu (A),\\mu (B))$}\\\\&(\\min (\\mu (A),\\mu (B))-\\tau )^2\\qquad &&\\text{if $\\sigma > \\min (\\mu (A),\\mu (B))$.",
"}\\end{alignedat} \\right.$ In particular, if $\\mu (A)+\\mu (B)\\le 1+\\tau $ then $\\int _\\tau ^\\infty \\mu (S_{A,B}(t))\\,dt\\le \\int _\\tau ^\\infty m(S_{{A^\\star },{B^\\star }}(t))\\,dt -h.$ The form of the right-hand side of (REF ) is unnatural when $\\mu (A)+\\mu (B)>1+\\tau $ , in the sense that $\\int _\\tau ^\\infty m(S_{{A^\\star },{B^\\star }}(t))\\,dt=\\mu (A)\\mu (B)-\\tau $ is strictly smaller than $(\\mu (A)-\\tau )(\\mu (B)-\\tau )$ for such values of $\\tau $ .",
"Write $S(t)=S_{A,B}(t)$ to simplify notation.",
"With $\\sigma $ as defined above, the hypothesis $\\mu (S(\\tau )) +\\mu (A) + \\mu (B)\\le 2$ can be equivalently written as $\\mu (A)+\\mu (B)-1\\le \\sigma $ .",
"This is one of two conditions needed to apply (REF ) to $\\int _\\sigma ^\\infty \\mu (S(t))\\,dt$ .",
"The second condition is that $\\sigma \\le \\min (\\mu (A),\\mu (B))$ , which need not hold under the hypotheses of Theorem REF , in general.",
"The proof is consequently organized into cases.",
"If $\\sigma \\le \\tau $ then indeed $\\sigma \\le \\min (\\mu (A),\\mu (B))$ , so (REF ) may be applied to obtain $\\int _\\tau ^\\infty \\mu (S(t))\\,dt&=\\int _\\sigma ^\\infty \\mu (S(t))\\,dt- \\int _\\sigma ^\\tau \\mu (S(t))\\,dt\\\\&\\le (\\mu (A)-\\sigma )(\\mu (B)-\\sigma ) - (\\tau -\\sigma ) \\mu (S(\\tau ))\\\\&= (\\mu (A)-\\tau )(\\mu (B)-\\tau ) -(\\sigma -\\tau )^2$ as follows by expanding $\\tau = \\sigma - (\\tau -\\sigma )$ in the product $(\\mu (A)-\\tau )(\\mu (B)-\\tau )$ and invoking the relation $\\mu (A)+\\mu (B) = 2\\sigma + \\mu (S(\\tau ))$ .",
"If $\\tau \\le \\sigma $ and if $\\sigma $ does satisfy $\\sigma \\le \\min (\\mu (A),\\mu (B))$ , then again by (REF ), $\\int _\\tau ^\\infty \\mu (S(t))\\,dt&=\\int _\\sigma ^\\infty \\mu (S(t))\\,dt+ \\int _\\tau ^\\sigma \\mu (S(t))\\,dt\\\\&\\le (\\mu (A)-\\sigma )(\\mu (B)-\\sigma )+(\\sigma -\\tau ) \\mu (S(\\tau ))$ which we have already stated to be equal to $(\\mu (A)-\\tau )(\\mu (B)-\\tau ) -(\\sigma -\\tau )^2$ .",
"If on the other hand $\\sigma \\ge \\min (\\mu (A),\\mu (B))$ then by permutation invariance, we may assume without loss of generality that $\\mu (A)\\le \\mu (B)$ .",
"Thus $\\tfrac{1}{2}(\\mu (A)+\\mu (B)-\\mu (S(\\tau )))=\\sigma \\ge \\mu (A)$ , so $\\mu (S(\\tau )) \\le \\mu (B)-\\mu (A)$ .",
"Since ${\\mathbf {1}}_A*{\\mathbf {1}}_B\\le \\mu (A)$ , $\\int _\\tau ^\\infty \\mu (S(t))\\,dt= \\int _\\tau ^{\\mu (A)} \\mu (S(t))\\,dt\\le (\\mu (A)-\\tau ) \\mu (S(\\tau ))$ since the integrand is a nonincreasing function of $t$ .",
"The right-hand side is $\\le (\\mu (A)-\\tau )(\\mu (B)-\\mu (A))= (\\mu (A)-\\tau )(\\mu (B)-\\tau ) - (\\mu (A)-\\tau )^2.$ Corollary 2.2 Let $G$ be a compact connected Abelian topological group, equipped with Haar measure $\\mu $ satisfying $\\mu (G)=1$ .",
"Let $A,B\\subset G$ be measurable sets.",
"Suppose that $\\mu (A)+\\mu (B)-1 < t < \\min (\\mu (A),\\mu (B)).$ If $(A,B,t)$ achieves equality in (REF ) (equivalently in any or all of (REF ), (REF ), (REF )), then $ \\mu (S_{A,B}(t)) = \\mu (A)+\\mu (B) -2t.$ We remark that $\\mu (A)+\\mu (B) -2\\tau $ is not an extremal value for $\\mu (S_{A,B}(\\tau ))$ for any single value of $\\tau $ ; $\\mu (S_{A,B}(\\tau ))$ can in general be either larger, or smaller.",
"If $\\mu (A)+\\mu (B)+\\mu (S_{A,B}(t))\\le 2$ then all hypotheses of Theorem REF are satisfied, and (REF ) follows from its conclusion since $t$ is strictly less than $\\min (\\mu (A),\\mu (B))$ .",
"We claim that $\\mu (S_{A,B}(t))\\le 1-t$ , whence $\\mu (A)+\\mu (B)+\\mu (S_{A,B}(t))\\le 1+t+1-t =2$ , completing the proof of the corollary.",
"Suppose to the contrary that $\\mu (S_{A,B}(t)) > 1-t$ .",
"Define $\\tau \\in (0,t)$ by $\\mu (A)+\\mu (B)=1+\\tau $ .",
"For every $x\\in G$ , ${\\mathbf {1}}_A*{\\mathbf {1}}_B(x)=\\mu \\big (A\\cap (x-B)\\big )\\ge \\mu (A)+\\mu (B)-1=\\tau $ .",
"Thus, for every $r\\in [0,\\tau )$ , $S_{A,B}(r)=G$ , so $\\mu (S_{A,B}(r))=1\\text{ for every }r\\in [0,\\tau ).",
"$ For any $r\\in [\\tau ,t]$ , $S_{A,B}(r)\\supset S_{A,B}(t)$ , so $\\mu (S_{A,B}(r))\\ge \\mu (S_{A,B}(t))> 1-t\\text{ for every }r\\in [\\tau ,t].",
"$ The assumption that $(A,B,t)$ satisfies equality in (REF ) means that $\\int _G \\min (1_A*1_B,t)\\,d\\mu =t(\\mu (A)+\\mu (B)-t)=t(1+\\tau -t).",
"$ Substituting $\\int _G \\min (1_A*1_B,t)\\,d\\mu = \\int _{0}^{t}\\mu (S_{A,B}(r))\\,dr$ in the left-hand side and invoking (REF ) and (REF ) gives $t(1+\\tau -t)=\\int _{0}^{t}\\mu (S_{A,B}(r))\\,dr=\\int _{0}^{\\tau }\\mu (S_{A,B}(r))\\,dr+\\int _{\\tau }^t\\mu (S_{A,B}(r))\\,dr\\\\> \\int _{0}^{\\tau }1+\\int _{\\tau }^t(1-t)\\,dr=\\tau +(t-\\tau )(1-t)=t(1+\\tau -t),$ which is a contradiction.",
"Therefore $\\mu (S_{A,B}(t))\\le 1-t$ , and the proof of the corollary is complete."
],
[
"On the Riesz-Sobolev for $G$",
"In this section we prove the Riesz-Sobolev inequality (REF ) for $G$ using Theorem REF .",
"The sharpened form (REF ) of (REF ) for $\\sigma \\le \\min (\\mu (A),\\mu (B))$ is exactly what is needed in this derivation.",
"Also, for the defects $\\mathcal {D}$ and $\\mathcal {D}^{\\prime }$ corresponding to these inequalities, defined below, we discuss approximation of the set $C$ by superlevel sets $S_{A,B}(t)$ , under the assumption that ${\\mathcal {D}}(A,B,C)$ is small.",
"We also discuss majorization of ${\\mathcal {D}}(A,B,C)$ by ${\\mathcal {D}}^{\\prime }(A,B,\\tau )$ and vice versa, under appropriate hypotheses linking $C$ to $\\tau $ .",
"The defects ${\\mathcal {D}}(A,B,C)$ and ${\\mathcal {D}}^{\\prime }(A,B,\\tau )$ are defined as follows.",
"Definition 3.1 $&{\\mathcal {D}}(A,B,C) = \\int _{C^\\star }{\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star }\\,dm- \\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu .\\\\&{\\mathcal {D}}^{\\prime }(A,B,\\tau ) = \\int _{\\mathbb {T}}\\max ({\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star }-\\tau ,0)\\,dm- \\int _G \\max ({\\mathbf {1}}_A*{\\mathbf {1}}_B-\\tau ,0)\\,d\\mu .$ Theorem REF states that ${\\mathcal {D}}(A,B,C)\\ge 0$ for any ordered triple, while inequality (REF ) states that ${\\mathcal {D}}^{\\prime }(A,B,\\tau )\\ge 0$ for all $\\tau \\in [0,\\min (\\mu (A),\\mu (B))]$ .",
"These defects can usefully be expressed in terms of distribution functions $\\mu (S_{A,B}(t))$ , as discussed in §.",
"The following quantity arises throughout our analysis.",
"Definition 3.2 To sets $A,B,C\\subset G$ satisfying $\\mu (C)\\le \\mu (A)+\\mu (B)$ is associated $\\tau _C = \\tfrac{1}{2} (\\mu (A)+\\mu (B)-\\mu (C)).$ This quantity satisfies $m(S_{{A^\\star },{B^\\star }}(\\tau _C))=m({C^\\star })= \\mu (C)$ ; it represents the parameter $\\tau $ for which ${C^\\star }$ equals the superlevel set $S_{{A^\\star },{B^\\star }}(\\tau )$ , provided that $(\\mu (A),\\mu (B),\\mu (C))$ is admissible.",
"Lemma 3.1 Suppose that $A,B\\subset G$ and $\\tau \\in [0,1]$ satisfy $0\\le \\tau \\le \\min (\\mu (A),\\mu (B)),$ $\\mu (A)+\\mu (B) + \\mu (S_{A,B}(\\tau ))\\le 2.$ Then $ \\tau \\mu (S_{A,B}(\\tau )) + \\int _\\tau ^\\infty \\mu (S_{A,B}(\\alpha ))\\,d\\alpha \\le \\mu (A)\\mu (B)-\\tfrac{1}{4} (\\mu (A)-\\mu (B)-\\mu (S_{A,B}(\\tau )))^2.$ That is, $(A,B,C)= (A,B,S_{A,B}(\\tau ))$ satisfies (REF ) (and thus (REF ), under some additional hypotheses).",
"Define $\\sigma = \\tfrac{1}{2}(\\mu (A)+\\mu (B)-\\mu (S_{A,B}(\\tau )))$ .",
"Equivalently, $\\mu (S_{A,B}(\\tau )) = \\mu (A)+\\mu (B)-2\\sigma $ .",
"Calculate $(\\mu (A)-\\tau )(\\mu (B)-\\tau ) - \\big (\\mu (A)\\mu (B)-\\sigma ^2\\big )&= -\\tau (\\mu (A)+\\mu (B))+\\tau ^2+\\sigma ^2\\\\&= -\\tau (\\mu (A)+\\mu (B)-2\\sigma ) + (\\sigma -\\tau )^2\\\\&= -\\tau \\mu (S(\\tau )) + (\\sigma -\\tau )^2.$ Thus $ \\tau \\mu (S_{A,B}(\\tau ))=-(\\mu (A)-\\tau )(\\mu (B)-\\tau ) + \\big (\\mu (A)\\mu (B)-\\sigma ^2\\big ) + (\\sigma -\\tau )^2.$ Note that $(A,B,\\tau )$ satisfies the hypotheses of Theorem REF .",
"Applying Theorem REF to the second term on the left-hand side of (REF ) and then invoking (REF ) gives the desired upper bound $\\tau \\mu (S_{A,B}(\\tau )) + (\\mu (A)-\\tau )(\\mu (B)-\\tau ) - (\\sigma -\\tau )^2= \\mu (A)\\mu (B) -\\sigma ^2.$ Let $A,B,C\\subset G$ .",
"Consider first the case in which $\\mu (A)+\\mu (B)+\\mu (C)\\ge 2$ .",
"Define $t$ by $\\mu (A)+\\mu (B)=1+t$ ; note that $t\\ge 0$ .",
"Then ${\\mathbf {1}}_A*{\\mathbf {1}}_B(x)\\ge t$ for every $x\\in G$ .",
"Indeed, ${\\mathbf {1}}_A*{\\mathbf {1}}_B(x) = \\mu (A\\cap (x-B)) \\ge \\mu (A)+\\mu (x-B)-\\mu (G)= \\mu (A)+\\mu (B)-1 = t.$ Therefore $ \\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu \\le \\int _G {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu -t\\mu (G\\setminus C)= \\mu (A)\\mu (B)-t(1-\\mu (C)).$ On the other hand, ${\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star }\\equiv t$ on ${\\mathbb {T}}\\setminus {C^\\star }$ , and so the same calculation gives $\\int _{C^\\star }{\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star }\\,dm= m({A^\\star })m({B^\\star })-t(1-m({C^\\star }))= \\mu (A)\\mu (B)-t(1-\\mu (C)).$ Thus the stated conclusion holds in this case.",
"If $\\mu (C)\\le |\\mu (A)-\\mu (B)|$ then, while ${\\mathbf {1}}_A*{\\mathbf {1}}_B\\le \\min (\\mu (A),\\mu (B))$ on $C$ , it also holds that ${\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star }\\equiv \\min (m({A^\\star }),m({B^\\star }))$ on ${C^\\star }$ .",
"Therefore (REF ) holds.",
"If $\\mu (C)\\ge \\mu (A)+\\mu (B)$ then either $\\mu (A)\\le |\\mu (B)-\\mu (C)|$ or $\\mu (B)\\le |\\mu (A)-\\mu (C)|$ .",
"(REF ) thus follows by permutation invariance from the case in which $\\mu (C)\\le |\\mu (A)-\\mu (B)|$ .",
"Assume henceforth that $\\mu (A)+\\mu (B)+\\mu (C) < 2$ , and that $|\\mu (A)-\\mu (B)| < \\mu (C) < \\mu (A)+\\mu (B)$ .",
"If there exists $t\\in [0,1]$ for which the superlevel set $S=S_{A,B}(t)$ satisfies $\\mu (S)=\\mu (C)$ , then the desired inequality (REF ) holds for $(A,B,C)$ .",
"More precisely, $\\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B \\le \\int _S {\\mathbf {1}}_A*{\\mathbf {1}}_B$ .",
"The parameter $t$ satisfies $t\\le \\min (\\mu (A),\\mu (B))$ , since $ \\Vert {\\mathbf {1}}_A*{\\mathbf {1}}_B \\Vert _{C^0}\\le \\min (\\mu (A),\\mu (B))$ and $\\mu (C)>0$ .",
"It also satisfies $\\mu (A)+\\mu (B)\\le 1+t$ .",
"Indeed, if $\\mu (A)+\\mu (B)>1+t$ then ${\\mathbf {1}}_A*{\\mathbf {1}}_B(x)>t$ for every $x\\in G$ as noted above, so $S =S_{A,B}(t)=G$ , so $\\mu (C)=\\mu (S)=1$ , forcing $\\mu (A)+\\mu (B)+\\mu (C)=\\mu (A)+\\mu (B)+1 > 2+t\\ge 2$ and thereby contradicting the assumption that $\\mu (A)+\\mu (B)+\\mu (C)< 2$ .",
"Thus the hypotheses of Lemma REF are satisfied by $A,B,t$ and $S_{A,B}(t)$ .",
"Applying that lemma to $S_{A,B}(t)$ gives the desired upper bound for $\\int _{S_{A,B}(t)} {\\mathbf {1}}_A*{\\mathbf {1}}_B$ , hence for $\\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B$ .",
"It remains to reduce the general case to that in which there exists $t\\in [0,1]$ satisfying $\\mu (S_{A,B}(t))=\\mu (C)$ , under the hypotheses $\\mu (A)+\\mu (B)+\\mu (C)< 2$ and $|\\mu (A)-\\mu (B)|<\\mu (C)<\\mu (A)+\\mu (B)$ .",
"We may also assume the auxiliary condition $ \\mu (\\lbrace x: {\\mathbf {1}}_A*{\\mathbf {1}}_B(x)>0\\rbrace )\\ge \\mu (C).$ Indeed, if this fails, set $\\tilde{C} = C\\cap \\lbrace x: {\\mathbf {1}}_A*{\\mathbf {1}}_B(x)>0\\rbrace $ .",
"The value of the integral $\\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu $ is unchanged when $C$ is replaced by $\\tilde{C}$ .",
"If $\\mu (\\tilde{C})<|\\mu (A)-\\mu (B)|$ then we have already observed that $\\int _{\\tilde{C}} {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu \\le \\int _{\\tilde{C}^\\star } {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star }\\,dm$ (that is, $(A,B,\\tilde{C})$ satisfies (REF )).",
"Since $\\mu (\\tilde{C})\\le \\mu (C)$ , the right-hand side is in turn majorized by $\\int _{C^\\star } {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star }\\,dm$ , so (REF ) holds for $(A,B,C)$ .",
"If $\\mu (\\tilde{C}) \\ge |\\mu (A)-\\mu (B)|$ then it suffices to prove that $(A,B,\\tilde{C})$ satisfies (REF ).",
"Thus matters are reduced to the case in which $(A,B,C)$ satisfies (REF ).",
"Given (REF ), a sufficient condition for the existence of $t$ satisfying $\\mu (C)=\\mu (S_{A,B}(t))$ is that all level sets of ${\\mathbf {1}}_A*{\\mathbf {1}}_B$ should be null sets, that is, for every $r>0$ , $\\mu (\\lbrace x: {\\mathbf {1}}_A*{\\mathbf {1}}_B(x)=r\\rbrace )=0$ .",
"Moreover, because $(A,B,C)\\mapsto \\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu $ is continuous in the sense that $\\int _{C_n}{\\mathbf {1}}_{A_n}*{\\mathbf {1}}_{B_n}\\,d\\mu \\rightarrow \\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu \\text{ if } \\mu (A_n\\,\\Delta \\,A) + \\mu (B_n\\,\\Delta \\,B) + \\mu (C_n\\,\\Delta \\,C)\\rightarrow 0,$ it would suffice to construct $(A_n,B_n,C_n)$ , converging to $(A,B,C)$ in this sense, such that all level sets of ${\\mathbf {1}}_{A_n}*{\\mathbf {1}}_{B_n}$ are $\\mu $ –null.",
"Such a construction does not necessarily exist in $G$ , but it does in the auxiliary group $\\tilde{G} = G\\times {\\mathbb {T}}$ with normalized Haar measure $\\tilde{\\mu }$ .",
"Consider a sequence of triples $(\\alpha _n,\\beta _n,\\gamma _n)$ of Lebesgue measurable subsets of ${\\mathbb {T}}$ satisfying $\\mu (\\alpha _n)\\rightarrow 1$ as $n\\rightarrow \\infty $ and likewise for $\\mu (\\beta _n),\\mu (\\gamma _n)$ , such that all level sets of ${\\mathbf {1}}_{\\alpha _n}*{\\mathbf {1}}_{\\beta _n}$ on ${\\mathbb {T}}$ are Lebesgue null sets.",
"The existence of such sequences can be proved in various ways.",
"Consider $(\\tilde{A},\\tilde{B},\\tilde{C}) = (A\\times \\alpha _n, B\\times \\beta _n,C\\times \\gamma _n)$ .",
"Then ${\\mathbf {1}}_{\\tilde{A}_n}*{\\mathbf {1}}_{\\tilde{B}_n}$ is the product function $G\\times {\\mathbb {T}}\\ni (x,y)\\mapsto ({\\mathbf {1}}_A*{\\mathbf {1}}_B(x))\\cdot ({\\mathbf {1}}_{\\alpha _n}*{\\mathbf {1}}_{\\beta _n}(y))$ , so $ \\int _{\\tilde{C}_n} {\\mathbf {1}}_{\\tilde{A}_n}*{\\mathbf {1}}_{\\tilde{B}_n}\\,d\\tilde{\\mu }= \\Big (\\int _{C_n} {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu \\big )\\cdot \\Big (\\int _{\\gamma _n} {\\mathbf {1}}_{\\alpha _n}*{\\mathbf {1}}_{\\beta _n}\\,dm\\big )$ converges to $\\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu $ as $n\\rightarrow \\infty $ .",
"Moreover, all level sets of ${\\mathbf {1}}_{\\tilde{A}_n}*{\\mathbf {1}}_{\\tilde{B}_n}$ are null sets; this is a simple consequence of Fubini's theorem and the corresponding property of ${\\mathbf {1}}_{\\alpha _n}*{\\mathbf {1}}_{\\beta _n}$ .",
"Therefore the conclusion of Theorem REF , or equivalently that of Theorem REF (whose hypotheses are satisfied by $(\\tilde{A}_n,\\tilde{B}_n,\\tilde{C}_n)$ for large $n$ ), holds for $(\\tilde{A}_n,\\tilde{B}_n,\\tilde{C}_n)$ for all sufficiently large $n$ .",
"Since $\\tilde{\\mu }(A_n)=\\mu (A)m(\\alpha _n)\\rightarrow \\mu (A)$ and likewise for $\\tilde{B}_n,\\tilde{C}_n$ , it follows from passage to the limit that the conclusion also holds for $(A,B,C)$ .",
"The proofs of the subsequent statements are not provided, as they are direct adaptations of proofs in [10].",
"The next lemma states that if $(A,B,C)$ nearly maximizes the Riesz-Sobolev functional $\\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu $ , then $C$ nearly coincides with a superlevel set $S_{A,B}(\\tau )$ (as long as $(A,B,C)$ is appropriately admissible).",
"Lemma 3.2 [10] Let $A,B,C\\subset G$ be measurable sets with $\\mu (A),\\mu (B),\\mu (C)>0$ .",
"Suppose that $\\big |\\,\\mu (A)-\\mu (B)\\,\\big | + 2{\\mathcal {D}}(A,B,C)^{1/2}< \\mu (C) < \\mu (A)+\\mu (B) -2{\\mathcal {D}}(A,B,C)^{1/2}\\\\\\mu (A)+\\mu (B)+\\mu (C) < 2-2{\\mathcal {D}}(A,B,C)^{1/2}.$ Define $\\tau $ by $\\mu (C) = \\mu (A)+\\mu (B)-2\\tau $ .",
"Then the superlevel set $S_{A,B}(\\tau )$ satisfies $ \\mu (S_{A,B}(\\tau )\\bigtriangleup C) \\le 4{\\mathcal {D}}(A,B,C)^{1/2}\\\\\\big |\\mu (S_{A,B}(\\tau ))-\\mu (C)\\big | \\le 2\\mathcal {D}(A,B,C)^{1/2}\\\\{\\mathcal {D}}(A,B,S_{A,B}(\\tau )) \\le {\\mathcal {D}}(A,B,C).$ The next result sharpens Theorem REF in the same way that Theorem REF sharpens (REF ).",
"It is simply a restatement of (REF ) in alternative terms.",
"Theorem 3.3 Under the hypotheses of Lemma REF , $\\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu \\le \\int _{C^\\star }{\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star }\\,dm- \\tfrac{1}{16} \\mu \\big (C\\,\\Delta \\,S_{A,B}(\\tau _C)\\big )^2$ where $\\tau _C = \\tfrac{1}{2} (\\mu (A)+\\mu (B)-\\mu (C))$ .",
"The next two lemmas relate the two defects ${\\mathcal {D}},{\\mathcal {D}}^{\\prime }$ to one another.",
"Lemma 3.4 [10] Let $A,B$ be measurable subsets of $G$ of positive Haar measures, and suppose that $\\tau \\in [0,\\min (\\mu (A),\\mu (B))]$ and $\\mu (A)+\\mu (B)<1+\\tau $ .",
"Then $\\mathcal {D}(A,B,S_{A,B}(\\tau ))\\le \\mathcal {D}^{\\prime }(A,B,\\tau ).",
"$ Lemma 3.5 [10] Let $A,B,C\\subset G$ be measurable sets with positive Haar measures.",
"Let $\\tau _C = \\tfrac{1}{2} (\\mu (A)+\\mu (B)-\\mu (C))$ .",
"If $ \\big |\\,\\mu (A)-\\mu (B)\\,\\big | + 2{\\mathcal {D}}(A,B,C)^{1/2}< \\mu (C) < \\mu (A)+\\mu (B)-2{\\mathcal {D}}(A,B,C)^{1/2}$ and $\\mu (A)+\\mu (B)+\\mu (C) \\le 2-2{\\mathcal {D}}(A,B,C)^{1/2}$ then $ {\\mathcal {D}}^{\\prime }(A,B,\\tau _C) \\le 2{\\mathcal {D}}(A,B,C).",
"$ Corollary 3.6 [10] Let $G$ be a compact connected Abelian topological group, equipped with normalized Haar measure $\\mu $ .",
"Let $A,B\\subset G$ be measurable sets with positive Haar measures.",
"Let $\\tau \\in [0,\\min (\\mu (A),\\mu (B))]$ , and suppose that $\\mu (A)+\\mu (B)\\le 1+\\tau $ and $\\big |\\,\\mu (A)-\\mu (B)\\,\\big | \\le \\mu (S_{A,B}(\\tau )),\\\\\\mu (A)+\\mu (B)+\\mu (S_{A,B}(\\tau ))\\le 2.$ Then $ \\big |\\mu (S_{A,B}(\\tau ))-(\\mu (A)+\\mu (B)-2\\tau )\\big |\\le 2{\\mathcal {D}}^{\\prime }(A,B,\\tau )^{1/2}.$ The hypotheses of Theorem REF are satisfied.",
"The hypothesis $\\big |\\,\\mu (A)-\\mu (B)\\,\\big | \\le \\mu (S_{A,B}(\\tau ))$ of the corollary is equivalent to $\\sigma \\le \\min (\\mu (A),\\mu (B))$ , where $\\sigma $ is defined by $\\mu (S_{A,B}(\\tau ))=\\mu (A)+\\mu (B)-2\\sigma $ .",
"Thus, (REF ) holds by being a restatement of the conclusion of Theorem REF for $\\sigma $ in this range."
],
[
"Two key principles",
"In analyzing near-maximizers $(A,B,C)$ of the Riesz-Sobolev functional, we have found it to be useful to transform $(A,B,C)$ in several different ways.",
"Two of these are based on the principles of submodularity and complementation, which are developed in this section as Proposition REF and Lemma REF , respectively.",
"A third is the transformation of $(A,B,C)$ to a triple $(A,B,\\tau )$ , based on the relationship between ${\\mathcal {D}}(A,B,S_{A,B}(\\tau ))$ and ${\\mathcal {D}}^{\\prime }(A,B,\\tau )$ explored in §.",
"A fourth is the flow $(A,B,C)\\mapsto (A(t),B(t),C(t))$ introduced in §.",
"A fifth arises when $C\\subset G$ is a rank one Bohr set or is well approximated by such a set, and relates $\\int _{C} {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu $ to a relaxed version of this functional for associated data on ${\\mathbb {T}}$ .",
"This connection is developed in §.",
"At certain stages of the analysis we will pass from a triple $(A,B,C)$ to a related triple $(A^{\\prime },B^{\\prime },C^{\\prime })$ with certain more advantageous properties, or from $(A,B,\\tau )$ to $(A^{\\prime },B^{\\prime },\\tau ^{\\prime })$ .",
"We want to do this without sacrificing smallness of ${\\mathcal {D}}(A,B,C)$ or of ${\\mathcal {D}}^{\\prime }(A,B,\\tau )$ , respectively.",
"Two principles that make this possible are submodularity and complementation.",
"Let $G$ be a compact connected Abelian group $G$ , with normalized Haar measure $\\mu $ .",
"Proposition 4.1 (Submodularity) (Tao [22]) Let $A, B_1, B_2$ be measurable subsets of $G$ , and let $\\tau \\in [ 0,\\min (\\mu (A),\\mu (B_1\\cap B_2))]$ with $\\mu (A)+\\mu (B_1\\cup B_2)-\\tau \\le 1$ .",
"Then $\\mathcal {D}^{\\prime }(A,B_1\\cap B_2,\\tau )+\\mathcal {D}^{\\prime }(A,B_1\\cup B_2,\\tau )\\le \\mathcal {D}^{\\prime }(A,B_1,\\tau )+\\mathcal {D}^{\\prime }(A,B_2,\\tau )$ and the above four quantities ${\\mathcal {D}}^{\\prime }$ are all nonnegative.",
"Lemma 4.2 Suppose that each of $A,B,C$ has Haar measure strictly $>0$ and strictly $<1$ .",
"$(A,B,C)$ is admissible and satisfies $\\mu (A)+\\mu (B)+\\mu (C)\\le 2$ if and only if $(G\\setminus A,G\\setminus B,C)$ is admissible and satisfies $\\mu (G\\setminus A)+\\mu (G\\setminus B)+\\mu (C)\\le 2$ .",
"The relation $\\mu (C) \\le \\mu (G\\setminus A) + \\mu (G\\setminus B)$ is equivalent to $\\mu (A)+\\mu (B)+\\mu (C)\\le 2$ , and by symmetry $\\mu (C) \\le \\mu (A) + \\mu (B)$ is equivalent to $\\mu (G\\setminus A)+\\mu (G\\setminus B)+\\mu (C)\\le 2$ .",
"The relation $\\mu (G\\setminus A)\\le \\mu (G\\setminus B)+\\mu (C)$ is equivalent to $\\mu (B)\\le \\mu (A)+\\mu (C)$ , and interchanging $A,B$ in this equivalence yields the equivalence of the remaining two relations.",
"Lemma 4.3 For each $\\eta >0$ there exists $\\eta ^{\\prime }>0$ with the following property.",
"Suppose that each of $A,B,C$ has Haar measure strictly $>0$ and strictly $<1$ , and that $(A,B,C)$ is $\\eta $ –strictly admissible and $\\eta $ -bounded.",
"Then $(G\\setminus A,G\\setminus B,C)$ is $\\eta ^{\\prime }$ –strictly admissible and $\\eta ^{\\prime }$ -bounded.",
"This is proved in the same way as Lemma REF .",
"$\\Box $ Lemma 4.4 Suppose that each of $A,B$ has Haar measure strictly $>0$ , that $\\mu (A)+\\mu (B)<1$ , and that $A+B$ is measurable.",
"Then $\\mu _*(A+\\tilde{B})-\\mu (A)-\\mu (\\tilde{B})\\le \\mu (A+B)-\\mu (A)-\\mu (B)$ where $\\tilde{B} = -\\big (G\\setminus (A+B)\\big )$ .",
"It holds that $(G\\setminus (A+B)) -A\\subset G\\setminus B$ .",
"Indeed, let $x\\in A$ and $z\\notin A+B$ .",
"If $y=z-x$ belongs to $B$ then $x+y=z$ , whence $z\\in A+B$ , a contradiction.",
"Therefore $\\mu _*\\big (A-G\\setminus (A+B) \\big )\\le 1-\\mu (B)$ and consequently $\\mu _*\\big (A-G\\setminus (A+B) \\big ) -\\mu (A)-\\mu (G\\setminus (A+B))\\\\ \\le 1-\\mu (B)-\\mu (A)-[1-\\mu (A+B)]\\\\= \\mu (A+B)-\\mu (A)-\\mu (B).$ Lemma 4.5 (Complementation) If $(A,B,C)$ is admissible and $\\mu (A)+\\mu (B)+\\mu (C)\\le 2$ then $ {\\mathcal {D}}(A,B,C) = {\\mathcal {D}}(G\\setminus A,\\,G\\setminus B,\\,C).$ Writing ${\\mathbf {1}}_{G\\setminus A} = 1-{\\mathbf {1}}_A$ and likewise for $B$ , then expanding the integrand, gives $\\int _C {\\mathbf {1}}_{G\\setminus A}*{\\mathbf {1}}_{G\\setminus B} \\,d\\mu &= \\int _C (1-\\mu (A)-\\mu (B)+{\\mathbf {1}}_A*{\\mathbf {1}}_B)\\,d\\mu \\\\&= \\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu + \\mu (C)(1-\\mu (A)-\\mu (B)).$ Since $(G\\setminus A)^\\star =\\lbrace \\tfrac{1}{2}-x: x\\in {\\mathbb {T}}\\setminus {A^\\star }\\rbrace $ and likewise for $(G\\setminus B)^\\star $ , and since ${A^\\star },{B^\\star },{C^\\star }$ are symmetric under $x\\mapsto -x$ , the same calculation gives $\\int _{C^\\star }{\\mathbf {1}}_{(G\\setminus A)^\\star }*{\\mathbf {1}}_{(G\\setminus B)^\\star }\\,dm= \\int _{C^\\star }{\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star }\\,dm+ \\mu (C)(1-\\mu (A)-\\mu (B))$ since $m({A^\\star })=\\mu (A)$ and likewise for $B$ .",
"Subtracting these two relations gives ${\\mathcal {D}}(A,B,C) = {\\mathcal {D}}(G\\setminus A,\\,G\\setminus B,\\,C)$ ."
],
[
"A link between Riesz-Sobolev and sumset inequalities",
"The next lemma lies at the heart of this part of the analysis.",
"It states that if $(A,B,C)$ is nearly a maximizer for the functional $\\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu $ , then a certain associated superlevel set $S=S_{A,B}(\\beta )$ has small sumset in the sense that $\\mu (S-S)$ is nearly equal to $2\\mu (S)$ .",
"A theorem of Tao [22] then implies that S is nearly a rank one Bohr set.",
"The same holds for $C$ , as, by Lemma REF , $C\\,\\Delta \\,S$ has small Haar measure.",
"However, the proof of Lemma REF requires its very restrictive hypothesis that $\\mu (A)=\\mu (B)$ .",
"In an analysis of the Riesz-Sobolev equality for ${\\mathbb {R}}^1$ in [10], this hypothesis was removed in a subsequent step, by a method that does not apply to compact groups $G$ .",
"In the present paper we will accomplish this removal for compact connected Abelian groups by an unrelated and somewhat lengthy alternative method based in part on ideas of Tao [22].",
"This necessitates the reductions carried out in §.",
"Lemma 5.1 [10] Let $(A,B,C)$ be an $\\eta $ –strictly admissible ordered triple of measurable subsets of $G$ with positive Haar measures.",
"Suppose that $ &\\mu (A)=\\mu (B) \\le \\tfrac{1}{2} ,\\\\ & \\mu (C) \\le \\mu (A)- 4{\\mathcal {D}}(A,B,C)^{1/2} ,\\\\ &{\\mathcal {D}}(A,B,C)^{1/2} < \\tfrac{1}{28}\\eta \\mu (A).",
"$ Let $\\beta = \\tfrac{1}{2}\\big (\\mu (A)+\\mu (B)-\\mu (C)\\big )$ .",
"Then $ \\mu \\big (S_{A,B}(\\beta )-S_{A,B}(\\beta )\\big )\\le 2\\mu (S_{A,B}(\\beta )) + 12{\\mathcal {D}}(A,B,C)^{1/2}.$ The proof of this lemma is essentially identical to the proof of the corresponding result in [10], so it is not included here.",
"$\\Box $ Under certain hypotheses, it can be concluded that the set $S_{A,B}(\\beta )$ above is nearly a rank one Bohr set.",
"Corollary 5.2 For each $\\varepsilon ,\\eta >0$ there exists $\\rho >0$ with the following property.",
"Let $(A,B,C)$ be an $\\eta $ –strictly admissible $\\eta $ –bounded ordered triple of measurable subsets of $G$ satisfying the hypotheses (REF ) and () of Lemma REF .",
"If ${\\mathcal {D}}(A,B,C)\\le \\rho $ then there exists a rank one Bohr set ${\\mathcal {B}}\\subset G$ satisfying $ \\mu (C\\bigtriangleup {\\mathcal {B}}) \\le \\varepsilon .",
"$ If $\\mu (C)\\le \\mu (A)=\\mu (B)\\le \\tfrac{1}{2}-\\eta $ and ${\\mathcal {D}}(A,B,C)=0$ , then there exists a rank one Bohr set ${\\mathcal {B}}\\subset G$ satisfying $ \\mu (C\\bigtriangleup {\\mathcal {B}}) =0.",
"$ Let $\\varepsilon >0$ .",
"By (REF ), $\\mu (S_{A,B}(\\beta )\\,\\Delta \\,C)\\le 4{\\mathcal {D}}(A,B,C)^{1/2}$ .",
"Moreover, if ${\\mathcal {D}}(A,B,C)$ is sufficiently small as a function of $\\eta $ , then the conclusion (REF ) of Lemma REF states that $S_{A,B}(\\beta )$ satisfies a strong form of the hypothesis of the theorems of Tao [22] and Griesmer [17] discussed in §1.",
"The conclusion of those theorems is the existence of a rank one Bohr set satisfying $\\mu ({\\mathcal {B}}\\,\\Delta \\,S_{A,B}(\\beta ))\\le \\varepsilon $ , where $\\varepsilon \\rightarrow 0$ as ${\\mathcal {D}}(A,B,C)\\rightarrow 0$ with $\\eta $ fixed.",
"Therefore $ \\mu ({\\mathcal {B}}\\,\\Delta \\,C)\\le \\mu ({\\mathcal {B}}\\,\\Delta \\,S_{A,B}(\\beta ))+\\mu (S_{A,B}(\\beta )\\,\\Delta \\,C)\\le \\varepsilon + 4{\\mathcal {D}}(A,B,C)^{1/2}.$ If $\\varepsilon =0$ and the measures of $A,B,C$ satisfy the indicated hypotheses, then by Lemma REF , $S=S_{A,B}(\\beta )$ satisfies $\\mu (\\setminus C)=0$ and $\\mu (S-S)\\le 2\\mu (S)$ .",
"Therefore $\\mu (S_{A,B}(\\beta ))=\\mu (C)\\le \\tfrac{1}{2}-\\eta $ , and $S$ achieves equality in Kneser's inequality.",
"Therefore by Kneser's inverse theorem, there exists a rank one Bohr set ${\\mathcal {B}}$ satisfying $\\mu ({\\mathcal {B}})=\\mu (S)$ and $\\mu ({\\mathcal {B}}\\setminus S)=0$ .",
"Thus $\\mu ({\\mathcal {B}}\\,\\Delta \\,C)=0$ also."
],
[
"Two reductions",
"This section is devoted to two auxiliary results, of limited if any intrinsic interest, whose purpose is to reduce the analysis of triples that nearly saturate the Riesz-Sobolev inequality to triples that satisfy the hypotheses of Corollary REF .",
"In particular, we show that if $(A,B,C)$ nearly maximizes the Riesz-Sobolev functional among triples of sets with specified Haar measures, then there exists a closely related near maximizing triple $(\\tilde{A},\\tilde{B},\\tilde{C})$ satisfying supplementary properties, including the hypotheses of Corollary REF .",
"Those properties will subsequently be used to deduce that $(\\tilde{A},\\tilde{B},\\tilde{C})$ is nearly a compatibly centered parallel triple of rank one Bohr sets.",
"From that we will deduce the same property for $(A,B,C)$ .",
"This will be achieved by ultimately applying this reasoning to a short chain of triples $(A_n,B_n,C_n)$ , with $(A_n,B_n,C_n)$ constructed recursively from $(A_{n-1},B_{n-1},C_{n-1})$ beginning with $(A_0,B_0,C_0)= (A,B,C)$ , and with conclusions propagated in reverse from $(A_{n},B_{n},C_{n})$ to $(A_{n-1},B_{n-1},C_{n-1})$ , Lemma 6.1 Let $(A,B,C)$ be an $\\eta $ -strictly admissible and $\\eta $ –bounded triple of $\\mu $ -measurable subsets of $G$ , satisfying $\\mu (C)\\le \\mu (A)\\le \\mu (B),$ $ \\mu (A) \\le \\tfrac{1}{2} ,$ $\\mathcal {D}(A,B,C)^{1/2}\\le \\tfrac{1}{400}\\eta ^2 \\mu (B).$ Define $\\tau $ by $\\mu (C)=\\mu (A)+\\mu (B)-2\\tau $ .",
"Then there exists a measurable set $B^{\\prime }\\subseteq G$ with $\\mu (A)=\\mu (B^{\\prime })$ such that $(A,B^{\\prime },S_{A,B^{\\prime }}(\\tau ))\\text{ is $\\eta /2$--strictly admissibleand $\\eta ^2/2$--bounded},\\\\\\mathcal {D}(A,B^{\\prime },S_{A,B^{\\prime }}(\\tau ))\\le \\frac{1}{\\eta }\\mathcal {D}(A,B,C).$ Moreover, if $\\mu (C) \\le (1-\\tfrac{\\eta }{50})\\mu (B)$ then $ \\mu (S_{A,B^{\\prime }}(\\tau )) \\le \\mu (A)-4{\\mathcal {D}}(A,B^{\\prime },S_{A,B^{\\prime }}(\\tau ))^{1/2}.$ The set $B^{\\prime }$ is constructed via an iterative process, in the course of which $B$ is recursively replaced by successively smaller sets $B_j$ , finally arriving at a set $B^{\\prime }$ with the same Haar measure as $A$ .",
"The quantity $\\mathcal {D}^{\\prime }(A,B_j,\\tau )$ is controlled by induction on $j$ , yielding control of $\\mathcal {D}^{\\prime }(A,B^{\\prime },\\tau )$ .",
"Before starting this process, recall that $C$ is essentially equal to $S_{A,B}(\\tau )$ in the sense that $\\big |\\mu \\big (S_{A,B}(\\tau )\\big )-\\mu (C)\\big |\\le 2\\mathcal {D}(A,B,C)^{1/2},\\\\{\\mathcal {D}}(A,B,S_{A,B}(\\tau ))\\le {\\mathcal {D}}(A,B,C),\\\\{\\mathcal {D}}^{\\prime }(A,B,\\tau ) \\le 2{\\mathcal {D}}(A,B,C),$ with these inequalities justified by Lemmas REF and REF .",
"The following lemma will be useful.",
"Lemma 6.2 Let $B$ be a measurable subset of $G$ .",
"For any $t\\in [\\mu (B)^2,\\mu (B)]$ , there exists $x_t\\in B$ satisfying $\\mu \\big (B\\cap (x_t+B)\\big )=t$ .",
"This is a direct consequence of the connectivity of $G$ , since $x\\mapsto \\mu (B\\cap (B+x))$ is a continuous function from $G$ to ${\\mathbb {R}}$ .",
"$\\Box $ Iteratively invoking Lemma REF , a nested sequence of subsets of $B$ will be constructed; the last set in the sequence will be the desired $B^{\\prime }$ .",
"The properties of this sequence are described in the following Claim, the proof of which is postponed until after the proof of Lemma REF .",
"Claim 6.1 There exists a nested sequence $B=:B_0\\supseteq B_1\\supseteq B_2\\supseteq \\ldots \\supseteq B_J$ of subsets of $G$ , with $\\mu (B_J)=\\mu (A),$ $\\mathcal {D}^{\\prime }(A,B_j,\\tau )\\le 2\\mathcal {D}^{\\prime }(A,B_{j-1},\\tau )\\text{ for each }j\\le J,$ $2^J\\le \\frac{2}{\\eta ^2}.$ It follows that $\\mathcal {D}^{\\prime }\\big (A,B^{\\prime },\\tau \\big )\\le 2^J\\cdot 2 \\mathcal {D}(A,B,C)\\le \\frac{4}{\\eta ^2}\\mathcal {D}(A,B,C), $ whence $\\mathcal {D}^{\\prime }\\big (A,B^{\\prime },\\tau \\big )^{1/2}\\le \\tfrac{1}{200}\\eta \\mu (B)$ by the hypothesis on ${\\mathcal {D}}(A,B,C)$ .",
"We claim that $(A,B^{\\prime },\\tau )$ satisfies the hypotheses of Corollary REF .",
"Firstly, $\\tau = \\tfrac{1}{2}(\\mu (A)+\\mu (B)-\\mu (C))\\le \\min (\\mu (A),\\mu (B^{\\prime }))=\\mu (A)$ is equivalent to $\\mu (B)\\le \\mu (A)+\\mu (C)$ , which holds since $(A,B,C)$ is admissible.",
"Secondly, the superlevel set $S_{A,B^{\\prime }}(\\tau )$ satisfies $|\\mu (A)-\\mu (B^{\\prime })|\\le \\mu (S_{A,B^{\\prime }}(\\tau ))$ , since $\\mu (A)-\\mu (B^{\\prime })=0$ .",
"Thirdly, $\\mu (A)+\\mu (B^{\\prime })\\le 1+\\tau = 1 + \\tfrac{1}{2}(\\mu (A)+\\mu (B)-\\mu (C))$ is equivalent to $ \\mu (A) + \\mu (C) + (2\\mu (B^{\\prime })-\\mu (B))\\le 2$ , which holds since $\\mu (A)+\\mu (B)+\\mu (C)\\le 2$ and $\\mu (B^{\\prime })\\le \\mu (B)$ .",
"Fourthly, $\\mu (A)+\\mu (B^{\\prime })+\\mu (S_{A,B^{\\prime }}(\\tau ))\\le 2$ , as $\\mu (A)=\\mu (B^{\\prime })\\le \\tfrac{1}{2}$ .",
"Invoking Corollary REF for the triple $(A,B^{\\prime },S_{A,B^{\\prime }}(\\tau ))$ gives $|\\mu (S_{A,B^{\\prime }}(\\tau ))-(\\mu (A)+\\mu (B^{\\prime })-2\\tau )|\\le 2\\mathcal {D}^{\\prime }(A,B^{\\prime },\\tau )^{1/2}.$ Since $\\mu (A)+\\mu (B^{\\prime })-2\\tau = \\mu (A)+\\mu (C)-\\mu (B)$ is $\\ge \\eta \\mu (B)\\ge \\eta \\mu (A)$ by the $\\eta $ –strict admissibility hypothesis, while also $2\\tau \\ge \\mu (B)$ , it follows from (REF ) that $(A,B^{\\prime },S_{A,B^{\\prime }}(\\tau ))$ is $\\eta /2$ -strictly admissible and satisfies the estimates $\\mu (A)+\\mu (B^{\\prime })+\\mu (S_{A,B^{\\prime }}(\\tau ))\\le 2-\\tfrac{1}{2}\\eta $ and $\\min (\\mu (A),\\mu (B^{\\prime }),\\mu (S_{A,B^{\\prime }}(\\tau )))\\ge \\eta ^2/2$ .",
"Moreover, if $\\mu (C)\\le \\mu (B)- \\tfrac{1}{50}\\eta \\mu (B)$ then $\\begin{aligned}\\mu (S_{A,B^{\\prime }}(\\tau ))&\\le \\mu (A)+\\mu (B^{\\prime })-2\\tau +2\\mathcal {D}^{\\prime }(A,B^{\\prime },\\tau )^{1/2}\\\\&= \\mu (A)+\\mu (C)-\\mu (B) + 2\\mathcal {D}^{\\prime }(A,B^{\\prime },\\tau )^{1/2}\\\\&\\le \\mu (A)- \\big (\\mu (B)-\\mu (C)\\big )+\\tfrac{1}{100}\\eta \\mu (B) \\\\& \\le \\mu (A)-\\tfrac{1}{50}\\eta \\mu (B) + \\tfrac{1}{100} \\eta \\mu (B).\\end{aligned}$ Therefore $\\mu (S_{A,B^{\\prime }}(\\tau )) \\le \\mu (A)-4{\\mathcal {D}}(A,B^{\\prime },\\tau )^{1/2}$ , establishing together with Lemma REF the final assertion of Lemma REF .",
"The sets $B_j$ will be constructed by an iterative use of Lemma REF , in such a way that Proposition REF can be invoked to control each $\\mathcal {D}^{\\prime }(A,B_j,\\tau )$ .",
"More precisely, for each $j=1,\\ldots ,J$ , define $B_j:=B_{j-1}\\cap (x_j+B_j),$ with $x_j\\in G$ chosen to ensure that $\\mu (B_j)=\\mu (B_{j-1})-b_j$ for appropriate quantities $b_j\\in [0,\\mu (B_{j-1})-\\mu (B_{j-1})^2]$ that will be specified later (such $x_j$ exists by Lemma REF ), where $J$ is defined as the smallest non-negative integer such that $\\mu (B_J)=\\mu (A)$ .",
"(The quantities $b_j$ will be such that such $J$ will exist.)",
"Now, the $b_j\\in [0,\\mu (B_{j-1})-\\mu (B_{j-1})^2]$ are chosen so that $\\mu (B_j)\\ge \\mu (A)$ for all $j$ , i.e.",
"$b_j\\le \\mu (B_{j-1})-\\mu (A),$ and so that Proposition REF can be applied for $(A,B_j,\\tau )$ and $\\big (A,B_{j-1}\\cup (x_j+B_{j-1}),\\tau \\big )$ , to deduce (REF ).",
"To that end, for each $j$ the estimate $\\mu (A)+\\mu \\big (B_{j-1}\\cup (x_j+B_{j-1})\\big )\\le 1+\\tau $ should hold, i.e.",
"$\\mu (A)+\\mu (B_{j-1})+b_j\\le 1+\\tau $ for all $j$ .",
"By (REF ), this is equivalent to $\\left\\lbrace \\begin{aligned}&\\mu (A)+\\mu (B)+b_1\\le 1+\\tau \\text{ (for }j=1\\text{)},\\\\&\\mu (A)+\\mu (B)-(b_1+b_2+\\ldots +b_{j-1})+b_j\\le 1+\\tau \\text{ for }j\\ge 2,\\\\\\end{aligned} \\right.$ that is $\\left\\lbrace \\begin{aligned}&b_1\\le d,\\\\&b_j-(b_1+b_2+\\ldots +b_{j-1})\\le d\\text{ for }j\\ge 2,\\\\\\end{aligned} \\right.$ where $d:=\\tfrac{1}{2}\\big (2-\\mu (A)-\\mu (B)-\\mu (C))\\big ).$ Therefore, it suffices to find $b_j\\in [0,\\mu (B_{j-1})-\\mu (B_{j-1})^2]$ that satisfy (REF ), that are small enough for (REF ) to hold, but also large enough for $\\mu (B_J)=\\mu (A)$ to hold for some $J$ with $2^J\\le \\tfrac{2}{\\eta ^2}$ .",
"Observe that, if not for the condition $b_j\\in [0,\\mu (B_{j-1})-\\mu (B_{j-1})^2]$ , the quantities $b_j=2^jd$ for all $j=1,\\ldots ,J-1$ and $b_J=\\mu (B_{J-1})-\\mu (A)$ , where $J$ is the smallest positive integer with $\\mu (B)-d-2d-\\ldots -2^Jd<\\mu (A)$ , would work as they satisfy (REF ) and (REF ), while also $2^J\\le \\tfrac{2}{\\eta ^2}$ .",
"In order to achieve the additional condition $b_j\\in [0,\\mu (B_{j-1})-\\mu (B_{j-1})^2]$ , more care needs to be taken.",
"For simplicity, once $B_{j-1}$ has been defined, denote $m_j:=\\min \\big (\\mu (B_{j-1})-\\mu (B_{j-1})^2,\\mu (B_{j-1})-\\mu (A)\\big ).$ Define $b_j:=2^jd\\text{ for all }j=1,\\ldots ,J_1-1,$ where $J_1$ is the smallest non-negative integer $j$ such that $2^jd>m_j$ .",
"Observe that the so far defined $b_j$ satisfy the required conditions.",
"If $2^{J_1}d\\le \\mu (B_{J_1-1})-\\mu (B_{J_1-1})^2$ , then $2^{J_1}d>\\mu (B_{J_1-1})-\\mu (A)$ , so $\\mu (B_{J_1-1})-\\mu (A)\\in [0, \\mu (B_{J_1-1})-\\mu (B_{J_1-1})^2]$ .",
"In this case, define $b_{J_1}:=\\mu (B_{J_1-1})-\\mu (A)$ and terminate the process.",
"The $b_j$ satisfy all the required conditions.",
"Otherwise, $2^{J_1}d>\\mu (B_{J_1-1})-\\mu (B_{J_1-1})^2$ .",
"Define $b_j:=m_j\\text{ for all }J=J_1+1,\\ldots , \\bar{J}_2-1,$ where $\\bar{J}_2$ is the smallest integer larger than $J_1$ with $2^{J_1}d\\le m_{\\bar{J}_2}$ , having terminated the process at the smallest $j$ along the way for which $m_j=0$ , if such a $j$ exists.",
"Observe that the so far defined $b_j$ satisfy the required conditions.",
"If the process has not been terminated, define $b_j:=2^{J_1+j-\\bar{J}_2}d\\text{ for all }j=\\bar{J}_2,\\ldots , J_2-1,$ where $J_2$ is the smallest integer $j$ larger than $\\bar{J}_2$ with $2^{J_1+j-\\bar{J}_2}d>m_j$ .",
"The so far defined $b_j$ satisfy the required conditions.",
"Now, working as above, if $2^{J_1+J_2-\\bar{J}_2}d\\le \\mu (B_{J_1-1})-\\mu (B_{J_1-1})^2$ define $b_{J_2}:=\\mu (B_{J_2-1})-\\mu (A)$ and terminate the process.",
"Otherwise, define $b_j:=m_j\\text{ for all }J=J_2+1,\\ldots , \\bar{J}_3-1,$ where $\\bar{J}_3$ is the smallest integer larger than $J_2$ with $2^{J_1+J_2-\\bar{J}_2}d\\le m_2$ , having terminated the process at the smallest $j$ along the way for which $m_j=0$ , if such a $j$ exists.",
"Continuing this way, one definitely finds $J\\in \\mathbb {N}$ with $\\mu (B_J)=\\mu (A)$ ; that is when the process terminates.",
"The $b_j$ satisfy (REF ) and (REF ).",
"Therefore, it remains to show that $2^J\\le \\tfrac{2}{\\eta ^2}$ .",
"Indeed, $b_1+\\ldots +b_J=\\mu (B)-\\mu (A)$ .",
"Now, let $\\mathcal {M}$ be the set of $j$ for which $b_j=m_j$ , and $\\mathcal {M}^{\\prime }:=\\lbrace 1,\\ldots ,J\\rbrace \\setminus \\mathcal {M}$ .",
"On the one hand, $\\sum _{j\\in \\mathcal {M}^{\\prime }}b_j=d+2d+2^2d+\\ldots +2^{m^{\\prime }}d\\ge 2^{m^{\\prime }}d,$ where $m^{\\prime }=\\#\\mathcal {M}^{\\prime }$ .",
"Therefore, $2^{m^{\\prime }}d\\le \\mu (B)-\\mu (A)$ , so $2^{m^{\\prime }}\\le \\tfrac{1}{\\eta }.$ On the other hand, $m$ equals at most the number of consecutive intervals of the form $[c^2,c]$ needed to cover $[\\mu (A),\\mu (B)]$ (with the right-most interval being $[\\mu (B)^2,\\mu (B)]$ ).",
"This in turn equals the smallest positive integer $k$ with $\\mu (B)^{2^k}\\le \\mu (A)$ .",
"Since $\\mu (B)^{2^{k-1}}\\ge \\mu (A)$ , it follows that $2^m\\le 2\\tfrac{\\ln \\big (\\tfrac{1}{\\mu (A)}\\big )}{\\ln \\big (\\tfrac{1}{\\mu (B)}\\big )}\\le 2\\tfrac{\\ln \\big (\\tfrac{1}{\\eta }\\big )}{\\ln 2}\\le \\tfrac{2}{\\eta }.$ So, $2^J=2^{m+m^{\\prime }}\\le \\frac{2}{\\eta ^2}$ .",
"The next lemma will be used to deduce properties of more general triples from properties of triples that satisfy the hypotheses of Lemma REF .",
"Lemma 6.3 Let $(A,B,C)$ be $\\eta $ -strictly admissible and $\\eta $ –bounded and satisfy $\\mu (C)\\le \\mu (A)\\le \\mu (B),\\\\\\mu (A)\\le \\tfrac{1}{2},\\\\\\mathcal {D}(A,B,C)^{1/2}\\le \\tfrac{1}{800} \\eta \\mu (B).$ Define $\\tau $ by $\\mu (B)=\\mu (A)+\\mu (C)-2\\tau $ .",
"If $\\mu (C) > (1-\\tfrac{\\eta }{50})\\mu (B)$ then there exist measurable sets $C^{\\prime }\\subseteq C$ and $A^{\\prime }\\subseteq A$ that satisfy $\\left\\lbrace \\begin{aligned}&(S_{C^{\\prime },A}(\\tau ),C^{\\prime },A)\\text{ is $\\eta /4$--strictly admissibleand $\\eta /4$--bounded}\\\\&\\mathcal {D}(S_{C^{\\prime },A}(\\tau ),C^{\\prime },A)\\le 16\\mathcal {D}(C,B,A)\\\\&\\mu (C^{\\prime })=\\mu (A^{\\prime })=\\mu (C)-\\tfrac{1}{10} \\eta \\mu (B),\\end{aligned} \\right.$ while $\\left\\lbrace \\begin{aligned}&(S_{C^{\\prime },A^{\\prime }}(\\tau ),C^{\\prime },A^{\\prime })\\text{ is $\\eta /2$--strictly admissible and $\\eta /2$--bounded}\\\\&\\mathcal {D}(S_{C^{\\prime },A^{\\prime }}(\\tau ),C^{\\prime },A^{\\prime })\\le 16\\mathcal {D}(C,B,A)\\\\&\\mu (S_{A^{\\prime },C^{\\prime }}(\\tau )) \\le (1-\\tfrac{\\eta /2}{50}) \\mu (C^{\\prime }).\\end{aligned} \\right.$ Define $\\tau = \\tfrac{1}{2} (\\mu (A)+\\mu (C)-\\mu (B))$ .",
"Then $\\tau \\ge \\tfrac{1}{2} \\eta \\mu (B)\\ge \\tfrac{1}{2} \\eta ^2$ by the $\\eta $ –strict admissibility hypothesis, while $\\tau \\le \\tfrac{1}{2}\\mu (C)\\le \\tfrac{1}{4}$ since $\\mu (B)\\ge \\mu (A)$ .",
"Since $(A,B,C)$ is $\\eta $ -strictly admissible and $\\mathcal {D}(A,B,C)$ is small relative to $\\eta \\mu (B)$ , Lemma REF gives $\\big |\\mu \\big (S_{C,A}(\\tau )\\big )-\\mu (B)\\big |\\le 2\\mathcal {D}(A,B,C)^{1/2},$ whence $(C,A,S_{C,A}(\\tau ))$ is $\\tfrac{1}{2}\\eta $ -strictly admissible.",
"Lemma REF also gives $\\mathcal {D}(C,A,S_{C,A}(\\tau ))\\le \\mathcal {D}(A,B,C).$ By Lemma REF , $\\mathcal {D}^{\\prime }(C,A,\\tau )\\le 2\\mathcal {D}(A,B,C).$ Now, there exist $x_C,x_A\\in G$ such that $C^{\\prime }:=C\\cap (x_C +C)$ and $A^{\\prime }:=A\\cap (x_A +A)$ satisfy $ \\left\\lbrace \\begin{aligned}\\mu (C^{\\prime })&=\\mu (C)-\\tfrac{\\eta }{10}\\mu (B) \\ \\in [\\mu (C)^2,\\mu (C)]\\\\\\mu (A^{\\prime })&=\\mu (C^{\\prime }) \\ \\in [\\mu (A)^2,\\mu (A)].\\end{aligned} \\right.",
"$ (Observe that $\\mu (C)-\\tfrac{\\eta }{10}\\mu (B)\\ge \\mu (A)^2$ ($\\ge \\mu (C)^2$ ) because $\\mu (A)\\le \\tfrac{1}{2}$ , thus $\\mu (A)^2\\le \\tfrac{1}{2}\\mu (A)\\le \\tfrac{1}{2}\\mu (B)$ ; combining this with the lower bound assumption on $\\mu (C)$ , one obtains $\\mu (C)-\\mu (A)^2\\ge (1-\\tfrac{\\eta }{50}-\\tfrac{1}{2})\\mu (B)\\ge \\tfrac{\\eta }{10}\\mu (B)$ .)",
"It holds that $0\\le \\tau \\le \\mu (C^{\\prime })=\\min \\big \\lbrace \\mu (C^{\\prime }),\\mu (A)\\big \\rbrace =\\min \\big \\lbrace \\mu (C^{\\prime }),\\mu (A^{\\prime })\\big \\rbrace $ and $\\mu (C^{\\prime })+\\mu (A\\cup A^{\\prime })-\\tau \\le \\mu (A)+\\mu (C\\cup C^{\\prime })-\\tau <1$ (as $2\\frac{\\eta }{10}\\mu (B)<2-(\\mu (A)+\\mu (B)+\\mu (C))$ ).",
"Therefore, $\\begin{aligned}0\\le \\mathcal {D}^{\\prime }(C^{\\prime },A^{\\prime },\\tau )\\le 2\\mathcal {D}^{\\prime }(C^{\\prime },A,\\tau )&\\le 4\\mathcal {D}^{\\prime }(C,A,\\tau )\\le 8\\mathcal {D}(A,B,C)\\end{aligned}$ by the submodularity principle, Proposition REF .",
"We apply Corollary REF to the triple $(A^{\\prime },C^{\\prime },\\tau )$ .",
"Its hypotheses are satisfied.",
"First, $0\\le \\tau \\le \\min (\\mu (C^{\\prime }),\\mu (A^{\\prime }))=\\mu (C^{\\prime })$ ; also, $\\mu (A^{\\prime })+\\mu (C^{\\prime })<1+\\tau $ holds, since $\\mu (C^{\\prime })=\\mu (A^{\\prime })\\le \\mu (A)\\le \\tfrac{1}{2}$ while $\\tau >0$ .",
"Second, $\\mu (S_{A^{\\prime },C^{\\prime }}(\\tau )) \\ge 0 = |\\mu (A^{\\prime })-\\mu (C^{\\prime })|$ .",
"Third, $\\mu (A^{\\prime })+\\mu (C^{\\prime })+\\mu (S_{A^{\\prime },C^{\\prime }}(\\tau ))\\le 2$ because $\\mu (A^{\\prime })=\\mu (C^{\\prime })\\le \\mu (A)\\le \\tfrac{1}{2}$ while $\\mu (S_{A^{\\prime },C^{\\prime }}(\\tau ))\\le 1$ .",
"Therefore the Corollary may be applied to obtain $\\begin{aligned}|\\mu (S_{C^{\\prime },A^{\\prime }}(\\tau ))-(\\mu (A^{\\prime })+\\mu (C^{\\prime })-2\\tau )|\\le 2\\mathcal {D}^{\\prime }(C^{\\prime },A^{\\prime },\\tau )^{\\frac{1}{2}}\\le \\tfrac{\\eta }{100}\\mu (B).\\end{aligned}$ We next show that $(S_{C^{\\prime },A^{\\prime }}(\\tau ),C^{\\prime },A^{\\prime })$ is $\\frac{\\eta }{2}$ -strictly admissible.",
"Inserting the definition of $\\tau $ into (REF ) gives $\\begin{aligned}\\mu (S_{C^{\\prime },A^{\\prime }}(\\tau ))&\\le \\mu (B)-\\big (\\mu (A)-\\mu (A^{\\prime })\\big )-\\big (\\mu (C)-\\mu (C^{\\prime })\\big )+\\tfrac{\\eta }{100}\\mu (B)\\\\&\\le \\mu (C)+\\tfrac{\\eta }{50}\\mu (B)-2\\cdot \\tfrac{\\eta }{10}\\mu (B)+\\tfrac{\\eta }{100}\\mu (B)\\\\&\\le \\mu (C^{\\prime })-\\tfrac{\\eta }{50}\\mu (B)\\\\&\\le (1-\\tfrac{\\eta }{50})\\mu (C^{\\prime }).\\end{aligned}$ Note that the last of the three conclusions stated for $(A^{\\prime },C^{\\prime },S_{A^{\\prime },B^{\\prime }}(\\tau ))$ has been verified.",
"On the other hand, $\\begin{aligned}\\mu (S_{C^{\\prime },A^{\\prime }}(\\tau ))&\\ge \\mu (B)-\\big (\\mu (A)-\\mu (A^{\\prime })\\big )-\\big (\\mu (C)-\\mu (C^{\\prime })\\big )-\\tfrac{\\eta }{100}\\mu (B)\\\\&\\ge \\mu (B)-\\left(\\tfrac{\\eta }{10}\\mu (B)+\\tfrac{\\eta }{100}\\mu (B)\\right)-\\tfrac{\\eta }{10}\\mu (B)-\\tfrac{\\eta }{100}\\mu (B)\\\\&\\ge \\mu (B)-\\tfrac{\\eta }{4}\\mu (B)\\\\&\\ge \\mu (C^{\\prime })-\\tfrac{\\eta }{4}\\mu (B)\\\\&>\\left(1-\\tfrac{\\eta }{50}-\\tfrac{\\eta }{4}\\right)\\mu (B)\\\\&>\\tfrac{\\eta }{2}\\mu (B).\\end{aligned}$ Since $\\mu (A^{\\prime })=\\mu (C^{\\prime })$ and $\\mu (B)\\ge \\max (\\mu (A^{\\prime }),\\mu (C^{\\prime }),\\mu (S_{A^{\\prime },C^{\\prime }}(\\tau )))$ , the triple $(A^{\\prime },C^{\\prime },S_{A^{\\prime },C^{\\prime }}(\\tau ))$ is $\\eta /2$ –strictly admissible.",
"We claim next that the intermediate triple $(S_{C^{\\prime },A}(\\tau ),C^{\\prime },A)$ is $\\tfrac{\\eta }{4}$ -strictly admissible.",
"Indeed, since $A^{\\prime }\\subseteq A$ and $C^{\\prime }\\subseteq C$ , $\\mu (S_{C^{\\prime },A^{\\prime }}(\\tau ))\\le \\mu (S_{C^{\\prime },A}(\\tau ))\\le \\mu (S_{C,A}(\\tau )),$ whence, by (REF ) and one of the inequalities in (REF ), $\\begin{aligned}\\mu (B)-\\tfrac{\\eta }{4}\\mu (B)\\le \\mu (S_{C^{\\prime },A}(\\tau ))\\le \\mu (B)+2\\mathcal {D}(A,B,C)^{1/2}\\le \\mu (B)+\\tfrac{\\eta }{400}\\mu (B).\\end{aligned}$ Therefore, $\\tfrac{\\eta }{4}$ -strict admissibility follows from the $\\eta $ –strict admissibility of $(A,B,C)$ and the inequalities $|\\mu (C^{\\prime })-\\mu (C)| \\le \\tfrac{\\eta }{10}\\mu (B)$ and $|\\mu (A)-\\mu (B)| \\le \\frac{\\eta }{50}\\mu (B)$ .",
"Finally, the $\\eta /2$ –boundedness of $(A^{\\prime },C^{\\prime },S_{A^{\\prime },C^{\\prime }}(\\tau ))$ and $\\eta /4$ –boundedness of $(A,C^{\\prime },S_{A,C^{\\prime }}(\\tau ))$ follow from estimates shown above."
],
[
"Relaxation",
"For function $g_j: G\\rightarrow [0,1]$ , define $g_j^{\\star \\star }:{\\mathbb {T}}\\rightarrow [0,\\infty )$ to be the indicator function of the interval centered at 0 whose Lebesgue measure is equal to $\\int _G g_j\\,d\\mu $ .",
"Define $\\mathbf {g}^{\\star \\star }= (g_1^{\\star \\star },g_2^{\\star \\star },g_3^{\\star \\star })$ .",
"Assuming that $g_j$ takes values in $[0,1]$ for each index $j$ , we say that $\\mathbf {g}$ is $\\eta $ –strictly admissible if the triple $(\\int _G g_j\\,d\\mu : 1\\le j\\le 3)$ is $\\eta $ –strictly admissible.",
"With these notations, Theorem REF can be equivalently stated as the inequality ${\\mathcal {T}}_G(\\mathbf {g}) \\le {\\mathcal {T}}_{\\mathbb {T}}(\\mathbf {g}^{\\star \\star })\\ \\text{ for all functions $g_j:G\\rightarrow [0,1]$.",
"}$ Notation 7.1 For any ordered triple ${\\mathbf {E}}$ of measurable subsets of $G$ , define $\\overline{{\\mathcal {D}}}({\\mathbf {E}})={\\mathcal {T}}_G({\\mathbf {E}}^\\star )-{\\mathcal {T}}_{\\mathbb {T}}({\\mathbf {E}}).$ More generally, for $g:G\\rightarrow [0,1]$ , define $\\overline{{\\mathcal {D}}}(\\mathbf {g}) = {\\mathcal {T}}_{\\mathbb {T}}(\\mathbf {g}^{\\star \\star })-{\\mathcal {T}}_G(\\mathbf {g}),$ and for $\\mathbf {g}= (g_j: j\\in \\lbrace 1,2,3\\rbrace )$ , define $\\mathbf {g}^{\\star \\star }= (g_j^\\star : j\\in \\lbrace 1,2,3\\rbrace )$ .",
"Then $ {\\mathcal {D}}(A,B,C)=\\overline{{\\mathcal {D}}}(A,B,-C)$ for any ordered triple $(A,B,C)$ of measurable subsets of $G$ .",
"That is, $\\langle {\\mathbf {1}}_{{A^\\star }}*{\\mathbf {1}}_{{B^\\star }},{\\mathbf {1}}_{{C^\\star }}\\rangle _{\\mathbb {T}}- \\langle {\\mathbf {1}}_A*{\\mathbf {1}}_B,{\\mathbf {1}}_C\\rangle _G= \\langle {\\mathbf {1}}_{{A^\\star }}*{\\mathbf {1}}_{{B^\\star }},{\\mathbf {1}}_{(-C)^\\star }\\rangle _{\\mathbb {T}}- \\langle {\\mathbf {1}}_A*{\\mathbf {1}}_B,{\\mathbf {1}}_{-C}\\rangle _G.$ Theorem REF can again be restated as $\\overline{{\\mathcal {D}}}({\\mathbf {E}})\\ge 0$ for every triple ${\\mathbf {E}}$ .",
"The function $h:{\\mathbb {T}}\\rightarrow [0,\\infty )$ is said to be symmetric if $h(-x)=h(x)$ for all $x\\in {\\mathbb {T}}$ .",
"If $h$ is symmetric, $h$ is said to be nonincreasing if its restriction to $[0,\\tfrac{1}{2}]\\subset {\\mathbb {T}}$ is nonincreasing, under the usual identification of ${\\mathbb {T}}$ with $[-\\tfrac{1}{2},\\tfrac{1}{2}]$ .",
"Lemma 7.1 Let $f_1,f_2,f_3:\\mathbb {T}\\rightarrow \\mathbb {R}$ be symmetric, nonincreasing functions satisfying $0\\le f_1,f_2,f_3\\le 1$ .",
"Let $I\\subset {\\mathbb {T}}$ be the interval centered at 0 of length $|I|=\\int _{\\mathbb {T}}f_1\\, dm$ .",
"Then $\\mathcal {T}_{\\mathbb {T}}(f_1,f_2,f_3)\\le \\mathcal {T}_{\\mathbb {T}}({\\mathbf {1}}_I,f_2,f_3).",
"$ Defining $F$ by $f_1={\\mathbf {1}}_I+F$ , one has $F\\le 0\\text{ on }I\\text{, }F\\ge 0\\text{ on }\\mathbb {T}\\setminus I\\text{ and }\\textstyle \\int _{\\mathbb {T}}F\\,dm=0.$ Since $\\mathcal {T}_{\\mathbb {T}}(f_1,f_2,f_3)=\\langle f_1, f_2*f_3\\rangle _{\\mathbb {T}}=\\langle {\\mathbf {1}}_I,f_2*f_3\\rangle _{\\mathbb {T}}+\\langle F, f_2*f_3\\rangle _{\\mathbb {T}},$ it suffices to show that $\\langle F, f_2*f_3\\rangle _{\\mathbb {T}}\\le 0$ .",
"Now, since $f_2, f_3$ are symmetric, non-increasing and non-negative, each can be approximated by a superposition of indicator functions of intervals centered at 0.",
"Therefore, it suffices to show that $\\langle F, {\\mathbf {1}}_J*{\\mathbf {1}}_K\\rangle _{\\mathbb {T}}\\le 0 $ for all intervals $J, K$ centered at 0.",
"This is in fact trivially true, due to (REF ) and the fact that ${\\mathbf {1}}_J*{\\mathbf {1}}_K$ is symmetric, non-increasing and non-negative.",
"Indeed, $\\begin{aligned}\\langle F, {\\mathbf {1}}_J*{\\mathbf {1}}_K\\rangle _{\\mathbb {T}}&=\\int _I {\\mathbf {1}}_J*{\\mathbf {1}}_K\\cdot F\\,dm + \\int _{\\mathbb {T}\\setminus I} {\\mathbf {1}}_J*{\\mathbf {1}}_K\\cdot F\\,dm\\\\&\\le \\int _I \\left(\\inf _I{\\mathbf {1}}_J*{\\mathbf {1}}_K\\right) F\\,dm+ \\int _{\\mathbb {T}\\setminus I} \\left(\\sup _{\\mathbb {T}\\setminus I}{\\mathbf {1}}_J*{\\mathbf {1}}_K\\right) F\\,dm\\\\&= c\\int _I F\\,dm+ c\\int _{\\mathbb {T}\\setminus I} F\\,dm=c\\int F\\,dm=0,\\end{aligned}$ where $c:={\\mathbf {1}}_J*{\\mathbf {1}}_K\\left(\\tfrac{m(I)}{2}\\right)$ .",
"By expressing each of $f,g,h$ as a superposition of indicator functions and invoking Theorem REF , we deduce that $\\langle f*g,h\\rangle _G \\le \\langle f^\\star *g^\\star ,h^\\star \\rangle _{\\mathbb {T}}.$ Express $h^\\star $ as a superposition $\\int _0^1 {\\mathbf {1}}_{D(t)}\\,dt$ where each $D(t)\\subset {\\mathbb {T}}$ is an interval centered at 0.",
"According to Lemma REF , $\\langle f^\\star ,g^\\star ,{\\mathbf {1}}_{D}\\rangle _{\\mathbb {T}}\\le \\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star },{\\mathbf {1}}_D\\rangle _{\\mathbb {T}}$ for any interval $D$ centered at 0.",
"Integrating with respect to $t\\in [0,1]$ yields $\\langle f^\\star *g^\\star ,h^\\star \\rangle _{\\mathbb {T}}\\le \\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star },h^\\star \\rangle _{\\mathbb {T}}.$ A repetition of this reasoning gives $\\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star },h^\\star \\rangle _{\\mathbb {T}}\\le \\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star },{\\mathbf {1}}_{C^\\star }\\rangle _{\\mathbb {T}}.$"
],
[
"The perturbative Riesz-Sobolev regime",
"In this section, we prove the following lemma, in which the sets in question are assumed to be moderately well approximated by appropriately related rank one Bohr sets, and are proved to be better approximated if ${\\mathcal {D}}({\\mathbf {E}}) = {\\mathcal {T}}_{\\mathbb {T}}({\\mathbf {E}}^\\star )-{\\mathcal {T}}_G({\\mathbf {E}})$ is sufficiently small.",
"The analysis is adapted from [11].",
"Lemma 8.1 For each $\\eta ,\\eta ^{\\prime }>0$ there exist $\\delta _0>0$ and ${\\mathbf {C}}<\\infty $ with the following property.",
"Let ${\\mathbf {E}}= (E_1,E_2,E_3)$ be an $\\eta $ –strictly admissible triple of measurable subsets of $G$ satisfying $ \\mu (E_1)+\\mu (E_2)+\\mu (E_3)\\le 2-\\eta ^{\\prime }.",
"$ Suppose that there exists a compatibly centered parallel ordered triple ${\\mathbf {B}}= ({\\mathcal {B}}_1,{\\mathcal {B}}_2,{\\mathcal {B}}_3)$ of rank one Bohr sets ${\\mathcal {B}}_j\\subset G$ satisfying $\\mu ({\\mathcal {B}}_j)=\\mu (E_j)$ and $\\max _j \\mu (E_j\\,\\Delta \\,{\\mathcal {B}}_j) \\le \\delta _0 \\max _k \\mu (E_k).$ Then there exists $\\mathbf {y}$ satisfying $y_1+y_2=y_3$ such that $\\max _j \\mu (E_j\\,\\Delta \\,({\\mathcal {B}}_j+y_j)) \\le {\\mathbf {C}}{\\mathcal {D}}({\\mathbf {E}})^{1/2}.$ Since $0<\\mu ({\\mathcal {B}}_j)<1=\\mu (G)$ , the homomorphism $\\phi $ does not vanish identically.",
"Definition 8.1 An ordered triple $({\\mathcal {B}}_1,{\\mathcal {B}}_2,{\\mathcal {B}}_3)$ of rank one Bohr sets is ${\\mathcal {T}}_G$ -compatibly centered if $({\\mathcal {B}}_1,{\\mathcal {B}}_2,-{\\mathcal {B}}_3)$ is compatibly centered.",
"All of our discussion of the Riesz-Sobolev inequality can be rephrased in terms of ${\\mathcal {T}}_G$ since ${\\mathcal {T}}_G({\\mathbf {E}}) = \\langle {\\mathbf {1}}_{E_1}*{\\mathbf {1}}_{E_2},{\\mathbf {1}}_{-E_3}\\rangle $ and $\\mu (-E_3)=\\mu (E_3)$ .",
"Theorem REF thus states that $ {\\mathcal {T}}_G({\\mathbf {E}}) \\le {\\mathcal {T}}_{\\mathbb {T}}({\\mathbf {E}}^\\star ) $ for all triples ${\\mathbf {E}}$ of measurable subsets of $G$ .",
"Another equivalent formulation is ${\\mathcal {T}}_G({\\mathbf {E}})\\le {\\mathcal {T}}_G({\\mathbf {B}})$ for any ${\\mathcal {T}}_G$ –compatibly centered ordered triple ${\\mathbf {B}}$ of parallel rank one Bohr sets satisfying $\\mu (E_j)=\\mu ({\\mathcal {B}}_j)$ for each $j\\in \\lbrace 1,2,3\\rbrace $ ; the right-hand side equals ${\\mathcal {T}}_{\\mathbb {T}}({\\mathbf {E}}^\\star )$ for any such triple ${\\mathbf {B}}$ .",
"Lemma REF can thus be equivalently formulated as follows.",
"Lemma 8.2 For each $\\eta ,\\eta ^{\\prime }>0$ there exist $\\delta _0>0$ and ${\\mathbf {C}}<\\infty $ with the following property.",
"Let ${\\mathbf {E}}= (E_1,E_2,E_3)$ be an $\\eta $ –strictly admissible triple of measurable subsets of $G$ satisfying $ \\mu (E_1)+\\mu (E_2)+\\mu (E_3)\\le 2-\\eta ^{\\prime }.",
"$ Suppose that there exists a ${\\mathcal {T}}_G$ -compatibly centered parallel ordered triple ${\\mathbf {B}}= ({\\mathcal {B}}_1,{\\mathcal {B}}_2,{\\mathcal {B}}_3)$ of rank one Bohr sets ${\\mathcal {B}}_j\\subset G$ satisfying $\\mu ({\\mathcal {B}}_j)=\\mu (E_j)$ and $\\max _j \\mu (E_j\\,\\Delta \\,{\\mathcal {B}}_j) \\le \\delta _0 \\max _k \\mu (E_k).$ Then there exists $\\mathbf {y}$ satisfying $y_1+y_2+y_3=0$ such that $\\max _j \\mu (E_j\\,\\Delta \\,({\\mathcal {B}}_j+y_j)) \\le {\\mathbf {C}}\\overline{\\mathcal {D}}({\\mathbf {E}})^{1/2}.$ Remark 8.2 One aspect of the conclusion may be unanticipated.",
"Suppose that ${\\mathbf {E}},{\\mathbf {B}}$ satisfy the hypotheses, and that $\\overline{{\\mathcal {D}}}({\\mathbf {E}})$ vanishes.",
"Then the conclusion is not only that ${\\mathbf {E}}$ is equivalent to some ordered triple of Bohr sets, but that it is equivalent to a translate of ${\\mathbf {B}}$ .",
"A consequence is that for any $\\eta >0$ , there exists $\\varepsilon >0$ with this property: If $B,B^{\\prime }$ are rank one Bohr sets satisfying $\\eta \\le \\mu (B)=\\mu (B^{\\prime })\\le 1-\\eta $ , and if $\\mu (B\\,\\Delta \\,B^{\\prime })<\\varepsilon $ , then $\\mu (B\\,\\Delta \\,B^{\\prime })=0$ .",
"There is no surprise in this consequence, but it is interesting that it is implicit in the lemma.",
"To deduce it, assume without loss of generality that $B,B^{\\prime }$ are centered at 0, that is, $B = \\lbrace x: \\Vert \\phi (x)\\Vert _\\mathbb {T}\\le r\\rbrace $ for some homomorphism $\\phi $ and $2r\\in [\\eta ,1-\\eta ]$ , and likewise for $B^{\\prime }$ with respect to a homomorphism $\\phi ^{\\prime }$ .",
"Set ${\\mathbf {B}}= (B,B,B)$ and ${\\mathbf {E}}= (B^{\\prime },B^{\\prime },B^{\\prime })$ .",
"The hypotheses of Lemma REF are satisfied, if $\\varepsilon $ is sufficiently small.",
"Moreover, $\\overline{{\\mathcal {D}}}({\\mathbf {E}})=0$ ; any ${\\mathcal {T}}$ –compatibly centered parallel family of rank one Bohr sets saturates the Riesz-Sobolev inequality.",
"The conclusion of the lemma is that $B^{\\prime }$ differs from some translate of $B$ by a $\\mu $ –null set.",
"$\\Box $ We will prove Lemma REF in the more general relaxed framework, in which indicator functions of sets are replaced by functions taking values in $[0,1]$ .",
"In the remainder of §, we study triples $\\mathbf {g}= (g_j: j\\in \\lbrace 1,2,3\\rbrace )$ with $g_j: G\\rightarrow [0,1]$ .",
"For functions $g:G\\rightarrow [0,1]$ , define $g^{\\star \\star }:{\\mathbb {T}}\\rightarrow [0,\\infty )$ to be the indicator function of the interval centered at $0\\in {\\mathbb {T}}$ whose Lebesgue measure is equal to $\\int _G g\\,d\\mu $ .",
"For triples $\\mathbf {g}$ , define $\\mathbf {g}^{\\star \\star }= (g_1^{\\star \\star },g_2^{\\star \\star },g_3^{\\star \\star })$ .",
"Recall the notation $\\overline{{\\mathcal {D}}}(\\mathbf {g}) = {\\mathcal {T}}_{\\mathbb {T}}(\\mathbf {g}^{\\star \\star })-{\\mathcal {T}}_G(\\mathbf {g})$ introduced in (REF ).",
"Assuming that $g_j$ takes values in $[0,1]$ for each index $j$ , we say that $\\mathbf {g}$ is $\\eta $ –strictly admissible if the triple $(\\int _G g_j\\,d\\mu : 1\\le j\\le 3)$ of positive scalars is $\\eta $ –strictly admissible.",
"The next lemma generalizes Lemma REF to the relaxed framework.",
"The remainder of this section will be devoted to its proof.",
"Lemma 8.3 For each $\\eta ,\\eta ^{\\prime }>0$ there exist $\\delta _0>0$ and ${\\mathbf {C}}<\\infty $ with the following property.",
"Let $\\mathbf {g}$ be an $\\eta $ –strictly admissible triple of measurable functions $g_j:G\\rightarrow [0,1]$ satisfying $ \\sum _{j=1}^3 \\int g_j\\,d\\mu \\le 2-\\eta ^{\\prime }.",
"$ Suppose that there exists a ${\\mathcal {T}}_G$ -compatibly centered parallel ordered triple ${\\mathbf {B}}= ({\\mathcal {B}}_1,{\\mathcal {B}}_2,{\\mathcal {B}}_3)$ of rank one Bohr sets ${\\mathcal {B}}_j\\subset G$ satisfying $\\mu ({\\mathcal {B}}_j)= \\int g_j\\,d\\mu $ and $\\max _j \\Vert g_j -{\\mathbf {1}}_{{\\mathcal {B}}_j} \\Vert _{L^1(G)} \\le \\delta _0 \\max _k \\int g_k\\,d\\mu .$ Then there exists $\\mathbf {y}\\in G^3$ satisfying $y_1+y_2+y_3=0$ such that $\\max _j \\Vert g_j - {\\mathbf {1}}_{{\\mathcal {B}}_j+y_j} \\Vert _{L^1(G)} \\le {\\mathbf {C}}\\overline{\\mathcal {D}}(\\mathbf {g})^{1/2}.$ Define the orbit ${\\mathcal {O}}({\\mathbf {A}})$ of the triple ${\\mathbf {A}}$ of subsets of $G$ to be the set of all triples ${\\mathbf {A}}+\\mathbf {y}=(A_j+y_j: j\\in \\lbrace 1,2,3\\rbrace )$ with $\\mathbf {y}\\in G^3$ satisfying $y_1+y_2+y_3=0$ .",
"For $g_j:G\\rightarrow [0,1]$ and ${\\mathbf {B}}= ({\\mathcal {B}}_j: 1\\le j\\le 3)$ satisfying $\\mu ({\\mathcal {B}}_j) = \\int g_j\\,d\\mu $ , define $\\operatorname{distance}\\,(\\mathbf {g},{\\mathcal {O}}({\\mathbf {B}}))= \\inf _\\mathbf {y}\\max _{j\\in \\lbrace 1,2,3\\rbrace } \\Vert g_j - {\\mathbf {1}}_{{\\mathcal {B}}_j+y_j} \\Vert _{L^1(G)},$ with the infimum taken over all $\\mathbf {y}\\in G^3$ satisfying $y_1+y_2+y_3=0$ .",
"With these definitions, Lemma REF states that if ${\\mathbf {B}},\\mathbf {g}$ satisfy its hypotheses then $ \\operatorname{distance}\\,(\\mathbf {g},{\\mathcal {O}}({\\mathbf {B}})) \\le {\\mathbf {C}}\\overline{{\\mathcal {D}}}(\\mathbf {g})^{1/2}.$ We use $c$ to denote a strictly positive constant that depends only on $\\eta $ , but whose value is permitted to change from one occurrence to the next.",
"We write $\\langle f,g\\rangle = \\int _G fg\\,d\\mu $ for functions $f,g:G\\rightarrow {\\mathbb {R}}$ .",
"Set $ \\delta = \\operatorname{distance}\\,(\\mathbf {g},{\\mathcal {O}}({\\mathbf {B}})).",
"$ Choose $\\mathbf {z}$ satisfying $z_1+z_2+z_3=0$ so that $ \\max _j \\Vert g_j - {\\mathbf {1}}_{{\\mathcal {B}}_j+z_j} \\Vert _{L^1(G)} = \\delta .",
"$ Such a minimizing $\\mathbf {z}$ must exist, since $ \\Vert g_j - {\\mathbf {1}}_{{\\mathcal {B}}_j+z_j} \\Vert _{L^1(G)}$ is a continuous function of $\\mathbf {z}$ with compact domain.",
"If $\\delta =0$ then the conclusion of the lemma certainly holds, so we may assume for the remainder of the proof that $\\delta >0$ .",
"The hypotheses and conclusion of the lemma are invariant under translation of each $g_j$ by $u_j\\in G$ , with $\\sum _j u_j=0$ .",
"By means of such a transformation, we may assume without loss of generality that ${\\mathcal {B}}_j=\\lbrace x\\in G: \\Vert \\phi (x) \\Vert _{{\\mathbb {T}}}\\le r_j\\rbrace $ , with $\\phi :G\\rightarrow {\\mathbb {T}}$ a continuous homomorphism independent of $j$ , and each $z_j=0$ .",
"Here, $0<r_j = \\tfrac{1}{2}\\mu ({\\mathcal {B}}_j) \\le \\tfrac{1}{2} (1-\\tilde{\\eta })$ with $\\tilde{\\eta }=\\tilde{\\eta }(\\eta ,\\eta ^{\\prime })>0$ .",
"Define functions $f_j$ by $g_j = {\\mathbf {1}}_{{\\mathcal {B}}_j} + f_j.$ These functions take values in $[-1,1]$ , and satisfy $\\int _G f_j\\,d\\mu =0$ .",
"Moreover, $\\max _{k\\in \\lbrace 1,2,3\\rbrace } \\Vert f_k \\Vert _{L^1} =\\delta $ by (REF ), $f_k\\le 0$ in ${\\mathcal {B}}_k$ , and $f_k\\ge 0$ in $G\\setminus {\\mathcal {B}}_k$ .",
"Regard $\\phi $ as a (discontinuous) mapping from $G$ to $(-\\tfrac{1}{2},\\tfrac{1}{2}]$ by identifying ${\\mathbb {T}}$ with $(-\\tfrac{1}{2},\\tfrac{1}{2}]$ in the usual way.",
"For each $k\\in \\lbrace 1,2,3\\rbrace $ , write $\\lbrace 1,2,3\\rbrace = \\lbrace i,j,k\\rbrace $ and define $ K_k(x) = {\\mathbf {1}}_{{\\mathcal {B}}_i}*{\\mathbf {1}}_{{\\mathcal {B}}_j}(x) \\ \\text{ for $x\\in G$.",
"}$ $K_k$ is continuous and nonnegative.",
"There exists $\\gamma _k>0$ such that $K_k(x)>\\gamma _k$ if $|\\phi (x)|<\\tfrac{1}{2}\\mu ({\\mathcal {B}}_k)$ , $K_k(x)<\\gamma _k$ if $|\\phi (x)|>\\tfrac{1}{2}\\mu ({\\mathcal {B}}_k)$ , and $K_k(x)=\\gamma _k$ when $|\\phi (x)|=\\tfrac{1}{2}\\mu ({\\mathcal {B}}_k)$ .",
"The $\\eta $ –strict admissibility hypothesis implies that there exists a small positive constant $c>0$ , depending only on $\\eta $ , such that $ {\\left\\lbrace \\begin{array}{ll}& |K_k(x)-\\gamma _k|= \\big |\\,|\\phi (x)|- \\tfrac{1}{2} \\mu (B_k)\\,\\big |\\ \\ \\text{whenever $\\big |\\,|\\phi (x)|- \\tfrac{1}{2} \\mu (B_k)\\,\\big | \\le c\\mu (B_k)$,}\\\\& |K_k(x)-\\gamma _k|\\ge c\\mu (B_k) \\text{ otherwise.}\\end{array}\\right.}",
"$ Let $\\lambda $ be a large positive constant, to be chosen below.",
"There exist a decomposition $ f_j = f_j^\\dagger + \\tilde{f}_j $ and consequently an expansion $g_j = {\\mathbf {1}}_{{\\mathcal {B}}_j} + f_j^\\dagger + \\tilde{f}_j$ , with the following properties: $\\int f_j^\\dagger \\,d\\mu = \\int \\tilde{f}_j\\,d\\mu =0\\\\\\text{$\\tilde{f}_j,f_j^\\dagger \\ge 0$ on $G\\setminus {\\mathcal {B}}_j$}\\\\\\text{$\\tilde{f}_j,f_j^\\dagger \\le 0$ on ${\\mathcal {B}}_j$}\\\\\\text{If$\\big |\\,|\\phi (x)| -\\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)\\,\\big |\\ge \\lambda \\delta $then $f_j^\\dagger (x)=0$.",
"}\\\\ \\Vert \\tilde{f}_j \\Vert _{L^1}\\le 2 \\int _{ \\big |\\,|\\phi (x)| -\\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)\\,\\big |\\ge \\lambda \\delta }|f_j(x)|\\,d\\mu (x).",
"$ To achieve this, set $\\tilde{f}_j(x)=f_j(x)$ whenever $\\big |\\,|\\phi (x)| -\\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)\\,\\big |\\ge \\lambda \\delta $ .",
"We do not simply set $\\tilde{f}_j(x)\\equiv 0$ otherwise (even though such an $\\tilde{f}_j$ clearly satisfies the desired condition (REF ) above), because the vanishing condition $\\int \\tilde{f}_j\\,d\\mu =0$ will be essential below.",
"Instead, for $x\\in G$ satisfying $\\big |\\,|\\phi (x)| -\\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)\\,\\big |<\\lambda \\delta $ , we define $\\tilde{f}_j(x) = f_j(x){\\mathbf {1}}_S(x)$ with the set $S$ chosen as follows.",
"If $\\int _{\\big |\\,|\\phi (x)| -\\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)\\,\\big |\\ge \\lambda \\delta } f_j\\,d\\mu \\ge 0$ , then $S\\subset {\\mathcal {B}}_j$ , and $S$ is chosen so that $\\int \\tilde{f}_j\\,d\\mu =0$ .",
"Such a subset exists because $\\int f_j\\,d\\mu =0$ , $f_j \\ge 0$ on $G\\setminus {\\mathcal {B}}_j$ and $\\le 0$ on ${\\mathcal {B}}_j$ , and $\\mu $ is nonatomic.",
"For our purpose, any such set $S$ suffices.",
"If $\\int _{\\big |\\,|\\phi (x)| -\\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)\\,\\big |\\ge \\lambda \\delta } f_j\\,d\\mu < 0$ , then instead choose $S\\subset G\\setminus {\\mathcal {B}}_j$ to ensure that $\\int \\tilde{f}_j\\,d\\mu =0$ .",
"In both cases, define $f_j^\\dagger = f_j-\\tilde{f}_j$ .",
"The resulting functions $\\tilde{f}_j,f_j^\\dagger $ enjoy all of the required properties.",
"Set $g_j^\\dagger = {\\mathbf {1}}_{{\\mathcal {B}}_j}+f_j^\\dagger $ .",
"These functions satisfy $g_j= g_j^\\dagger + \\tilde{f}_j$ , $0\\le g_j^\\dagger \\le 1$ , $-1\\le \\tilde{f}_j,f_j^\\dagger \\le 1$ , and (since $\\int \\tilde{f}_j=0$ ) $\\int g_j^\\dagger = \\int g_j$ .",
"Define $\\tilde{\\delta }= \\max _{j} \\Vert \\tilde{f}_j \\Vert _{L^1(G)} \\le \\delta .$ ${\\mathcal {T}}={\\mathcal {T}}_G$ satisfies $|{\\mathcal {T}}(h_1,h_2,h_3)|\\le \\Vert h_1 \\Vert _{L^1} \\Vert h_2 \\Vert _{L^1} \\Vert h_3 \\Vert _{L^\\infty }$ for arbitrary functions, and is invariant under permutation of $(h_1,h_2,h_3)$ .",
"Using the assumption that $ \\Vert g_j \\Vert _{L^\\infty }\\le 1$ , and for each $k$ writing $\\lbrace 1,2,3\\rbrace = \\lbrace i,j,k\\rbrace $ in some arbitrary manner, it follows that ${\\mathcal {T}}(\\mathbf {g})&= {\\mathcal {T}}(g_1^\\dagger + \\tilde{f}_1, g_2^\\dagger + \\tilde{f}_2, g_3^\\dagger + \\tilde{f}_3)\\\\&= {\\mathcal {T}}(\\mathbf {g}^\\dagger )+ \\sum _{k=1}^3 {\\mathcal {T}}(g_i^\\dagger ,g_j^\\dagger ,\\tilde{f}_k)+ O(\\tilde{\\delta }^2)\\\\&= {\\mathcal {T}}(\\mathbf {g}^\\dagger )+ \\sum _{k=1}^3 {\\mathcal {T}}({\\mathbf {1}}_{{\\mathcal {B}}_i},{\\mathbf {1}}_{{\\mathcal {B}}_j},\\tilde{f}_k)+ O(\\tilde{\\delta }\\cdot \\delta )\\\\&= {\\mathcal {T}}(\\mathbf {g}^\\dagger )+ \\sum _{k=1}^3 \\langle {\\mathcal {K}}_k,\\tilde{f}_k\\rangle + O(\\tilde{\\delta }\\cdot \\delta ).$ The constant implicit in the $O(\\tilde{\\delta }\\cdot \\delta )$ term is independent of the parameter $\\lambda $ .",
"Since $\\int \\tilde{f}_k\\,d\\mu =0$ , $\\langle {\\mathcal {K}}_k,\\tilde{f}_k\\rangle = \\langle {\\mathcal {K}}_k-\\gamma _k,\\tilde{f}_k\\rangle $ .",
"On the complement of ${\\mathcal {B}}_k$ , $\\tilde{f}_k\\ge 0$ and $K_k-\\gamma _k\\le 0$ ; on ${\\mathcal {B}}_k$ , both signs are reversed.",
"Therefore $\\langle K_k, \\tilde{f}_k\\rangle = \\int (K_k-\\gamma _k) \\tilde{f}_k\\,d\\mu = -\\int |K_k-\\gamma _k|\\cdot |\\tilde{f}_k|\\,d\\mu \\le -c\\lambda \\delta \\Vert \\tilde{f}_k \\Vert _{L^1}$ according to the properties (REF ) of ${\\mathcal {K}}_k$ and the relation $\\lambda \\delta \\le c\\mu ({\\mathcal {B}}_k)$ , which holds, for any particular choice of large constant $\\lambda $ , by the smallness hypothesis on $\\delta /\\mu ({\\mathcal {B}}_k)$ .",
"Therefore in all, ${\\mathcal {T}}(\\mathbf {g})\\le {\\mathcal {T}}(\\mathbf {g}^\\dagger ) - c\\lambda \\delta \\cdot \\tilde{\\delta }+ O(\\tilde{\\delta }\\cdot \\delta )$ with both $c$ and the implicit constant in the remainder term $O(\\delta ^2)$ independent of the parameter $\\lambda $ , but with $\\tilde{\\delta }$ dependent on $\\lambda $ .",
"Choosing $\\lambda $ sufficiently large gives ${\\mathcal {T}}(\\mathbf {g})\\le {\\mathcal {T}}(\\mathbf {g}^\\dagger ) - c\\lambda \\delta \\cdot \\tilde{\\delta }\\le \\min \\big ({\\mathcal {T}}(\\mathbf {g}^\\dagger ),{\\mathcal {T}}_{\\mathbb {T}}(\\mathbf {g}^{\\star \\star }) - c\\lambda \\delta \\cdot \\tilde{\\delta }\\big ),$ with $c>0$ independent of $\\lambda $ , and $\\lambda $ independent of $\\mathbf {g}$ .",
"We have used the bound ${\\mathcal {T}}(\\mathbf {g}^\\dagger )\\le {\\mathcal {T}}_{\\mathbb {T}}((\\mathbf {g}^\\dagger )^{\\star \\star })$ of Theorem REF , and the identity $(\\mathbf {g}^\\dagger )^{\\star \\star }= \\mathbf {g}^{\\star \\star }$ .",
"There are now two cases, depending on the magnitude of $\\tilde{\\delta }/\\delta $ .",
"If $\\tilde{\\delta }\\ge \\tfrac{1}{2}\\delta $ then ${\\mathcal {T}}(\\mathbf {g}) \\le {\\mathcal {T}}_{\\mathbb {T}}(\\mathbf {g}^{\\star \\star }) - \\tfrac{1}{2} c \\delta ^2$ .",
"This is the desired conclusion of Lemma REF .",
"In the second case, $\\tilde{\\delta }\\le \\tfrac{1}{2}\\delta $ .",
"From the triangle inequality in the form $ \\max _j \\Vert f_j^\\dagger \\Vert _{L^1}= \\max _j \\big ( \\Vert f_j \\Vert _{L^1} - \\Vert \\tilde{f}_j \\Vert _{L^1} \\big )\\ge \\delta -\\tilde{\\delta }\\ge \\tfrac{1}{2}\\delta ,$ it follows that $ \\max _j \\Vert g_j^\\dagger - {\\mathbf {1}}_{{\\mathcal {B}}_j} \\Vert _{L^1}= \\max _j \\Vert f_j^\\dagger \\Vert _{L^1} \\ge \\tfrac{1}{2}\\delta .$ In this case, we use the alternative bound ${\\mathcal {T}}(\\mathbf {g}) \\le {\\mathcal {T}}(\\mathbf {g}^\\dagger )$ from (REF ).",
"Thus it suffices to prove that $ {\\mathcal {T}}(\\mathbf {g}^\\dagger )\\le {\\mathcal {T}}_{\\mathbb {T}}(\\mathbf {g}^{\\star \\star }) - c\\max _j \\Vert g_j^\\dagger -{\\mathbf {1}}_{{\\mathcal {B}}_j} \\Vert _{L^1}^2,$ that is, to establish the conclusion of Lemma REF for $\\mathbf {g}^\\dagger $ .",
"The modified triple $\\mathbf {g}^\\dagger $ satisfies all hypotheses of the lemma, and enjoys the supplementary property that $g_j^\\dagger - {\\mathbf {1}}_{{\\mathcal {B}}_j} \\equiv 0$ whenever $\\big |\\,|\\phi (x)|-\\tfrac{1}{2} \\mu ({\\mathcal {B}}_j) \\,\\big | \\ge \\lambda \\delta $ .",
"Moreover, $\\tfrac{1}{2} \\operatorname{distance}\\,(\\mathbf {g},{\\mathcal {O}}({\\mathbf {B}}))\\le \\operatorname{distance}\\,(\\mathbf {g}^\\dagger ,{\\mathcal {O}}({\\mathbf {B}}))\\le \\tfrac{3}{2} \\operatorname{distance}\\,(\\mathbf {g},{\\mathcal {O}}({\\mathbf {B}}))$ by the triangle inequality for $L^1(G)$ norms, since $\\tilde{\\delta }\\le \\tfrac{1}{2} \\delta $ .",
"Therefore we have reduced matters to proving Lemma REF under the supplementary hypothesis that for every $j\\in \\lbrace 1,2,3\\rbrace $ , $ g_j- {\\mathbf {1}}_{{\\mathcal {B}}_j} \\equiv 0 \\text{ whenever }\\big |\\,|\\phi (x)|-\\tfrac{1}{2} \\mu ({\\mathcal {B}}_j) \\,\\big | \\ge C_0 \\delta .$ Here $C_0$ is some universal constant that is not at our disposal, but is dictated by our choice of $\\lambda $ .",
"For the remainder of the proof of Lemma REF we drop the superscripts $\\dagger $ , denoting by $\\mathbf {g}$ an ordered triple of functions that satisfies the hypotheses of the lemma, as well as (REF ) for $\\delta $ and ${\\mathbf {B}}$ such that $\\max _j\\Vert g_j-{\\mathbf {1}}_{{\\mathcal {B}}_j}\\Vert _1\\sim \\delta $ .",
"Redefine $f_j = g_j-{\\mathbf {1}}_{{\\mathcal {B}}_j}$ .",
"The perturbative term $f_j$ satisfies (REF ), that is, is supported where $\\big |\\,|\\phi (x)|-\\tfrac{1}{2} \\mu ({\\mathcal {B}}_j)\\,\\big | \\le C_0\\delta $ .",
"We claim that if $\\varepsilon _0$ is a sufficiently small constant multiple of $\\eta \\max _k \\mu ({\\mathcal {B}}_k)$ , and if $0 < \\delta \\le \\varepsilon _0$ , then this restriction on the support of $f_j$ ensures that $ {\\mathcal {T}}(f_1,f_2,f_3)=0.",
"$ Indeed, $f_1*f_2$ is supported where $\\phi $ differs by at most $2C_0 \\delta $ from some quantity $(\\pm \\tfrac{1}{2} \\mu ({\\mathcal {B}}_1)\\pm \\tfrac{1}{2} \\mu ({\\mathcal {B}}_2))$ , while $f_3$ is supported where $\\phi $ differs by at most $C_0 \\delta $ from $\\pm \\tfrac{1}{2} \\mu ({\\mathcal {B}}_3)$ .",
"The upper bound on $\\mu ({\\mathcal {B}}_1)+\\mu ({\\mathcal {B}}_2)+\\mu ({\\mathcal {B}}_3)$ and the $\\eta $ –strict admissibility of ${\\mathbf {B}}$ ensure that $ \\eta \\max _j\\mu ({\\mathcal {B}}_j)\\le \\big | \\pm \\mu ({\\mathcal {B}}_1)\\pm \\mu ({\\mathcal {B}}_2) \\pm \\mu ({\\mathcal {B}}_3)\\big |\\le 2-\\eta ^{\\prime }$ for all eight choices of signs, yielding (REF ) by the triangle inequality since $\\delta \\le \\varepsilon _0$ is assumed to be small relative to $\\eta \\max _k\\mu ({\\mathcal {B}}_k)$ .",
"For any $\\mathbf {y}=(y_1,y_2,y_3)\\in {\\mathbb {T}}^3$ satisfying $y_1+y_2+y_3=0$ , these constructions can be applied to the triple $\\mathbf {g}^{\\mathbf {y}}$ defined by replacing $g_j(x)$ by the translated function $g_j^{y_j}(x) = g_j(x-y_j)$ .",
"Then ${\\mathcal {T}}(\\mathbf {g})={\\mathcal {T}}(\\mathbf {g}^\\mathbf {y})$ , and $\\int g_j^{y_j}\\,d\\mu = \\int g_j\\,d\\mu $ .",
"Assume that $|\\phi (y_j)| =O(\\delta )$ for all three indices $j$ .",
"Then $ \\max _j \\Vert g_j^{y_j}-{\\mathbf {1}}_{{\\mathcal {B}}_j} \\Vert _{L^1}\\le \\max _j \\Vert g_j^{y_j}-{\\mathbf {1}}_{{\\mathcal {B}}_j^{y_j}} \\Vert _{L^1}+ \\max _j \\Vert {\\mathbf {1}}_{{\\mathcal {B}}_j^{y_j}}- {\\mathbf {1}}_{{\\mathcal {B}}_j} \\Vert _{L^1}= O(\\delta ).$ On the other hand, $ \\max _j \\Vert g_j^{y_j}-{\\mathbf {1}}_{{\\mathcal {B}}_j} \\Vert _{L^1} \\ge \\operatorname{distance}\\,(\\mathbf {g}^{\\mathbf {y}},{\\mathcal {O}}({\\mathbf {B}}))= \\operatorname{distance}\\,(\\mathbf {g},{\\mathcal {O}}({\\mathbf {B}}))\\ge c\\delta $ by $\\mathbf {y}$ –translation invariance of the orbit and translation invariance of $\\mu $ .",
"Each translated function $g_j^{y_j}-{\\mathbf {1}}_{{\\mathcal {B}}_j}$ remains supported in $\\lbrace x: \\big |\\,|\\phi (x)|-\\mu (E_j)/2\\,\\big | )\\le O(\\delta ) \\rbrace $ .",
"Each $f_j=g_j-{\\mathbf {1}}_{{\\mathcal {B}}_j}$ has a unique additive decomposition $f_j = f_j^+ + f_j^-$ , with $f_j^\\pm $ supported where $\\big |\\,\\phi (x) \\mp \\tfrac{1}{2} \\mu (B_j)\\,\\big |=O(\\delta )$ , respectively.",
"It will be advantageous to work instead with $\\mathbf {g}^\\mathbf {y}$ , with $\\mathbf {y}$ chosen so that the summands corresponding to $g_j^{y_j}-{\\mathbf {1}}_{{\\mathcal {B}}_j}$ satisfy certain vanishing properties which the summands $f_j^\\pm $ potentially lack.",
"In particular, define functions $f_{j,y_j}^\\pm $ by first setting $f_{j,y_j} = g_j^{y_j}-{\\mathbf {1}}_{{\\mathcal {B}}_j}$ , and then expressing $f_{j,y_j} = f_{j,y_j}^+ + f_{j,y_j}^-$ , with $f_{j,y_j}^\\pm $ supported where $|\\phi (x)\\mp \\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)| = O(\\delta )$ .",
"Lemma 8.4 For each index $j$ , there exists $y_j\\in G$ satisfying $|\\phi (y_j)| \\le C_0\\delta $ and $ \\int f_{j,y_j}^+\\,d\\mu = \\int f_{j,y_j}^-\\,d\\mu =0.",
"$ $f_{j,y}^+$ is that portion of $g_j^{y}-{\\mathbf {1}}_{{\\mathcal {B}}_j}$ that is supported where $|\\phi (x)-\\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)|$ is small.",
"Since $|\\phi (y)| \\le C_0\\delta $ and $g_j(x)={\\mathbf {1}}_{{\\mathcal {B}}_j}(x)$ wherever $|\\phi (x)-\\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)|>C_0\\delta $ , $f_{j,y}^+$ is supported where $|\\phi (x)-\\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)|\\le 2C_0\\delta $ .",
"Consider the function that maps $z\\in [-C_0\\delta ,C_0\\delta ]$ to $ \\int f_{j,y}^+(x)\\,d\\mu (x)= \\int _{|\\phi (x)-\\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)| \\le 2C_0\\delta }\\big (g_{j}^y - {\\mathbf {1}}_{{\\mathcal {B}}_j}\\big )(x)\\,d\\mu (x),$ with $y= y(z)$ satisfying $\\phi (y)=z$ .",
"While $y$ is not uniquely determined by $z$ via this equation, the integral nonetheless depends only on $z$ .",
"Indeed, the contribution of the term ${\\mathbf {1}}_{{\\mathcal {B}}_j}$ to the integral does not involve $y$ .",
"Substituting $x=u+y$ allows us to rewrite the contribution of $g_{j}^y(x)=g_j(x-y)$ as $ \\int _{|\\phi (x)- \\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)|\\le 2C_0\\delta } g_{j}(x-y)\\,d\\mu (x)= \\int _{|\\phi (u)+z - \\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)| \\le 2C_0\\delta } g_{j}(u)\\,d\\mu (u)$ which likewise depends on $z$ alone.",
"This function of $z$ is nonnegative when $z=C_0\\delta $ .",
"Indeed, if $\\phi (x) \\in [\\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)-2C_0\\delta ,\\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)]$ then $g_j(x-y)=1$ , since $\\phi _j(x-y) = \\phi _j(x)-C_0\\delta \\le \\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)-C_0\\delta $ and (by virtue of the reduction to the case $g_j=g_j^\\dagger $ made above) $g_j(u)\\equiv 1$ when $\\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)-O(\\delta ) \\le \\phi (u)\\le \\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)-C_0\\delta $ .",
"Thus $g_{j}^y(x)-{\\mathbf {1}}_{{\\mathcal {B}}_j}(x)=1-1=0$ for these values of $x$ .",
"On the other hand, if $\\phi (x) \\in [\\tfrac{1}{2}\\mu ({\\mathcal {B}}_j),\\tfrac{1}{2}\\mu ({\\mathcal {B}}_j)+2C_0\\delta ]$ then ${\\mathbf {1}}_{{\\mathcal {B}}_j}(x)=0$ , so $g_j^y(x)-{\\mathbf {1}}_{{\\mathcal {B}}_j}(x)\\ge 0$ .",
"The same reasoning shows that this function of $z$ is nonpositive when $z=-C_0\\delta $ .",
"Therefore we may apply the Intermediate Value Theorem on $[-C_0\\delta ,C_0\\delta ]$ to conclude that there exists $y_j$ with $\\phi (y_j)=z\\in [-C_0\\delta ,C_0\\delta ]$ satisfying $\\int f_{j,y_j}^+\\,d\\mu =0$ .",
"It follows at once that $\\int f_{j,y_j}^-\\,d\\mu = \\int f_{j,y_j}\\,d\\mu -\\int f_{j,y_j}^+\\,d\\mu =0$ .",
"Choose $y_1,y_2$ to ensure (REF ) for $j=1,2$ , but then define $y_3$ by $y_1+y_2+y_3=0$ .",
"With such a choice of $\\mathbf {y}$ fixed henceforth, simplify notation by suppressing $y_j$ and writing again $g_j,f_j,f_j^\\pm $ , continuing to use the notation $\\mathbf {g}$ for this modified triple.",
"The quantities ${\\mathcal {T}}(\\mathbf {g})$ and $\\operatorname{distance}\\,(\\mathbf {g},{\\mathcal {O}}({\\mathbf {B}}))$ are unchanged.",
"The functions $f_3^\\pm $ need not have vanishing integrals.",
"Nonetheless, $ \\int f_i^\\pm *f_j^\\pm \\,d\\mu =0\\ \\text{ for any distinct indices $i\\ne j\\in \\lbrace 1,2,3\\rbrace $,}$ for all four possible choices of $\\pm $ signs, since $\\int (f_i^\\pm *f_j^\\pm )\\,d\\mu = \\int f_i^\\pm \\,d\\mu \\,\\cdot \\,\\int f_j^\\pm \\,d\\mu $ and at least one of the two indices $i,j$ must belong to $\\lbrace 1,2\\rbrace $ .",
"Expand ${\\mathcal {T}}(\\mathbf {g}) = {\\mathcal {T}}({\\mathbf {1}}_{{\\mathcal {B}}_j}+f_j: j\\in \\lbrace 1,2,3\\rbrace )$ into eight terms, using the multilinearity of ${\\mathcal {T}}$ .",
"The simplest term is ${\\mathcal {T}}(f_1,f_2,f_3)$ .",
"Provided that $\\delta $ is sufficiently small relative to $\\max _j \\mu ({\\mathcal {B}}_j)$ , with constant of proportionality depending on $\\eta ,\\eta ^{\\prime }$ , this term vanishes for the modified triple $\\mathbf {g}$ , just as it was shown in (REF ) to vanish for the original triple.",
"The vanishing of ${\\mathcal {T}}(f_1,f_2,f_3)$ simplifies the expansion of ${\\mathcal {T}}(\\mathbf {g})$ to $ {\\mathcal {T}}(\\mathbf {g})={\\mathcal {T}}({\\mathbf {B}})+ \\sum _{k=1}^3 \\langle K_k,f_k\\rangle + \\sum _{i<j} \\langle {\\mathbf {1}}_{{\\mathcal {B}}_l},\\,f_i*f_j\\rangle .$ In the final sum, $i<j\\in \\lbrace 1,2,3\\rbrace $ and $l$ is defined by $\\lbrace 1,2,3\\rbrace = \\lbrace i,j,l\\rbrace $ .",
"We next discuss the terms $\\langle K_k,f_k\\rangle = -\\int |f_k(x)|\\,\\big |\\,|\\phi (x)|- \\tfrac{1}{2} \\mu ({\\mathcal {B}}_k)\\,\\big |\\,d\\mu (x) \\le 0.$ There exist an absolute constant $c_0>0$ and $n\\in \\lbrace 1,2,3\\rbrace $ such that $ \\Vert f_n \\Vert _{L^1}\\ge c_0\\delta $ .",
"Let $c_1 = \\tfrac{1}{8} c_0$ .",
"Because $ \\Vert f_n \\Vert _{L^\\infty } \\le 1$ and $ \\mu (\\lbrace x\\in G: \\big |\\,|\\phi (x)|-\\tfrac{1}{2} \\mu ({\\mathcal {B}}_n)\\,\\big |\\le c_1\\delta \\rbrace )= 4c_1\\delta ,$ necessarily $\\int _{ |\\,|\\phi (x)|-\\tfrac{1}{2} \\mu ({\\mathcal {B}}_n)\\,|\\ge c_1\\delta }|f_n|\\,d\\mu \\ge \\Vert f_n \\Vert _{L^1}-4c_1\\delta \\ge \\tfrac{1}{2} c_0\\delta .$ Therefore $\\langle K_n,f_n\\rangle \\le - \\int _{ |\\,|\\phi (x)|-\\tfrac{1}{2} \\mu ({\\mathcal {B}}_n)\\,|\\ge c_1\\delta }|f_n(x)| \\cdot \\big |\\,|\\phi (x)|- \\tfrac{1}{2} \\mu ({\\mathcal {B}}_n)\\,\\big |\\,d\\mu (x)\\le -c_1\\delta \\cdot \\tfrac{1}{2} c_0\\delta ,$ which is comparable to $\\max _j \\Vert f_j \\Vert _{L^1}^2$ and therefore to $\\operatorname{distance}\\,(\\mathbf {g},{\\mathcal {O}}({\\mathbf {B}}))^2$ .",
"Thus $ \\sum _k \\langle K_k,f_k\\rangle \\le -c^{\\prime } \\delta ^2.",
"$ To complete the proof, we next show that $ \\langle {\\mathbf {1}}_{{\\mathcal {B}}_l},f_i*f_j\\rangle =0\\ \\text{ for any three distinct indices $i,j,l$.}",
"$ For any of the four possible choices of $\\pm $ signs, the support of the convolution $f_i^\\pm *f_j^\\pm $ is contained in the sum of the supports of the two factors, hence consists of points $x$ at which $\\phi (x)= \\tfrac{1}{2} (\\pm \\mu (B_i) \\pm \\mu (B_j)) + O(\\delta )$ .",
"On the other hand, ${\\mathcal {B}}_l$ is the set of $x$ satisfying $|\\phi (x)|\\le \\tfrac{1}{2}\\mu ({\\mathcal {B}}_l)$ , and the $\\eta $ –strict admissibility hypothesis says that $|\\pm \\mu ({\\mathcal {B}}_l)\\pm \\mu ({\\mathcal {B}}_i)\\pm \\mu ({\\mathcal {B}}_j)| \\ge c\\eta \\max _k \\mu ({\\mathcal {B}}_k).$ A hypothesis of Lemma REF is that $\\delta $ is small relative to $\\eta \\max _k \\mu ({\\mathcal {B}}_k)$ .",
"Therefore for any choice of $\\pm $ signs, the support of $f_i^\\pm *f_j^\\pm $ is either entirely contained in ${\\mathcal {B}}_l$ , or entirely contained in its complement.",
"Therefore in the integral $ \\langle {\\mathbf {1}}_{{\\mathcal {B}}_l},f_i^\\pm *f_j^\\pm \\rangle = \\int {\\mathbf {1}}_{{\\mathcal {B}}_l}\\cdot (f_i^\\pm *f_j^\\pm )\\,d\\mu ,$ the factor ${\\mathbf {1}}_{{\\mathcal {B}}_1}$ is constant.",
"Since $\\int f_i^\\pm *f_j^\\pm \\,d\\mu =0$ by (REF ), this integral vanishes.",
"Summing over all four possible choices of signs gives (REF ).",
"Inserting these results into the expansion (REF ), we conclude that when the supplementary hypothesis (REF ) is satisfied, ${\\mathcal {T}}(\\mathbf {g}) \\le {\\mathcal {T}}(\\mathbf {g}^{\\star \\star })-c\\delta ^2$ , that is, $ {\\mathcal {T}}(\\mathbf {g}) \\le {\\mathcal {T}}(\\mathbf {g}^{\\star \\star })-c\\operatorname{distance}\\,({\\mathbf {E}},{\\mathcal {O}}({\\mathbf {B}}))^2, $ as was to be shown."
],
[
"The perturbative regime for sumsets",
"In this section we prove Theorem REF , the quantitative stability result for the inequality $\\mu _*(A+B)\\ge \\min (\\mu (A)+\\mu (B),\\mu (G))$ .",
"We begin with a small lemma needed in the analysis.",
"Lemma 9.1 Let $K$ be a compact Abelian group with Haar measure $\\nu $ .",
"Let $A,B\\subset K$ be compact.",
"Suppose that $B\\ne \\emptyset $ and that $\\nu (A)>\\tfrac{1}{2}\\nu (K)$ .",
"Then $\\nu (A+B) \\ge \\min (\\nu (B) + \\tfrac{1}{2}\\nu (A),\\nu (K)).$ $K$ is not assumed to be connected.",
"The conclusion is false in general, without the hypothesis that $\\nu (A)>\\tfrac{1}{2}\\mu (K)$ .",
"It fails, for instance, if there exists a subgroup $H$ of $K$ satisfying $\\nu (H)=\\tfrac{1}{2}\\nu (K)$ and $A=B=H$ .",
"According to a theorem of Kneser [18], either $\\nu (A+B)\\ge \\nu (A)+\\nu (B)$ or there exists a subgroup $H$ of $K$ of positive Haar measure satisfying $A+B+H=A+B$ and $\\nu (A+B) = \\nu (A+H)+\\nu (B+H)-\\nu (H)$ .",
"In the first case, the conclusion of the lemma holds.",
"In the second case, if $\\nu (H)=\\nu (K)$ then $\\nu (A+H)+\\nu (B+H)-\\nu (H) = \\nu (K)+\\nu (K)-\\nu (K)=\\nu (K)$ and again the conclusion holds.",
"Now, suppose that $\\nu (H)<\\nu (K)$ .",
"$A+H$ is a union of cosets of $H$ .",
"It cannot be a single coset, for $\\nu (H)<\\nu (K)$ implies $\\nu (H)\\le \\tfrac{1}{2}\\nu (K)<\\nu (A)$ .",
"Therefore $A+H$ is a union of at least two cosets of $H$ , so $\\nu (A+H) \\ge 2\\nu (H)$ , so $ \\nu (A+H)-\\nu (H)\\ge \\tfrac{1}{2}\\nu (A+H) \\ge \\tfrac{1}{2} \\nu (A).$ Thus $\\nu (A+B) = \\nu (A+H) -\\nu (H) + \\nu (B+H)\\ge \\tfrac{1}{2}\\nu (A) + \\nu (B),$ as was to be shown.",
"Let $G$ be a compact Abelian group with Haar measure $\\mu $ , satisfying $\\mu (G)=1$ .",
"Let $|{\\mathcal {A}}|$ denote the Lebesgue measure of any set ${\\mathcal {A}}\\subset {\\mathbb {T}}$ .",
"Proposition 9.2 There exists $\\delta _0>0$ with the following property.",
"Let $A,B\\subset G$ be compact sets of positive measures satisfying $\\mu (A)+\\mu (B)\\le 1-200\\delta _0\\min (\\mu (A),\\mu (B)).$ Suppose that $ \\Vert \\phi (x) \\Vert _{\\mathbb {T}}&\\le \\tfrac{1}{2}\\mu (A)+\\delta _0\\min (\\mu (A),\\mu (B))\\ \\text{ for all $x\\in A$},\\\\ \\Vert \\phi (x) \\Vert _{\\mathbb {T}}&\\le \\tfrac{1}{2}\\mu (B)+\\delta _0\\min (\\mu (A),\\mu (B))\\ \\text{ for all $x\\in B$},\\\\\\mu (A+B) &\\le \\mu (A)+\\mu (B) +\\delta \\min (\\mu (A),\\mu (B))\\text{ for some }0<\\delta \\le \\delta _0.$ Then $\\phi (A)$ is contained in some interval in ${\\mathbb {T}}$ of length $\\mu (A)+100\\delta \\min (\\mu (A),\\mu (B))$ .",
"Likewise for $B$ .",
"Define ${\\mathcal {A}}=\\phi (A)$ and ${\\mathcal {B}}= \\phi (B)$ in ${\\mathbb {T}}$ .",
"For $t\\in {\\mathbb {T}}$ define $ A_t = \\lbrace x\\in A: \\phi (x)=t\\rbrace \\subset A\\subset G.$ $A_t$ will be regarded sometimes as a subset of a coset of $K=\\operatorname{Kernel}(\\phi )$ , and sometimes as a subset of $K$ itself (by translating by any appropriate element of $G$ ).",
"Likewise define $B_t\\subset B$ .",
"Let $\\nu $ be Haar measure on $H=\\operatorname{Kernel}(\\phi )$ , normalized to satisfy $\\nu (H)=1$ .",
"Each slice $\\phi ^{-1}(\\lbrace t\\rbrace )\\subset G$ is a coset of $H$ .",
"By translation, $\\nu $ also defines a measure on each such coset, which will also be denoted by $\\nu $ .",
"Thus we may write $\\nu (A_t)$ , even though there is no canonical identification of $A_t$ with a subset of $H$ .",
"The hypotheses allow us to regard $\\phi $ as a mapping from $A+B$ to ${\\mathbb {R}}$ , rather than to ${\\mathbb {T}}$ .",
"Indeed, denoting $\\eta :=\\delta _0\\min (\\mu (A),\\mu (B))$ , each element of $\\phi (a)\\in \\phi (A)$ is represented by some element $\\tilde{\\phi }(a)\\in [-\\tfrac{1}{2}\\mu (A)-\\eta ,\\tfrac{1}{2}\\mu (A)+\\eta ]$ , and correspondingly for $\\phi (B)$ .",
"Therefore, for any $a\\in A$ and $b\\in B$ , $\\phi (a+b)$ is represented by some element $\\tilde{\\phi }(a+b)\\in (-\\tfrac{1}{2},\\tfrac{1}{2})$ .",
"These satisfy $\\tilde{\\phi }(a+b)=\\tilde{\\phi }(a)+\\tilde{\\phi }(b)$ , where addition on the right-hand side is performed in ${\\mathbb {R}}$ rather than in ${\\mathbb {T}}$ .",
"These three mappings, all denoted by the common symbol $\\tilde{\\phi }$ , are measure-preserving bijections.",
"Define $\\rho _A,\\rho _B\\in [0,1)$ by $ 1- \\rho _A = \\sup _t \\nu (A_t) \\text{ and } 1- \\rho _B = \\sup _s \\nu (B_s).$ The hypothesis $ \\Vert \\phi (x) \\Vert _{\\mathbb {T}}\\le \\tfrac{1}{2}\\mu (A)+\\delta _0\\min (\\mu (A),\\mu (B))$ for $x\\in A$ implies that $|{\\mathcal {A}}| \\le (1+2\\delta _0)\\mu (A)$ .",
"On the other hand, $\\mu (A)\\le (1-\\rho _A)|{\\mathcal {A}}|$ .",
"Therefore $\\rho _A \\le 1-(1+2\\delta _0)^{-1}$ .",
"Thus if $\\delta _0$ is sufficiently small then $\\rho _A < \\tfrac{1}{4}$ .",
"Likewise $\\rho _B < \\tfrac{1}{4}$ .",
"Therefore $\\min (1-2\\rho _A,\\rho _B)=\\rho _B$ .",
"This relation will be used momentarily.",
"Let $\\varepsilon \\in (0,\\rho _A)$ be sufficiently small so that $1-\\rho _A-\\epsilon >\\tfrac{3}{4}$ , and choose $\\tau \\in {\\mathcal {A}}$ satisfying $ \\nu (A_\\tau )>1-\\rho _A-\\varepsilon .$ Set $A_-=\\lbrace a\\in A: \\phi (a)<\\tau \\rbrace \\text{ and } A_+=\\lbrace a\\in A: \\phi (a)>\\tau \\rbrace $ (where $\\mathcal {A}$ is seen as a subset of ${\\mathbb {R}}$ rather than of ${\\mathbb {T}}$ ).",
"Regarding ${\\mathcal {B}}$ as a subset of ${\\mathbb {R}}$ , let $b_-,b_+\\in {\\mathbb {R}}$ be its minimum and maximum elements, respectively.",
"Now $A+B \\supset (A_\\tau +B) + (A_- +B_{b_-}) + (A_+ + B_{b_+})$ and these three sets are pairwise disjoint.",
"Therefore $\\mu (A+B) \\ge \\mu (A_\\tau +B) + \\mu (A_- +B_{b_-}) + \\mu (A_+ + B_{b_+}).$ $A_- + B_{b_-}$ contains a translate of $A_-$ , so $\\mu (A_-+B_{b_-}) \\ge \\mu (A_-)$ .",
"Likewise $\\mu (A_+ + B_{b_+})\\ge \\mu (A_+)$ .",
"Therefore $ \\mu (A+B) \\ge \\mu (A_\\tau +B) + \\mu (A).",
"$ One application of (REF ) is the relation $ \\max (\\rho _A,\\rho _B)\\le \\delta .",
"$ To prepare for its proof recall that according to Lemma REF , $ \\nu (A_\\tau +B_t)\\ge \\min \\big (\\tfrac{1}{2}\\nu (A_\\tau ) + \\nu (B_t),1\\big )\\ge \\min \\big (\\nu (B_t) + \\tfrac{3}{8},1\\big )$ for any $t\\in \\phi (B)$ , since $\\nu (A_\\tau ) > \\tfrac{3}{4}$ .",
"Therefore $\\mu (A_\\tau +B)= \\int _{\\mathcal {B}}\\nu (A_\\tau +B_t)\\,dt\\\\\\ge \\int _{\\mathcal {B}}\\min (\\nu (B_t)+\\tfrac{3}{8},1)\\,dt= \\mu (B) + \\int _{\\mathcal {B}}\\big [ \\min (\\tfrac{3}{8},1-\\nu (B_t))\\big ]\\,dt\\\\\\ge \\mu (B) + \\int _{\\mathcal {B}}\\big [ \\min (\\tfrac{3}{8},\\rho _B)\\big ]\\,dt= \\mu (B) + \\rho _B|{\\mathcal {B}}|$ since $\\rho _B<\\tfrac{1}{4}$ .",
"Since $|{\\mathcal {B}}|\\ge \\mu (B)$ , inserting this bound into (REF ) gives $ \\mu (A+B) \\ge \\mu (A)+\\mu (B) + \\rho _B\\mu (B).$ Since $\\mu (A+B)\\le \\mu (A)+\\mu (B)+\\delta \\mu (B)$ , we may conclude that $\\rho _B\\le \\delta $ .",
"The roles of $A,B$ can be interchanged, so $\\rho _A\\le \\delta $ also.",
"Let ${\\mathcal {A}}^{\\prime }=\\lbrace t\\in {\\mathcal {A}}: \\nu (A_t)>\\tfrac{1}{2}\\rbrace $ .",
"Likewise define ${\\mathcal {B}}^{\\prime }\\subset {\\mathcal {B}}$ .",
"We claim that $ \\mu (A+B) \\ge \\mu (A)+\\mu (B) + (\\tfrac{1}{2}-\\rho _A) |{\\mathcal {B}}\\setminus {\\mathcal {B}}^{\\prime }|.$ The proof will use the fact that for any subsets $S,T$ of a compact group $H$ satisfying $\\mu (S)+\\mu (T)>\\mu (H)$ , the associated sumset $S+T$ is all of $H$ .",
"Connectivity of $H$ is not required for this conclusion; it is valid for the kernel $H$ of $\\phi $ .",
"Indeed, for any $z\\in H$ it holds that $\\lbrace z-x: x\\in S\\rbrace \\cap T\\ne \\emptyset $ , since the intersection of these sets has measure equal to $\\mu (S)+\\mu (T)-\\mu (H)>0$ .",
"To prove the claim, majorize $ \\mu (A_\\tau +B)\\ge \\int _{\\mathcal {B}}\\nu (A_\\tau +B_t) \\,dt.$ One has $\\nu (A_\\tau +B_t)\\ge \\nu (B_t)$ for all $t$ .",
"Moreover, if $\\nu (B_t) \\le \\tfrac{1}{2}$ then $\\nu (A_\\tau +B_t) \\ge \\nu (A_\\tau )\\ge 1-\\rho _A-\\varepsilon \\ge \\nu (B_t) + \\tfrac{1}{2}-\\rho _A-\\varepsilon .",
"$ Therefore $ \\mu (A_\\tau +B)\\ge \\int _{\\mathcal {B}}\\nu (B_t)\\,dt+ \\int _{{\\mathcal {B}}\\setminus {\\mathcal {B}}^{\\prime }} (\\tfrac{1}{2}-\\rho _A-\\varepsilon )\\,dt= \\mu (B) + (\\tfrac{1}{2}-\\rho _A-\\varepsilon ) |{\\mathcal {B}}\\setminus {\\mathcal {B}}^{\\prime }|.",
"$ Letting $\\varepsilon \\rightarrow 0$ and combining this with (REF ) gives (REF ).",
"$\\Box $ From (REF ) together with the hypothesis $\\mu (A+B)\\le \\mu (A)+\\mu (B)+\\delta \\min (\\mu (A),\\mu (B))$ and the bound $\\max (\\rho _A,\\rho _B)\\le \\delta $ we deduce that $|{\\mathcal {B}}\\setminus {\\mathcal {B}}^{\\prime }|\\le (2+O(\\delta ))\\delta \\min (\\mu (A),\\mu (B)).$ Since the roles of $A,B$ can be freely interchanged in this reasoning, $|{\\mathcal {A}}\\setminus {\\mathcal {A}}^{\\prime }|$ satisfies the same inequality.",
"For every $s\\in {\\mathcal {A}}^{\\prime }$ and $t\\in {\\mathcal {B}}^{\\prime }$ , $\\nu (A_s+B_t) \\ge \\min (\\nu (A_s)+\\nu (B_t),1) =1$ since $\\nu (A_s)>\\tfrac{1}{2}$ and likewise $\\nu (B_t)> \\tfrac{1}{2}$ .",
"Therefore $\\nu ((A+B)_x)=1$ for every $x\\in {\\mathcal {A}}^{\\prime }+{\\mathcal {B}}^{\\prime }$ .",
"Therefore $|{\\mathcal {A}}^{\\prime }+{\\mathcal {B}}^{\\prime }| \\le \\mu (A+B)$ , and consequently $ |{\\mathcal {A}}^{\\prime }+{\\mathcal {B}}^{\\prime }|&\\le \\mu (A)+\\mu (B)+\\delta \\min (\\mu (A),\\mu (B))\\\\&\\le |{\\mathcal {A}}|+|{\\mathcal {B}}|+\\delta \\min (\\mu (A),\\mu (B))\\\\&< |{\\mathcal {A}}^{\\prime }|+|{\\mathcal {B}}^{\\prime }|+ 6\\delta \\min (\\mu (A),\\mu (B)).$ On the other hand, $|{\\mathcal {A}}^{\\prime }|&\\ge |{\\mathcal {A}}| -(2+O(\\delta ))\\delta \\min (\\mu (A),\\mu (B))\\\\&\\ge \\mu (A) -(2+O(\\delta ))\\delta \\min (\\mu (A),\\mu (B))\\\\&> \\mu (A)-3\\delta \\min (\\mu (A),\\mu (B))$ and likewise for $|{\\mathcal {B}}^{\\prime }|$ .",
"A straightforward adaptation to ${\\mathbb {R}}$ (see [10]) of a theorem of Freĭman states that if $S,S^{\\prime }\\subset {\\mathbb {R}}$ are nonempty Lebesgue measurable sets satisfying $|S+S^{\\prime }|_* < |S|+|S^{\\prime }|+ \\min (|S|,|S^{\\prime }|)$ , then $S$ is contained in an interval of length $\\le |S+S^{\\prime }|-|S^{\\prime }|$ .",
"Regarding ${\\mathcal {A}}^{\\prime },{\\mathcal {B}}^{\\prime }$ as subsets of ${\\mathbb {R}}$ , as we may, this result allows us to conclude that if $\\delta _0$ is less than some absolute constant, then ${\\mathcal {A}}^{\\prime }$ is contained in an interval $I$ of length $\\le |{\\mathcal {A}}^{\\prime }|+(6+O(\\delta ))\\delta \\min (\\mu (A),\\mu (B))$ .",
"Similarly, ${\\mathcal {B}}^{\\prime }$ is contained in an interval $J$ of length $\\le |{\\mathcal {B}}^{\\prime }|+(6+O(\\delta ))\\delta \\min (\\mu (A),\\mu (B))$ .",
"The following claim completes the proof of Proposition REF .",
"Claim 9.1 The full sets ${\\mathcal {A}}=\\phi (A)$ and ${\\mathcal {B}}=\\phi (B)$ are contained in intervals of lengths $\\mu (A)+100\\delta \\min (\\mu (A),\\mu (B))$ and $\\mu (B)+100\\delta \\min (\\mu (A),\\mu (B))$ , respectively.",
"The reasoning in the following proof of this claim will be used again below.",
"Suppose that some point $z\\in {\\mathcal {A}}$ were to lie to the left of the left endpoint of $I$ by a distance $\\ge C_1\\delta \\min (\\mu (A),\\mu (B))$ .",
"If $y\\in {\\mathcal {B}}^{\\prime }$ lies within distance $C_1\\delta \\min (\\mu (A),\\mu (B))$ of the left endpoint of $J$ , then $A_z + B_y$ lies outside $I+J$ .",
"The set of all $y\\in {\\mathcal {B}}^{\\prime }$ with this property has Lebesgue measure $ \\ge |{\\mathcal {B}}^{\\prime }| - \\big (|J|-C_1\\delta \\min (\\mu (A),\\mu (B))\\big ) \\ge (C_1-6)\\delta \\min (\\mu (A),\\mu (B)).$ The sum of $A_z$ with the union of all such $B_y$ therefore has Haar measure $\\ge \\tfrac{1}{2} (C_1-6)\\delta \\min (\\mu (A),\\mu (B))$ .",
"This sumset is disjoint from $\\phi ^{-1}({\\mathcal {A}}^{\\prime }+{\\mathcal {B}}^{\\prime })= \\phi ^{-1}({\\mathcal {A}}^{\\prime }) + \\phi ^{-1}({\\mathcal {B}}^{\\prime })$ .",
"Therefore $ \\mu (A+B)&\\ge \\mu (\\phi ^{-1}({\\mathcal {A}}^{\\prime }))+ \\mu (\\phi ^{-1}({\\mathcal {B}}^{\\prime }))+ \\tfrac{1}{2} (C_1-6)\\delta \\min (\\mu (A),\\mu (B))\\\\&\\ge (\\mu (A)-|{\\mathcal {A}}\\setminus {\\mathcal {A}}^{\\prime }|)+ (\\mu (B)-|{\\mathcal {B}}\\setminus {\\mathcal {B}}^{\\prime }|)+ \\tfrac{1}{2} (C_1-6)\\delta \\min (\\mu (A),\\mu (B))\\\\&\\ge \\mu (A)+\\mu (B) + \\tfrac{1}{2} (C_1-18)\\delta \\min (\\mu (A),\\mu (B)).$ Choosing $C_1=21$ yields a contradiction for all sufficiently small $\\delta $ .",
"Thus ${\\mathcal {A}}=\\phi (A)$ is contained in an interval of length less than $|I|+ 40\\delta \\min (\\mu (A),\\mu (B))\\le |{\\mathcal {A}}^{\\prime }|+46\\delta \\min (\\mu (A),\\mu (B))\\le \\mu (A) + 100\\delta \\min (\\mu (A),\\mu (B)).$ Likewise for ${\\mathcal {B}}$ .",
"The conclusions of Proposition REF hold if $A,B$ satisfy the same hypotheses but are merely assumed to be measurable, rather than compact, except that the constant 200 is replaced by a sufficiently large finite constant ${\\mathbf {C}}$ .",
"To prove this, choose compact subsets $A^{\\prime },B^{\\prime }$ of $A,B$ whose Haar measures are nearly those of $A,B$ respectively, and invoke Proposition REF to obtain parallel rank one Bohr sets ${\\mathcal {B}}_{A^{\\prime }}\\supset A^{\\prime }$ and ${\\mathcal {B}}_{B^{\\prime }}\\supset B^{\\prime }$ satisfying $\\mu ({\\mathcal {B}}_{A^{\\prime }})\\le \\mu (A)+{\\mathbf {C}}\\delta \\min (\\mu (A),\\mu (B))$ with the corresponding bound for $\\mu ({\\mathcal {B}}_{B^{\\prime }})$ .",
"Then repeat the reasoning in the proof of the claim above to deduce that there exist slightly larger parallel rank one Bohr sets, associated to the same homomorphism $\\phi $ , which contain all of $A,B$ respectively, and whose measures satisfy the required upper bounds with a larger constant factor ${\\mathbf {C}}$ .",
"With a small modifications, the proof of Proposition REF establishes an extension: Setting $M = \\min (\\mu (A),\\mu (B))$ to simplify notation, the hypotheses that $ \\Vert \\phi (x) \\Vert _{\\mathbb {T}}\\le \\tfrac{1}{2}\\mu (A)+\\delta _0M$ for all $x\\in A$ and analogously for $B$ can be relaxed to $\\left\\lbrace \\begin{aligned}& \\Vert \\phi (x) \\Vert _{\\mathbb {T}}\\le \\tfrac{1}{2}\\mu (A)+\\delta _0M\\ \\text{ $\\forall \\, x\\in A$ outside a set of Haar measure $\\le \\delta _0M$}\\\\& \\Vert \\phi (x) \\Vert _{\\mathbb {T}}\\le \\tfrac{1}{2}\\mu (B)+\\delta _0M\\ \\text{ $\\forall \\,x\\in B$ outside a set of Haar measure $\\le \\delta _0M$,}\\end{aligned} \\right.",
"$ to conclude that, provided that $\\delta _0$ is sufficiently small, $\\phi (A)$ is contained in some interval in ${\\mathbb {T}}$ of length $\\mu (A)+{\\mathbf {C}}\\delta _0\\min (\\mu (A),\\mu (B))$ , and likewise for $\\phi (B)$ .",
"To prove this, define $A^{\\prime },B^{\\prime }$ to be the subsets of $A,B$ , respectively, specified by these inequalities.",
"We will use the proof of Claim REF to control $A,B$ in terms of $A^{\\prime },B^{\\prime }$ , demonstrating that $A,B$ satisfy the hypotheses of Proposition REF with $\\delta _0$ replaced by $\\varepsilon _0$ , where $\\varepsilon _0$ depends only on $\\delta _0$ and tends to zero as $\\delta _0\\rightarrow 0$ .",
"Thus the extension will be proved.",
"The only change to the reasoning in the proof of the claim is that we may no longer conclude that, in the notation of that discussion, $A_z+B_y$ is disjoint from $I+J=\\lbrace x\\in {\\mathbb {T}}: \\Vert x \\Vert _{\\mathbb {T}}\\le \\tfrac{1}{2}(\\mu (A)+\\mu (B))+O(\\delta _0)M\\rbrace $ whenever $\\phi (z)\\in [\\tfrac{1}{2}\\mu (A)+2{\\mathbf {C}}\\delta _0M,\\tfrac{1}{2}]$ and $\\phi (y)\\in [\\tfrac{1}{2}\\mu (B)-{\\mathbf {C}}\\delta _0M,\\tfrac{1}{2}\\mu (B)]$ .",
"Under the hypotheses of this extension, it is not permissible to regard $\\phi (A),\\phi (B)$ as subsets of ${\\mathbb {R}}$ , and the desired disjointness could fail due to periodicity.",
"Instead, we claim that if $C_1$ is a sufficiently large constant and $\\delta _0$ is sufficiently small then for any $x\\in A$ satisfying $ \\tfrac{1}{2} \\mu (A) + C_1\\delta _0 M\\le |\\phi (x) |\\le \\tfrac{1}{2},$ the set of all $y\\in B^{\\prime }$ satisfying $\\phi (x+y)\\notin I+J$ has Haar measure $\\ge C_2\\delta _0 M$ , where $C_2$ depends on $C_1$ but not on $\\delta _0$ , and $C_2\\rightarrow \\infty $ as $C_1\\rightarrow \\infty $ .",
"We may assume without loss of generality that $\\phi (x)\\in [0,\\tfrac{1}{2}]$ by replacing $(A,B)$ by $(-A,-B)$ if necessary.",
"The two desired conditions for $y$ are that $w=\\phi (y)$ should satisfy $ w\\ge \\tfrac{1}{2} \\mu (B)+\\tfrac{1}{2}\\mu (A)-\\phi (x) +O(\\delta _0 M)$ and $ w \\le 1-\\tfrac{1}{2}\\mu (A)-\\tfrac{1}{2}\\mu (B) -\\phi (x) -O(\\delta _0 M).$ The set of all $w\\in [-\\tfrac{1}{2}\\mu (B),\\tfrac{1}{2}\\mu (B)]$ that satisfy both inequalities has Lebesgue measure $\\ge {\\mathbf {C}}_1\\delta _0 M$ provided that $\\delta _0$ is sufficiently small.",
"The inverse image under $\\phi $ of this set of elements $w$ has Haar measure $\\ge {\\mathbf {C}}_1\\delta _0 M$ .",
"The complement of the intersection with $B$ of this inverse image has Haar measure $O(\\delta _0 M)$ , with the constant in the $O(\\cdot )$ notation independent of the choice of ${\\mathbf {C}}_1$ .",
"The result therefore follows.",
"This completes the proof of the extension of Proposition REF .",
"$\\Box $ Let $\\eta >0$ .",
"Let $A,B\\subset G$ satisfy $\\min (\\mu (A),\\mu (B))\\ge \\eta $ and $\\mu (A)+\\mu (B)\\le 1-\\eta $ .",
"Suppose that $\\mu _*(A+B)\\le \\mu (A)+\\mu (B)+\\delta \\min (\\mu (A),\\mu (B))$ .",
"By the same reasoning as the one used to extend the statement of Proposition REF to measurable sets, it suffices to treat the case in which $A,B$ are compact.",
"If $\\delta $ is sufficiently small, as a function of $\\eta $ alone, then the theorems of Tao [22] and/or Griesmer [17] can be applied.",
"The conclusion is that there exist parallel rank one Bohr sets ${\\mathcal {B}}_A,{\\mathcal {B}}_B$ such that $\\mu (A\\,\\Delta \\,{\\mathcal {B}}_A)\\le \\varepsilon (\\delta )\\min (\\mu (A),\\mu (B))$ and likewise for $\\mu (B\\,\\Delta \\,{\\mathcal {B}}_B)$ .",
"The quantity $\\varepsilon (\\delta )$ tends to 0 as $\\delta \\rightarrow 0$ , provided that $\\eta $ remains fixed.",
"The reasoning in the proof of the claim above now shows that the full sets $A,B$ are contained in parallel rank one Bohr sets ${\\mathcal {B}}^\\sharp _A,{\\mathcal {B}}^\\sharp _B$ , respectively, satisfying $\\mu ({\\mathcal {B}}^\\sharp _A)\\le \\mu (A)+\\varepsilon ^\\sharp \\min (\\mu (A),\\mu (B))$ where $\\varepsilon ^\\sharp \\rightarrow 0$ as $\\delta \\rightarrow 0$ .",
"Likewise for $B,{\\mathcal {B}}^\\sharp _B$ .",
"This is not the desired conclusion, since it includes no quantitative bound for the dependence of $\\varepsilon ^\\sharp $ on $\\delta $ .",
"However, since $\\varepsilon ^\\sharp \\rightarrow 0$ as $\\delta \\rightarrow 0$ , it follows that if $\\delta $ is sufficiently small then the pair $(A,B)$ satisfies the hypotheses of Proposition REF .",
"Invoking that proposition completes the proof of the theorem."
],
[
"A special case on ${\\mathbb {T}}$",
"In this section, we discuss our functionals for $G={\\mathbb {T}}$ , in the special situation in which one of the sets is an interval.",
"In particular, our next result ensures that, if $(A,B,C)$ satisfies near equality in the Riesz-Sobolev inequality on $\\mathbb {T}$ , and $C$ is an interval, then $A$ and $B$ are nearly intervals.",
"When discussing the special case $G={\\mathbb {T}}$ , we will often use $|E|$ , rather than $m(E)$ , to denote the Lebesgue measure of $E$ .",
"Proposition 10.1 Let $\\eta >0$ .",
"There exists a constant ${\\mathbf {C}}<\\infty $ , depending only on $\\eta $ , with the following property.",
"Let $(A,B,C)$ be an $\\eta $ –strictly admissible and $\\eta $ –bounded triple of measurable subsets of ${\\mathbb {T}}$ .",
"Suppose that $C$ is an interval with center $x_C$ .",
"Then $\\inf _{x+y=x_C}\\big (|A\\,\\Delta \\,(A^\\star +x)|+|B\\,\\Delta \\,(B^\\star +y)|\\big )\\le {\\mathbf {C}}{\\mathcal {D}}(A,B,C)^{1/2}.$ We outline here a proof based on a method relying on reflection symmetry and a two-point inequality of Baernstein and Taylor [5].",
"This technique does not otherwise appear in this paper.",
"It is also used by O'Neill [21] to analyze the corresponding issue for the sphere $S^d$ , $d\\ge 2$ .",
"The proof will consist of three steps.",
"Step 1.",
"If ${\\mathcal {D}}(A,B,C)=0$ and $C$ is an interval, then $A,B$ differ from intervals by Lebesgue null sets, and these three intervals are compatibly centered.",
"Assume without loss of generality that $C$ is centered at 0.",
"Thus $C=C^*$ .",
"By the complementation principle described in §, it may also be assumed that $m(A)\\le \\tfrac{1}{2}$ , $m(B)\\le \\tfrac{1}{2}$ .",
"Identify ${\\mathbb {T}}$ with the unit circle in ${\\mathbb {C}}\\leftrightarrow {\\mathbb {R}}^2$ via the mapping $x\\mapsto e^{2\\pi i (x+ \\tfrac{\\pi }{2})}$ .",
"For each $x=(x_1,x_2)\\in {\\mathbb {T}}$ let $R(x)=(x_1,-x_2)$ be the reflection of $x$ about the horizontal axis.",
"To any $E\\subset {\\mathbb {T}}$ associate $E^\\sharp \\subset {\\mathbb {T}}$ , defined as follows.",
"For each pair of points $\\lbrace x,R(x)\\rbrace $ with $x=(x_1,x_2)$ with $x_2\\ne 0$ , let $x_+=(x_1,|x_2|)$ and $x_-=(x_1,-|x_2|)$ .",
"If both $x_+,x_-\\in E$ then both $x_+,x_-\\in E^\\sharp $ ; if neither belongs to $E$ then neither belongs to $E^\\sharp $ ; and if exactly one belongs to $E$ then $x_+\\in E^\\sharp $ and $x_-\\notin E^\\sharp $ .",
"If $x_2=0$ then $x\\in E^\\sharp $ if and only if $x\\in E$ .",
"Define ${\\mathbb {T}}_+=\\lbrace x=(x_1,x_2)\\in {\\mathbb {T}}: x_2>0\\rbrace $ .",
"For $y\\in {\\mathbb {T}}$ define $R_y E = (E+y)^\\sharp $ , where addition is in the additive group ${\\mathbb {T}}={\\mathbb {R}}/{\\mathbb {Z}}$ .",
"Assume without loss of generality that the interval $C$ is centered at 0.",
"The following hold: for any measurable sets $A,B\\subset {\\mathbb {T}}$ , $m(A^\\sharp )=m(A),\\; m(B^\\sharp )=m(B),\\\\m(A^\\sharp \\,\\Delta \\,B^\\sharp ) \\le m(A \\,\\Delta \\,B), \\\\\\langle {\\mathbf {1}}_{A^\\sharp } * {\\mathbf {1}}_C,{\\mathbf {1}}_{B^\\sharp }\\rangle \\ge \\langle {\\mathbf {1}}_{A} * {\\mathbf {1}}_C,{\\mathbf {1}}_{B}\\rangle .",
"\\qquad \\mathrm {(a,b,c)}$ Consequently the above conclusions hold with $A^\\sharp ,B^\\sharp $ replaced by $R_yA, R_yB$ , respectively, for any $y\\in {\\mathbb {T}}$ .",
"(a) and (b) are direct consequences of the definition of the $\\sharp $ operation, while (c) is an almost equally direct consequence [5].",
"Observe that $\\langle {\\mathbf {1}}_{A^\\sharp } * {\\mathbf {1}}_C,{\\mathbf {1}}_{B^\\sharp }\\rangle \\ > \\ \\langle {\\mathbf {1}}_{A} * {\\mathbf {1}}_C,{\\mathbf {1}}_{B}\\rangle \\qquad \\mathrm {(d)}$ if the set of all points $(x_+,y_+)\\in {\\mathbb {T}}_+^2$ satisfying $x_+\\in A$ , $x_-\\notin A$ , $y_+\\notin B$ , $y_-\\in B$ and $ \\Vert x_+-y_+ \\Vert _{\\mathbb {T}}< \\tfrac{1}{2}m(C)$ and $ \\Vert x_+-y_- \\Vert _{\\mathbb {T}}> \\tfrac{1}{2}m(C)$ has positive Lebesgue measure in ${\\mathbb {T}}^2$ .",
"The same holds if the set of all points $(x_+,y_+)\\in {\\mathbb {T}}_+^2$ satisfying $x_+\\notin A$ , $x_-\\in A$ , $y_+\\in B$ , $y_-\\notin B$ and the above two inequalities has positive Lebesgue measure in ${\\mathbb {T}}^2$ .",
"Moreover, if $A\\subset {\\mathbb {T}}$ is a finite union of closed intervals then there exists a finite sequence $y_1,\\dots ,y_N$ of elements of ${\\mathbb {T}}$ such that $R_{y_N}R_{y_{N-1}}\\cdots R_1A = A^\\star .",
"\\qquad \\mathrm {(e)}$ This is elementary, and its proof is left to the reader.",
"If $A\\subset {\\mathbb {T}}$ is Lebesgue measurable then there exists an infinite sequence $y_n\\in {\\mathbb {T}}$ such that $\\lim _{N\\rightarrow \\infty } m\\big (R_{y_N}R_{y_{N_1}}\\cdots R_1 A\\,\\Delta \\,A^\\star \\big ) = 0; \\qquad \\mathrm {(f)}$ (f) follows by combining (e) with the contraction property (b).",
"Consider any pair of measurable sets $A,B\\subset {\\mathbb {T}}$ that satisfy $\\langle {\\mathbf {1}}_A*{\\mathbf {1}}_C,{\\mathbf {1}}_B\\rangle = \\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_C,{\\mathbf {1}}_{B^\\star }\\rangle $ .",
"Choose a sequence $(y_n)$ such that the sets defined recursively by $A_0=A$ and $A_{n} = R_{y_n}A_{n-1}$ for $n\\ge 1$ satisfy $m(A_n\\,\\Delta \\,{A^\\star })\\rightarrow 0.$ Define $B_n$ recursively by $B_0=B$ and $B_{n} = R_{y_n}B_{n-1}$ for $n\\ge 1$ .",
"Then $\\langle {\\mathbf {1}}_{A_n}*{\\mathbf {1}}_C,{\\mathbf {1}}_{B_n}\\rangle = \\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_C,{\\mathbf {1}}_{B^\\star }\\rangle $ for every $n$ .",
"Choose $n_\\nu $ so that the sequence ${\\mathbf {1}}_{B_{n_\\nu }}$ converges weakly in $L^2({\\mathbb {T}})$ to some $h\\in L^2({\\mathbb {T}})$ , with $0\\le h\\le 1$ , $\\int h\\,dm=m(B)$ .",
"Denoting $(A_{\\nu _n},B_{\\nu _n})$ by $(A_n,B_n)$ for simplicity, the above implies $\\langle {\\mathbf {1}}_{A_n}*{\\mathbf {1}}_C,{\\mathbf {1}}_{B_n}\\rangle \\rightarrow \\langle {\\mathbf {1}}_{{A^\\star }}*{\\mathbf {1}}_C,h\\rangle .$ From this and the admissibility hypothesis it follows that $h={\\mathbf {1}}_{{B^\\star }}$ .",
"Thus ${\\mathbf {1}}_{B_n}\\rightarrow {\\mathbf {1}}_{B^\\star }$ weakly.",
"Since $m({\\mathbb {T}})$ is finite, this forces $m(B_n\\,\\Delta \\,{B^\\star })\\rightarrow 0.$ By (a), $A_n^\\star =A^\\star $ and $B_n^\\star =B^*$ for all $n\\in \\mathbb {N}$ ; therefore, for all $\\epsilon >0$ there exists $N=N(\\epsilon )\\in \\mathbb {N}$ for which $m(A_n\\,\\Delta \\,A_n^\\star )<\\epsilon \\text{ and }m(B_n\\,\\Delta \\,B_n^\\star )<\\epsilon ,$ while also $\\langle {\\mathbf {1}}_{A_N}*{\\mathbf {1}}_C,{\\mathbf {1}}_{B_N}\\rangle =\\langle {\\mathbf {1}}_{A_N^*}*{\\mathbf {1}}_{C^*},{\\mathbf {1}}_{B_N^*} \\rangle $ by (c).",
"Therefore, fixing $\\varepsilon $ to be sufficiently small as a function of $\\eta $ alone, then the perturbative theory of Lemma REF can be applied, implying that $ \\text{there exists $y_N$ such that $A_N={A^\\star }+y_N$ and $B_N={B^\\star }+y_N$.",
"}$ Denote by ${\\mathcal {R}}:{\\mathbb {T}}\\rightarrow {\\mathbb {T}}$ the reflection ${\\mathcal {R}}(x_1,x_2)=(x_1,-x_2)$ .",
"Consider any measurable $A,B\\subset {\\mathbb {T}}$ such that the triple $(A,B,C)$ (for our fixed $C$ ) satisifes the hypotheses of the proposition.",
"Claim 10.1 If $A^\\sharp ={A^\\star }$ and $B^\\sharp ={B^\\star }$ , and if $\\langle {\\mathbf {1}}_A*{\\mathbf {1}}_C,{\\mathbf {1}}_B\\rangle =\\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_C,{\\mathbf {1}}_{B^\\star }\\rangle $ , then either $(A,B) = ({A^\\star },{B^\\star })$ or $(A,B) = ({\\mathcal {R}}{A^\\star },{\\mathcal {R}}{B^\\star })$ .",
"The strict admissibility hypothesis guarantees that there exists $\\varepsilon >0$ such that for any $x\\in {A^\\star }$ there exists $y=b(x)\\in {B^\\star }$ such that whenever $x^{\\prime },y^{\\prime }\\in {\\mathbb {T}}$ satisfy $ \\Vert x-x^{\\prime } \\Vert _{\\mathbb {T}}\\le \\varepsilon $ and $ \\Vert y-y^{\\prime } \\Vert _{\\mathbb {T}}\\le \\varepsilon $ , one has $ \\Vert x_+^{\\prime }-y_+^{\\prime } \\Vert _{\\mathbb {T}}<\\tfrac{1}{2}m(C) \\ \\text{ but } \\ \\Vert x_+^{\\prime }-y_-^{\\prime } \\Vert _{\\mathbb {T}}>\\tfrac{1}{2}m(C).$ Indeed, identifying ${\\mathbb {T}}$ with $[-\\frac{1}{2},\\frac{1}{2})$ , it suffices to prove the above for $x\\in A^\\star $ with $x\\ge 0$ .",
"The $\\eta $ -strict admissibility and $\\eta $ -boundedness of $(A,B,C)$ ensure that $\\bar{p}:=\\tfrac{1}{2}m(A)-\\tfrac{1}{2}m(C)$ has the property $-\\tfrac{1}{2}m(B)+\\tfrac{\\eta ^2}{2}\\le \\bar{p} \\le \\tfrac{1}{2}m(B)-\\tfrac{\\eta ^2}{2}$ (in particular $\\bar{p} \\in B^*$ ), while the right endpoint $\\tfrac{1}{2}m(A)$ of $A^*$ satisfies $\\big |\\tfrac{1}{2}m(A)-\\bar{p}_-\\big |_{{\\mathbb {T}}}=\\big |\\tfrac{1}{2}m(A)-\\big (\\tfrac{1}{2}-\\tfrac{1}{2}m(A)+\\tfrac{1}{2}m(C)\\big )\\big |=\\tfrac{1}{2}-m(A)\\ge \\tfrac{1}{2}m(C),$ as $m(A)\\le \\tfrac{1}{2}$ .",
"Therefore, if $\\bar{p}\\le 0$ , define $b(x):=\\bar{p}-\\tfrac{\\eta ^2}{4}$ for all $x\\in A^\\star $ .",
"If $\\bar{x}>0$ , define $b\\left(t\\tfrac{m(A)}{2}\\right):=\\bar{p}-\\tfrac{\\eta ^2}{4}$ for every element $t\\tfrac{m(A)}{2}$ of $A^\\star $ , for all $0<t\\le 1$ .",
"Denoting by $N_x$ the $\\epsilon $ -neighbourhood on ${\\mathbb {T}}$ of any $x\\in {\\mathbb {T}}$ , it follows by the above and (d) that, for any $x\\in A^\\star $ , $\\begin{aligned}\\text{either }&m\\left(\\lbrace x^{\\prime }\\in A^\\star \\cap N_x: x_+^{\\prime }\\notin A, x_-^{\\prime }\\in A\\rbrace \\right)=0\\\\\\text{or }&m\\left(\\lbrace y^{\\prime }\\in B^\\star \\cap N_{b(x)}: y_+^{\\prime }\\in B, y_-^{\\prime }\\notin B\\rbrace \\right)=0\\end{aligned}$ and $\\begin{aligned}\\text{either }&m\\left(\\lbrace x^{\\prime }\\in A^\\star \\cap N_x: x_+^{\\prime }\\in A, x_-^{\\prime }\\notin A\\rbrace \\right)=0\\\\\\text{or }&m\\left(\\lbrace y^{\\prime }\\in B^\\star \\cap N_{b(x)}: y_+^{\\prime }\\notin B, y_-^{\\prime }\\in B\\rbrace \\right)=0.\\end{aligned}$ The second conclusion of (REF ) and the second conclusion of (REF ) cannot simultaneously hold, therefore the first conclusion of either (REF ) or (REF ) holds.",
"That is, $\\text{for every }x\\in A^\\star \\text{, either }m({\\mathcal {R}}A\\cap N_x)=0\\text{ or }m(A\\cap N_x)=0.$ Now assume that, for some $x\\in A^\\star $ with $N_x\\subset A^\\star $ , it holds that $m({\\mathcal {R}}A\\cap N_x)=0$ .",
"It will be shown that $m({\\mathcal {R}}A\\cap N_y)=0\\text{ for all }y\\in A^\\star \\text{ with }\\Vert x-y\\Vert _{\\mathbb {T}}<\\epsilon \\text{ and }N_y\\subset A^\\star $ (and therefore, by the connectivity of $A^\\star $ , $m({\\mathcal {R}}A\\cap A^\\star )=0$ , i.e.",
"$A=A^\\star $ up to a Lebesgue null set).",
"Indeed, let $y\\in A^\\star $ as above, and suppose that $m({\\mathcal {R}}A\\cap N_y)>0$ .",
"Due to the fact that $A^\\sharp =A^\\star $ , it holds that $m({\\mathcal {R}}A\\cap N_z)+m(A\\cap N_z)=m(A^\\star \\cap N_z)$ for every $z\\in A^\\star $ .",
"Therefore, $m({\\mathcal {R}}A\\cap N_y)=m(A\\cap N_x)=\\epsilon $ .",
"Since the sets ${\\mathcal {R}}A$ and $A$ share at most two points (as $m(A)\\le \\tfrac{1}{2}$ ), it follows that $m\\big (({\\mathcal {R}}A\\cap N_y)\\cup (A\\cap N_x)\\big )=2\\epsilon .$ This is a contradiction, as the set $({\\mathcal {R}}A\\cap N_y)\\cup (A\\cap N_x)$ is contained in the arc $N:=N_y\\cup N_x$ of ${\\mathbb {T}}$ , of length $<\\tfrac{\\epsilon }{2}+\\epsilon +\\tfrac{\\epsilon }{2}<2\\epsilon $ .",
"Therefore, if $x$ as above exists, then $A=A^\\star $ up to a Lebesgue null set.",
"In a similar manner it can be shown that if there exists $x\\in A^\\star $ with $m(A\\cap N_x)=0$ and $N_x\\subset A^\\star $ , then ${\\mathcal {R}}(A)=A^\\star $ up to a Lebesgue null set.",
"Thus, either $A=A^*$ or $A={\\mathcal {R}}A^\\star $ up to a Lebesgue null set.",
"Without loss of generality, it is assumed that the former holds (the functional $\\langle {\\mathbf {1}}_A*{\\mathbf {1}}_C,{\\mathbf {1}}_B\\rangle $ is invariant under simultaneous translations of $A$ and $B$ ).",
"Then, the fact that ${\\mathcal {D}}(A,B,C)=0$ means that $\\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{C},{\\mathbf {1}}_{-B}\\rangle =\\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{C},{\\mathbf {1}}_{-B^\\star }\\rangle ;$ since $A^\\star ,C$ are intervals, the above implies that $|B\\,\\Delta \\,B^\\star |=0$ .",
"It has thus been shown that either $(A,B)=(A^\\star ,B^\\star )$ or $(A,B)=({\\mathcal {R}}A^\\star ,{\\mathcal {R}}B^\\star )$ (up to Lebesgue null sets).",
"This completes the proof of the claim.",
"$\\Box $ We are now in a position to complete the proof for Step 1.",
"Return to the sequence of pairs $(A_n,B_n)$ , for $n=1,\\ldots ,N$ .",
"By (REF ), $(A_N,B_N)=({A^\\star }+y_N,{B^\\star }+y_N)$ up to Lebesgue null sets.",
"Now, $(A_N,B_N) =((A_{N-1}-y)^\\sharp ,(B_{N-1}-y)^\\sharp )$ for some $y\\in \\mathbb {T}$ .",
"Therefore, either $(A_{N-1},B_{N-1})$ equals either $({A^\\star }+y_N+y,{B^\\star }+y_N+y)$ or $({\\mathcal {R}}{A^\\star }+y_N+y,{\\mathcal {R}}{B^\\star }+y_N+y)$ , up to Lebesgue null sets.",
"Repeating this reasoning recursively for $n=N-2,N-3,\\dots $ , we get a similar conclusion for $(A,B)$ .",
"This completes the discussion of Step 1.",
"Step 2.",
"For any $\\varepsilon >0$ , there exists $\\delta =\\delta (\\eta )>0$ such that if ${\\mathcal {D}}(A,B,C)^{1/2}\\le \\delta $ then $ \\inf _{x+y=x_C}\\big (|A\\,\\Delta \\,(A^\\star +x)| +|B\\,\\Delta \\,(B^\\star +y)|\\big ) \\le \\varepsilon .$ We argue by contradiction.",
"If the conclusion fails to hold then there exists an $\\eta $ –strictly admissible $\\eta $ –bounded sequence $(A_k,B_k,C_k)$ such that $\\lim _{k\\rightarrow \\infty } {\\mathcal {D}}(A_k,B_k,C_k) = 0$ , but $ \\inf _{x+y=x_{C_k}}\\big (|A_k\\,\\Delta \\,(A_k^\\star +x)| +|B_k\\,\\Delta \\,(B_k^\\star +y)|\\big ) \\ge \\varepsilon .$ It may be assumed without loss of generality that each $x_{C_k}=0$ .",
"By passing to subsequences we may assume that ${\\mathbf {1}}_{A_k}, {\\mathbf {1}}_{B_k}$ converge weakly in $L^2({\\mathbb {T}})$ to $f,g\\in L^2({\\mathbb {T}})$ , respectively.",
"Then $0\\le f,g\\le 1$ , $\\lim _{k\\rightarrow \\infty } |A_k|$ exists and is equal to $\\int _{\\mathbb {T}}f$ , and likewise $|B_k|\\rightarrow \\int _{\\mathbb {T}}g$ .",
"Moreover, after a further diagonal argument, ${\\mathbf {1}}_{C_k}\\rightarrow {\\mathbf {1}}_C$ weakly for some interval $C$ centered at 0.",
"Because the $C_k$ are intervals, a simple compactness argument shows that ${\\mathbf {1}}_{A_k}*{\\mathbf {1}}_{C_k}$ converges strongly in $L^2({\\mathbb {T}})$ .",
"Therefore $\\lim _{k\\rightarrow \\infty } {\\mathcal {T}}(A_k,B_k,C_k) = {\\mathcal {T}}({A^\\star },{B^\\star },C)$ where ${A^\\star },{B^\\star }$ denote here the intervals centered at 0 of lengths $\\int _{\\mathbb {T}}f,\\int _{\\mathbb {T}}g$ , respectively.",
"By continuity, the limiting triple $({A^\\star },{B^\\star },C)$ satisfies ${\\mathcal {D}}({A^\\star },{B^\\star },C)=0$ .",
"By Lemma REF , $\\int _C f*g \\le \\int _C f*{\\mathbf {1}}_{B^\\star }\\le \\langle f^\\star ,{\\mathbf {1}}_{B^\\star }*{\\mathbf {1}}_C\\rangle .$ The triple $(\\int _{\\mathbb {T}}f^\\star ,|{B^\\star }|,|C|)$ is $\\eta $ –strictly admissible.",
"Because $0\\le f^\\star \\le 1$ and $\\int _{\\mathbb {T}}f^\\star =|{A^\\star }|$ , $\\eta $ –strict admissibility ensures that ${\\mathbf {1}}_{{B^\\star }}*{\\mathbf {1}}_C$ , which is is symmetric and nonincreasing, is also strictly decreasing with derivative identically equal to $-1$ in $\\lbrace x: |\\,x-|{A^\\star }|/2\\,|\\le r\\rbrace $ for some $r>0$ which depends only on $\\eta $ .",
"Therefore $\\langle f^\\star ,{\\mathbf {1}}_{B^\\star }*{\\mathbf {1}}_C\\rangle =\\langle {\\mathbf {1}}_{A^\\star },{\\mathbf {1}}_{B^\\star }*{\\mathbf {1}}_C\\rangle $ if and only if $f^\\star ={\\mathbf {1}}_{A^\\star }$ almost everywhere.",
"Thus $f^\\star $ is the indicator function of a set.",
"Since $f$ has the same distribution function as $f^\\star $ , we conclude that $f={\\mathbf {1}}_A$ for some $A\\subset {\\mathbb {T}}$ .",
"Likewise, $g={\\mathbf {1}}_B$ for some set $B$ .",
"Thus ${\\mathbf {1}}_{A_k}\\rightarrow {\\mathbf {1}}_A$ and ${\\mathbf {1}}_{B_k}\\rightarrow {\\mathbf {1}}_B$ weakly in $L^2({\\mathbb {T}})$ .",
"Therefore $|A_k\\,\\Delta \\,A| + |B_k\\,\\Delta \\,B|\\rightarrow 0$ , and ${\\mathcal {D}}(A,B,C)=0$ .",
"Step 1 now applies, allowing us to conclude that $A$ and $B$ differ from intervals by Lebesgue null sets, and that the centers of $A,$ satisfy $x_A+x_B=0$ .",
"This contradicts (REF ), completing Step 2.",
"$\\Box $ Step 3.",
"Let $(A,B,C)$ be a triple satisfying the hypotheses of the proposition.",
"Let $\\delta _0$ be the constant appearing in the statement of the perturbative Lemma REF .",
"By Step 2, there exists $\\delta =\\delta (\\eta )>0$ , such that if $\\mathcal {D}(A,B,C)^{1/2}\\le \\delta \\max (|A|,|B|,|C|)$ then $\\inf _{x+y=x_C} \\big (|A\\,\\Delta \\,(A^\\star +x)|+ |B\\,\\Delta \\,(B^\\star +y)| \\le \\delta _0\\eta \\le \\delta _0 \\max (|A|,|B|,|C|),$ where the last inequality is due to the $\\eta $ -boundedness of $(A,B,C)$ .",
"Therefore, by Lemma REF , there exist $x^{\\prime },y^{\\prime },z^{\\prime }\\in \\mathbb {T}$ with $x^{\\prime }+y^{\\prime }=z^{\\prime }$ such that $|A\\,\\Delta \\,(A^\\star +x^{\\prime })|, |B\\,\\Delta \\,(B^\\star +y^{\\prime })|, |C\\,\\Delta \\,(C^\\star +z^{\\prime })|\\le {\\mathbf {C}}\\mathcal {D}(A,B,C)^{1/2},$ for some ${\\mathbf {C}}>0$ depending only on $\\eta $ .",
"This would be the desired result if $z^{\\prime }=x_C$ , something which does not necessarily follow from Lemma REF .",
"However, it can be proved that $z^{\\prime }$ is very close to $x_C$ ; so close that, perturbing $z^{\\prime }$ to become $x_C$ and perturbing $x^{\\prime }$ by the same amount, the truth of (REF ) is not violated, up to multiplication by constant factors.",
"More precisely, it holds that $\\Vert z^{\\prime }-x_C\\Vert _\\mathbb {T}\\le {\\mathbf {C}}\\mathcal {D}(A,B,C)^{1/2}$ .",
"Indeed, first observe that $C\\cap (C^\\star +z^{\\prime })\\ne \\emptyset $ , as otherwise (REF ) would imply $\\mathcal {D}(A,B,C)^{1/2}\\ge \\tfrac{1}{{\\mathbf {C}}}|C\\,\\Delta \\,(C^\\star +z^{\\prime })|=\\tfrac{2}{{\\mathbf {C}}}|C|\\ge \\tfrac{2\\eta }{{\\mathbf {C}}}\\max (|A|,|B|,|C|)$ by the $\\eta $ -strict admissibility of $(A,B,C)$ , a contradiction for $\\delta $ sufficiently small.",
"Thus, since $C,C^\\star +z^{\\prime }$ are intervals centered at $x_C,z^{\\prime }$ , respectively, it holds that $\\Vert z^{\\prime }-x_C\\Vert _\\mathbb {T}=\\tfrac{1}{2}|C\\,\\Delta \\,(C^\\star +z^{\\prime })|\\le {\\mathbf {C}}\\mathcal {D}(A,B,C)^{1/2}$ .",
"Therefore, $\\bar{x}^{\\prime }:=x^{\\prime }+(x_C-z^{\\prime })$ satisfies $\\bar{x}^{\\prime }+y^{\\prime }=x_C$ and $\\begin{aligned}\\big |A\\,\\Delta \\,(A^\\star +\\bar{x}^{\\prime })\\big |&\\le \\big |A\\,\\Delta \\,(A^\\star +x^{\\prime })\\big |+\\big | (A^\\star +x^{\\prime })\\,\\Delta \\,(A^\\star +\\bar{x}^{\\prime })\\big |\\\\& \\le {\\mathbf {C}}\\mathcal {D}(A,B,C)^{1/2}+\\Vert x^{\\prime }-\\bar{x}^{\\prime }\\Vert _{\\mathbb {T}}\\\\&\\le 2{\\mathbf {C}}\\mathcal {D}(A,B,C)^{1/2};\\end{aligned}$ likewise for $B$ .",
"Therefore, the triple $(A,B,C)$ satisfies (REF ) with constant depending only on $\\eta $ .",
"As long as the quantity $\\delta $ in the argument above is chosen sufficiently small, the complementary situation in which $\\mathcal {D}(A,B,C)^{1/2}>\\delta \\max (|A|,|B|,|C|)$ also leads to (REF ) with constant ${\\mathbf {C}}=2\\delta ^{-1}$ , simply because, for all $x\\in \\mathbb {T}$ , $|A\\,\\Delta \\,(A^\\star +x)|\\le 2|A|\\le 2\\max (|A|,|B|,|C|).",
"$ Likewise for $B$ ."
],
[
"When one set is nearly rank one Bohr",
"The aim of this section is to establish for general groups $G$ that if $(A,B,C)$ is a strictly admissible triple with ${\\mathcal {D}}(A,B,C)$ small, if $(A,B,C)$ satisfies appropriate auxiliary hypotheses, and if one of the three sets $A,B,C$ is nearly a rank one Bohr set, then the other two are also nearly rank one Bohr sets (parallel to the first, with the triple compatibly centered).",
"Proposition 11.1 Let $G$ be a compact connected Abelian topological group with normalized Haar measure $\\mu $ .",
"For any $\\eta ,\\eta ^{\\prime }>0$ , there exist $c=c(\\eta ,\\eta ^{\\prime })>0$ and ${\\mathbf {C}}={\\mathbf {C}}(\\eta ,\\eta ^{\\prime },c)<\\infty $ such that the following holds.",
"Let $(A,B,C)$ be an $\\eta $ -strictly admissible triple of $\\mu $ -measurable subsets of $G$ , with $\\min (\\mu (A),\\mu (B),\\mu (C))\\ge \\eta $ and $\\mu (A)+\\mu (B)+\\mu (C)\\le 2-\\eta ^{\\prime }$ .",
"If there exists a rank one Bohr set $\\mathcal {B}$ with $\\mu (C\\,\\Delta \\,\\mathcal {B})\\le c(\\eta ,\\eta ^{\\prime })\\max \\big (\\mu (A),\\mu (B),\\mu (C)\\big ),$ then there exists a compatibly centered parallel ordered triple $(\\mathcal {B}_A,\\mathcal {B}_B,\\mathcal {B}_C)$ of rank one Bohr sets satisfying $\\mu (A\\,\\Delta \\,\\mathcal {B}_A)\\le {\\mathbf {C}}\\mathcal {D}(A,B,C)^{1/2},$ and likewise for $\\mu (B\\,\\Delta \\,\\mathcal {B}_B)$ and $\\mu (C\\,\\Delta \\,\\mathcal {B}_C)$ .",
"Let $\\eta ,\\eta ^{\\prime }>0$ and $(A,B,C)$ be as in the statement of the proposition.",
"We may assume that $(A,B,C)$ satisfy the supplementary hypothesis $\\mathcal {D}(A,B,C)<c(\\eta ,\\eta ^{\\prime })\\max \\big (\\mu (A),\\mu (B),\\mu (C)\\big )^2$ for a small constant $c(\\eta ,\\eta ^{\\prime })$ .",
"Indeed, otherwise $\\mu (A\\,\\Delta \\,\\mathcal {B}_A)\\le {\\mathbf {C}}(\\eta ,\\eta ^{\\prime })\\mathcal {D}(A,B,C)^{1/2}$ holds trivially for any rank one Bohr set $\\mathcal {B}_A$ with $\\mu (\\mathcal {B}_A)=\\mu (A)$ ; likewise for $B$ and $C$ .",
"First, consider the case in which $C$ is a rank one Bohr set.",
"That is, $C=\\phi ^{-1}(C^\\star )+x$ , for some continuous homomorphism $\\phi :G\\rightarrow \\mathbb {T}$ and some $x\\in G$ .",
"We assume without loss of generality that $C=\\phi ^{-1}(C^\\star )$ .",
"Define $\\phi _*: L^1(G)\\rightarrow L^1({\\mathbb {T}})$ by $ \\int _E \\phi _*(f)\\,dm = \\int _{\\phi ^{-1}(E)} f\\,d\\mu \\ \\text{ for all measurable $E\\subset {\\mathbb {T}}$.}",
"$ Then $\\phi _*({\\mathbf {1}}_A*{\\mathbf {1}}_B)=\\phi _*({\\mathbf {1}}_A)*\\phi _*({\\mathbf {1}}_B), $ and consequently $\\int _C{\\mathbf {1}}_A*{\\mathbf {1}}_B\\,d\\mu ={\\mathcal {T}}_G({\\mathbf {1}}_A,{\\mathbf {1}}_B,{\\mathbf {1}}_C) = {\\mathcal {T}}_{\\mathbb {T}}(\\phi _*({\\mathbf {1}}_A),\\phi _*({\\mathbf {1}}_B), {\\mathbf {1}}_{C^\\star })= {\\mathcal {T}}_{\\mathbb {T}}(f,g,{\\mathbf {1}}_{{C^\\star }}),$ where the functions $f:=\\phi _*({\\mathbf {1}}_A)\\text{ and }g:=\\phi _*({\\mathbf {1}}_B)$ from $G$ to $[0,\\infty )$ satisfy $0\\le f,g\\le 1, \\; \\textstyle \\int _{\\mathbb {T}}f\\,dm=\\mu (A), \\; \\int _{\\mathbb {T}}g\\,dm =\\mu (B).",
"$ Thus, by the Riesz-Sobolev inequality on $\\mathbb {T}$ , ${\\mathcal {T}}_G({\\mathbf {1}}_A,{\\mathbf {1}}_B,{\\mathbf {1}}_C)= {\\mathcal {T}}_{\\mathbb {T}}(f,g,{\\mathbf {1}}_{C^\\star })\\le {\\mathcal {T}}_{\\mathbb {T}}(f^\\star ,g^\\star , {\\mathbf {1}}_{C^\\star }).$ Applying Lemma REF to the functions $f^\\star , g^\\star , {\\mathbf {1}}_{{C^\\star }}$ gives $\\begin{aligned}{\\mathcal {T}}_G({\\mathbf {1}}_A,{\\mathbf {1}}_B,{\\mathbf {1}}_C)&\\le {\\mathcal {T}}_\\mathbb {T}(f^\\star ,g^\\star ,{\\mathbf {1}}_{{C^\\star }})\\\\&\\le \\max \\lbrace {\\mathcal {T}}_\\mathbb {T}(f^\\star , {\\mathbf {1}}_{B^\\star },{\\mathbf {1}}_{C^\\star }),{\\mathcal {T}}_\\mathbb {T}({\\mathbf {1}}_{A^\\star }, g^\\star ,{\\mathbf {1}}_{C^\\star })\\rbrace \\\\&\\le {\\mathcal {T}}_\\mathbb {T}({\\mathbf {1}}_{A^\\star },{\\mathbf {1}}_{B^\\star },{\\mathbf {1}}_{C^\\star }).\\end{aligned}$ Moreover, since $f^\\star $ , $g^\\star $ are non-increasing functions with $0\\le f^\\star ,g^\\star \\le 1$ , $\\int _{\\mathbb {T}}f\\,dm=m(A^\\star )$ and $\\int _{\\mathbb {T}}g\\,dm=m(B^\\star )$ , the following holds.",
"Claim 11.1 There exists ${\\mathbf {C}}<\\infty $ , depending only on $\\eta $ , such that $ \\Vert f^\\star -{\\mathbf {1}}_{A^\\star }\\Vert _{L^1({\\mathbb {T}})}+\\Vert g^\\star -{\\mathbf {1}}_{B^\\star }\\Vert _{L^1({\\mathbb {T}})}\\le {\\mathbf {C}}{\\mathcal {D}}(A,B,C)^{1/2}.$ By (REF ) and because $\\int _{\\mathbb {T}}({\\mathbf {1}}_{A^\\star }-f^\\star )\\,dm=0$ , $\\mathcal {D}(A,B,C)\\ge \\int _{\\mathbb {T}}({\\mathbf {1}}_{A^\\star }-f^\\star )\\cdot ({\\mathbf {1}}_{B^\\star }*{\\mathbf {1}}_{C^\\star }) \\,dm= \\int _{\\mathbb {T}}({\\mathbf {1}}_{A^\\star }-f^\\star )\\cdot ({\\mathbf {1}}_{B^\\star }*{\\mathbf {1}}_{C^\\star } -\\gamma ) \\,dm$ for any constant $\\gamma $ , and in particular for $\\gamma = {\\mathbf {1}}_{{B^\\star }}*{\\mathbf {1}}_{C^\\star }\\left(\\tfrac{\\mu (A)}{2}\\right)$ .",
"The function $K(x)= {\\mathbf {1}}_{B^\\star }*{\\mathbf {1}}_{C^\\star } -\\gamma $ is nonnegative on ${A^\\star }$ and nonpositive on ${\\mathbb {T}}\\setminus {A^\\star }$ , as is ${\\mathbf {1}}_{A^\\star }-f^\\star $ , so $ {\\mathcal {D}}(A,B,C)\\ge \\int _{\\mathbb {T}}|{\\mathbf {1}}_{A^\\star }-f^\\star |\\cdot |K|\\,dm.$ Let $a=\\mu (A)/2$ .",
"Obtaining a lower bound for the right-hand side would be simpler if $|K|$ enjoyed a strictly positive lower bound, but $K(a)=0$ .",
"$K$ does satisfy $|K(x)|=|x-a|$ for $x\\in [0,\\tfrac{1}{2}]$ with $|x-a|\\le \\tfrac{1}{2} \\min (\\mu (B)+\\mu (C)-\\mu (A),\\, \\mu (A)-|\\mu (B)-\\mu (C)|)$ , and the $\\eta $ –strict admissibility hypothesis ensures that this holds whenever $|x-a| \\le \\tfrac{1}{2}\\eta \\max (\\mu (A),\\mu (B),\\mu (C))$ .",
"Since ${\\mathbf {1}}_{B^\\star }*{\\mathbf {1}}_{C^\\star }$ is nonincreasing, we find that, for $x\\in [0,\\tfrac{1}{2}]$ , $ |K(x)| \\ge \\left\\lbrace \\begin{aligned}&|x-a| \\qquad \\text{if $|x-a|\\le \\tfrac{\\eta }{2} \\max (\\mu (A),\\mu (B),\\mu (C))$} \\\\&\\tfrac{\\eta }{2} \\max (\\mu (A),\\mu (B),\\mu (C)) \\qquad \\text{ otherwise}.\\end{aligned} \\right.",
"$ It is elementary that if $0\\le \\psi \\le 1$ then $\\int _{\\mathbb {R}}|x| \\psi (x)\\,dx \\ge \\tfrac{1}{4} \\Vert \\psi \\Vert _{L^1(\\mathbb {R})}^2$ .",
"Therefore from the lower bound for $K$ and the upper bound $ \\Vert {\\mathbf {1}}_{A^\\star }-f^\\star \\Vert _{C^0} \\le 1$ it follows that $\\int _{\\mathbb {T}}|{\\mathbf {1}}_{A^\\star }-f^\\star |\\cdot |K|\\,dm\\ge c\\min \\Big ( \\Vert {\\mathbf {1}}_{A^\\star }-f^\\star \\Vert _{L^1({\\mathbb {T}})}, \\ \\eta \\max (\\mu (A),\\mu (B),\\mu (C)) \\Big )\\cdot \\Vert {\\mathbf {1}}_{A^\\star }-f^\\star \\Vert _{L^1({\\mathbb {T}})}$ for a certain absolute constant $c>0$ .",
"Now $ \\Vert {\\mathbf {1}}_{A^\\star }-f^\\star \\Vert _{L^1({\\mathbb {T}})}\\le 2\\mu (A)$ , so, provided that $\\eta \\le 1$ , this implies that $\\int _{\\mathbb {T}}|{\\mathbf {1}}_{A^\\star }-f^\\star |\\cdot |K|\\,dm\\ge c \\Vert {\\mathbf {1}}_{A^\\star }-f^\\star \\Vert _{L^1({\\mathbb {T}})}^2,$ for a constant $c>0$ that only depends on $\\eta $ .",
"The indicated conclusion for ${\\mathbf {1}}_{A^\\star }-f^\\star $ follows directly from this and (REF ).",
"The same holds for ${\\mathbf {1}}_{B^\\star }-g^\\star $ since the roles of $A,B$ can be interchanged.",
"Since $f,f^\\star $ have identical distribution functions and likewise for $g,g^\\star $ , there exist $\\tilde{A},\\tilde{B}\\subset {\\mathbb {T}}$ satisfying $ \\Vert f-{\\mathbf {1}}_{\\tilde{A}} \\Vert _{L^1({\\mathbb {T}})}= \\Vert f^\\star -{\\mathbf {1}}_{A^\\star }\\Vert _{L^1({\\mathbb {T}})}$ and $ \\Vert g-{\\mathbf {1}}_{\\tilde{B}} \\Vert _{L^1({\\mathbb {T}})}=\\Vert g^\\star -{\\mathbf {1}}_{B^\\star }\\Vert _{L^1({\\mathbb {T}})}$ , with $m(\\tilde{A}) = \\mu (A)=\\int _{\\mathbb {T}}f\\,dm$ and $m(\\tilde{B}) = \\mu (B)=\\int _{\\mathbb {T}}g\\,dm$ .",
"Therefore, if $c(\\eta ,\\eta ^{\\prime })$ is sufficiently small, the triple $(\\tilde{A},\\tilde{B},C^*)$ is $\\tfrac{\\eta }{2}$ -strictly admissible, $\\min (\\eta ,\\eta ^{\\prime })$ -bounded and $m(\\tilde{A})+m(\\tilde{B})+m(C^*)\\le 2-\\tfrac{\\eta ^{\\prime }}{2}$ .",
"Since $C^*$ is an interval, Proposition REF states that there exists $\\bar{x}\\in {\\mathbb {T}}$ satisfying $ m(\\tilde{A}\\,\\Delta \\,({A^\\star }+\\bar{x}))+ m(\\tilde{B}\\,\\Delta \\,({B^\\star }-\\bar{x}))\\le {\\mathbf {C}}\\mathcal {D}(\\tilde{A},\\tilde{B},C^*)^{1/2},$ for a constant ${\\mathbf {C}}$ depending only on $\\eta ,\\eta ^{\\prime }$ .",
"Now $\\mathcal {D}(\\tilde{A},\\tilde{B},C^\\star )\\le {\\mathbf {C}}{\\mathcal {D}}(A,B,C)^{1/2}\\max (m(A),m(B),m(C)).$ Indeed, since $m(\\tilde{A})=m(A)$ and $m(\\tilde{B})=m(B)$ , it follows that $\\tilde{A}^\\star =A^\\star $ and $\\tilde{B}^\\star =B^\\star $ , so ${\\mathcal {T}}_\\mathbb {T}(\\tilde{A}^\\star ,\\tilde{B}^\\star ,C^\\star )={\\mathcal {T}}_\\mathbb {T}(A^\\star ,B^\\star ,C^\\star ),$ while $\\begin{aligned}{\\mathcal {T}}_\\mathbb {T}(\\tilde{A},\\tilde{B},C^\\star )&={\\mathcal {T}}_\\mathbb {T}\\big (f+({\\mathbf {1}}_{\\tilde{A}}-f),g+({\\mathbf {1}}_{\\tilde{B}}-g),{\\mathbf {1}}_{C^\\star }\\big ) \\\\&\\ge {\\mathcal {T}}_\\mathbb {T}(f,g,{\\mathbf {1}}_{C^\\star })\\\\& -(\\Vert {\\mathbf {1}}_{\\tilde{A}}-f\\Vert _{L^1({\\mathbb {T}})}+\\Vert {\\mathbf {1}}_{\\tilde{B}}-g\\Vert _{L^1({\\mathbb {T}})})m(C^\\star )+\\Vert {\\mathbf {1}}_{\\tilde{A}}-f\\Vert _{L^1({\\mathbb {T}})}\\Vert {\\mathbf {1}}_{\\tilde{B}}-g\\Vert _{L^1({\\mathbb {T}})}\\\\&\\ge {\\mathcal {T}}_G(A,B,C)-{\\mathbf {C}}{\\mathcal {D}}(A,B,C)^{1/2}\\max (m(A),m(B),m(C))\\end{aligned}$ by Claim REF .",
"Thus, (REF ) follows by (REF ).",
"The homomorphism $\\phi $ preserves measure in the sense that $\\mu (\\phi ^{-1}(E))=m(E)$ for any measurable $E\\subset {\\mathbb {T}}$ .",
"Therefore, since $f=\\phi _*({\\mathbf {1}}_A)$ , $\\mu \\big (A\\,\\Delta \\,\\phi ^{-1}(\\tilde{A})\\big ) = \\Vert f-{\\mathbf {1}}_{\\tilde{A}} \\Vert _{L^1({\\mathbb {T}})}\\le {\\mathbf {C}}{\\mathcal {D}}(A,B,C)^{1/2}.$ Moreover, (REF ) and (REF ) together with this property of $\\phi $ yield $\\mu \\big (\\phi ^{-1}(\\tilde{A}) \\,\\Delta \\,\\phi ^{-1}(A^\\star + \\bar{x})\\big )\\le {\\mathbf {C}}{\\mathcal {D}}(A,B,C)^{1/4}\\max (\\mu (A),\\mu (B),\\mu (C))^{1/2}$ In all, $\\begin{aligned}\\mu (A\\,\\Delta \\,{\\mathcal {B}}_A) &\\le {\\mathbf {C}}{\\mathcal {D}}(A,B,C)^{1/4}\\max (\\mu (A),\\mu (B),\\mu (C))^{1/2}\\\\&\\le {\\mathbf {C}}c(\\eta ,\\eta ^{\\prime })^{1/4}\\max (\\mu (A),\\mu (B),\\mu (C))\\end{aligned}$ with ${\\mathcal {B}}_A = \\phi ^{-1}({A^\\star })+x$ for some $x\\in G$ , and likewise for $B$ , with $x$ replaced by $-x$ .",
"The last inequality above is due to (REF ), and it ensures that, as long as $c(\\eta ,\\eta ^{\\prime })$ is sufficiently small, the perturbative Lemma REF can be applied, yielding the desired conclusion for $(A,B,C)$ .",
"The analysis of the case in which the set $C$ coincides with a rank one Bohr set is now complete.",
"Suppose next that $\\mu (C\\,\\Delta \\,\\bar{C}\\big )\\le c(\\eta ,\\eta ^{\\prime })\\max \\big (\\mu (A),\\mu (B),\\mu (C)\\big ),$ where $\\bar{C}=\\phi ^{-1}(C^*)$ for some continuous homomorphism $\\phi :G\\rightarrow \\mathbb {T}$ .",
"If $c(\\eta ,\\eta ^{\\prime })$ is sufficiently small, then the triple $(A,B,\\bar{C})$ is $\\tfrac{\\eta }{2}$ -strictly admissible and satisfies $\\mu (A)+\\mu (B)+\\mu (\\bar{C})\\le 2-\\tfrac{\\eta }{2}$ , while, by (REF ), $\\begin{aligned}\\mathcal {D}(A,B,\\bar{C})&\\le {\\mathbf {C}}c(\\eta ,\\eta ^{\\prime })\\max \\big (\\mu (A),\\mu (B),\\mu (C)\\big )^2\\\\&\\le {\\mathbf {C}}c(\\eta ,\\eta ^{\\prime })\\big (1+c(\\eta ,\\eta ^{\\prime })\\big )^2\\max \\big (\\mu (A),\\mu (B),\\mu (\\bar{C})\\big )^2.\\end{aligned}$ Therefore, since $\\bar{C}$ is a rank one Bohr subset of $G$ , if $c(\\eta ,\\eta ^{\\prime })$ is sufficiently small then the partial result proved above can be applied to $(A,B,\\bar{C})$ , ensuring that there exists a compatibly centered parallel ordered triple $(\\mathcal {B}_A,\\mathcal {B}_B,\\mathcal {B}_{\\bar{C}})$ of rank one Bohr sets, such that $\\mu (A\\,\\Delta \\,\\mathcal {B}_A)&\\le {\\mathbf {C}}(\\eta ,\\eta ^{\\prime })\\mathcal {D}(A,B,\\bar{C})^{1/2}\\\\&\\le {\\mathbf {C}}(\\eta ,\\eta ^{\\prime }) c(\\eta ,\\eta ^{\\prime }) \\max \\big (\\mu (A),\\mu (B),\\mu (C)\\big ),$ and likewise for $B$ and $\\bar{C}$ .",
"Now, this further implies that $\\begin{aligned}\\mu (C\\,\\Delta \\,\\mathcal {B}_{\\bar{C}})&\\le \\mu (C\\,\\Delta \\,\\bar{C})+\\mu (\\bar{C}\\,\\Delta \\,\\mathcal {B}_{\\bar{C}})\\\\&\\le {\\mathbf {C}}(\\eta ,\\eta ^{\\prime })c(\\eta ,\\eta ^{\\prime })\\max \\big (\\mu (A),\\mu (B),\\mu (C)\\big ).\\end{aligned}$ Therefore, if $c(\\eta ,\\eta ^{\\prime })$ is sufficiently small then the triple $(A,B,C)$ satisfies the hypotheses of the perturbative Lemma REF , the conclusion of which implies the desired estimate for $(A,B,C)$ ."
],
[
"Stability of the Riesz-Sobolev inequality",
"In this section we complete the proof of Theorem REF .",
"This proof consists of five main steps.",
"Firstly, given ${\\mathbf {E}}$ with small discrepancy ${\\mathcal {D}}({\\mathbf {E}})$ for the Riesz-Sobolev functional, an associated triple ${\\mathbf {E}}^{\\prime }$ is constructed, also with small discrepancy but with altered Haar measures $\\mu (E^{\\prime }_j)$ satisfying a supplementary condition.",
"Secondly, under this supplementary condition, small Riesz-Sobolev discrepancy for ${\\mathbf {E}}^{\\prime }$ implies that $E^{\\prime }_3$ nearly saturates Kneser's sumset inequality.",
"Thirdly, the inverse theorems of Griesmer and/or Tao imply that any saturator $E^{\\prime }_3$ nearly coincides with a rank one Bohr set.",
"Fourthly, this conclusion for $E^{\\prime }_3$ implies that the given triple ${\\mathbf {E}}$ nearly coincides with a parallel compatibly centered triple of rank one Bohr sets, with $o_{{\\mathcal {D}}({\\mathbf {E}})}(1)$ control.",
"In the fifth step, this crude bound is refined to $O({\\mathcal {D}}({\\mathbf {E}})^{1/2})$ .",
"All of the ingredients have been developed in preceding sections.",
"Here, we link them together.",
"Let $\\eta >0$ .",
"Let $\\delta _0>0$ be a sufficiently small positive constant, which will depend only on $\\eta $ .",
"Let $(A,B,C)$ be an $\\eta $ –strictly admissible $\\eta $ –bounded ordered triple of measurable subsets of $G$ satisfying $\\mathcal {D}(A,B,C)\\le \\delta _0.$ In this discussion, ${\\mathbf {C}}_\\eta $ will denote positive constants that depend only on $\\eta $ , not on $(A,B,C)$ .",
"${\\mathbf {C}}_\\eta $ is allowed to change in value from one occurrence to the next.",
"Assume without loss of generality that $\\mu (C)\\le \\mu (A)\\le \\mu (B)$ .",
"The proof is organized into three cases, reflecting the analysis in §.",
"Case 1: $\\mu (A)\\le \\tfrac{1}{2}$ and $\\mu (C)\\le (1-\\tfrac{\\eta }{50})\\mu (B)$ .",
"In this case, the lower bound assumption $\\min (\\mu (A),\\mu (B),\\mu (C))\\ge \\eta $ implies that, for $\\delta _0$ sufficiently small, $(A,B,C)$ satisfies the hypotheses of Lemma REF .",
"Define $\\tau $ by $\\mu (C)=\\mu (A)+\\mu (B)-2\\tau $ , and define $C^{\\prime } = S_{A,B^{\\prime }}(\\tau )$ .",
"According to Lemma REF , there exists a measurable set $B^{\\prime }\\subset G$ such that the triple $(A,B^{\\prime },C^{\\prime })$ also nearly saturates the Riesz-Sobolev inequality, in the sense that $\\mathcal {D}(A,B^{\\prime },C^{\\prime })\\le \\eta ^{-1} \\mathcal {D}(A,B,C)\\le \\delta _0 \\eta ^{-1},$ satisfies the key supplementary condition $\\mu (A)=\\mu (B^{\\prime }),$ and satisfies the technical conditions $(A,B^{\\prime },C^{\\prime })\\text{ is $\\eta /2$--strictly admissible and $\\eta ^2/2$--bounded},\\\\\\mu (C^{\\prime }) \\le \\mu (A)-4{\\mathcal {D}}(A,B^{\\prime },C^{\\prime })^{1/2}.$ Therefore, if $\\delta _0$ is sufficiently small then the triple $(A,B^{\\prime },C^{\\prime })$ satisfies the hypotheses of Lemma REF , whose conclusion is that $C^{\\prime }$ nearly coincides with a superlevel set: $\\mu (C^{\\prime } \\,\\Delta \\,S_{A,B^{\\prime }}(\\beta ))\\le 4\\mathcal {D}(A,B^{\\prime },C^{\\prime })^{1/2}\\le 4(\\delta _0/\\eta )^{1/2}$ with $\\beta =\\tfrac{1}{2}\\big (\\mu (A)+\\mu (B^{\\prime })-\\mu (C^{\\prime })\\big )$ .",
"Moreover, $(A,B^{\\prime },C^{\\prime })$ satisfies the hypotheses of the key Lemma REF (in particular, $\\mu (A)=\\mu (B^{\\prime })$ ), whose conclusion is that the superlevel set $S_{A,B^{\\prime }}(\\beta )$ has small difference set: $\\begin{aligned}\\mu \\big (S_{A,B^{\\prime }}(\\beta )-S_{A,B^{\\prime }}(\\beta )\\big )&\\le 2\\mu (S_{A,B^{\\prime }}(\\beta )) + 12{\\mathcal {D}}(A,B^{\\prime },S_{A,B^{\\prime }}(\\beta ))^{1/2}\\\\&\\le 2\\mu (S_{A,B^{\\prime }}(\\beta ))+12 (\\delta _0/\\eta )^{1/2}.\\end{aligned}$ So long as $\\delta _0$ is appropriately small, $S_{A,B^{\\prime }}(\\beta )$ satisfies the hypotheses of Corollary REF , whose proof relied on the stability theorems of Tao [22] and/or Griesmer [17] for Kneser's inequality.",
"Its conclusion is that there exists a rank one Bohr set $\\mathcal {B}_{\\beta }$ satisfying $\\mu (\\mathcal {B}_{\\beta }\\,\\Delta \\,S_{A,B^{\\prime }}(\\beta )) \\le {\\mathbf {C}}_\\eta \\delta _0$ .",
"Combining this with (REF ) yields $\\mu (\\mathcal {B}_{\\beta }\\,\\Delta \\,C^{\\prime }) \\le {\\mathbf {C}}_\\eta \\delta _0.",
"$ Therefore for sufficiently small $\\delta _0$ , the triple $(A,B^{\\prime },C^{\\prime })$ satisfies the hypotheses of Proposition REF , with parameters that depend only on $\\eta $ ; $C^{\\prime }$ nearly coincides with a rank one Bohr set, and ${\\mathcal {D}}(A,B^{\\prime },C^{\\prime })$ is small.",
"The proposition states that $A$ and $B^{\\prime }$ consequently also nearly coincide with rank one Bohr sets; in particular, there exists a rank one Bohr set $\\mathcal {B}^{\\prime }_A$ satisfying $\\mu (\\mathcal {B}^{\\prime }_A\\,\\Delta \\,A)\\le {\\mathbf {C}}_\\eta \\mathcal {D}(A,B^{\\prime },C^{\\prime })^{1/2}\\le {\\mathbf {C}}_\\eta (\\delta _0/\\eta )^{1/2}$ for some finite constant ${\\mathbf {C}}_\\eta $ .",
"The last inequality is (REF ).",
"With this control of $A$ we return to the originally given triple $(A,B,C)$ .",
"For sufficiently small $\\delta _0$ , the $\\eta $ -strictly admissible, $\\eta $ -bounded triple $(A,B,C)$ satisfies the hypotheses of Proposition REF , since $A$ is now known to nearly coincide with a rank one Bohr set.",
"The proposition states that there exists a compatibly centered parallel ordered triple $(\\mathcal {B}_A,\\mathcal {B}_B,\\mathcal {B}_C)$ of rank one Bohr sets satisfying $\\mu (A\\,\\Delta \\,\\mathcal {B}_A)+\\mu (B\\,\\Delta \\,\\mathcal {B}_B)+\\mu (C\\,\\Delta \\,\\mathcal {B}_C)\\le {\\mathbf {C}}_\\eta \\mathcal {D}(A,B,C)^{1/2}.$ This completes the proof in Case 1.",
"$\\Box $ Case 2: $\\mu (A)\\le \\tfrac{1}{2}$ and $\\mu (C)> (1-\\tfrac{\\eta }{50})\\mu (B)$ .",
"In this case, $\\eta $ –strict admissibility and $\\eta $ –boundedness together with sufficient smallness of $\\delta _0$ ensure that $(A,B,C)$ satisfies the hypotheses of Lemma REF .",
"Therefore, with $\\tau $ defined by $\\mu (C)=\\mu (A)+\\mu (B)-2\\tau $ , there exist measurable sets $C^{\\prime }\\subset C$ and $A^{\\prime }\\subset A$ that satisfy $\\left\\lbrace \\begin{aligned}&(S_{C^{\\prime },A}(\\tau ),C^{\\prime },A)\\text{ is $\\eta /4$--strictly admissibleand $\\eta /4$--bounded}\\\\&\\mathcal {D}(S_{C^{\\prime },A}(\\tau ),C^{\\prime },A)\\le 16\\mathcal {D}(C,B,A)\\\\&\\mu (C^{\\prime })=\\mu (A^{\\prime })=\\mu (C)-\\tfrac{1}{10} \\eta \\mu (B),\\end{aligned} \\right.$ while $\\left\\lbrace \\begin{aligned}&(S_{C^{\\prime },A^{\\prime }}(\\tau ),C^{\\prime },A^{\\prime })\\text{ is $\\eta /2$--strictly admissible and $\\eta /2$--bounded}\\\\&\\mathcal {D}(S_{C^{\\prime },A^{\\prime }}(\\tau ),C^{\\prime },A^{\\prime })\\le 16\\mathcal {D}(C,B,A)\\\\&\\mu (S_{A^{\\prime },C^{\\prime }}(\\tau )) \\le (1-\\tfrac{\\eta /2}{50}) \\mu (C^{\\prime }).\\end{aligned} \\right.$ The triple $(S_{A^{\\prime },C^{\\prime }}(\\tau ),C^{\\prime },A^{\\prime })$ falls into Case 1 above, with parameters that depend only on $\\eta $ .",
"Therefore, if $\\delta _0$ is sufficiently small then there exists a rank one Bohr set $\\mathcal {B}_{C^{\\prime }}$ satisfying $\\mu (C^{\\prime }\\,\\Delta \\,\\mathcal {B}_{C^{\\prime }})\\le {\\mathbf {C}}_\\eta \\mathcal {D}(S_{A^{\\prime },C^{\\prime }}(\\tau ),A^{\\prime },C^{\\prime })^{1/2}\\le {\\mathbf {C}}_{\\eta } \\delta _0^{1/2}.$ Setting $F:=S_{C^{\\prime },A}(\\tau )$ , the $\\eta /4$ -strict admissibility and $\\eta /4$ -boundedness of the triple $(F,C^{\\prime },A)$ ensure that, for sufficiently small $\\delta _0$ , $(F,C^{\\prime },A)$ satisfies the hypotheses of Proposition REF .",
"Therefore there exists a rank one Bohr set $\\mathcal {B}_A$ satisfying $\\mu (\\mathcal {B}_A\\,\\Delta \\,A)\\le {\\mathbf {C}}_\\eta \\mathcal {D}(F,C^{\\prime },A)^{1/2}\\le {\\mathbf {C}}_\\eta \\delta _0^{1/2}.$ By $\\eta $ –admissibility and $\\eta $ –boundeness, $(A,B,C)$ satisfies the hypotheses of Proposition REF provided that $\\delta _0$ is sufficiently small.",
"Therefore there exists a compatibly centered parallel ordered triple $(\\mathcal {B}_A^{\\prime },\\mathcal {B}_B,\\mathcal {B}_C)$ of rank one Bohr sets satisfying $\\mu (A\\,\\Delta \\,\\mathcal {B}_A^{\\prime })+\\mu (B\\,\\Delta \\,\\mathcal {B}_B)+\\mu (C\\,\\Delta \\,\\mathcal {B}_C)\\le {\\mathbf {C}}_\\eta \\mathcal {D}(A,B,C)^{1/2}.$ $\\Box $ Case 3: $\\mu (A)>\\tfrac{1}{2}$ .",
"As discussed in §, the triple $(C,G\\setminus A, G\\setminus B)$ is $\\tfrac{\\eta }{4}$ -strictly admissible and $\\tfrac{\\eta }{4}$ -bounded.",
"Moreover, since $\\tfrac{1}{2} < \\mu (A)\\le \\mu (B)$ , $\\mu (G\\setminus A)<\\tfrac{1}{2}$ and $\\mu (G\\setminus B)<\\tfrac{1}{2}$ .",
"Therefore, $(C,G\\setminus A, G\\setminus B)$ falls in the range of one of the two cases already analyzed above.",
"Thus there exists a compatibly centered parallel ordered triple $(\\mathcal {B}_C,\\mathcal {B}_{G\\setminus A},\\mathcal {B}_{G\\setminus B})$ of rank one Bohr sets satisfying $\\begin{aligned}\\mu \\big ((G\\setminus A)\\,\\Delta \\,\\mathcal {B}_{G\\setminus A}\\big )\\le {\\mathbf {C}}_\\eta \\mathcal {D}(C,G\\setminus A, G\\setminus B)^{1/2}={\\mathbf {C}}_\\eta \\mathcal {D}(A,B,C)^{1/2}\\le {\\mathbf {C}}_\\eta \\delta ^{1/2}\\end{aligned}$ and likewise for $\\mu \\big ((G\\setminus B)\\,\\Delta \\,\\mathcal {B}_{G\\setminus B}\\big )$ and for $\\mu (C\\,\\Delta \\,\\mathcal {B}_C)$ .",
"The equality of $\\mathcal {D}(C,G\\setminus A, G\\setminus B)^{1/2}$ with $\\mathcal {D}(A,B,C)^{1/2}$ was established in Lemma REF .",
"For any measurable subsets $E_1,E_2$ of $G$ , $\\mu (E_1\\,\\Delta \\,E_2)=\\mu \\big ((G\\setminus E_1)\\,\\Delta \\,(G\\setminus E_2)\\big )$ .",
"Therefore the compatibly centered parallel ordered triple $(\\mathcal {B}_A,\\mathcal {B}_B,\\mathcal {B}_C)$ of rank one Bohr sets with $\\mathcal {B}_A:=G\\setminus \\mathcal {B}_{G\\setminus A}$ , $\\mathcal {B}_B:=G\\setminus \\mathcal {B}_{G\\setminus B}$ satisfies $\\mu (A\\,\\Delta \\,\\mathcal {B}_A)+\\mu (B\\,\\Delta \\,\\mathcal {B}_B)+\\mu (C\\,\\Delta \\,\\mathcal {B}_C)\\le {\\mathbf {C}}_\\eta {\\mathcal {D}}(A,B,C)^{1/2}.$ The proof of Theorem REF is complete."
],
[
"Cases of equality in the Riesz-Sobolev inequality",
"Theorem REF states that if ${\\mathcal {T}}_G({\\mathbf {E}}) = {\\mathcal {T}}_{\\mathbb {T}}({\\mathbf {E}}^\\star )$ , and if ${\\mathbf {E}}$ is admissible, then there exists a ${\\mathcal {T}}_G$ –compatibly centered ordered triple of parallel rank one Bohr sets satisfying $\\mu (E_j\\,\\Delta \\,{\\mathcal {B}}_j)=0$ for every $j\\in \\lbrace 1,2,3\\rbrace $ .",
"There are two cases.",
"If ${\\mathbf {E}}$ is strictly admissible, then there exists $\\eta >0$ such that ${\\mathbf {E}}$ is $\\eta $ –strictly admissible and $\\eta $ –bounded.",
"Therefore ${\\mathbf {E}}$ satisfies the hypotheses of Theorem REF , the quantitative stability theorem, with $\\delta =0$ .",
"That theorem, whose proof has been completed above, gives the required conclusion.",
"If ${\\mathbf {E}}$ is admissible but not strictly admissible, then after appropriate permutation of the three indices, $\\mu (E_1)+\\mu (E_2) = \\mu (E_3)<1$ , and $\\langle {\\mathbf {1}}_{E_1}*{\\mathbf {1}}_{E_2},{\\mathbf {1}}_{-E_3}\\rangle = \\mu (E_1)\\mu (E_2)= \\langle {\\mathbf {1}}_{E_1}*{\\mathbf {1}}_{E_2},{\\mathbf {1}}_{G}\\rangle .$ Therefore ${\\mathbf {1}}_{E_1}*{\\mathbf {1}}_{E_2}=0$ $\\mu $ –almost everywhere on the complement of $-E_3$ , that is, $E_1+_0 E_2$ is contained in the union of $-E_3$ with a nullset.",
"Thus $\\mu (E_1+_0 E_2) \\le \\mu (E_3)$ .",
"The converse inequality holds by Kneser's theorem, so $\\mu (\\,\\Delta \\,(E_1+_0 E_2,\\,-E_3)=0$ .",
"It is a corollary of more quantitative results of Griesmer [17] and Tao [22] that equality of $\\mu (E_1+_0 E_2)$ with $\\mu (E_1)+\\mu (E_2)$ implies existence of a parallel pair of rank one Bohr sets satisfying $\\mu (E_j\\,\\Delta \\,{\\mathcal {B}}_j)=0$ for $j=1,2$ .",
"Set ${\\mathcal {B}}_3 = {\\mathcal {B}}_1 +{\\mathcal {B}}_2$ .",
"Then $({\\mathcal {B}}_1,{\\mathcal {B}}_2,{\\mathcal {B}}_3)$ is an ordered triple of rank one Bohr sets with all required properties.",
"$\\Box $"
],
[
"Stability in the relaxed framework",
"Theorem REF , a stability theorem for the Riesz-Sobolev inequality in the situation in which indicator functions of sets are replaced by functions taking values in $[0,1]$ , follows from slight modification of the proof of Theorem REF .",
"Let $f,g,h$ be as in the statement of the theorem.",
"To simplify notation, set $ M = \\max \\big (\\textstyle \\int f\\,d\\mu ,\\textstyle \\int g\\,d\\mu ,\\textstyle \\int h\\,d\\mu \\big ).",
"$ With the notation of §, $\\langle f*g,h\\rangle _G\\le \\langle f^\\star *g^\\star ,h^\\star \\rangle _{\\mathbb {T}}\\le \\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star },h^\\star \\rangle _{\\mathbb {T}}\\le \\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star },{\\mathbf {1}}_{C^\\star }\\rangle _{\\mathbb {T}}.$ As shown in §, this implies that $ \\Vert h^\\star -{\\mathbf {1}}_{C^\\star } \\Vert _{L^1({\\mathbb {T}})} \\le {\\mathbf {C}}{\\mathcal {D}}^{1/2}.$ Since $h$ has the same distribution function as $h^\\star $ , there exists a set $C\\subset G$ satisfying $ \\Vert h-{\\mathbf {1}}_C \\Vert _{L^1(G,\\mu )}= \\Vert h^\\star -{\\mathbf {1}}_{C^\\star } \\Vert _{L^1({\\mathbb {T}})}\\le {\\mathbf {C}}{\\mathcal {D}}^{1/2}.$ The same reasoning applies to $f$ and to $g$ , yielding corresponding sets $A,B\\subset {\\mathbb {T}}$ , respectively.",
"Now $ \\big |\\,\\langle f*g,h\\rangle _G - \\langle {\\mathbf {1}}_A*{\\mathbf {1}}_B,{\\mathbf {1}}_C\\rangle _G\\,\\big |\\le {\\mathbf {C}}M {\\mathcal {D}}^{1/2}, $ so, for ${\\mathcal {D}}$ sufficiently small as a function of $\\eta $ alone, $ \\langle {\\mathbf {1}}_A*{\\mathbf {1}}_B,{\\mathbf {1}}_C\\rangle _G& \\ge \\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star },{\\mathbf {1}}_{C^\\star }\\rangle _{\\mathbb {T}}- {\\mathbf {C}}M {\\mathcal {D}}^{1/2}\\\\&= \\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star },{\\mathbf {1}}_{C^\\star }\\rangle _{\\mathbb {T}}- {\\mathbf {C}}\\max (\\mu (A),\\mu (B),\\mu (C)){\\mathcal {D}}^{1/2}\\\\& \\ge \\langle {\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star },{\\mathbf {1}}_{C^\\star }\\rangle _{\\mathbb {T}}- {\\mathbf {C}}M {\\mathcal {D}}^{1/2}$ with the convention that the constant ${\\mathbf {C}}\\in (0,\\infty )$ may change from one occurrence to the next.",
"In the final line we have used the fact that $\\max (\\mu (A),\\mu (B),\\mu (C))$ is comparable to $M$ .",
"Therefore according to Theorem REF , there exists a compatibly centered parallel triple $(\\tilde{A}, \\tilde{B}, \\tilde{C})$ of rank one Bohr subsets of $G$ such that $ |A\\,\\Delta \\,\\tilde{A}| \\le {\\mathbf {C}}M^{1/2} {\\mathcal {D}}^{1/4},$ with the same bound for $|B\\,\\Delta \\,\\tilde{B}|$ and $|C\\,\\Delta \\,\\tilde{C}|$ .",
"In combination with (REF ) and the corresponding results for $f,g$ , this gives $ \\max \\big (\\, \\Vert f-{\\mathbf {1}}_{\\tilde{A}} \\Vert _{L^1},\\, \\Vert g-{\\mathbf {1}}_{\\tilde{B}} \\Vert _{L^1},\\, \\Vert h-{\\mathbf {1}}_{\\tilde{C}} \\Vert _{L^1}\\,\\big )\\le {\\mathbf {C}}M^{1/2} {\\mathcal {D}}^{1/4}.$ It is given in the hypotheses of Theorem REF that ${\\mathcal {D}}/M^2$ is less than some small absolute constant that is at our disposal, but no lower bound is given.",
"Therefore this conclusion is weaker than the desired bound ${\\mathbf {C}}{\\mathcal {D}}^{1/2}$ .",
"However, any bound of the form $o_{{\\mathcal {D}}/M^2}(1)\\cdot M$ is sufficient to place us in the perturbative context of Lemma REF , which gives the desired bound, completing the proof of Theorem REF ."
],
[
"A flow of subsets of ${\\mathbb {T}}$",
"This section and the next develop an alternative approach which, as it now stands, applies directly only for $G={\\mathbb {T}}$ , but which yields slightly superior results for that group; the bounds remain appropriately uniform as the measures of $A,B,C$ tend to zero.",
"It is based on monotonicity of the functional $(A,B,C)\\mapsto {\\mathcal {T}}_{\\mathbb {T}}(A,B,C)$ under a certain continuous one-parameter deformation.",
"Such a monotonicity phenomenon is well-known for $G={\\mathbb {R}}$ [11].",
"The variant developed here, which applies to ${\\mathbb {T}}$ , is less effective but nonetheless useful.",
"In the present section we develop the deformation and its basic properties for Kneser's inequality and the Riesz-Sobolev inequality.",
"In the following section we apply it to establish an improved stability theorem for ${\\mathbb {T}}$ .",
"In the present section and in §, the Lebesgue measure of a subset $E\\subset {\\mathbb {T}}$ is denoted by $|E|$ .",
"All integrals over ${\\mathbb {T}}$ are formed with respect to Lebesgue measure.",
"Let ${\\mathcal {L}}({\\mathbb {T}})$ be the class of all equivalence classes of Lebesgue measurable sets $E\\subset {\\mathbb {T}}$ with $|E|>0$ , and $E$ equivalent to $E^{\\prime }$ if and only if $|E\\,\\Delta \\,E^{\\prime }|=0$ .",
"Assuming that $|E|>0$ , define $ T_E = -\\ln (|E|)>0.$ In the next theorem, $E$ and $E_j$ denote arbitrary equivalence classes of Lebesgue measurable subsets of ${\\mathbb {T}}$ .",
"For equivalence class $A,B$ , the notation $A\\subset B$ means of course that any two representatives of these classes satisfy $|B\\setminus A|=0$ .",
"Recall that $ A+_0 B = \\lbrace x: {\\mathbf {1}}_A*{\\mathbf {1}}_B(x)>0\\rbrace .$ The inequality $|A+B|_* \\ge \\min (|A|+|B|,1)$ for all measurable $A,B\\subset G$ implies that $ |A+_0 B| \\ge \\min (|A|+|B|,1)\\ \\text{ for all measurable $A,B\\subset \\mathbb {T}$.}",
"$ Indeed, given $A,B\\subset \\mathbb {T}$ , denote by $A^\\dagger \\subset A$ and $B^\\dagger \\subset B$ the sets of Lebesgue points of $A,B$ , respectively.",
"From the fact that almost every point is a Lebesgue point, it follows easily that $A^\\dagger +_0B^\\dagger =A^\\dagger +B^\\dagger .",
"$ Therefore, since $A^\\dagger \\subset A$ and $B^\\dagger \\subset B$ , it follows that $|A+_0B|\\ge |A^\\dagger +_0B^\\dagger |=|A^\\dagger +B^\\dagger |\\ge \\min (|A^\\dagger |+|B^\\dagger |,1)=\\min (|A|+|B|,1),$ establishing (REF ) for $A,B$ .",
"Theorem 15.1 There exists a flow $(t,E)\\mapsto E(t)$ of elements of ${\\mathcal {L}}({\\mathbb {T}})$ , defined for $t\\in [0,T_E]$ , having the following properties.",
"$E(0)=E$ and $E(T_E) = {\\mathbb {T}}$ .",
"$E(s)\\subset E(t)$ whenever $s\\le t$ .",
"$|E(t)| = e^t|E|$ for all $t\\in [0,T_E]$ .",
"$|E(s)\\,\\Delta \\,E(t)|\\rightarrow 0$ as $s\\rightarrow t$ .",
"If $E\\subset \\tilde{E}$ then $E(s)\\subset \\tilde{E}(s)$ for all $s\\in [0,T_{\\tilde{E}}]$ .",
"$e^{-t}|E_1(t)\\,\\Delta \\,E_2(t)|\\le e^{-s} |E_1(s)\\,\\Delta \\,E_2(s)|$ for all $E_1,E_2$ and every $0\\le s\\le t\\le \\min (T_{E_1},T_{E_2})$ .",
"If $0\\le s\\le t\\le T_E$ then $E(t) = (E(s))(t-s)$ If $E$ is the rank one Bohr set $\\lbrace x: \\Vert \\phi (x) \\Vert \\le r\\rbrace $ associated to a nonconstant homomorphism $\\phi :{\\mathbb {T}}\\rightarrow {\\mathbb {T}}$ then $E(t) = \\lbrace x: \\Vert \\phi (x) \\Vert \\le e^t r\\rbrace $ .",
"$(E+y)(t) = E(t)+y$ for every $E\\in {\\mathcal {L}}({\\mathbb {T}})$ , $y\\in {\\mathbb {T}}$ , and $t\\le T_E$ .",
"Likewise, $(-E)(t)=-E(t)$ .",
"The function $t\\mapsto e^{-t}|E_1(t)\\,+_0\\,E_2(t)|$ is nonincreasing on $[0,\\min (T_{E_1},T_{E_2})]$ .",
"The function $t\\mapsto e^{-2t}{\\mathcal {T}}(E_1(t),E_2(t),E_3(t))$ is nondecreasing on $[0,\\tau ]$ provided that $\\tau \\le \\min _{j\\in \\lbrace 1,2,3\\rbrace }(T_{E_j})$ and $\\sum _{j=1}^3 |E_j(\\tau )|\\le 2$ .",
"Each conclusion is to be interpreted in terms of equivalence classes of measurable sets.",
"Thus, for instance, $A\\subset B$ means $|B\\setminus A|=0$ .",
"As mentioned earlier, a flow with variants of these properties is known for ${\\mathbb {R}}$ .",
"See for instance a discussion in [11].",
"Such a flow acting on a dense class of sets, namely finite unions of intervals, is discussed in [19].",
"That it extends to arbitrary sets has been known to experts [7], though it seems not to have been extensively discussed in the literature.",
"The flow for $\\mathbb {R}$ [11] preserves Lebesgue measures, whereas that of Theorem REF does not.",
"There exists no flow for ${\\mathbb {T}}$ that mimics all properties of the flow for ${\\mathbb {R}}$ .",
"Indeed, a rank one Bohr set $E\\subset {\\mathbb {T}}$ is a union of small intervals centered at the elements of a finite cyclic subgroup $H$ of ${\\mathbb {T}}$ , or at elements of a coset of $H$ .",
"$E$ satisfies $|E+E|=2|E|$ if $|E|<\\tfrac{1}{2}$ , so $E$ realizes equality in the sumset inequality.",
"There is no way to continuously deform one such set $E$ to another, through sets satisfying $|E(t)+E(t)|=2|E(t)|$ with $|E(t)|$ independent of $t$ , if the two sets in question are associated to subgroups $H$ having different numbers of elements.",
"The flow of Theorem REF lacks another key property of its analogue for ${\\mathbb {R}}$ , a lack which may appear to severely limit its utility, although we will show in the next section that it is nonetheless a valuable tool.",
"The functionals $e^{-2t}{\\mathcal {T}}(E_1(t),E_2(t),E_3(t))$ and $e^{-t}|E_1(t) +_0 E_2(t)|$ are only defined with desired monotonicity properties for $t\\le T$ , for a certain terminal time $T$ .",
"The defect is that the terminal time sets $E_j(T)$ need not possess any particular structure (such as $E_j(T) = E_j(T)^\\star $ up to translation, or $E_j(T)={\\mathbb {T}}$ ).",
"In contrast, the corresponding flow for ${\\mathbb {R}}$ deforms all three sets to their symmetrizations $E_j^\\star $ .",
"The proof is nearly identical in many respects to that of a corresponding result for ${\\mathbb {R}}$ proved in [11], with the exception of the conclusion concerning $|E_1(t)+_0 E_2(t)|$ .",
"We will provide only a sketch which deals with those points at which differences arise.",
"One begins by defining $t\\mapsto E(t)$ in the special case in which $E$ is a finite union of closed intervals.",
"One verifies the stated properties in that case, then uses these properties to show that the flow extends to ${\\mathcal {L}}({\\mathbb {T}})$ via uniform continuity with respect to the metric $\\rho (E,E^{\\prime }) = |E\\,\\Delta \\,E^{\\prime }|$ .The flow of Theorem REF acts on equivalence classes of sets.",
"Its restriction to finite unions of closed intervals agrees with the preliminary flow defined on finite unions of intervals, up to Lebesgue null sets.",
"Let $E=\\cup _j I_j$ (a finite union), where $I_j\\subset {\\mathbb {T}}$ is a closed arc of length $|I_j|$ with center $c_j$ , and these closed arcs are pairwise disjoint.",
"Define $E(t)=\\cup _j I_j(t)$ , where $I_j(t)$ is the arc with center $c_j$ and length $e^t|I_j|$ , for all $0\\le t\\le T_1$ , where $T_1$ is the smallest $t$ for which some pair of arcs $I_i(t),I_j(t)$ intersect.",
"Any two arcs that do intersect share only an endpoint (or two endpoints, in the case in which the union has length 1).",
"Thus $E(T_1)$ may be expressed in a unique way as a disjoint union of finitely many closed arcs, with certain centers.",
"The number of such arcs is strictly smaller than the number of arcs comprising the initial set $E$ .",
"Repeat the first step for this new collection of arcs, stopping at the first time $T_2>T_1$ at which intersection occurs.",
"Again reorganize $E(T_2)$ as a union of finitely many pairwise disjoint closed arcs, and repeat until a single arc remains.",
"This occurs, because the number of arcs is reduced with each iteration, and it is not possible for the number of arcs to exceed 1 if the measure of their union equals 1.",
"Continue until $|E(t)|=1$ .",
"We claim that if $E_j$ is a finite union of $N_j$ pairwise disjoint closed arcs for each index $j\\in \\lbrace 1,2,3\\rbrace $ , and if $\\tau >0$ is sufficiently small that $E_j(t)$ is defined for $t\\in [0,\\tau ]$ and is a union of exactly $N_j$ pairwise disjoint closed arcs for every $t\\in [0,\\tau )$ for each index $j$ , then $e^{-2t}{\\mathcal {T}}({\\mathbf {E}}(t))$ is a nondecreasing function of $t\\in [0,\\tau ]$ .",
"It suffices to prove this for $t\\in [0,T_1]$ .",
"Write ${\\mathbf {1}}_{E_j(t)} = \\sum _{n=1}^{N_j}{\\mathbf {1}}_{I_{j,n}(t)}$ with the natural notations.",
"Then $|I_{j,n}(t)| = e^t|I_{j,n}(0)|$ for all indices $j,n$ .",
"By linearity of ${\\mathcal {T}}$ , it suffices to show that $t\\mapsto e^{-2t}{\\mathcal {T}}({\\mathbf {I}}(t))$ is a nondecreasing function for any triple ${\\mathbf {I}}(t)=(I_j(t): j\\in \\lbrace 1,2,3\\rbrace )$ of intervals, with centers $c_j$ of $I_j(t)$ independent of $t$ and with lengths $|I_j(t)|=e^t|I_j(0)|$ .",
"By translation-invariance, we may assume that $c_1=c_2=0$ .",
"By reflecting about 0 if necessary, we may assume that the center $\\bar{c}_3:=-c_3$ of $-I_3$ satisfies $e^t \\bar{c}_3\\in [0,\\tfrac{1}{2}]$ .",
"Set $l_j=|I_j(0)|/2$ .",
"Now ${\\mathcal {T}}({\\mathbf {I}}(t)) = \\iint _{{\\mathbb {T}}^2}{\\mathbf {1}}_{ \\Vert x \\Vert \\le e^t l_1} {\\mathbf {1}}_{ \\Vert y \\Vert \\le e^t l_2} {\\mathbf {1}}_{ \\Vert x+y-\\bar{c}_3 \\Vert \\le e^t l_3}\\,dx\\,dy.$ Define $K(x) = {\\mathbf {1}}_{\\tilde{I}_1}*{\\mathbf {1}}_{\\tilde{I}_2}(x)$ for $x\\in {\\mathbb {R}}$ , where $\\tilde{I}_j = [-\\tfrac{1}{2} l_j,\\tfrac{1}{2} l_j]\\subset {\\mathbb {R}}$ .",
"Then, since $|I_1(t)|+|I_2(t)|<1$ (as $t<T_1$ ), ${\\mathcal {T}}({\\mathbf {I}}(t))$ can be expressed as $ {\\mathcal {T}}({\\mathbf {I}}(t)) = \\int _{{\\mathbb {R}}}e^t (K(e^{-t}u) + K(e^{-t}(u-1))) {\\mathbf {1}}_{|u-\\bar{c}_3|\\le e^tl_3}(u)\\,du.",
"$ Splitting this as a sum of two integrals and substituting $u=e^t x$ in one and $u = e^ty+1$ in the other gives $e^{-2t} {\\mathcal {T}}({\\mathbf {I}}(t))= \\int _{\\mathbb {R}}K(x){\\mathbf {1}}_{-I_3-\\bar{c}_3}(x-e^{-t}\\bar{c}_3)\\,dx+\\int _{\\mathbb {R}}K(y){\\mathbf {1}}_{-I_3-\\bar{c}_3}(y+e^{-t}(1-\\bar{c}_3))\\,dy.$ Because $K$ is nonnegative, even, and is nonincreasing on $[0,\\infty )$ , each of the two integrals above represents a nondecreasing function of $t$ for any interval $I_3$ .",
"This completes the proof of monotonicity.",
"The conclusions of Theorem REF now follow in the same way as in [11], with the exception of monotonicity of $e^{-t}|E_1(t) +_0 E_2(t)|$ , which was not discussed there.",
"Set $E_3 = -(E_1 +_0 E_2)$ .",
"Then ${\\mathcal {T}}(E_1,E_2,E_3) = |E_1||E_2|$ .",
"We have shown above that $t\\mapsto e^{-2t} {\\mathcal {T}}(E_1(t),E_2(t),E_3(t))$ is a nondecreasing function of $t$ .",
"In particular, $ e^{-2t} {\\mathcal {T}}(E_1(t),E_2(t),E_3(t)) \\ge {\\mathcal {T}}(E_1(0),E_2(0),E_3(0)) ={\\mathcal {T}}(E_1,E_2,E_3) = |E_1|\\cdot |E_2|.",
"$ But $ \\int _{\\mathbb {T}}{\\mathbf {1}}_{E_1(t)}* {\\mathbf {1}}_{E_2(t)}\\le |E_1(t)|\\cdot |E_2(t)| = e^{2t}|E_1|\\cdot |E_2|.$ Therefore $\\int _{-E_3(t)} {\\mathbf {1}}_{E_1(t)}* {\\mathbf {1}}_{E_2(t)}=\\int _{\\mathbb {T}}{\\mathbf {1}}_{E_1(t)}* {\\mathbf {1}}_{E_2(t)},$ forcing $\\lbrace x: {\\mathbf {1}}_{E_1(t)}*{\\mathbf {1}}_{E_2(t)}(x)>0\\rbrace \\subset -E_3(t)$ up to a Lebesgue null set.",
"Therefore $ e^{-t}|E_1(t) +_0 E_2(t)| \\le e^{-t}|E_3(t)|= |E_1+_0 E_2|.$ If $0\\le s\\le t$ then $E_j(t) = (E_j(s))(t-s)$ , so the general relation $ e^{-s}|E_1(s) +_0 E_2(s)| \\le e^{-t}|E_1(t) +_0 E_2(t)|$ follows from the case $s=0$ .",
"Remark 15.1 An equivalent formulation of the monotonicity of $e^{-2t}{\\mathcal {T}}(E_1(t),E_2(t),E_3(t))$ is that $t\\mapsto e^{-2t}{\\mathcal {D}}(E_1(t),E_2(t),E_3(t))$ is nonincreasing on $[0,\\tau ]$ , provided that $\\tau \\le \\min _{j\\in \\lbrace 1,2,3\\rbrace }T_{E_j}$ and $\\sum _{j=1}^3|E_j(\\tau )|\\le 2$ .",
"The monotonicity will be invoked in this form.",
"Indeed, for $t\\in [0,\\tau ]$ , $\\begin{aligned}e^{-2t}{\\mathcal {D}}(E_1(t),&E_2(t),E_3(t))\\\\&=e^{-2t}{\\mathcal {T}}\\big (E_1(t)^\\star ,E_2(t)^\\star ,(-E_3(t))^\\star \\big )-e^{-2t}{\\mathcal {T}}\\big (E_1(t),E_2(t),-E_3(t)\\big )\\\\&=e^{-2t}{\\mathcal {T}}\\big (E_1^\\star (t),E_2^\\star (t),(-E_3)^\\star (t)\\big )-e^{-2t}{\\mathcal {T}}\\big (E_1(t),E_2(t),(-E_3)(t)\\big )\\\\&={\\mathcal {T}}\\big (E_1^\\star ,E_2^\\star ,(-E_3)^\\star \\big )-e^{-2t}{\\mathcal {T}}\\big (E_1(t),E_2(t),(-E_3)(t)\\big ).\\end{aligned}$ Now $e^{-2t}{\\mathcal {T}}\\big (E_1(t),E_2(t),(-E_3)(t)\\big )$ is nondecreasing by the final conclusion of Theorem REF ; its hypotheses are satisfied since $|(-E_3)(\\tau )|=|E_3(\\tau )|$ and $T_{-E_3}=T_{E_3}$ .",
"The following remark, which will not be used in this paper but which may nonetheless be of interest, also follows in the same way as in [11].",
"Proposition 15.2 Let $E\\subset {\\mathbb {R}}^1$ be a Lebesgue measurable set with finite measure.",
"For each $t\\in (0,T_E]$ , $E(t)$ equals a union of intervals, up to a Lebesgue null set.",
"That is, there exists a countable family of pairwise disjoint intervals $I_n(t)$ such that $|E(t) \\,\\Delta \\,\\bigcup _n I_n(t)|=0$ .",
"The next lemma makes it possible to propagate control of a triple ${\\mathbf {E}}(t)$ backwards in time, with respect to the flow $t \\mapsto {\\mathbf {E}}(t)$ , in the analysis of inequality (REF ) for ${\\mathbb {T}}$ .",
"Lemma 15.3 (Time reversal) For each $\\eta ,\\eta ^{\\prime }>0$ there exist $\\delta _1>0$ and ${\\mathbf {C}}<\\infty $ with the following property.",
"Let ${\\mathbf {E}}$ be an $\\eta $ –strictly admissible ordered triple of measurable subsets of ${\\mathbb {T}}$ , satisfying $\\sum _j |E_j|\\le 2-\\eta ^{\\prime }$ .",
"Let $0<t\\le \\min _{1\\le j\\le 3}T_{E_j}$ with $e^t-1\\le \\delta _1$ .",
"Suppose that there exists $\\mathbf {y}=(y_1,y_2,y_3)\\in {\\mathbb {T}}^3$ satisfying $y_1+y_2=y_3$ such that $ |E_j(t)\\,\\Delta \\,(E_j(t)^\\star +y_j)| \\le \\delta _1 \\max _j |E_j(t)|\\qquad \\forall \\,j\\in \\lbrace 1,2,3\\rbrace .",
"$ Then there exists $\\mathbf {z}= (z_1,z_2,z_3)\\in {\\mathbb {T}}^3$ satisfying $z_1+z_2=z_3$ such that $ |E_j\\,\\Delta \\,(E_j^\\star +z_j)| \\le {\\mathbf {C}}{\\mathcal {D}}({\\mathbf {E}})^{1/2}\\qquad \\forall \\,j\\in \\lbrace 1,2,3\\rbrace .$ Requiring $\\delta _1\\le 1$ , as we may, yields $|E_j\\,\\Delta \\,(E_j^\\star +y_j)|&\\le |E_j\\,\\Delta \\,E_j(t)|+ |E_j(t)\\,\\Delta \\,(E_j(t)^\\star +y_j)|+ |(E_j(t)^\\star +y_j)\\,\\Delta \\,(E_j^\\star +y_j)|\\\\&\\le (e^t-1)|E_j|+ \\delta _1e^t\\max _k|E_k|+ (e^t-1) |E_j|\\\\&\\le (2(e^t-1)+\\delta _1)\\max _k|E_k|\\\\&=O(\\delta _1\\max _k|E_k|).$ Therefore, if $\\delta _1$ is sufficiently small, then ${\\mathbf {E}}$ satisfies the hypotheses of Lemma REF .",
"Its conclusion is the desired inequality (REF )."
],
[
"Concluding steps for ${\\mathbb {T}}$",
"In this section we prove the following slight improvement of Theorem REF in the case $G={\\mathbb {T}}$ .",
"The improvement lies in the absence of any lower bound for $\\min (m(A),m(B),m(C))$ .",
"That no lower bound is needed, is to be expected after the work of Bilu [1] on the sumset inequality.",
"Theorem 16.1 For each $\\eta >0$ there exist $\\delta _0>0$ and ${\\mathbf {C}}<\\infty $ with the following property.",
"Let $(A,B,C)$ be an $\\eta $ –strictly admissible ordered triple of Lebesgue measurable subsets of ${\\mathbb {T}}$ satisfying $m(A)+m(B)+m(C)\\le 2-\\eta $ .",
"Let $\\delta \\le \\delta _0$ .",
"If $\\int _C {\\mathbf {1}}_A*{\\mathbf {1}}_B\\,dm \\ge \\int _{C^\\star }{\\mathbf {1}}_{A^\\star }*{\\mathbf {1}}_{B^\\star }\\,dm-\\delta \\max (m(A),m(B),m(C))^2$ then there exists a compatibly centered parallel ordered triple $({\\mathcal {B}}_A,{\\mathcal {B}}_B,{\\mathcal {B}}_C)$ of rank one Bohr subsets of ${\\mathbb {T}}$ satisfying $m(A\\,\\Delta \\,{\\mathcal {B}}_A) \\le {\\mathbf {C}}\\delta ^{1/2} \\max (m(A),m(B),m(C))$ and likewise for $(B,{\\mathcal {B}}_B)$ and $(C,{\\mathcal {B}}_C)$ .",
"By Theorem REF , the desired conclusion holds for all triples $(A,B,C)$ that additionally satisfy $\\min (m(A),m(B),m(C))\\ge \\tfrac{1}{3}\\eta ^2$ .",
"Now, let $(A,B,C)$ be a triple satisfying the hypotheses of the theorem, but with $\\min (m(A),m(B),m(C))<\\tfrac{1}{3}\\eta ^2.$ Set ${\\mathbf {E}}= (E_1,E_2,E_3) = (A,B,C)$ and consider the flowed triples ${\\mathbf {E}}(t)$ for $0\\le t\\le T$ , with $T$ chosen so that $\\min _{j=1,2,3} m(E_j(t))=\\tfrac{1}{3}\\eta ^2.$ That is, $\\tfrac{1}{3}\\eta ^2=e^Tm$ , for $m:=\\min _{j=1,2,3}m(E_j)$ .",
"For all $t\\in [0,T]$ , the triple ${\\mathbf {E}}(t)$ is $\\eta $ -strictly admissible.",
"Setting $M := \\max _{j=1,2,3}m(E_j)$ , the $\\eta $ -strict admissibility of ${\\mathbf {E}}$ ensures that $\\max _{j=1,2,3}m(E_j(T))=e^TM\\le e^Tm\\eta ^{-1}=\\tfrac{1}{3}\\eta ,$ whence $\\sum _{j=1}^3m(E_j(t))\\le \\eta \\le 2-\\eta \\text{ for all }t\\in [0,T].$ Moreover, the assumption ${\\mathcal {D}}({\\mathbf {E}})\\le \\delta M^2$ together by the monotonicity of the Riesz-Sobolev functional under the flow (discussed in §) imply that ${\\mathcal {D}}({\\mathbf {E}}(t))\\le e^{2t}{\\mathcal {D}}({\\mathbf {E}})\\le e^{2t}\\delta M^2=\\delta \\max _{j=1,2,3}m(E_j(t))^2\\text{ for all }t\\in [0,T].$ The triple ${\\mathbf {E}}(T)$ enjoys the additional property that it is $\\eta ^2$ -bounded, and therefore satisfies the hypotheses of by Theorem REF with parameters depending only on $\\eta $ .",
"It follows that, provided that $\\delta _0$ is sufficiently small as a function of $\\eta $ alone, there exists a compatibly centered parallel ordered triple ${\\mathbf {B}}:=({\\mathcal {B}}_1,{\\mathcal {B}}_2,{\\mathcal {B}}_3)$ of rank one Bohr sets with $m\\big ({\\mathcal {B}}_j\\,\\Delta \\,E_j(T)\\big )\\le {\\mathbf {C}}\\delta ^{1/2}\\max _{j=1,2,3}m(E_j(T)).$ Assuming again that $\\delta _0$ is sufficiently small as a function of $\\eta $ , the time reversal Lemma REF can be applied in a straightforward series of reverse time steps to conclude that there exists a compatibly centered triple $({\\mathcal {B}}_1^{\\prime },{\\mathcal {B}}_2^{\\prime },{\\mathcal {B}}_3^{\\prime })$ of rank one Bohr sets such that $ m({\\mathcal {B}}_j^{\\prime }\\,\\Delta \\,E_j)\\le {\\mathbf {C}}{\\mathcal {D}}({\\mathbf {E}})^{1/2} $ for each $j\\in \\lbrace 1,2,3\\rbrace $ ."
]
] | 1808.08368 |
[
[
"Circulant matrices and Galois-Togliatti systems"
],
[
"Abstract The goal of this article is to compare the coefficients in the expansion of the permanent with those in the expansion of the determinant of a three-lines circulant matrix.",
"As an application we prove a conjecture concerning the minimality of Galois-Togliatti systems."
],
[
"Introduction",
"Circulant matrices appear naturally in many areas of mathematics.",
"In the last decades, for instance, they have been related to holomorphic mappings ([7]), cryptography, coding theory ([10]), digital signal processing ([9]), image compression ([22]), physics ([2]), engineering simulations, number theory, theory of statistical designs ([12]), etc.",
"Even if the basic facts about these matrices can be proved in elementary way, many questions about them are subtle and remain still open (see, for instance, [11]).",
"Our interest in this topic was originally motivated by its connections, exposed in [17], with a class of homogeneous ideals of a polynomial ring failing the Weak Lefschetz Property.",
"In that context, the first question relevant to us was that of determining which monomials in the entries of a “generic” circulant matrix appear explicitly in the development of its determinant.",
"More precisely, let us denote by $\\operatorname{Circ}(x_{0},x_{1}\\cdots , x_{d-1})$ the circulant matrix of the form $\\begin{pmatrix}x_0&x_1&x_2&\\cdots &x_{d-1}\\\\x_{d-1}&x_0&x_1&\\cdots &x_{d-2}\\\\x_{d-2}&x_{d-1}&x_0&\\cdots &x_{d-3}\\\\\\vdots &\\vdots &\\vdots &\\ddots &\\vdots \\\\x_1&x_2&x_3&\\cdots &x_0\\end{pmatrix},$ where $x_0, \\cdots , x_{d-1}$ are complex numbers, or more generally elements of a ring.",
"Every summand of the determinant $\\det \\operatorname{Circ}(x_{0},x_{1}\\cdots , x_{d-1})$ is of the form $c_{i_0, \\cdots ,i_{d-1}}x_0^{i_0}\\cdots x_{d-1}^{i_{d-1}},$ where $c_{i_0, \\cdots ,i_{d-1}}\\in \\mathbb {Z}$ and $i_0+\\cdots +i_{d-1}=d$ .",
"The question is: for which indices $i_0, \\cdots ,i_{d-1}$ is the coefficient $c_{i_0, \\cdots ,i_{d-1}}$ different from zero?",
"An analogous question can be posed for the permanent of $\\operatorname{Circ}(x_{0},x_{1}\\cdots , x_{d-1})$ .",
"In this case the answer was given in [4], where it was proved that the monomials appearing with non-zero coefficient are precisely those whose exponents satisfy the two conditions: ${\\left\\lbrace \\begin{array}{ll}i_0+\\cdots +i_{d-1}=d\\\\0i_0+1i_1+2i_2+\\cdots +(d-1)i_{d-1}\\equiv 0 \\pmod {d}.\\end{array}\\right.",
"}$ Clearly, conditions (REF ) are necessary for the non-vanishing of the coefficient $c_{i_0, \\cdots ,i_{d-1}}$ in $\\det \\operatorname{Circ}(x_{0},x_{1}\\cdots , x_{d-1})$ .",
"It has been recently proved that they are also sufficient if and only if $d>1$ is a power of a prime number ([25], [6]).",
"A more general question is to find a formula for the coefficient $c_{i_0, \\cdots ,i_{d-1}}$ .",
"This problem had already been considered in 1951 by Ore [23], who gave an explicit expression.",
"Other expressions were given more recently in [14], [29].",
"However, they are not always easy to apply in order to decide if $c_{i_0, \\cdots ,i_{d-1}}$ vanishes or not.",
"In this article, we are interested in the so-called $r$ -lines circulant matrices, i.e.",
"circulant matrices $\\operatorname{Circ}(x_{0},x_{1}\\cdots , x_{d-1})$ of order $d>r$ , where $d-r$ among $x_0, \\cdots , x_{d-1}$ are specialized to 0.",
"We ask if for some pairs $(r,d)$ , $r<d$ , conditions (REF ) are sufficient for the non-vanishing of the corresponding coefficient in the determinant.",
"Our main result is Theorem REF , where we prove that conditions (REF ) are sufficient in the case of 3-lines circulant matrices of order $d$ , of the form $\\operatorname{Circ}(x,0,\\cdots , 0, y, 0, \\cdots , 0, z, 0, \\cdots , 0),$ where $y$ appears in position $a$ (counting from zero), $z$ appears in position $b$ , and $\\operatorname{GCD}(a,b,d)=1$ .",
"We also give examples of: 3-lines circulant matrices with $\\operatorname{GCD}(a,b,d)\\ne 1$ , $r$ -lines circulant matrices with $r\\ge 4$ and similar $\\operatorname{GCD}$ equal to one, for which the analogous property fails.",
"Moreover, we prove that the coefficient of any specific monomial in a 3-lines circulant determinant is always equal, up to the sign, to the analogous coefficient in the permanent of the same matrix under the assumption $\\operatorname{GCD}(a,b,d)=1$ .",
"Our results are inspired by and extend a previous result, concerning 3-lines circulant matrices of the special form $\\operatorname{Circ}(x, y, 0, \\cdots , 0, z, 0, \\cdots , 0)$ , given by Loehr, Warrington and Wilf [13].",
"This case, i.e.",
"$a=1$ , was also studied by Codenotti and Resta [5] who gave an expression for twice the permanent as a sum of four related determinants.",
"This setting easily extends to the cases when at least one of $\\operatorname{GCD}(a,d)$ , $\\operatorname{GCD}(b,d)$ , $\\operatorname{GCD}(b-a,d)$ equals 1.",
"The second part of this article is devoted to describing applications of Theorem REF ; it concerns mainly the minimality of Galois-Togliatti systems in three variables.",
"These systems, abbreviated GT-systems, form a class of ideals of a polynomial ring introduced and studied in [17].",
"A GT-system in three variables is the homogeneous artinian ideal $I^d_{a,b}$ of $R:=\\mathbb {K}[x_0, x_1, x_2]$ , generated by all forms of degree $d$ , that are invariant under the action of a diagonal matrix $M_{a,b}:=\\begin{pmatrix}1&0&0\\\\0&e^a&0\\\\0&0&e^b\\end{pmatrix}$ , where $e$ is a $d$ -th root of the unit.",
"Note that the group generated by $M_{a,b}$ is the cyclic group of order $d$ provided $\\operatorname{GCD}(a,b,d)=1$ , and that all actions of this group on $R$ can be represented by a matrix of the form $M_{a,b}$ .",
"In [17] it was proved that, if $\\operatorname{GCD}(a,b,d)=1$ , then the ideal $I^d_{a,b}$ is a Togliatti system.",
"This means that it fails the Weak Lefschetz Property in degree $d-1$ , i.e.",
"for a general linear form $L$ (equivalently, for all linear forms), the multiplication map $\\times L\\colon (R/I^d_{a,b})_{d-1}\\longrightarrow (R/I^d_{a,b})_{d}$ is not injective.",
"The authors then conjectured that $I^d_{a,b}$ is a minimal Togliatti system.",
"As an application of Theorem REF we prove this conjecture.",
"Next we outline the structure of this paper.",
"Section contains our main results about circulant matrices.",
"After introducing $r$ -lines circulant matrices, we give a precise formulation of the problems we want to study (Question REF ).",
"We then state our main theorems (Theorems REF and REF ) and produce some examples showing that our results are optimal (Examples REF and REF ).",
"Subsection REF contains the proofs of the two theorems; it relies on a series of lemmas, aiming to describe the structure, in the symmetric group on $d$ elements, of the permutations that contribute non-trivially in the development of the circulant determinants we study.",
"Section is devoted to the application of the results of Section to Togliatti systems.",
"We first recall the background about the Weak Lefschetz Property and Togliatti systems, in particular minimal monomial Togliatti systems and GT-systems, and the conjecture on the minimality of GT-systems in three variables.",
"Finally, Theorem REF shows how the conjecture follows from the results of Section .",
"In Section we indicate a computational complexity application of our main result.",
"Notation Throughout this paper $\\mathbb {K}$ will be an algebraically closed field of characteristic zero, $R=\\mathbb {K}[x_0,x_1,\\cdots ,x_n]$ and $\\mathbb {P}^n=\\operatorname{Proj}(\\mathbb {K}[x_0,x_1,\\cdots ,x_n])$ .",
"For any polynomial $F\\in R$ , we denote by $[F]_{i_0,i_1,\\cdots ,i_n}$ the coefficient of the monomial $x_0^{i_0}x_1^{i_1}\\cdots x_n^{i_n}$ in $F$ .",
"Hence, we have $F=\\sum _{i_0,i_1,\\cdots ,i_n} [F]_{i_0,i_1,\\cdots ,i_n}x_0^{i_0}x_1^{i_1}\\cdots x_n^{i_n}$ .",
"Let $S_d$ denote the symmetric group on $d$ elements.",
"Acknowledgments This work was started at the workshop “Lefschetz Properties and Jordan Type in Algebra, Geometry and Combinatorics,” held at Levico (Trento) in June 2018.",
"The authors thank the Centro Internazionale per la Ricerca Matematica (CIRM) for its support.",
"We also thank Nati Linial and Amir Shpilka for helpful discussions on the computational complexity aspects and for pointing us to [5]."
],
[
"Three-lines circulant matrices",
"This section is devoted to the study of circulant matrices and their determinant and permanent.",
"They have been previously studied by Ore [23], Kra and Simanca [11], Wyn-Jones [29] and Malenfant [14]; we will mainly follow Loehr, Warrington and Wilf [13].",
"Let us start by recalling their definition: Definition 2.1 Let $M=(y_{i,j})$ be a $d\\times d$ matrix.",
"$M$ is a circulant matrix if, and only if $y_{i,j}=y_{k,l}$ whenever $j-i\\equiv l-k\\pmod {d}$ .",
"That is, $M$ is of the type $\\begin{pmatrix}x_0&x_1&x_2&\\cdots &x_{d-1}\\\\x_{d-1}&x_0&x_1&\\cdots &x_{d-2}\\\\x_{d-2}&x_{d-1}&x_0&\\cdots &x_{d-3}\\\\\\vdots &\\vdots &\\vdots &\\ddots &\\vdots \\\\x_1&x_2&x_3&\\cdots &x_0\\end{pmatrix}$ where successive rows are circular permutations of the first row.",
"It is a particular form of a Toeplitz matrix, i.e.",
"a matrix whose elements are constant along the diagonals.",
"For short we denote such matrices as $\\operatorname{Circ}_d(x_{0},x_{1}\\cdots , x_{d-1})$ or simply $\\operatorname{Circ}_d$ .",
"We now define an $r$ -lines circulant matrix as follows: we fix an integer $r\\le d$ and an $r-$ tuple of integers $0\\le \\alpha _{0}<\\cdots <\\alpha _{r-1}\\le d-1$ and define $\\operatorname{Circ}_{(d;\\alpha _{0},\\cdots ,\\alpha _{r-1})}:=(\\operatorname{Circ}_{d})(0,\\cdots ,0,x_{\\alpha _{0}},0,\\cdots ,0,x_{\\alpha _{i}},0,\\cdots ,0,x_{\\alpha _{r-1}},0\\cdots ,0)$ where $x_{\\alpha _{i}}$ is located at the $\\alpha _{i}+1$ position.",
"Notice that $\\operatorname{Circ}_{(d;\\alpha _{0},\\cdots , \\alpha _{r-1})}$ is nothing but the specialization of $\\operatorname{Circ}_{d}$ to $\\lbrace x_{i}=0\\mid i\\notin \\lbrace \\alpha _{0},\\cdots ,\\alpha _{r-1}\\rbrace \\rbrace $ .",
"Let us denote by $D_{(d;\\alpha _{0},\\cdots ,\\alpha _{r-1})}$ (resp.",
"$P_{(d;\\alpha _{0},\\cdots ,\\alpha _{r-1})}$ ) the number of different monomials that appear with non-zero coefficient in the expansion of the determinant $\\det (\\operatorname{Circ}_{(d;\\alpha _{0},\\cdots ,\\alpha _{r-1})})$ (resp.",
"permanent $\\operatorname{per}(\\operatorname{Circ}_{(d;\\alpha _{0},\\cdots ,\\alpha _{r-1})})$ ).",
"We always have $D_{(d;\\alpha _{0},\\cdots ,\\alpha _{r-1})}\\le P_{(d;\\alpha _{0},\\cdots ,\\alpha _{r-1})}$ and we are lead to pose the following questions: Question 2.2 Fix integers $r\\le d$ and an $r-$ tuple of integers $0\\le \\alpha _{0}<\\cdots <\\alpha _{r-1}\\le d-1$ .",
"(1) Is $D_{(d;\\alpha _{0},\\cdots ,\\alpha _{r-1})}=P_{(d;\\alpha _{0},\\cdots ,\\alpha _{r-1})}$ ?",
"(2) More strongly, comparing coefficients, is $[\\det (\\operatorname{Circ}_{(d;\\alpha _{0},\\cdots ,\\alpha _{r-1})})]_{d-A_1-\\cdots -A_{r-1},A_1,\\cdots ,A_{r-1}}=$ $\\pm [\\operatorname{per}(\\operatorname{Circ}_{(d;\\alpha _{0},\\cdots ,\\alpha _{r-1})})]_{d-A_1-\\cdots -A_{r-1},A_1,\\cdots ,A_{r-1}}?$ In this paper, we deal with 3-lines circulant matrices.",
"Without loss of generality we can always assume $\\alpha _0=0$ and we set $a=\\alpha _1$ and $b=\\alpha _2$ .",
"We answer both questions affirmatively under the condition $\\operatorname{GCD}(a,b,d)=1$ : Theorem 2.3 Fix integers $d\\ge 3$ and $1\\le a<b\\le d-1$ .",
"Assume $\\operatorname{GCD}(a,b,d)=1$ .",
"Then, $[\\det (\\operatorname{Circ}_{(d;0,a,b))})]_{d-A-B,A,B}=\\pm [\\operatorname{per}(\\operatorname{Circ}_{(d;0,a,b))})]_{d-A-B,A,B}$ .",
"In particular, $D_{(d;0,a,b)}=P_{(d;0,a,b)}$ .",
"The case $a=1$ in the above theorem was established in [13], and our proof strategy largely follows theirs.",
"In fact, our proof gives the sign, as well as a combinatorial interpretation of the magnitude, for the coefficients of $\\det (\\operatorname{Circ}_{(d;0,a,b))})$ : Theorem 2.4 Fix integers $d\\ge 3$ and $1\\le a<b\\le d-1$ such that $\\operatorname{GCD}(a,b,d)=1$ .",
"Then, for any nonnegative integers $A,B$ such that $A+B\\le d$ , (1) $[\\det (\\operatorname{Circ}_{(d;0,a,b)})]_{d-A-B,A,B}\\ne 0$ if and only if $d|(aA+bB)$ .",
"Further, assuming $d|(aA+bB)$ , then: (2) the sign of $[\\det (\\operatorname{Circ}_{(d;0,a,b)})]_{d-A-B,A,B}$ is $+$ if and only if at least one of $\\operatorname{GCD}(A,B,\\frac{aA+bB}{d})$ and $A+B-1$ is even; and (3) the magnitude of $[\\det (\\operatorname{Circ}_{(d;0,a,b)})]_{d-A-B,A,B}$ equals the number of permutations in $S_d$ with cycle decomposition $C_1\\circ C_2\\circ \\cdots \\circ C_k$ , where $k=\\operatorname{GCD}(A,B,\\frac{aA+bB}{d})$ , and each $C_i$ has length $A+B$ and consists of exactly $A$ elements $j$ with $C_i(j)\\equiv j+a$ and $B$ elements $j$ with $C_i(j)\\equiv j+b$ , modulo $d$ .",
"Remark 2.5 For $r=2$ we may assume $\\alpha _0=0$ and $\\alpha _1=a$ divides $d$ , with $x$ on the main diagonal and $y$ on another nontrivial diagonal of the circulant $d\\times d$ -matrix.",
"One easily verifies that all coefficients of the determinant $\\det (\\operatorname{Circ}_{(d;0,a)})$ equal up to sign to the corresponding coefficients in the permanent $\\operatorname{per}(\\operatorname{Circ}_{(d;0,a)})$ , and are given by the explicit formula $\\det (\\operatorname{Circ}_{(d;0,a)})=\\sum _{s=0}^{a} (-1)^{(a-s)(\\frac{d}{a}-1)} \\binom{a}{s} x^{\\frac{ds}{a}}y^{\\frac{d(a-s)}{a}}.$ However, for $r=3$ , the following example shows that the assumption $\\operatorname{GCD}(a,b,d)=1$ cannot be dropped from Theorem REF : Example 2.6 Indeed, we take $(a,b,d)=(2, 6, 12)$ .",
"We compute the permanent and the determinant of $\\operatorname{Circ}_{(12;0,2,6)}$ and get: $\\det (\\operatorname{Circ}_{(12;0,2,6)})= x^{12}-6x^{10}y^2+15x^8y^4-20x^6y^6+15x^4y^8-6x^2y^{10}+\\\\+y^{12}-12x^8yz^3+32x^6y^3z^3-24x^4y^5z^3+4y^9z^3-2x^6z^6+42x^4y^2z^6+18x^2y^4z^6+\\\\+6y^6z^6+12x^2yz^9+4y^3z^9+z^{12}$ and $\\operatorname{per}(\\operatorname{Circ}_{(12;0,2,6)})=x^{12}+6x^{10}y^2+15x^8y^4+20x^6y^6+15x^4y^8+6x^2y^{10}+\\\\+y^{12}+12x^8yz^3+40x^6y^3z^3+48x^4y^5z^3+24x^2y^7z^3+4y^9z^3+2x^6z^6+42x^4y^2z^6+\\\\+30x^2y^4z^6+6y^6z^6+12x^2yz^9+4y^3z^9+z^{12}.$ Therefore, we have $D_{(12;0,2,6)}=18<P_{(12;0,2,6)}=19$ and $[\\det (\\operatorname{Circ}_{(12;0,2,6))})]_{6,3,3}=32 \\ne [\\operatorname{per}(\\operatorname{Circ}_{(12;0,2,6))})]_{6,3,3}=40.$ Example 2.7 For $r$ -lines circulant matrices with $r\\ge 4$ Theorem REF is no longer true.",
"In fact, For $r=4$ , we have $D_{(6;0,2,4,5)}<P_{(6;0,2,4,5)}$ since the monomial $xz^2uv^2$ appears in $\\operatorname{per}(\\operatorname{Circ}(x,0,z,0,u,v))$ but it does not appear in $\\det (\\operatorname{Circ}(x,0,z,0,u,v))$ (see [6] and [5]).",
"Assume $r\\ge 5$ .",
"We choose two prime integers $p$ and $q$ such that $p<q$ and $r\\le pq$ .",
"Set $d=pq$ .",
"We will first prove that $D_{(d;0,1,\\cdots , d-1)}<P_{(d;0,1,\\cdots , d-1)}$ .",
"To this end we exhibit a $d$ -tuple $A_0,A_1,\\cdots ,A_{d-1}$ such that $A_0+2A_1+\\cdots +dA_{d-1}&\\equiv 0 \\pmod {d}\\\\A_0+A_1+\\cdots +A_{d-1}&=d$ and $[\\det (\\operatorname{Circ}_{(d;0,1,\\cdots ,d-1)})]_{A_0,A_1,\\cdots ,A_{d-1}}=0.$ We apply Bezout's theorem and we write $\\lambda q=1+\\mu p$ with $1\\le \\lambda , \\mu $ and $\\lambda q< d$ .",
"We define $A_0=d-\\mu p-2$ , $A_1=\\mu p-1$ , $A_{d-\\mu p}=A_{\\mu p -\\mu \\lambda +1}=A_{d-\\mu p+\\lambda \\mu }=1$ , and $A_{i}=0$ for $i\\ne 0,1, d-\\mu p,\\mu p -\\mu \\lambda +1,d-\\mu p+\\lambda \\mu $ .",
"By the proof of [6], $[\\det (\\operatorname{Circ}_{(d;0,1,\\cdots , d-1)})]_{A_0,A_1,\\cdots ,A_{d-1}}=0$ , i.e.",
"the monomial $x_0^{d-\\mu p-2}x_1^{\\mu p -1}x_{d-\\mu p}x_{\\mu p- \\lambda \\mu +1}x_{d-\\mu p+\\lambda \\mu }$ appears in $\\operatorname{per}(\\operatorname{Circ}(x_{0},x_{1}\\cdots , x_{d-1}))$ but it does not appear in $\\det (\\operatorname{Circ}(x_{0},x_{1}\\cdots , x_{d-1}))$ .",
"Therefore, $D_{(d;0,1,\\cdots , d-1)}<P_{(d;0,1,\\cdots , d-1)}$ .",
"Since $5\\le r\\le d$ , for any choice of an $r$ -tuple $(a_0,a_1,\\cdots ,a_{r-1})$ containing $\\lbrace 0,1, d-\\mu p,\\mu p -\\mu \\lambda +1,d-\\mu p+\\lambda \\mu \\rbrace $ , the monomial $x_0^{d-\\mu p-2}x_1^{\\mu p -1}x_{d-\\mu p}x_{\\mu p- \\lambda \\mu +1}x_{d-\\mu p+\\lambda \\mu }$ appears in the permanent of the $d\\times d$ $r$ -lines circulant matrix $\\operatorname{Circ}_{(d;a_0,a_1,\\cdots , a_{r-1})}$ but it does not appear in the determinant.",
"Therefore, $D_{(d;a_0,a_1,\\cdots ,a_{r-1})}<P_{(d;a_0, a_1,\\cdots ,a_{r-1})}$ and we are done."
],
[
"Notation",
"For a permutation $\\sigma $ of $d$ elements $I:=\\lbrace 0,\\cdots ,d-1\\rbrace $ and an integer $q$ , we define $S_{q,\\sigma }:=\\lbrace i\\in I\\mid \\sigma (i)\\equiv i+q \\pmod {d}\\rbrace .$ Let: $P_{a,b,d,A,B}:=\\lbrace \\sigma \\in S_d\\mid \\vert S_{a,\\sigma }\\vert =A, \\vert S_{b,\\sigma }\\vert =B, \\vert S_{0,\\sigma }\\vert =d-A-B\\rbrace .$ The permutations in $P_{a,b,d,A,B}$ can be characterised as those in $S_d$ where each $i$ is either fixed, i.e.",
"$\\sigma (i)=i$ , or translated $a$ or $b$ steps forward, i.e.",
"$\\sigma (i)\\equiv i+a$ or $\\sigma (i)\\equiv i+b\\pmod {d}$ , and further, the second situation $\\sigma (i)\\equiv i+a$ happens exactly $A$ times and the last one exactly $B$ times.",
"Clearly, Lemma 2.8 The following equalities hold: $[\\det (\\operatorname{Circ}_{(d;0,a,b)})]_{d-A-B,A,B}&=\\sum _{\\sigma \\in P_{a,b,d,A,B}}\\operatorname{sgn}(\\sigma ),\\\\[\\operatorname{per}(\\operatorname{Circ}_{(d;0,a,b)})]_{d-A-B,A,B}&=\\sum _{\\sigma \\in P_{a,b,d,A,B}}\\vert \\operatorname{sgn}(\\sigma )\\vert =\\vert P_{a,b,d,A,B}\\vert .$ $\\Box $ We often work with cyclic indices, say modulo $d$ : an element in $\\mathbb {Z}/d\\mathbb {Z}$ is uniquely determined by an element in $s\\in I=\\lbrace 0,\\cdots ,d-1\\rbrace $ ; by abuse of notation, we frequently identify in what follows this integer $s$ with its class in $\\mathbb {Z}/d\\mathbb {Z}$ .",
"With this identification, we will write $s_1<s_2<s_3<\\cdots $ , with $s_i\\in \\mathbb {Z}/d\\mathbb {Z}$ if, for the corresponding elements in $I$ we have $0<s_2-s_1<s_3-s_1<\\cdots $ .",
"Example 2.9 We have $4<1<2$ modulo 5, as $0<2<3$ .",
"On the other hand it is not true that $1<3<2$ ."
],
[
"Structure of permutations",
"In this subsection we fix a permutation $\\sigma \\in P_{a,b,d,A,B}$ and its canonical cycle decomposition $\\sigma =C_1\\circ C_2\\circ \\cdots \\circ C_k$ .",
"We set $A_i=\\vert S_{a,C_i}\\vert $ , $B_i=\\vert S_{b,C_i}\\vert $ and “winding number” $\\ell _i=\\frac{A_ia+B_ib}{d}$ .",
"Our aim is to prove, in steps, that the canonical cycle decomposition $\\sigma =C_1\\circ C_2\\circ \\cdots \\circ C_k$ must be of a very special type, as described in Theorem REF (3), and in particular all permutations $\\sigma \\in P_{a,b,d,A,B}$ have the same cycle structure, hence the same sign, implying Theorem REF .",
"The following lemma is a straightforward generalization of [13], where the case $a=1$ was considered.",
"We include the proof for the sake of completeness.",
"Lemma 2.10 For any integer $i$ , $1\\le i \\le k$ , we have $\\operatorname{GCD}(A_i,B_i,\\ell _i)=1$ .",
"Let $g:=\\operatorname{GCD}(A_i,B_i,\\ell _i)$ .",
"We consider the cycle $C_i=(s_1,\\cdots ,s_w)$ , where $w=A_i+B_i$ and $s_j\\in \\mathbb {Z}/d\\mathbb {Z}$ .",
"Let $w=gw^{\\prime }$ .",
"To simplify notation we assume that the indexes $j$ of each $s_j$ are considered modulo $w$ .",
"We know that $s_{j+1}-s_{j}$ is congruent either to $a$ or $b$ .",
"Hence, we may represent $C_i$ as a word $W$ of length $w$ with letters $a,b$ : $W:=s_2-s_1,s_3-s_2,\\cdots ,s_w-s_{w-1},s_1-s_w=:p_1\\cdots p_w.$ Claim: There exists a subword $W^{\\prime }=p_{w_0+1},p_{w_0+2},\\cdots ,p_{w_0+w^{\\prime }}$ of the word $W$ such that: the number of $a$ 's in $W^{\\prime }$ equals $A_i/g$ and the number of $b$ 's in $W^{\\prime }$ equals $B_i/g$ .",
"As the length of the word $W^{\\prime }$ is fixed to be $w^{\\prime }$ , the number of letters $b$ is determined by the number of letters $a$ in it.",
"Hence, it is enough to find $W^{\\prime }$ that satisfies the first condition.",
"The word $W$ is a concatenation of $g$ words of length $w/g$ : $W=W_1\\cdots W_g.$ If one of the $W_j$ 's has $A_i/g$ letters $a$ the claim follows.",
"Hence, we assume each of them has either strictly more or strictly less letters $a$ then $A_i/g$ .",
"As the total number of letters $a$ in $W$ equals $A_i$ , it is not possible that all words $W_j$ have simultaneously more or simultaneously less letters $a$ then $A_i/g$ .",
"Thus, we may find two consecutive words $W_j$ , $W_{j+1}$ and assume without loss of generality that $W_j$ has less and $W_{j+1}$ has more than $A_i/g$ letters $a$ .",
"Consider the sequence of all subwords of $W_jW_{j+1}$ of length $w^{\\prime }$ ordering them by the starting index: $W_j=W^{\\prime }_1,W^{\\prime }_2,\\cdots ,W^{\\prime }_{w^{\\prime }+1}=W_{j+1}.$ The word $W^{\\prime }_{s+1}$ is shifted by one index to the right, with respect to the word $W^{\\prime }_s$ .",
"In particular, $W^{\\prime }_{s+1}$ is getting exactly one additional letter and looses exactly one letter.",
"Hence, the number of letters $a$ in $W^{\\prime }_s$ and $W^{\\prime }_{s+1}$ may differ by at most one.",
"In particular, as the starting word $W_j=W^{\\prime }_1$ has less than $A_i/g$ letters $a$ and the last word $W_{j+1}=W^{\\prime }_{w^{\\prime }+1}$ has more than $A_i/g$ letters $a$ , there must exist a word $W^{\\prime }=W^{\\prime }_s$ with precisely $A_i/g$ letters $a$ .",
"We may cyclically permute the entries $(s_1,\\cdots ,s_w)$ of $C_i$ .",
"By the Claim we may assume that in the multiset $\\lbrace s_2-s_1,s_3-s_2,\\cdots , s_{w^{\\prime }}-s_{w^{\\prime }-1},s_{w^{\\prime }+1}-s_{w^{\\prime }}\\rbrace $ there are precisely $A_i/g$ differences $a$ and $B_i/g$ differences $b$ .",
"Summing up all the elements of this multiset modulo $d$ we obtain: $s_{w^{\\prime }+1}-s_1=a\\frac{A_i}{g}+b\\frac{B_i}{g}=\\frac{aA_i+bB_i}{g}=\\frac{\\ell _id}{g}=\\frac{\\ell _i}{g}d\\equiv 0 \\pmod {d}.$ Hence, $s_{w^{\\prime }+1}\\equiv s_1$ .",
"As the cycle $C_i$ was assumed to be primitive, the $s_j$ 's must be all distinct.",
"Hence, $w^{\\prime }+1=1$ modulo $w$ , which means that $w^{\\prime }=w$ and $g=1$ .",
"Lemma 2.11 Suppose $\\ell _1=\\ell _2=1$ .",
"Then $A_1=A_2$ and $B_1=B_2$ .",
"First we treat the special case when one of the numbers $A_1,A_2,B_1,B_2$ equals 0, say $B_1=0$ .",
"As $\\ell _1=1$ we have $d=a\\cdot A_1$ and without loss of generality we may assume that $C_1$ consists of all numbers divisible by $a$ .",
"If $B_2=0$ we are done.",
"We assume $B_2\\ne 0$ in order to obtain a contradiction.",
"Let $C_2=(s_1,\\cdots ,s_w)$ for $w=A_2+B_2$ .",
"As $C_1$ and $C_2$ are disjoint, we know that each $s_i$ is not divisible by $a$ .",
"As $a|d$ and $\\operatorname{GCD}(a,b,d)=1$ we have $\\operatorname{GCD}(a,b)=1$ .",
"Let $0\\le t<a$ be such that $tb\\equiv -s_1$ modulo $a$ .",
"We have $A_2a+B_2b=d$ thus $a|B_2b$ and hence $a|B_2$ .",
"In particular $B_2\\ge a>t$ .",
"Hence, there exists such $s_{i_0}$ that $\\vert \\lbrace 1\\le i<i_0:s_{i+1}=s_i+b\\rbrace \\vert =t,$ i.e.",
"until $i_0$ , the cycle $C_2$ made exactly $t$ jumps of size $b$ .",
"Modulo $a$ we have: $s_{i_0}=s_1+tb=s_1-s_1=0,$ which gives the contradiction.",
"Hence, from now on we assume that all $A_1,A_2,B_1,B_2$ are nonzero.",
"Without loss of generality we may assume $A_1+B_1\\le A_2+B_2$ .",
"Our proof is inductive on the length of the cycle $C_1$ , i.e.",
"on $A_1+B_1$ .",
"Base of induction: $A_1+B_1=2$ .",
"As $A_1,B_1\\ne 0$ we have $A_1=B_1=1$ .",
"Thus $d=a+b$ .",
"We also have $A_2a+B_2b=d=a+b$ .",
"As $A_2,B_2\\ne 0$ we must have $A_2=B_2=1=A_1=B_1$ .",
"Inductive step: We start by proving the following statement illustrated on Figure REF .",
"Figure: Two cycles C 1 C_1 and C 2 C_2 with i<j<C 1 (i)i<j<C_1(i) as in the Claim in proof of Lemma .Claim: There exist $i,j$ such that: $C_1(i)-i=C_2(j)-j\\ne 0,$ $i< j< C_1(i)$ or $j<i<C_2(j)$ with a cyclic ordering modulo $d$ .",
"We know that $A_1,B_1\\ne 0$ thus without loss of generality we may consider $i^{\\prime }$ such that $C_1(i^{\\prime })=i^{\\prime }+a$ and $C_1(i^{\\prime }+a)=i^{\\prime }+a+b$ .",
"Presenting $C_2=(s_1,\\cdots ,s_w)$ for $w=A_2+B_2$ , we may find $s_{j^{\\prime }}<i^{\\prime }+a<s_{j^{\\prime }+1}$ .",
"If $s_{j^{\\prime }+1}-s_{j^{\\prime }}=a$ , then setting $i=i^{\\prime }$ and $j=s_{j^{\\prime }}$ we obtain the desired indices.",
"If $s_{j^{\\prime }+1}-s_{j^{\\prime }}=b$ , then we set $j=s_{j^{\\prime }}$ and $i=i^{\\prime }+a$ to conclude.",
"By interchanging $C_1$ and $C_2$ and $a$ and $b$ if needed, by the above Claim we may assume that there are indices $i,j$ such that $C_1(i)-i=C_2(j)-j=b$ and $i< j< C_1(i)$ , cf.",
"Figure REF .",
"As $\\ell _1=1$ , for $i<k<j$ we have $C_1(k)=k$ .",
"Next, we show how to conclude in a special Case I, depicted on Figure REF when there exist $i,j$ such that: for all $i<k<j$ we have $C_2(k)=k$ ; in other words $k$ does not belong to any of the two cycles.",
"Figure: Two cycles C 1 C_1 and C 2 C_2 with i<j<C 1 (i)i<j<C_1(i) as in Case I in proof of Lemma .",
"The black dots do not belong to either cycle.",
"The dashed lines indicate the parts of the cycles that will be contracted in the inductive proof.Let $d^{\\prime }=d-b$ .",
"We will define two disjoint cycles $C_1^{\\prime }$ and $C_2^{\\prime }$ that will be permutations of $\\lbrace 1,2,\\ldots ,d^{\\prime }\\rbrace $ .",
"The cycle $C_i^{\\prime }$ will have exactly $A_i$ steps of size $a$ and $B_i-1$ steps of size $b$ .",
"As $\\operatorname{GCD}(d^{\\prime },a,b)=\\operatorname{GCD}(d,a,b)=1$ this will allow us to conclude by induction.",
"Let $C_2=(s_1,\\cdots , j,j+b,\\cdots ,s_w)$ for $w=A_2+B_2$ be presented in such a way that $s_1<s_2\\cdots <s_w$ .",
"We define $C_2^{\\prime }$ by removing $j+b$ and decreasing all indices after it by $b$ : $C_2^{\\prime }=(s_1,\\cdots ,j,\\cdots ,s_w-b)$ .",
"Similarly, we present $C_1$ as $(s_1^{\\prime },\\cdots ,i,i+b,\\cdots s^{\\prime }_{w^{\\prime }})$ for $w^{\\prime }=A_1+B_1$ .",
"The cycle $C_1^{\\prime }$ removes $i+b$ and decreases all indices after it by $b$ : $C_1^{\\prime }=(s_1^{\\prime },\\cdots ,i,\\cdots , s^{\\prime }_{w^{\\prime }}-b)$ .",
"The only nontrivial claim is that $C_1^{\\prime }$ and $C_2^{\\prime }$ are disjoint.",
"Consider $C_1^{\\prime }$ .",
"The indices $s_1^{\\prime },\\cdots , i$ are distinct from $s_1,\\cdots ,j$ as $C_1$ and $C_2$ were disjoint.",
"Also $s_1^{\\prime },\\cdots , i$ are distinct from $s_\\ell -b>j$ for $s_\\ell >j$ , as $i<j$ .",
"It remains to consider elements of $C_1^{\\prime }$ that are larger than $i$ .",
"Consider such an element $s_\\ell ^{\\prime }-b>i$ .",
"By assumption the index preceding $j$ in $C_1$ is smaller than $i$ .",
"Thus $s_\\ell ^{\\prime }-b$ could only coincide with the second part of elements of $C_2$ : $j=j+b-b,\\cdots , s_w-b$ .",
"However, $s_\\ell ^{\\prime }-b= s_q-b$ would imply that $s_\\ell ^{\\prime }=s_q$ , which is not possible.",
"This finishes the proof in the special Case I.",
"Figure: Two cycles C 1 C_1 and C 2 C_2 with i<j<C 1 (i)i<j<C_1(i) as in Case II in proof of Lemma .",
"The dashed lines indicate the parts of the cycles that will be contracted in the inductive proof.Case II: If we are not in the special Case I, we must have $i<(C_2)^{-1}(j)<j$ , cf.",
"Figure REF .",
"This implies $j-(C_2)^{-1}(j)=a$ and $a<b$ .",
"By inverting the direction of both cycles we see that we may also assume that $C_1(i+b)=i+b+a$ .",
"In analogy to Case I, we set $d^{\\prime }=d-a$ and we will define two cycles $C_1^{\\prime }$ and $C_2^{\\prime }$ .",
"Precisely if $C_2=(s_1,\\cdots ,j-a,j,j+b,\\cdots , s_w)$ let $C_2^{\\prime }=(s_1,\\cdots ,j-a,j+b-a,\\cdots , s_w-a)$ , and if $C_1=(s_1^{\\prime },\\cdots ,\\textbf {i},i+b,i+b+a,\\cdots ,s_{w^{\\prime }})$ let $C_1^{\\prime }=(s_1^{\\prime },\\cdots ,i,i+b,\\cdots ,s_{w^{\\prime }}-a)$ .",
"It is straightforward to check, as in the Case I, that the cycles $C_1^{\\prime }$ and $C_2^{\\prime }$ are disjoint.",
"We conclude by induction.",
"Remark 2.12 Lemma REF may not hold when $\\operatorname{GCD}(a,b,d)\\ne 1$ .",
"Indeed, consider $d=8$ , $a=2$ , $b=4$ .",
"Let $\\sigma =(0,2,4,6)\\circ (1,5)$ .",
"Clearly the two cycles in the decomposition do not satisfy the conclusion of the lemma.",
"Lemma 2.13 For any $1\\le i,j\\le k$ we have $A_i=A_j$ and $B_i=B_j$ .",
"Without loss of generality we assume $i=1$ and $j=2$ .",
"Our aim is to rearrange the cycles $C_1$ and $C_2$ , possibly changing $a,b,d$ , in such a way that we can apply Lemma REF .",
"Reduction 1: We first reduce to the situation when $a|d$ and $\\operatorname{GCD}(a,b)=1$ .",
"Precisely, we construct new cycles, without changing $A_1,A_2,B_1,B_2$ and $d$ , however with new $a^{\\prime }$ and $b^{\\prime }$ , such that $\\operatorname{GCD}(a^{\\prime },b^{\\prime })=1$ and $a^{\\prime }|d$ .",
"Let $a^{\\prime }:=\\operatorname{GCD}(a,d)$ and $a=za^{\\prime }$ .",
"Let $z^{\\prime }<d$ be such that $z\\cdot z^{\\prime }\\equiv 1$ modulo $d$ .",
"Let $d>b^{\\prime }:=bz^{\\prime }$ modulo $d$ .",
"We rearrange the rests $\\lbrace 0,1,\\cdots ,d-1\\rbrace $ modulo $d$ by multiplying them by $z^{\\prime }$ .",
"Precisely, we change the cycle $C_1=(s_1,\\cdots ,s_{A_1+B_1})$ and $C_2= (s_1^{\\prime },\\cdots ,s_{A_2+B_2}^{\\prime })$ respectively to $C_1^{\\prime }=(z^{\\prime }s_1,\\cdots ,z^{\\prime }s_{A_1+B_1})$ and $C_2^{\\prime }= (z^{\\prime }s_1^{\\prime },\\cdots ,z^{\\prime }s_{A_2+B_2}^{\\prime })$ .",
"These are clearly two disjoint cycles with possible differences $a^{\\prime }$ and $b^{\\prime }$ .",
"Further $a^{\\prime }|d$ and $\\operatorname{GCD}(a^{\\prime },b^{\\prime },d)=\\operatorname{GCD}(a^{\\prime },bz^{\\prime },d)$ .",
"Note that $\\operatorname{GCD}(a^{\\prime },b,d)=1$ (as $a^{\\prime }|a$ ) and $\\operatorname{GCD}(a^{\\prime },z^{\\prime },d)=1$ (as $z^{\\prime }$ and $d$ are coprime), hence $\\operatorname{GCD}(a^{\\prime },b^{\\prime },d)=1$ .",
"Reduction 2: Let $C_i^{\\prime }$ be as above, $i=1,2$ .",
"We now reduce to the case of winding numbers $\\ell _1=\\ell _2=1$ .",
"Let $m:=\\operatorname{GCD}(\\ell _1,\\ell _2)$ and $\\ell _1=m\\ell _1^{\\prime }$ , $\\ell _2=m\\ell _2^{\\prime }$ .",
"Let $d^{\\prime }=dm\\ell _1^{\\prime }\\ell _2^{\\prime }=d\\operatorname{LCM}(\\ell _1,\\ell _2)$ .",
"We extend the cycle $C_1^{\\prime }$ (that is a permutation of $d$ elements) to a cycle $C_1^{\\prime \\prime }$ (that is a permutation of $d^{\\prime }$ elements) as follows.",
"Say $C_1^{\\prime }=(c_1,\\cdots ,c_{A_1+B_1})$ , where $c_1$ is the smallest integer appearing in $C_1^{\\prime }$ .",
"We may encode it as a word, with letters $a^{\\prime }$ and $b^{\\prime }$ , of length $A_1+B_1$ : $w:=c_2-c_1,\\cdots ,c_{A_1+B_1}-c_{A_1+B_1-1}.$ We concatenate the word $w$ with itself $\\ell _2^{\\prime }$ times, obtaining a word $w^{\\circ \\ell _2^{\\prime }}$ with exactly $A_1\\ell _2^{\\prime }$ letters $a^{\\prime }$ and $B_1\\ell _2^{\\prime }$ letters $b^{\\prime }$ .",
"The word $w^{\\circ \\ell _2^{\\prime }}$ encodes the cycle $C_1^{\\prime \\prime }$ , that starts at $c_1$ with differences $c_{i+1}-c_i$ equal to either $a^{\\prime }$ or $b^{\\prime }$ , according to $w^{\\circ \\ell _2^{\\prime }}$ .",
"In the analogous way we obtain a cycle $C_2^{\\prime \\prime }$ with $A_2\\ell _1^{\\prime }$ differences $a^{\\prime }$ and $B_2\\ell _2^{\\prime }$ differences $b^{\\prime }$ .",
"We are now in position to apply Lemma REF .",
"As $a^{\\prime }|d$ , we must have $\\operatorname{GCD}(a^{\\prime },b^{\\prime })=1$ .",
"In particular, $\\operatorname{GCD}(a^{\\prime },b^{\\prime },d^{\\prime })=1$ .",
"Further, the cycles $C_1^{\\prime \\prime }$ and $C_2^{\\prime \\prime }$ have their winding numbers $\\ell _1^{\\prime \\prime }=\\ell _2^{\\prime \\prime }=1$ .",
"They are also disjoint, as their reductions modulo $d$ coincide with $C_1$ and $C_2$ that are disjoint.",
"Hence, by Lemma REF , $A_1\\ell _2^{\\prime }=A_2\\ell _1^{\\prime }$ and $B_1\\ell _2^{\\prime }=B_2\\ell _1^{\\prime }$ .",
"In particular, $\\ell _1^{\\prime }|A_1\\ell _2^{\\prime }$ and as $\\operatorname{GCD}(\\ell _1^{\\prime },\\ell _2^{\\prime })=1$ we have $\\ell _1^{\\prime }|A_1$ .",
"In the same way $\\ell _1^{\\prime }|B_1$ and by definition $\\ell _1^{\\prime }|\\ell _1$ .",
"Thus, by Lemma REF we must have $\\ell _1^{\\prime }=1$ .",
"Analogously $\\ell _2^{\\prime }=1$ .",
"Hence, indeed $A_1=A_2$ and $B_1=B_2$ .",
"Recall $A=\\sum _{i=1}^{k} A_i$ , $B=\\sum _{i=1}^{k} B_i$ , and let $\\ell :=\\frac{aA+bB}{d}$ .",
"Lemma 2.14 We have $k=\\operatorname{GCD}(A,B,\\ell )$ .",
"By Lemma REF we know that $A=kA_1$ , $B=kB_1$ and $\\ell =k\\ell _1$ .",
"We conclude by Lemma REF : $k=k\\operatorname{GCD}(A_1,B_1,\\ell _1)=\\operatorname{GCD}(kA_1,kB_1,k\\ell _1)=\\operatorname{GCD}(A,B,\\ell ).$ By Lemma REF , the results of this subsection imply Theorems REF and REF .",
"$\\square $"
],
[
"On the minimality of GT-systems",
"In this section, we will apply the results on the determinant of a three-lines circulant matrix obtained in the previous section to study the minimality of GT-systems and to solve a conjecture stated by Mezzetti and Miró-Roig in [17].",
"To state this conjecture we need first to introduce some definitions.",
"Definition 3.1 Let $I\\subset R $ be a homogeneous artinian ideal.",
"We say that $I$ has the Weak Lefschetz Property (WLP, for short) if there is a $L \\in [R/I]_1$ such that, for all integers $j$ , the multiplication map $\\times L\\colon [R/I]_{j-1} \\rightarrow [R/I]_j$ has maximal rank, i.e.",
"it is either injective or surjective.",
"To establish whether an ideal $I\\subset R$ has the WLP is a difficult and challenging problem and even in simple cases, such as complete intersections, much remains unknown about the presence of the WLP.",
"Recently the failure of the WLP has been connected to a large number of problems, that appear to be unrelated at first glance.",
"For example, in [15], Mezzetti, Miró-Roig and Ottaviani proved that the failure of the WLP is related to the existence of varieties satisfying at least one Laplace equation of order greater than 2; we recall that a $k$ -dimensional variety $X\\subset \\mathbb {P}^n$ satisfies $r$ Laplace equations of order $d$ if for any parametrization $F=F(t_1,\\cdots ,t_k)$ of $X$ around a smooth, general point, $F$ satisfies a system of $r$ (linearly independent) PDE's with constant coefficients of order $d$ .",
"Their result is the following: Theorem 3.2 ([15]) Let $I\\subset R$ be an artinian ideal generated by $r$ homogeneous polynomials $F_1,\\cdots ,F_{r}$ of degree $d$ and let $I^{-1}$ be its Macaulay inverse system.",
"If $r\\le \\binom{n+d-1}{n-1}$ , then the following conditions are equivalent: (1) the ideal $I$ fails the WLP in degree $d-1$ ; (2) the homogeneous forms $F_1,\\cdots ,F_{r}$ become $k$ -linearly dependent on a general hyperplane $H$ of $\\mathbb {P}^n$ ; (3) the $n$ -dimensional variety $X=\\overline{\\operatorname{Im}(\\varphi )}$ where $\\varphi \\colon \\mathbb {P}^n \\dashrightarrow \\mathbb {P}^{\\binom{n+d}{d}-r-1}$ is the rational map associated to $(I^{-1})_d$ , satisfies at least one Laplace equation of order $d-1$ .",
"The above result motivated the following definitions: Definition 3.3 Let $I \\subset R$ be an artinian ideal generated by $r$ forms of degree $d$ , and $r \\le \\binom{n+d-1}{n-1}$ .",
"We will say: (i) $I$ is a Togliatti system if it satisfies one of three equivalent conditions in Theorem REF .",
"(ii) $I$ is a monomial Togliatti system if, in addition, $I$ can be generated by monomials.",
"(iii) $I$ is a smooth Togliatti system if, in addition, the rational variety $X$ is smooth.",
"(iv) A monomial Togliatti system $I$ is minimal if there is no proper subset of the set of generators defining a monomial Togliatti system.",
"These definitions were introduced in [15] and [16] and the names are in honor of Eugenio Togliatti who proved that for $n = 2$ the only smooth Togliatti system of cubics is $I = (x_0^3,x_1^3,x_2^3,x_0x_1x_2)\\subset \\mathbb {K}[x_0,x_1,x_2]$ (see [3], [26] and [27]).",
"The systematic study of Togliatti systems was initiated in [15] and for recent results the reader can see [18], [16], [1], [20] and [17].",
"Precisely in the latter reference the authors introduced the notion of GT-system which we recall now.",
"Definition 3.4 A GT-system is an artinian ideal $I\\subset \\mathbb {K}[x_0, x_1,\\cdots ,x_n]$ generated by $r$ forms $F_{1},\\cdots ,F_{r}$ of degree $d$ such that: $I$ is a Togliatti system.",
"The regular map $\\phi _{I}\\colon \\mathbb {P}^{n}\\rightarrow \\mathbb {P}^{r-1}$ defined by $(F_{1},\\cdots ,F_{r})$ is a Galois covering of degree $d$ with cyclic Galois group $\\mathbb {Z}/d\\mathbb {Z}$ .",
"Any representation of the cyclic group $\\mathbb {Z}/d\\mathbb {Z}$ as subgroup of $GL(n+1,\\mathbb {K})$ can be diagonalized.",
"In particular it is represented by a matrix of the form $M:=M_{\\alpha _0,\\alpha _1,\\cdots ,\\alpha _{n}}=\\begin{pmatrix} e^{\\alpha _0} & 0 & \\cdots & 0 \\\\ 0 & e^{\\alpha _1} & \\cdots & 0 \\\\& & \\cdots & \\\\ 0 & 0 &\\cdots & e^{\\alpha _{n}}\\end{pmatrix}$ where $e$ is a primitive $d$ th root of 1 and $\\alpha _0,\\alpha _1,\\cdots ,\\alpha _{n}$ are integers with $\\operatorname{GCD}(\\alpha _0,\\alpha _1,\\cdots ,\\alpha _{n},d)=1.$ It follows (see [6]) that the above definition is equivalent to the next one: Definition 3.5 Fix integers $3\\le d\\in \\mathbb {Z}$ , $2\\le n\\in \\mathbb {Z}$ , with $n\\le d$ , and $0\\le \\alpha _0\\le \\alpha _1\\le \\cdots \\le \\alpha _{n}\\le d$ , $e$ a primitive $d$ -th root of 1 and $ M_{\\alpha _0,\\alpha _1,\\cdots ,\\alpha _{n}}$ a representation of $\\mathbb {Z}/d\\mathbb {Z}$ in $GL(n+1,\\mathbb {K})$ .",
"A GT-system will be an ideal $I^d_{\\alpha _0,\\cdots ,\\alpha _{n}}\\subset \\mathbb {K}[x_0,x_1,\\cdots ,x_{n}]$ generated by all forms of degree $d$ invariant under the action of $ M_{\\alpha _0,\\alpha _1,\\cdots ,\\alpha _{n}}$ provided the number of generators $\\mu (I^d_{\\alpha _0,\\cdots ,\\alpha _{n}})\\le \\binom{n+d-1}{n-1}$ .",
"Remark 3.6 It is an immediate consequence of the above description that the ideal $I^d_{\\alpha _0,\\cdots ,\\alpha _n}$ is always monomial, i.e.",
"a GT-system is a monomial Togliatti system.",
"Remark 3.7 Indeed in the proof of [6], the authors observed that $I:=I^d_{\\alpha _0,\\cdots ,\\alpha _{n}}$ fails the Weak Lefschetz Property from degree $d-1$ to degree $d$ , because for any linear form $\\ell \\in R$ the induced map $\\times \\ell :[R/I]_{d-1}\\longrightarrow [R/I]_d$ is not injective.",
"By [19], since $I$ is monomial, it is enough to check it for $\\ell = x_0+x_1+\\cdots +x_{n}$ .",
"This is equivalent to prove that there exists a form $F_{d-1}\\in R$ of degree $d-1$ such that $(x_0+x_1+\\cdots +x_{n})\\cdot F_{d-1}\\in I$ .",
"Consider $F_{d-1}=(e^{\\alpha _0}x_0+e^{\\alpha _1}x_1+\\cdots +e^{\\alpha _{n}}x_{n})(e^{2\\alpha _0}x_0+e^{2\\alpha _1}x_1+\\cdots +e^{2\\alpha _{n}}x_{n})\\cdots (e^{(d-1)\\alpha _0}x_0+e^{(d-1)\\alpha _1}x_1+\\cdots +e^{(d-1)\\alpha _{n}}x_{n}).$ The homogeneous form of degree $d$ , $F=(x_0+x_1+\\cdots +x_{n})\\cdot F_{d-1}$ is invariant under the action of $M_{\\alpha _0,\\alpha _1,\\cdots ,\\alpha _{n}}$ , hence, it belongs to $I$ .",
"In the following, since we are interested in the projective space and in the action of $M_{\\alpha _0,\\alpha _1,\\cdots ,\\alpha _{n}}$ up to proportionality, we will assume that the first exponent $\\alpha _0$ is equal to zero.",
"Moreover, $R$ will always denote the polynomial ring in three variables: $R=\\mathbb {K}[x,y,z]$ .",
"To determine the minimality of a GT-system is a subtle problem.",
"In [17], in the case of three variables, the second and fourth authors proved that the ideal $I^d_{0,a,b}$ always satisfies the condition on the number of generators $\\mu (I)\\le d+1$ , and conjectured the following, which we now prove using Theorem REF : Theorem 3.8 ([17]) Let $d\\ge 3$ be an integer and $M_{a,b}$ be a $3\\times 3$ -matrix representing the cyclic group $\\mathbb {Z}/d\\mathbb {Z}$ with $1\\le a< b\\le d-1$ such that $\\operatorname{GCD}(a,b,d)=1$ .",
"Let $I=I^d_{0,a,b}\\subset R=\\mathbb {K}[x,y,z]$ be the ideal generated by all the monomials of degree $d$ invariant under the action of $M_{a,b}$ .",
"Then $I$ is a minimal GT-system.",
"As observed in Remark REF , the form $F_{d-1}:=(x+e^{a}y+e^{b}z)\\cdots (x+e^{(d-1)a}y+e^{(d-1)b}z)$ is in the kernel of $\\times (x+y+z):[R/I]_{d-1}\\rightarrow [R/I]_{d}$ , so the dimension of the kernel is $\\ge 1$ , thus $I$ is a GT-system.",
"We now work towards showing its minimality.",
"Claim: The dimension of the kernel $K_{d-1}$ of $\\times (x+y+z):[R/I]_{d-1}\\rightarrow [R/I]_{d}$ is one.",
"We will prove that $F_{d-1}$ generates $K_{d-1}$ .",
"Assume that $G_{d-1}(x,y,z)$ is a form of degree $d-1$ which belongs to $K_{d-1}$ .",
"We will prove that $(x+e^ay+e^bz)$ divides $G_{d-1}(x,y,z)$ .",
"Analogously the other factors of $F_{d-1}$ divide $G_{d-1}(x,y,z)$ , and we are done.",
"Since $G_{d-1}(x,y,z)$ belongs to $K_{d-1}$ , we have $(x+y+z)G_{d-1}(x,y,z) \\in I$ .",
"So the form $H(x,y,z):=(x+y+z)G_{d-1}(x,y,z)$ of degree $d$ is in $I$ .",
"Since $H(x,y,z)$ belongs to $I$ , it is invariant under the action of $M_{a,b}$ and we have $(x+e^ay+e^bz)G_{d-1}(x,e^ay,e^bz)=H(x,e^ay,e^bz)=H(x,y,z)=(x+y+z)G_{d-1}(x,y,z),$ which allows us to conclude that $(x+e^ay+e^bz)$ divides $G_{d-1}(x,y,z)$ .",
"Our next claim is that the Togliatti system $I$ is minimal if and only if all monomials of degree $d$ , which are invariant under the action of $M_{a,b}$ , appear with non-zero coefficient in the form $C_{d;a,b}:=(x+y+z)(x+e^{a}y+e^{b}z)\\cdots (x+e^{(d-1)a}y+e^{(d-1)b}z)=(x+y+z)F_{d-1}.$ One implication is obvious.",
"For the other, assume that $I$ is not a minimal Togliatti system: this means that there is an ideal $J$ , strictly contained in $I$ , which is again a Togliatti system.",
"Let $G_1, \\cdots , G_s$ be a system of generators of $J$ .",
"Then for any linear form $\\ell $ there is a form $G$ such that $\\ell G$ is a linear combination of $G_1,\\cdots , G_s$ .",
"In particular, $(x+y+z) G$ belongs to $I$ , therefore $G$ is in the kernel of the map $\\times (x+y+z)\\colon [R/I]_{d-1}\\rightarrow [R/I]_{d}$ .",
"Since the kernel has dimension one, by the Claim, it follows that $G$ is a scalar multiple of $F_{d-1}$ .",
"We conclude that not all invariant monomials appear with non-zero coefficient in $C_{d;a,b}$ .",
"We easily observe that $C_{d;a,b}$ coincides with the determinant of the circulant matrix $\\operatorname{Circ}_{(d;0,a,b)}$ .",
"On the other hand, a monomial $x^{\\alpha }y^{\\beta }z^{\\gamma }$ of degree $d$ is invariant under the action of $M_{a,b}$ if, and only if, it satisfies the following system of equations: ${\\left\\lbrace \\begin{array}{ll}\\alpha +\\beta +\\gamma =d&\\\\a\\beta +b\\gamma \\equiv 0&\\pmod {d}\\end{array}\\right.",
"}$ or, equivalently, the monomials appearing in the permanent $\\operatorname{per}(\\operatorname{Circ}_{(d;0,a,b)})$ with non-zero coefficient are exactly all the monomials of degree $d$ invariant by the action of $M_{a,b}$ .",
"Thus, to conclude the proof we need the equality $D_{(d;0,a,b)}=P_{(d;0,a,b)}$ to hold, which is indeed the case by Theorem REF .",
"Remark 3.9 It is worthwhile to underline the following interpretation of our results in terms of representation theory of cyclic matrix groups.",
"In the proof of Theorem REF we have proved that a monomial of degree $d$ in three variables is invariant under the action of the cyclic matrix group of order $d$ , generated by $M_{a,b}$ , if and only if it appears with non-zero coefficient in the form $C_{d;a,b}$ of (REF ).",
"So we get information about a minimal system of generators for the homogeneous component of degree $d$ of the ring of invariants of these representations.",
"Up to a coefficient $d$ , the polynomial $C_{d;a,b}$ is the image of the linear form $x+y+z$ under the Reynolds operator.",
"This is in fact the same point of view of Emmy Noether, when she proved the finiteness of the ring of invariants of a polynomial ring under the action of a finite matrix group [21].",
"From Example REF it follows that a similar result is not true for polynomials in $r$ variables, with $r\\ge 4$ ."
],
[
"Computational complexity",
"The task of computing the permanent of a $d$ by $d$ $(0/1)$ -matrix is computationally hard ($\\#P$ -complete), by a celebrated theorem of Valiant [28], and remains so even when there are only 3 nonzero entries per row [8].",
"The best known upper bounds to compute the permanent are exponential in $d$ , by Ryser [24].",
"Theorem REF immediately tells us that computing the permanent of $\\operatorname{Circ}_{(d;0,a,b)}$ , when $\\operatorname{GCD}(a,b,d)=1$ , can be done in polynomial time in $d$ , say by computing $\\det (\\operatorname{Circ}_{(d;0,a,b)})\\in \\mathbb {Z}[x,y,z]$ via polynomial interpolation (evaluating it on $O(d^2)$ points suffices)."
]
] | 1808.08387 |
[
[
"Rule Module Inheritance with Modification Restrictions"
],
[
"Abstract Adapting rule sets to different settings, yet avoiding uncontrolled proliferation of variations, is a key challenge of rule management.",
"One fundamental concept to foster reuse and simplify adaptation is inheritance.",
"Building on rule modules, i.e., rule sets with input and output schema, we formally define inheritance of rule modules by incremental modification in single inheritance hierarchies.",
"To avoid uncontrolled proliferation of modifications, we introduce formal modification restrictions which flexibly regulate the degree to which a child module may be modified in comparison to its parent.",
"As concrete rule language, we employ Datalog+/- which can be regarded a common logical core of many rule languages.",
"We evaluate the approach by a proof-of-concept prototype."
],
[
"Introduction",
"In data- and knowledge-intensive systems it is good practice to separate explicit knowledge, elicited from domain experts and translated into declarative expressions by rule developers, from application code developed by application developers.",
"Rule-based knowledge representation and reasoning build the core of systems for business rule engines [26], web data extraction [15], data wrangling [19], knowledge graph management [3], and information tailoring [8].",
"With the increasing number and complexity of rules, their maintenance and their adaptation to different settings and contexts become key challenges of rule developers.",
"In this paper we present an approach employing rule modules and inheritance to cope with these challenges.",
"In the following we sketch challenges and our approach along a self-contained business rule example which is used throughout the paper.",
"Subsequently, we zoom out to give the big picture of potential application areas where the presented approach may serve as a central building block.",
"In this paper, we present an approach which builds on rule modules – i.e, rule sets with interfaces describing the schema of input and output data – to provide a clear separation and interfaces between rule sets and data-intensive applications.",
"These interfaces shield application developers from the intricacies of rule sets as well as rule developers from application code and clarify which knowledge and data schemata are internal to the module and which interface either as input or output.",
"— For example, a bank clerk is responsible for processing a set of mortgage applications, i.e., assessing the applications' credit worthiness in order of their priority.",
"Domain knowledge regarding assessment and prioritization of mortgage applications is encoded in a rule module MortgageApps.",
"This rule module is employed by a data-intensive application collecting and managing mortgage applications and their assessment.",
"Rule module MortgageApps takes as input a set of mortgage applications each described by the mortgage value and the estimated values of real estate securities.",
"As output it produces a preliminary assessment of the credit worthiness, either good or bad, of each application together with a prioritization of the applications for detailed assessment.",
"Any issues with applications, e.g., having a mortgage value below the specified minimum loan value, are also output.",
"Organizing business rules into rule modules entails the danger of redundancy: One and the same rule may be relevant in different settings and thus introduced and maintained separately in different modules.",
"This duplicates human effort in developing and maintaining modules and makes it difficult to keep rules synchronized across modules.",
"— For example, our bank decides to offer private loans as well, creating a rule module PrivateLoanApps.",
"Some rules, such as the minimum loan value, apply in module MortgageApps as well as module PrivateLoanApps.",
"Thus, any changes to this rule need to be performed in both rule modules.",
"In this paper we introduce inheritance of parent modules to child modules by incremental modification as one way to mitigate these problems without sacrificing flexibility.",
"In child modules, rule developers may introduce additional rules, remove inherited rules, as well as extend and/or reduce the input and output interfaces.",
"This reduces redundancy and thus should ease maintenance of modules adapted to different business settings.",
"— For example, extracting common rules, such as the minimum loan value rule, into a parent module LoanApps we can remove redundancy.",
"Any changes to a common rule are made in the parent module and by inheritance propagated to all child modules.",
"Moreover, since we allow modifications to inherited rules, we can define default rules for loan application assessment and ranking in module LoanApps.",
"Allowing arbitrary modifications in child rule modules, however, would undermine the benefits of inheritance and would pave the way for uncontrolled proliferation of variations: a rule or application developer trying to get an overall picture of the interfaces and the behavior, i.e., the derived knowledge, encoded in a hierarchy of rule modules would still have to inspect all modules.",
"Furthermore, undesired changes to inherited rules and interfaces would be possible.",
"— For example, consider our minimum loan value rule in module LoanApps.",
"Since arbitrary modifications are allowed we can simply remove this rule in child modules; any problematic loan applications would not be output anymore.",
"In this paper, we introduce a set of modification restrictions to flexibly regulate the degree to which a module's interfaces and behavior may be modified in comparison to its parent module's interfaces and behavior.",
"Treating rule modules as black boxes, modification restrictions set boundaries within which a module may be modified.",
"Thus, it should be sufficient to inspect the root module to get an abstract overview of the behavior implemented in a hierarchy of modules.",
"Structural restrictions restrain allowed modifications to the input and output schema of a module.",
"— For example, module LoanApps specifies the basic input for loan application assessments.",
"Child module MortgageApps requires further inputs like real estates provided as securities.",
"Thus, we define LoanApps' inputs as non-omitable.",
"Moreover, module MortgageApps fixes the output for all mortgage application modules.",
"Thus, we define its output as non-extensible.",
"Behavioral restrictions constrain modifications changing the output at instance level (for a particular output schema element a behavioral modification may lead to different instances in the output).",
"— For example, we want to prohibit child modules from deriving a subset of minimum loan value issues compared to module loanApps.",
"Thus, we employ restriction non-shrinkable.",
"Similarly, child modules must not have weaker requirements for good credit worthiness, i.e., must not derive good credit worthiness for a superset of loan applications compared to module loanApps.",
"Thus, we employ restriction non-growable."
],
[
"Potential Application Areas",
"Potential application areas, besides business rules as shown in the examples, are rule-based systems for information tailoring [8], web data extraction [15], and knowledge graph management [3], where rule modules encode knowledge for tasks such as data extraction, transformation, cleansing, and filtering and need to be adapted to different settings.",
"For example, in the DIADEM system for web data extraction [15], multiple rule modules, each responsible for a particular task such as web form understanding or form filling, are dynamically orchestrated in networks where one module's output is another module's input.",
"Some of the encoded knowledge necessary for web form understanding is generic to 'all' web sites, while other encoded knowledge is specific to domains such as 'real estate' websites.",
"Inheritance from the rule module for generic web form understanding to a rule module for web form understanding for 'real estate' websites should help to reduce redundancy and thus ease maintenance.",
"Structural modification restrictions can be used to ensure that child rule modules remain orchestrable, similar to co- and contravariance in object-orientation which can be used to ensure type-safety and substitutability.",
"Behavioral modification restrictions should help to keep the overall rule base understandable and thus maintainable, e.g., looking at a parent module and its behavioral restrictions gives an overview of possible behavior in child modules.",
"Regarding Web developments, e.g., Internet of Things, Semantic Web, or Smart Things, rules become a vital and integral part [33].",
"Due to the number of rules and their context-dependent applicability, efficient rule management is essential.",
"Rule modules and module inheritance are means to manage such large rule sets and can be extended to manage contexts of application (c.f.",
"[9]).",
"We expect our approach to be applicable and beneficial in these areas – an evaluation is yet to be performed.",
"In this paper we focus on rule modules, their inheritance, and modification restrictions independent of specific applications."
],
[
"Contributions and Overview",
"Currently, work on rule inheritance and related work on contextualized knowledge representation is fragmented and there exists no approach for inheritance of rule modules where modifications can be flexibly restricted.",
"— In this paper we introduce an overall approach to rule module inheritance and specifically make the following contributions: formal definition of (1) downward rule and interface inheritance in single inheritance hierarchies of modules, and (2) modification restrictions and inheritance of modification restrictions discussion of conformance checks for detecting violations of defined structural and behavioral modification restrictions proof-of-concept prototype implementing formal definitions in Datalog, publicly available for further experimentation The remainder of this paper is structured as follows: Sect.",
"presents our Datalog$^\\pm $ -based rule language and rule modules.",
"In Sect.",
"we present our basic inheritance mechanism.",
"Sect.",
"introduces modification restrictions and delineates inheritance of modification restrictions.",
"In Sect.",
"we present a proof-of-concept prototype.",
"Sect.",
"discusses related work.",
"Sect.",
"concludes the paper."
],
[
"Rule Modules",
"First, we discuss expressing rules with Datalog$^\\pm $ where a rule is constructed from predicates and operators like logical conjunction.",
"Subsequently, we introduce rule modules and discuss their structure and behavior."
],
[
"Underlying Rule Language",
"This section delineates our notion of rules based on a formal language.",
"In order to focus on rule module inheritance and modification restrictions, the underlying rule language and data model should be simple, i.e., have few constructs and operators, to avoid unnecessary complexity.",
"A fitting family of formal languages is Datalog$^\\pm $ [11] which has clearly defined formal semantics and employs a relational data model.",
"Datalog$^\\pm $ extends plain Datalog by existentially quantified variables in rule heads, negative constraints, and equality-generating dependencies while restricting the language so as to achieve decidability and, more particularly, good performance.",
"Vadalog [4] is a practical implementation of Datalog$^\\pm $ that adds many features needed in commercial use, including a wide range of built-ins.",
"Thus, Datalog$^\\pm $ /Vadalog is quite expressive, e.g., encompassing full Datalog with no restriction on recursion and SPARQL under the OWL 2 QL entailment regime, while still being efficient.",
"Datalog$^\\pm $ /Vadalog is a versatile rule language family also employed for big data wrangling [19] and knowledge graph management [3].",
"A plain Datalog rule comprises a body (premise) and a head (conclusion) where the conclusion is derived if the premise holds.",
"Both conclusion and premise are conjunctions of atoms (i.e., predicates with arguments).",
"Datalog$^\\pm $ allows use of existentially quantified variables in conclusions enabling value creation, truth value false in conclusions (negative constraints), and equality-generating dependencies, e.g., Y=Z:- r1(X,Y), r2(X,Z).",
"[Rule Structure] Predicates (a.k.a.",
"relations) are taken from a universe of predicates $P$ .",
"Rules are taken from a universe of rules $R$ .",
"A rule $r \\in R$ has body predicates $B_r \\subseteq P$ and head predicates $H_r \\subseteq P$ .",
"Our bank deems mortgage loans of less than 10,000 Euro as not worth the organizational costs (rule R0).",
"We translate this natural language rule to Datalog$^\\pm $ : lowLValue(X,V) :- lValue(X,V), V < 10000.",
"The mortgage value applied for is represented by lValue/2 (predicate lValue with arity two) relating loan applications with loan values while lowLValue/2 annotates mortgage applications below the defined mortgage value threshold.",
"Rule R0 has body predicates $B_{{\\tt R0}} = \\lbrace $lValue/2$\\rbrace $ and head predicates $H_{{\\tt R0}} = \\lbrace $lowLValue/2$\\rbrace $ ."
],
[
"Module Structure and Behavior",
"A collection of rules and facts (i.e., rules without variables and premise true) in Datalog is called a Datalog program.",
"In the following we extend Datalog programs with input schema and output schema to rule modules, where the Datalog program itself is the implementation of the rule module.",
"We derive structural aspects of rule modules from Datalog and Vadalog.",
"Datalog splits the predicates of a program into extensional database (EDB) and intensional database (IDB).",
"EDB contains predicates asserted in the knowledge base, IDB predicates defined by rules.",
"A similar distinction is made in DMN's rule representation [25].",
"Vadalog [4] extends this idea to inputs provided by external sources (input predicates) and derived predicates output to external sinks (output predicates).",
"These two sets are disjoint.",
"Predicates derived and potentially used in rule bodies but not exported are auxiliary predicates.",
"[Rule Modules] Rule modules are taken from a universe of rule modules $M$ .",
"A rule module $m \\in M$ is defined by a set of rules $R_m$ , a set of input predicates $I_m \\subseteq P$ , and a set of output predicates $O_m \\subseteq P$ .",
"The sets of input and output predicates are disjoint.",
"The predicates of a module $m$ ($P_m$ ) are the union of its rule head and body, input, and output predicates.",
"We now discuss the development of a rule module from an organizational perspective: Domain experts elicit and determine rules, often in natural language, and organize them into rule sets, e.g., a rule set may cover a specific business case.",
"Furthermore, domain experts determine the necessary data input and derived output for the elicited set of rules.",
"Once the domain experts consider a rule set and its inputs and outputs complete, rule developers translate the elicited rules and interfaces into a rule module with a Datalog$^\\pm $ program as implementation.",
"We want to elicit for the process of assessing mortgage loan applications all relevant rules.",
"In addition to rule R0 (see Example REF ) we identified: Credit worthiness of a mortgage application is deemed good (predicate cwGood/1) if the value of provided properties exceeds 80 % of the mortgage's value (R1).",
"In all other cases credit worthiness is deemed bad (R2 – output cwBad/1).",
"Loan applications of higher loan values have priority over those with lower loan values (R3 – priorityOver/2).",
"For each provided property, its value needs to be calculated (sValue/2) and associated with the loan application (R4 – property/1, properties/2).",
"Associated properties are securities (R5 – securities/2, security/1).",
"Properties of a value below 30,000 Euro are problematic as securities (R6 – lowPropValue/2).",
"The predicates derived by rules R0-R6 form module MortgageApps's output predicates $O_{\\tt MortgageApps}$ .",
"From these natural language rules our domain experts derive the necessary input.",
"The rules employ information given with each mortgage application (loan/1).",
"Such applications need to state the intended mortgage value (lValue/2), the duration (duration/2), the applying customer (customer/2), and any real estate properties which may be used as securities (mProperty/2) as well as their value (pValue/2).",
"These predicates form the input predicates $I_{\\tt MortgageApps}$ .",
"Furthermore, properties may be hierarchically organized (hasPart/2), e.g., a property may consist of an area containing buildings.",
"A visual summary including the Datalog$^\\pm $ representation is given in REF .",
"Figure: Visual summary of module MortgageApps as described in Example and comprising three compartments: the Input and Output compartments containing input and output declarations of predicate names and their arity respectively and the rules compartment containing Datalog ± ^\\pm rules with rule identifiers in brackets.Besides structure, a rule module exhibits behavior when executed.",
"We regard rule module behavior as observable effects, i.e., derived facts, when applying the set of rules in a module to a dataset containing facts for its input predicates.",
"Multiple facts for each predicate may be provided.",
"A data set providing facts for all input predicates of a module is called applicable.",
"[Module Execution] Data sets are taken from a universe of data sets $D$ .",
"A data set $d \\in D$ with schema $P_d \\subseteq P$ contains extensions, i.e., sets of facts, for all $p \\in P_d$ .",
"A data set $d$ is applicable input for module $m$ if the module's input schema is a subset of the data set's schema, i.e., if $I_m \\subseteq P_d$ .",
"The execution of a module $m$ on applicable input data $d$ results, for each output predicate $p \\in O_m$ , in a set of derived facts, denoted as $p^d_m$ ($p^d_m$ is the set of derived $p$ -facts resulting from executing $m$ on $d$ ).",
"Executing module MortgageApps on a dataset of two mortgage applications L1 and L2, its rules are evaluated and derived facts of output predicates (behavior) are returned (see REF ).",
"Figure: Visualization of the execution of module MortgageApps on a dataset containing facts describing the two mortgage applications L1 and L2 and its output."
],
[
"Inheritance",
"In the following, we shortly summarize different aspects of inheritance in general.",
"We describe the scope of our approach regarding the found aspects and subsequently define our notion of inheritance hierarchy for rule modules, rule inheritance therein, and abstract predicates and modules.",
"A common notion of inheritance is inheritance as incremental modification, that is, “reusing a conceptual or physical entity in constructing an incrementally similar one” [32].",
"From related work we extracted various aspects and their options: All found inheritance mechanisms are transitive.",
"Furthermore, inheritance is often discussed regarding certain foci, i.e., signatures (schema of inputs and outputs), behavior, or implementation [32], [27], [20], [30], [23], [17], [5].",
"Besides these aspects, inheritance is usually distinguished into single-inheritance and multi-inheritance [12], [20], [27], [30], [32], [17], i.e., child entities inherit from a single or multiple parent entities respectively.",
"Direction of inheritance is distinguished into downward [12], [17], e.g., inheritance in Java, upward [12], [17], e.g., an array inheriting properties from its entries, and lateral [12], e.g., a motorized trike is a motorcycle except that is has three wheels.",
"Granularity [32], [17] regards whether an inheritance mechanism is specified for groups of entities or single entities.",
"We restrict our investigation of rule module inheritance to the following inheritance options: Focus is on signature and behavior inheritance.",
"Multi-inheritance is not investigated.",
"Direction of inheritance is downward, from parent module to child module.",
"Granularity is rule modules.",
"[Inheritance Hierarchy] Rule modules are arranged in an inheritance hierarchy $H \\subset M \\times M$ which forms a forest (i.e., a set of trees).",
"We say module $m^{\\prime }$ inherits from $m$ if $(m^{\\prime },m) \\in H$ , with $m^{\\prime }$ playing the role of child and $m$ playing the role of parent.",
"Rules and sets of interface predicates (input and output predicates) are propagated from parent module to its child modules.",
"A child module is modified by introducing additional rules or interface predicates and/or by deleting inherited rules or interface predicates.",
"Thus, rules and interface predicates can only be overridden by deletion and addition.",
"Modifying rules implies modification of a rule module's behavior.",
"[Inheritance by Incremental Modification] When a module $m^{\\prime }$ inherits from module $m$ , $(m^{\\prime },m) \\in H$ , the child module $m^{\\prime }$ inherits the parent's rule set $R_m$ from which it may remove a set of rules $R^-_{m^{\\prime }} \\subseteq R_m$ and to which it may add a set of rules $R^+_{m^{\\prime }} \\subseteq R$ , which results in the child module's rule set $R_{m^{\\prime }} \\stackrel{\\textup {\\tiny def}}{=}R_m \\cup R^+_{m^{\\prime }} \\setminus R^-_{m^{\\prime }}$ .",
"Inheritance and incremental modification of the sets of input and output predicates are defined analogously, i.e., $I_{m^{\\prime }} \\stackrel{\\textup {\\tiny def}}{=}I_m \\cup I^+_{m^{\\prime }} \\setminus I^-_{m^{\\prime }}$ and $O_{m^{\\prime }} \\stackrel{\\textup {\\tiny def}}{=}O_m \\cup O^+_{m^{\\prime }} \\setminus O^-_{m^{\\prime }}$ .",
"Often discussed, in particular regarding object-orientation, are abstract entities.",
"For instance, an abstract method is a method for which the signature is defined but no implementation is available [23], [32], [30].",
"Methods fully defined and implemented are usually called concrete.",
"Analogously, we distinguish predicates in a module into concrete and abstract.",
"We define concrete predicates as the union of predicates which are in the input interface, truth value true (represented as a nullary predicate), and predicates which contain only concrete predicates in the body of any rule having them in the rule head.",
"Abstract predicates are defined as predicates in a module which are not concrete.",
"In object-oriented design, a class is abstract if it contains abstract elements.",
"Analogously, we call a rule module abstract if it contains abstract predicates and concrete otherwise.",
"Similar to abstract classes which cannot be instantiated, abstract modules should not be applied as their behavior is incomplete.",
"Consequently, abstract predicates and modules should always be concreted in descendant modules.",
"Leaf modules in the module hierarchy should always be concrete.",
"[Abstract Predicates and Modules] A predicate $p$ depends on a predicate $p^{\\prime }$ in module $m$ , denoted as $dep_m(p,p^{\\prime })$ , if there is a rule $r$ in $R_m$ which has $p$ in the head and $p^{\\prime }$ in the body, i.e., $dep_m(p,p^{\\prime }) \\stackrel{\\textup {\\tiny def}}{=}\\exists r \\in R_m : p \\in H_r \\wedge \\, p^{\\prime } \\in B_r$ .",
"A predicate $p$ is concrete for a module $m$ if it is nullary predicate true or an input predicate or depends on some and only concrete predicates, i.e., if $(p = $ true$) \\vee \\, (p \\in I_m) \\vee ((\\forall p^{\\prime } : dep_m(p,p^{\\prime }) \\rightarrow concrete_m(p^{\\prime })) \\wedge (\\exists p^{\\prime } : dep_m(p,p^{\\prime })))$ .",
"A predicate $p \\in P_m$ is abstract for a module $m$ if it is not concrete.",
"A module is abstract if it has an abstract predicate.",
"From an organizational viewpoint, inheritance of rule modules can be employed for rule elicitation, definition, and organization: (a) An existing module can be adapted to a more specific setting and context by constructing a child module.",
"Therefore, rule developers have to know which rules and interface predicates are contained in a module.",
"To this end, they look at resolved modules, i.e., modules for which all inheritance relations have been resolved.",
"(b) A parent module can be constructed by extracting common/similar rules and interface predicates from child modules.",
"Abstract modules are of importance to the latter approach allowing to extract common predicates although the rules defining those predicates are different in child modules.",
"Hierarchical organization of modules eases management, in particular maintenance, as rule redundancy can be reduced and rule reuse is promoted.",
"Recently, our bank decided to extend its services to private loan applications.",
"Therefore, we elicited relevant rules for private loan applications.",
"We uncovered overlaps with module MortgageApps, in particular rules R0-R3.",
"Rule R0 overlaps with “Private loan values must exceed 12,000” (for now RX), R1 with “Credit worthiness of a private loan application is deemed good (predicate cwGood/1) if the value of provided securities exceeds 60 % of the loan's value” (for now RY), and rules R2 and R3 are identical.",
"Non-overlapping are: Regarding income, private loans are reported if the customer earns less than 600 Euro per month (R7).",
"The provided attachable income (attachableIncome/1) is calculated as 30 % of the income earned over the loan's duration and associated with the loan application (R8 – incomes/2) as security (R9 – securities/2, security/1) allowing to reuse rule R1 for deriving credit worthiness.",
"Thus, $O_{{\\tt PrivateLoanApps}} =$ $\\lbrace $cwGood/1, cwBad/1, attachableIncome/1, incomes/2, priorityOver/2, lowLValue/2, sValue/2, lowIncome/2$\\rbrace $ .",
"For these rules we identified the required inputs private loan application (loan/1), the intended loan value (lValue/2), the duration (duration/2), the applying customer (customer/2), and any incomes which may be used as securities (income/2).",
"These predicates form the input predicates $I_{{\\tt PrivateLoanApps}}$ which considerably overlap with $I_{{\\tt MortgageApps}}$ .",
"A subsequent discussion with domain experts revealed that rules R0, R2, R3, and RY are actually default rules applying to all existing and future loan types.",
"We extract the default rules and MortgageApps's and PrivateLoanApps's common interface predicates into a parent module LoanApps.",
"Rule RX is renamed to R0.1, R1 to R1.1, and RY to R1.",
"Regarding interfaces we have $I_{{\\tt LoanApps}} = \\lbrace $loan/1, lValue/2, duration/2, customer/2$\\rbrace $ and $O_{{\\tt LoanApps}} =$ $\\lbrace $cwGood/1, cwBad/1, priorityOver/2, lowLValue/1, sValue/2, securities/2, security/1$\\rbrace $ .",
"Since predicates securities/2, security/1, and sValue/2 are not derived in LoanApps but in its child modules, module LoanApps is abstract.",
"A visual summary of our use case including restrictions and Datalog$^\\pm $ representations is given in REF .",
"There, rule R0.1 overrides rule R0 and rule R1.1 rule R1, i.e., $R^-_{{\\tt PrivateLoanApps}}=\\lbrace $ R0$\\rbrace $ and $R^+_{{\\tt PrivateLoanApps}} = \\lbrace $R0.1, R7, R8, R9_1, R9_2$\\rbrace $ .",
"REF depicts module PrivateLoanApps with inheritance resolved.",
"Figure: Visual summary of module PrivateLoanApps with inheritance of rules, input declarations, and output declarations resolved according to Def.",
"and inheritance of modification restrictions resolved according to Def.",
"."
],
[
"Modification Restrictions",
"Of particular interest to this paper are modification restrictions constraining the allowed modifications by child modules.",
"We identified the following kinds of modifications: extension, elimination, and redefinition.",
"Extension adds features to child entities [32], [30], [29], [20], [27], [6], [17].",
"The contrary modification is elimination (also called reduction) removing features [32], [29], [30].",
"Redefinition redefines inherited features but does not eliminate features [32], [27], [20], [17].",
"In this paper, in order to keep the approach simple and compact, we focus on extension and elimination which can be used to simulate redefinition.",
"We introduce restrictions for module structure prohibiting (a) to extend interfaces and (b) to remove specific predicates from interfaces.",
"Regarding module behavior we introduce restrictions prohibiting: (c) to extend a module's behavior and (d) to reduce a module's behavior.",
"[Modification Restrictions] A module $m$ may define a set of modification restrictions $S_m$ of the following forms: no_additional_input, non_omitable_input($p$ ) with $p \\in I_m$ , no_additional_output, non_omitable_output($p$ ) with $p \\in O_m$ , non_growable($p$ ) with $p \\in O_m$ , and non_shrinkable($p$ ) with $p \\in O_m$ ."
],
[
"Structural Modification Restrictions",
"In order to regulate modification operations on module interfaces, we introduce four restrictions.",
"The restrictions no_additional_input and no_additional_output prohibit the addition of predicates to the input and output interface respectively.",
"The restrictions non_omitable_input and non_omitable_output prohibit to remove the specified predicate from the respective interface.",
"[Consistent Structural Modification] Let module $m^{\\prime }$ inherit from module $m$ , $(m^{\\prime },m) \\in H$ .",
"Structural modifications in child module $m^{\\prime }$ are consistent with modification restrictions in parent module $m$ if the following conditions hold: if $no\\_additional\\_input \\in S_m$ then $I^+_{m^{\\prime }} = \\emptyset $ , if $non\\_omitable\\_input(p) \\in S_m$ then $p \\notin I^-_{m^{\\prime }}$ , if $no\\_additional\\_output \\in S_m$ then $O^+_{m^{\\prime }} = \\emptyset $ , and if $non\\_omitable\\_output(p) \\in S_m$ then $p \\notin O^-_{m^{\\prime }}$ .",
"During the rule elicitation process our domain experts determined several modification restrictions regarding module LoanApps: Any application for a specific loan type must contain at least the same information as an application for a generic loan.",
"Consequently, we define: $non\\_omitable\\_input($loan$),$ $non\\_omitable\\_input($lValue$),$ $non\\_omitable\\_input($duration$),$ $non\\_omitable\\_input($customer$)$ .",
"Furthermore, any loan application module must output at least the predicates for credit worthiness, priority, low loan values, and security values.",
"Output securities may be replaced by more specific forms of outputs.",
"Thus, we define cwGood, cwBad, priorityOver, lowLValue, and sValue as $non\\_omitable\\_output$ .",
"In REF these restrictions are listed under $\\lnot omitable$ in the respective interface.",
"Module MortgageApps finalizes the output for any rule module inheriting from it, i.e., such a module may employ more input predicates for determining securities and their value but may not output additional predicates.",
"Therefore, we define $no\\_additional\\_output$ for module MortgageApps.",
"In REF we denoted this as $\\lnot extensible$ written next to the output of module MortgageApps.",
"From an organizational perspective, determining structural restrictions is part of rule elicitation, definition, and organization.",
"Conformance checking of structural restrictions is necessary whenever an interface of a rule module is changed, a new rule module is added to the module hierarchy, or any structural restrictions of an existing module are modified.",
"In the latter case, conformance of any child module of the modified module with the modified module's restrictions needs to be checked as well.",
"A prerequisite for structural conformance checks is the identification of performed modifications.",
"This is achieved by simple interface comparisons, e.g., parent's input schema with child's input schema."
],
[
"Behavioral Modification Restrictions",
"Modifying rules contained in a rule module influences the module's behavior with respect to derived facts of output predicates.",
"A child module, when applied on a dataset (see Def.",
"REF ), may return for a specific predicate the same, a superset, a subset, or a subset of a superset of derived facts compared to its parent module.",
"To regulate behavioral modifications, we introduce the restrictions non_growable and non_shrinkable for output predicates.",
"The restriction non_growable prohibits the derivation a superset of facts for a predicate in child modules whereas non_shrinkable prohibits child modules from deriving a subset of the facts derived by the parent module for a predicate.",
"[Consistent Behavioral Modification] Let module $m^{\\prime }$ inherit from module $m$ , $(m^{\\prime },m) \\in H$ .",
"Behavioral modifications in child module $m^{\\prime }$ are consistent with modification restrictions in parent module $m$ , if the following conditions hold for every data set $d \\in D$ which is applicable to both $m$ and $m^{\\prime }$ , as well as for every predicate $p$ which is in the output of $m$ and $m^{\\prime }$ : if $non\\_growable(p) \\in S_m$ then $p^d_{m^{\\prime }} \\subseteq p^d_m$ and if $non\\_shrinkable(p) \\in S_m$ , then $p^d_{m^{\\prime }} \\supseteq p^d_{m}$ .",
"Regarding behavior, domain experts reported several restrictions for module LoanApps: the basic rule for good credit worthiness (R1) is the minimum requirement, i.e., its condition may only be stricter.",
"Thus, we define $non\\_growable($cwGood$)$ meaning that a loan application not deemed credit worthy according to the rules in parent module LoanApps may not be derived as credit worthy by child modules.",
"Since every loan application is classified either cwGood/1 or cwBad/1 we state $non\\_shrinkable($cwBad$)$ .",
"Furthermore, the loan value in R0 is the minimum threshold for loan values.",
"Consequently, the value may only be increased when specializing module LoanApps, represented as $non\\_shrinkable($lowLValue$)$ .",
"As every loan application must be prioritized, behavior regarding priorityOver/2 must not change, represented as $non\\_growable($priorityOver$),$ $ non\\_shrinkable($priorityOver$)$ .",
"From an organizational viewpoint, behavioral restrictions are determined during rule elicitation, definition, and organization.",
"To perform behavioral conformance checks, performed behavioral modifications are compared with behavioral modification restrictions.",
"This check can be performed during rule module testing or during execution: (a) Before modifications to a module are disseminated and deployed, they need to be thoroughly tested, i.e., the module, any parent module, and any child modules are tested with various input data sets.",
"In addition to traditional testing, we then employ modification detection (see below) to check conformance to defined modification restrictions.",
"(b) We can also compare the behavior of parent and child module at runtime when the child module is executed.",
"Any violated behavioral restrictions are reported."
],
[
"Detection of Behavioral Modifications",
"To determine performed behavioral modifications we propose: (a) asking responsible rule developer(s) to state his/her performed behavioral modifications manually, and (b) to automatically detect performed behavioral modification operations employing static or dynamic detection.",
"We expect complete static detection (by automatic reasoning over Datalog programs) of behavioral modification operations to be undecidable due to rule dependencies as well as references and predicates within input data.",
"Nevertheless, partial detection is feasible by comparing a child module's with its parent module's rule dependency graph.",
"A proof that this problem is undecidable as well as an algorithm for partial static detection are beyond this paper.",
"We now introduce dynamic detection of behavioral modifications which can be used as part of testing or during rule execution.",
"Therefore, a child module and its parent module are executed on the same input data and their output facts compared.",
"For a specific parent module $m$ and child module $m^{\\prime }$ we select a set of data sets from $D$ applicable to both the child and the parent module.",
"For each data set $d$ , we execute both modules and compare the derived facts for concrete predicates; abstract predicates are not considered in conformance checks as their behavior is incomplete.",
"For each predicate $p$ concrete in the parent and child module, we compare $p^{d}_{m^{\\prime }}$ with $p^{d}_{m}$ .",
"If $p^{d}_{m^{\\prime }} \\subset p^{d}_{m}$ the set of output facts has shrunk, if $p^{d}_{m^{\\prime }} \\supset p^{d}_{m}$ the set of output facts has grown, if the sets of output facts has neither shrunk nor grown and $p^{d}_{m} \\ne p^d_{m^{\\prime }}$ it has shrunk and grown, and lastly $p^{d}_{m} = p^{d}_{m^{\\prime }}$ implies no changes.",
"The overall modification in behavior for a specific predicate is the union of all detected modifications in behavior.",
"These detected behavioral modifications must not violate specified behavioral modification restrictions.",
"The more data sets from $D$ are employed, the more reliable the detected modification(s) in behavior is/are."
],
[
"Inheritance of Modification Restrictions",
"In order to achieve transitive conformance to modification restrictions we need to introduce inheritance of modification restrictions.",
"Basically, child rule modules must not remove any modification restrictions imposed on their ancestors.",
"[Inheritance of Modification Restrictions] Let module $m^{\\prime }$ inherit directly from module $m$ , $(m^{\\prime },m) \\in H$ .",
"The child module's set of modification restrictions $S_{m^{\\prime }}$ is the union of the set of modification restrictions $S_m$ inherited from parent module $m$ and the set of modification restrictions $S^+_{m^{\\prime }}$ added by the child module $m^{\\prime }$ , i.e., $S_{m^{\\prime }} \\stackrel{\\textup {\\tiny def}}{=}S_m \\cup S^+_{m^{\\prime }}$ .",
"Module PrivateLoanApps inherits from module LoanApps.",
"Besides rules and interface predicates, the defined modification restrictions are inherited.",
"Consequently, any modification restriction defined in module LoanApps must hold in module PrivateLoanApps as well.",
"This is depicted in REF where inheritance has been resolved for module PrivateLoanApps, e.g., $non\\_omitable($cwGood$)$ defined in module LoanApps must also hold in module PrivateLoanApps."
],
[
"Proof-of-Concept Prototype",
"Our proof-of-concept prototype implements the presented formal definitions, structural and behavioral conformance checks, and proposed modification detections.",
"To this end, we need meta-representations of rule modules including their interfaces, inheritance relations, modification operations, and modification restrictions.",
"Therefore, we embedded our prototype in an environment able to generate meta representations of Datalog$^\\pm $ programs (e.g.",
"Vadalog [4]).",
"To facilitate widespread experimentation we provide a downloadhttp://files.dke.uni-linz.ac.at/publications/burgstaller/ODBASEPrototype.zip containing a Datalog implementation of our prototype (terms.datalog) and Datalog meta-representations for the rule modules in our use case (<module>Meta.datalog).",
"The meta-representations do not include built-ins.",
"While Datalog code is, beyond this use-case, in general severely limited in expressive power, it allows experimenting with our prototype independently of the concrete Datalog$^\\pm $ engine.",
"Executing our prototype on meta-representations of rule modules resolves inheritance, i.e., determines rules, facts, and modification restrictions applying in each module, reports abstract predicates and modules, and detects violations of structural modification restrictions.",
"In order to test dynamic behavioral modification detection, we provide meta-representations of the output facts resulting from executing module LoanApps and module PrivateLoanApps on the same input facts (result<module>.datalog).",
"Executing the prototype for our use case using DLV yields reasonable performance considering that the average times include DLV start-up and no performance optimizations have been carried out.",
"Resolving inheritance, reporting abstract predicates and modules, detecting behavioral modification operations, and structural and behavioral conformance checking take on average 0.017 s (sd = 0.001 s) on a standard notebook (Intel Core i5-6200U, DDR4, 16 GB)."
],
[
"Related Work",
"In the following we discuss inheritance of single rules; rule sets, rule modules, and their inheritance; as well as contextual knowledge and its inheritance.",
"Several researchers proposed inheritance of pre- and postconditions of operations (rule premises) [23], [24], [30] where modifications to conditions are restricted to strengthening or weakening.",
"Other research regards inheritance of triggers (which can be considered event-condition-action rules) in object-oriented databases [5], where a trigger's premises may become weaker and trigger actions may be extended.",
"$\\mathcal {F}LORA$ -2 [17] combines rules and object-oriented features where methods are specified as rules and can be inherited."
],
[
"Rule Sets, Modules, and Inheritance",
"Modularization is employed in many fields, e.g., software engineering or ontologies, enabling controlled and structured development of large systems [28].",
"Regarding knowledge, a knowledge base may either be partitioned into non-overlapping modules or relevant portions extracted into possibly overlapping modules [28].",
"The concrete modularization strategy depends on the use case.",
"Regarding ontologies, a triple containing an ontology, a query language, and a vocabulary can be considered a module, where the query language and the vocabulary serve as interface allowing interaction by querying [18].",
"A simple extension mechanism enabling addition of ontological statements is described [18].",
"Rule modules with relational schemas as interface specifications are a simplified variant of relational transducers [1].",
"In the original proposal, relational transducers serve as “declarative specifications of business models, using an approach in the spirit of active databases”.",
"Relational transducers transform sequences of input into sequences of output relations.",
"In addition to input and output relations, a relational transducer specifies database, state, and log relations, where the log relations are the semantically relevant subset of input and output relations.",
"Regarding inheritance, they discuss customization of relational transducers and with regard to restricted modification they discuss “containment and equivalence of relational transducers relative to a specified log”.",
"Modular Web rule bases [2] separate interfaces, i.e., predicates used and predicates defined, from the logic program, i.e., rules.",
"Each predicate in a module defines its reasoning mode, its scope, and its origin rule bases or rule bases it is visible to.",
"Modular rule base extension is supported, allowing to add new rules to existing rule bases and to add new rule bases to the set of rule bases.",
"Inheritance of (business) rules is touched in [10], [9] (see contextualized knowledge below) and discussed specific to situation-condition-actions rules in [20].",
"Inheritance of rule sets is discussed in [27].",
"Lang [20] identifies the rule of origin as prerequisite, i.e., a feature may only be defined in a single point.",
"Moreover, he utilizes the abstract parent class rule.",
"Situations, conditions, and actions may be specialized if the occurrence of a situation implies occurrence of its child situations and rule conditions are only weakened.",
"Based on these conditions modification operations are proposed [20]: extension denotes addition of rules or redefining rules to fire more often, refinement denotes concreting abstract rules, i.e., concreting rule-triggering interval classes to event classes, and redefinition is constrained to specialized actions and events as well as weakened conditions.",
"Pachet [27] describes rule set inheritance as inheriting all rules from parent rule sets and allows for unconstrained redefinition of rules.",
"A related field are business rule management systems (BRMSs) like IBM's JRules, JBoss Drools, FICO Blaze Advisor, or Oracle Business Rules, which organize business rules into rule sets.",
"JRules supports inheritance of rules and rule sets where rules may be overridden.",
"Drools and Blaze both support inheritance of rule conditions.",
"Oracle does not report support for rule (set) inheritance.",
"Many of the above approaches support rule sets ([28], [18], [10], [9], [20], [27], [17], BRMSs) but only Abiteboul et al.",
"[1], Analyti et al.",
"[2], and Konev et al.",
"[18] describe modules with explicit definition of input and output interfaces as supported by our approach.",
"Rule set inheritance is supported by quite a few approaches, where some allow no or only predefined modification types ([1], [18], [20], [9], Drools, Blaze) and others allow arbitrary modifications ([10], [27], JRules).",
"Nevertheless, none of them provides any means to assert fine-grained control over the types of modifications to inherited rule sets as our approach does with the presented modification restrictions.",
"By these modification restrictions, our approach allows, unlike the ones discussed above, to flexibly adapt the inheritance mechanism to the specific needs at hand."
],
[
"Contextualized Knowledge Representation and Inheritance",
"A related field are contextual knowledge and its inheritance.",
"Contextualized Knowledge repositories (CKR) [31] for the Semantic Web organize ontological concepts employing hierarchically ordered contexts.",
"A context is defined by a set of values for various but fixed context dimensions.",
"Allowed dimension values form a subsumption hierarchy from which the hierarchy of contexts is derived.",
"Concepts propagate along this hierarchy from general contexts to more specific contexts.",
"Concepts may assume different meanings in different contexts.",
"A similar idea was proposed for knowledge organization in CYC [21] where it is explicitly allowed to contradict or override inherited knowledge.",
"McCarthy [22] views contexts as generalizing collections of assumptions where a child context must have at least the same assumptions as its parent context.",
"Knowledge from child contexts must be translatable into meaningful knowledge in the parent context, i.e., a root context would contain all knowledge in decontextualized form.",
"Building on these approaches, we introduced a contextualized (business) rule repository [10] allowing to organize rule sets into multi-dimensional context hierarchies.",
"Regarding inheritance, we differentiate additive and most-specific inheritance of rule sets.",
"With the former inheritance mechanism all rules of a parent rule set apply in its child rule sets as well.",
"The latter approach allows redefinition of inherited rules.",
"Previous work on contextualized rule repositories for the Semantic Web [9] employed an additive inheritance semantics only.",
"Since contexts can be considered modules, all of the above approaches support modules.",
"Nevertheless, none of them explicitly defines clear input and output interfaces for modules as our approach does.",
"Regarding module inheritance, all of the above approaches support inheritance of knowledge, some [21], [10] allow redefinition of knowledge whereas others do not [22], [31], [9].",
"Compared to our approach, none of the above approaches supports modification restrictions, i.e., one cannot control modifications to the knowledge in a module."
],
[
"Conclusion",
"We investigated inheritance of rule modules to foster reuse of rules, simplify adaptation, and ease maintenance.",
"Therefore, we introduced rule modules and proposed a formal inheritance mechanism.",
"We presented modification restrictions regulating modifications with corresponding conformance checks as mechanisms for keeping child modules aligned with parent modules.",
"In ongoing and future work we investigate: extending our proposed approach to multi-inheritance and integrating it into context-aware business rule management [7] and Vadalog [4].",
"rule modules as part of derivation chains or module networks in big data wrangling [19], [16] and knowledge graph management [3].",
"There, rule set adaptation is necessary to cope with the variety of relevant subject domains and the variety of integrated sources of data and knowledge.",
"One such scenario that we want to apply our approach to is building knowledge graphs on huge elections in the area of computational social choice [13], another one is in data extraction, where inheritance hierarchies are essential for managing diverse ontologies [15], [14].",
"Furthermore, the clean interfaces of our rule modules support derivation chains and networks."
]
] | 1808.08634 |
[
[
"An Upper Bound on the Number of $(132,213)$-Avoiding Cyclic Permutations"
],
[
"Abstract We show a $n^2 \\cdot 2^{n/2}$ upper bound on the number of $(132,213)$ avoiding cyclic permutations.",
"This is the first nontrivial upper bound on the number of such permutations.",
"We also construct an algorithm to determine whether a $(132,213)$ avoiding permutation is cyclic that references only the permutation's layer lengths."
],
[
"Introduction",
"The theory of pattern avoidance in permutations has been widely studied since its introduction by Knuth in 1968 [9].",
"The classical form of this problem asks to count the number of permutations of $[n] = \\lbrace 1,\\ldots ,n\\rbrace $ avoiding a given pattern $\\sigma $ .",
"Since then, many variations of this problem have been proposed and studied in the literature.",
"We will focus on the problem of pattern avoidance among permutations consisting of a single cycle.",
"This problem was first posed by Stanley in 2007 at the Permutation Patterns Conference, and was subsequently studied by Archer and Elizalde [2] and Bóna and Cory [4].",
"Let us first recall the definition of pattern avoidance.",
"Let ${\\mathfrak {S}}_n$ denote the set of permutations of $[n]$ .",
"Definition 1.1 Let $\\sigma \\in {\\mathfrak {S}}_k$ .",
"A permutation $\\pi \\in {\\mathfrak {S}}_n$ contains $\\sigma $ if there exist indices $1\\le i_1< \\ldots < i_k\\le n$ such that the sequence $\\pi _{i_1}\\pi _{i_2} \\cdots \\pi _{i_k}$ is in the same relative order as $\\sigma _1\\sigma _2\\cdots \\sigma _k$ .",
"Otherwise, $\\pi $ avoids $\\sigma $ .",
"In either case, $\\sigma $ is the pattern that $\\pi $ contains or avoids.",
"Example 1.1 The permutation 12534 contains the pattern 132 and avoids the pattern 321.",
"In classical pattern avoidance, the central objects of study are the numbers $A_n(\\sigma )$ , defined as follows.",
"Definition 1.2 Let $\\sigma \\in {\\mathfrak {S}}_k$ be a pattern.",
"Then, $A_n(\\sigma )$ is the number of permutations $\\pi \\in {\\mathfrak {S}}_n$ avoiding the pattern $\\sigma $ .",
"Similarly, for $(\\sigma _1,\\sigma _2)\\in {\\mathfrak {S}}_{k_1}\\times {\\mathfrak {S}}_{k_2} $ , $A_n(\\sigma _1,\\sigma _2)$ is the number of permutations $\\pi \\in {\\mathfrak {S}}_n$ avoiding both $\\sigma _1$ and $\\sigma _2$ .",
"In a classical result, Knuth [9] showed that $A_n(\\sigma ) = {n+1} \\binom{2n}{n}$ , the $n^{\\text{th}}$ Catalan number, for any pattern $\\sigma \\in {\\mathfrak {S}}_3$ .",
"In 1985, Simion and Schmidt [11] proved analogous results for permutations avoiding two patterns, computing the value of $A_n(\\sigma _1,\\sigma _2)$ for any pair of distinct patterns $(\\sigma _1,\\sigma _2)\\in {\\mathfrak {S}}_3\\times {\\mathfrak {S}}_3$ .",
"For an overview of related results in classical pattern avoidance, see the book by Linton, Rušcuk, and Vatter [10].",
"In classical pattern avoidance, we think of permutations only as linear orders.",
"We can also think of permutations algebraically, in terms of their cycle decompositions.",
"For example, the permutation whose one-line notation is 24513 has cycle decomposition $(124)(35)$ .",
"With this perspective, we can discuss pattern avoidance among permutations whose cycle decompositions consist of a single cycle.",
"Definition 1.3 Let $\\sigma \\in {\\mathfrak {S}}_k$ be a pattern.",
"Then, $C_n(\\sigma )$ is the number of permutations $\\pi \\in {\\mathfrak {S}}_n$ avoiding $\\sigma $ that consist of a single $n$ -cycle.",
"Similarly, for $(\\sigma _1,\\sigma _2)\\in {\\mathfrak {S}}_{k_1}\\times {\\mathfrak {S}}_{k_2}$ , $C_n(\\sigma _1,\\sigma _2)$ is the number of permutations $\\pi \\in {\\mathfrak {S}}_n$ avoiding both $\\sigma _1$ and $\\sigma _2$ that consist of a single $n$ -cycle.",
"In 2007, Richard Stanley asked for the determination of the value of $C_n(\\sigma )$ for any $\\sigma \\in {\\mathfrak {S}}_3$ .",
"All cases of this problem remain open; this problem is difficult because it requires considering both views of permutations described above.",
"Work on the analogous problem for cyclic permutations avoiding two patterns was begun by Archer and Elizalde in 2014.",
"They showed the following result.",
"Theorem 1.1 [2] Let $\\mu $ be the number-theoretic Möbius function.",
"For all positive integers $n$ , $C_n(132,321) ={2n}\\sum _{\\begin{array}{c}d|n \\\\ \\text{$d$ odd}\\end{array}} \\mu (d) 2^{n/d}.$ This result was proved by studying permutations realized by shifts.",
"For more results about such permutations, see [1], [5], [6].",
"Bóna and Cory [4] determined $C_n(\\sigma _1,\\sigma _2)$ for several pairs $(\\sigma _1,\\sigma _2)\\in {\\mathfrak {S}}_3\\times {\\mathfrak {S}}_3$ .",
"Their most significant result is as follows.",
"Theorem 1.2 [4] Let $\\phi $ be the Euler totient function.",
"For all positive integers $n$ , $C_n(123,231) ={\\left\\lbrace \\begin{array}{ll}\\phi (n̑2) & n\\equiv 0\\pmod {4}, \\\\\\phi ({4}) + \\phi (n̑2) & n\\equiv 2 \\pmod {4}, \\\\\\phi ({2}) & n\\equiv 1\\pmod {2}.\\end{array}\\right.",
"}$ Bóna and Cory also proved the following formulae for other pairs $(\\sigma _1,\\sigma _2) \\in {\\mathfrak {S}}_3\\times {\\mathfrak {S}}_3$ .",
"Theorem 1.3 [4] The following identities hold.",
"For all $n\\ge 5$ , $C_n(123,321)=0$ .",
"For all $n\\ge 3$ , $C_n(231,312)=0$ .",
"For all positive integers $n$ , $C_n(231,321)=1$ .",
"For all positive integers $n$ , $C_n(132,321)=\\phi (n)$ .",
"For all positive integers $n$ , $C_n(123,132)=2^{\\lfloor (n-1)/2\\rfloor }$ .",
"Other results regarding pattern avoiding permutations with given cyclic structure can be found in [7], [8], [12], [13].",
"The above formulae by [2], [4] and analogous formulae obtained from them by symmetry determine the values of $C_n(\\sigma _1,\\sigma _2)$ for each pair of distinct patterns $(\\sigma _1,\\sigma _2)\\in {\\mathfrak {S}}_3\\times {\\mathfrak {S}}_3$ except $(132,213)$ .",
"This motivates the following problem.",
"Problem 1.1 Determine an explicit formula for $C_n(132,213)$ .",
"Solving this problem would complete the enumeration of cyclic permutations avoiding pairs of patterns of length 3.",
"The main result of this paper is the following bound.",
"Theorem 1.4 For all positive integers $n$ , $C_n(132,213) \\le n^2 \\cdot 2^{n/2}$ .",
"To our knowledge, this is the first nontrivial upper bound of $C_n(132,213)$ , though computer experiments suggest this bound is not asymptotically tight.",
"A composition of $n$ is a tuple $(a_1,\\ldots ,a_k)$ of positive integers with sum $n$ .",
"The $(132,213)$ avoiding permutations of size $n$ are the so-called reverse layered permutations, which correspond to the compositions of $n$ .",
"We refer to compositions corresponding to cyclic permutations as cyclic compositions.",
"The second main result of this paper is the following algorithm, which determines, without reference to the associated permutation, whether a composition $C=(a_1,\\ldots ,a_k)$ is cyclic.",
"Algorithm 1.1 Take as input a composition $C$ .",
"Run the repeated reduction algorithm $\\operatorname{Rred}$ on the equalization $\\operatorname{Eq}(C)$ .",
"If $\\operatorname{Rred}$ outputs that $\\operatorname{Eq}(C)$ is cyclic, output that $C$ is cyclic.",
"Otherwise, output that $C$ is not cyclic.",
"The operations $\\operatorname{Rred}$ and $\\operatorname{Eq}$ are defined in Sections and , respectively.",
"This result is interesting in its own right, and is the key step in the proof of Theorem REF ; it implies that any cyclic composition has one or two odd terms, which gives the bound in Theorem REF .",
"The rest of this paper is structured as follows.",
"In Section we formalize the connection between $(132,213)$ -avoiding permutations and compositions of $n$ .",
"We introduce the notions of balanced compositions and cycle diagrams, tools that will be useful in our analysis.",
"In Section we prove our results for balanced permutations.",
"In Section we generalize our results to all permutations.",
"Finally, in Section we present some directions for further research."
],
[
"Reverse Layered Permutations",
"The skew sum $\\pi \\ominus \\tau $ of permutations $\\pi \\in {\\mathfrak {S}}_m$ , $\\tau \\in {\\mathfrak {S}}_n$ is defined by $(\\pi \\ominus \\tau )(i) ={\\left\\lbrace \\begin{array}{ll}\\pi (i)+n & i \\le m \\\\\\tau (i-m) & i>m\\end{array}\\right.",
"}$ for all $i\\in [m+n]$ .",
"Note that $\\ominus $ is an associative operation.",
"Moreover, let $I_n=123\\cdots n$ denote the identity permutation on $n$ elements.",
"A reverse layered permutation is a permutation of the form $\\pi = I_{a_1}\\ominus I_{a_2}\\ominus \\cdots \\ominus I_{a_k},$ for some positive integers $a_1,\\ldots ,a_k$ summing to $n$ .",
"Explicitly, a reverse layered permutation is of the form $\\pi = (n-a_1+1)\\cdots (n) ~ (n-a_1-a_2+1)\\cdots (n-a_1) ~ \\cdots ~ (1)(2)\\cdots (a_k).$ It is known that the $(132,213)$ avoiding permutations are the reverse layered permutations [3].",
"We can bijectively identify these permutations with the compositions $(a_1,\\ldots ,a_k)$ of $n$ .",
"As an immediate consequence, there are $2^{n-1}$ reverse layered permutations of length $n$ , corresponding to the $2^{n-1}$ compositions of $n$ .",
"It remains, therefore, to determine the number of these permutations that are also cyclic.",
"Recall that a composition of $n$ is cyclic if its associated reverse layered permutation is cyclic.",
"Determining $C_n(132,213)$ is therefore equivalent to counting the cyclic compositions of $n$ ."
],
[
"Balanced Compositions",
"We say a composition $C=(a_1,\\ldots ,a_k)$ of $n$ is balanced if some prefix $a_1,\\ldots ,a_j$ has sum $n̑2$ .",
"We denote such compositions with the notation $(a_1,\\ldots ,a_j | a_{j+1},\\ldots , a_k)$ .",
"Otherwise, we say $C$ is unbalanced.",
"Note that compositions of odd $n$ are all unbalanced.",
"Equivalently, $C$ is balanced if its associated reverse layered permutation $\\pi $ has the property that $\\pi (i) > n̑2$ if and only if $i\\le n̑2$ .",
"We say a reverse layered permutation is balanced if it has this property, and unbalanced otherwise.",
"Example 2.1 The composition $C=(1,2,1,2)$ is balanced, and corresponds to the reverse layered permutation $\\pi =645312$ .",
"The permutation $\\pi $ has the property that $\\pi (i)>3$ if and only if $i\\le 3$ .",
"The composition $C^{\\prime }=(1,3,2)$ is unbalanced, and corresponds to the reverse layered permutation $\\pi ^{\\prime }=634512$ .",
"The permutation $\\pi ^{\\prime }$ does not have this property."
],
[
"Cycle Diagrams",
"Define the graph of a permutation $\\pi \\in {\\mathfrak {S}}_n$ as the collection of points $(i, \\pi (i))$ , for $i\\in [n]$ .",
"The cycle diagram of $\\pi $ is obtained from the graph of $\\pi $ by drawing vertical and horizontal line segments, called wires, from each point in the graph to the line $y=x$ .",
"We say the points $(i, \\pi (i))$ are the points of the cycle diagram.",
"By slight abuse of notation, we say the cycle diagram of a composition $C$ is the cycle diagram of its associated permutation.",
"The wires in the cycle diagram of a permutation $\\pi $ form one or more contiguous loops.",
"Each loop has the property that, when followed clockwise, the column it visits after the $i^{\\text{th}}$ column is the $\\pi (i)^{\\text{th}}$ column.",
"Thus, the loops of the cycle diagram of $\\pi $ coincide with the cycle decomposition of $\\pi $ ; in particular, a permutation is cyclic if and only if the wires in its cycle diagram make a single closed loop.",
"Moreover, a permutation is balanced if and only if each point in its cycle diagram is, along the wire path, adjacent to two points on the opposite side of the line $y=x$ .",
"Each layer $I_{a_i}$ in a reverse layered permutation corresponds to a layer of $a_i$ diagonally-adjacent points in the cycle diagram running from bottom left to top right; successive layers are ordered from top left to bottom right.",
"Example 2.2 Figure REF shows the cycle diagrams of the balanced cyclic permutation 645312, the unbalanced cyclic permutation 53412, and the balanced noncyclic permutation 456321.",
"In the cycle diagram of 645312, the wire forms a single loop, and each point (marked with $\\times $ ) is, along the wire path, adjacent to two points on the opposite side of the line $y=x$ .",
"In the cycle diagram of 53412, the wire still forms a single loop, but the two points in the middle layer are adjacent to each other on the wire path, and not to points on the opposite side of the line $y=x$ .",
"In the cycle diagram of 456321, the wire does not form a single loop.",
"Figure: Cycle diagrams for permutations 645312, 53412, and 456321,with associated compositions (1,2|1,2)(1,2|1,2), (1,2,2)(1,2,2), and (3|1,1,1)(3|1,1,1), respectively.The line y=xy=x is shown by a dashed line."
],
[
"Determining if a Balanced Composition is Cyclic",
"In this section, we will prove specialized versions of our main results for balanced compositions.",
"We will generalize these results to all compositions in the next section."
],
[
"Reducing Cycles",
"The key result in this subsection is the Cycle Reduction Lemma, which is Lemma REF .",
"Lemma 3.1 Suppose the balanced composition $(a_1,\\ldots ,a_k | b_m,\\ldots ,b_1)$ is cyclic, and $a_k = b_m$ .",
"Then $a_k=b_m=k=m=1$ and $n=2$ .",
"If $a_k=b_m$ , then the associated permutation $\\pi $ contains the 2-cycle $\\left(n̑2, n̑2+b_m\\right)$ .",
"Since the composition is cyclic, this 2-cycle is the only cycle, and thus $n=2$ and $a_k=b_m=k=m=1$ .",
"For a balanced composition $C = (a_1,\\ldots ,a_k | b_m,\\ldots ,b_1)$ with $a_k \\ne b_m$ , we define the reduction operation $\\operatorname{Red}$ as follows.",
"Let $u &=& a_k \\mod {|}a_k - b_m| \\\\v &=& |a_k-b_m| - u.$ If $a_k>b_m$ , define $\\operatorname{Red}(C) = (a_1,\\ldots ,a_{k-1},u,v | b_{m-1},\\ldots ,b_1).$ Analogously, if $a_k<b_m$ , define $\\operatorname{Red}(C) = (a_1,\\ldots ,a_{k-1}| v,u,b_{m-1},\\ldots ,b_1).$ In both cases, if $u=0$ , omit $u$ from the composition.",
"Note that $\\operatorname{Red}$ always decreases a composition's sum, because $u+v=|a_k-b_m|<a_k+b_m$ .",
"Lemma 3.2 (Cycle Reduction Lemma) Suppose $a_k\\ne b_m$ .",
"Then, the composition $C = (a_1,\\ldots ,a_k | b_m,\\ldots ,b_1)$ is cyclic if and only if $\\operatorname{Red}(C)$ is cyclic.",
"Before proving this lemma, we give an example that captures the spirit of the proof.",
"Example 3.1 Let $a_k=5$ , $b_m=2$ , so $u=2$ and $v=1$ .",
"The Cycle Reduction Lemma states that a balanced composition $C = (a_1,\\ldots ,a_{k-1},5|2,b_{m-1},\\ldots ,b_1)$ is cyclic if and only if $\\operatorname{Red}(C) = (a_1,\\ldots ,a_{k-1},2,1|b_{m-1},\\ldots ,b_1)$ is cyclic.",
"Let us focus on the layers corresponding to $a_k=5$ and $b_m=2$ , which are the innermost layers of the cycle diagram on either side of the line $y=x$ .",
"The wires incident to these two layers connect with each other, leaving three pairs of loose ends, as shown in the left diagram in Figure REF .",
"The wires induce a natural pairing on these loose ends: two loose ends are paired if they are connected by these wires.",
"These loose ends are paired the same way by two layers of size $u=2$ and $v=1$ , as shown in the right diagram in Figure REF .",
"Figure: Cycle diagrams for a layer of size 5 opposite a layer of size 2 (left),and for consecutive layers of size 2 and 1 on the same side of theline y=xy=x (right), with loose ends.The line y=xy=x is shown by a dashed line, and loose ends are circled.Consider any configuration of the remaining wires in the cycle diagram, which we call the outer wiring.",
"These wires also form three pairs of loose ends, which connect with the three pairs of loose ends from the innermost wires.",
"Thus, once we fix an outer wiring, whether the cycle diagram is cyclic depends only on the pairing of loose ends induced by the innermost wires.",
"Since the wirings of the innermost layers of $C$ and $\\operatorname{Red}(C)$ pair their loose ends the same way, the cycle diagram of $C$ is a single cycle if and only if the cycle diagram of $\\operatorname{Red}(C)$ is a single cycle.",
"Let us assume $a_k > b_m$ ; the case $a_k < b_m$ is symmetric.",
"Let $D = a_k - b_m$ .",
"Thus $u = a_k \\mod {D}$ and $v = D - u$ .",
"Let us write $a_k = qD+u$ .",
"Consider the cycle diagram of $C$ .",
"The terms $a_k$ and $b_m$ correspond to the two innermost layers of the cycle diagram on either side of the line $y=x$ .",
"Call these layers $L_a$ and $L_b$ .",
"When we connect the wires incident to these layers, each point in $L_b$ is connected to two points in $L_a$ , which are diagonally $D$ cells apart.",
"The rightward wires from the upper-right $a_k-b_m$ points in $L_a$ and the downward wires from the lower-left $a_k-b_m$ points in $L_a$ are loose ends.",
"Call these sets the upper-right loose ends and the lower-left loose ends.",
"The wiring of $L_a$ and $L_b$ induces a pairing between these sets of loose ends, as described in Example REF .",
"Let us determine this pairing by starting at one of the upper-right loose ends and following its wire.",
"Every time the wire winds around the center of the cycle diagram, it passes through one point in $L_a$ and one point in $L_b$ ; because each point in $L_b$ is connected to two points in $L_a$ that are diagonally $D$ cells apart, successive points in $L_a$ and on this wire are diagonally $D$ cells apart.",
"Let us examine the points this wire passes through in $L_a$ .",
"If we started at one of the upper-rightmost $u$ loose ends, the wire passes through $q+1$ points in $L_a$ ; thus the last point in $L_a$ on the wire is diagonally $qD$ cells from the first point in $L_a$ on the wire.",
"Moreover, this implies that the upper-rightmost $u$ loose ends are paired with the lower-leftmost $u$ loose ends.",
"The remaining $D-u=v$ upper-right loose ends pass through $q$ points in $L_a$ ; thus the last point in $L_a$ on their wires is diagonally $(q-1)D$ cells from the first point in $L_a$ on their wires.",
"This pairing is the same as the pairing produced by two consecutive layers of $u$ and $v$ cells, with the layer of size $v$ on the inside.",
"Therefore, any fixed outer wiring makes a cyclic wiring when connected with opposing layers of size $a_k, b_m$ if and only it makes a cyclic wiring when connected with consecutive layers of size $u, v$ .",
"So, the cycle diagram of $C$ consists of a single cycle if and only if the cycle diagram of $\\operatorname{Red}(C)$ consists of a single cycle."
],
[
"The Repeated Reduction Algorithm",
"Lemmas REF and REF imply the following algorithm to determine if a balanced composition is cyclic.",
"This is Algorithm REF specialized to balanced compositions.",
"Algorithm 3.1 (Repeated Reduction Algorithm) Take as input a balanced composition $C$ .",
"Repeatedly apply $\\operatorname{Red}$ to $C$ until $a_k=b_m$ .",
"If this procedure stops at the composition $(1|1)$ , output that $C$ is cyclic.",
"Otherwise, output that $C$ is not cyclic.",
"We denote this algorithm by $\\operatorname{Rred}$ .",
"Note that this algorithm must terminate, because each application of $\\operatorname{Red}$ decreases the sum of the composition.",
"We can now prove a necessary condition for a balanced composition to be cyclic.",
"Proposition 3.3 Every cyclic balanced composition has exactly two odd entries.",
"The operation $\\operatorname{Red}$ , and therefore the algorithm $\\operatorname{Rred}$ , preserves the number of odd entries in a composition.",
"The cyclic balanced compositions all reduce to $(1|1)$ , so they must themselves have exactly two odd entries."
],
[
"The General Setting",
"In this section, we will generalize the results of the previous section to all compositions."
],
[
"Equalization",
"The Repeated Reduction Algorithm, developed in the previous section, determines whether a balanced composition is cyclic.",
"We now address the problem of determining whether any composition is cyclic.",
"We do this via equalization, an operation that transforms any composition into a balanced composition.",
"The Equalization Lemma, which is Lemma REF , reduces this new problem to the one addressed in the previous section.",
"In an unbalanced composition $C=(a_1,\\ldots ,a_k)$ , no prefix sums to $n̑2$ .",
"Therefore there is $i$ such that $a_1+\\cdots +a_{i-1} < n̑2\\quad \\text{and}\\quad a_1+\\cdots +a_i > n̑2.$ We call this $i$ the dividing index of the composition.",
"For unbalanced compositions, we will develop notions of nearly-equal division and unequalness.",
"Definition 4.1 The nearly-equal division of an unbalanced composition $C$ with dividing index $i$ is defined as follows.",
"If $a_1+\\cdots +a_{i-1} \\le a_{i+1}+\\cdots +a_k$ , then the nearly equal division is $((a_1,\\ldots ,a_i),(a_{i+1},\\ldots ,a_k))$ .",
"If $a_1+\\cdots +a_{i-1} > a_{i+1}+\\cdots +a_k$ , then the nearly equal division is $((a_1,\\ldots ,a_{i-1}),(a_i,\\ldots ,a_k))$ .",
"Example 4.1 The nearly-equal division of $(1,1,3,1)$ is $((1,1),(3,1))$ ; the 3 belongs to the second part of the division because $1+1>1$ .",
"The nearly-equal division of $(3,4,2,2)$ is $((3,4),(2,2))$ ; the 4 belongs to the first part of the division because $3\\le 2+2$ .",
"Definition 4.2 The unequalness of an unbalanced composition $C$ with nearly-equal division $((a_1,\\ldots ,a_j),(a_{j+1},\\ldots ,a_k))$ is $U(C) = \\left|\\sum _{\\ell =1}^j a_\\ell - \\sum _{\\ell =j+1}^k a_\\ell \\right|.$ Definition 4.3 The equalization of an unbalanced composition $C$ with nearly-equal division $((a_1,\\ldots ,a_j),(a_{j+1},\\ldots ,a_k))$ is $\\operatorname{Eq}(C) = (a_1,\\ldots ,a_j,U(C),a_{j+1},\\ldots ,a_k),$ The equalization of a balanced composition $C$ is $\\operatorname{Eq}(C) = C$ .",
"Note that $\\operatorname{Eq}(C)$ is always a balanced composition; if $C$ is unbalanced, the term $U(C)$ gets added to the smaller side of the nearly-equal division of $C$ , thereby balancing it.",
"Remark 4.1 The nearly-equal division is defined non-symmetrically; when $a_1+\\cdots +a_{i-1} = a_{i+1}+\\cdots +a_k,$ we arbitrarily define the nearly-equal division to be $((a_1,\\ldots ,a_i),(a_{i+1},\\ldots ,a_k))$ .",
"However, the definition of equalization is still symmetric.",
"When the above equality holds, regardless of whether we define the nearly-equal division $C$ as $((a_1,\\ldots ,a_i),(a_{i+1},\\ldots ,a_k))$ or $((a_1,\\ldots ,a_{i-1}),(a_i,\\ldots ,a_k)),$ the equalization is $\\operatorname{Eq}(C)=(a_1,\\ldots ,a_i,a_i,\\ldots ,a_k).$ The following lemma allows us to reduce the question of whether an unbalanced composition is cyclic to the question of whether a balanced composition is cyclic.",
"Lemma 4.1 (Equalization Lemma) The unbalanced composition $C$ is cyclic if and only if $\\operatorname{Eq}(C)$ is cyclic.",
"Once again, we first demonstrate the lemma with an example.",
"Example 4.2 Suppose $a_1+\\cdots +a_{i-1} + 3 = a_{i+1}+\\cdots +a_k$ , and $a_i=5$ .",
"Then the nearly-equal division of $C$ is $((a_1,\\ldots ,a_{i-1},5),(a_{i+1},\\ldots ,a_k)),$ and $U(C) = \\left| \\sum _{\\ell =1}^i a_\\ell - \\sum _{\\ell =i+1}^k a_\\ell \\right| = 2.$ Thus, $\\operatorname{Eq}(C) = (a_1,\\ldots ,a_{i-1},5 | 2,a_{i+1},\\ldots ,a_k).$ The left and right diagrams of Figure REF show, respectively, the central layer of the cycle diagram of $C$ and the two innermost layers of the cycle diagram of $\\operatorname{Eq}(C)$ .",
"Note that equalization slides the layer of size 5 upward and to the left, and inserts a layer of size 2 opposite it, while keeping the connectivity of the remaining wires unchanged.",
"Figure: Cycle diagram for a central layer of size 5 in a composition with unequalness 2 (left),and for a layer of size 5 opposite a layer of size 2 (right).The main diagonal is shown by a dotted line, and loose ends are circled.In both diagrams, there are three pairs of loose ends; the loose ends are paired by connectivity in the same way.",
"Therefore, $C$ is cyclic if and only if $\\operatorname{Eq}(C)$ is cyclic.",
"Assume $C$ is unbalanced.",
"Let $C=(a_1,\\ldots ,a_k)$ have dividing index $i$ .",
"If $a_1+\\cdots +a_{i-1} = a_{i+1}+\\cdots +a_k,$ then $\\operatorname{Eq}(C) = (a_1,\\ldots ,a_i,a_i,\\ldots ,a_k).$ So, the middle $a_i$ entries of the permutation associated to $C$ are fixed points, and the middle $2a_i$ entries of the permutation associated to $\\operatorname{Eq}(C)$ form $a_i$ 2-cycles.",
"Therefore if $C=(1)$ , $C$ and $\\operatorname{Eq}(C)$ are simultaneously cyclic, and otherwise they are simultaneously not cyclic.",
"Otherwise, we further assume $a_1+\\cdots +a_{i-1} \\ne a_{i+1}+\\cdots +a_k.$ The remaining proof is, once again, a matter of following loose ends.",
"Let us assume, without loss of generality, that $a_1+\\cdots +a_{i-1} < a_{i+1}+\\cdots +a_k.$ Then the nearly-equal division of $C$ is $((a_1,\\ldots ,a_i),(a_{i+1},\\ldots ,a_k))$ .",
"Let $U=U(C)$ .",
"Note that $U < a_i$ , because otherwise $\\sum _{\\ell =1}^i a_\\ell < \\sum _{\\ell =i+1}^k a_\\ell ,$ contradicting that $i$ is the dividing index.",
"Let us call the layer corresponding to $a_i$ in the cycle diagram of $C$ the central layer of the cycle diagram; we denote this layer $L_c$ .",
"Note that points not in $L_c$ are adjacent, along the wire, to two points on the opposite side of the line $y=x$ .",
"This property may only fail to hold for points in $L_c$ .",
"Since $\\sum _{\\ell =i+1}^k a_\\ell - \\sum _{\\ell =1}^{i-1} a_\\ell = a_i - U,$ all points in $L_c$ are $a_i-U$ cells above the main diagonal.",
"Therefore, points in $L_c$ that are diagonally $a_i-U$ apart are adjacent along the wire.",
"Moreover, these connections leave $a_i-U$ pairs of loose ends: $a_i-U$ rightward-pointing loose ends incident to the upper-rightmost $a_i-U$ points in $L_c$ , and equally many downward-pointing loose ends incident to the lower-leftmost $a_i-U$ points in $L_c$ .",
"In $\\operatorname{Eq}(C)$ , an additional layer $L_u$ of size $U$ is added just below and to the right of $L_c$ , resulting in a balanced composition where the two innermost layers have size $a_i$ and $U$ .",
"Each point in $L_u$ is adjacent along the wire to two points in $L_c$ ; it is easy to see these points are diagonally $a_i-U$ apart.",
"Thus, if in the cycle diagram of $C$ , two points in $L_c$ are adjacent along the wire, in the cycle diagram of $\\operatorname{Eq}(C)$ , they are two apart along the wire, separated by a point in $L_u$ .",
"It follows that the loose ends in $L_c$ are paired the same way in the cycle diagrams of $C$ and $\\operatorname{Eq}(C)$ .",
"So, the cycle diagram of $C$ is a single cycle if and only if the cycle diagram of $\\operatorname{Eq}(C)$ is a single cycle."
],
[
"Completion of the Proof",
"Lemma REF , in conjunction with Lemma REF , immediately implies the validity of Algorithm REF .",
"We will now prove Theorem REF , restated below for clarity.",
"Theorem REF For all positive integers $n$ , $C_n(132,213) \\le n^2 \\cdot 2^{n/2}$ .",
"We first note the following structural result.",
"Proposition 4.2 All cyclic compositions of $n$ have exactly one odd term if $n$ is odd and two odd terms if $n$ is even.",
"Suppose $C$ is a cyclic composition of $n$ .",
"If $n$ is odd, the unequalness $U(C)$ is odd; if $n$ is even, the unequalness $U(C)$ is even.",
"By Proposition REF , $\\operatorname{Eq}(C)$ has exactly two odd terms.",
"Therefore, $C$ has one odd term if $n$ is odd, and two odd terms if $n$ is even.",
"Suppose $n$ is odd.",
"By Proposition REF , cyclic compositions of $n$ have exactly one odd term.",
"We can obtain any such composition by decrementing one term of a composition of $n+1$ with only even terms.",
"There are $2^{(n-1)/2}$ compositions of $n+1$ with only even terms.",
"Since each has at most 2 terms, the number of cyclic compositions of $n$ is bounded above by ${2} \\cdot 2^{(n-1)/2} \\le n^2 \\cdot 2^{n/2}.$ Otherwise, suppose $n$ is even.",
"By Proposition REF , cyclic compositions of $n$ have exactly two odd terms.",
"We can obtain any such composition by decrementing two terms of a composition of $n+2$ with only even terms.",
"There are $2^{n/2}$ compositions of $n+2$ with only even terms.",
"Since each has at most $n̑2 + 1$ terms, we can decrement two terms in at most $\\binom{n/2 + 1}{2}$ ways.",
"So, the number of cyclic compositions of $n$ is bounded by $\\binom{n/2 + 1}{2} \\cdot 2^{n/2} \\le n^2 \\cdot 2^{n/2}$ Remark 4.2 The proof of Theorem REF implies an upper bound of $n^2 \\cdot 2^{(n-1)/2}$ , and this bound can be further refined by a constant factor.",
"For odd $n$ , the proof achieves a tighter bound by a factor of $n$ .",
"Since, as we discuss in the next section, we do not believe our bound's exponential term is tight, we are content to drop these factors."
],
[
"Future Directions",
"This paper makes progress towards Problem REF , the enumeration of the $(132,213)$ -avoiding cyclic permutations.",
"This problem is still open.",
"Leveraging Algorithm REF , computer experiments done by the author have computed $C_n(132,213)$ for $n$ up to 75.Implementation detail: this data was computed by a dynamic programming algorithm, where the subproblems were to count the number of balanced cyclic compositions $(a_1,\\ldots ,a_k | b_m,\\ldots ,b_1)$ of $n$ where the sequence $a_1,\\ldots ,a_k$ has a specified suffix.",
"The runtime of this algorithm is exponential, but grows slowly enough that data collection for $n$ up to 75 is possible.",
"This data is shown in Table REF .",
"Table: Values for C n (132,213)C_n(132,213) for nn up to 75.Some observations are apparent from this data.",
"First, the values of $C_n(132,213)$ show different behavior for even and odd $n$ .",
"For $n\\ge 8$ , the values for even $n$ are larger than the values for adjacent odd $n$ .",
"This is expected, because cyclic compositions with odd sum have exactly one odd term, whereas cyclic compositions with even sum have exactly two odd terms; the former condition is more restrictive.",
"Second, the growth of $C_n(132,213)$ , for both even and odd $n$ , appears to be asymptotically slower than $2^{n/2}$ .",
"From fitting the data, we get an asymptotic estimate of $C_n(132,213) = 2^{\\alpha n + o(n)},$ where $\\alpha \\approx 0.38$ .",
"We believe this $\\alpha $ is the same for even and odd $n$ ; this is because the discrepancy $\\frac{C_{n}(132,213)}{C_{n-1}(132,213)}$ for even $n$ appears, empirically, to be subexponential (and in fact, sublinear).",
"Thus, we do not believe the upper bound in Theorem REF is asymptotically tight.",
"Of course, this prompts the following problem.",
"Problem 5.1 What is the correct value of $\\alpha $ in the above asymptotic?",
"Theorem REF implies $\\alpha \\le 0.5$ .",
"Both an improvement of this bound and a nontrivial lower bound would be interesting results.",
"One approach for future research is to study the number-theoretic properties of Algorithm REF , to obtain results in the spirit of Proposition REF .",
"If one can determine more number-theoretic structure of cyclic compositions, it may be possible to refine the upper bound in Theorem REF .",
"Another more algebraic approach is to obtain recursive identities or inequalities for values of $C_n(132,213)$ .",
"This approach involves breaking $C_n(132,213)$ into subproblems, perhaps by suffixes of the sequence $a_1,\\ldots ,a_k$ in the balanced composition $\\operatorname{Eq}(C) = (a_1,\\ldots ,a_k|b_m,\\ldots ,b_1)$ , and finding injective or bijective mappings among the subproblems.",
"It may be possible to obtain a lower bound for $C_n(132,213)$ in this manner.",
"Because the first step of Algorithm REF reduces all compositions to a balanced composition, it would be of independent interest to enumerate the balanced cyclic compositions of $n$ .",
"These correspond to the balanced reverse layered permutations of length $n$ .",
"Let $C^B_n(132,213)$ be the number of such compositions and permutations.",
"For even $n$ up to 74, computer experiments give the values of $C^B_n(132,213)$ in Table REF .",
"Table: Values for C n B (132,213)C^B_n(132,213) for even nn up to 74.The data suggests the following conjecture.",
"Conjecture 5.1 For even $n$ , ${C_n(132,213)} = \\Omega (1).$ That is, the proportion of $(132,213)$ -avoiding cyclic permutations that are balanced is bounded below by a constant, which is empirically about $0.33$ ."
],
[
"Acknowledgements",
"This research was completed in the 2018 Duluth Research Experience for Undergraduates (REU) program, and was funded by NSF/DMS grant 1650947 and NSA grant H98230-18-1-0010.",
"The author gratefully acknowledges Joe Gallian suggesting the problem and supervising the research.",
"The author thanks Colin Defant and Levent Alpoge for useful discussions over the course of this work, and Joe Gallian and Danielle Wang for comments on early drafts of this paper.",
"The author thanks the anonymous reviewers for useful comments and suggestions."
]
] | 1808.08462 |
[
[
"Intraband divergences in third order optical response of 2D systems"
],
[
"Abstract The existence of large nonlinear optical coefficients is one of the preconditions for using nonlinear optical materials in nonlinear optical devices.",
"For a crystal, such large coefficients can be achieved by matching photon energies with resonant energies between different bands, and so the details of the crystal band structure play an important role.",
"Here we demonstrate that large third-order nonlinearities can also be generally obtained by a different strategy: As any of the incident frequencies or the sum of any two or three frequencies approaches zero, the doped or excited populations of electronic states lead to divergent contributions in the induced current density.",
"We refer to these as intraband divergences, by analogy with the behavior of Drude conductivity in linear response.",
"Physically, such resonant processes can be associated with a combination of inraband and interband optical transitions.",
"Current-induced second order nonlinearity, coherent current injection, and jerk currents are all related to such divergences, and we find similar divergences in degenerate four wave mixing and cross-phase modulation under certain conditions.",
"These divergences are limited by intraband relaxation parameters, and lead to a large optical response from a high quality sample; we find they are very robust with respect to variations in the details of the band structure.",
"To clearly track all of these effects, we analyze gapped graphene, describing the electrons as massive Dirac fermions; under the relaxation time approximation, we derive analytic expressions for the third order conductivities, and identify the divergences that arise in describing the associated nonlinear phenomena."
],
[
"Introduction",
"Motivated by the novel optical properties of graphene[1], [2], [3], many researchers have turned their attention to the linear and nonlinear optical response of 2D systems more generally [4], [5].",
"While there are certainly strong-field excitation circumstances under which a perturbative treatment will fail [6], [7], [8], [9], for many materials a useful first step towards understanding the optical response is the calculation of the conductivities that arise in an expansion of the response of the induced current density in powers of the electric field [10].",
"In materials where inversion symmetry is present, or its lack can be neglected, the first non-vanishing nonlinear response coefficient in the long-wavelength limit arises at third order, and that is our focus in this paper.",
"The simplest approach one can take to calculate such response coefficients is to treat the electrons in an independent particle approximation [11], describing any electron-electron scattering effects and interactions with phonons by the introduction of phenomenological relaxation rates.",
"Such a strategy certainly has its limitations, but at least it identifies many of the qualitative features of the optical response, and in particular it identifies what we call “divergences” in that response.",
"We use this term to refer to the infinite optical response coefficients that are predicted at certain frequencies or sets of frequencies in the so-called “clean limit,” where all scattering effects, including carrier-carrier scattering, carrier-phonon scattering, and carrier-impurity scattering, are ignored by omitting any relaxation rates from the calculation.",
"Under these conditions the actual predicted magnitude of a response depends critically on the values chosen for the phenomenological relaxation rates.",
"These “divergences” are of particular interest to experimentalists because they indicate situations where the optical response can be expected to be large; they are also of particular interest to theorists since they indicate conditions under which a more sophisticated treatment of scattering within the material, or perhaps a treatment of the response more sophisticated than the perturbative one, is clearly in order.",
"The optical response of a crystal arises due to interband and intraband transitions [11].",
"Resonances can be associated with both transitions.",
"For linear optical response, only a single optical transition is involved.",
"A single interband transition can be on resonance for a large range of photon energies, as long as the photon energy is above the band gap.",
"But the resonant electronic states are limited to those with the energy difference matching the photon energy, which depend on the details of the band structure.",
"A single intraband transition can be on resonance only for zero photon energy, and for electronic states at the Fermi surface.",
"Thus, whether or not these resonances lead to a divergent optical response can depend strongly on the material being considered.",
"For the nonlinear optical response, intraband and interband transitions can be combined, leading to complicated nonlinear optical transitions [12], [13].",
"As with single intraband transitions, when the nonlinear optical transitions involve the same initial and final electronic states the resonant frequency, which is the sum of all involved frequencies, is also zero.",
"This is analogous to the Drude conductivity in linear response, which diverges at zero frequency in the “clean limit.” However, due to the interplay of interband transitions, the incident frequencies need not necessarily all be zero for there to be a divergence, and the involved electronic states need not necessarily be around the Fermi surface.",
"By explicitly deriving the general expressions for the third order nonlinear conductivities in the clean limit, we show that the existence and characteristics of such divergences are of a more general nature.",
"To highlight them in a clear and tractable way, we apply our approach to 2D gapped graphene, for which the perturbative third order conductivities can be analytically obtained from the Dirac-like band structure in the single particle approximation.",
"Although our discussion is in the context of such 2D systems, the underlying physics is the same for systems of different dimension.",
"Because the nonlinear transitions involve a number of frequencies, these divergences can be classified into different types, associated with different types of nonlinear phenomena.",
"Several of these have been widely studied in the literature, usually within the context of a particular material or model or excitation condition; yet the connection with the general nonlinear conductivities is seldom discussed.",
"Our goal here is to demonstrate the general nature of the expressions for the response across a range of materials.",
"The first type of divergence can be called “current-induced second order nonlinearity” (CISNL).",
"It arises when free electrons in the system are driven by an applied DC field; the induced DC current breaks the initial inversion symmetry, and thus the material exihibits an effective second-order response to applied optical fields, leading to phenomena such as sum and difference frequency generation.",
"The nature of the divergence here is in the response to the DC field, similar to a single intraband resonance, which would be infinite if phenomenological relaxation terms were not introduced; however, when written as proportional to the induced DC current, the effective second-order response coefficients are finite.",
"This phenomenon has been investigated extensively in different materials, both experimentally [14], [15], [16], [17], [18] and theoretically [19], [20], [21] .",
"A second type is “coherent current injection” (CCI) [22], [23], [24], where the presence of fields at $\\omega $ and $2\\omega -\\delta $ leads to a divergent DC response as $\\delta \\rightarrow 0$ if the excitation at $2\\omega $ is able to create free carriers; the divergent response signals the injection of current by the interference of one-photon absorption and degenerate two-photon absorption amplitudes.",
"This is the most widely studied process, both experimentally [25], [26] and theoretically [27], [28], [29], [30].",
"Recent theoretical work has also identified an injection process associated with one-photon absorption and the stimulated Raman process [31], [28].",
"A third type is the jerk current [32], [33], which is a new type of one color CCI with the assistance of a static electric field.",
"It is a high order divergence involving both a static electric field and an optical field.",
"The static DC field can change the carrier injection rate induced by the optical field, as well as a hydrodynamic acceleration of these optically injected carriers; thus, as opposed to the usual two-color CCI, the injection rate of the jerk current increases with the injection time.",
"We can also identify new divergences, which have not been well recognized in the literature, for two familiar third order nonlinear phenomena.",
"The first arises in cross-phase modulation (XPM) when fields at $\\omega _{p}$ and $\\omega _{s}$ are present.",
"The response for the field at $\\omega _{s}$ due to the field at $\\omega _{p}$ can diverge when $\\omega _{p}$ is above or near the energy gap, leading to a phase modulation of the field at $\\omega _{s}$ that is limited by a relaxation rate.",
"The second also involves excitation with fields at $\\omega _{p}$ and $\\omega _{s}$ , but focuses on the degenerate four-wave mixing (DFWM) field generated at $2\\omega _{p}-\\omega _{s}$ .",
"As $\\omega _{s}\\rightarrow \\omega _{p}$ this term diverges for $\\omega _{p}$ above or near the energy gap.",
"These cases merge as $\\omega _{p}\\rightarrow \\omega _{s}$ , which corresponds to the most widely studied nonlinear phenomenon of Kerr effects and two-photon absorption [34], [35], [36], [37], [38], [39], [40], [41], [42].",
"The very large variation of the extracted values of the nonlinear susceptibilities associated with these phenomena [41], [5], [43] may be related to such divergences.",
"In section II we review the general expressions for the third order optical response in the independent particle approximation, and identify in general the divergences that appear associated with the nonlinear optical transitions with a vanishing total frequency.",
"In section III we specialize to the case of gapped graphene, and use it as an example to illustrate the divergences.",
"In section IV we point out the differences between the divergent behavior of gapped and ungapped graphene.",
"In section V we conclude."
],
[
"The third order response conductivities",
"The general third order nonlinear susceptibility have been well studied in the literature for a cold intrinsic semiconductor [11], with a large effort devoted to working out many subtle features.",
"In this section we mainly repeat the same procedure for a general band system, and classify the expression in a way that the divergent term can be easily identified.",
"Writing the electric field $E(t)$ as $& E(t)=\\sum _{i}E(\\omega _{i})e^{-i\\omega t},$ and other fields similarly, to third order in the electric field the induced current density $J^{(3)}(t)$ is characterized by the response coefficients $\\sigma ^{(3);dabc}(\\omega _{1},\\omega _{2},\\omega _{3})$ , $& J^{(3);d}(\\omega _{1}+\\omega _{2}+\\omega _{3})=\\sum _{\\omega _{1},\\omega _{2},\\omega _{3}}\\sigma ^{(3);dabc}(\\omega _{1},\\omega _{2},\\omega _{3})E^{a}(\\omega _{1})E^{b}(\\omega _{2})E^{c}(\\omega _{3})$ where superscripts $a,b,...$ indicate Cartesian components and are summed over when repeated.",
"The coefficients $\\sigma ^{(3);abcd}(\\omega _{1},\\omega _{2},\\omega _{3})$ can be taken to be symmetric under simultaneous permutation of $(bcd)$ and $(\\omega _{1},\\omega _{2},\\omega _{3})$ , and since the sums over the $\\omega _{i}$ are over all frequencies there are “degeneracy factors” that arises under certain combinations of frequencies.",
"For example, if fields at $\\omega _{p}$ and $\\omega _{s}$ are present we have $& J^{d}(2\\omega _{p}-\\omega _{s})=3\\sigma ^{(3);dabc}(\\omega _{p},\\omega _{p},-\\omega _{s})E^{a}(\\omega _{p})E^{b}(\\omega _{p})E^{c}(-\\omega _{s}).$ Often the response coefficient $\\sigma ^{(3);dabc}(\\omega _{1},\\omega _{2},\\omega _{3})$ is written as $\\sigma ^{(3);dabc}(-\\omega _{1}-\\omega _{2}-\\omega _{3};\\omega _{1},\\omega _{2},\\omega _{3})$ .",
"We do not do that here to avoid cluttering the notation, but we consider it implicit in that when we picture these response coefficients we draw arrows associated with all four of the variables appearing in $\\sigma ^{(3);dabc}(-\\omega _{1}-\\omega _{2}-\\omega _{3};\\omega _{1},\\omega _{2},\\omega _{3})$ , upward arrows associated with positive variables and downward arrows associated with negative variables.",
"To calculate the response coefficients in the independent particle approximation we label the bands by lower case letters $n,m,$ etc., and the wave vectors in the first Brillouin zone by $k$ .",
"The density operator elements $\\rho _{nmk}$ associated with bands $n,m$ and wave vector $k$ satisfy the equation of motion [11] $i\\hbar \\frac{\\partial \\rho _{nm\\mathbf {k}}(t)}{\\partial t} & =(\\varepsilon _{n\\mathbf {k}}-\\varepsilon _{m\\mathbf {k}})\\rho _{nm\\mathbf {k}}-e\\mathbf {E}(t)\\cdot \\left[\\sum _{l\\ne n}\\mathbf {\\xi }_{nl\\mathbf {k}}\\rho _{lm\\mathbf {k}}(t)-\\sum _{l\\ne m}\\rho _{nl\\mathbf {k}}(t)\\mathbf {\\xi }_{lm\\mathbf {k}}\\right]\\nonumber \\\\& -ie\\mathbf {E}(t)\\cdot \\left[\\mathbf {\\nabla }_{\\mathbf {k}}\\rho _{nm\\mathbf {k}}(t)-i(\\mathbf {\\xi }_{nn\\mathbf {k}}-\\mathbf {\\xi }_{mm\\mathbf {k}})\\rho _{nm\\mathbf {k}}(t)\\right]\\nonumber \\\\& +i\\hbar \\left.\\frac{\\partial \\rho _{nm\\mathbf {k}}}{\\partial t}\\right|_{\\text{scat}}\\,.$ Here we describe the interaction of light with the matter using the “$r\\cdot E$ ” approach rather than the “$p\\cdot A$ ” approach involving the vector potential $A(t)$ , for the latter can lead to false divergences associated with the violation of sum rules when the number of bands are inevitably truncated to make any calculation.",
"The coefficients $\\xi _{nmk}$ are the Berry connections, using the definition in the work by Aversa and Sipe [11].",
"The interband optical transitions are identified by the off-diagonal terms of $\\mathbf {\\xi }_{nm\\mathbf {k}}$ , while the rest of terms associated with $\\mathbf {E}(t)$ are associated with the intraband optical transitions.",
"The last term, $i\\hbar \\left.\\frac{\\partial \\rho _{nm\\mathbf {k}}}{\\partial t}\\right|_{\\text{scat}}$ , describes the relaxation processes.",
"In our approach, we take a relaxation time approximation, and specify different relaxation time for different transitions, as given below.",
"We solve Eq.",
"(REF ) perturbatively by setting $\\rho (t)=\\sum \\limits _{j\\ge 0}\\rho ^{(j)}(t)$ , where $\\rho ^{(0)}(t)=\\rho ^{0}$ stands for the density matrix in the thermal equilibrium, and $\\rho ^{(j)}(t)\\propto [\\mathbf {E}]^{j}$ .",
"The iteration for each order is given by $i\\hbar \\frac{\\partial \\rho _{nm\\mathbf {k}}^{(j+1)}(t)}{\\partial t} & =(\\varepsilon _{n\\mathbf {k}}-\\varepsilon _{m\\mathbf {k}})\\rho _{nm\\mathbf {k}}^{(j+1)}-e\\mathbf {E}(t)\\cdot \\left[\\sum _{l\\ne n}\\mathbf {\\xi }_{nl\\mathbf {k}}\\rho _{lm\\mathbf {k}}^{(j)}(t)-\\sum _{l\\ne m}\\rho _{nl\\mathbf {k}}^{(j)}(t)\\mathbf {\\xi }_{lm\\mathbf {k}}\\right]\\nonumber \\\\& -ie\\mathbf {E}(t)\\cdot \\left[\\mathbf {\\nabla }_{\\mathbf {k}}\\rho _{nm\\mathbf {k}}^{(j)}(t)-i(\\mathbf {\\xi }_{nn\\mathbf {k}}-\\mathbf {\\xi }_{mm\\mathbf {k}})\\rho _{nm\\mathbf {k}}^{(j)}(t)\\right]+i\\hbar \\left.\\frac{\\partial \\rho _{nm\\mathbf {k}}^{(j+1)}}{\\partial t}\\right|_{\\text{scat}}\\,,$ where we take the relaxation terms [44], as $\\hbar \\left.\\frac{\\partial \\rho _{nn\\mathbf {k}}^{(j)}}{\\partial t}\\right|_{\\text{scat}} & =-\\Gamma _{a}^{(j)}\\rho _{nn\\mathbf {k}}^{(j)}\\,,\\\\\\hbar \\left.\\frac{\\partial \\rho _{nm\\mathbf {k}}^{(j)}}{\\partial t}\\right|_{\\text{scat}} & =-\\Gamma _{e}^{(j)}\\rho _{nm\\mathbf {k}}^{(j)}\\,,\\text{ for }n\\ne m\\,.$ Here the $\\Gamma _{e}^{(j)}$ and $\\Gamma _{a}^{(j)}$ are phenomenological relaxation parameters associated with interband and intraband motion respectively, with the superscript $(j)$ indicating the order of perturbation at which they are introduced.",
"At optical frequencies, the presence of relaxation parameters removes divergences associated with the resonances, and the values of the relaxation parameters are important for evaluating the nonlinear conductivities when resonant transitions exist.",
"It is natural to choose different values of $\\Gamma _{e/a}^{(j)}$ for resonant and non-resonant transitions.",
"A general perturbative solution are presented in Appendix .",
"With the evolution of the density matrix, the current density can be obtained through $\\mathbf {J}(t)=e\\sum _{nm}\\int \\frac{d\\mathbf {k}}{(2\\pi )^{2}}\\mathbf {v}_{mn\\mathbf {k}}\\rho _{nm\\mathbf {k}}(t)$ , where $\\mathbf {v}_{mn\\mathbf {k}}$ are the matrix elements for a velocity operator.",
"A third order response calculation [10] leads to the result $\\sigma ^{(3);dabc}(\\omega _{1},\\omega _{2},\\omega _{3}) & = & \\frac{1}{6}\\bigg [\\widetilde{\\sigma }^{(3);dabc}(\\omega ,\\omega _{2}+\\omega _{3},\\omega _{3})+\\widetilde{\\sigma }^{(3);dacb}(\\omega ,\\omega _{2}+\\omega _{3},\\omega _{2})\\nonumber \\\\& + & \\widetilde{\\sigma }^{(3);dbac}(\\omega ,\\omega _{1}+\\omega _{3},\\omega _{3})+\\widetilde{\\sigma }^{(3);dbca}(\\omega ,\\omega _{1}+\\omega _{3},\\omega _{1})\\nonumber \\\\& + & \\widetilde{\\sigma }^{(3);dcba}(\\omega ,\\omega _{1}+\\omega _{2},\\omega _{1})+\\widetilde{\\sigma }^{(3);dcab}(\\omega ,\\omega _{1}+\\omega _{2},\\omega _{2})\\bigg ]\\,,$ with $\\omega =\\omega _{1}+\\omega _{2}+\\omega _{3}$ , and the unsymmetrized coefficients $\\widetilde{\\sigma }^{(3);dabc}(\\omega ,\\omega _{0},\\omega _{3})$ take the form $\\widetilde{\\sigma }^{(3);dabc}(\\omega ,\\omega _{0},\\omega _{3}) & = & \\frac{1}{vv_{0}v_{3}}S_{1}^{dabc}+\\frac{1}{vv_{0}}S_{2}^{dabc}(w_{3})+\\frac{1}{vv_{3}}S_{3}^{dabc}(w_{0})+\\frac{1}{v}S_{4}^{dabc}(w_{0},w_{3})\\nonumber \\\\& + & \\frac{1}{v_{0}v_{3}}S_{5}^{dabc}(w)+\\frac{1}{v_{0}}S_{6}^{dabc}(w,w_{3})+\\frac{1}{v_{3}}S_{7}^{dabc}(w,w_{0})+S_{8}^{dabc}(w,w_{0},w_{3})\\,,$ with $v & =\\hbar \\omega +i\\Gamma _{a}^{(3)}\\,, & w & =\\hbar \\omega +i\\Gamma _{e}^{(3)}\\,,\\\\v_{0} & =\\hbar \\omega _{0}+i\\Gamma _{a}^{(2)}\\,, & w_{0} & =\\hbar \\omega _{0}+i\\Gamma _{e}^{(2)}\\,,\\\\v_{3} & =\\hbar \\omega _{3}+i\\Gamma _{a}^{(1)}\\,, & w_{3} & =\\hbar \\omega _{3}+i\\Gamma _{e}^{(1)}\\,.$ Table: Illustration of the resonant processes of of thedivergences associated with the intrabandmotion.",
"The row “Divergent contributions”illustrates the divergent transitions, by the magenta arrows andlabels in the diagram.",
"The magenta dots show the doped orexcited electronic states.",
"The row “Resonant conditions” givesthe conditions for these resonances.Note that the actual value of the energy appearing in, for example, $v_{0}$ depends on the corresponding frequency (here $\\omega _{0}$ , a sum of two of the incident frequencies) appearing in $\\widetilde{\\sigma }^{(3);dabc}(\\omega ,\\omega _{0},\\omega _{3})$ .",
"The quantities $v$ , $v_{0}$ , and $v_{3}$ are associated with the intraband motions (for carriers or excited carriers).",
"The coefficients $S_{i}^{dabc}$ are associated with interband transitions; we give expressions for them, and for the expressions to which they reduce for the particular models we consider, in the Appendix .",
"Any divergences they contain are associated with interband motion, and thus all the intraband divergences are explicitly indicated by the $v_{i}$ in the denominators appearing in Eq.",
"(REF ).",
"Thus it is the $\\Gamma _{a}^{(i)}$ that will be of importance to us.",
"Typically one $\\Gamma _{a}^{(i)}$ is important for a given divergence; the other $\\Gamma _{a}^{(j)}$ , and the $\\Gamma _{e}^{(j)}$ , to which the process is not sensitive, are all set equal to a nominal value $\\Gamma $ .",
"These divergent processes are summarized in Table REF .",
"Table: A short summary of the structure of the divergences in the nonlinearoptical phenomena discussed in this work.",
"Here 2E c 2E_c is thegap or the chemical potential induced gap; all the A i A_i and B i B_iare expansion coefficients.",
"The third columnlists the condition when the divergences can occur.",
"(*) Here we set all the relaxation parameters Γ a/e (j) \\Gamma _{a/e}^{(j)} exceptΓ a (2) \\Gamma _{a}^{(2)} equal to Γ\\Gamma .The general expression for the conductivity in Eq.",
"(REF ) immediately indicates the possibilities of the nonlinear phenomena discussed in the Introduction.",
"For CISNL, the conductivities $\\sigma ^{(3);dabc}(\\omega _{1},\\omega _{2},0)$ include divergences associated with $v_{3}\\rightarrow 0$ ; for coherent current injection, the conductivities $\\sigma ^{(3);dabc}(-2\\omega ,\\omega ,\\omega )$ include divergences associated with $v\\rightarrow 0$ ; the jerk current is a special case of CISNL, described by $\\sigma ^{(3);dabc}(\\omega ,-\\omega ,0)$ , with divergences associated with $v_{3}\\rightarrow 0$ and $v\\rightarrow 0$ ; for XPM and DFWM, the conductivities $\\sigma ^{(3);dabc}(\\omega ,\\omega _{p},-\\omega _{p})$ and $\\sigma ^{(3);dabc}(-\\omega _{s},\\omega _{p},\\omega _{p})$ include divergences associated with $v_{0}\\rightarrow 0$ .",
"In special cases, there may be extra divergences identified by a combination of these limits, and the detailed divergences types are determined by the values of $S_{i}$ .",
"Of course, for finite relaxation times we will not have a vanishing $v_{3}$ , $v_{0}$ , or $v$ .",
"Nonetheless, for frequencies where the real part of one of these quantities vanishes the term(s) in Eq.",
"(REF ) containing this quantity will make the largest contribution, and we refer to them as the “divergent contributions.” Our focus is the identification of these divergent contributions.",
"Before getting into the details of these effects, it is helpful to isolate the divergent contributions in these conductivities, as shown in Table REF ."
],
[
"Nonlinear optical conductivity of\ngapped graphene",
"We apply the approach to gapped graphene, a two dimensional system.",
"The low energy excitations exist in two valleys, which can be described by a simplified two band model for the unperturbed Hamiltonian $H_{\\tau \\mathbf {k}}^{0}(\\Delta )=\\begin{pmatrix}\\Delta & \\hbar v_{F}(ik_{x}+\\tau k_{y})\\\\\\hbar v_{F}(-ik_{x}+\\tau k_{y}) & -\\Delta \\end{pmatrix}\\,.$ The quantity $\\mathbf {k}=k_{x}\\hat{\\mathbf {x}}+k_{y}\\hat{\\mathbf {y}}$ is a two dimensional wave vector, $\\Delta \\ge 0$ is a mass parameter to induce a band gap $2\\Delta $ , $\\tau =\\pm $ stands for a valley index, and $v_{F}$ is the Fermi velocity.",
"Ungapped graphene corresponds to the limit $\\Delta \\rightarrow 0$ .",
"All the necessary quantities for the calculation of the third order conductivity are given [46] by $\\varepsilon _{\\tau s\\mathbf {k}} & =s\\epsilon _{k}\\,,\\text{ with }\\epsilon _{k}=\\sqrt{(\\hbar v_{F}k)^{2}+\\Delta ^{2}}\\,,\\\\\\mathbf {\\xi }_{\\tau +-\\mathbf {k}} & =\\frac{\\hbar v_{F}(ik_{x}+\\tau k_{y})}{2\\epsilon _{\\mathbf {k}}k^{2}}\\left(-i\\frac{\\Delta }{\\epsilon _{\\mathbf {k}}}\\mathbf {k}+\\tau \\mathbf {k}\\times \\hat{\\mathbf {z}}\\right)\\,,\\\\\\mathbf {\\xi }_{\\tau ++\\mathbf {k}}-\\mathbf {\\xi }_{\\tau --\\mathbf {k}} & =\\frac{1}{k^{2}}\\left(1-\\frac{\\Delta }{\\epsilon _{\\mathbf {k}}}\\right)\\tau \\mathbf {k}\\times \\hat{\\mathbf {z}}\\,,\\\\\\mathbf {v}_{\\tau +-\\mathbf {k}} & =2i\\hbar ^{-1}\\epsilon _{\\mathbf {k}}\\xi _{\\tau +-\\mathbf {k}}\\,,\\\\\\mathbf {v}_{\\tau ss\\mathbf {k}} & =\\hbar ^{-1}\\mathbf {\\nabla }_{\\mathbf {k}}\\varepsilon _{\\tau s\\mathbf {k}}\\,.$ For the special two band system we use $s=\\pm $ to indicate the upper ($+$ ) and lower ($-$ ) bands.",
"In this approximation for the dispersion relation of the bands, all the $S_{j}^{dabc}$ can be analytically obtained, and are listed in Appendix .",
"Furthermore, the unsymmetrized conductivity $\\widetilde{\\sigma }^{(3);dabc}(\\omega ,\\omega _{0},\\omega _{3})$ can be written as the sum of two terms, $& \\widetilde{\\sigma }^{(3);dabc}(\\omega ,\\omega _{0},\\omega _{3})=\\widetilde{\\sigma }_{f}^{(3);dabc}(\\omega ,\\omega _{0},\\omega _{3})+\\widetilde{\\sigma }_{t}^{(3);dabc}(\\omega ,\\omega _{0},\\omega _{3}),$ where $\\widetilde{\\sigma }_{f}^{(3);dabc}$ includes all terms in Eq.",
"(REF ) in which $v_{3}$ appears, and $\\widetilde{\\sigma }_{t}^{(3);dabc}$ includes all remaining terms.",
"Note that $\\widetilde{\\sigma }_{f}^{(3);dabc}$ is non-zero only for a doped system.",
"Induced currents in this model flow in the plane in which the gapped graphene is assumed to lie, which we take to be the $(xy)$ plane; $\\sigma ^{(3);dabc}$ , $\\widetilde{\\sigma }^{(3);dabc}$ , and $S_{j}^{dabc}$ are fourth rank tensors, each with 16 components, and the independent components can be taken to be those with the components $xxyy$ , $xyxy$ , and $xyyx$ .",
"The other nonzero components can be obtained through $S_{j}^{xxxx}=S_{j}^{xxyy}+S_{j}^{xyxy}+S_{j}^{xyyx}$ , and the symmetry $\\lbrace x\\leftrightarrow y\\rbrace $ .",
"We list the independent nonzero components of these tensors, taking $S_{j}^{dabc}(\\cdots )$ as an example, as a column vector $S_{j}=\\begin{bmatrix}S_{j}^{xxyy}\\\\S_{j}^{xyxy}\\\\S_{j}^{xyyx}\\end{bmatrix}$ .",
"For gapped graphene, the expressions for $S_{i}$ show features similar to the corresponding expressions for graphene [44], : They are functions of the effective gap parameter $E_{c}=\\text{max}\\lbrace \\Delta ,|\\mu |\\rbrace $ and $\\Delta $ , in addition to the energies $w$ , $w_{0}$ , and $w_{3}$ .",
"In all these expressions, $E_{c}$ appears only in functions of $E_{c}^{-5}$ , $E_{c}^{-3}$ , $E_{c}^{-1}$ , ${\\cal I}(E_{c};w)$ , ${\\cal H}(E_{c};w)$ , ${\\cal H}(E_{c};w_{0})$ , ${\\cal G}(E_{c};w)$ , ${\\cal G}(E_{c};w_{0})$ , and ${\\cal G}(E_{c};w_{3})$ , where ${\\cal G}(x;w) & = & \\ln \\left|\\frac{w+2x}{w-2x}\\right|+i\\left(\\pi +\\arctan \\frac{\\text{Re}[w]-2x}{\\text{Im}[w]}-\\arctan \\frac{\\text{Re}[w]+2x}{\\text{Im}[w]}\\right)\\,,\\\\{\\cal H}(x;w) & = & \\frac{1}{2x+w}+\\frac{1}{2x-w}\\,,\\\\{\\cal I}(x;w) & = & \\frac{1}{(2x+w)^{2}}-\\frac{1}{(2x-w)^{2}}\\,.$ We can write each $S_{j}^{dabc}$ as a linear combination of these functions, with their arguments themselves being functions of $w$ , $w_{0}$ , $w_{3}$ , and $\\Delta $ .",
"All terms can be expanded in even orders of $\\Delta $ , particularly as functions proportional to $\\Delta ^{0}$ , $\\Delta ^{2}$ , and $\\Delta ^{4}$ , as shown in Appendix .",
"With all these analytic expressions in hand, any third order nonlinear conductivity can be calculated and studied directly.",
"In the following, we consider the intraband divergences that are of interest here, by giving the leading contributions to the conductivities."
],
[
"Current-Induced Second order Nonlinearity",
"We begin by considering the response coefficient $\\sigma ^{(3)dabc}(\\omega _{1},\\omega _{2},0)$ , where we assume that neither $\\omega _{1}$ nor $\\omega _{2}$ vanishes, and that their sum is finite.",
"This describes a second-order optical nonlinearity induced by a DC field ($\\omega _{3}=0),$ and if there are free carriers we would expect a divergent contribution because of the large response of the free carriers to the DC field.",
"A picture of the excitation scenario is given in Fig.",
"REF .",
"In our expression for $\\widetilde{\\sigma }^{(3);dabc}(\\omega ,\\omega _{0},\\omega _{3})$ the divergence of interest is clearly signaled by $v_{3}\\rightarrow 0$ , and the divergent response term can be obtained from the analytic expressions.",
"When this is isolated, the most important relaxation parameter is $\\Gamma _{a}^{(1)}$ , which we treat differently than we treat the other relaxation parameters; for $j=2,3$ we set $\\Gamma _{a}^{(j)}=\\Gamma _{e}^{(j)}=\\Gamma $ to give $v=w$ and $v_{0}=w_{0}$ in the expression for $\\widetilde{\\sigma }^{(3);dabc}$ .",
"The divergent term can be written as $\\widetilde{\\sigma }^{(3)}(\\omega ,\\omega _{0},\\omega _{3}\\rightarrow 0)\\approx \\frac{i\\sigma _{3}}{\\Delta ^{5}}\\frac{E_{c}^{2}-\\Delta ^{2}}{\\hbar \\omega _{3}+i\\Gamma _{a}^{(1)}}{\\cal Z}_{1}\\left(\\frac{w}{\\Delta },\\frac{w_{0}}{\\Delta };\\frac{E_{c}}{\\Delta }\\right)\\,,$ with $\\sigma _{3}=\\sigma _{0}(\\hbar v_{F}e)^{2}/\\pi $ , $\\sigma _{0}=e^{2}/(4\\hbar )$ , and ${\\cal Z}_{1}(x,x_{0};\\alpha ) & = & \\frac{-16}{\\alpha xx_{0}\\left(x^{2}-4\\alpha ^{2}\\right)^{2}\\left(x_{0}^{2}-4\\alpha ^{2}\\right)}\\left\\lbrace \\begin{bmatrix}4\\alpha ^{2}x\\left(2x+x_{0}\\right)\\\\-x^{3}x_{0}\\\\x_{0}\\left(x^{3}+x_{0}\\left(3x^{2}-4\\alpha ^{2}\\right)\\right)\\end{bmatrix}\\right.\\nonumber \\\\& + & \\left.\\left[4\\alpha ^{2}\\left(\\alpha ^{2}+3\\right)-\\left(3\\alpha ^{2}+1\\right)x^{2}-\\left(x^{2}+4\\right)x_{0}^{2}-2\\left(\\alpha ^{2}+1\\right)x_{0}x\\right]\\begin{bmatrix}1\\\\1\\\\1\\end{bmatrix}\\right\\rbrace \\,.$ These expressions have a number of interesting features: (1) The divergent term in Eq.",
"(REF ) does not include any contributions from the ${\\cal G}$ functions, indicating immediately that it is the existence of free carriers that is important.",
"Indeed, this can be immediately confirmed, for if the system is undoped we have $E_{c}=\\Delta $ , and the term in Eq.",
"(REF ) vanishes.",
"(2) The divergent term is inversely proportional to the relaxation parameter.",
"Since in a very rough approximation the DC current satisfies $\\mathbf {J}_{\\text{DC}}\\propto \\left(\\Gamma _a^{(1)}\\right)^{-1}E_{\\text{DC}}$ with the DC field $E_{\\text{DC}}$ , the divergent contribution to $\\sigma ^{(3)dabc}(\\omega _{1},\\omega _{2},0)$ can be written as a finite term proportional to $J_{\\text{DC}}$ , hence the identification of this response as “current-induced second order nonlinearity.” (3) The expression in Eq.",
"(REF ) for ${\\cal Z}_{1}(x,x_{0};\\alpha )$ can exhibit further divergences as $w\\rightarrow 2E_{c}$ and $w_{0}\\rightarrow 2E_{c}$ , indicating that interband divergences can arise in CISNL as well, depending on the values of the optical frequencies $\\omega _{1}$ and $\\omega _{2}$ .",
"Figure: Conductivity for second harmonic generation (a) μ=0\\mu =0 and (b) μ/Δ=1.4\\mu /\\Delta =1.4for Γ/Δ=0.05\\Gamma /\\Delta =0.05 and Γ a (3) /Δ=0.01\\Gamma _{a}^{(3)}/\\Delta =0.01.",
"In(b), the current induced contribution is also plotted.We refer to the terms in $\\sigma ^{(3)dabc}(\\omega _{1},\\omega _{2},0)$ that are $not$ divergent as $v_{3}\\rightarrow 0$ as the “field-induced second order nonlinearity.” They exist in the absence of free carriers, and have an analogue in the field induced second order nonlinearity of usual semiconductors, which leads to processes such as electric-field induced second harmonic generation (EFISH) [44], , [47].",
"Adopting this perspective, we can write the effective second order conductivity $\\sigma ^{(2);dab}(\\omega _{1},\\omega _{2})=3\\sigma ^{(3);dabc}(\\omega _{1},\\omega _{2},0)E_{\\text{DC}}^{c}$ with $\\sigma ^{(3);dabc}(\\omega _{1},\\omega _{2},0)=\\sigma _{J}^{(3);dabc}(\\omega _{1},\\omega _{2},0)+\\sigma _{E}^{(3);dabc}(\\omega _{1},\\omega _{2},0)\\,.$ The first term characterizes the current-induced second order response coefficient, and arises from the divergent contribution in Eq.",
"(REF ), $\\sigma _{J}^{(3);dabc}(\\omega _{1},\\omega _{2},0) & = & \\frac{\\sigma _{3}E_{\\text{dc}}^{c}}{6\\Gamma _{a}^{(1)}}\\left[\\left(\\frac{E_{c}}{\\Delta }\\right)^{2}-1\\right]\\left[{\\cal Z}_{1}^{dabc}\\left(\\frac{\\hbar (\\omega _{1}+\\omega _{2})+i\\Gamma }{\\Delta },\\frac{\\hbar \\omega _{2}+i\\Gamma }{\\Delta };\\frac{E_{c}}{\\Delta }\\right)\\right.\\\\& & \\left.+{\\cal Z}_{1}^{dbac}\\left(\\frac{\\hbar (\\omega _{1}+\\omega _{2})+i\\Gamma }{\\Delta },\\frac{\\hbar \\omega _{1}+i\\Gamma }{\\Delta };\\frac{E_{c}}{\\Delta }\\right)\\right]\\,.$ All the remaining contributions are collected in the field induced response coefficient $\\sigma _{E}^{(3);dabc}(\\omega _{1},\\omega _{2},0)$ .",
"Figure REF shows the spectrum of $\\sigma ^{(3);xxxx}(\\omega ,\\omega ,0)$ with a relaxation parameter $\\Gamma /\\Delta =0.05$ for (a) an undoped system with no free carriers, $\\mu /\\Delta =0$ , and (b) a doped system with $\\mu /\\Delta =1.4$ .",
"For both systems, there are obvious resonant peaks at $\\hbar \\omega =nE_{c}$ for $n=1,2$ associated with interband transitions; in the response of the doped system, there is an additional peak as $\\hbar \\omega \\rightarrow 0$ , and here the contribution from the current-induced SHG dominates for photon energies away from the resonances.",
"This divergent process results in a qualitatively larger conductivity $\\sigma ^{(3)dabc}(\\omega ,\\omega ,0)$ for a doped system (b) than for an undoped system (a)."
],
[
"Cross-Phase Modulation",
"In cross-phase modulation the propagation of light at frequency $\\omega _{s}$ is modified by the presence of an intense optical field at a different frequency, $\\omega _{p}$ ; it is characterized by the response coefficient $\\sigma ^{(3)}(\\omega _{s},-\\omega _{p},\\omega _{p})$ .",
"This conductivity exhibits a divergence as $w_{0}\\rightarrow 0$ .",
"Unlike CISNL, the divergence arises from the nonlinear response of the system and a static field is not required.",
"The process is pictured in Fig.",
"REF ; we will see below that the divergent term vanishes unless $\\hbar \\omega _{p}$ is near or above the effective band gap $2E_{c}$ .",
"Cross-phase modulation of a signal frequency $\\omega _{s}$ can be effected not just by a CW beam at $\\omega _{p}$ , but by a pulse of light centered at such a frequency.",
"So in general we seek $\\sigma ^{(3)}(\\omega _{s},-\\omega _{p}+\\delta ,\\omega _{p})$ , where knowledge of this expression for a small range of frequencies $\\omega _{p}$ centered about a nominal pump frequency, and for a small range of detunings $\\delta $ centered around zero, will allow us calculated the cross-phase modulation of a signal by a pump pulse.",
"As $\\delta ,\\Gamma _{a}^{(2)}\\rightarrow 0$ , the leading term of the relevant unsymmetrized conductivity is $\\widetilde{\\sigma }^{(3)}(\\omega _{s}+\\delta ,\\delta \\rightarrow 0,\\omega _{p})\\approx \\frac{i\\sigma _{3}}{\\Delta ^{3}}\\frac{2}{\\hbar \\delta +i\\Gamma _{a}^{(2)}}{\\cal Z}_{2}\\left(\\frac{w}{\\Delta },\\frac{w_{3}}{\\Delta };\\frac{E_{c}}{\\Delta }\\right)\\,,$ with $w=v=\\hbar (\\omega _{s}+\\delta )+i\\Gamma $ , $w_{3}=v_{3}=\\hbar \\omega _{p}+i\\Gamma $ , where we set all first order and third order relaxation energies to be the same for simplicity, and ${\\cal Z}_{2}(x,x_{3};\\alpha ) & = & \\frac{8(A_{0}+2xx_{3}A_{4})}{x^{3}x_{3}^{3}\\alpha }+\\frac{8A_{0}}{3x^{3}x_{3}\\alpha ^{3}}+\\frac{(x^{2}-4)^{2}A_{0}+4x^{2}(4-x^{2})A_{5}}{2x^{4}x_{3}}{\\cal I}(\\alpha ;x)\\nonumber \\\\& + & \\frac{(x^{2}-4)^{2}A_{0}+32x^{2}A_{5}}{2x^{5}x_{3}}{\\cal H}(\\alpha ;x)\\nonumber \\\\& + & Z_{1}(x,x_{3}){\\cal G}(\\alpha ;x)+Z_{2}(x,x_{3}){\\cal G}(\\alpha ;x_{3})\\,.$ and $Z_{1}(x,x_{3}) & = & \\frac{x_{3}(x^{2}-4)^{2}A_{0}+32x^{2}(x_{3}A_{5}+xA_{4})}{2x^{6}(x^{2}-x_{3}^{2})}\\,,\\\\Z_{2}(x,x_{3}) & = & -\\frac{x(x_{3}^{2}-4)^{2}A_{0}+32xx_{3}(x_{3}A_{5}+xA_{4})}{2x^{2}x_{3}^{4}(x^{2}-x_{3}^{2})}\\,.$ Here we used $A_0=-(A_1+A_2+A_3)$ , $A_{4}=\\frac{1}{4}(A_{2}-A_{3})$ and $A_{5}=-\\frac{1}{4}(2A_{1}+A_{2}+A_{3})$ , where $A_{1} & =\\begin{bmatrix}-3\\\\1\\\\1\\end{bmatrix}\\,,\\quad A_{2}=\\begin{bmatrix}1\\\\-3\\\\1\\end{bmatrix}\\,,\\quad A_{3}=\\begin{bmatrix}1\\\\1\\\\-3\\end{bmatrix}\\,.$ We note that $Z_{1}$ and $Z_{2}$ themselves diverge as $x\\rightarrow -x_{3}$ (i.e., $\\omega _{p}\\rightarrow \\omega _{s})$ , which will be discussed in next section.",
"Using the expression above we can write $\\sigma ^{(3)}(\\omega _{s},-\\omega _{p}+\\delta ,\\omega _{p})=\\sigma _{\\text{xpm};d}^{(3)}(\\omega _{s},\\omega _{p};\\delta )+\\sigma _{\\text{xpm}}^{(3)}(\\omega _{s},\\omega _{p};\\delta )\\,,$ where $\\sigma _{\\text{xpm};d}^{(3)}(\\omega _{s},\\omega _{p};\\delta )$ contains a divergence with respect to $\\left(\\hbar \\delta +i\\Gamma _{a}^{(2)}\\right)^{-1}$ $\\sigma _{\\text{xpm};d}^{(3);dabc}(\\omega _{s},\\omega _{p};\\delta ) & = & \\frac{i\\sigma _{3}}{3\\Delta ^{4}}\\left(\\frac{\\hbar \\delta +i\\Gamma _{a}^{(2)}}{\\Delta }\\right)^{-1}\\left[{\\cal Z}_{2}^{(3);dabc}\\left(\\frac{\\hbar \\omega _{s}+i\\Gamma }{\\Delta },\\frac{-\\hbar \\omega _{p}+i\\Gamma }{\\Delta };\\frac{E_{c}}{\\Delta }\\right)\\right.\\\\& & \\left.+{\\cal Z}_{2}^{(3);dacb}\\left(\\frac{\\hbar \\omega _{s}+i\\Gamma }{\\Delta },\\frac{\\hbar \\omega _{p}+i\\Gamma }{\\Delta };\\frac{E_{c}}{\\Delta }\\right)\\right],$ and $\\sigma _{\\text{xpm}}^{(3)}(\\omega _{s},\\omega _{p};\\delta )$ contains the non-divergent response.",
"Note that the divergent term is independent of the sequences of the limits $\\delta \\rightarrow 0$ and $\\Gamma \\rightarrow 0$ .",
"Figure: Signal frequency ω s \\omega _{s} dependence of the conductivity σ 3 -1 Δ 4 σ xpm (3);xxxx (ω s ,ω p ;0)\\sigma _{3}^{-1}\\Delta ^{4}\\sigma _{\\text{xpm}}^{(3);xxxx}(\\omega _{s},\\omega _{p};0)at different chemical potential and pump frequencies for Γ/Δ=0.05\\Gamma /\\Delta =0.05and Γ a (2) /Δ=0.01\\Gamma _{a}^{(2)}/\\Delta =0.01.",
"The parameters of μ Δ,ℏω p E c \\left(\\frac{\\mu }{\\Delta },\\frac{\\hbar \\omega _{p}}{E_{c}}\\right)are (a) (0,1.5)(0,1.5), (b) (0,2.5)(0,2.5), (c) (1.4,1.5)(1.4,1.5), and (d) (1.4,2.5)(1.4,2.5).The divergent contribution σ xpm;d (3);xxxx (ω s ,ω p ;0)\\sigma _{\\text{xpm};d}^{(3);xxxx}(\\omega _{s},\\omega _{p};0)are also shown for comparison.Figure REF gives the spectrum of $\\sigma _{\\text{xpm}}^{(3);xxxx}(\\omega _{s},\\omega _{p};0)$ , appropriate for CW illumination with the pump; as above we have taken a relaxation parameter $\\Gamma /\\Delta =0.05$ and $\\Gamma _{a}^{(2)}/\\Delta =0.01$ , and again consider an undoped system $\\left(\\mu /\\Delta =0\\right)$ and a doped system with $\\mu /\\Delta =1.4$ .",
"The complicated dependence of both the real and imaginary parts on the frequencies indicates the rich nature of the nonlinear response.",
"The real parts show additional divergences as $\\hbar \\omega _{s}\\rightarrow 0$ and $\\hbar \\omega _{p}\\rightarrow 0$ , as would be expected from the discussion of CISNL above, and as $\\hbar \\omega _{s}\\rightarrow \\hbar \\omega _{p}$ , as we discuss in the section below.",
"As would be expected from the discussion of CISNL above, there are other divergences or resonant peaks associated with either $\\hbar \\omega _{s}\\rightarrow 0$ or interband transitions; the latter are not our focus in this work.",
"Away from these resonances, the values of the conductivity for the cases $\\hbar \\omega _{p}/E_{c}=1.5$ are much smaller than those for the cases $\\hbar \\omega _{p}/E_{c}=2.5$ , which include the intraband divergences, and the contribution from $\\sigma _{\\text{xpm};d}^{(3);xxxx}(\\omega _{s},\\omega _{p};0)$ is generally the largest part of $\\sigma ^{(3)}(\\omega _{s},-\\omega _{p},\\omega _{p})$ away from these other divergences.",
"We can get some insight into the importance of pump frequency by noting that as $\\Gamma \\rightarrow 0$ the divergent term $\\sigma _{\\text{xpm};d}^{(3);dabc}(\\omega _{s},\\omega _{p};\\delta )$ becomes $\\sigma _{\\text{xpm};d}^{(3)}(\\omega _{s},\\omega _{p};\\delta )\\rightarrow \\frac{i\\sigma _{3}}{\\Delta ^{4}}\\left(\\frac{\\hbar \\delta +i\\Gamma _{a}^{(2)}}{\\Delta }\\right)^{-1}h_{1}\\left(\\frac{\\hbar \\omega _{p}}{\\Delta },\\frac{\\hbar \\omega _{s}}{\\Delta }\\right)T\\left(\\frac{E_{c}}{\\Delta },\\frac{\\hbar \\omega _{p}}{\\Delta }\\right)$ with $h_{1}(x,x_{1})=\\frac{(x^{2}-4)^{2}A_{0}+32x(xA_{5}+x_{1}A_{4})}{6x^{4}x_{1}(x^{2}-x_{1}^{2})}$ and $& T(y,x)=\\mathcal {G}(y;x+i0^{+})+\\mathcal {G}(y;-x+i0^{+})=2i\\pi \\theta (x-2y).$ So in the clean limit, where all $\\Gamma _{a/e}^{(j)}\\rightarrow 0$ , $\\sigma _{\\text{xpm};d}^{(3)}(\\omega _{s},\\omega _{p};\\delta )$ vanishes if $\\hbar \\omega _{p}$ is below the effective gap $2E_{c}$ .",
"The frequency variation $\\delta $ comes from the pumping beam, which is negligible for very long pulse duration; then $\\sigma _{\\text{xpm};d}^{(3)}$ is pure imaginary, and the modulation is associated with refraction rather than absorption.",
"Deviations from this arise from the finite relaxation rate for other transition processes, but clearly this divergence is associated with resonant excitation from the valence to conduction band at $\\hbar \\omega _{p}$ , as sketched in Fig.",
"REF ."
],
[
"Degenerate Four Wave Mixing",
"In Eqs.",
"(REF ) and (REF ) there appear to be divergences as $v\\rightarrow \\pm v_{3}$ .",
"Working out the expression in detail it can be immediately seen that in fact no divergence results as $v\\rightarrow v_{3}$ , but a divergence does indeed arise as $v\\rightarrow -v_{3}$ .",
"The leading divergence is associated with terms of the form $v_{0}^{-1}(v+v_{3})^{-1}$ .",
"It gives a higher order divergence with a Lorentz type lineshape, as shown in Figs.",
"REF (b) and (d).",
"Such a divergence also exists in the widely studied nonlinear phenomena of four-wave mixing, which is characterized by the response coefficient $\\sigma ^{(3)}(\\omega _{p},\\omega _{p},-\\omega _{s})$ .",
"As for XPM, the divergent point is at $\\omega _{s}=\\omega _{p}$ , but since the generated field is at $2\\omega _{p}-\\omega _{s}$ rather than at $\\omega _{s}$ the path to the divergence is different.",
"As $\\omega _{s}\\rightarrow \\omega _{p}$ , the divergent term arises in the unsymmetrized conductivities $\\widetilde{\\sigma }^{(3)}(2\\omega _{p}-\\omega _{s},\\omega _{p}-\\omega _{s},-\\omega _{s})$ and $\\widetilde{\\sigma }^{(3)}(2\\omega _{p}-\\omega _{s},\\omega _{p}-\\omega _{s},\\omega _{p})$ .",
"Taking $\\delta =\\omega _{s}-\\omega _{p}$ and all $\\Gamma _{a/e}^{(j)}$ as small quantities, the conductivity can be approximated as ${\\sigma }^{(3)}(\\omega _{p},\\omega _{p},-\\omega _{s})&\\approx &\\frac{i\\sigma _{3}}{\\Delta ^{4}}\\frac{\\Delta }{\\hbar \\delta +i\\Gamma _{a}^{(2)}}\\left[\\frac{\\Delta }{\\hbar \\delta +i\\Gamma }{\\cal Z}_{3}\\left(\\frac{\\hbar \\omega _{p}}{\\Delta },\\frac{\\Gamma }{\\Delta };\\frac{E_c}{\\Delta }\\right)\\right.\\\\&& \\left.+{\\cal Z}_{4}\\left(\\frac{\\hbar \\omega _{p}}{\\Delta },\\frac{\\Gamma }{\\Delta };\\frac{E_c}{\\Delta }\\right)\\right]\\,.$ Here we set all the relaxation parameters $\\Gamma _{a/e}^{(j)}$ except $\\Gamma _{a}^{(2)}$ equal to $\\Gamma $ .",
"The exact expression can be obtained following Eq.",
"(REF ), or even from the full expression in Appendix. .",
"The physics of this divergence can be revealed if we consider the clean limit for ${\\cal Z}_{3}$ and ${\\cal Z}_{4}$ .",
"They are ${\\cal Z}_j(x,0;\\alpha )=Z_j(x)T(\\alpha ,x)$ for $j=3, 4$ with $Z_{3}(x) & =\\frac{1}{12x^{6}}\\left[(x^{4}+24x^{2}+16)A_{0}-48x^{2}A_{6}\\right]\\,\\\\Z_{4}(x) & =\\frac{1}{12x^{7}}\\left[(x^{4}-56x^{2}-48)A_{0}+160x^{2}A_{6}\\right]\\,,$ with $A_{6}=-(A_{1}+A_{2}+2A_{3})/4$ .",
"Here $T(\\alpha ;x)$ survives only for $x>2\\alpha $ (see Eq.",
"(REF )), where one photon absorption exists.",
"Thus a relevant picture for this kind of resonance is shown in Fig.",
"REF .",
"Figure: Conductivity σ (3);xxxx (ω p ,ω p ,-ω s ){\\sigma }^{(3);xxxx}(\\omega _{p},\\omega _{p},-\\omega _{s})for ω s \\omega _{s} around ω p \\omega _{p} at different chemical potential|μ|/Δ=0|\\mu |/\\Delta =0 and 1.41.4 and different pump frequency ℏω p /E c =1.5\\hbar \\omega _{p}/E_{c}=1.5and 2.52.5.",
"The relaxation parameters are taken as Γ a (2) /Δ=0.01\\Gamma _{a}^{(2)}/\\Delta =0.01and Γ/Δ=0.05\\Gamma /\\Delta =0.05, and the gap parameter is taken as Δ=1\\Delta =1 eV.We illustrate these divergences in Fig.",
"REF for two different pump frequencies, $\\hbar \\omega _{p}/E_{c}=1.5$ and $2.5$ .",
"We plot the results for an undoped system, $|\\mu |/\\Delta =0$ , and a doped system with $\\mu |/\\Delta =1.4$ .",
"The conductivity strongly depends on the ratio ${\\hbar \\omega }/{E_{c}}$ .",
"For $\\hbar \\omega /(2E_{c})<1$ , the divergent term vanishes in the clean limit and only the non-divergent term survives.",
"The features in Fig.",
"REF (a) as $\\hbar \\omega _{s}$ is around $\\hbar \\omega _{p}$ are induced by the nonzero relaxation parameters.",
"For $\\hbar \\omega /(2E_{c})>1$ the divergent term survives and to leading order varies as $\\left[\\hbar \\delta +i\\Gamma _{a}^{(2)}\\right]^{-1}(\\hbar \\delta +i\\Gamma )^{-1}$ , where we separate out the second order intraband relaxation parameter, and put the rest equal to $\\Gamma $ .",
"The near degenerate FWM is approximately determined by $\\sigma ^{(3)}(\\omega _{p},\\omega _{p},-\\omega _{p}+\\delta ) &\\approx &-\\frac{2\\pi \\sigma _{3}}{\\Delta ^{2}[\\hbar \\delta +i\\Gamma _{a}^{(2)}](\\hbar \\delta +i\\Gamma )}Z_{3}\\left(\\frac{\\hbar \\omega _p}{\\Delta }\\right)\\\\&&\\times \\theta \\left(\\hbar \\omega _p-2E_{c}\\right)\\,.$ Note that the nonlinear response to a pulse of light can expected to be very complicated due to the strong dependence on the detuning."
],
[
"Coherent Current Injection",
"Now we turn to another divergence associated with $v\\rightarrow 0$ in Eq.",
"(REF ), which exists in the conductivity $\\sigma ^{(3)}(-2\\omega +\\delta ,\\omega ,\\omega )$ as $\\delta \\rightarrow 0$ .",
"This divergence describes coherent current injection.",
"Setting all the relaxation parameters $\\Gamma _{a/e}^{(j)}$ except $\\Gamma _{a}^{(3)}$ equal to $\\Gamma $ , we have the leading contribution with respect to the small quantity $\\Gamma _{a}^{(3)}$ and $\\delta $ given by $\\sigma ^{(3)}(-2\\omega +\\delta ,\\omega ,\\omega )=\\frac{\\eta _{\\text{cci}}(\\omega ,\\Gamma )}{\\hbar ^{-1}\\Gamma _{a}^{(3)}-i\\delta }+\\cdots $ While the full expressions of $\\eta _{\\text{cci}}(\\omega ,\\Gamma )$ can be obtained from our general analytic expressions, the underlying physics can be easily shown at the clean limit; setting $\\Gamma \\rightarrow 0^{+}$ , we find $\\eta _{\\text{cci}}(\\omega ,0) & = & \\frac{\\sigma _{3}}{\\hbar \\Delta ^{3}}\\left[g_{1}\\left(\\frac{\\hbar \\omega }{\\Delta }\\right)T\\left(\\frac{E_{c}}{\\Delta },\\frac{2\\hbar \\omega }{\\Delta }\\right)+g_{2}\\left(\\frac{\\hbar \\omega }{\\Delta }\\right)T\\left(\\frac{E_{c}}{\\Delta },\\frac{\\hbar \\omega }{\\Delta }\\right)\\right]\\,,$ where the functions $g_{1}$ and $g_{2}$ are given by $g_{1}(x) & = & \\frac{1}{3x^{7}}\\begin{bmatrix}(1-x^{2})^{2}\\,, & 1-x^{4}\\,, & 1-x^{4}\\end{bmatrix}^{T}\\,,\\\\g_{2}(x) & = & \\frac{1}{12x^{7}}\\begin{bmatrix}-16+24x^{2}-5x^{4}\\,, & -16-8x^{2}+3x^{4}\\,, & -16-8x^{2}+3x^{4}\\end{bmatrix}^{T}\\,.$ The coefficient $\\eta _{\\text{cci}}$ describes the two-color coherent current injection coefficient.",
"In the clean limit, it includes two terms: the one involving $g_{2}$ is nonzero only for $\\hbar \\omega >2E_{c}$ , and the one involving $g_{1}$ is nonzero for $\\hbar \\omega >E_{c}$ .",
"Both terms arises because of the interference of one-photon and two-photon absorption processes.",
"However, the contribution from the term involving $g_{1}$ comes from the interference between the pathways of one-photon absorption and degenerate two-photon absorption, as illustrated in the transition process on the left side of Fig.",
"REF .",
"While the term involving $g_{2}$ comes from the interference between one-photon absorption and the stimulated Raman process, as shown in the transition process on the right side of Fig.",
"REF .",
"In experiments involving typical semiconductors $2\\hbar \\omega $ is usually greater than the band gap energy but $\\hbar \\omega $ is not; however, if $\\hbar \\omega $ also is greater than the gap there can be a contribution due to stimulated electronic Raman scattering.",
"In the clean limit, a direct calculation of the functions $g_{i}(x)$ gives $g_{1}(1)=g_{2}(2)=0$ .",
"For the component $\\sigma ^{(3);xxxx}$ , their contributions are maximized at $x\\approx 1.2$ for $g_{1}(x)$ and $x\\approx 2.4$ for $g_{2}(x)$ .",
"However, at $x\\approx 2.4$ , the contribution from the stimulated electronic Raman scattering is no more than 20% of the contribution from the usual injection process; the total contribution has no maximum around $x\\approx 2.4$ .",
"A direct consequence of the divergence $v\\rightarrow 0$ is the form of the current response to a pulse light.",
"For simplicity, we only take the beam at $2\\omega $ as a pulse, and then the time evolution of the current density is $J^{(3);d}(t)=\\int \\frac{d\\delta }{2\\pi }\\sigma ^{(3);dabc}(-2\\omega +\\delta ,\\omega ,\\omega )E^{a}(-2\\omega +\\delta )E^{b}(\\omega )E^{c}(\\omega )e^{-i\\delta t}\\,.$ With substitution of the leading term in Eq.",
"(REF ) we can get $\\frac{dJ^{(3);d}(t)}{dt}=\\eta _{cci}^{dabc}(\\omega ,\\Gamma )E_{2\\omega }^{a}(t)E_{\\omega }^{b}(t)E_{\\omega }^{c}(t)-\\hbar ^{-1}\\Gamma _{a}^{(3)}J^{(3);d}(t)\\,.$ Here $\\mathbf {E}_{2\\omega }(t)$ and $\\mathbf {E}_{\\omega }(t)$ are the time evolution of pulses with center frequencies at $2\\omega $ and $\\omega $ , respectively.",
"The first term at the right hand side is a source term, while the second term describes the damping.",
"This equation shows exactly how the injection process occurs.",
"Figure: Conductivity σ (3);xxxx (-2ω,ω,ω){\\sigma }^{(3);xxxx}(-2\\omega ,\\omega ,\\omega ) for differentchemical potential |μ|/Δ=0|\\mu |/\\Delta =0, 1.11.1, 1.21.2, and 1.41.4.The relaxation parameters are taken as Γ a (3) /Δ=0.01\\Gamma _{a}^{(3)}/\\Delta =0.01and Γ/Δ=0.05\\Gamma /\\Delta =0.05, the gap parameter is taken as Δ=1\\Delta =1 eV.As an illustration, we plot the ${\\sigma }^{(3);xxxx}(-2\\omega ,\\omega ,\\omega )$ as a function of $\\omega $ for different chemical potentials $|\\mu |/\\Delta =0$ , $1.1$ , $1.2$ , and $1.4$ ; the other parameters are taken as $\\Gamma _{a}^{(3)}/\\Delta =0.01$ , $\\Gamma /\\Delta =0.05$ , and $\\Delta =1$ eV.",
"In the clean limit, ${\\sigma }^{(3);xxxx}$ is purely imaginary, and although with the inclusion of damping the real parts do not vanish, they are about an order of magnitude smaller than the imaginary parts, as shown in Fig.",
"REF .",
"In the calculations at finite relaxation parameters, both the real and the imaginary parts show obvious peaks/valleys around $\\hbar \\omega \\sim E_{c}$ and $\\hbar \\omega \\sim 2E_{c}$ , which correspond to the contributions from the terms including $g_{1}$ and $g_{2}$ , respectively.",
"In contrast to the situation discussed after Eq.",
"(REF ) for clean limits, the appearance of the peaks for $\\hbar \\omega >2E_{c}$ arises because of the inclusion of the finite relaxation parameters.",
"There also exists increases of the conductivity values as $\\hbar \\omega \\rightarrow 0$ , mostly for nonzero $\\mu /\\Delta $ .",
"They are associated with a divergence induced by the free-carriers, which is not our focus in this work."
],
[
"Jerk Current",
"Some of the divergences considered here can appear simultaneously.",
"A good example is the recently discussed jerk current [32], which can be treated as a special case of XPM for a zero signal frequency, or a special case of CISNL for current induced one-photon current injection.",
"The corresponding conductivity is $\\sigma ^{(3);dabc}(\\omega ,-\\omega ,0)$ , which involves the unsymmetrized conductivities $\\widetilde{\\sigma }^{(3);dabc}(0,-\\omega ,0)$ , $\\widetilde{\\sigma }^{(3);dacb}(0,-\\omega ,-\\omega )$ , $\\widetilde{\\sigma }^{(3);dbac}(0,\\omega ,0)$ , $\\widetilde{\\sigma }^{(3);dbca}(0,\\omega ,\\omega )$ , $\\widetilde{\\sigma }^{(3);dcab}(0,0,-\\omega )$ , and $\\widetilde{\\sigma }^{(3);dcba}(0,\\omega ,0)$ .",
"They all include intraband divergences, and the highest order is described by the limiting behavior as $v,v_{0}\\rightarrow 0$ or $v,v_{3}\\rightarrow 0$ .",
"In general, the leading orders are $\\sigma ^{(3);dabc}(\\omega ,-\\omega ,0) & \\approx \\frac{i\\sigma _{3}}{\\Delta ^{4}}\\frac{\\Delta }{i\\Gamma _{a}^{(3)}}\\left[\\frac{\\Delta }{i\\Gamma _{a}^{(2)}}Q_{1}\\left(\\frac{\\hbar \\omega }{\\Delta };\\left\\lbrace \\frac{\\Gamma _{a/e}^{(j)}}{\\Delta }\\right\\rbrace \\right)+\\frac{\\Delta }{i\\Gamma _{a}^{(1)}}Q_{2}\\left(\\frac{\\hbar \\omega }{\\Delta };\\left\\lbrace \\frac{\\Gamma _{a/e}^{(j)}}{\\Delta }\\right\\rbrace \\right)\\right]\\,.$ The clean limits of $Q_{1}$ and $Q_{2}$ are $Q_{1}(x;0) & =\\frac{1}{6x^{6}}\\left[(x^{2}-4)^{2}A_{0}+32x^{2}A_{6}\\right]T(\\alpha ;x)\\,,\\\\Q_{2}(x;0) & =\\frac{(\\alpha ^{2}-1)(\\alpha ^{2}+3)}{3\\alpha ^{5}}\\frac{1}{x}A_{0}\\,,$ with $\\alpha =E_{c}/\\Delta $ .",
"Here $Q_{1}(x;0)$ is nonzero only when the one-photon absorption exists as $\\hbar \\omega >2E_{c}$ , while $Q_{2}(x;0)$ exists only for a doped system where $\\alpha >1$ .",
"These two terms have a different power dependence, and the term involving $Q_{2}(x;0)$ can exists for any optical field frequency.",
"Thus the frequency dependence of the response can be used to distinguish between them."
],
[
"Comparison with graphene",
"Some of the phenomena discussed above have been considered earlier for doped graphene [19], [44], , [48], .",
"Although the expressions we derived above are normalized to the gap parameter $\\Delta $ , it is safe to take the limit as $\\Delta \\rightarrow 0$ .",
"This is because our general expressions of the conductivity can be safely reduced to the case of graphene with taking $\\Delta =0$ , as shown in Appendix .",
"The results of CISNL, DFWM, and CCI in graphene have been discussed earlier [19], [44], , [48], in the clean limit and with finite relaxation parameters.",
"Here we give a brief discussion for XPM and the jerk current in graphene.",
"In the clean limit, the XPM for graphene can be found from Eq.",
"(REF ) to be $\\sigma _{\\text{xpm};d}^{(3)}(\\omega _{s},\\omega _{p};\\delta )\\rightarrow \\frac{i\\sigma _{3}}{\\hbar \\delta }\\frac{T\\left(|\\mu |;\\hbar \\omega _{p}\\right)}{12\\hbar \\omega _{s}[(\\hbar \\omega _{p})^{2}-(\\hbar \\omega _{s})^{2}]}A_{0}\\,,$ Here the chemical potential induced gap plays a role similar to that of the gap parameter in gapped graphene.",
"For the jerk current, the leading term becomes $\\sigma ^{(3);dabc}(\\omega ,-\\omega ,0) & \\rightarrow \\frac{\\sigma _{3}}{3i\\Gamma _{a}^{(3)}}\\left[\\frac{1}{\\Gamma _{a}^{(2)}}\\frac{T(|\\mu |;\\hbar \\omega )}{2(\\hbar \\omega )^{2}}+\\frac{1}{\\Gamma _{a}^{(1)}}\\frac{1}{|\\mu |(\\hbar \\omega )}\\right]A_{0}\\,.$"
],
[
"Conclusions",
"We have systematically discussed intraband divergences in the third order optical response, and identified the leading terms in the corresponding third order conductivity.",
"Due to the combination of intraband and interband transitions, these divergences can appear at optical frequencies, and lead to large nonlinear conductivities.",
"We have shown that the existence of such divergences is very general, independent of the details of the band structure.",
"We illustrated these divergences in gapped graphene, with analytic expressions obtained for the third order conductivities in the relaxation time approximation.",
"Such divergences are of interest to experimentalists, because within the independent particle treatment presented here the optical response is limited only by the phenomenological relaxation times introduced in the theory, and thus that optical response can be expected to be large.",
"As well, at a qualitative level the predicted nature of the divergent behavior is robust against approximations made in describing the details of the interband transitions.",
"The divergences are also of interest to theorists, because one can expect that under such conditions the kind of treatment presented here is too naive.",
"This could be both because more realistic treatments of relaxation processes are required, and as well because the large optical response predicted could be an indication that in a real experimental scenario the perturbative approach itself is insufficient.",
"Thus the identification of these divergences identifies regions of parameter space where experimental and theoretical studies can be expected to lead to new insights into the nature of the interaction of light with matter.",
"More generally, we can expect that the calculation of other response coefficients involving perturbative expressions of the density matrix response to an electric field will reveal similar divergences in the nonlinear contributions to the response of other physical quantities, such as carrier density, spin/valley polarization, and spin/valley current.",
"A deep understanding of these divergences can lead to new ways to probe these quantities, and to study new effects in the optical response of materials that depend on them.",
"This work has been supported by CAS QYZDB-SSW-SYS038, NSFC Grant No.",
"11774340 and 61705227.",
"S.W.W.",
"is supported by the National Basic Research Program of China under Grant No.",
"2014CB921601S.",
"J.E.S.",
"is supported by the Natural Sciences and Engineering Research Council of Canada."
],
[
"Perturbative conductivity for general band structure",
"In this appendix we give the formal derivation of the third order conductivities in terms of the electron energy, velocity matrix elements, and Berry connections.",
"The density matrix can be expanded in terms of electric fields as $\\rho _{nm\\mathbf {k}}(t) & =\\sum _{\\omega _{3}}\\widetilde{{\\cal P}}_{nm\\mathbf {k}}^{(1);c}(\\omega _{3})E^{c}(\\omega _{3})e^{-i\\omega _{3}t}\\nonumber \\\\& +\\sum _{\\omega _{2}\\omega _{3}}\\widetilde{{\\cal P}}_{nm\\mathbf {k}}^{(2);bc}(\\omega _{0},\\omega _{3})E^{b}(\\omega _{2})E^{c}(\\omega _{3})e^{-i\\omega _{0}t}\\nonumber \\\\& +\\sum _{\\omega _{1}\\omega _{2}\\omega _{3}}\\widetilde{{\\cal P}}_{nm\\mathbf {k}}^{(3);abc}(\\omega ,\\omega _{0},\\omega _{3})E^{a}(\\omega _{1})E^{b}(\\omega _{2})E^{c}(\\omega _{3})e^{-i\\omega t}+\\cdots \\,,$ with $\\omega =\\omega _{1}+\\omega _{2}+\\omega _{3}$ and $\\omega _{0}=\\omega _{2}+\\omega _{3}$ .",
"Using the iteration in Eq.",
"(REF ), we can get the density matrix at different perturbation orders.",
"The first order terms of the density matrix are $\\widetilde{{\\cal P}}_{nn\\mathbf {k}}^{(1);c}(\\omega _{3}) & =\\frac{1}{v_{3}}A_{1;nn\\mathbf {k}}^{(1);c}\\,, & A_{1;nn\\mathbf {k}}^{(1);c} & =i\\frac{\\partial f_{n\\mathbf {k}}}{\\partial k_{a}}\\,,\\\\\\widetilde{{\\cal P}}_{nm\\mathbf {k}}^{(1);c}(\\omega _{3}) & =B_{1;nm\\mathbf {k}}^{(1);c}(w_{3})\\,, & B_{1;nm\\mathbf {k}}^{(1);c}(w_{3}) & =\\frac{r_{nm\\mathbf {k}}^{a}f_{mn\\mathbf {k}}}{w_{3}-(\\varepsilon _{n\\mathbf {k}}-\\varepsilon _{m\\mathbf {k}})}\\,.$ The second order terms are $\\widetilde{{\\cal P}}_{nn\\mathbf {k}}^{(2);bc}(\\omega _{2},\\omega _{3}) & = & \\frac{1}{v_{0}v_{3}}A_{1;nn\\mathbf {k}}^{(2);bc}+\\frac{1}{v_{0}}A_{2;nn\\mathbf {k}}^{(3);bc}(w_{3})\\,,\\\\\\widetilde{{\\cal P}}_{nm\\mathbf {k}}^{(2);bc}(\\omega _{2},\\omega _{3}) & = & \\frac{1}{v_{3}}B_{1;nm\\mathbf {k}}^{(2);bc}(w_{0})+B_{2;nm\\mathbf {k}}^{(2);bc}(w_{0},w_{3})\\,,$ with $A_{1;nn\\mathbf {k}}^{(2);bc} & = & i\\frac{\\partial A_{1;nn\\mathbf {k}}^{(1);c}}{\\partial k_{b}}\\,,\\\\A_{2;nn\\mathbf {k}}^{(2);bc}(w_{3}) & = & \\sum _{m}[r_{nm\\mathbf {k}}^{b}B_{1;mn\\mathbf {k}}^{(1);c}(w_{3})-B_{1;nm\\mathbf {k}}^{(1);c}(w_{3})r_{mn\\mathbf {k}}^{b}]\\,,\\\\B_{1;nm\\mathbf {k}}^{(2);bc}(w_{0}) & = & \\frac{r_{nm\\mathbf {k}}^{b}[A_{1;mm\\mathbf {k}}^{(1);c}-A_{1;nn\\mathbf {k}}^{(1);c}]}{w_{0}-(\\varepsilon _{n\\mathbf {k}}-\\varepsilon _{m\\mathbf {k}})}\\,,\\\\B_{2;nm\\mathbf {k}}^{(2);bc}(w_{0},w_{3}) & = & \\frac{[r_{\\mathbf {k}}^{b},B_{1;\\mathbf {k}}^{(1);c}(w_{3})]_{nm}+i\\left(B_{1;\\mathbf {k}}^{(1);c}(w_{3})\\right)_{;nmk_{b}}}{w_{0}-(\\varepsilon _{n\\mathbf {k}}-\\varepsilon _{m\\mathbf {k}})}\\,.$ Here we have used the notation $\\mathbf {r}_{nm\\mathbf {k}}=(1-\\delta _{nm})\\mathbf {\\xi }_{nm\\mathbf {k}}$ and $\\left(B_{1;\\mathbf {k}}^{(1);c}(w_{3})\\right)_{;nmk_{b}} & = & \\frac{\\partial B_{1;nm\\mathbf {k}}^{(1);c}(w_{3})}{\\partial k_{b}}-i(\\xi _{nn\\mathbf {k}}^{b}-\\xi _{mm\\mathbf {k}}^{b})B_{1;nm\\mathbf {k}}^{(1);c}(w_{3})\\,.$ It shows that the diagonal terms of the Berry connections $\\mathbf {\\xi }_{nn\\mathbf {k}}$ appear always with the derivative $\\mathbf {\\nabla }_{\\mathbf {k}}$ to form a gauge invariant term [11].",
"The third order terms are $\\widetilde{{\\cal P}}_{nn\\mathbf {k}}^{(3);abc}(\\omega _{1},\\omega _{2},\\omega _{3}) & = & \\frac{1}{vv_{0}v_{3}}A_{1;nn\\mathbf {k}}^{(3);abc}+\\frac{1}{vv_{0}}A_{2;nn\\mathbf {k}}^{(3);abc}(w_{3})\\nonumber \\\\& + & \\frac{1}{vv_{3}}A_{3;nn\\mathbf {k}}^{(3);abc}(w_{0})+\\frac{1}{v}A_{4;nn\\mathbf {k}}^{(3);abc}(w_{0},w_{3})\\,,\\\\\\widetilde{{\\cal P}}_{nm\\mathbf {k}}^{(3);abc}(\\omega _{1},\\omega _{2},\\omega _{3}) & = & \\frac{1}{v_{0}v_{3}}B_{1;nm\\mathbf {k}}^{(3);abc}(w)+\\frac{1}{v_{0}}B_{2;nm\\mathbf {k}}^{(3);abc}(w,w_{3})\\nonumber \\\\& + & \\frac{1}{v_{3}}B_{3;nm\\mathbf {k}}^{(3);abc}(w,w_{0})+B_{4;nm\\mathbf {k}}^{(3);abc}(w,w_{0},w_{3})\\,,$ with $A_{1;nn\\mathbf {k}}^{(3);abc} & = & i\\frac{\\partial A_{1;nn\\mathbf {k}}^{(2);bc}}{\\partial k_{a}}\\,,\\\\A_{2;nn\\mathbf {k}}^{(3);abc}(w_{3}) & = & i\\frac{\\partial A_{2;nn\\mathbf {k}}^{(2);bc}(w_{3})}{\\partial k_{a}}\\,,\\\\A_{3;nn\\mathbf {k}}^{(3);abc}(w_{0}) & = & \\sum _{m}[r_{nm\\mathbf {k}}^{a}B_{1;mn\\mathbf {k}}^{(2);bc}(w_{0})-B_{1;nm\\mathbf {k}}^{(2);bc}(w_{0})r_{mn\\mathbf {k}}^{a}]\\,,\\\\A_{4;nn\\mathbf {k}}^{(3);abc}(w_{0},w_{3}) & = & \\sum _{m}[r_{nm\\mathbf {k}}^{a}B_{2;mn\\mathbf {k}}^{(2);bc}(w_{0},w_{3})-B_{2;nm\\mathbf {k}}^{(2);bc}(w_{0},w_{3})r_{mn\\mathbf {k}}^{a}]\\,,$ and $B_{1;nm\\mathbf {k}}^{(3);abc}(w) & = & \\frac{r_{nm\\mathbf {k}}^{b}[A_{1;mm\\mathbf {k}}^{(2);bc}-A_{1;nn\\mathbf {k}}^{(2);bc}]}{w-(\\varepsilon _{n\\mathbf {k}}-\\varepsilon _{m\\mathbf {k}})}\\,,\\\\B_{2;nm\\mathbf {k}}^{(3);abc}(w,w_{3}) & = & \\frac{r_{nm\\mathbf {k}}^{b}[A_{2;mm\\mathbf {k}}^{(2);bc}(w_{3})-A_{2;nn\\mathbf {k}}^{(2);bc}(w_{3})]}{w-(\\varepsilon _{n\\mathbf {k}}-\\varepsilon _{m\\mathbf {k}})}\\,,\\\\B_{3;nm\\mathbf {k}}^{(3);abc}(w,w_{0}) & = & \\frac{[r_{\\mathbf {k}}^{a},B_{1;\\mathbf {k}}^{(2);bc}(w_{0})]_{nm}+i\\left(B_{1;\\mathbf {k}}^{(2);bc}(w_{0})\\right)_{;nmk_{a}}}{w-(\\varepsilon _{n\\mathbf {k}}-\\varepsilon _{m\\mathbf {k}})}\\,,\\\\B_{4;nm\\mathbf {k}}^{(3);abc}(w,w_{0},w_{3}) & = & \\frac{[r_{\\mathbf {k}}^{a},B_{2;\\mathbf {k}}^{(2);bc}(w_{0},w_{3})]_{nm}+i\\left(B_{2;\\mathbf {k}}^{(2);bc}(w_{0})\\right)_{;nmk_{a}}}{w-(\\varepsilon _{n\\mathbf {k}}-\\varepsilon _{m\\mathbf {k}})}$ The current density is then calculated through $\\mathbf {J}(t)=e\\sum _{nm}\\int \\frac{d\\mathbf {k}}{(2\\pi )^{2}}v_{mn\\mathbf {k}}^{d}\\rho _{nm\\mathbf {k}}(t)$ , and the $S_{j}$ terms are given by $S_{i}^{dabc}(\\cdots ) & =-e^{4}\\sum _{n}\\int \\frac{d\\mathbf {k}}{(2\\pi )^{2}}v_{nn\\mathbf {k}}^{a}A_{i;nn\\mathbf {k}}^{(3);abc}(\\cdots )\\,,\\text{ for }i=1,2,3,4,\\\\S_{i}^{dabc}(\\cdots ) & =-e^{4}\\sum _{nm}\\int \\frac{d\\mathbf {k}}{(2\\pi )^{2}}v_{mn\\mathbf {k}}^{a}B_{i;nm\\mathbf {k}}^{(3);abc}(\\cdots )\\,,\\text{ for }i=5,6,7,8\\,.$"
],
[
"$S_{j}^{dabc}$ for a gapped graphene",
"The calculation of $S_{j}^{dabc}$ is straightforward.",
"In the linear dispersion approximation and relaxation time approximation, analytic expressions can be obtained.",
"By listing the nonzero components of $S_{j}^{dabc}(\\cdots )$ as a column vector $S_{j}=\\begin{bmatrix}S_{j}^{xxyy} \\\\ S_{j}^{xyxy} \\\\ S_{j}^{xyyx}\\end{bmatrix}$ , we can write $S_{j} & =i\\sigma _{3}\\left[W_{j}^{(0)}+2\\Delta ^{2}W_{j}^{(2)}+2\\Delta ^{4}W_{j}^{(4)}\\right]\\theta (|\\mu |-\\Delta )\\,, & \\text{ for }j=1,3,5,7\\,,\\nonumber \\\\S_{j} & =i\\sigma _{3}\\left[W_{j}^{(0)}+2\\Delta ^{2}W_{j}^{(2)}+2\\Delta ^{4}W_{j}^{(4)}\\right]\\,, & \\text{ for }j=2,4,6,8\\,.$ with $\\sigma _{3}=\\sigma _{0}\\frac{(\\hbar v_{F}e)^{2}}{\\pi }\\,, \\quad \\sigma _{0}=\\frac{e^{2}}{4\\hbar }\\,.$ Note that $W^{(j)}(\\cdots )$ depends on these functions $\\begin{array}{rrr}|E_{c}|^{-5}\\,, & |E_{c}|^{-3}\\,, & |E_{c}|^{-1}\\,,\\\\{\\cal I}(E_{c};w)\\,, & {\\cal H}(E_{c};w)\\,, & {\\cal H}(E_{c};w_{0})\\,,\\\\{\\cal G}(E_{c};w)\\,, & {\\cal G}(E_{c};w_{0})\\,, & {\\cal G}(E_{c};w_{3})\\,.\\end{array}$ These functions depend on the photon energies and the effective gap parameter $E_{c}$ ."
],
[
"$W_{j}^{(0)}$",
"Gapped graphene reduces to graphene [44], as $\\Delta \\rightarrow 0$ and $E_{c}\\rightarrow |\\mu |$ , and thus $W_{j}^{(0)}$ should give the results for graphene, as $W_{1}^{(0)} & = & \\frac{1}{E_{c}}A_{0}\\,,\\\\W_{3}^{(0)}(w_{0}) & = & {\\cal H}(E_{c};w_{0})\\frac{A_{3}}{w_{0}}-\\frac{1}{E_{c}}\\frac{A_{3}}{w_{0}}\\,,\\\\W_{5}^{(0)}(w) & = & {\\cal H}(E_{c};w)\\frac{A_{0}}{w}+{\\cal I}(E_{c};w)A_{1}-\\frac{1}{E_{c}}\\frac{A_{0}}{w}\\,,\\\\W_{7}^{(0)}(w,w_{0}) & = & {\\cal H}(E_{c};w_{0})\\left(\\frac{A_{2}}{w_{1}^{2}}-\\frac{A_{3}}{w_{0}w_{1}}\\right)+{\\cal H}(E_{c};w)\\left(\\frac{A_{3}}{ww_{1}}-\\frac{A_{2}}{w_{1}^{2}}\\right)\\nonumber \\\\& & -{\\cal I}(E_{c};w)\\frac{A_{2}}{w_{1}}+\\frac{1}{E_{c}}\\frac{A_{3}}{ww_{0}}\\,,$ $W_{2}^{(0)}(w_{3}) & = & {\\cal G}\\left(E_{c};w_{3}\\right)\\frac{A_{0}}{w_{3}^{2}}-\\frac{1}{E_{c}}\\frac{A_{0}}{w_{3}}\\,,\\\\W_{4}^{(0)}(w_{0},w_{3}) & = & -{\\cal G}(E_{c};w_{3})\\frac{w_{3}A_{2}+w_{2}A_{3}}{w_{2}^{2}w_{3}^{2}}+{\\cal G}(E_{c};w_{0})\\frac{(w_{0}+w_{2})A_{2}+w_{2}A_{3}}{w_{0}^{2}w_{2}^{2}}\\nonumber \\\\& & -{\\cal H}(E_{c};w_{0})\\frac{A_{2}}{w_{0}w_{2}}+\\frac{1}{E_{c}}\\frac{A_{3}}{w_{0}w_{3}}\\,,\\\\W_{6}^{(0)}(w,w_{3}) & = & -{\\cal G}(E_{c};w_{3})\\frac{wA_{0}}{w_{3}^{2}(w^{2}-w_{3}^{2})}+{\\cal G}(E_{c};w)\\frac{w_{3}A_{0}}{w^{2}(w^{2}-w_{3}^{2})}+\\frac{1}{E_{c}}\\frac{A_{0}}{ww_{3}}\\,,$ and $& & W_{8}^{(0)}(w,w_{0},w_{3})\\nonumber \\\\& = & {\\cal G}(E_{c};w_{3})\\left[\\frac{A_{2}}{(w-w_{3})w_{2}^{2}w_{3}}+\\frac{w^{2}w_{2}+w_{3}^{3}+ww_{3}(-3w_{0}+2w_{3})}{(w-w_{3})^{3}w_{2}^{2}w_{3}^{2}}A_{3}\\right]\\nonumber \\\\& + & {\\cal G}(E_{c};w_{0})\\left[-\\frac{w_{0}w_{1}+w_{1}w_{2}-w_{0}w_{2}}{w_{0}^{2}w_{1}^{2}w_{2}^{2}}A_{2}-\\frac{w_{1}w_{2}-w_{0}^{2}-w_{0}w_{2}}{w_{1}^{2}w_{0}^{2}w_{2}^{2}}A_{3}\\right]\\nonumber \\\\& + & {\\cal G}(E_{c};w)\\left[-\\frac{1}{ww_{1}^{2}(w-w_{3})}A_{2}-\\frac{5w^{2}+w_{3}(w_{0}+w_{3})-w(3w_{0}+4w_{3})}{ww_{1}^{2}(w-w_{3})^{3}}A_{3}\\right]\\nonumber \\\\& + & {\\cal H}(E_{c};w_{0})\\left(\\frac{A_{2}}{w_{0}w_{1}w_{2}}-\\frac{A_{3}}{w_{1}^{2}w_{2}}\\right)+{\\cal H}(E_{c};w)\\frac{4w^{2}-3ww_{0}-2ww_{3}+w_{0}w_{3}}{ww_{1}^{2}(w-w_{3})^{2}}A_{3}\\nonumber \\\\& + & {\\cal I}(E_{c};w)\\frac{A_{3}}{w_{1}(w-w_{3})}-\\frac{1}{E_{c}}\\frac{A_{3}}{ww_{0}w_{3}}\\,.$ Here we have used $w_{2}=w_{0}-w_{3}$ , $w_{1}=w-w_{0}$ , and $A_i$ defined in Eq.",
"(REF )."
],
[
"$W_{j}^{(2)}$",
"The $\\Delta ^{2}$ terms are given as $W_{1}^{(2)} & = & \\frac{1}{E_{c}^{3}}A_{0}\\,,\\\\W_{3}^{(2)}(w_{0}) & = & \\left[-\\frac{1}{E_{c}^{3}}\\frac{1}{2w_{0}}-\\frac{1}{E_{c}}\\frac{2}{w_{0}^{3}}+{\\cal H}_{E_{c}}(w_{0})\\frac{2}{w_{0}^{3}}\\right](A_{1}+A_{2})\\,,\\\\W_{5}^{(2)}(w) & = & -\\frac{1}{E_{c}^{3}}\\frac{A_{0}}{w}-{\\cal I}(E_{c};w)\\frac{2A_{0}+2A_{1}}{w^{2}}+\\frac{1}{E_{c}}\\frac{4A_{1}}{w^{3}}-{\\cal H}(E_{c};w)\\frac{4A_{1}}{w^{3}}\\,,\\\\W_{7}^{(2)}(w,w_{0}) & = & \\frac{1}{E_{c}^{3}}\\frac{A_{1}+A_{2}}{2ww_{0}}+\\frac{1}{E_{c}}\\left[\\frac{2(w^{2}-w_{0}^{2})(A_{1}+A_{3})}{w^{3}w_{0}^{3}}+\\frac{2(w^{3}-w_{0}^{3})(A_{2}-A_{3})}{w_{1}w^{3}w_{0}^{3}}\\right]\\nonumber \\\\& & -{\\cal I}(E_{c};w)\\frac{2(A_{1}+A_{3})}{w^{2}w_{1}}+{\\cal H}(E_{c};w)\\frac{(-4w+2w_{0})(A_{1}+A_{3})+2w_{1}(A_{2}-A_{3})}{w^{3}w_{1}^{2}}\\nonumber \\\\& & +{\\cal H}(E_{c};w_{0})\\frac{-2w_{1}(A_{1}+A_{2})+2w_{0}(A_{1}+A_{3})}{w_{1}^{2}w_{0}^{3}}\\,,$ and $W_{2}^{(2)}(w_{3}) & = & \\left\\lbrace \\frac{1}{E_{c}^{3}}\\frac{1}{3w_{3}}+\\frac{1}{E_{c}}\\frac{4}{w_{3}^{3}}-{\\cal G}(E_{c};w_{3})\\frac{4}{w_{3}^{3}}\\right\\rbrace A_{1}\\,,\\\\W_{4}^{(2)}(w_{0},w_{3}) & = & \\frac{1}{E_{c}^{3}}\\frac{-A_{1}+A_{2}-2A_{3}}{6w_{0}w_{3}}\\nonumber \\\\& & +\\frac{1}{E_{c}}\\left[-\\frac{2(w_{0}^{2}-w_{3}^{2})(A_{1}+A_{3})}{w_{0}^{3}w_{3}^{3}}+\\frac{2(w_{0}^{3}-w_{3}^{3})(A_{2}-A_{3})}{w_{2}w_{0}^{3}w_{3}^{3}}\\right]\\nonumber \\\\& & -{\\cal H}(E_{c};w_{0})\\frac{A_{1}+A_{3}}{w_{0}w_{2}}\\nonumber \\\\& & +{\\cal G}(E_{c};w_{0})\\frac{2w_{0}(A_{1}+A_{3})+2w_{2}(A_{1}-A_{0})}{w_{0}^{4}w_{2}^{2}}\\nonumber \\\\& & +{\\cal G}(E_{c};w_{3})\\frac{(2w_{0}-4w_{3})(A_{1}+A_{3})-2w_{2}(A_{2}-A_{3})}{w_{2}^{2}w_{3}^{4}}\\,,\\\\W_{6}^{(2)}(w,w_{3}) & = & -\\frac{1}{E_{c}^{3}}\\frac{A_{1}}{3ww_{3}}+\\frac{1}{E_{c}}\\left[-\\frac{4(w^{2}+w_{3}^{2})A_{1}}{w^{3}w_{3}^{3}}+\\frac{4(A_{2}-A_{3})}{w^{2}w_{3}^{2}}\\right]\\nonumber \\\\& & +{\\cal G}(E_{c};w)\\frac{-4w_{3}A_{1}+4w(A_{2}-A_{3})}{w^{4}(w^{2}-w_{3}^{2})}\\nonumber \\\\& & +{\\cal G}(E_{c};w_{3})\\frac{4wA_{1}-4w_{3}(A_{2}-A_{3})}{w_{3}^{4}(w^{2}-w_{3}^{2})}\\,,$ and $& & W_{8}^{(2)}(w,w_{0},w_{3})\\nonumber \\\\& = & \\frac{1}{E_{c}^{3}}\\frac{A_{1}-A_{2}+2A_{3}}{6ww_{0}w_{3}}+\\frac{1}{E_{c}}\\left\\lbrace \\frac{2[w_{0}^{2}w_{3}^{2}+w^{2}(w_{0}^{2}-w_{3}^{2})](A_{1}-A_{2}+2A_{3})}{w^{3}w_{0}^{3}w_{3}^{3}}\\right.\\nonumber \\\\& & \\left.+\\frac{2[w_{0}(2w_{0}+w_{3})+w(w_{0}+2w_{3})](-A_{2}+A_{3})}{w^{2}w_{0}^{3}w_{3}^{2}}\\right\\rbrace \\nonumber \\\\& & +{\\cal I}(E_{c};w)\\frac{2(A_{1}+A_{2})}{w^{2}w_{1}(w-w_{3})}\\nonumber \\\\& & +{\\cal H}(E_{c};w)\\frac{2(6ww_{1}+ww_{2}-3w_{1}w_{3})(A_{1}+A_{2})}{w^{3}w_{1}^{2}(w-w_{3})^{2}}\\nonumber \\\\& & -{\\cal G}(E_{c};w)\\left[\\frac{2(6ww_{1}+ww_{2}-3w_{1}w_{3})(A_{1}+A_{2})}{w^{3}w_{1}^{2}(w-w_{3})^{3}}-\\frac{2(3w-2w_{0})}{w^{4}w_{1}^{2}(w-w_{3})}(A_{0}-A_{1})\\right]\\nonumber \\\\& & -{\\cal H}(E_{c};w_{0})\\frac{2w_{1}(A_{1}+A_{3})-2w_{0}(A_{1}+A_{2})}{w_{1}^{2}w_{0}^{3}w_{2}}\\nonumber \\\\& & +{\\cal G}(E_{c};w_{0})\\frac{2ww_{0}(A_{0}-2A_{1}-A_{3})+2(3w_{0}^{2}+ww_{3}-2w_{0}w_{3})(A_{1}-A_{0})}{w_1^2w_0^4w_2^2}\\nonumber \\\\& & +{\\cal G}(E_{c};w_{3})\\left[\\frac{2w(3ww_{0}-4ww_{3}-5w_{0}w_{3}+6w_{3}^{2})(-A_{1}+A_{2}-2A_{3})}{3(w-w_{3})^{3}w_{2}^{2}w_{3}^{4}}\\right.\\nonumber \\\\& & \\left.+\\frac{2(w-3w_{3})(w+2w_{0}-3w_{3})(A_{1}-A_{0})}{3(w-w_{3})^{3}w_{2}^{2}w_{3}^{3}}\\right]\\,.$"
],
[
"$W_{j}^{(4)}$",
"All terms proportional to $\\Delta ^{4}$ can be written as $W_{j}^{(4)}={\\cal W}_{j}^{(4)}A_{0}$ , with all quantity ${\\cal W}_{j}^{(4)}$ giving by ${\\cal W}_{1}^{(4)} & = & -\\frac{1}{E_{c}^{5}}\\frac{3}{2}\\,,\\\\{\\cal W}_{3}^{(4)}(w_{0}) & = & -\\frac{1}{E_{c}^{5}}\\frac{1}{2w_{0}}-\\frac{1}{E_{c}^{3}}\\frac{2}{w_{0}^{3}}-\\frac{1}{E_{c}}\\frac{8}{w_{0}^{5}}+{\\cal H}(E_{c};w_{0})\\frac{8}{w_{0}^{5}}\\,,\\\\{\\cal W}_{5}^{(4)}(w) & = & \\frac{1}{E_{c}^{5}}\\frac{3}{2w}+\\frac{1}{E_{c}^{3}}\\frac{2}{w^{3}}-\\frac{1}{E_{c}}\\frac{8}{w^{5}}+{\\cal I}(E_{c};w)\\frac{8}{w^{4}}+{\\cal H}(E_{c};w)\\frac{8}{w^{5}}\\,,\\\\{\\cal W}_{7}^{(4)}(w,w_{0}) & = & \\frac{1}{E_{c}^{5}}\\frac{1}{2ww_{0}}+\\frac{1}{E_{c}^{3}}\\frac{2(w^{2}-w_{0}^{2})}{w^{3}w_{0}^{3}}+\\frac{1}{E_{c}}\\frac{8(w^{4}-w^{2}w_{0}^{2}-2ww_{0}^{3}-3w_{0}^{4})}{w^{5}w_{0}^{5}}\\nonumber \\\\& & -{\\cal I}(E_{c};w)\\frac{8}{w^{4}w_{1}}-{\\cal H}(E_{c};w)\\frac{8(w+3w_{1})}{w^{5}w_{1}^{2}}-{\\cal H}(E_{c};w_{0})\\frac{8(w-2w_{0})}{w_{1}^{2}w_{0}^{5}}\\,.$ ${\\cal W}_{2}^{(4)}(w_{3}) & = & -\\frac{1}{E_{c}^{5}}\\frac{1}{10w_{3}}-\\frac{1}{E_{c}^{3}}\\frac{2}{3w_{3}^{3}}-\\frac{1}{E_{c}}\\frac{8}{w_{3}^{5}}+{\\cal G}(E_{c};w_{3})\\frac{8}{w_{3}^{6}}\\,,\\\\{\\cal W}_{4}^{(4)}(w_{0},w_{3}) & = & -\\frac{1}{E_{c}^{5}}\\frac{3}{10w_{0}w_{3}}-\\frac{1}{E_{c}^{3}}\\frac{2(3w_{0}^{2}+2w_{0}w_{3}+w_{3}^{2})}{3w_{0}^{3}w_{3}^{3}}-\\frac{1}{E_{c}}\\frac{8(3w_{0}^{4}+2w_{0}^{3}w_{3}+w_{0}^{2}w_{3}^{2}-w_{3}^{4})}{w_{0}^{5}w_{3}^{5}}\\nonumber \\\\& & -{\\cal H}(E_{c};w_{0})\\frac{8}{w_{0}^{5}w_{2}}+{\\cal G}(E_{c};w_{0})\\frac{8(3w_{0}-2w_{3})}{w^{6}w_{2}^{2}}+{\\cal G}(E_{c};w_{3})\\frac{8(3w_{0}-4w_{3})}{w_{2}^{2}w_{3}^{6}}\\,,\\\\{\\cal W}_{6}^{(4)}(w,w_{3}) & = & \\frac{1}{E_{c}^{5}}\\frac{1}{10ww_{3}}+\\frac{1}{E_{c}^{3}}\\frac{2(w^{2}+w_{3}^{2})}{3w^{3}w_{3}^{3}}+\\frac{1}{E_{c}}\\frac{8(w^{4}+w^{2}w_{3}^{2}+w_{3}^{4})}{w^{5}w_{3}^{3}}\\nonumber \\\\& & +{\\cal G}(E_{c};w)\\frac{8w_{3}}{w^{6}(w^{2}-w_{3}^{2})}-{\\cal G}(E_{c};w_{3})\\frac{8w}{w_{3}^{6}(w^{2}-w_{3}^{2})}\\,,$ and ${\\cal W}_{8}^{(4)}(w,w_{0},w_{3}) & = & \\frac{1}{E_{c}^{5}}\\frac{3}{10ww_{0}w_{3}}+\\frac{1}{E_{c}^{3}}\\frac{2\\left[w^{2}\\left(3w_{0}^{2}+2w_{0}w_{3}+w_{3}^{2}\\right)-w_{0}^{2}w_{3}^{2}\\right]}{3w^{3}w_{0}^{3}w_{3}^{3}}\\nonumber \\\\& & +\\frac{1}{E_{c}}\\frac{8\\left(\\left(3w_{0}^{4}+2w_{3}w_{0}^{3}+w_{3}^{2}w_{0}^{2}-w_{3}^{4}\\right)w^{4}+\\left(w_{3}^{2}-w_{0}^{2}\\right)w^{2}w_{0}^{2}w_{3}^{2}+w_{0}^{3}w_{3}^{4}(2w+3w_{0})\\right)}{w^{5}w_{0}^{5}w_{3}^{5}}\\nonumber \\\\& & +{\\cal I}(E_{c};w)\\frac{8}{w^{4}w_{1}(w-w_{3})}+{\\cal H}(E_{c};w)\\frac{8(8w^{2}-7ww_{0}-6ww_{3}+5w_{0}w_{3})}{w^{5}w_{1}^{2}(w-w_{3})^{2}}\\nonumber \\\\& & -{\\cal G}(E_{c};w)\\frac{8[18w^{3}-8w_{0}w_{3}^{2}+ww_{3}(21w_{0}+10w_{3})-w^2(15w_0+26w_3)]}{w^{6}w_{1}^{2}(w-w_{3})^{3}}\\nonumber \\\\& & +{\\cal H}(E_{c};w_{0})\\frac{8\\left(w-2w_{0}\\right)}{\\left(w-w_{0}\\right){}^{2}w_{0}^{5}\\left(w_{0}-w_{3}\\right)}-{\\cal G}(E_{c};w_{0})\\frac{8\\left(w-2w_{0}\\right)(3w_0-2w_3)}{\\left(w-w_{0}\\right){}^{2}w_{0}^{6}\\left(w_{0}-w_{3}\\right)^2}\\nonumber \\\\& & +{\\cal G}(E_{c};w_{3})\\frac{8\\left[\\left(4w_{3}-3w_{0}\\right)w^{2}+3\\left(3w_{0}-4w_{3}\\right)w_{3}w+2w_{3}^{2}\\left(5w_{3}-4w_{0}\\right)\\right]}{\\left(w-w_{3}\\right){}^{3}\\left(w_{0}-w_{3}\\right){}^{2}w_{3}^{6}}\\,.\\quad $"
]
] | 1808.08354 |
[
[
"On the motive of Kapustka-Rampazzo's Calabi-Yau threefolds"
],
[
"Abstract Kapustka and Rampazzo have exhibited pairs of Calabi-Yau threefolds $X$ and $Y$ that are L-equivalent and derived equivalent, without being birational.",
"We complete the picture by showing that $X$ and $Y$ have isomorphic Chow motives."
],
[
"Introduction",
"Let $\\hbox{Var}($ denote the category of algebraic varieties over the field $.The Grothendieck ring $ K0(Var()$ encodes fundamental properties of the birational geometry of varieties.",
"The intricacy of the ring $ K0(Var()$is highlighted by the result of Borisov \\cite {Bor}, showing that the class of the affine line $ L$ is a zero--divisor in $ K0(Var()$.",
"Following on Borisov^{\\prime }s pioneering result, a great many people have been hunting for Calabi--Yau varieties$ X, Y$ that are {\\em not} birational (and so $ [X]=[Y]$ in the Grothendieck ring), but$$ ([X] -[Y]) \\mathbb {L}^r=0\\ \\ \\ \\hbox{in}\\ K_0(\\hbox{Var}() \\ ,$$i.e., $ X$ and $ Y$ are ``L--equivalent^{\\prime \\prime } in the sense of \\cite {KS}.In many cases, the captured varieties $ X$ and $ Y$ are also derived equivalent \\cite {IMOU}, \\cite {IMOU2}, \\cite {Mar}, \\cite {Kuz}, \\cite {OR}, \\cite {BCP}, \\cite {KS}, \\cite {HL}, \\cite {Man}, \\cite {KR}, \\cite {KKM}.$ According to a conjecture made by Orlov [26], derived equivalent smooth projective varieties should have isomorphic Chow motives.",
"This conjecture is true for $K3$ surfaces [12], but is still open for Calabi–Yau varieties of dimension $\\ge 3$ .",
"In [20], I verified Orlov's conjecture for the Calabi–Yau threefolds of Ito–Miura–Okawa–Ueda [13].",
"The aim of the present note is to check that Orlov's conjecture is also true for the threefolds constructed recently by Kapustka–Rampazzo: -1mmTheorem (=theorem REF ) Let $X, Y$ be two derived equivalent Calabi–Yau threefolds as in [17].",
"Then there is an isomorphism of Chow motives $ h(X)\\cong h(Y)\\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat}\\ .$ To prove theorem REF , we exploit the “homological projective duality–style” diagram given in [17] relating $X$ and $Y$ .",
"One key ingredient in the proof that might be of independent interest is a result (theorem REF ) concerning higher Chow groups of certain fibrations; this is a variant of a result of Vial's [33].",
"-1mmConventions In this note, the word variety will refer to a reduced irreducible scheme of finite type over the field of complex numbers $.",
"For any variety $ X$, we will denote by $ Aj(X)$ the Chow group of dimension $ j$ cycles on $ X$with $ Q$--coefficients.For $ X$ smooth of dimension $ n$, the notations $ Aj(X)$ and $ An-j(X)$ will be used interchangeably.$ The notation $A^j_{hom}(X)$ will be used to indicate the subgroups of homologically trivial cycles.",
"For a morphism between smooth varieties $f\\colon X\\rightarrow Y$ , we will write $\\Gamma _f\\in A^\\ast (X\\times Y)$ for the graph of $f$ , and ${}^t \\Gamma _f\\in A^\\ast (Y\\times X)$ for the transpose correspondence.",
"The contravariant category of Chow motives (i.e., pure motives with respect to rational equivalence as in [30], [25]) will be denoted $\\mathcal {M}_{\\rm rat}$ ."
],
[
"The Calabi–Yau threefolds",
"Theorem 2.1 (Kapustka–Rampazzo [17]) Let $X,Y$ be a general pair of Calabi–Yau threefolds in the family $\\bar{\\mathcal {X}}_{25}$ that are dual to one another (in the sense of [17]).",
"Then $X$ and $Y$ are not birational, and so $ [X]\\ne [Y]\\ \\ \\ \\hbox{in}\\ K_0(\\hbox{Var}() \\ .$ However, one has $ ([X] -[Y]) \\mathbb {L}^2=0\\ \\ \\ \\hbox{in}\\ K_0(\\hbox{Var}() \\ .$ Moreover, $X$ and $Y$ are derived equivalent, i.e.",
"there is an isomorphism of bounded derived categories $ D^b(X)\\cong D^b(Y)\\ .$ In particular, there is an isomorphism of polarized Hodge structures $ H^3(X,\\mathbb {Z})\\ \\cong \\ H^3(Y,\\mathbb {Z})\\ .$ Everything but the last phrase is in [17].",
"The isomorphism of Hodge structures is a corollary of the derived equivalence, in view of [27].",
"Remark 2.2 As explained in [17], the threefolds $X,Y$ in the family $\\bar{\\mathcal {X}}_{25}$ are a limit case of the Calabi–Yau threefolds in the family $\\mathcal {X}_{25}$ studied in [8], [27].",
"A pair of dual varieties $X,Y$ in the family $\\mathcal {X}_{25}$ are also derived equivalent and L-equivalent (the exponent of $\\mathbb {L}$ is, however, higher than in theorem REF )."
],
[
"Higher Chow groups and fibrations",
"Definition 3.1 (Bloch [4], [5]) Let $\\Delta ^j\\cong \\mathbb {A}^j($ denote the standard $j$ –simplex.",
"For any quasi–projective variety $M$ and any $i\\in \\mathbb {Z}$ , let $z_i^{simp}(M,\\ast )$ denote the simplicial complex where $z_i(X,j)$ is the group of $(i+j)$ –dimensional algebraic cycles in $M\\times \\Delta ^j$ that meet the faces properly.",
"Let $z_i^{}(M,\\ast )$ denote the single complex associated to $z_i^{simp}(M,\\ast )$ .",
"The higher Chow groups of $M$ are defined as $ A_i(M,j):= H^j( z_i^{}(M,\\ast )\\otimes \\mathbb {Q})\\ .$ Remark 3.2 Clearly one has $A_i(M,0)\\cong A_i(M)$ .",
"Higher Chow groups are related to higher algebraic $K$ –theory: there are isomorphisms $ \\hbox{Gr}_\\gamma ^{n-i} K_j(M)_\\mathbb {Q}\\cong A_i(M,j) \\ \\ \\ \\hbox{for\\ all\\ }i,j $ where $K_j(M)$ is Quillen's higher $K$ –theory group associated to the category of coherent sheaves on $M$ , and $\\hbox{Gr}^\\ast _\\gamma $ is a graded for the $\\gamma $ –filtration [4].",
"Higher Chow groups are also related to Voevodsky's motivic cohomology (defined as hypercohomology of a certain complex of Zariski sheaves) [9], [24].",
"For later use, we establish the following result, which is a variant of a result of Vial's [33]: Theorem 3.3 Let $\\pi \\colon M\\rightarrow B$ be a flat projective morphism between smooth quasi–projective varieties of relative dimension $m$ .",
"Assume that for every $b\\in B$ , the fibre $M_b:=\\pi ^{-1}(b)$ has $ A_i(M_b)=\\mathbb {Q}\\ \\ \\ \\forall i\\ .$ (i) The maps $ \\Phi _\\ast := \\sum _{k=0}^{m} h^{m -k}\\circ \\pi ^\\ast \\colon \\ \\ \\ \\bigoplus _{k=0}^{m} A_{\\ell -k}(B,j)\\ \\rightarrow \\ A_{\\ell }(M,j) $ and $ \\Psi _\\ast := \\sum _{k=0}^{m} \\pi _\\ast \\circ h^{k}\\colon \\ \\ \\ A_{\\ell }(M,j)\\ \\rightarrow \\ \\bigoplus _{k=0}^{m} A_{\\ell -k}(B,j) $ are both isomorphisms, for any $\\ell $ and $j$ .",
"(Here $h^k$ denotes the operation of intersecting with the $k$ –th power of a hyperplane section $h\\in A^1(M)$ .)",
"(ii) Set $V_k:=(\\Psi _\\ast )^{-1} A_{\\ell -k}(B,j)\\ \\subset A_\\ell (M,j)$ .",
"Then $ (\\Phi _\\ast \\Psi _\\ast )\\vert _{V_{m}} = \\lambda \\, \\hbox{id}\\ ,$ for some non–zero $\\lambda \\in \\mathbb {Q}$ .",
"(i) For $j=0$ (i.e., for usual Chow groups), this is exactly [33].",
"For arbitrary $j$ (i.e., for higher Chow groups), a straightforward although laborious proof would consist in convincing the reader that everything Vial does in the proof of [33] also applies to higher Chow groups.",
"Indeed, all formal properties of Chow groups exploited in loc.",
"cit.",
"also hold for higher Chow groups.",
"Under the simplifying assumption that all fibres $M_b$ are isomorphic to $\\mathbb {P}^m$ (which will be the case when we apply theorem REF in this note), a quick proof could be as follows.",
"Let $H\\subset M$ be a general hyperplane section, and let $U=\\subset B$ be the open over which the fibres of the restricted morphism $\\pi \\vert _H\\colon H\\rightarrow B$ are isomorphic to $\\mathbb {P}^{m-1}$ .",
"Let $M_U:=\\pi ^{-1}(U)$ , and let us consider the restricted morphism $\\pi \\vert _U\\colon M_U\\rightarrow U\\ .$ Using the localization sequence for higher Chow groups and noetherian induction, we are reduced to proving (i) for $\\pi \\vert _U$ .",
"Let us consider the open $M^\\prime _U:=M_U\\setminus (H\\cap M_U)$ .",
"The fibres of the morphism $\\pi ^\\prime \\colon M^\\prime _U\\rightarrow U$ are isomorphic to $\\mathbb {A}^m$ .",
"There is a commutative diagram with exact rows $ \\begin{array}[c]{ccccccc}\\rightarrow & A_i(M^\\prime _U,j+1) &\\rightarrow & A_i(H,j) &\\rightarrow & A_i(M_U,j) &\\!\\!\\!\\rightarrow \\\\&&&&&&\\\\& \\uparrow {\\scriptstyle (\\pi ^\\prime )^\\ast } &&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\uparrow {\\scriptstyle \\sum _{k=0}^{m-1} h^{m-1-k}\\circ (\\pi \\vert _H)^\\ast }&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\uparrow {\\scriptstyle \\sum _{k=0}^m h^{m-k}\\circ (\\pi \\vert _U)^\\ast }&\\\\&&&&&&\\\\\\rightarrow & A_i(U,j+1)&\\rightarrow & {\\displaystyle \\bigoplus _{k=0}^{m-1}} A_{k}(U,j) &\\rightarrow & {\\displaystyle \\bigoplus _{k=0}^{m}} A_{k}(U,j) &\\!\\!\\!\\rightarrow \\\\\\end{array}$ Doing an induction on the fibre dimension $m$ , it will suffice to prove that $(\\pi ^\\prime )^\\ast $ is an isomorphism for all $i,j$ .",
"But this follows from the corresponding result for $K$ –theory [29], in view of the isomorphism (REF ) and the fact that the pullback $(\\pi ^\\prime )^\\ast \\colon K_j(U)\\rightarrow K_j(M^\\prime _U)$ respects the $\\gamma $ –filtration.",
"This proves that $\\Phi _\\ast $ is an isomorphism.",
"The argument for $\\Psi _\\ast $ is similar.",
"(ii) The direct summand $V_{m}$ can be identified as $ V_{m}=\\bigcap _{k=0}^{m-1} \\ker \\bigl ( \\pi _\\ast \\circ h^{k}\\bigr )\\ \\ \\ \\ \\subset \\ A_\\ell (M,j)\\ .$ Using this description, it is readily checked that $ V_{m}= \\pi ^\\ast A_{\\ell -m}(X,j)\\ .$ This implies (ii).",
"Remark 3.4 In case $B$ and $M$ are smooth projective, theorem REF can be upgraded to a relation of Chow motives [33].",
"In the more general case where $B$ and $M$ are smooth but only quasi–projective, perhaps one can relate $B$ and $M$ in the category $DM^{eff}_{gm}$ of Voevodsky motives ?",
"If so, the relation of higher Chow groups obtained in theorem REF would be an immediate consequence, since higher Chow groups (with $\\mathbb {Q}$ –coefficients) can be expressed as Hom–groups in $DM^{eff}_{gm}$ [9], [24]."
],
[
"Main result",
"Theorem 4.1 Let $X,Y$ be a pair of Calabi–Yau threefolds in the family $\\bar{\\mathcal {X}}_{25}$ that are dual to one another, in the sense of [17].",
"Then there is an isomorphism $ h(X)\\cong h(Y)\\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat}\\ .$ First, to simplify matters, let us slightly cut down the motives of $X$ and $Y$ .",
"It is known [17] that $X$ and $Y$ have Picard number 1.",
"A routine argument gives a decomposition of the Chow motives $ \\begin{split} h(X)&= {1} \\oplus {1}(1)\\oplus h^3(X) \\oplus {1}(2) \\oplus {1}(3)\\ ,\\\\h(Y)&= {1} \\oplus {1}(1)\\oplus h^3(Y) \\oplus {1}(2) \\oplus {1}(3)\\ \\ \\ \\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat}\\ ,\\\\\\end{split} $ where 1 is the motive of the point $\\hbox{Spec}(k)$ .",
"(The gist of this “routine argument” is as follows: let $H\\in A^1(X)$ be a hyperplane section.",
"Then $ \\pi ^{2i}_X:= c_i H^{3-i}\\times H\\ \\ \\ \\in \\ A^3(X\\times X)\\ , \\ \\ \\ 0\\le i\\le 3\\ ,$ defines an orthogonal set of projectors lifting the Künneth components, for appropriate $c_i\\in \\mathbb {Q}$ .",
"One can then define $\\pi ^3_X=\\Delta _X-\\sum _i \\pi ^{2i}_X\\in A^3(X\\times X)$ , and $h^j(X)=(X,\\pi ^i_X,0)\\in \\mathcal {M}_{\\rm rat}$ , and ditto for $Y$ .)",
"To prove the theorem, it will thus suffice to prove there is an isomorphism of motives $ h^3(X)\\cong h^3(Y)\\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat}\\ .$ We observe that the above decomposition (plus the fact that $H^\\ast (h^3(X))=H^3(X)$ is odd–dimensional) implies equality $ A^\\ast (h^3(X)) = A^\\ast _{hom}(X)\\ ,$ and similarly for $Y$ .",
"To construct the isomorphism (REF ), we need look no further than the construction of the threefolds $X, Y$ .",
"As explained in [17], the Calabi–Yau threefolds $X, Y$ are related via a diagram $ \\begin{array}[c]{ccccccccc} && D & \\xrightarrow{} & M & \\xleftarrow{} & E && \\\\&&&&&&&&\\\\&{}^{\\scriptstyle f_X} \\swarrow \\ \\ && {\\scriptstyle f} \\swarrow \\ \\ \\ &\\downarrow & \\ \\ \\ \\searrow {\\scriptstyle g} & & \\ \\ \\searrow {}^{\\scriptstyle g_Y} & \\\\&&&&&&&&\\\\X & \\hookrightarrow & G(2,5) & \\xleftarrow{} & F & \\xrightarrow{}& G(3,5) & \\hookleftarrow & Y\\\\\\end{array}$ Here $G(j,5)$ denotes the Grassmannian of $j$ –dimensional subspaces in a 5–dimensional vector space.",
"The variety $F$ is the flag variety parametrizing pairs $(V,W)\\in G(2,5)\\times G(3,5)$ such that $V\\subset W$ .",
"The variety $M\\subset F$ is a hyperplane section.",
"The Calabi–Yau varieties $X,Y$ are closed subvarieties of $G(2,5)$ resp.",
"$G(3,5)$ , and the closed subvarieties $D,E$ are defined as $f^{-1}(X)$ resp.",
"$g^{-1}(Y)$ .",
"The morphisms $f,g$ are $\\mathbb {P}^1$ –fibrations over the opens $G(2,5)\\setminus X$ resp.",
"$G(3,5)\\setminus Y$ , but the restrictions $f_X, g_Y$ are $\\mathbb {P}^2$ –fibrations.",
"The flag variety $F$ has trivial Chow groups (i.e.",
"$A^\\ast _{hom}(F)=0$ ), and so $F$ has a Chow–Künneth decomposition (this is a general fact for any smooth projective variety with trivial Chow groups: since all cohomology is algebraic, a Künneth decomposition exists; since $F\\times F$ again has trivial Chow groups, the Künneth decomposition is a Chow–Künneth decomposition).",
"By a standard trick (cf.",
"for instance [15]), this induces a Chow–Künneth decomposition $\\lbrace \\pi ^j_M\\rbrace $ for the hyperplane section $M\\subset F$ , with the property that $ (M,\\pi ^j_M)\\cong \\oplus {1}(\\ast )\\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat}\\ \\ \\ \\hbox{for\\ all\\ }j\\ne 7=\\dim M\\ .$ In particular, we have that $ A^i_{hom}(M)= A^i(h^7(M)):= (\\pi ^7_M)_\\ast A^i(M)\\ \\ \\ \\hbox{for\\ all\\ }i\\ .$ We now make the following claim: Claim 4.2 There are isomorphisms $ \\begin{split} \\Gamma _1\\colon \\ \\ h^3(X)\\ &\\xrightarrow{}\\ h^7(M)(2)\\ ,\\\\\\Gamma _2\\colon \\ \\ h^3(Y)\\ &\\xrightarrow{}\\ h^7(M)(2)\\ \\ \\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat}\\ .\\ \\\\\\end{split}$ This claim obviously suffices to prove (REF ).",
"To prove the claim, let us treat the isomorphism $\\Gamma _1$ in detail (the same argument applies to $\\Gamma _2$ , upon replacing $X$ and $G(2,5)$ by $Y$ resp.",
"$G(3,5)$ ).",
"To prove the claim for $\\Gamma _1$ , it will suffice to find correspondences $\\Gamma _1\\in A^{5}(X\\times M), \\Psi _1\\in A^{5}(M\\times X)$ with the property that $ \\begin{split} (\\Xi _1)_\\ast (\\Gamma _1)_\\ast =\\hbox{id}\\colon \\ \\ \\ A^i_{hom}(X)\\ &\\rightarrow \\ A^i_{hom}(X)\\ ,\\\\(\\Gamma _1)_\\ast (\\Xi _1)_\\ast =\\hbox{id}\\colon \\ \\ \\ A^i_{hom}(M)\\ &\\rightarrow \\ A^i_{hom}(M)\\ .\\\\\\end{split}$ (Indeed, let us assume one has correspondences $\\Gamma _1,\\Xi _1$ satisfying (REF ).",
"By what we have said above, this means that $ \\begin{split} & (\\pi ^3_X\\circ \\Xi _1\\circ \\pi ^7_M\\circ \\Gamma _1\\circ \\pi ^3_X)_\\ast =(\\pi ^3_X)_\\ast \\colon \\ \\ \\ A^i_{}(X)\\ \\rightarrow \\ A^i_{}(X)\\ ,\\\\& (\\pi ^7_M\\circ \\Gamma _1\\circ \\pi ^3_X\\circ \\Xi _1\\circ \\pi ^7_M)_\\ast =(\\pi ^7_M)_\\ast \\colon \\ \\ \\ A^i(M)\\ \\rightarrow \\ A^i_{}(M)\\ .\\\\\\end{split}$ There exists a field $k\\subset , finitely generated over $ Q$, such that $ X,M,jX,Mj1,1$ are defined over $ k$.",
"Because $ is a universal domain, for any finitely generated field extension $K\\supset k$ , there is an inclusion $K\\subset .",
"Thus, the natural maps $ Ai(XK)Ai(X$ and $ Ai(MK)Ai(M$ are injections \\cite [Appendix to Lecture 1]{B}.",
"This implies that the relations (\\ref {iso2}) also hold over $ K$.",
"Manin^{\\prime }s identity principle then gives that$$ \\Gamma _1\\colon \\ \\ \\ h^3(X_k)\\ \\rightarrow \\ h^7(M_k)(2)\\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat} $$is an isomorphism, and so $ 1$ induces an isomorphism of motives over $ as claimed.)",
"Before proving the claim, let us introduce some lemmas.",
"Lemma 4.3 Set–up as above.",
"The composition $ A^i_{hom}(X)\\ \\xrightarrow{}\\ A^i_{hom}(D)\\ \\xrightarrow{}\\ A^{i+2}_{hom}(M) $ is surjective, for any $i$ .",
"Let us write $U:=M\\setminus D$ , and $G:=G(2,5)$ .",
"By assumption, $U$ is a $\\mathbb {P}^1$ –fibration over $V:=G\\setminus X$ .",
"For any $i$ , there is a commutative diagram with exact rows $ \\begin{array}[c]{cccccccc}\\ \\ \\rightarrow & A_i(V,1) &\\rightarrow & A_i(X) &\\rightarrow & A_i(G) &\\rightarrow & A_i(V) \\\\&\\downarrow &&\\downarrow &&\\downarrow &&\\downarrow {}\\\\0 \\rightarrow & W^{-2i} H_{2i-1}(V,\\mathbb {Q})\\cap F_{-i} &\\rightarrow & H_{2i}(X,\\mathbb {Q})\\cap F_{-i} &\\rightarrow & H_{2i}(G,\\mathbb {Q})&\\rightarrow & W^{-2i} H_{2i}(V,\\mathbb {Q})\\\\\\end{array}$ where vertical arrows are (higher) cycle class maps into Borel–Moore homology, and $W^\\ast , F_\\ast $ denote the weight filtration resp.",
"the Hodge filtration on Borel–Moore homology [28].",
"(The upper row is exact thanks to localization for higher Chow groups [4], [5], [21].",
"The lower row is exact because the category of polarizable pure Hodge structures is semisimple [28].",
"For the cycle class map from higher Chow groups into Borel–Moore homology, cf.",
"[31].)",
"The Grassmannian $G$ has trivial Chow groups.",
"Using the fact that the Hodge conjecture is true for the threefold $X$ , this implies that the cycle class map induces isomorphisms $ A^i(V)\\ \\xrightarrow{}\\ W ^{-2i} H^{2i}(V,\\mathbb {Q})\\ ,$ and the higher cycle class map induces a surjection $ A_i(V,1)\\ \\twoheadrightarrow \\ W^{1-2i}H_{2i-1}(V,\\mathbb {Q})\\cap F_{1-2i}\\ .$ (These two facts together can be paraphrased by saying that $V$ satisfies a variantIt is a variant, because in [31] only the weight filtration and not the Hodge filtration is taken into account.",
"This works fine for the linear varieties considered in [31], but not for the varieties $U,V$ under consideration here.",
"of the “strong property” of Totaro's [31].)",
"Using theorem REF , plus the corresponding property of cohomology, this implies that $U$ has the same property (i.e., $U$ satisfies the strong property).",
"For any $i$ , there is a commutative diagram with exact rows $ \\begin{array}[c]{cccccccc}\\rightarrow & A_i(U,1) &\\rightarrow & A_i(D) &\\rightarrow & A_i(M) &\\rightarrow & A_i(U) \\\\&\\downarrow &&\\downarrow &&\\downarrow &&\\downarrow {}\\\\\\rightarrow & W^{-2i} H_{2i-1}(U,\\mathbb {Q})\\cap F_{-i} &\\rightarrow & H_{2i}(D,\\mathbb {Q})\\cap F_{-i} &\\rightarrow & H_{2i}(M,\\mathbb {Q})&\\rightarrow & W^{-2i} H_{2i}(U,\\mathbb {Q})\\\\\\end{array}$ By what we have just observed (the strong property for $U$ ), the left vertical arrow is a surjection and the right vertical arrow is an isomorphism.",
"A quick diagram chase then reveals that pushforward induces a surjection $ i_\\ast \\colon \\ \\ \\ A_i^{hom}(D)\\ \\twoheadrightarrow \\ A_i^{hom}(M) \\ \\ \\ \\forall i\\ .$ Next, since $f_X\\colon D\\rightarrow X$ is a $\\mathbb {P}^2$ –fibration, theorem REF ensures that there are isomorphisms $ \\begin{split} \\Phi _\\ast := \\sum _{ k=0}^2 h^{2-k}\\circ (f_X)^\\ast \\colon \\ \\ \\bigoplus _{k=0}^2 A_{i-k}^{hom}(X)\\ &\\xrightarrow{}\\ A_i^{hom}(D)\\ ,\\\\\\Psi _\\ast := \\sum _{ k=0}^2 (f_X)_\\ast \\circ h^k\\colon \\ \\ A_i^{hom}(D)\\ &\\xrightarrow{}\\ \\bigoplus _{k=0}^2 A_{i-k}^{hom}(X)\\ .\\end{split}$ We write $ \\begin{split} A_i^{hom}(D)&= V_0\\oplus V_1\\oplus V_2\\\\ &:= (\\Psi _\\ast )^{-1}A_i^{hom}(X) \\oplus (\\Psi _\\ast )^{-1}A_{i-1}^{hom}(X) \\oplus (\\Psi _\\ast )^{-1}A_{i-2}^{hom}(X)\\ .\\\\\\end{split}$ To prove the lemma, it remains to understand the pushforward map (REF ).",
"Precisely, we will show that one summand of the decomposition (REF ) already surjects onto $A_i^{hom}(M)$ : $\\ \\hbox{Im}\\bigl ( V_0\\oplus V_1\\ \\xrightarrow{}\\ A_i^{hom}(M)\\bigr ) \\ \\subset \\ \\hbox{Im}\\bigl ( V_2\\ \\xrightarrow{}\\ A_i^{hom}(M)\\bigr )\\ \\ \\ \\hbox{for\\ all\\ }i\\ .$ To see this, we observe that there is a commutative diagram of complexes $ \\begin{array}[c]{ccccc}z_i(D,\\ast ) &\\rightarrow & z_i(M,\\ast ) &\\rightarrow & z_i(U,\\ast )\\\\\\downarrow &&\\downarrow &&\\downarrow \\\\z_i(X,\\ast )&\\rightarrow & z_i(G(2,5),\\ast )&\\rightarrow &z_i(V,\\ast )\\\\\\end{array}$ (where the vertical arrows are proper pushforward maps).",
"This gives rise to a commutative diagram with long exact rows $ \\begin{array}[c]{ccccccc}\\rightarrow &A_i(U,1) &\\xrightarrow{}& A_i(D) &\\rightarrow & A_i(M)&\\rightarrow \\\\&\\ \\ \\ \\ \\ \\ \\downarrow {\\scriptstyle (f_U)_\\ast }&&\\ \\ \\ \\ \\ \\downarrow {\\scriptstyle (f_X)_\\ast }&&\\ \\ \\ \\ \\downarrow {\\scriptstyle f_\\ast }&\\\\\\rightarrow &A_i(V,1) &\\xrightarrow{}& A_i(X) &\\rightarrow & A_i(G(2,5))&\\rightarrow \\\\\\end{array}$ Let us now assume $b\\in A_i^{hom}(D)$ lies in the summand $V_0$ of the decomposition (REF ).",
"Then $(f_X)_\\ast (b)$ is in $A_i^{hom}(X)$ .",
"Since $A_i^{hom}(G(2,5))=0$ , this means that $(f_X)_\\ast (b)$ is in the image of the map $\\delta ^\\prime $ , say $(f_X)_\\ast (b)=\\delta ^\\prime (c^\\prime )$ .",
"In view of theorem REF , the element $c^\\prime \\in A_i(V,1)$ comes from an element $c\\in A_i(U,1)$ lying in the direct summand (isomorphic to) $A_i(V,1)$ .",
"Using sublemma REF below, this means that there is equality $ \\delta (c) = b - b_2\\ \\ \\ \\hbox{in}\\ A_i(D)\\ ,$ for some $b_2\\in A_i(D)$ lying in the summand (isomorphic to) $A_{i-2}(X)$ .",
"It follows that $ i_\\ast (b)= i_\\ast (b_2)\\ \\ \\in \\ \\hbox{Im}\\Bigl ( A_{i-2}(X)\\rightarrow A_i(D)\\rightarrow A_i(M)\\Bigr )\\ .$ As $i_\\ast (b_2)=i_\\ast (b)$ is homologically trivial, the surjection (REF ) above shows that we may suppose $b_2$ is homologically trivial, and so we have found $b_2$ lying in the summand denoted $V_2$ (isomorphic to $A_{i-2}^{hom}(X)$ ).",
"This shows that $ i_\\ast (b)\\ \\ \\in \\ \\hbox{Im}\\Bigl ( A_{i-2}^{hom}(X)\\rightarrow A_i(D)\\rightarrow A_i(M)\\Bigr )=: \\hbox{Im}\\bigl ( V_2\\ \\rightarrow \\ A_i(M)\\bigr ) \\ .$ Let us next assume that $b\\in A_i^{hom}(D)$ lies in the summand $V_1$ of the decomposition (REF ).",
"The commutative diagram of complexes up to quasi–isomorphism $ \\begin{array}[c]{ccccc}z_i(D,\\ast ) &\\rightarrow & z_i(M,\\ast ) &\\rightarrow & z_i(U,\\ast )\\\\\\downarrow {\\scriptstyle h} &&\\downarrow {\\scriptstyle h}&&\\downarrow {\\scriptstyle h}\\\\z_{i-1}(D,\\ast ) &\\rightarrow & z_{i-1}(M,\\ast ) &\\rightarrow & z_{i-1}(U,\\ast )\\\\\\downarrow &&\\downarrow &&\\downarrow \\\\z_{i-1}(X,\\ast )&\\rightarrow & z_{i-1}(G(2,5),\\ast )&\\rightarrow &z_{i-1}(V,\\ast )\\\\\\end{array}$ gives rise to a commutative diagram with exact rows $ \\begin{array}[c]{ccccccc}\\rightarrow &A_i(U,1) &\\xrightarrow{}& A_i(D) &\\rightarrow & A_i(M)&\\rightarrow \\\\&&&&&&\\\\& \\ \\ \\ \\ \\ \\ \\downarrow { \\scriptstyle (f\\vert _U)_\\ast \\circ h} &&\\ \\ \\ \\ \\ \\ \\downarrow {\\scriptstyle (f_X)_\\ast \\circ h} &&\\ \\ \\ \\ \\ \\ \\downarrow {\\scriptstyle f_\\ast \\circ h} &\\\\&&&&&&\\\\\\rightarrow &A_{i-1}(V,1) &\\xrightarrow{}& A_{i-1}(X) &\\rightarrow & A_{i-1}(G(2,5))&\\rightarrow \\\\\\end{array}$ Reasoning just as above, we can find $c\\in A_i(U,1)$ lying in the summand (isomorphic to) $A_{i-1}(V,1)$ such that $ \\delta (c) = b - b_2\\ \\ \\ \\hbox{in}\\ A_i(D)\\ ,$ where $b_2\\in A_i(D)$ is in the summand (isomorphic to) $A_{i-2}(X)$ .",
"It follows once more that $ i_\\ast (b)= i_\\ast (b_2)\\ \\ \\in \\ \\hbox{Im}\\Bigl ( A_{i-2}(X)\\rightarrow A_i(D)\\rightarrow A_i(M)\\Bigr )\\ ,$ and (using the surjectivity (REF )) that $ i_\\ast (b)\\ \\ \\in \\ \\hbox{Im}\\Bigl ( A_{i-2}^{hom}(X)\\rightarrow A_i(D)\\rightarrow A_i(M)\\Bigr )=: \\hbox{Im}\\bigl ( V_2\\ \\rightarrow \\ A_i(M)\\bigr ) \\ .$ We have now proven the inclusion (REF ).",
"Combining (REF ), (REF ) and theorem REF (ii), we see that there is a surjection $ A_{i-2}^{hom}(X) \\ \\twoheadrightarrow \\ A^{hom}_i(M)\\ ,$ which is given by $i_\\ast (f_X)^\\ast $ .",
"This proves the lemma.",
"In the proof of lemma REF we have used the following sublemma: Sublemma 4.4 Given $i\\in \\mathbb {Z}$ , let $ \\Psi _\\ast \\colon \\ \\ A_i(D)=\\bigoplus _{k=0}^2 A_{i-k}(X)\\ ,\\ \\ \\ \\Psi _\\ast \\colon \\ \\ A_i(U,1)=\\bigoplus _{k=0}^1 A_{i-k}(V,1)\\ $ be the decompositions of theorem REF .",
"Let $\\delta \\colon A_i(U,1)\\rightarrow A_i(D)$ be the boundary map of the localization exact sequence for the inclusion $D\\subset M$ .",
"Then $ \\begin{split} \\delta \\bigl ( A_i(V,1)\\bigr )\\ &\\subset \\ A_i(X)\\oplus A_{i-2}(X) \\ ,\\\\\\delta \\bigl ( A_{i-1}(V,1)\\bigr )\\ &\\subset \\ A_{i-1}(X)\\oplus A_{i-2}(X) \\ .\\\\ \\end{split} $ For the first inclusion, we consider the commutative diagram (REF ).",
"In view of theorem REF , the direct summand of $A_i(U,1)$ isomorphic to $A_i(V,1)$ is exactly the kernel of the map $(f\\vert _U)_\\ast \\circ h$ .",
"As such, the image under $\\delta $ is contained in $ \\ker \\Bigl ( A_i(D)\\ \\xrightarrow{}\\ A_{i-1}(X)\\Bigr )\\ .$ Again applying theorem REF , this kernel coincides with the two summands isomorphic to $A_i(X)$ resp.",
"to $A_{i-2}(X)$ , as claimed.",
"The second inclusion is proven similarly, reasoning in the diagram (REF ).",
"Lemma 4.5 Set–up as above.",
"There is equality $ D =\\lambda h^2 + h\\cdot f^\\ast (d_1) + f^\\ast (d_2) \\ \\ \\ \\hbox{in}\\ A^2(M)\\ ,$ for some non–zero $\\lambda \\in \\mathbb {Q}$ and some $d_i\\in A^i(G(2,5))$ , $i=1,2$ .",
"Let us consider the restriction $h^2\\vert _U$ of $h^2\\in A^2(M)$ to the open $U:=M\\setminus D$ .",
"Let $f_U\\colon U\\rightarrow V$ be the restriction of the morphism $f$ , where $V:=G(2,5)\\setminus X$ .",
"As we have seen, $f_U$ is a $\\mathbb {P}^1$ –fibration.",
"It thus follows from theorem REF that $ h^2\\vert _U = h\\cdot (f_U)^\\ast (c_1) + (f_U)^\\ast (c_2)\\ \\ \\ \\hbox{in}\\ A^2(U)\\ ,$ for some $c_i\\in A^i(V)$ , $i=1,2$ .",
"Let $\\bar{c}_i\\in A^i(G(2,5))$ be elements such that $\\bar{c}_i\\vert _U=c_i$ for $i=1,2$ .",
"The localization exact sequence (plus the fact that $D$ is irreducible of codimension 2 in $M$ ) then implies that $ h^2 = h\\cdot f^\\ast (\\bar{c}_1) + f^\\ast (\\bar{c}_2) + \\mu D \\ \\ \\ \\hbox{in}\\ A^2(M) \\ ,$ for some $\\mu \\in \\mathbb {Q}$ .",
"Let us assume, for a moment, that $\\mu =0$ .",
"Then relation (REF ) would imply in particular that $ h^2\\vert _D = \\Bigl (h\\cdot f^\\ast (\\bar{c}_1) + f^\\ast (\\bar{c}_2) \\Bigr )\\vert _D\\ \\ \\ \\hbox{in}\\ A^2(D)\\ .$ But this is absurd, for the right hand side maps to 0 under pushforward $(f_X)_\\ast $ whereas the left hand side maps to a non–zero multiple of $[X]\\in A_3(X)$ under pushforward $(f_X)_\\ast $ .",
"It follows that $\\mu \\ne 0$ .",
"Relation (REF ) proves the lemma; it suffices to define $\\lambda :=1/\\mu $ and $d_i:= \\lambda \\, \\bar{c}_i\\in A^i(G(2,5))$ , $i=1,2$ .",
"Armed with these lemmas, we are now ready to prove the claim REF (and hence close the proof of the theorem).",
"Let $d\\in \\mathbb {Z}$ be the non–zero integer such that $(f_X)_\\ast (h^2)=d[X]$ in $A_3(X)$ .",
"We define correspondences $\\Gamma _1, \\Xi _1$ as follows: $ \\begin{split} \\Gamma _1&:= \\Gamma _i\\circ {}^t \\Gamma _{f_X} \\ \\ \\ \\in \\ A^5(X\\times M)\\ ,\\\\\\Xi _1&:={1\\over d\\lambda } \\, {}^t \\Gamma _1 = {1\\over d\\lambda }\\, \\Gamma _{f_X}\\circ {}^t \\Gamma _i \\ \\ \\ \\in \\ A^{5}(M\\times X)\\ \\\\\\end{split} $ (where $\\lambda $ is the non–zero constant of lemma REF ).",
"Let us show these correspondences $\\Gamma _1, \\Xi _1$ verify the relations (REF ).",
"By construction, the composition $\\Xi _1\\circ \\Gamma _1$ acts on Chow groups in the following way: $ (\\Xi _1\\circ \\Gamma _1)_\\ast \\colon \\ \\ \\ A_i^{}(X)\\xrightarrow{} A_{i+2}^{}(D) \\xrightarrow{} A_{i+2}^{}(M)\\xrightarrow{} A_i^{}(D)\\xrightarrow{} A_i^{}(X)\\ .$ Thanks to lemma REF , the map $ {1\\over d\\lambda }i^\\ast i_\\ast \\colon \\ \\ A_{i+2}(D)\\ \\rightarrow \\ A_i(D) $ is the same as intersecting with $ {1\\over d}\\, \\Bigl ( h^2 + {1\\over \\lambda }(f_X)^\\ast (d_1\\vert _X)\\cdot h + {1\\over \\lambda }(f_X)^\\ast (d_2\\vert _X)\\Bigr )\\ \\ \\ \\in \\ A^2(D)\\ .$ In particular, if $b\\in A_i(X)$ then $ {1\\over d\\lambda }i^\\ast i_\\ast (f_X)^\\ast (b) ={1\\over d}\\, \\Bigl (h^2\\circ (f_X)^\\ast (b) + {1\\over \\lambda }h\\circ (f_X)^\\ast (b\\cdot d_1\\vert _X) +{1\\over \\lambda }(f_X)^\\ast (b\\cdot d_2\\vert _X)\\Bigr ) \\ \\ \\ \\hbox{in}\\ A_{i}(D)\\ .$ But then, it follows that $ \\begin{split} (f_X)_\\ast {1\\over d\\lambda }i^\\ast i_\\ast (f_X)^\\ast (b) &= {1\\over d}\\, (f_X)_\\ast \\Bigl (h^2\\circ (f_X)^\\ast (b)+ {1\\over \\lambda }h\\circ (f_X)^\\ast (b\\cdot d_1\\vert _X) +{1\\over \\lambda }(f_X)^\\ast (b\\cdot d_2\\vert _X) \\Bigr )\\\\&= {1\\over d} (f_X)_\\ast \\bigl (h^2\\circ (f_X)^\\ast (b)\\bigr )\\\\&= {1\\over d} (f_X)_\\ast (h^2)\\cdot b =b\\ \\ \\ \\hbox{in}\\ A_{i}(X)\\ .\\\\\\end{split}$ That is, $\\Xi _1\\circ \\Gamma _1$ acts as the identity on $A_i(X)$ , which proves the first half of the claimed result (REF ).",
"It remains to prove the second half of (REF ).",
"The composition $\\Gamma _1\\circ \\Xi _1$ acts on Chow groups in the following way: $ (\\Gamma _1\\circ \\Xi _1)_\\ast \\colon \\ \\ \\ A_i^{hom}(M)\\xrightarrow{}A_{i-2}^{hom}(D)\\xrightarrow{} A_{i-2}^{hom}(X)\\xrightarrow{}A_i^{hom}(D)\\xrightarrow{} A_i^{hom}(M)\\ .$ Let $a\\in A_i^{hom}(M)$ .",
"In view of lemma REF , we may suppose $a=i_\\ast (f_X)^\\ast (b)$ , for some $b\\in A_{i-2}^{hom}(X)$ .",
"But we have just checked that $(\\Xi _1\\circ \\Gamma _1)_\\ast (b)=b$ for any $b\\in A_{i-2}(X)$ , which means that $ (f_X)_\\ast {1\\over d\\lambda } i^\\ast (a)= (f_X)_\\ast {1\\over d\\lambda } i^\\ast i_\\ast (f_X)^\\ast (b) = b\\ \\ \\ \\hbox{in}\\ A_{i-2}^{hom}(X)\\ .$ Applying $i_\\ast (f_X)^\\ast $ on both sides, we conclude that $ (\\Gamma _1\\circ \\Xi _1)_\\ast (a)= i_\\ast (f_X)^\\ast (f_X)_\\ast {1\\over d\\lambda } i^\\ast i_\\ast (f_X)^\\ast (b) = i_\\ast (f_X)^\\ast b=a\\ \\ \\ \\hbox{in}\\ A_{i}^{hom}(M)\\ ,$ i.e., $\\Gamma _1\\circ \\Xi _1$ acts as the identity on $A^{hom}_i(M)$ as claimed.",
"We have now established the equalities (REF ), and so we have proven the first half of claim REF .",
"The second half of claim REF (i.e., the existence of the isomorphism $\\Gamma _2$ ) is proven by the same argument, the only difference being that $X$ and $G(2,5)$ should be replaced by $Y$ resp.",
"$G(3,5)$ .",
"Remark 4.6 It would be interesting to refine theorem REF to an isomorphism with $\\mathbb {Z}$ –coefficients.",
"Is it true that there are isomorphisms $ \\ \\ \\ A^i_{}(X)_{\\mathbb {Z}}\\ \\xrightarrow{}\\ A^i_{}(Y)_{\\mathbb {Z}}\\ \\ \\ \\forall i $ of Chow groups with $\\mathbb {Z}$ –coefficients ?",
"The problem, in proving this, is that the fibration result (theorem REF ) is a priori only valid for (higher) Chow groups with rational coefficients.",
"Remark 4.7 It would also be interesting to prove theorem REF for a dual pair $(X,Y)$ of Calabi–Yau threefolds in the family $\\mathcal {X}_{25}$ of [8], [27].",
"In the absence of a nice diagram like (REF ) linking $X$ and $Y$ , this seems considerably more difficult than theorem REF ."
],
[
"A corollary",
"Corollary 5.1 Let $X,Y$ be the Calabi–Yau threefolds constructed as in [17].",
"Let $M$ be any smooth projective variety.",
"Then there are isomorphisms $ N^j H^i(X\\times M,\\mathbb {Q})\\cong N^j H^i(Y\\times M,\\mathbb {Q})\\ \\ \\ \\hbox{for\\ all\\ }i,j\\ .$ (Here, $N^\\ast $ denotes the coniveau filtration [6].)",
"Theorem REF implies there is an isomorphism of Chow motives $h(X\\times M)\\cong h(Y\\times M)$ .",
"As the cohomology and the coniveau filtration only depend on the motive [2], [32], this proves the corollary.",
"Remark 5.2 It is worth noting that for any derived equivalent threefolds $X,Y$ , there are isomorphisms $ N^j H^i(X,\\mathbb {Q})\\cong N^j H^i(Y,\\mathbb {Q})\\ \\ \\ \\hbox{for\\ all\\ }i,j\\ ;$ this is proven in [1].",
"-1mmAcknowledgements This note was written during a stay at the Schiltigheim Math Research Institute.",
"Thanks to its director, Mrs. Ishitani, for running the institute with iron hands gloved in velvet."
]
] | 1808.08338 |
[
[
"K-stability, Futaki invariants and cscK metrics on orbifold resolutions"
],
[
"Abstract In this paper we compute the Futaki invariant of adiabatic Kaehler classes on resolutions of Kaehler orbifolds with isolated singularities.",
"Combined with previous existence results of extremal metrics by Arezzo-Lena-Mazzieri, this gives a number of new existence and non-existence results for cscK metrics."
],
[
"Introduction",
"In this paper we address the question of existence (and non-existence) of constant scalar curvature Kähler metrics (cscK from now on) in adiabatic Kähler classes on resolutions of compact cscK orbifolds with isolated singularities.",
"Form a purely conceptual point of view the basic existence result for extremal Kähler metrics proved in [2] can be reinterpreted in the following form in the spirit of Szekelyhidi's work on blow ups of smooth points [16], [17]: Theorem 1.1 Let $M$ be a Kähler orbifold of dimension $m$ with finite singular set $S \\subset M$ , and let $\\pi : M^{\\prime } \\rightarrow M$ be a resolution of singularities with local model $\\pi _p : X_p \\rightarrow \\mathbf {C}^m/\\Gamma _p$ at each $p \\in S$ .",
"Assume that $M$ admits a Kähler metric $\\omega $ with constant scalar curvature and that each $X_p$ admits a scalar flat ALE Kähler metric $\\eta _p$ .",
"Then there exists $\\varepsilon _0 >0$ such that for all $ \\varepsilon < \\varepsilon _0$ , the following are equivalent $M^{\\prime }$ has a Kcsc metric in the class $\\pi ^{*}[\\omega ]+ \\sum _{p\\in S}\\varepsilon [ \\eta _p ]$ $(M^{\\prime }, \\pi ^{*}[\\omega ]+ \\sum _{p\\in S}\\varepsilon [ \\eta _p] )$ is $K$ -stable.",
"$1 \\Rightarrow 2$ is proved in [17], while $2\\Rightarrow 1$ is a simple consequence of [2].",
"While this result settles the celebrated Tian-Yau-Donaldson Conjecture [7], [8] for these type of manifolds and classes, because of the known difficulty in checking $K$ -stability for a polarized manifold, it remains of great interest to have some effective method to give some geometric conditions on $S$ which guarantee the existence of cscK metrics.",
"This is the primary aim of this paper.",
"In [1] and [2] partial results have been obtained in this direction by performing a careful analysis of the PDE, very much in the spirit of the analogue results of Arezzo-Pacard [3], [4] for blow ups of smooth points.",
"This approach produces a variety of sufficient conditions for the existence of a cscK metric, all of which follow from the more general results of the present work.",
"The main result of this paper, Theorem REF , is the computation of the Futaki invariant of adiabatic Kähler classes on resolution of singularities in terms of corresponding objects on the base orbifold and the geometry of $X_p$ .",
"In fact, the nonuniqueness of the resolution one decides to consider, prevents from using the algebraic techniques already employed in the analogue situation for the blow ups of smooth points by many authors (Stoppa [15], [6], Odaka [14] and Szekelyhidi [16]).",
"What the PDE analysis showed is that a critical difference in the behaviour of this problem comes from the ADM mass of the local model of the resolution.",
"While it has been longly known how to relate this number to the behaviour at infinity of ALE metrics, only very recently Hein-LeBrun [10] have discovered some very elegant interpretation of this quantity in purely cohomological terms for scalar flat metrics.",
"Objects coming into the computation of the Futaki invariant are different from theirs, yet their work has been a key source of inspiration to bypass the problem of non-uniqueness of the resolution.",
"A number of consequences follows from our Theorem REF both getting a new proof of the results of [1] and [2], but more importantly of new existence and various nonexistence results, which are discussed in details in Section below.",
"Using this approach, we can distinguish three different situations (it is worth recalling for the convenience of the reader that an ALE manifold is allowed to have zero ADM mass without being isometric to the flat Euclidean space, as pointed out in [12]): For all $p\\in S$ , the local model $X_p$ has a scalar flat metric with zero ADM mass; There exist $p,q\\in S$ such that the local model $X_p$ has a scalar flat metric with zero ADM mass, the local model $X_q$ has a scalar flat metric with non-zero ADM mass, and the adiabatic classes have the same scales of volumes of exceptional divisors; There exist $p,q\\in S$ such that local model $X_p$ has a scalar flat metric with zero ADM mass, the local model $X_q$ has a scalar flat metric with non-zero ADM mass, and the adiabatic classes have different scales of volumes of exceptional divisors.",
"In each of these cases we give a sufficient condition which generalizes the ones found in [1] and [2] in terms of the position of the singular points to be resolved, which we prove to be essentially also necessary in Theorem REF .",
"This is done in Theorems REF , REF and REF respectively."
],
[
"The Futaki invariant of an orbifold resolution",
"Let $(M,\\omega )$ be a compact Kähler orbifold of complex dimension $m$ with finite singular set $S \\subset M$ .",
"This means that $M$ is a compact Hausdorff topological space endowed with a structure of $n$ -dimensional complex manifold on the subset $M\\setminus S$ such that for each singular point $p \\in S$ there exist the following data: a neighborhood $U_p$ intersecting $S$ just at $p$ , a non-trivial finite subgroup $\\Gamma _p \\subset U(m)$ , a homeomorphism between $U_p$ and a ball $B(r)/\\Gamma _p \\subset \\mathbf {C}^m/\\Gamma _p$ which restricts to a biholomorphism between $U_p \\setminus \\lbrace p\\rbrace $ and the punctured ball $B^{\\prime }(r)/\\Gamma _p$ Moreover, $\\omega $ restricts to a Kähler form of a genuine Kähler metric on $M \\setminus S$ , and the restriction of $\\omega $ to $U_p$ lifts to a $\\Gamma _p$ -invariant Kähler form on the ball $B(r) \\subset \\mathbf {C}^m$ .",
"The quotient $\\mathbf {C}^m/\\Gamma _p$ is called the local model for the singularity at $p$ .",
"We stress that different singular points may have different local models.",
"Finally note that, in principle, the radius $r$ may depend on $p$ , but taking the minimum as $p$ varies on $S$ we can suppose that $r$ is indeed independent of the point."
],
[
"Resolution of orbifold singularities",
"Let $p \\in S$ be a singular point of $M$ , and let $\\pi _p : X_p \\rightarrow \\mathbf {C}^m/\\Gamma _p$ be a resolution of singularities of the local model at $p$ .",
"In view of our applications, we will always assume that $X_p$ admits a Kähler metric.",
"By definition, $\\pi _p$ is a proper birational morphism from a $m$ -dimensional complex manifold $X_p$ to $\\mathbf {C}^m/\\Gamma _p$ which restricts to a biholomorphism on the complement of $\\pi _p^{-1}(0)$ .",
"It follows by definition that $\\pi _p^{-1}(0)$ is a union of compact complex submanifolds of $X$ .",
"The biholomorphism between $U_p \\setminus \\lbrace p\\rbrace $ and the punctured ball $B^{\\prime }(r)/\\Gamma _p$ existing by definition of complex orbifold, and the fact that $\\pi _p$ is a biholomorphism on a neighborhood of $\\partial B(r)/ \\Gamma _p$ also allow to replace each neighborhood $U_p \\subset M$ with the resolved ball $\\pi _p^{-1} (B(r)/\\Gamma _p)$ and obtain a complex manifold $M^{\\prime }$ and a resolution of singularities $\\pi : M^{\\prime } \\rightarrow M.$ This map collects all maps $\\pi _p$ as $p$ varies in $S$ .",
"More precisely, $\\pi $ acts on $\\pi _p^{-1}(B(r)/\\Gamma _p)$ as the composition of $\\pi _p$ together with the homeomorphism from $B(r)/\\Gamma _p$ to $U_p$ .",
"Moreover $\\pi $ is the identity on the complement of $\\pi ^{-1}(U)$ , where $U$ is the union of all $U_p$ as $p$ varies in $S$ .",
"In particular $\\pi $ turns out to be a biholomorphism when restricted to the complement of $\\pi ^{-1}(S)$ ."
],
[
"A Kähler metric on $M^{\\prime }$",
"In this subsection, we construct a Kähler metric on $M^{\\prime }$ which is, in some respect, a deformation of the Kähler metric $\\omega $ on $M$ .",
"For any $p \\in S$ , let $\\eta _p$ be a Kähler metric on the model resolution $X_p$ of the form $\\eta _p = \\xi _p + dd^c \\varphi _p,$ where $\\xi _p$ is a $(1,1)$ -form supported in $\\pi _p^{-1}(B(r)/\\Gamma _p)$ , and $\\varphi _p$ is a smooth function.",
"Since we constructed $M^{\\prime }$ by replacing each singular ball $U_p$ with the resolved ball $\\pi _p^{-1}(B(r)/\\Gamma _p)$ , we can think of each $\\xi _p$ as a $(1,1)$ -form on $M^{\\prime }$ .",
"Thus, for all real $\\varepsilon $ , we can consider the following $(1,1)$ -form on $M^{\\prime }$ $\\omega _\\varepsilon = \\pi ^* \\omega + \\varepsilon \\sum _{p\\in S} \\xi _p.$ Note that $\\omega _\\varepsilon $ defines a Kähler metric on $M^{\\prime }$ for all $\\varepsilon >0$ sufficiently small.",
"This can be seen by considering the restriction of $\\omega $ on $\\pi ^{-1}(U)$ and on $M^{\\prime } \\setminus \\pi ^{-1}(U)$ .",
"The latter is positive since there the map $\\pi $ restricts to a biholomorphism and $\\xi $ vanishes.",
"Thus it remains to check that for $\\varepsilon $ sufficiently small $\\omega _\\varepsilon $ is positive around the resolution $\\pi ^{-1}(U_p)$ of any singular point $p \\in S$ .",
"Over $\\pi ^{-1}(U_p)$ the form $\\omega _\\varepsilon $ restricts to $\\pi _p^*\\omega + \\varepsilon \\xi _p$ (here, for ease of notation, we wrote $\\omega $ instead of the pullback of $\\omega |_{U_p}$ to the ball $B(r)/\\Gamma _p$ ).",
"Since $\\omega $ comes from a $\\Gamma _p$ -invariant form on $B(r)$ , we can suppose $\\omega = dd^c h_p$ for some smooth function $h_p$ on the ball $B(r)/\\Gamma _p$ .",
"As a consequence, on $\\pi ^{-1}(U_p)$ we have $\\omega _\\varepsilon = dd^c (\\pi _p^* h_p - \\varepsilon \\varphi _p) + \\varepsilon \\eta _p.$ Note that $\\eta _p$ is positive in any neighborhood of the exceptional set $\\pi _p^{-1}(p)$ .",
"On the other hand, for $\\varepsilon $ sufficiently small the function $\\pi _p^* h_p - \\varepsilon \\varphi _p$ is plurisubharmonic on the complement of the exceptional set.",
"This shows that $\\omega _\\varepsilon $ is positive on any $\\pi ^{-1}(U_p)$ provided $\\varepsilon $ is sufficiently small, as claimed."
],
[
"Pushing down vector fields",
"In this section we show that any holomorphic vector field on $M^{\\prime }$ induces a holomorphic vector field on $M$ .",
"If the vector field on $M^{\\prime }$ is also Hamiltonian, the induced vector field on $M$ is Hamiltonian too.",
"Lemma 2.1 Any holomorphic vector field $V$ on $M^{\\prime }$ descends to a holomorphic vector field $\\pi _*V$ on $M$ which vanishes at all points of $S$ .",
"Since $\\pi $ is a biholomorphism on the complement of $\\pi ^{-1}(S)$ , pushing down the restriction of $V$ to that set defines a vector field $V^{\\prime }$ on $M \\setminus S$ .",
"Given $p \\in S$ , the restriction of $V^{\\prime }$ to $U_p \\setminus \\lbrace p\\rbrace $ lifts to a $\\Gamma _p$ -invariant vector field on the punctured ball $B^{\\prime }(r)$ of $\\mathbf {C}^m$ .",
"By Hartog's theorem such a vector field extends to a holomorphic vector field on the whole ball $B(r)$ .",
"Of course such a vector field is $\\Gamma _p$ -invariant, and so it gives a holomorphic vector field on $U_p$ which is equal to $V^{\\prime }$ on $U_p \\setminus \\lbrace p\\rbrace $ .",
"Therefore one ends up with a holomorphic vector field $\\pi _*V$ on $M$ .",
"It remains to show that $\\pi _*V$ vanishes at any $p \\in S$ .",
"To this end, note that the fiber $\\pi ^{-1}(p)$ is a union of compact complex submanifolds.",
"Therefore $V$ must be tangent to it, and consequently $\\pi _*V$ must tend to zero as approaching to $p$ .",
"By continuity we can then conclude that $\\pi _*V$ actually vanishes at $p$ .",
"Lemma 2.2 If $V$ is a holomorphic vector field on $M^{\\prime }$ and is Hamiltonian with respect to $\\omega _\\varepsilon $ , then $\\pi _*V$ is Hamiltonian with respect to $\\omega $ on $M$ .",
"Moreover, if $u_\\varepsilon $ and $u$ are Hamiltonian potentials for $V$ and $\\pi _*V$ respectively, then one has $u_\\varepsilon = \\pi ^*u + \\varepsilon \\sum _{p \\in S} u_p + c(\\varepsilon ),$ where $u_p$ is a smooth function supported in $\\pi ^{-1}(U_p)$ satisfying $ d u_p = i_V \\xi _p $ , and $c(\\varepsilon )$ is a constant.",
"On the complement of $\\pi ^{-1}(U)$ the Kähler form $\\omega _\\varepsilon $ is equal to $\\omega $ and the vector field $V$ is equal to $\\pi _*V$ .",
"Therefore the Hamiltonian potential $u_\\varepsilon $ of $V$ restricts to a function on $M^{\\prime } \\setminus \\pi ^{-1}(U)$ , say $u$ , which does not depend on $\\varepsilon $ and is a Hamiltonian potential for $\\pi _*V$ with respect to $\\omega $ on $M \\setminus U$ .",
"Now let $p \\in S$ .",
"Identifying $\\pi ^{-1}(U_p)$ with the resolved ball $\\pi _p^{-1} (B(r)/\\Gamma _p)$ in the local model $X_p$ , one can write $\\omega _\\varepsilon = \\pi _p^* dd^c h_p + \\varepsilon \\xi _p,$ where $h_p$ is a Kähler potential for $\\omega $ in the ball $B(r)/\\Gamma _p$ .",
"Since, by hypothesis, $V$ is Hamiltonian with potential $u_\\varepsilon $ , one has $d u_\\varepsilon = i_V dd^c \\pi _p^*h_p + \\varepsilon i_V \\xi _p.$ On the other hand, $V$ is holomorphic, therefore Cartan formula yields $d u_\\varepsilon = d^c V(\\pi _p^*h_p) +\\frac{1}{4} d JV (\\pi _p^*h_p) + \\varepsilon i_V \\xi _p,$ whence it follows that $i_V \\xi _p = d u_p - \\frac{1}{\\varepsilon }d^c V(\\pi _p^*h_p),$ for some smooth function $u_p$ .",
"Note that the last summand does not depend on $\\xi _p$ , but it is forced to vanish where $\\xi _p$ does.",
"Therefore, by arbitrariness of the metric $\\eta _p$ we started with, it must vanish everywhere.",
"As a consequence the support of $d u_p$ is contained in the support of $\\xi _p$ .",
"In particular, up to adding a suitable constant, we can suppose that $u_p$ is supported in $\\pi ^{-1}(U_p)$ .",
"Thus $u_p$ has all the properties stated above.",
"Moreover we proved that on $\\pi ^{-1}(U_p)$ it holds $u_\\varepsilon = \\frac{1}{4} JV (\\pi _p^*h_p) + \\varepsilon u_p + c(p,\\varepsilon ),$ where $c(p,\\varepsilon )$ is a constant.",
"Since the support of $\\xi _p$ is compactly contained in the resolved ball $\\pi _p^{-1}(B(r)/\\Gamma _p)$ , on a neighborhood of the boundary $\\pi ^{-1}(\\partial U_p)$ it holds $u_\\varepsilon = \\pi _p^* \\left( \\frac{1}{4} (J\\pi _*V) (h_p) + c(p,\\varepsilon ) \\right).$ Therefore $u$ extends to $\\frac{1}{4} (J\\pi _*V) (h_p) + c(p,\\varepsilon )$ on $U_p$ .",
"Finally note that it holds $du=i_{\\pi _*V}\\omega $ , that is $u$ is a Hamiltonian potential for $\\pi _*V$ .",
"Since Hamiltonian potentials are defined just up to an additive constant, equation (REF ) then follows."
],
[
"The Futaki invariant",
"Given a holomorphic vector field $V$ on the resolution $M^{\\prime }$ , and supposing that $V$ is Hamiltonian with respect to $\\omega _\\varepsilon $ with potential $u_\\varepsilon $ , one can form the Futaki invariant $\\operatorname{Fut}(V,\\omega _\\varepsilon )= \\int _{M^{\\prime }} (u_\\varepsilon - \\underline{u_\\varepsilon }) \\frac{\\rho _\\varepsilon \\wedge \\omega _\\varepsilon ^{m-1}}{(m-1)!",
"},$ where $\\rho _\\varepsilon $ is the Ricci form of $\\omega _\\varepsilon $ , and $\\underline{u_\\varepsilon } = \\int u_\\varepsilon \\omega _\\varepsilon ^m / \\int \\omega _\\varepsilon ^m$ is the mean value of $u_\\varepsilon $ with respect to $\\omega _\\varepsilon $ .",
"On the other hand, thanks to Lemmata REF and REF , $V$ descends to a holomorphic vector field $\\pi _*V$ on $M$ wich is Hamiltonian with respect to $\\omega $ with potential, say, $u$ .",
"Thus one can also consider the Futaki invariant $\\operatorname{Fut}(\\pi _*V,\\omega )= \\int _M (u - \\underline{u}) \\frac{\\rho \\wedge \\omega ^{m-1}}{(m-1)!",
"},$ where $\\rho $ is the Ricci form of $\\omega $ , and $\\underline{u} = \\int u \\,\\omega ^m / \\int \\omega ^m$ .",
"The Futaki invariants $\\operatorname{Fut}(V,\\omega _\\varepsilon )$ and $\\operatorname{Fut}(\\pi _*V,\\omega )$ are relate by the following Theorem 2.3 As $\\varepsilon \\rightarrow 0$ one has $\\operatorname{Fut}(V,\\omega _\\varepsilon )= \\operatorname{Fut}(\\pi _*V,\\omega )+ \\varepsilon ^{m-1} \\sum _{p \\in S} (u(p) - \\underline{u}) \\int _{X_p} \\frac{\\rho _p \\wedge \\xi _p^{m-1}}{(m-1)!}",
"\\\\- \\varepsilon ^m \\sum _{p \\in S} \\left(\\underline{s} (u(p) - \\underline{u}) + \\Delta u(p)\\right) \\int _{X_p} \\frac{\\xi _p^m}{m!}",
"+ O(\\varepsilon ^{m+1}).$ where $\\rho _p$ is the Ricci form of the chosen ALE Kähler metric $\\eta _p$ on the model resolution $X_p$ , and $\\underline{s} = m \\int \\rho \\wedge \\omega ^{m-1} / \\int \\omega ^m$ is the mean scalar curvature of $\\omega $ .",
"The Futaki invariant of the vector field $V$ can be written as $\\operatorname{Fut}(V,\\omega _\\varepsilon )= \\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\frac{(\\omega _\\varepsilon + u_\\varepsilon )^m}{m!",
"}- \\underline{u_\\varepsilon } \\int _{M^{\\prime }} \\rho _\\varepsilon \\wedge \\frac{\\omega _\\varepsilon ^{m-1}}{(m-1)!",
"},$ where $\\Delta _\\varepsilon $ denotes the Laplacian of the Kähler metric on $M^{\\prime }$ associated to $\\omega _\\varepsilon $ .",
"The first integral and the average of $u_\\varepsilon $ perhaps could be calculated using equivariant cohomology theory.",
"However one can avoid that theory and prove the statement by means of more elementary arguments.",
"In order to understand the formula above, note that $\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon $ and $\\omega _\\varepsilon + u_\\varepsilon $ are non-homogeneous differential forms on $M^{\\prime }$ .",
"Their wedge product is the usual one, and so the integrand is a sum of even degree differential forms.",
"The integral of such a form is, by definition, the integral of its $2m$ -degree component.",
"Now consider the differential operator $d_V = d - i_V$ acting on differential froms on $M^{\\prime }$ .",
"The fact that $u_\\varepsilon $ is a Hamiltonian potential for $V$ with respect to $\\omega _\\varepsilon $ can be stated as $d_V (\\omega _\\varepsilon + u_\\varepsilon ) = 0.$ In other words, $\\omega _\\varepsilon + u_\\varepsilon \\in \\Omega ^*(M)$ is a $d_V$ -closed differential form.",
"After expanding $\\omega _\\varepsilon + u_\\varepsilon = \\pi ^*(\\omega +u) + \\varepsilon \\sum _{p\\in S} \\xi _p+u_p,$ one sees that $\\pi ^*(\\omega +u)$ and $\\xi _p+u_p$ are $d_V$ -closed as well.",
"Finally, by a standard local calculation one can check that $d_V (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) = 0.$ Now pick a singular point $p \\in S$ and let $\\pi ^{-1}(p)$ be the exceptional divisor over $p$ .",
"The proof of the statement of the Theorem rests essentially on the following Claim Any $d_V$ -closed differential form $\\alpha \\in \\Omega ^*(M^{\\prime })$ which restricts to the zero form on the exceptional divisor $\\pi ^{-1}(p)$ satisfies $\\int _{M^{\\prime }} \\alpha \\wedge (\\xi _p + u_p) = 0.$ In order to prove the claim note that by Poincaré-Lelong equation one can find a path $F_t$ of smooth functions on $M^{\\prime }$ , with $t>0$ , such that $\\xi _p + dd^c F_t$ weekly converges to a current supported on $\\pi ^{-1}(p)$ as $t \\rightarrow 0$ .",
"A moment's thought should show that $\\xi _p + u_p + d_Vd^c F_t$ also converges to the same current.",
"On the other hand, note that by Stokes' Theorem the integral of any $d_V$ -closed form on $M^{\\prime }$ vanishes.",
"As a consequence one has $\\int _{M^{\\prime }} \\alpha \\wedge (\\xi _p + u_p) = \\int _{M^{\\prime }} \\alpha \\wedge (\\xi _p + u_p + d_V d^c F_t) \\rightarrow 0$ as $t \\rightarrow 0$ thanks to the hypothesis that $\\alpha $ restricts to the zero form on $\\pi ^{-1}(p)$ .",
"Now we can proceed with the proof of the Theorem.",
"We need to expand (REF ) in powers of $\\varepsilon $ .",
"First of all consider the average $\\underline{u_\\varepsilon } = \\int u_\\varepsilon \\omega _\\varepsilon ^m / \\int \\omega _\\varepsilon ^m$ .",
"Note that $(m+1) \\int _{M^{\\prime }} u_\\varepsilon \\omega _\\varepsilon ^m = \\int _{M^{\\prime }} (\\omega _\\varepsilon + u_\\varepsilon )^{m+1}.$ Substituting (REF ) yields $\\int _{M^{\\prime }} (\\omega _\\varepsilon + u_\\varepsilon )^{m+1}= \\int _M (\\omega +u)^{m+1} + \\sum _{\\ell =1}^m \\varepsilon ^\\ell \\binom{m+1}{\\ell } \\sum _{p\\in S} \\int _{M^{\\prime }} \\pi ^*(\\omega +u)^{m+1-\\ell } \\wedge (\\xi _p + u_p)^\\ell \\\\+ \\varepsilon ^{m+1} \\sum _{p\\in S} \\int _{M^{\\prime }} (\\xi _p + u_p)^{m+1}.$ Focus on the middle summands of the right hand side.",
"Note that for all $1 \\le \\ell \\le m$ and $p \\in S$ the differential form $\\alpha = \\pi ^*(\\omega +u)^{m+1-\\ell } \\wedge (\\xi _p + u_p)^{\\ell -1} - u(p)^{m+1-\\ell } (\\xi _p + u_p)^{\\ell -1}$ satisfies the hypotheses of the Claim, whence it follows $\\int _{M^{\\prime }} \\pi ^*(\\omega +u)^{m+1-\\ell } \\wedge (\\xi _p + u_p)^\\ell = u(p)^{m+1-\\ell } \\int _{M^{\\prime }} (\\xi _p + u_p)^\\ell .$ Since $\\ell $ runs from 1 to $m$ , the right hand side integral vanishes unless $\\ell =m$ , in which case it reduces to the integral of $\\xi _p^m$ .",
"Therefore, substituting in (REF ) and (REF ) we get the expansion $\\int _{M^{\\prime }} u_\\varepsilon \\omega _\\varepsilon ^m= \\int _M u \\, \\omega ^m + \\varepsilon ^m \\sum _{p\\in S} u(p) \\int _{M^{\\prime }} \\xi _p^m+ O(\\varepsilon ^{m+1}).$ In order to get the expansion of $\\underline{u_\\varepsilon }$ we need to divide the expression above by the total volume of $\\omega _\\varepsilon $ .",
"Recalling that $\\omega _\\varepsilon = \\pi ^*\\omega + \\varepsilon \\sum \\xi _p$ , one has $\\int _{M^{\\prime }} \\omega _\\varepsilon ^m= \\int _M \\omega ^m + \\sum _{\\ell =1}^m \\varepsilon ^\\ell \\binom{m}{\\ell } \\sum _{p\\in S} \\int _{M^{\\prime }} \\pi ^*\\omega ^{m-\\ell } \\wedge \\xi _p^\\ell .$ Since $\\xi _p$ is supported on $\\pi ^{-1}(U_p)$ and $\\pi ^*\\omega $ is exact on there, Stokes' Theorem yields $\\int _{M^{\\prime }} \\omega _\\varepsilon ^m= \\int _M \\omega ^m + \\varepsilon ^m \\sum _{p\\in S} \\int _{X_p} \\xi _p^m.$ Note that we replaced the integral of $\\xi _p$ over $M^{\\prime }$ with the integral over the model resolution $X_p$ .",
"This is possible since the support of $\\xi _p$ is contained in $\\pi ^{-1}(U_p)$ , which in turn we have identified with a neighborhood of the exceptional divisor of $X_p$ .",
"Dividing (REF ) by (REF ) finally gives $\\underline{u_\\varepsilon }= \\underline{u} + \\varepsilon ^m \\sum _{p\\in S} (u(p) - \\underline{u}) \\frac{\\int _{X_p} \\xi _p^m}{\\int _M \\omega ^m}+ O(\\varepsilon ^{m+1}).$ Now we pass to consider the total scalar curvature of $\\omega _\\varepsilon $ .",
"Arguing as above, after susbstituting $\\omega _\\varepsilon = \\pi ^*\\omega + \\varepsilon \\sum \\xi _p$ and applying Stokes' Theorem, one gets $\\int _{M^{\\prime }} \\rho _\\varepsilon \\wedge \\omega _\\varepsilon ^{m-1}= \\int _{M^{\\prime }} \\rho _\\varepsilon \\wedge \\pi ^*\\omega ^{m-1}+ \\varepsilon ^{m-1} \\sum _{p \\in S} \\int _{M^{\\prime }} \\rho _\\varepsilon \\wedge \\xi _p^{m-1}.$ Consider the two summands separately.",
"Adding and subctracting $\\pi ^* \\rho $ in the first summand yields $\\int _{M^{\\prime }} \\rho _\\varepsilon \\wedge \\pi ^*\\omega ^{m-1}= \\int _{M^{\\prime }} \\pi ^*\\rho \\wedge \\pi ^*\\omega ^{m-1} + \\int _{M^{\\prime }} (\\rho _\\varepsilon - \\pi ^* \\rho ) \\wedge \\pi ^*\\omega ^{m-1}.$ The first summand reduces to the integral of $\\rho \\wedge \\omega ^{m-1}$ over $M$ , and the second summand vanishes once again by Stokes' Theorem.",
"Indeed $\\omega _\\varepsilon = \\omega $ on the complement of $U$ .",
"As a consequence $\\rho _\\varepsilon - \\pi ^*\\rho $ vanishes on that set.",
"On the other hand $\\pi ^* \\omega $ is exact on any connected component of $U$ .",
"Therefore $\\int _{M^{\\prime }} \\rho _\\varepsilon \\wedge \\pi ^*\\omega ^{m-1}= \\int _M \\rho \\wedge \\omega ^{m-1}.$ Now consider the second summand of the right hand side of (REF ).",
"Since the support of $\\xi _p$ is contained in $\\pi ^{-1}(U_p)$ , which we identified with a neighborhood of the exceptional divisor of the model resolution $X_p$ , we can consider the Ricci form $\\rho _p$ of the chosen ALE Kähler metric $\\eta _p$ on $X_p$ and thought of $\\rho _p \\wedge \\xi _p^{m-1}$ as a differential form on $M^{\\prime }$ .",
"After noting that $\\rho _\\varepsilon -\\rho _p = dd^c \\log (\\eta _p^m/\\omega _\\varepsilon ^m)$ , Stokes' Theorem yields $\\int _{M^{\\prime }} (\\rho _\\varepsilon -\\rho _p) \\wedge \\xi _p^{m-1} = 0.$ On the other hand, we can also consider $\\rho _p \\wedge \\xi _p^{m-1}$ as a differential form on the local resolution $X_p$ .",
"As a consequence we can rewrite equation above in the form $\\int _{M^{\\prime }} \\rho _\\varepsilon \\wedge \\xi _p^{m-1}= \\int _{X_p} \\rho _p \\wedge \\xi _p^{m-1}.$ Therefore substituting (REF ) together with (REF ) into (REF ) yields $\\int _{M^{\\prime }} \\rho _\\varepsilon \\wedge \\omega _\\varepsilon ^{m-1}= \\int _M \\rho \\wedge \\omega ^{m-1} + \\varepsilon ^{m-1} \\sum _{p\\in S} \\int _{X_p} \\rho _p \\wedge \\xi _p^{m-1}.$ Thanks to (REF ) and (REF ), the second summand of the right hand side of (REF ) expands as $\\underline{u_\\varepsilon } \\int _{M^{\\prime }} \\rho _\\varepsilon \\wedge \\frac{\\omega _\\varepsilon ^{m-1}}{(m-1)!",
"}= \\underline{u} \\int _M \\rho \\wedge \\frac{\\omega ^{m-1}}{(m-1)!",
"}+ \\varepsilon ^{m-1} \\sum _{p\\in S} \\underline{u} \\int _{X_p} \\frac{\\rho _p \\wedge \\xi _p^{m-1}}{(m-1)!}",
"\\\\+ \\varepsilon ^m \\sum _{p\\in S} \\underline{s} (u(p) - \\underline{u}) \\int _{X_p} \\frac{\\xi _p^m}{m!",
"}+ O(\\varepsilon ^{m+1}).$ Finally it remains to consider the first summand of (REF ).",
"Substituting (REF ) yields $\\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\frac{(\\omega _\\varepsilon + u_\\varepsilon )^m}{m!",
"}= \\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\frac{\\pi ^*(\\omega + u)^m}{m!}",
"\\\\+ \\sum _{\\ell =1}^m \\varepsilon ^\\ell \\sum _{p \\in S} \\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\frac{\\pi ^*(\\omega + u)^{m-\\ell }}{(m-\\ell )!}",
"\\wedge \\frac{\\left(\\xi _p + u_p\\right)^\\ell }{\\ell !",
"}.$ Focus on the summands of the second line.",
"Fix $p \\in S$ and suppose $0 < \\ell <m$ , so that the differential form $\\alpha = (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\frac{\\pi ^*(\\omega + u)^{m-\\ell } - u(p)^{m-\\ell }}{(m-\\ell )!}",
"\\wedge \\frac{\\left(\\xi _p + u_p\\right)^{\\ell -1}}{\\ell !",
"}$ satisfies the hypotheses of the Claim above, whence it follows that $\\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\frac{\\pi ^*(\\omega + u)^{m-\\ell }}{(m-\\ell )!}",
"\\wedge \\frac{\\left(\\xi _p + u_p\\right)^\\ell }{\\ell !",
"}= \\frac{u(p)^{m-\\ell }}{(n-\\ell )!}",
"\\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\frac{\\left(\\xi _p + u_p\\right)^\\ell }{\\ell !",
"}.$ If $\\ell <m-1$ , the integrand differential form of the right hand side has no component of degree $2m$ , therefore the integral is zero.",
"On the other hand, for $\\ell = m-1$ , equation above together with (REF ) give $\\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\pi ^*(\\omega + u) \\wedge \\frac{\\left(\\xi _p + u_p\\right)^{m-1}}{(m-1)!",
"}= u(p) \\int _{X_p} \\frac{\\rho _p \\wedge \\xi _p^{m-1}}{(m-1)!",
"}.$ By discussion above, (REF ) reduces to $\\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\frac{(\\omega _\\varepsilon + u_\\varepsilon )^m}{m!",
"}= \\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\frac{\\pi ^*(\\omega + u)^m}{m!}",
"\\\\+ \\varepsilon ^{m-1} \\sum _{p \\in S} u(p) \\int _{X_p} \\frac{\\rho _p \\wedge \\xi _p^{m-1}}{(m-1)!",
"}+ \\varepsilon ^m \\sum _{p \\in S} \\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\frac{\\left(\\xi _p + u_p\\right)^m}{m!",
"}.$ In order to treat the first summand of the right hand side, note that the difference $\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon - \\pi ^*(\\rho - \\Delta u)$ is a $d_V$ -closed differential form on $M^{\\prime }$ which is compactly supported in the union of all $\\pi ^{-1}(U_p)$ as $p$ varies in $S$ .",
"The proof of the Claim above works also replacing the form $\\xi _p + u_p$ with $\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon - \\pi ^*(\\rho - \\Delta u)$ .",
"As a consequence, since $\\pi ^*(\\omega +u)^m$ restricts to a constant function on any exceptional divisor, one then has $\\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\frac{\\pi ^*(\\omega + u)^m}{m!",
"}= \\int _{M^{\\prime }} \\pi ^*(\\rho - \\Delta u) \\wedge \\frac{\\pi ^*(\\omega + u)^m}{m!",
"}.$ Therefore (REF ) simplifies to $\\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\frac{(\\omega _\\varepsilon + u_\\varepsilon )^m}{m!",
"}= \\int _M (\\rho - \\Delta u) \\wedge \\frac{(\\omega + u)^m}{m!",
"}+ \\varepsilon ^{m-1} \\sum _{p \\in S} u(p) \\int _{X_p} \\frac{\\rho _p \\wedge \\xi _p^{m-1}}{(m-1)!}",
"\\\\+ \\varepsilon ^m \\sum _{p \\in S} \\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\frac{\\left(\\xi _p + u_p\\right)^m}{m!",
"}.$ Finally consider the last summand, which can be rewritten in the form $\\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\frac{\\left(\\xi _p + u_p\\right)^m}{m!",
"}= \\int _{M^{\\prime }} \\pi ^*(\\rho - \\Delta u) \\wedge \\frac{\\left(\\xi _p + u_p\\right)^m}{m!}",
"\\\\+ \\int _{M^{\\prime }} \\left(\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon - \\pi ^*(\\rho - \\Delta u)\\right) \\wedge \\frac{\\left(\\xi _p + u_p\\right)^m}{m!",
"}.$ The first summand of the right hand side can be trated once again by the Claim above and the Stokes' Theorem.",
"Indeed the differential form $\\alpha = \\left(\\pi ^*(\\rho - \\Delta u) + \\Delta u(p)\\right) \\wedge \\left(\\xi _p + u_p\\right)^{m-1}$ on $M^{\\prime }$ satisfies the hypothesis of the Claim, whence arguing as above yields $\\int _{M^{\\prime }} \\pi ^*(\\rho - \\Delta u) \\wedge \\frac{\\left(\\xi _p + u_p\\right)^m}{m!",
"}= -\\Delta u (p) \\int _{X_p} \\frac{\\xi _p^m}{m!",
"}.$ On the other hand, the second summand of (REF ) is $O(\\varepsilon )$ , as follows by (REF ).",
"As a consequence, (REF ) reduces to $\\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\left(\\xi _p + u_p\\right)^m= -\\Delta u (p) \\int _{X_p} \\xi _p^m + O(\\varepsilon )$ .",
"Substituting this into (REF ) yields $\\int _{M^{\\prime }} (\\rho _\\varepsilon - \\Delta _\\varepsilon u_\\varepsilon ) \\wedge \\frac{(\\omega _\\varepsilon + u_\\varepsilon )^m}{m!",
"}= \\int _M (\\rho - \\Delta u) \\wedge \\frac{(\\omega + u)^m}{m!}",
"\\\\+ \\varepsilon ^{m-1} \\sum _{p \\in S} u(p) \\int _{X_p} \\frac{\\rho _p \\wedge \\xi _p^{m-1}}{(m-1)!",
"}- \\varepsilon ^m \\sum _{p \\in S} \\Delta u (p) \\int _{X_p} \\frac{\\xi _p^m}{m!}",
"+ O(\\varepsilon ^{m+1}).$ Finally the thesis follows by plugging this and (REF ) into (REF )."
],
[
"ALE resolutions",
"In this section we introduce Kähler metrics on local models having behavior at infinity suitable for applications in next section.",
"Let $\\Gamma $ be a finite subgroup of the unitary group $U(m)$ and suppose that $\\Gamma $ acts freely on the complement of $0 \\in \\mathbf {C}^m$ .",
"The quotient $(\\mathbf {C}^m \\setminus \\lbrace 0\\rbrace )/\\Gamma $ is therefore a complex manifold and the Euclidean metric on $\\mathbf {C}^m$ descend to a Kähler metric $\\eta _0$ on the quotient.",
"Such a Kähler metric serves as a model at infinity for ALE resolutions.",
"By ALE resolution (of $\\mathbf {C}^m/\\Gamma $ ) we mean a non-compact complex manifold $X$ equipped with a complete Kähler metric $\\eta $ satisfying the following requirements: There exists a finite subgroup $\\Gamma \\subset U(m)$ acting freely on $\\mathbf {C}^m \\setminus \\lbrace 0\\rbrace $ and a proper birational morphism $\\pi : X \\rightarrow \\mathbf {C}^m/\\Gamma $ which restricts to a biholomorphism of $X \\setminus \\pi ^{-1}(0)$ onto $(\\mathbf {C}^m \\setminus \\lbrace 0\\rbrace )/ \\Gamma $ .",
"The metric $\\eta $ approximates smoothly the model metric $\\eta _0$ at infinity.",
"More specifically, for all integers $k \\ge 0$ one has $\\nabla ^k (\\pi _*\\eta - \\eta _0) = O(|z|^{2-2m-k}) \\quad \\mbox{ as } |z| \\rightarrow \\infty ,$ where $\\nabla $ denotes the Euclidean connection.",
"In particular, an ALE resolution turns out to be an ALE Kähler manifold subject to a couple of more restricting requirements.",
"Firstly, here we assume that $\\pi $ is a biholomorphism at all smooth points of the quotient $\\mathbf {C}^m / \\Gamma $ (hence the name resolution), wheras an ALE Kähler manifold (with one end) is just required to contain a compact subset $K$ whose complement is biholomorphic to the complement of a ball centered at the origin in $\\mathbf {C}^m / \\Gamma $ .",
"Secondly, ALE Kähler metrics are commonly allowed to have quite permissive fall-off order at infinity.",
"However, in second point above the order $2-2m$ is chosen in accordance with the well-known decay of scalar flat Kähler metrics on $\\mathbf {C}^m / \\Gamma $ [3], being metrics of that kind our main interest for applications.",
"On ALE resolutions one can develop Hodge theory as for compact Kähler manifolds [11].",
"The upshot is that any cohomology class in $H^{1,1}(X)$ can be represented by a closed compactly supported $(1,1)$ -form on $X$ [11].",
"Moreover one can suppose that $\\eta = \\xi + dd^c \\varphi $ for some $(1,1)$ -from $\\xi $ compactly supported around the exceptional locus $\\pi ^{-1}(0)$ and some smooth real function $\\varphi $ on $X$ [11].",
"Clearly $\\xi $ and $\\varphi $ are not uniquely defined.",
"In particular $\\varphi $ can be added by a function in the kernel of $dd^c$ operator.",
"However we rule out this indeterminacy by requiring that $\\varphi - |z|^2/4 = O(|z|^{4-2m})$ for large $z$ .",
"Therefore by (REF ) the derivative $\\nabla ^k(\\varphi - |z|^2/4)$ must be $O(|z|^{4-2m-k})$ for all $k \\ge 0$ .",
"Finally, here we recall an elementary result that will be useful in the following.",
"Lemma 3.1 Let $B(R) \\subset \\mathbf {C}^m$ be the ball centered at zero of radius $R$ .",
"One has $\\int _{\\partial B(R)} d^c|z|^2 \\wedge (dd^c|z|^2)^{m-1} = (4\\pi )^m R^{2m}.$ First of all note that $dd^c|z|^2 = 2i \\sum _{j=1}^m dz_j \\wedge d\\bar{z}_j$ , whence it follows that $(dd^c|z|^2)^m$ is $4^mm!$ times the Euclidean volume form $\\Omega _E$ .",
"Therefore, by Stokes theorem one has $\\int _{\\partial B(R)} d^c|z|^2 \\wedge (dd^c|z|^2)^{m-1} = 4^m m!",
"\\int _{B(R)} \\Omega _E,$ whence the thesis follows by $ \\int _{B(R)} \\Omega _E = \\operatorname{vol}(S^{2m-1})\\int _0^R t^{2m-1}dt = \\frac{2\\pi ^m}{\\Gamma (m)} \\frac{R^{2m}}{2m}$ ."
],
[
"Asymptotic formulae for volume and total scalar curvature",
"Let $m>1$ be an integer and consider the real function $f$ on $(0,+\\infty )$ defined by $f(t) = \\frac{1}{4}t + e \\frac{1-t^{2-m}}{2-m} + ct^{1-m}$ for some real constants $e,c$ .",
"In this section we consider an $m$ -dimensional ALE resolution whose Kähler metric is of the form $\\eta = \\xi + dd^c \\varphi $ with $\\xi $ compactly supported around $\\pi ^{-1}(0)$ and $\\varphi $ satisfying $\\nabla ^k \\varphi = \\nabla ^k f(|z|^2) + O(|z|^{-2m-k}) \\quad \\mbox{ as } |z| \\rightarrow +\\infty $ for all integer $k \\ge 0$ .",
"This ensures that $\\eta $ satisfies the fall-off requirement (REF ) of the third point of definition of ALE resolution.",
"Moreover, note that the second summand of $f$ is chosen so that $f$ depends smoothly (in fact analytically) on the dimension $m$ and for $m=2$ one has $f(t) = t/4 - e\\log (t) + ct^{-1}$ .",
"We shall compute, at least up to some controlled error, the volume and the total scalar curvature of some subsets of $X$ with respect to $\\eta $ .",
"More specifically we shall give an asymptotic formula for the volume and the total scalar curvature of $\\pi ^{-1}(B(R)/\\Gamma )$ for large $R$ .",
"Proposition 3.2 For $R \\rightarrow +\\infty $ one has $\\int _{\\pi ^{-1}(B(R)/\\Gamma )} \\frac{\\eta ^m}{m!}",
"= \\frac{\\pi ^m}{m!|\\Gamma |} R^{2m} - \\frac{4\\pi ^m e}{(m-1)!|\\Gamma |} R^2 - \\frac{4\\pi ^m(c-2e^2R^{4-2m})}{(m-2)!|\\Gamma |} + \\int _X \\frac{\\xi ^m}{m!}",
"+ O(R^{-1})$ (Note that the term of order $R^{4-2m}$ is not infinitesimal just in dimension $m=2$ ).",
"Since $\\eta = \\xi + dd^c \\varphi $ , the volume form of $\\eta $ is given by $\\eta ^m = \\xi ^m + d\\left( \\sum _{\\ell =1}^m \\binom{m}{\\ell } \\xi ^{m-\\ell } \\wedge d^c \\varphi \\wedge (dd^c \\varphi )^{\\ell -1}\\right).$ Recall that $\\xi $ is supported in $K$ , which in turns is compactly contained in $\\pi ^{-1}(B(R)/\\Gamma )$ .",
"Therefore, integrating formula above and applying Stokes' theorem gives $\\int _{\\pi ^{-1}(B(R)/\\Gamma )} \\eta ^m = \\int _X \\xi ^m + \\int _{\\partial B(R)/\\Gamma } d^c \\varphi \\wedge (dd^c \\varphi )^{m-1}.$ Equation (REF ) yields $d^c \\varphi = d^c f + O(|z|^{-2m-1})$ and $dd^c \\varphi = dd^c f + O(|z|^{-2m-2})$ , whence by easy calculations one gets $d^c \\varphi \\wedge (dd^c \\varphi )^{m-1} = (f^{\\prime })^m d^c|z|^2 \\wedge (dd^c|z|^2)^{m-1} + O(|z|^{-2m-1}).$ Integrating over $\\partial B(R)/\\Gamma $ and applying lemma REF gives $\\int _{\\partial B(R)/\\Gamma } d^c \\varphi \\wedge (dd^c \\varphi )^{m-1}= \\frac{(4\\pi )^m R^{2m}f^{\\prime }(R^2)^m}{|\\Gamma |} +O(R^{-1}).$ By (REF ) one calculates $f^{\\prime }(t)^m = 4^{-m}\\left(1 - 4me t^{1-m} - 4m(m-1)(ct^{-m}-2e^2t^{2-2m})\\right) + O(t^{1-2m})$ , whence the thesis follows after substituting in (REF ) and the result in (REF ).",
"We now aim to determine an asymptotic formula for the total scalar curvature of $\\pi ^{-1}(B(R)/\\Gamma )$ as $R$ growths.",
"Our main interest is in Corollary REF , which also follows by [10] and the classical fact (see for example [12]) that $e$ is, up to a positive normalization constant depending on the dimension $m$ and the order of $\\Gamma $ , the ADM mass of the ALE metric associated with $\\eta $ .",
"Nevertheless we include a complete, direct proof for the reader convenience.",
"Proposition 3.3 Let $s$ be the scalar curvature of $\\eta $ .",
"For $R \\rightarrow +\\infty $ one has $\\int _{\\pi ^{-1}(B(R)/\\Gamma )} s \\frac{\\eta ^m}{m!}",
"= \\int _X \\frac{\\rho \\wedge \\xi ^{m-1}}{(m-1)!}",
"+ \\frac{16\\pi ^me}{(m-2)!|\\Gamma |} + O(R^{-2}).$ The Ricci form $\\rho $ of $\\eta $ satisfies $s \\eta ^m = m \\rho \\wedge \\eta ^{m-1}$ .",
"Thanks to (REF ) one has $\\eta ^{m-1} = \\xi ^{m-1} + d\\left( \\sum _{\\ell =1}^{m-1} \\binom{m-1}{\\ell } \\xi ^{m-1-\\ell } \\wedge d^c \\varphi \\wedge (dd^c \\varphi )^{\\ell -1}\\right).$ Since $\\xi $ is supported in $K$ , which in turns is compactly contained in $\\pi ^{-1}(B(R)/\\Gamma )$ , integrating $m \\rho \\wedge \\eta ^{m-1}$ and applying Stokes' theorem gives $\\int _{\\pi ^{-1}(B(R)/\\Gamma )} s\\eta ^m = \\int _X m \\rho \\wedge \\xi ^{m-1} + \\int _{\\partial B(R)/\\Gamma } m \\rho \\wedge d^c \\varphi \\wedge (dd^c \\varphi )^{m-2}.$ The fact that $\\xi $ is supported in $K$ , also implies that $\\eta = dd^c \\varphi $ in a neighborhood $U$ of $\\partial B(R)/\\Gamma $ .",
"On the other hand the two-form $\\tilde{\\eta }= dd^c f(|z|^2)$ defines a Kähler metric on $U$ at least if $R$ is sufficiently large.",
"Of course how large has to be $R$ depends on the value of the constants $e$ and $c$ .",
"However, we are interested in large $R$ asymptotic, therefore we can suppose that $\\tilde{\\eta }$ is Kähler with no loss of generality.",
"Let $\\tilde{\\rho }$ be the Ricci form of $\\tilde{\\eta }$ , so that on $U$ one has $\\rho = \\tilde{\\rho }- dd^c \\log ((dd^c \\varphi )^m / (dd^c f)^m).$ By (REF ) derivatives of $\\varphi $ equals derivatives of $f(|z|^2)$ up to a controlled error.",
"In particular one has $dd^c \\log ((dd^c \\varphi )^m / (dd^c f)^m)= O(|z|^{-2m-4}).$ On the other hand, since $\\tilde{\\eta }= f^{\\prime }dd^c|z|^2 + f^{\\prime \\prime }d|z|^2 \\wedge d^c|z|^2$ one can readily calculate $\\tilde{\\rho }= - dd^c \\log \\left[ (f^{\\prime })^{m-1}\\left( f^{\\prime }+ |z|^2f^{\\prime \\prime }\\right)\\right].$ The second summand of (REF ) is then given by $\\int _{\\partial B(R)/\\Gamma } m \\rho \\wedge d^c \\varphi \\wedge (dd^c \\varphi )^{m-2} \\\\= \\int _{\\partial B(R)/\\Gamma } m d^c \\log \\left[ (f^{\\prime })^{m-1}\\left( f^{\\prime }+ |z|^2f^{\\prime \\prime }\\right)\\right] \\wedge (dd^c \\varphi )^{m-1} + O(R^{-4}).$ Again by (REF ) one has $dd^c \\varphi = dd^c f + O(|z|^{-2m-2})$ so that $\\int _{\\partial B(R)/\\Gamma } m \\rho \\wedge d^c \\varphi \\wedge (dd^c \\varphi )^{m-2} \\\\= \\int _{\\partial B(R)/\\Gamma } m d^c \\log \\left[ (f^{\\prime })^{m-1}\\left( f^{\\prime }+ |z|^2f^{\\prime \\prime }\\right)\\right] \\wedge (dd^c f)^{m-1} + O(R^{-2}).$ Since $f$ depends only on $|z|^2$ , one has $d^c \\log \\left[ (f^{\\prime })^{m-1}\\left( f^{\\prime }+ |z|^2f^{\\prime \\prime }\\right)\\right] \\wedge (dd^c f)^{m-1} = h(|z|^2)d^c|z|^2 \\wedge (dd^c|z|^2)^{m-1}$ where $h = \\left[ (f^{\\prime })^{m-1}\\left( f^{\\prime }+ tf^{\\prime \\prime }\\right)\\right]^{\\prime } / \\left( f^{\\prime }+ tf^{\\prime \\prime }\\right)$ .",
"By definition of $f$ and elementary computations one gets $f^{\\prime }(t)^{m-1} = 4^{1-m}\\left(1 - 4(m-1)e t^{1-m} - 4(m-1)^2 c t^{-m}\\right) + O(t^{2-2m})$ where the error term vanishes in dimension $m=2$ .",
"Moreover one calculates $f^{\\prime }(t)+ tf^{\\prime \\prime }(t) = \\frac{1}{4} + (m-2)e t^{1-m} + (m-1)^2 c t^{-m},$ so that $f^{\\prime }(t)^{m-1}(f^{\\prime }(t)+ tf^{\\prime \\prime }(t)) = 4^{-m}(1 - 4e t^{1-m}) + O(t^{2-2m}),$ and deriving yields $[f^{\\prime }(t)^{m-1}(f^{\\prime }(t)+ tf^{\\prime \\prime }(t))]^{\\prime } = 4^{1-m}(m-1)e t^{-m} + O(t^{1-2m}).$ Dividing (REF ) by (REF ) one finally gets the following expansion $h(t) = 4^{2-m}(m-1)e t^{-m} + O(t^{1-2m}).$ Therefore substituting (REF ) into (REF ) and applying lemma REF yields $\\int _{\\partial B(R)/\\Gamma } m \\rho \\wedge d^c \\varphi \\wedge (dd^c \\varphi )^{m-2} \\\\= \\frac{m(m-1)(4\\pi )^me}{|\\Gamma |} + O(R^{-2}).$ whence the thesis follows after substituting in (REF ).",
"A straightforward consequence of Proposition REF is that the scalar curvature is integrable on $X$ .",
"More specifically it holds the following Corollary 3.4 ([10]) The total scalar curvature of $\\eta $ is given by $\\int _X s \\frac{\\eta ^m}{m!}",
"= \\int _X \\frac{\\rho \\wedge \\xi ^{m-1}}{(m-1)!}",
"+ \\frac{16\\pi ^me}{(m-2)!|\\Gamma |}.$ Among the consequences of Proposition REF and Corollary REF there is the possibility of expressing cohomological quantities on $X$ like $\\int _X \\xi ^m/m!$ and $\\int _X \\rho \\wedge \\xi ^{m-1}/(m-1)!$ in terms of geometric quantities of the Kähler metric $\\eta $ like the total scalar curvature and the volume growth of balls, together with the constants $e,c$ in the expansion of the Kähler potential at infinity.",
"Note that mentioned cohomological quantities appear in the formula (REF ) for the Futaki invariant of an orbifold resolution.",
"Thus they can be recovered by any ALE Kähler metric $\\eta _p = \\xi _p + dd^c \\varphi _p$ representing the fixed Kähler class on the model resolution $X_p$ of the singular point $p$ .",
"In order to get an explicit expression, write $\\varphi _p$ as $\\varphi _p = \\frac{1}{4}|z|^2 + e_p \\frac{1-|z|^{4-2m}}{2-m} + c_p |z|^{2-2m} + O(|z|^{-2m}),$ for some real constants $e_p$ and $c_p$ as $|z| \\rightarrow +\\infty $ .",
"Thanks to Corollary REF then one has $\\int _{X_p} \\frac{\\rho _p \\wedge \\xi _p^{m-1}}{(m-1)!",
"}= \\int _{X_p} s_p \\frac{\\eta _p^m}{m!}",
"- \\frac{16\\pi ^m e_p}{(m-2)!|\\Gamma _p|},$ where $s_p$ is the scalar curvature of $\\eta _p$ .",
"Moreover, proposition REF yields $\\int _{X_p} \\frac{\\xi _p^m}{m!",
"}= \\frac{4\\pi ^mc_p}{(m-2)!|\\Gamma _p|} \\\\+ \\lim _{R \\rightarrow +\\infty } \\int _{\\pi _p^{-1}(B(R)/\\Gamma _p)} \\frac{\\eta _p^m}{m!}",
"- \\frac{\\pi ^m}{m!|\\Gamma _p|} \\left(R^{2m} - 4me_p R^2 + 8m(m-1)e_p^2 R^{4-2m}\\right).$ Note that formula above reduces drastically whenever $\\eta _p$ is Ricci-flat.",
"In this case, the volume form of $\\eta _p$ equals the euclidean volume form [1] and moreover $e_p=0$ , hence $\\int _{X_p} \\frac{\\xi _p^m}{m!}",
"= \\frac{4\\pi ^mc_p}{(m-2)!|\\Gamma _p|}.$"
],
[
"Constant scalar curvature Kähler resolutions",
"In this section we prove the existence and non-existence results for constant scalar curvature Kähler metrics on resolutions of Kähler orbifolds under suitable hypotheses.",
"The set-up is similar to that of the previous sections.",
"In particular, we assume that $\\pi : M^{\\prime } \\rightarrow M$ is a resolution of a $m$ -dimensional orbifold $M$ having finite singular set $S \\subset M$ .",
"More precisely, around each singular point $p \\in S$ , the map $\\pi $ is equal to a resolution $\\pi _p : X_p \\rightarrow \\mathbf {C}^m / \\Gamma _p$ restricted to some neighborhood of the exceptional set $\\pi _p^{-1}(0)$ .",
"Moreover we assume given a Kähler metric $\\omega $ on $M$ with constant scalar curvature $s$ and we denote by $\\mu : M \\rightarrow \\mathfrak {g}^*$ a moment map for the action of the group of holomorphic Hamiltonian diffeomorphisms of $(M,\\omega )$ , normalized so that $\\int _M \\mu \\omega ^n = 0$ .",
"We also assume given for each singular point $p \\in S$ a scalar-flat ALE metric $\\eta _p$ on the model resolution $X_p$ of the form $\\eta _p = \\xi _p + dd^c \\varphi _p$ , with $\\xi _p$ compactly supported around the exceptional set of $\\pi _p$ .",
"Note that by Corollary REF and Proposition REF the scalar-flatness hypothesis of $\\eta _p$ affects the expansion of the ALE Kähler potential $\\varphi _p$ for large $z$ .",
"More precisely, it gives a cohomological formula for the ADM mass of $\\eta _p$ .",
"In particular one has $\\varphi _p =\\frac{1}{4}|z|^2 + e_p \\frac{1-|z|^{4-2m}}{2-m} + c_p |z|^{2-2m} + O(|z|^{-2m})$ with ADM mass $e_p = - \\frac{|\\Gamma _p|}{16\\pi ^m(m-1)} \\int _{X_p} \\rho _p \\wedge \\xi _p^{m-1}$ .",
"Finally, consider for each $p \\in S$ the positive constant $a_p = \\frac{1}{16\\pi ^mm(m-1)}\\int _{X_p} \\xi _p^m$ .",
"Note that $a_p$ is related to $c_p$ and $e_p$ by formula (REF ) and that, by discussion after that formula, one has $a_p=\\frac{c_p}{4|\\Gamma _p|}$ whenever $\\eta _p$ is Ricci-flat.",
"Recalling that the Futaki invariant is an obstruction for the existence of constant scalar curvature Kähler metrics, the asymptotic formula for the Futaki invariant of Theorem REF and discussion above on the ADM mass readily give the following non-existence result Theorem 4.1 If one of the following conditions holds $\\sum _{p \\in S} \\frac{e_p}{|\\Gamma _p|}\\mu (p) \\ne 0$ , $\\sum _{p \\in S} a_p (s\\mu + \\Delta \\mu )(p) \\ne 0$ , then for all $\\varepsilon >0$ sufficiently small the Kähler class $\\pi ^*[\\omega ] + \\varepsilon \\sum _{p \\in S} [\\xi _p] \\in H^{1,1}(M^{\\prime })$ contains no constant scalar curvature Kähler metrics.",
"The starting point for our existence results is the following theorem on existence of extremal Kähler metrics on resolutions [2]: Theorem 4.2 With the notation above, assume that $\\omega $ is an extremal Kähler metric on the orbifold $M$ .",
"Then for all $\\varepsilon >0$ sufficiently small there is an extremal Kähler metric on $M^{\\prime }$ which in the Kähler class $\\pi ^*[\\omega ] + \\varepsilon \\sum _{p \\in S} [\\xi _p] \\in H^{1,1}(M^{\\prime })$ .",
"At this point we are in position to state and prove our main existence results.",
"Theorem 4.3 Let $e_p$ be the ADM mass of $\\eta _p$ , and let $Q \\subset S$ be the subset of singular points with non-zero ADM mass.",
"If $Q$ is non-empty and $\\sum _{q \\in Q} \\frac{e_q}{|\\Gamma _q|}\\mu (q) = 0 \\qquad \\mbox{and} \\qquad \\operatorname{span}\\lbrace \\mu (q) \\,|\\, q \\in Q\\rbrace = \\mathfrak {g}^*$ then, after identifying each $\\xi _p$ with a $(1,1)$ -form on $M^{\\prime }$ as above, for all $\\varepsilon >0$ sufficiently small there exists $\\lambda _q(\\varepsilon ) >0$ and a constant scalar curvature Kähler metric $\\omega ^{\\prime }_\\varepsilon $ such that $[\\omega ^{\\prime }_\\varepsilon ] = \\pi ^* [\\omega ] + \\sum _{q \\in Q} \\lambda _q(\\varepsilon ) [\\xi _q] + \\varepsilon \\sum _{p \\in S \\setminus Q} [\\xi _p] \\in H^{1,1}(M^{\\prime })$ and $\\lambda _q(\\varepsilon ) \\sim \\varepsilon $ as $\\varepsilon $ tends to 0.",
"An analytic proof of this result has been given in [2].",
"We skip the new proof being similar and simpler to the one of the next result.",
"Theorem 4.4 Suppose that $e_p = 0$ for all $p \\in S$ .",
"If $\\sum _{p \\in S} a_p(s \\mu + \\Delta \\mu )(p) = 0 \\qquad \\mbox{and} \\qquad \\operatorname{span}\\lbrace (s \\mu + \\Delta \\mu )(p) \\,|\\, p \\in S\\rbrace = \\mathfrak {g}^*$ then, after identifying each $\\xi _p$ with a $(1,1)$ -form on $M^{\\prime }$ as above, for all $\\varepsilon >0$ sufficiently small there exists $\\lambda _p(\\varepsilon ) >0$ and a constant scalar curvature Kähler metric $\\omega ^{\\prime }_\\varepsilon $ such that $[\\omega ^{\\prime }_\\varepsilon ] = \\pi ^* [\\omega ] + \\sum _{p \\in S} \\lambda _p(\\varepsilon ) [\\xi _p] \\in H^{1,1}(M^{\\prime })$ and $\\lambda _p(\\varepsilon ) \\sim \\varepsilon $ as $\\varepsilon $ tends to 0.",
"This result extends an analogue one in [1], proved under the additional assumption of Ricci-flatness of the local resolutions, in which case $a_p=\\frac{c_p}{4|\\Gamma _p|}$ , as remarked above.",
"For each $p \\in S$ and for all real $t_p$ such that $|t_p|<1$ , take $\\varepsilon >0$ sufficiently small and consider on $M^{\\prime }$ the Kähler metric $\\omega _{t,\\varepsilon }= \\pi ^* \\omega + \\varepsilon \\sum _{p \\in S} (1+t_p)^{1/m} \\xi _p.$ Note that this metric is invariant with respect to any holomorphic vector field with zeroes $V$ on $M^{\\prime }$ since $\\pi ^*\\omega $ and each $\\xi _p$ are.",
"Moreover note that any such vector field is Hamiltonian with respect to $\\omega _{t,\\varepsilon }$ .",
"In particular, by theorem REF , one has $\\operatorname{Fut}(V,\\omega _{t,\\varepsilon })= - \\frac{16\\pi ^m\\varepsilon ^m}{(m-2)!}",
"\\sum _{p \\in S} (1+t_p) a_p \\left(s (u(p) - \\underline{u}) + \\Delta u(p)\\right) + O(\\varepsilon ^{m+1}),$ where we used the hypothesis that $\\omega $ has constant scalar curvature (hence vanishing Futaki invariant), for each $p\\in S$ the model metric $\\eta _p$ is Ricci flat (hence $\\rho _p=0$ ), and the scalar curvature $s$ of $\\omega $ is constant.",
"On the other hand, note that by general theory of Futaki invariant, $\\operatorname{Fut}(V,\\omega _{t,\\varepsilon })$ depends polynomially on the cohomology class of $\\omega _{t,\\varepsilon }$ , hence on $\\varepsilon $ and $(1+t_p)^{1/m}$ .",
"Finally, observe that the normalized Hamiltonian potential $u - \\underline{u}$ of $\\pi _*V$ is equal to $\\langle \\mu ,V \\rangle $ .",
"Therefore, letting $F(t,\\varepsilon ) (V) = - \\frac{(m-2)!",
"}{16\\pi ^m\\varepsilon ^m} \\operatorname{Fut}(V,\\omega _{t,\\varepsilon })$ defines a smooth function $F$ on $(-1,1)^{|S|} \\times \\mathbf {R}$ with values in $\\mathfrak {g}^*$ , being $|S|$ the cardinality of the singular set $S$ .",
"By (REF ), for small $\\varepsilon $ one has $F(t,\\varepsilon )= \\sum _{p \\in S} (1+t_p) a_p \\left(s \\mu + \\Delta \\mu \\right)(p) + O(\\varepsilon ).$ Therefore, hypotheses (REF ) ensure that $F(0,0)=0$ , and the Jacobian $\\partial F/ \\partial t$ at the point $(0,0)$ has rank equal to $\\dim \\mathfrak {g}$ .",
"As a consequence, by implicit function theorem one can find $\\varepsilon _0>0$ and a smooth function $t$ on $(-\\varepsilon _0,\\varepsilon _0)$ with values to $(-1,1)^{|S|}$ such that $F(t(\\varepsilon ),\\varepsilon ) = 0$ , and $t(0)=0$ .",
"Clearly there are $|S|-\\dim \\mathfrak {g}$ free parameters in doing this, but we don't need this extra flexibility for our purposes.",
"By discussion above, for all holomorphic vector field with zeroes on $M^{\\prime }$ then one has $\\operatorname{Fut}(V,\\omega _{t(\\varepsilon ),\\varepsilon })=0$ if $0 < \\varepsilon < \\varepsilon _0$ .",
"Therefore, letting $\\lambda _p(\\varepsilon ) = \\varepsilon (1+t(\\varepsilon )_p)^{1/m}$ yields a family of Kähler classes $\\pi ^* [\\omega ] + \\sum _{p \\in S} \\lambda _p(\\varepsilon ) [\\xi _p] \\in H^{1,1}(M^{\\prime })$ with vanishing Futaki invariant and approaching the class $[\\pi ^*\\omega ]$ as $\\varepsilon \\rightarrow 0$ .",
"By the elementary remark that extremal Kähler metrics with vanishing Futaki invariant have constant scalar curvature [5], [9], in order to get the thesis we are now reduced to show that classes as in (REF ) contain an extremal Kähler metric, at least when $\\varepsilon $ is sufficiently small.",
"This follows by Theorem REF and openness of the extremal cone [13].",
"Indeed, by a standard perturbation argument these two theorems imply that under our hypotheses there is a small open ball $H^{1,1}(M^{\\prime })$ centered at $[\\pi ^*\\omega ]$ whose intersection $A$ with the Kähler cone of $M^{\\prime }$ is constituted by extremal Kähler classes, i.e.",
"Kähler classes representable by extremal Kähler metrics.",
"Since classes as in (REF ) are contained in $A$ for $\\varepsilon $ sufficiently small, it follows that all these classes contain extremal Kähler metrics.",
"Comparing formulae for Kähler classes (REF ) and (REF ) somehow suggests that one can still get cscK metrics in adiabatic classes of $M^{\\prime }$ in cases not covered by theorems above.",
"This can be done by choosing different scaling volumes with respect to $\\varepsilon $ to excepional divisors, according they project via $\\pi $ to a singular point with zero or non-zero ADM mass.",
"More precisely one has the following result which is not covered by previous works.",
"Theorem 4.5 Let $Q \\subset S$ be the subset constituted by those singular points of $M$ which have non-zero ADM mass and let $P$ its complement.",
"If $P$ and $Q$ are both not empty and $\\sum _{q \\in Q} \\frac{e_q}{|\\Gamma _q|}\\mu (q) + \\sum _{p \\in P} a_p(s \\mu + \\Delta \\mu )(p) = 0\\quad \\mbox{and} \\quad \\operatorname{span}\\lbrace \\mu (q), (s \\mu + \\Delta \\mu )(p) \\,|\\, q \\in Q, \\, p\\in P\\rbrace = \\mathfrak {g}^*$ then, after identifying each $\\xi _p$ with a $(1,1)$ -form on $M^{\\prime }$ as above, for all $\\varepsilon >0$ sufficiently small there exist $\\lambda _p(\\varepsilon )>0$ and a constant scalar curvature Kähler metric $\\omega ^{\\prime }_\\varepsilon $ such that $[\\omega ^{\\prime }_\\varepsilon ] = \\pi ^* [\\omega ] + \\sum _{q \\in Q} \\lambda _q(\\varepsilon ) [\\xi _q] + \\sum _{p \\in P} \\lambda _p(\\varepsilon ) [\\xi _p] \\in H^{1,1}(M^{\\prime }).$ Moreover, as $\\varepsilon $ tends to 0, one has $\\lambda _q(\\varepsilon ) \\sim \\varepsilon $ for all $q \\in Q$ and $\\lambda _p(\\varepsilon ) \\sim \\varepsilon ^\\frac{m-1}{m}$ for all $p \\in P$ .",
"The statement follows exactly from the same line of arguments as in the proof of theorem REF once one starts with the Kähler metric on $M^{\\prime }$ defined by $\\omega _{t,\\varepsilon }= \\pi ^* \\omega + \\varepsilon \\sum _{q \\in Q} (1+t_q)^\\frac{1}{m-1} \\xi _q + \\varepsilon ^\\frac{m-1}{m} \\sum _{p \\in P} (1+t_p)^\\frac{1}{m} \\xi _p$ with $t_p,t_q \\in \\mathbf {R}$ such that $|t_p|,|t_q|<1$ , and $\\varepsilon >0$ sufficiently small."
]
] | 1808.08420 |
[
[
"Revisiting the Galaxy Shape and Spin Alignments with the Large-Scale\n Tidal Field: An Effective Practical Model"
],
[
"Abstract An effective practical model with two characteristic parameters is presented to describe both of the tidally induced shape and spin alignments of the galactic halos with the large-scale tidal fields.",
"We test this model against the numerical results obtained from the Small MultiDark Planck simulation on the galactic mass scale of $0.5\\le M/(10^{11}\\,h^{-1}\\,M_{\\odot})\\le 50$ at redshift $z=0$.",
"Determining empirically the parameters from the numerical data, we demonstrate how successfully our model describes simultaneously and consistently the amplitudes and behaviors of the probability density functions of three coordinates of the shape and spin vectors in the principal frame of the large scale tidal field.",
"Dividing the samples of the galactic halos into multiple subsamples in four different mass ranges and four different types of the cosmic web, and also varying the smoothing scale of the tidal field from $5\\,h^{-1}$Mpc to $10,\\ 20,\\ 30\\,h^{-1}$Mpc, we perform repeatedly the numerical tests with each subsample at each scale.",
"Our model is found to match well the numerical results for all of the cases of the mass range, smoothing scale and web type and to properly capture the scale and web dependence of the spin flip phenomenon."
],
[
"Introduction",
"The physical properties of the observed galaxies in the universe is a reservoir of information on the conditions under which they formed, the evolutionary processes which they went through, and the interactions in which they are involved.",
"Although the local conditions and processes at the galactic scales must have had the most dominant impact on the galaxies, the non-local effects beyond the galactic scales are also believed to have contributed partly to their physical properties [48].",
"Subdominant as its contribution is, the non-local effects on the galaxies are worth investigating, since it may contain valuable independent information on the galaxy formation and the background cosmology as well.",
"The non-local effects on the galaxies are manifested by the correlations between the galaxy properties and the large-scale environments.",
"Among various properties of the galaxies that have been found correlated with the large-scale environments, the shape and spin alignments of the galaxies with the large-scale structures (collectively called the galaxy intrinsic alignments) have lately drawn considerable attentions, inspiring vigorous extensive studies [34], [31], [32].",
"It is partially because the galaxy intrinsic alignments, if present and significant, could become another systematics in the measurements of the extrinsic counterparts caused by the weak gravitational lensing [61].",
"The other important motivation for the recent flurry of research on this topic is that the origin of the galaxy intrinsic alignments is amenable to the first order perturbation theory and thus a rather fundamental approach to this topic is feasible [25], [37], [12], [17], [38], [52], [28], [8], [9], [64].",
"In the first order Lagrangian perturbation theory [75], [11], the minor (major) eigenvectors of the inertia momentum tensors of the proto-galactic regions are perfectly aligned with the major (minor) eigenvectors of the local tidal tensors around the regions.",
"Several $N$ -body simulations have indeed detected the existence of strong correlations between the inertia momentum and local tidal tensors at the proto-galactic sites [37], [52], [42].",
"Since the tidal fields smoothed on different scales are cross correlated, the eigenvectors of the inertia momentum tensors of the proto-galactic regions are expected to be aligned with those of the large-scale tidal fields.",
"The major eigenvectors of the inertia momentum tensors of the proto-galaxies correspond to the most elongated axes of their shapes, while the minor eigenvectors of the large-scale tidal tensors correspond to the directions along which the surrounding matter become minimally compressed.",
"Henceforth, this expectation based on the first order Lagrangian perturbation theory basically translates into the possible alignments between the galaxy shapes and the most elongated axes of the large-scale structures such as the axes of the filaments, the signals of which have been detected by several numerical and observational studies [1], [23], [72], [73], [13].",
"In the linear tidal torque (LTT) theory that [18] formulated by combining the first order Lagrangian perturbation theory with the Zel'dovich approximation [75], the anisotropic tidal field of the surrounding matter distribution originates the spin angular momentum of a proto-galaxy provided that its shape departs from a spherical symmetry.",
"The generic and unique prediction of this LTT theory is the inclinations of the spin vectors of the proto-galaxies toward the intermediate eigenvectors of the large scale tidal field [37], which has also garnered several numerical and observational supports [46], [63], [22], [40], [67], [74], [13].",
"The recently available large high-resolution $N$ -body simulations that covered a broad mass range, however, limited the validity of the LTT prediction to the mass scale of $M\\ge M_{\\rm t}\\sim 10^{12}\\,h^{-1}\\,M_{\\odot }$ , showing that on the mass scale below $M_{t}$ the spin vectors of dark matter halos at $z=0$ are aligned not with the intermediate but rather with the minor eigenvectors of the large scale tidal field, similar to the axes of the halo shapes [2], [23], [49], [72], [15], [45], [62], [19], [65].",
"This difference in the spin alignment tendency between the low and high mass scales were found most conspicuous in the filament environments: the spin axes of the galactic halos with masses lower (higher) than $M_{\\rm t}$ measured at $z=0$ tend to be parallel (perpendicular) to the elongated axes of their host filaments, in contradiction with the LTT prediction.",
"The transition of the spin alignment tendency at $M_{\\rm t}$ is often called \"spin flip\" phenomenon [15] and the break-down of the LTT prediction below $M_{\\rm t}$ has also been witnessed in recent observations [57], [58], [27], [14].",
"The detection of this spin-flip phenomenon puzzled the community and urged it to find a proper answer to the critical question of what the origin of this phenomenon is.",
"What has so far been suggested as a possible origin includes the major merging events, mass dependence of the merging and accretion processes, assembly bias, vorticity generation inside filaments, web-dependence of the galaxy formation epochs, nonlinear tidal interactions, geometrical properties of the host filaments and etc [5], [35], [15], [45], [69], [16], [36], [6], [68], [65].",
"Although these previously suggested factors were believed to play some roles for the occurrence of the spin-flip phenomenon, none of them are fully satisfactory in explaining all aspects of the spin-flip phenomenon including the dependence of the transition mass scale $M_{\\rm t}$ on the types of the cosmic web, redshifts, and scales of the filaments.",
"The occurrence of the spin-flip phenomenon basically implies that for the case of the galaxies with masses $M\\le M_{t}$ , the tendency of the spin alignments with the large scale tidal field becomes similar to that of the shape alignments.",
"Thus, it is suspected that whatever caused the spin-flip phenomenon, it should be linked to the shape alignments with the large scale tidal field.",
"To address these remaining issues, what is highly desired is an effective model that can describe consistently and simultaneously both of the galaxy shape and spin alignments.",
"Here, we attempt to construct such a model by modifying the original LTT theory and to explore if the shapes of the galaxies also show any transition of the alignment tendency like the spin counterparts The organization of this Paper is as follows.",
"A refined analytic model for the galaxy shape alignments is presented in Section REF and tested against the numerical results in Section REF .",
"An effective model for the tidally induced spin alignments is presented in Section REF and tested against the numerical results in Section REF .",
"A discussion over the possible application of this model as well as a summary of the results is presented in Section .",
"Throughout this Paper, we will assume a Planck universe whose total energy density is dominantly contributed by the cosmological constant ($\\Lambda $ ) and the cold dark matter (CDM) [51].",
"Suppose a galactic halo located in a region where a tidal tensor ${\\bf T}$ has its major, intermediate and minor eigenvectors ($\\hat{\\bf u}_{1},\\ \\hat{\\bf u}_{2}$ and $\\hat{\\bf u}_{3}$ , respectively), corresponding to the largest, second to the largest, smallest eigenvalues ($\\lambda _{1},\\ \\lambda _{2}$ and $\\lambda _{3}$ , respectively).",
"The tidal tensor ${\\bf T}$ depends on the smoothing scale, $R_{f}$ , as ${\\bf T}({\\bf x})\\propto \\partial _{i}\\partial _{j}\\int d{\\bf x}^{\\prime }\\,\\Phi ({\\bf x^{\\prime }})W(\\vert {\\bf x}-{\\bf x}^{\\prime }\\vert ;R_{f})$ , where $\\Phi ({\\bf x})$ is the perturbation potential field and $W(\\vert {\\bf x}-{\\bf x}^{\\prime }\\vert ;R_{f})$ is a window function with a filtering radius $R_{f}$ .",
"In the current analysis, we adopt a Gaussian window function.",
"As mentioned in Section , the first order Lagrangian perturbation theory [75], [11] predicts a strong anti-correlation between the principal axes of the inertia momentum tensor of a galactic halo and the local tidal tensor in the Lagrangian regime.",
"According to this theory, the correlation between the two tensors is strongest if the two tensors are defined on the same scale (i.e., the virial radius of the halo, $R_{g}$ ), becomes weaker if $R_{f}$ is larger than $R_{g}$ .",
"If the shape of this galactic halo can be approximated by an ellipsoid, then the direction of the coordinate vector of the largest shape ellipsoid, ${\\bf e}=({e}_{1}, {e}_{2}, {e}_{3})$ (i.e., the major principal axis of the inertia momentum tensor) is expected to be aligned with $\\hat{\\bf u}_{3}$ (i.e., the minor principal axis of the local tidal tensor) along which the surrounding matter is least compressed, provided that $\\lambda _{1}\\ne \\lambda _{2}\\ne \\lambda _{3}$ .",
"This alignment tendency can be statistically quantified by the conditional joint probability density function of three coordinates of the largest shape ellipsoid axis, $p({e}_{1}, {e}_{2}, {e}_{3}\\vert \\hat{\\bf T})$ , where $\\hat{\\bf T}$ is a unit traceless tidal tensor defined as $\\hat{\\bf T}\\equiv ({\\bf T}-{\\rm Tr}({\\bf T})/3)/\\vert {\\bf T}-{\\rm Tr}({\\bf T})/3\\vert $ with ${\\rm Tr}({\\bf T})$ denoting the trace of ${\\bf T}$ .",
"As [38] and [39] did, we assume here that $p({e}_{1}, {e}_{2}, {e}_{3}\\vert \\hat{\\bf T})$ follows a multivariate Gaussian distribution of $p({e}_{1},\\ {e}_{2},\\ {e}_{3}\\vert \\hat{\\bf T}) = \\frac{1}{[(2\\pi )^3 {\\rm det}({\\bf \\Sigma })]^{1/2}}\\exp \\left[-\\frac{1}{2}\\left({\\bf e}\\cdot {\\bf \\Sigma }^{-1}\\cdot {\\bf e}\\right)\\right]\\, ,$ where the components of the covariance matrix, ${\\bf \\Sigma }=({\\Sigma }_{ij})$ , are defined as the conditional ensemble averages, ${\\Sigma }_{ij} \\equiv \\langle {e}_{i}{e}_{j}\\vert \\hat{\\bf T}\\rangle $ .",
"Here, we suggest the following practical formula for $\\langle {e}_{i}{e}_{j}\\vert \\hat{\\bf T}\\rangle $ : $\\langle {e}_{i}{e}_{j}\\vert \\hat{\\bf T}\\rangle = \\frac{1+d_{t}}{3}\\delta _{ij} - d_{t}\\hat{T}_{ij}\\, ,$ where $d_{t}$ is the shape correlation parameter that measures the alignment strength between ${\\bf e}$ and $\\hat{\\bf u}_{3}$ .",
"Note that this formula describes a linear dependence of the covariance, $\\langle {e}_{i}{e}_{j}\\vert \\hat{\\bf T}\\rangle $ , on ${\\bf T}$ [12], [41], [28], unlike a spin vector whose covariance has a quadratic dependence on ${\\bf T}$ [70], [37], [38].",
"Focusing only on the direction of ${\\bf e}$ , we marginalize $p({e}_{1}, {e}_{2}, {e}_{3}\\vert {\\bf T})$ over $e\\equiv \\vert {\\bf e}\\vert $ to have $p(\\hat{\\bf e}\\vert \\hat{\\bf T})&=&\\int \\,p({\\bf e}\\vert \\hat{\\bf T}) e^{2}de \\nonumber \\\\&=&\\frac{1}{4\\pi {\\rm det}({\\bf \\Sigma })^{1/2}}\\left(\\hat{\\bf e}\\cdot {\\bf \\Sigma }^{-1}\\cdot \\hat{\\bf e}\\right)^{-3/2}\\, ,$ where $\\hat{\\bf e}\\equiv {\\bf e}/e$ denotes the unit vector in the direction of the largest shape ellipsoid axis.",
"While $\\hat{\\bf T}$ shares the same orthonormal eigenvectors with ${\\bf T}$ , its eigenvalues, $\\hat{\\lambda }_{1},\\ \\hat{\\lambda }_{2},\\ \\hat{\\lambda }_{3}$ , are subject to two additional constraints of $\\sum _{i=1}^{3}\\hat{\\lambda }_{i}=0$ and $\\sum _{i=1}^{3}\\hat{\\lambda }^{3}_{i}=1$ [38].",
"Putting Equation (REF ) into Equation (REF ) leads to the following analytic expression $p(\\hat{\\bf e}\\vert \\hat{\\bf T})&=& = \\frac{1}{2\\pi }\\left[\\prod _{n=1}^{3}\\left(1+d_{t}-3d_{t}\\hat{\\lambda }_{n}\\right)\\right]^{-\\frac{1}{2}}\\left(\\sum _{l=1}^{3}\\frac{\\vert \\hat{\\bf u}_{l}\\cdot \\hat{\\bf e}\\vert }{1+d_{t}-3d_{t}\\hat{\\lambda }_{l}} \\right)^{-\\frac{3}{2}}\\, ,$ since $\\hat{T}_{ij}=\\hat{\\lambda }_{i}\\delta _{ij}$ in the principal axis frame of $\\hat{\\bf T}$ .",
"Now, the conditional probability density function, $p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf e}\\vert )$ , for $i\\in \\lbrace 1,2,3\\rbrace $ , can be obtained as $p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf e}\\vert )=\\int _{0}^{2\\pi }p(\\hat{\\bf e}\\vert \\hat{\\bf T})\\,d\\phi _{jk}\\, ,$ where $\\phi _{jk}$ is the azimuthal angle of $\\hat{\\bf e}$ in the plane spanned by $\\hat{\\bf u}_{j}$ and $\\hat{\\bf u}_{k}$ perpendicular to $\\hat{\\bf u}_{i}$ .",
"Equation (REF ) indicates that the completion of this analytic model requires us to determine the value of $d_{t}$ .",
"If $\\hat{\\bf e}$ is perfectly aligned with $\\hat{\\bf u}_{3}$ , then $d_{t}$ would be unity.",
"Whereas, the zero value of $d_{t}$ would correspond to the case that $\\hat{\\bf e}$ is completely random having no correlation with $\\hat{\\bf u}_{3}$ .",
"As done in [38], for the determination of $d_{t}$ , we first evaluate the conditional ensemble average, $\\langle \\hat{e}_{i}\\hat{e}_{j}\\vert \\hat{\\bf T}\\rangle $ , under the assumption of $d_{t}\\ll 1$ $\\langle \\hat{e}_{i}\\hat{e}_{j}\\vert \\hat{\\bf T}\\rangle &=& \\int \\hat{e}_{i}\\hat{e}_{j} p(\\hat{\\bf e}\\vert \\hat{\\bf T}) d\\hat{\\bf e}\\, ,\\\\&\\approx & \\left[\\left(\\frac{1}{3}+\\frac{3}{5}d_{t}\\right) - \\frac{3}{5}d_{t}\\hat{\\lambda }_{i}\\right]\\delta _{ij}\\, ,$ Note that the off-diagonal elements vanish in Equation () since $\\hat{T}_{ij}=\\hat{\\lambda }_{i}\\delta _{ij}$ in its principal frame.",
"Multiplying Equation () by $\\hat{\\lambda }_{i}$ and summing over the three components, we finally derive a simple analytic formula for $d_{t}$ : $d_{t} = -\\frac{5}{3}\\sum _{i=1}^{3}\\hat{\\lambda }_{i}\\langle \\hat{e}^{2}_{i}\\vert \\hat{\\bf T}\\rangle \\, .$ The constraints of $\\sum _{i=1}^{3}\\hat{\\lambda }_{i}=0$ and $\\sum _{i=1}^{3}\\hat{\\lambda }^{2}_{i}=1$ are used to derive the above formula.",
"Equation (REF ) implies that once the values of $\\hat{\\lambda }_{i}$ and $\\langle \\hat{e}^{2}_{i}\\vert \\hat{\\bf T}\\rangle $ are measured, the shape correlation parameter, $d_{t}$ , can be empirically determined.",
"It is worth recalling that the shape correlation parameter, $d_{t}$ , depends on the smoothing scale, $R_{f}$ .",
"It is expected to have the highest value when $R_{f}=R_{g}$ , as mentioned in the above.",
"In the Eulerian regime, however, the approximation of $p({\\bf e}\\vert {\\bf T})$ as a multivariate Gaussian distribution and ${\\bf T}$ as a Gaussian random field used for Equation (REF ) are not valid on the scale of $R_{g}$ due to the nonlinear evolution of $\\hat{\\bf T}$ on the galactic scale.",
"Thus, we consider the scales $R_{f}$ much larger than $R_{g}$ where these approximations still hold true.",
"Since $\\hat{\\bf T}$ on two different scales of $R_{g}$ and $R_{f}$ are cross-correlated, it is expected that $\\hat{\\bf e}$ is still correlated with $\\hat{\\bf T}$ smoothed on the scale of $R_{f}\\gg R_{g}$ .",
"The larger the difference between $R_{f}$ and $R_{g}$ is, the lower the value of $d_{t}$ is."
],
[
"Numerical Tests",
"Our numerical analysis is based on the data set from the Small MultiDark Planck simulationdoi:10.17876/cosmosim/smdpl/(SMDPL), a DM only $N$ -body simulation conducted in a periodic box with a side length of $400\\,h^{-1}$ Mpc [33] as a part of the MultiDark simulation project [54] for a Planck universe [51].",
"The SMDPL tracks down the gravitational evolution of $3840^{3}$ DM particles each of which has individual mass of $9.63\\times 10^{7}\\,h^{-1}\\,M_{\\odot }$ , starting from $z=120$ down to $z=0$ [33].",
"The virialized DM halos were identified via the Rockstar halo-finding algorithm [7] from the spatial distributions of the DM particles at various snapshots of the SMDPL.",
"Through the CosmoSim database that stores all the experimental results from the MultiDark simulations, we first extract the Rockstar catalog, which provides information on a diverse set of the physical properties of the DM halos.",
"For the current analysis, we use such information as the parent id (pId), comoving position vector (${\\bf r}$ ), spatial grid index, virial mass ($M$ ), coordinate vector of the largest shape ellipsoid axis (${\\bf e}$ ) of each Rockstar halo.",
"The integer value of pId is used to exclude the subhalos from our analysis.",
"For the case of a distinct halo that is not a subhalo hosted by any other larger halo, the parent id has the value of pId=-1.",
"The coordinate vector, ${\\bf e}$ , is a measure of the most elongated axis of an ellipsoid to which the shape of a given Rockstar halo was fitted.",
"From here on, the unit coordinate vector of the largest shape ellipsoid axis, $\\hat{\\bf e}\\equiv {\\bf e}/e$ , will be called a shape vector.",
"We make a sample of the distinct galactic halos by selecting only those from the Rockstar catalog which meet two conditions of pId$=-1$ and $0.5\\le M/(10^{11}\\,h^{-1}\\,M_{\\odot })<50$ .",
"Then, we divide this sample into four subsamples which cover four different mass ranges: $0.5\\le M/(10^{11}\\,h^{-1}\\,M_{\\odot })<1$ (lowest-mass galactic halos), $1\\le M/(10^{11}\\,h^{-1}\\,M_{\\odot })<5$ (low-mass galactic halos), $5\\le M/(10^{11}\\,h^{-1}\\,M_{\\odot })<10$ (medium-mass galactic halos) and $10\\le M/(10^{11}\\,h^{-1}\\,M_{\\odot })<50$ (high-mass galactic halos), respectively.",
"We exclude the subhalos from the analysis, since the effect of the large-scale tidal field on the subhalos are likely to be negligible compared with that of the internal nonlinear tidal fields inside the host halos Those halos with $M<0.5\\times 10^{11}\\,h^{-1}\\,M_{\\odot }$ are excluded on the ground that the measurements of the shape and spin vectors of those halos are likely to be contaminated by the shot noise due to the small number of the component DM particles [4].",
"The group and cluster size halos with $M\\ge 5\\times 10^{12}\\,h^{-1}\\,M_{\\odot }$ are also excluded since the measurements of their shapes and spins should be severely affected by their dynamical states, internal structures and recent merging events.",
"We also retrieve the cloud-in-cell density field, $\\rho ({\\bf r})$ , defined on the $512^{3}$ grids at $z=0$ via the CosmoSim database.",
"Then, we calculate the dimensionless density contrast field, $\\delta ({\\bf r})=\\left[\\rho ({\\bf r})-\\langle \\rho \\rangle \\right]/\\langle \\rho \\rangle $ , where the ensemble average, $\\langle \\rho \\rangle $ , is taken over all the grids.",
"With the help of the numerical recipe code that performs the Fast Fourier Transformation (FFT) [53], we compute the Fourier amplitude of the density contrast field, $\\tilde{\\delta }({\\bf k})$ , where ${\\bf k}=k(\\hat{k}_{i})$ is the wave vector in the Fourier space.",
"The inverse FFT of $\\tilde{T}_{ij} = \\hat{k}_{i}\\hat{k}_{j}\\tilde{\\delta }({\\bf k})\\exp \\left(-k^{2}R^{2}_{f}/2\\right)$ for $i,j\\in \\lbrace 1,2,3\\rbrace $ leads us to have the tidal field, ${\\bf T}({\\bf r})$ , smoothed on the scale of $R_{f}$ .",
"For each subsample, we take the following steps.",
"First, at the grid where each halo is placed, we perform a similarity transformation of ${\\bf T}({\\bf r})$ , to find its eigenvectors $\\lbrace \\hat{\\bf u}_{i}\\rbrace _{i=1}^{3}$ as well as the eigenvalues $\\lbrace \\lambda _{i}\\rbrace _{i=1}^{3}$ .",
"Second, we calculate, $\\lbrace \\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf e}\\vert \\rbrace _{i=1}^{3}$ , whose values lie in the range of $[0,1]$ .",
"Breaking this unit interval $[0,1]$ into seven bins with equal length of $\\Delta =1/7$ , we count the number of the galactic halos, $n_{h,i}$ , whose values of $\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf e}\\vert $ fall in each bin for each $i\\in \\lbrace 1,2,3\\rbrace $ .",
"Third, the probability densities of $\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf e}\\vert $ at each bin are determined as $p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf e}\\vert )=n_{h,i}/(N_{t}\\,\\Delta )$ where $N_{t}$ is the total number of the galactic halos contained in each subsample.",
"Figure REF plots $p(\\vert \\hat{\\bf u}_{3}\\cdot \\hat{\\bf e}\\vert )$ (left panel), $p(\\vert \\hat{\\bf u}_{2}\\cdot \\hat{\\bf e}\\vert )$ (middle panels) and $p(\\vert \\hat{\\bf u}_{1}\\cdot \\hat{\\bf e}\\vert )$ (right panels) as filled circular dots for the cases of the lowest-mass (top panels), low-mass (second from the top panels), medium mass (second from the bottom panels), and high-mass (bottom panels) galactic halos.",
"To obtain these results, we smooth $\\hat{\\bf T}({\\bf r})$ on the scale of $R_{f}=5\\,h^{-1}$ Mpc.",
"As can be seen, for all four subsamples, the shape vector, $\\hat{\\bf e}$ , shows a strong inclination (anti-inclination) toward the minor (major) eigenvector, $\\hat{\\bf u}_{3}$ ($\\hat{\\bf u}_{1}$ ), while it shows no alignment with the intermediate eigenvector, $\\hat{\\bf u}_{2}$ .",
"Note also that the higher-mass galactic halos exhibit stronger alignment (anti-alignment) tendency between $\\hat{\\bf e}$ and $\\hat{\\bf u}_{3}$ ($\\hat{\\bf e}$ and $\\hat{\\bf u}_{1}$ ), which are consistent with the previously reported numerical and observational results [23], [72], [29], [73], [13], [26], [71], [50].",
"To compare the analytic model presented in Section REF against these numerical results, we first calculate the mean values of $\\hat{\\lambda }_{i}$ and $d_{t}$ averaged over the galactic halos contained in each subsample as $\\langle \\hat{\\lambda }_{i}\\rangle &=& \\frac{1}{N_{t}}\\sum _{\\alpha =1}^{N_{t}}\\hat{\\lambda }_{\\alpha ,i}=\\frac{1}{N_{t}}\\sum _{\\alpha =1}^{N_{t}}\\, ,\\left(\\frac{\\tilde{\\lambda }_{\\alpha ,i}}{\\sqrt{\\sum _{j=1}^{3}\\tilde{\\lambda }^{2}_{\\alpha ,j}}}\\right)\\, , \\quad i\\in \\lbrace 1,2,3\\rbrace \\, , \\\\\\tilde{\\lambda }_{\\alpha ,i} &=& \\lambda _{\\alpha ,i} - \\frac{1}{3}\\sum _{j=1}^{3}\\lambda _{\\alpha ,j}\\, , \\quad i\\in \\lbrace 1,2,3\\rbrace \\, , \\\\\\langle d_{t}\\rangle &=& \\frac{1}{N_{t}}\\sum _{\\alpha =1}^{N_{t}}\\left[-\\frac{5}{3}\\sum _{i=1}^{3}\\hat{\\lambda }_{\\alpha ,i}\\vert \\hat{\\bf u}_{\\alpha ,i}\\cdot \\hat{\\bf e}_{\\alpha }\\vert ^{2}\\right]\\, ,$ where $\\lbrace {\\lambda }_{\\alpha ,i}\\rbrace _{i=1}^{3}$ denotes a set of the three eigenvalues of ${\\bf T}$ at the grid where the $\\alpha $ th DM halo of a given subsample is located, and $\\hat{\\bf e}_{\\alpha }$ is the shape vector of the $\\alpha $ th DM halo.",
"Note that $\\langle \\hat{e}^{2}_{i}\\vert \\hat{\\bf T}\\rangle $ in Equation (REF ) is approximated by $\\vert \\hat{\\bf u}_{\\alpha ,i}\\cdot \\hat{\\bf e}_{\\alpha }\\vert ^{2}$ in Equation () since the measured values in numerical realizations are believed to be close to the expectation values in theory.",
"Substituting these mean values of $\\langle \\hat{\\lambda }_{i}\\rangle $ and $\\langle d_{t}\\rangle $ for $\\hat{\\lambda }_{i}$ and $d_{t}$ respectively in Equations (REF )-(REF ), we evaluate the analytical model and plot them as red solid lines in Figure REF .",
"As can be seen, for all of the four cases of the halo mass ranges, the analytic model with the empirically determined parameter $d_{t}$ describes very well not only the alignments of $\\hat{\\bf e}$ with $\\hat{\\bf u}_{3}$ but also simultaneously its anti-alignment with $\\hat{\\bf u}_{1}$ and no correlation with $\\hat{\\bf u}_{2}$ as well, even though no fitting process is involved.",
"Figure REF plots $\\langle d_{t}\\rangle $ for the four different cases of the mass ranges, showing quantitatively how the strength of the shape alignments increases with the increment of $M$ .",
"Smoothing $\\hat{\\bf T}$ on three larger scales, $R_{f}=10,\\ 20$ and $30\\,h^{-1}$ Mpc, we repeat the whole calculations, the results of which are shown in Figure REF for the case of the high-mass galactic halos.",
"The analytic model with the empirically determined parameter $d_{t}$ agrees quite well with the numerical results for all of the three cases of $R_{f}$ .",
"Figure REF shows quantitatively how the increment of $R_{f}$ weakens the shape alignments.",
"Although the alignment tendency becomes weaker as $R_{f}$ increases, the shape vector, $\\hat{\\bf e}$ , still shows significant alignment (anti-alignment) with $\\hat{\\bf u}_{3}$ ($\\hat{\\bf u}_{1}$ ) even for the case of $R_{f}=30\\,h^{-1}$ Mpc, which is consistent with the findings of the previous works [71].",
"True as it is that our analytic model shows good quantitive agreements with the numerical results for the case of the high-mass galactic halos, it is not perfect.",
"Some discrepancies are found in the behaviors of $p(\\vert \\hat{\\bf u}_{3}\\cdot \\hat{\\bf e}\\vert )$ and $p(\\vert \\hat{\\bf u}_{1}\\cdot \\hat{\\bf e}\\vert )$ between the analytic model and the numerical results, as can be seen in the bottom panel of Figure REF .",
"The former describes a slightly milder increase of $p(\\vert \\hat{\\bf u}_{3}\\cdot \\hat{\\bf e}\\vert )$ with $\\vert \\hat{\\bf u}_{3}\\cdot \\hat{\\bf e}\\vert $ and a slightly milder decrease of $\\vert \\hat{\\bf u}_{1}\\cdot \\hat{\\bf e}\\vert $ with $\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf e}\\vert $ than the latter especially for the case of the high-mass galactic halos.",
"However, Figure REF shows that the increment of $R_{f}$ improves the agreements between the analytic model and the numerical results, which in turn implies that the discrepancies may be caused by the uncertainties associated with the approximations of $p({\\bf e}\\vert {\\bf T})$ as a multivariate Gaussian distribution and ${\\bf T}$ as a Gaussian random field made to derive the analytic model.",
"The larger the scales are, the more valid these assumptions become.",
"It explains why the analytic model works better at $R_{f}>20\\,h^{-1}$ Mpc."
],
[
"Effect of the Cosmic Web",
"Now, we would like to investigate whether or not the strength of the alignments between the shapes of the galactic halos and the tidal eigenvectors depend on the types of the cosmic web.",
"Following the conventional scheme [22], we classify the galactic halos of each subsample into the knot, filament, sheet and void halos according to the signs of the eigenvalues of ${\\bf T}$ at the grids where the halos are located: $\\lambda _{3}>0 &\\rightarrow & {\\rm knot}\\, ,\\\\\\lambda _{2}>0\\, , \\lambda _{3}<0 &\\rightarrow & {\\rm filament}\\, , \\\\\\lambda _{1}>0\\, , \\lambda _{2}<0 &\\rightarrow & {\\rm sheet}\\, , \\\\\\lambda _{1}<0 &\\rightarrow & {\\rm void}\\, .$ Using only those galactic halos embedded in the same type of the cosmic web, we redo the whole analysis described in Section REF .",
"Figures REF -REF show the same as Figures REF but only with the knot, filament, sheet and void halos, respectively, showing how the shape alignment depends on the web environment.",
"Figure REF plots $\\langle d_{t}\\rangle $ versus $M$ for the four different cases of the web type.",
"As can be seen, the value of $\\langle d_{t}\\rangle $ increases more sharply with the increment of $M$ for the cases of the sheet and void halos than for the cases of the knot and filament counterparts, which indicates that the shapes of the galactic halos in the relatively low-density regions tend to be more strongly aligned with those in the relatively high-density regions.",
"Given that the galactic halos located in the knot and filament regions are expected to have formed earlier and undergone more severe nonlinear evolutions than those in the sheet and void regions [21], the results shown in Figure REF imply that the nonlinear evolution in denser environments will play a decisive role in diminishing the strength of the tidally induced shape alignments of the galactic halos.",
"It is interesting to note that the results shown in Figures REF -REF are in direct contradiction with that of [71] who found the strongest shape alignments of the halos in the knot environments.",
"We think that this apparent inconsistency between our and their results may be related the difference in the web classification scheme.",
"In their analysis, the types of the cosmic web are classified according to the signs of the eigenvalues of the Hessian matrix of the density field.",
"Whereas in our analysis the eigenvalues of the Hessian matrix of the gravitational potential field (i.e, tidal field) are used for the web classification.",
"Figures REF -REF show the same as Figures REF but with only those high-mass galactic halos located in the knot, filament, sheet and void environments, respectively.",
"Figure REF plots $\\langle d_{t}\\rangle $ versus $R_{f}$ for the four different cases of the web type.",
"The decrement of the alignment strength with the increment of $R_{f}$ is found for all of the four types of the cosmic web.",
"The void (knot) galactic halos show the most (least) rapid change of $\\langle d_{t}\\rangle $ with $R_{f}$ .",
"The web-dependence of the rate of the change of $\\langle d_{t}\\rangle $ with $R_{f}$ shown in Figure REF implies that the strength of the tidally induced shape alignments of the galactic halos is determined not only by the difference between $R_{g}$ and $R_{f}$ but also by the strength of the cross correlations between the tidal fields smoothed on different scales.",
"In the denser knot and filament environments, although the nonlinearity diminishes the strength of the initially induced shape alignments with the large-scale tidal fields, the stronger cross correlations between the tidal fields smoothed on different scales slow down the rate of the decrement of the strength of the shape alignments with the increment of $R_{f}$ .",
"Whereas, in the less dense sheet and void regions where the strongest signals of the shape alignments are found on the scale of $R_{f}=5\\,h^{-1}$ Mpc, the weaker cross-correlations between the tidal fields on different scales cause the strengths of the shape alignments to decrease quite rapidly as $R_{f}$ increases.",
"Figures REF -REF clearly demonstrate that our analytic model with the empirically determined parameter, Equations (REF )-(REF ), makes a quantitative success in describing simultaneously and consistently the amplitudes and behaviors of the three probability density functions, $\\lbrace p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf e}\\vert )\\rbrace _{i=1}^{3}$ , for all of the cases of the galactic mass ranges $M$ , the smoothing scales $R_{f}$ and the types of the cosmic web.",
"This notable success of our analytic model confirms the validity of the key assumption made for Equation (REF ) that the covariances of the shapes of the galactic halos have a linear dependence on the large-scale tidal fields [12], [41], [28]."
],
[
"Analytic Models",
"Employing the analytic model based on the LTT theory developed by [37], [38], [39] derived the probability density functions of the coordinates of the unit spin vectors, $\\hat{\\bf s}$ , given $\\hat{\\bf T}$ [44]: $p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf s}\\vert )= \\frac{1}{2\\pi }\\int _{0}^{2\\pi }\\,\\left[\\prod _{n=1}^{3}\\left(1+c_{t}-3c_{t}\\hat{\\lambda }^{2}_{n}\\right)\\right]^{-\\frac{1}{2}}\\left(\\sum _{l=1}^{3}\\frac{\\vert \\hat{\\bf u}_{l}\\cdot \\hat{\\bf s}\\vert }{1+c_{t}-3c_{t}\\hat{\\lambda }^{2}_{l}} \\right)^{-\\frac{3}{2}}\\,d\\phi _{jk}\\, ,$ where $c_{t}$ is the spin correlation parameter in the range of $[0,1]$ [38].",
"The larger value of $c_{t}$ is translated into the stronger $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignment.",
"Although Equation (REF ) is quite similar to Equation (REF ), there is an obvious difference: the former is expressed in terms of $\\hat{\\lambda }^{2}_{i}$ , while the latter in terms of $\\hat{\\lambda }_{i}$ .",
"This difference originates from the fact that the covariances of the spin vectors of the galactic halos have a quadratic dependence on $\\hat{\\bf T}$ according to the LTT theory [18], [70].",
"The core assumption that underlies Equation (REF ) is that the rescaled covariance, $\\langle {s}_{i}{s}_{j}\\vert \\hat{\\bf T}\\rangle $ , can be written as [37] $\\langle {s}_{i}{s}_{j}\\vert \\hat{\\bf T}\\rangle = \\frac{1+c_{t}}{3}\\delta _{ij} - c_{t}\\sum _{k=1}^{3}\\hat{T}_{ik}\\hat{T}_{kj}\\, .$ Solving Equation (REF ) for $c_{t}$ in the principal frame of $\\hat{\\bf T}$ gives In [44], there was a typo in the formula.",
"It is corrected here.",
"[38] $c_{t} = \\frac{10}{3} - 10\\sum _{i=1}^{3}\\hat{\\lambda }_{i}^{2}\\langle \\hat{s}^{2}_{i}\\vert \\hat{\\bf T}\\rangle \\, .$ Equation (REF ) enables us to evaluate the value of $c_{t}$ directly from the values of $\\hat{\\bf s}$ , $\\lbrace \\hat{\\lambda }_{i}\\rbrace _{i=1}^{3}$ and $\\lbrace \\hat{\\bf u}_{i}\\rbrace _{i=1}^{3}$ .",
"Several observational and numerical studies showed that this analytic model, Equations (REF )-(REF ), was indeed useful and adequate in describing the tidally induced spin alignments especially in the sheet environments [46], [63], [40], [44].",
"As mentioned in Section , however, the LTT theory breaks down on the mass scale below $M_{\\rm t}\\sim 10^{12}\\,h^{-1}\\,M_{\\odot }$ .",
"The numerical analyses based on recent large high-resolution $N$ -body simulations found that the spin flip, a transition of the tendency from the $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ ($M> M_{\\rm t}$ ) alignments to the $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ alignments ($M\\le M_{\\rm t}$ ) occurs [2], [15], [65] and that the value of the transition mass scale, $M_{\\rm t}$ , depends on the type of the cosmic web [45].",
"Now, we would like to construct a new model that might describe quantitatively the transition of the spin alignment tendency at $M_{\\rm t}$ and its dependence on the type of the cosmic web.",
"In the light of the previous studies which claimed that the nset of the non-Gaussianity of the tidal fields even on large scales would cause the covariance, $\\langle {s}_{i}{s}_{j}\\vert \\hat{\\bf T}\\rangle $ , to scale linearly with $\\hat{\\bf T}$ [28], [41], we first modify Equation (REF ) into $\\langle {s}_{i}{s}_{j}\\vert \\hat{\\bf T}\\rangle = \\frac{(1+c_{t}+d_{t})}{3}\\delta _{ij} -c_{t}\\sum _{k=1}^{3}\\hat{T}_{ik}\\hat{T}_{kj} - d_{t}\\hat{T}_{ij}\\, ,$ where two spin correlation parameters, $c_{t}$ and $d_{t}$ , both lying in the range of $[0,1]$ , are introduced to correlate $\\hat{\\bf s}$ to $\\hat{\\bf u}_{2}$ and to $\\hat{\\bf u}_{3}$ , respectively.",
"If the first spin correlation parameter, $c_{t}$ , is close to zero and the second spin correlation parameter, $d_{t}$ , is close to unity, then the spin vectors $\\hat{\\bf s}$ will show strong alignments with $\\hat{\\bf u}_{3}$ just like the shape vectors, $\\hat{\\bf e}$ .",
"If $c_{t} $ is close to unity and $d_{t}$ is close to zero, then it will be reduced to the original model, Equation (REF ), which describes the $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignments.",
"If both of the parameters are close to zero, then the spin vectors of the galactic halos will be random having no correlations with the large-scale tidal fields.",
"Replacing Equation (REF ) by Equations (REF ) in the original derivation of Equation (REF ), it is straightforward to show that the probability density functions, $p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf s}\\vert )$ , can be expressed as $p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf s}\\vert )&=& \\frac{1}{2\\pi }\\int _{0}^{2\\pi }\\,\\left[\\prod _{n=1}^{3}\\left(1+c_{t}-3c_{t}\\hat{\\lambda }^{2}_{n}+d_{t}-3d_{t}\\hat{\\lambda }_{n}\\right)\\right]^{-\\frac{1}{2}}\\times \\,\\nonumber \\\\&&\\left[\\sum _{l=1}^{3}\\left(\\frac{\\vert \\hat{\\bf u}_{l}\\cdot {\\bf s}\\vert }{1+c_{t}-3c_{t}\\hat{\\lambda }^{2}_{l}+d_{t}-3d_{t}\\hat{\\lambda }_{l}}\\right)\\right]^{-\\frac{3}{2}}\\,d\\phi _{jk}\\, .$ Equation (REF ), which was originally derived in the LTT theory, holds true even when the covariance, $\\langle \\hat{s}_{i}\\hat{s}_{j}\\vert \\hat{\\bf T}\\rangle $ , has an additional term, since the second and third terms in Equation (REF ) are uncorrelated due to $\\langle \\hat{T}_{ik}\\hat{T}_{kl}\\hat{T}_{lj}\\rangle =0$ [38].",
"Thus, the same formula as Equation (REF ) can be used to obtain the value of $c_{t}$ for this new model.",
"Likewise, the same formula as Equation (REF ) but with $\\hat{\\bf e}$ replaced by $\\hat{\\bf s}$ can be used to obtain the value $d_{t}$ as $d_{t} = -\\frac{5}{3}\\sum _{i=1}^{3}\\hat{\\lambda }_{i}\\langle \\hat{s}^{2}_{i}\\vert \\hat{\\bf T}\\rangle \\, .$ In Section REF , we will numerically test three models for the galaxy spin alignments, model I, model II and model III.",
"The model III is Equation (REF ) with two non-zero parameters, $c_{t}$ and $d_{t}$ .",
"The model I is Equation (REF ) with $d_{t}=0$ .",
"It is identical to the original model based on the LTT theory, Equation (REF ).",
"The model II is Equation (REF ) with $c_{t}=0$ .",
"It has the same functional form as Equation (REF ) for the tidally induced shape alignments."
],
[
"Numerical Tests",
"To numerically obtain three probability density functions, $\\lbrace p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf s}\\vert )\\rbrace _{i=1}^{3}$ , we perform the exactly same calculations as presented in Section REF , but with $\\hat{\\bf e}$ replaced by $\\hat{\\bf s}$ .",
"For the evaluation of the three analytic models, we first determine the ensemble values of $\\langle c_{t}\\rangle $ and $\\langle d_{t}\\rangle $ for each subsample as, $\\langle c_{t}\\rangle &=& \\frac{1}{N_{t}}\\sum _{\\alpha =1}^{N_{t}}\\left[\\frac{10}{3}-10\\sum _{i=1}^{3}\\hat{\\lambda }^{2}_{\\alpha ,i}\\vert \\hat{\\bf u}_{\\alpha ,i}\\cdot \\hat{\\bf s}_{\\alpha }\\vert ^{2}\\right]\\, ,\\\\\\langle d_{t}\\rangle &=& \\frac{1}{N_{t}}\\sum _{\\alpha =1}^{N_{t}}\\left[-\\frac{5}{3}\\sum _{i=1}^{3}\\hat{\\lambda }_{\\alpha ,i}\\vert \\hat{\\bf u}_{\\alpha ,i}\\cdot \\hat{\\bf s}_{\\alpha }\\vert ^{2}\\right]\\, ,$ and put these ensemble average values into Equation (REF ) to evaluate the model III.",
"Putting $\\langle c_{t}\\rangle $ ($\\langle d_{t}\\rangle $ ) into Equation (REF ) and setting $\\langle d_{t}\\rangle $ ($\\langle c_{t}\\rangle $ ) at zero, we evaluate the mode I (model II).",
"Figure REF plots the numerically obtained probability density functions, $\\lbrace p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf s}\\vert )\\rbrace _{i=1}^{3}$ (filled dots), and compares them with the model I (blue lines), model II (green lines) and model III (red lines).",
"As can be seen, the three functions, $\\lbrace p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf s}\\vert )\\rbrace _{i=1}^{3}$ , have much lower amplitudes than $\\lbrace p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf e}\\vert )\\rbrace _{i=1}^{3}$ displayed in Figure REF .",
"It indicates that the spin vectors of the galactic halos are much less strongly aligned with the large-scale tidal fields than the shape vectors, which is consistent with the results of the previous numerical and observational studies [23], [20], [74].",
"The occurrence of the spin-flip phenomenon is indeed witnessed: For the case of the lower mass galactic halos with $M< 10^{12}\\,h^{-1}M_{\\odot }$ , the unit spin vectors, $\\hat{\\bf s}$ , tend to be aligned not with the intermediate eigenvectors, $\\hat{\\bf u}_{2}$ , but with the minor eigenvectors, $\\hat{\\bf u}_{3}$ , while the high-mass galactic halos with $M\\ge 10^{12}\\,h^{-1}M_{\\odot }$ , exhibit the stronger alignments of $\\hat{\\bf s}$ with $\\hat{\\bf u}_{2}$ rather than with $\\hat{\\bf u}_{3}$ , which is quite consistent with the previous numerical results [2], [23], [15], [45], [19], [13], [65].",
"The strength of the $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ alignments tends to decrease with $M$ , while the strengths of the $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignments increases with $M$ .",
"These opposite trends can be quantitatively described by the variation of the first and second spin correlation parameters with $M$ as shown in the top and bottom panels of Figure REF , respectively.",
"As can be seen, $\\langle d_{t}\\rangle $ is larger than $\\langle c_{t}\\rangle $ in the lower mass range of $M<10^{12}\\,h^{-1}M_{\\odot }$ but drops below $\\langle c_{t}\\rangle $ in the higher mass range of $M\\ge 10^{12}\\,h^{-1}M_{\\odot }$ .",
"The transition mass scale of the spin-flip corresponds to the moment when $\\langle d_{t}\\rangle $ becomes lower than $\\langle c_{t}\\rangle $ .",
"For the case of the lowest and low-mass galactic halos with $M<5\\times 10^{11}\\,h^{-1}M_{\\odot }$ , both of the models II and III succeed in matching simultaneously the amplitudes and behaviors of the three numerically obtained probability density functions.",
"The model II is almost identical to the model III in these low-mass ranges, since the values of $\\langle c_{t}\\rangle $ obtained via Equation (REF ) are low for these cases.",
"It is also worth noting that the signal of the strong $\\hat{\\bf u}_{1}$ -$\\hat{\\bf s}$ anti-alignments is found to increase with $M$ whose behavior is well described by both of the model II and III.",
"The success of the model II and model III and the failure of the model I in describing the amplitudes and behaviors of $p(\\vert \\hat{\\bf u}_{3}\\cdot \\hat{\\bf s}\\vert )$ and $p(\\vert \\hat{\\bf u}_{1}\\cdot \\hat{\\bf s}\\vert )$ are also found for the case of the medium-mass halos (second from the bottom panels in Figure REF ).",
"It is, however, interesting to note that in this medium-mass range the spin vectors, $\\hat{\\bf s}$ , exhibit a weak but non-negligible alignment with the intermediate eigenvectors, $\\hat{\\bf u}_{2}$ , which tendency is properly described by both of model I and model III but not by the model II.",
"For the case of the high-mass galactic halos, the unit spin vectors, $\\hat{\\bf s}$ , turn out to be more strongly aligned with $\\hat{\\bf u}_{2}$ than with $\\hat{\\bf u}_{3}$ , which cannot be described by the model II.",
"But, the alignments of $\\hat{\\bf s}$ with $\\hat{\\bf u}_{3}$ and its anti-alignments with $\\hat{\\bf u}_{1}$ are still well described by the model II and III but not by the model I.",
"Thus, it is only the model III that agrees concurrently and consistently with the numerically obtained three probability density functions, $p(\\vert \\hat{\\bf u}_{3}\\cdot \\hat{\\bf s}\\vert )$ , $p(\\vert \\hat{\\bf u}_{2}\\cdot \\hat{\\bf s}\\vert )$ , and $p(\\vert \\hat{\\bf u}_{1}\\cdot \\hat{\\bf s}\\vert )$ , in all of the four mass ranges.",
"Figure REF shows the same as the bottom panels of Figure REF but for the cases that the tidal fields are smoothed on three larger scales of $R_{f}=10,\\ 20$ and $30\\,h^{-1}$ Mpc, in the top, middle and bottom panels, respectively.",
"As can be seen, the increment of $R_{f}$ decreases the alignment strengths even more rapidly than for the case of the shape alignments.",
"Note that it is only the model III that succeeds in making good simultaneous descriptions of the amplitudes and behaviors of $\\lbrace p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf s}\\vert )\\rbrace _{i=1}^{3}$ for all of the three cases of $R_{f}$ .",
"Although the model III achieves overall good agreements with the numerical results, some discrepancies between its description and the numerical results are found.",
"As can be seen in Figures REF -REF , the numerically obtained three probability functions display substantially fluctuating behaviors especially for the case of the high-mass galactic halos.",
"However, the increment of $R_{f}$ reduces these discrepancies as shown in Figure REF , which implies that the inaccuracies associated with the approximations of ${\\bf T}$ as a Gaussian random field and $p({\\bf s}\\vert {\\bf T})$ as a multivariate Gaussian distribution in the derivation of Equation (REF ) should be largely responsible for these discrepancies.",
"The uncertainties involved in the measurements of the spin vectors of the galactic halos may be another source of the discrepancies.",
"Since the spin direction of a galactic halo is dominantly determined by the positions and velocities of the outmost DM particles from the halo center, its measurement would depend sensitively on the dynamical state of the galactic halo, halo-finding algorithm and definition of the virial radius.",
"If a high-mass galactic halo has yet to be fully relaxed and/or in the middle of merging, containing multiple substructures, the measurement of its spin direction is likely to suffer from substantial uncertainties, which in turn would cause mismatches between the analytical and the numerical results on the spin alignments with the large-scale tidal field.",
"Figure REF shows how the first and second spin parameters vary with $R_{f}$ for the high-mass galactic halos in the top and bottom panels, respectively.",
"As can be seen, both of the parameters decrease with the increment of $R_{f}$ .",
"The two parameters, however, show different variations with $M$ .",
"The first spin parameter, $\\langle c_{t}\\rangle $ , decreases more rapidly with the increment of $R_{f}$ than the second spin parameter, $\\langle d_{t}\\rangle $ .",
"It is found that $\\langle c_{t}\\rangle >\\langle d_{t}\\rangle $ at $R_{f}\\le 20\\,h^{-1}$ Mpc, while $c_{t}<d_{t}$ at $R_{f}=30\\,h^{-1}$ Mpc.",
"This result implies that the occurrence of the spin flip phenomenon is contingent on the sizes of the large-scale structures.",
"Suppose that the galaxies with masses in the range of $0.5\\le M/(10^{11}\\,h^{-1}\\,M_{\\odot })\\le 50$ embedded in a coherent large-scale structure like a filament with size $R_{f}\\ge 30\\,h^{-1}$ Mpc.",
"According to our results, the spin vectors of those galaxies would not flip, with their spins always aligned with the elongated axes of the host filament since $\\langle d_{t}\\rangle $ is always higher than $\\langle c_{t}\\rangle $ in the given mass range (see Section REF )."
],
[
"Effect of the Cosmic Web",
"Following the same procedure as presented in Section REF , we investigate how the probability density functions, $\\lbrace p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf s}\\vert )\\rbrace _{i=1}^{3}$ , depend on the type of the cosmic web.",
"Figures REF -REF show the same as Figure REF but only with the galactic halos located in the knot, filament, sheet and void environments, respectively.",
"In the knot environments (Figure REF ), the unit spin vectors, $\\hat{\\bf s}$ , of the galactic halos are found strongly aligned with the minor eigenvector $\\hat{\\bf u}_{3}$ in all of the four mass ranges (i.e., no spin-flip).",
"For the cases of the lowest-mass, low-mass and medium-mass knot galactic halos, we find $\\hat{\\bf s}$ to be slightly anti-aligned rather than aligned with $\\hat{\\bf u}_{2}$ , while the high-mass knot galactic halos show weak $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignments.",
"Both of the model II and model III describe well the $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ alignment and the $\\hat{\\bf u}_{1}$ -$\\hat{\\bf s}$ anti-alignment.",
"However, model II cannot describe the observed tendency of the $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ anti-alignment in the mass scale of $5\\le M/(10^{11}\\,h^{-1}\\,M_{\\odot })< 10$ while the model III can.",
"It is interesting to see that the model I describes better the observed $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ anti-alignments in the medium-mass range better than the model II although it still notoriously fails in describing the observed strong $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ alignments and $\\hat{\\bf u}_{1}$ -$\\hat{\\bf s}$ anti-alignments.",
"The filament galactic halos yield much stronger $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ alignment and $\\hat{\\bf u}_{1}$ -$\\hat{\\bf s}$ anti-alignment in all of the four mass ranges than the knot counterpart, although the behaviors of $\\lbrace p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf s}\\vert )\\rbrace _{i=1}^{3}$ between the two cases are quite similar to each other (Figure REF ).",
"The high-mass filament galactic halos show a substantial $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignment whose strength is comparable to that of the $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ (i.e., the occurrence of the spin flip).",
"Although the model III works quite well in matching the numerically obtained probability density functions, it is interesting to note that the model II gives a better description of $p(\\vert \\hat{\\bf u}_{2}\\cdot \\hat{\\bf s}\\vert )$ than the model III in the mass range of $M< 10^{12}\\,h^{-1}M_{\\odot }$ .",
"The sheet galactic halos exhibit a different trend (Figure REF ).",
"Their spin vectors tend to lie in the plane spanned by $\\hat{\\bf u}_{2}$ and $\\hat{\\bf u}_{3}$ , being orthogonal to $\\hat{\\bf u}_{1}$ .",
"The increment of $M$ leads to the stronger $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignment and $\\hat{\\bf u}_{1}$ -$\\hat{\\bf s}$ anti-alignment but weaker $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ alignment.",
"For the case of the lowest-mass and low-mass sheet galactic halos, the $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignment tendency is weaker than the $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ alignment.",
"For the case of the medium-mass sheet galactic halos, the $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignment begins to exceed in strength the $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ alignment (i.e., occurrence of the spin flip).",
"The strongest signal of the $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignments is found from the high-mass sheet galactic halos, which result is consistent with the previous numerical finding of [22].",
"As can be seen, only the Model III succeeds in describing simultaneously and consistently the behaviors of $\\lbrace p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf s}\\vert )\\rbrace _{i=1}^{3}$ , fairly well for the case of the sheet galactic halos in all of the four mass ranges.",
"This result is inconsistent with the observational finding of [74] that the galaxies in the knot environments exhibited the strongest spin alignments with the tidal fields.",
"We suspect that two factors may have caused this inconsistency between the numerical and observational results on the web dependence of the spin alignments.",
"First, the difference in the way in which the tidal fields were constructed.",
"In the work of [74], the tidal fields, $\\hat{\\bf T}$ , were constructed from the spatial distributions of the galaxy groups, while in the SMDPL the spatial distribution of the DM particles were used.",
"Second, the difference in the measurements of $\\hat{\\bf s}$ : In the observational analysis of [74], the unit spin vectors $\\hat{\\bf s}$ were determined from the luminous parts of the galaxies while in the current numerical analyses, all of the constituent DM particles determine $\\hat{\\bf s}$ .",
"The weakest spin alignments with the large-scale tidal fields are found in the void environments (Figure REF ).",
"Although the signals are quite lower than those yielded by the sheet galactic halos, the behaviors of $\\lbrace p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf s}\\vert )\\rbrace _{i=1}^{3}$ obtained from the void galactic halos are quite similar to those from the sheet galactic halos: the alignments of $\\hat{\\bf s}$ with $\\hat{\\bf u}_{2}$ and $\\hat{\\bf u}_{3}$ .",
"The former (the latter) alignment become stronger (weaker) with the increment of $M$ .",
"For the lowest-mass and low-mass void galactic halos (top two panels), the $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ alignment is slightly stronger than the $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignment.",
"Only the Model III pulls it off to describe simultaneously the behaviors of $\\lbrace p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf s}\\vert )\\rbrace _{i=1}^{3}$ .",
"For the case of the medium-mass and high-mass void galactic halos, however, the large errors make it difficult to interpret the numerical results and to make a fair comparison of them with the three models.",
"Figures REF and REF plot $\\langle c_{t}\\rangle $ and $\\langle d_{t}\\rangle $ versus $M$ for the four different web types, respectively.",
"Although the increment of the first spin correlation parameter, $\\langle c_{t}\\rangle $ , with $M$ is universally shown, the increment rate sensitively depends on the web type.",
"The most (least) rapid change of $\\langle c_{t}\\rangle $ with $M$ is found from the sheet (knot) galactic halos.",
"Meanwhile, the second spin correlation parameter, $\\langle d_{t}\\rangle $ , does not show strong variations with $M$ .",
"For the case of the high-mass filament and void galactic halos, however, it shows an abrupt decrement with $M$ .",
"Defining the transition mass, $M_{t}$ , as the one beyond which $\\langle c_{t}\\rangle $ exceeds $\\langle d_{t}\\rangle $ , we expect the galactic halos with $M>M_{t}$ ($M\\le M_{t}$ ) to exhibit the preferential $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ ($\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ ) alignment.",
"The results shown in Figures REF and REF imply that the value of $M_{t}$ depends on the web type.",
"as shown in [45].",
"For the case of the knot galactic halos, no spin flip occurs in the given whole mass range since $\\langle d_{t}\\rangle $ is always larger than $\\langle c_{t}\\rangle $ .",
"The spin flip of the filament (sheet) galactic halos is expected to occur around $M_{t}\\sim 5\\times 10^{12}\\,h^{-1}M_{\\odot }$ ($M_{t}\\sim 10^{12}\\,h^{-1}M_{\\odot }$ ), while the void galactic halos show the lowest transition mass scale, $M_{t}\\sim 5\\times 10^{11}\\,h^{-1}M_{\\odot })$ .",
"As done in Section REF , smoothing the tidal fields on three larger scales $R_{f}$ and repeating the whole calculation for each case of $R_{f}$ , we investigate the dependence of the tendency and strength of the spin alignments on $R_{f}$ for the case of the high-mass galactic halos.",
"Figures REF -REF plot the same as the bottom panels of Figures REF -REF , respectively, but for the cases of $R_{f}=10,\\ 20,\\ 30\\,h^{-1}$ Mpc.",
"As can be seen, whatever type of the cosmic web the galactic halos are embedded in, the increment of $R_{f}$ always decreases the alignment strength, which is well described by the model III.",
"For the cases of the high-mass galactic halos in the knot and filament regions, the increment of $R_{f}$ just decreases the strength of the spin alignments but does not change its tendency (Figures REF -REF ).",
"However, for the case of the high-mass sheet galactic halos (Figure REF ), it changes both of the strength and the tendency of the spin alignments.",
"On the scales of $R_{f}=10$ and $20\\,h^{-1}$ Mpc, the high-mass sheet galactic halos show the stronger $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignments than the $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ alignments.",
"But, on the larger scale of $R_{f}=30\\,h^{-1}$ Mpc, we witness a different tendency, the $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ alignments seem slightly stronger than the $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignments.",
"In other words, if the sheet environment is defined on the scale equal to or larger than $30\\,h^{-1}$ Mpc, no spin-flip will occur in the given mass range.",
"Since both of $\\hat{\\bf u}_{2}$ and $\\hat{\\bf u}_{3}$ span the plane of a sheet [75], our result shown in Figure REF supports the claim of [23] that the spin vectors of the DM halos have a universal tendency of lying in the plane of the sheet, regardless of the halo mass."
],
[
"Summary and Discussion",
"To study the large-scale tidal effect on the spin and shape orientations of the galaxies and the spin-flip phenomenon, we have considered three different analytic models, the model I, model II and model III.",
"The model I, Equation (REF ), which was originally developed by [37] based on the LTT theory, describes the alignment tendency between the galaxy spin vectors, $\\hat{\\bf s}$ , and the intermediate eigenvectors, $\\hat{\\bf u}_{2}$ , of the large-scale tidal field, ${\\bf T}$ .",
"The model II, Equation (REF ), has been constructed here to describe the alignments (anti-alignments) of the galaxy shapes, $\\hat{\\bf e}$ , with the minor (major) eigenvectors, $\\hat{\\bf u}_{3}$ ($\\hat{\\bf u}_{1}$ ) of ${\\bf T}$ .",
"This model is based on the first order Lagrangian perturbation theory according to which the major principal axes of the inertia momentum tensors of the galactic halos are perfectly aligned with the minor principal axes of the local tidal tensors in the Lagrangian regime.",
"The model III, Equation (REF ), is a practical formula constructed by combining the model I and model II to describe simultaneously the tidally induced shape and spin alignments.",
"The model I (model II) carries a single parameter, $c_{t}$ ($d_{t}$ ), which measures the strength of the alignment with $\\hat{\\bf u}_{2}$ ($\\hat{\\bf u}_{3}$ ).",
"The model III carries two parameters, $c_{t}$ and $d_{t}$ , whose relative ratio determines the transition mass scale for the occurrence of the spin-flip.",
"The first parameter, $c_{t}$ , would reach the maximum value of unity, if the inertia momentum tensors of the galaxies are uncorrelated with the surrounding tidal tensors, while the second parameter, $d_{t}$ , will attain the value of unity if the two tensors are perfectly correlated.",
"These parameters can be empirically determined by Equation (REF ) directly from the measured values of $\\hat{\\bf e}$ and $\\hat{\\bf s}$ in the principal frame of $\\hat{\\bf T}$ without resorting to any fitting procedure.",
"To numerically test the three analytic models, we have utilized the density fields and the Rockstar halo catalogs extracted from the SMDPL simulations [33].",
"Constructing the unit traceless tidal tensor, $\\hat{\\bf T}$ , smoothed on the scale of $R_{f}=5\\,h^{-1}$ Mpc from the density fields given on the $512^{3}$ grids that constitute the simulation box of volume $400^{3}\\,h^{-3}\\,{\\rm Mpc}^{3}$ and selecting the galactic halos in the mass range of $0.5\\le M/(10^{11}\\,h^{-1}\\,M_{\\odot })\\le 50$ from the Rockstar catalog, we have first numerically obtained the probability density functions of the tidally induced shape alignments, $\\lbrace p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf e}\\vert )\\rbrace _{i=1}^{3}$ (see Figures REF -REF ).",
"The numerical results have clearly shown that $\\hat{\\bf e}$ has a tendency to be strongly aligned (anti-aligned) with $\\hat{\\bf u}_{3}$ ($\\hat{\\bf u}_{1}$ ) but no correlation with $\\hat{\\bf u}_{2}$ .",
"Investigating the dependence of the strength of the tidally induced shape alignments on $M$ , $R_{f}$ , and the type of the cosmic web, it has been found that the more massive galactic halos yield stronger $\\hat{\\bf u}_{3}$ -$\\hat{\\bf e}$ alignments ($\\hat{\\bf u}_{1}$ -$\\hat{\\bf e}$ anti-alignments) and that the increment of $R_{f}$ weakens the alignment tendency (see Figures REF ).",
"These numerical results are consistent with what the previous works already found [29], [73], [13], [26], [71], [50].",
"The strongest (weakest) $\\hat{\\bf u}_{3}$ -$\\hat{\\bf e}$ alignments are found from the void (knot) galactic halos (see Figures REF -REF ), which seem inconsistent with the previous numerical result that the DM halos showed the strongest shape alignments in the knot environments [71].",
"This inconsistency has been ascribed to the different classification schemes used in the two analyses.",
"The sheet galactic halos yield much stronger shape alignment tendency than the knot and filament galactic halos in the whole mass range, which result is consistent with what [22] found.",
"In the lowest and low mass range ($0.5\\le M/[10^{11}\\,h^{-1}\\,M_{\\odot }]<5)$ , the knot and filament galactic halos show similar strengths of the shape alignments.",
"In the medium-mass ($5\\le M/[10^{11}\\,h^{-1}\\,M_{\\odot }]<10$ ) and high-mass ($10\\le M/[10^{11}\\,h^{-1}\\,M_{\\odot }]<50$ ) ranges, the shape alignments of the filament galactic halos become stronger than the knot counterparts (Figure REF ).",
"These numerical results imply that the void and sheet galactic halos retain best the tidally induced shape alignments, while the evolution of the galactic halos in the dense environments like the knots and filaments has an effect of deviating the directions of their shapes from the tidally induced inclinations.",
"The comparison with the numerical results revealed the success of the model II in describing the amplitudes and behaviors of $\\lbrace p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf e}\\vert )\\rbrace _{i=1}^{3}$ , for all of the cases of $M$ , $R_{f}$ and the type of the cosmic web.",
"For the shape alignments, the model III turns out to be identical to the model II.",
"For all of the four cases of the web type, the increment of $R_{f}$ has been found to decrease the strength of the tidally induced shape alignments but improve the agreements between the model III and the numerical results (Figure REF -REF ).",
"We interpret this result as an evidence supporting the scenario that the nonlinear evolution has an effect of diminishing the strength of the tidally induced shape alignments.",
"In a similar manner, we have numerically determined the probability density functions of the tidally induced spin alignments, $\\lbrace p(\\vert \\hat{\\bf u}_{i}\\cdot \\hat{\\bf s}\\vert )\\rbrace _{i=1}^{3}$ , explored their dependences on $M$ , $R_{f}$ and the web type, and compared the results with the three analytic models.",
"The tidally induced spin alignments have been found significant but quite weak compared with the shape alignments (Figures REF -REF ), consistent with the results from the previous works [23], [20], [74].",
"The occurrence of the spin-flip phenomenon has been witnessed.",
"For the case of $R_{f}=5\\,h^{-1}$ Mpc, the lowest-mass, low-mass and medium-mass galactic halos show strong $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ alignments and negligible $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignments, while the high-mass galactic halos exhibit strong $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignments, which results have confirmed the claims of the previous works [2], [49], [72], [15], [45], [62], [19], [13], [65].",
"However, we have noted that the spin-flip does not occur abruptly at a certain fixed transition mass scale.",
"Rather it is a gradual transition of the spin alignment tendency that proceeds over a broader mass range, depending on $R_{f}$ (Figures REF -REF ).",
"For the case of $R_{f}=5\\,h^{-1}$ Mpc, the high-mass galactic halos have been found to yield stronger $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ and weaker but significant $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ alignments, while the medium-mass galactic halos exhibit strong $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ and much weaker $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignments.",
"For the case of $R_{f}\\ge 10\\,h^{-1}$ Mpc, however, the high-mass galactic halos exhibit stronger $\\hat{\\bf u}_{3}$ -$\\hat{\\bf s}$ and weaker but significant $\\hat{\\bf u}_{2}$ -$\\hat{\\bf s}$ alignments.",
"The strengths of the tidally induced spin alignments have been also found to sensitively vary with the types of the cosmic web (see Figures REF -REF ), which supports the claim of [45].",
"The strongest (weakest) signals of the tidally induced spin alignments have been found from the sheet (void) galactic halos, while the filament galactic halos have been found to have stronger spin alignments than the knot counterparts in the whole mass range ( Figures REF -REF ).",
"These results are inconsistent with the observational finding of [74] that the knot galaxies exhibited the strongest signals of the spin alignments.",
"We have suspected that this inconsistency might be related to the construction of the tidal field from the galaxy groups and the determination of the spin axes of the galaxies from their stellar components in the observational analysis.",
"Determining empirically $\\langle c_{t}\\rangle $ and $\\langle d_{t}\\rangle $ from the numerical data (Figures REF -REF ) and defining the condition for the occurrence of the spin flip as $\\langle c_{t}\\rangle > \\langle d_{t}\\rangle $ , we have quantitatively investigated how the occurrence and the transition mass scale, $M_{t}$ , of the spin-flip phenomenon depend on the size and type of the cosmic web and found the following: Regardless of the web type, the transition mass scale, $M_{t}$ , of the spin-flip increases with the increment of $R_{f}$ .",
"The knot galactic halos do no show any spin-flip phenomenon.",
"That is, the unit spin vectors, $\\hat{\\bf s}$ , of the knot galactic halos are always preferentially aligned with $\\hat{\\bf u}_{3}$ rather than with $\\hat{\\bf u}_{2}$ in the whole mass range, regardless of the value of $R_{f}$ (Figure REF ).",
"For the case of the filament galactic halos, the spin flip occurs around $M_{\\rm t}\\sim 5\\times 10^{12}\\,h^{-1}\\,M_{\\odot }$ when $R_{f}=5\\,h^{-1}$ Mpc.",
"At the larger scale of $R_{f}>5\\,h^{-1}$ Mpc, the value of $M_{\\rm t}$ exceeds the galactic mass scales, i.e, $M_{\\rm t}> 5\\times 10^{12}\\,h^{-1}\\,M_{\\odot }$ (Figure REF ).",
"In the sheet environment, the transition mass scale has a lower value than in the filaments: $M_{\\rm t}\\sim 10^{12}\\,h^{-1}\\,M_{\\odot }$ when $R_{f}=5\\,h^{-1}$ Mpc.",
"Only when $R_{f}$ reaches $30\\,h^{-1}$ Mpc, the value of $M_{\\rm t}$ becomes larger than the galactic mass scale (Figure REF ).",
"The void galactic halos yield the lowest transition mass scale, $M_{\\rm t}\\sim 5\\times 10^{11}\\,h^{-1}\\,M_{\\odot }$ when $R_{f}=5\\,h^{-1}$ Mpc.",
"At the larger scales, the number of the void galactic halos is too low to produce any significant signals (Figure REF ).",
"It is interesting to note that our results on the web and mass dependence of the spin-flip phenomenon are consistent with the theoretical explanation of [16], according to which the misalignments between the inertia momentum and tidal tensors in the anisotropic environments like the filaments and sheets are largely responsible for the occurrence of the spin flip.",
"In line with their theoretical explanation, we interpret no occurrence of the spin flip in the knot environments as an evidence for the stronger alignments between the two tensors in the dense environments.",
"In other words, in the knot regions where the tidal tensors are more isotropic, the inertia momentum and tidal tensors may be more strongly aligned with each other, which plays a role in suppressing the occurrence of the spin-flip of the knot galaxies.",
"It has also been clearly demonstrated in the current work that the model III succeeds in describing consistently and simultaneously the numerical results of the tidally induced shape and spin alignments for all of the cases of $M$ , $R_{f}$ and type of the cosmic web, while the model I and model II fail.",
"Showing that the model III works better as $R_{f}$ increases, we have ascribed the slight mismatches between the numerical results and the model III to the inaccuracies caused by the approximations of $p({\\bf s}\\vert {\\bf T})$ as a multivariate Gaussian distribution and $\\hat{\\bf T}$ as a Gaussian random field made in the construction of the model III.",
"We also suspect that the uncertainties in the measurements of $\\hat{\\bf s}$ and $\\hat{\\bf e}$ caused by the simple assumptions of each galactic halo having a perfect ellipsoidal shape and no substructure in a completely relaxed dynamical state must contribute to the mismatches.",
"We conclude that the model III is an effective practical model for the spin and the shape alignments of the galactic halos with the large-scale tidal fields, providing an analytic tool with which the condition of the spin flip occurrence as well as its dependence on the properties of the large-scale structures can be quantitatively described.",
"Its good accord with the numerical results supports the scenario that the occurrence of the spin flip phenomenon is associated more with the geometrical properties of the large-scale tidal field as well as the interactions of the galactic halos with the cosmic web rather than with the physical processes during the nonlinear evolution [45], [16], [68], [65].",
"Given that the model III is expressed in terms of the linear quantities, it may provide another independent probe of the background cosmology.",
"For this purpose, however, a couple of back-up works will have to be done.",
"First, as suspected in our analysis, differences in the schemes used to to construct the tidal fields, to measure the shape and spin axes of the galaxies, and to classify the cosmic web would yield different patterns in the dependence of the tidally induced shape and spin alignments on the sizes and types of the cosmic web.",
"Thus, it will be necessary to test the robustness of the model III against the variations of the schemes.",
"Second, it will be also essential to examine its validity using the numerical results for alternative cosmologies such as models with modified gravity, coupled dark energy, massive neutrinos, primordial non-Gaussianity, anisotropic inflation and so forth.",
"Our future work is in this direction.",
"ÒThe CosmoSim database used in this paper is a service by the Leibniz-Institute for Astrophysics Potsdam (AIP).",
"The MultiDark database was developed in cooperation with the Spanish MultiDark Consolider Project CSD2009-00064.Ó I gratefully acknowledge the Gauss Centre for Supercomputing e.V.",
"(www.gauss-centre.eu) and the Partnership for Advanced Supercomputing in Europe (PRACE, www.prace-ri.eu) for funding the MultiDark simulation project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, www.lrz.de).",
"The Bolshoi simulations have been performed within the Bolshoi project of the University of California High-Performance AstroComputing Center (UC-HiPACC) and were run at the NASA Ames Research Center.",
"I thank an anonymous referee for providing very helpful suggestions and constructive criticisms.",
"I acknowledge the support of the Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Education (NO.",
"2016R1D1A1A09918491).",
"I was also partially supported by a research grant from the NRF of Korea to the Center for Galaxy Evolution Research (No.2017R1A5A1070354)."
]
] | 1808.08559 |
[
[
"Design and Characterization of a Balloon-Borne Diffraction-Limited\n Submillimeter Telescope Platform for BLAST-TNG"
],
[
"Abstract The Next Generation Balloon-borne Large Aperture Submillimeter Telescope (BLAST-TNG) is a submillimeter mapping experiment planned for a 28 day long-duration balloon (LDB) flight from McMurdo Station, Antarctica during the 2018-2019 season.",
"BLAST-TNG will detect submillimeter polarized interstellar dust emission, tracing magnetic fields in galactic molecular clouds.",
"BLAST-TNG will be the first polarimeter with the sensitivity and resolution to probe the $\\sim$0.1 parsec-scale features that are critical to understanding the origin of structures in the interstellar medium.",
"With three detector arrays operating at 250, 350, and 500 $\\mu$m (1200, 857, and 600 GHz), BLAST-TNG will obtain diffraction-limited resolution at each waveband of 30, 41, and 59 arcseconds respectively.",
"To achieve the submillimeter resolution necessary for its science goals, the BLAST-TNG telescope features a 2.5 m aperture carbon fiber composite primary mirror, one of the largest mirrors flown on a balloon platform.",
"Successful performance of such a large telescope on a balloon-borne platform requires stiff, lightweight optical components and mounting structures.",
"Through a combination of optical metrology and finite element modeling of thermal and mechanical stresses on both the telescope optics and mounting structures, we expect diffraction-limited resolution at all our wavebands.",
"We expect pointing errors due to deformation of the telescope mount to be negligible.",
"We have developed a detailed thermal model of the sun shielding, gondola, and optical components to optimize our observing strategy and increase the stability of the telescope over the flight.",
"We present preflight characterization of the telescope and its platform."
],
[
"Introdution",
"The Next Generation Balloon-borne Large Aperture Submillimeter Telescope (BLAST-TNG) is a submillimeter mapping experiment which features three microwave kinetic inductance detector (MKID) arrays operating over 30% bandwidths centered at 250, 350, and 500 $$ m (1200, 857, and 600 GHz).",
"These highly-multiplexed, high-sensitivity arrays, featuring 918, 469, and 272 dual-polarization pixels, for a total of 3,318 detectors, are coupled to a 2.5 m diameter primary mirror and a cryogenic optical system providing diffraction-limited resolution of 30$^{\\prime \\prime }$ , 41$^{\\prime \\prime }$ , and 50$^{\\prime \\prime }$ respectively.",
"The arrays are cooled to $\\sim $ 275 mK in a liquid-helium-cooled cryogenic receiver which will enable observations over the course of a 28-day stratospheric balloon flight from McMurdo Station in Antarctica as part of NASA's long-duration-balloon (LDB) program, planned for the 2018/2019 winter campaign.",
"BLAST-TNG is the successor to the BLASTPol and BLAST balloon-borne experiments which flew five times between 2005 and 2012[1], [2].",
"Achieving diffraction-limited, sub-arcminute resolution and telescope pointing accuracy is one of the highest priorities for the success of the BLAST-TNG mission.",
"Although the science goals of BLAST-TNG are similar to the 2012 BLASTPol mission, most of the major instrument systems have been rebuilt and improved since the last flight.",
"A new 2.5 m aperture Cassegrain telescope, featuring a lightweight composite carbon fiber reinforced polymer (CFRP) primary mirror designed and built by Alliance Spacesystems,4398 Corporate Center Dr, Los Alamitos, CA 90720 will enable an increase in resolution to 30$^{\\prime \\prime }$ at 250 $$ m, from BLASTPol's 2.5$^{\\prime }$ at the same band.",
"With improved detector sensitivity and a increase in detector count by a factor of 12, we expect BLAST-TNG will have more than six times the mapping speed of BLASTPol.",
"The new cryostat has demonstrated a 28 day hold-time, enabling observations of many more targets at greater depth than were possible during the $\\sim $ 13 day BLASTPol flight in 2012.",
"The primary science goal of BLAST-TNG is to map the polarized thermal emission from galactic interstellar dust around star-forming regions and in the diffuse interstellar medium (ISM).",
"These maps will yield $\\sim $ 250,000 polarization vectors on the sky, allowing us to explore correlations between the magnetic field dispersion, polarization fraction, cloud temperature, and column density.",
"Quantifying the relationships between these variables over a large sample of clouds will yield testable relationships which can be fed back into numerical simulations.",
"The Planck satellite has observed strong correlations between the orientation of Galactic magnetic fields and large-scale ISM structures [3], as well as the interior of giant molecular clouds (GMCs)[4].",
"While BLASTPol was able to observe the magnetic fields within GMCs at higher resolution than Planck [2], [5], BLAST-TNG will be the first experiment to probe the fields within the characteristic filamentary structures within GMCs observed by Herschel [6].",
"Combining the BLAST-TNG data with molecular cloud simulations, [7] and numerical models of dust emission [8] and grain properties, [9] will give unprecedented insight into the interplay between the gravitational, turbulent and magnetic field contributions to star and cloud formation, as well as the physics of grain alignment and mass flow within the interstellar medium.",
"Polarized dust emission is also the dominant foreground for observations of the cosmic microwave background (CMB).",
"Characterization of these foregrounds is one of the most important requirements in the search for the gravitational wave signature of cosmic inflation [10].",
"While the power spectrum from polarized dust foregrounds is thought to be lowest at small angular scales, there is limited high-resolution observational data of the diffuse ISM [4], [11].",
"BLAST-TNG will be able to make the deepest maps to date of the dust emission in the types of dark, diffuse regions of the sky favored by state of the art CMB polarization experiments.",
"BLAST-TNG will probe angular scales not well-characterized to date, and explore correlations between diffuse dust emission and structures in the cold neutral medium [12] at submillimeter wavelengths where the intensity of the thermal dust signal dominates.",
"With its high pixel count and photon-noise-limited detectors, BLAST-TNG will produce maps of diffuse ISM with higher fidelity than the highest frequency Planck polarization maps at 353 GHz."
],
[
"BLAST-TNG Composite Optics Technology",
"In order to meet our angular resolution requirements, BLAST-TNG must feature large-aperture, lightweight telescope mirrors.",
"Building-large aperture balloon-borne telescope optics is particularly challenging.",
"With inadequate support, gravitationally-induced sag can introduce serious aberrations, yet support structures must be small enough to fit on NASA launch vehicles, and light enough not to compromise altitude during flight.",
"Achieving diffraction-limited performance in the submillimeter requires highly accurate optics with rms wavefront errors of order $\\sim $ 10$$ m. Traditional mirror fabrication techniques are inadequate to meet all the requirements for BLAST-TNG within the cost restrictions of a balloon mission.",
"To date, most balloon payloads operating in the millimeter/submillimeter wavebands have used aluminum mirrors, but have been limited by mechanical constraints to less than 2 m in diameter, including BLASTPol (1.8 m) [1] and EBEX (1.5 m) [13].",
"The SOFIA instrument features the largest sub-orbital primary mirror to date, at 2.7 m [14], made out of Zerodur, a ceramic silicate material that is lighter and stiffer than aluminum.",
"However, while SOFIA's 808 kg primary mirror is well-suited for a large airplane, it is unacceptably heavy for a balloon experiment.",
"Low-density metals, such as the beryllium alloy used in the construction of the JWST mirrors [15] are extremely expensive.",
"Mirrors made from these types of materials are cost-effective only if they are able to be reused for multiple flights, making them risky to use on a balloon mission where they may not be recovered at all.",
"The CFRP mirror which will fly on BLAST-TNG represents a significant technological development and research effort.",
"CRFP composites have many advantages over traditional metal mirrors.",
"They have a strength-to-weight ratio many times that of aluminum, and a near-zero coefficient of thermal expansion, which is especially desirable for a balloon platform, as the thermal environment in flight can be unstable.",
"Composite primary mirrors are produced via replication from a positive mold.",
"High-surface accuracy molds are much cheaper to produce than lightweight mirrors, and can be reused to make subsequent mirrors with equivalent optical quality without additional polishing.",
"While composite materials are expensive, the reduced recurring costs for replicating extremely lightweight mirrors make them well-suited to balloon experiments which plan on making repeated flights with no guarantee of recovering the telescope components.",
"Large-aperture composite mirrors with high surface accuracy have been demonstrated in the submillimeter.",
"The first launch of the BLAST experiment featured a 2 m composite primary mirror developed for the Herschel space telescope [16], although its performance in flight was degraded due to a lack of active focusing control.",
"With a 2.5 m aperture, the BLAST-TNG primary mirror will be both the largest mirror ever flown on a balloon experiment, and the largest CFRP telescope mirror operating at submillimeter wavelengths (THz frequencies).",
"This mirror was designed in partnership with a commercial collaborator, Alliance Spacesystems, under a NASA Small Business Innovation Research (SBIR) grant."
],
[
"BLAST-TNG Optical Architecture",
"The BLAST-TNG optics design is based on a 2.5 m aperture on-axis Cassegrain telescope, with a CFRP composite primary mirror and an aluminum secondary mirror.",
"The secondary mirror is mounted on three linear actuators which can move the secondary in piston/tip/tilt to account for changes in telescope focus due to differential thermal contraction of the telescope mirrors, CFRP support struts, and the aluminum gondola.",
"The optical design is shown in Fig.",
"REF .",
"Figure: Ray trace of the BLAST-TNG optics design from Zemax design software.",
"The left-hand side shows the on-axis Cassegrain telescope formed by the primary (M1) and secondary mirrors (M2).",
"The Cassegrain focus lies within the 4 K optics box, shown in the small rectangle, as well as the blown up inset.",
"Light enters the optics box towards the top left side of the enclosure where it passes through the window of the cryogenic receiver and a series of filters.",
"After these filters, the first optical element is the broadband achromatic half wave plate (AHWP), followed by the modified Offner relay formed by the three mirrors M3, M4 (the Lyot stop), and M5.",
"The location of the three focal plane arrays are shown schematically.The telescope feeds a cold (4 K) reimaging optics system which refocuses the beam onto three focal plane arrays.",
"The cold optics are arranged in a modified Offner relay configuration, shown in the inset of Fig.",
"REF .",
"A similar configuration was flown in the BLAST/BLASTPol optics box.",
"This configuration has several advantages, namely (1) it is compact, a necessary condition for running the optics in a liquid-helium-cooled cryostat, (2) the main optical elements all lie in a single plane, allowing all the elements to be mounted to a single sturdy optics bench, (3) the modified relay can be used to illuminate the focal planes with a different F/# than the telescope beam, and (4) the cryogenic Lyot stop allows us to limit the illumination of the primary mirror which reduces the thermal loading on the detectors.",
"The cold optics simultaneously illuminate three focal plane arrays of Microwave Kinetic Inductance Detectors, which are optically coupled via single-mode feedhorns.",
"The feedhorns were designed and machined at Arizona State University, based on a modified Potter design [17], and were drilled from a monolithic aluminum block with custom-manufactured drill bits.",
"Details of the cryogenic receiver design can be found in refs.",
"louriecryo,tyrblasttngspie.",
"The cold Lyot stop is placed at the image of the primary mirror and acts as the limiting aperture of the system.",
"The Lyot stop limits the central beam illumination of the primary mirror to 2.33 m. The detector feedhorns provide a near-Gaussian beam which overfills the Lyot stop, and tapers the illumination by 4.6 dB at the edge of the Lyot to reduce ringing in the beam.",
"The illumination of the system pupil is shown in Fig.",
"REF .",
"While all of the feedhorn beams from each of the focal plane arrays overlap on the Lyot stop, they do not illuminate the same area of the primary mirror.",
"A summary of the BLAST-TNG telescope optical design is given in Table REF .",
"Figure: Diagrams of the telescope pupil including obscuration from the struts and secondary mirror, the image of the Lyot stop on the primary mirror, the near-Gaussian illumination of the primary mirror by the Feedhorns.",
"The illumination of the effective system pupil is shown at right.Table: BLAST-TNG Telescope Optical Prescription"
],
[
"Telescope Optical Requirements",
"The BLAST-TNG telescope must maintain diffraction-limited performance under all combined stresses and loading conditions throughout the anticipated 28-day LDB flight.",
"The telescope must be operational on the ground for pre-flight integration and characterization, and must operate in flight over a broad range of temperatures and pointing angles.",
"To ensure diffraction-limited performance, the driving requirement for the telescope design was that the total wavefront error (WFE) of the telescope be no greater than 10 $$ m rms under all combined loading conditions.",
"This requirement would ensure that the telescope Strehl ratio be no less than $\\approx $ 90% across all wavebands.",
"A summary of the telescope performance specifications is given in Table REF .",
"There are three major loads that the telescope was designed to operate under.",
"These loads were studied extensively with finite element modeling (FEM) trade studies which ultimately drove the final design of the telescope optics and support system: Gravity Sag Gravity-induced sag is the dominant loading stress for the BLAST-TNG telescope.",
"Ground-based telescopes do not have strict limitations on the mass of telescope support structures, and space-borne missions which which have extremely demanding mass limits are not affected by gravity sag at all.",
"Sub-orbital flights occupy the worst of both worlds – having both gravitational stresses and mass constraints.",
"Thermal Soak/Gradients During the flight, the telescope will operate with a thermal shroud or Sun shield which should help control the thermal environment and block direct solar illumination.",
"Even so, we anticipate large thermal gradients across the structure, in particular between the primary and secondary mirrors.",
"These gradients were measured to be up to 15-20 $^\\circ $ C during the 2010 and 2012 BLASTPol flights.",
"To account for this, the ability to refocus the telescope in flight is critical.",
"Hygroscopic Strain The resin in CFRP composites exhibits temperature-dependent absorption of water vapor.",
"The amount of water vapor absorbed by the composite depends on the relative humidity (RH) of the environment.",
"The rate at which the moisture content of the composite changes is inversely proportional to the environment temperature [20].",
"As the mirror absorbs (desorbs) water it will grow (shrink) changing the radius of curvature and the conic constant similar to a change in temperature.",
"Table: BLAST-TNG Telescope Performance Specifications"
],
[
"Telescope Design and Fabrication",
"The BLAST-TNG telescope design is comprised of three major components: the primary and secondary mirrors which form the telescope beam, and a sturdy optical bench.",
"The design of each of these components is detailed in the sections below.",
"The final design of each component is the result of exhaustive finite element modeling, trade studies, and incorporates lessons learned from previous mirror designs.",
"The design team had experience building and designing previous generations of BLAST [1] as well as composite optics and reflectors for NASA space missions including Herschel [21], WMAP space telescope [22], and MAVEN [23].",
"The main elements of the telescope are detailed in Fig.",
"REF .",
"Figure: The completed BLAST-TNG telescope during assembly at Alliance Spacesystems, showing the primary and secondary mirrors, the CFRP struts and optical bench, and secondary mirror actuators.",
"The aluminum flexures mount to the aluminum gondola inner frame.The optical bench, or reaction structure, supports the primary mirror, the secondary mirror struts, and mounts to the balloon gondola.",
"It is responsible for holding the mirrors in a low-stress, kinematic configuration to maintain precise alignment between the mirrors and the cryogenic receiver while not transmitting bending stresses to the optical surfaces.",
"The optical bench is built of flat composite laminates bonded together in a rigid, box-like structure with internal webbing.",
"The laminates are formed by laying down layers of 0.13 mm-thick, layers of carbon fiber preimpregnated with epoxy, known as prepreg.",
"The laminates are layers of uni-directional tape where all the fibers are oriented in a single direction, as opposed to woven material.",
"To achieve nearly isotropic stiffness and thermal expansion in the plane of the laminate, the orientation of each layer of the laminate is rotated by 45$^{\\circ }$ from the previous layer.",
"The layers are then vacuum-pressed together and heat-cured.",
"The bench itself is roughly 15 cm thick.",
"The strength and rigidity of the bench is proportional to the thickness.",
"In order to increase the thickness of the bench without altering the location of the primary mirror surface, the front facesheet has a cutout so that the rear surface of the primary mirror sits below the front of the bench.",
"The primary mirror is supported by a pseudo-kinematic mount comprised of three bipods.",
"Each bipod is made of two composite tubes bonded to aluminum fittings on each end.",
"On one end these fittings bolt to aluminum fittings on the side of the optics bench, and on the other are bolted to Invar fittings bonded to the interior webbing of the primary mirror.",
"FEM showed that making the metallic fittings out of Invar was critical for managing the deformation of the primary under thermal stresses.",
"Invar, like CFRP composites, has a low CTE.",
"Bonding higher CTE materials like aluminum directly to the primary mirror caused significant thermal deformation around the bond sites.",
"FEM showed that under the -20 $^\\circ $ C uniform soak expected during flight, switching from aluminum to Invar greatly reduced the rms surface error from 51.9 to 11.1 $$ m, and the peak-to-valley surface error from 3.4 to 0.6 $$ m. The optical bench also supports three long struts which support the secondary mirror assembly.",
"Like the primary mirror struts, the secondary mirror struts are made from all-composite tubes bonded to aluminum end fittings.",
"The strut tubes have a rectangular cross section, with their long sides oriented radially outward from the optical axis.",
"This cross-section provides sufficient stiffness while reducing the obscuration of the primary mirror.",
"The far end of the struts is bolted to an aluminum triangular structure, known as the “push-plate\" which supports the secondary mirror and the focusing actuators.",
"The secondary mirror assembly is shown in Fig.",
"REF .",
"Figure: CAD Render of the secondary mirror assembly and support system.",
"The aluminum secondary mirror is shown as a copper color.",
"The push plate (yellow) is the aluminum mirror support which is bolted to the secondary struts (grey).",
"The three focusing actuators are shown in teal.",
"The actuators are mounted to the push plate, and actuate the mirror by pushing against a v-groove block with a ball-end.",
"The mirror is pulled against the ball/groove fittings by the six extension springs shown."
],
[
"Primary Mirror",
"The BLAST-TNG composite primary mirror features a monolithic front facesheet which is coated with a thin layer of vapor-deposited aluminum (VDA).",
"The mirror's structural strength comes from an interior CFRP core.",
"The core is formed by a honeycomb-like composite modules with flat laminate strips forming triangular voids.",
"This isogrid structure provides a high bending modulus while maintaining low mass, both of which minimize the effect of gravitational sag.",
"Additional stiffness is provided by a segmented rear facesheet which helps distribute bending stresses across the core.",
"The material selected for all the structural components was a woven prepreg combining K63712Mitsubishi Chemical Carbon Fiber and Composites, Inc. graphite fiber and BT250E low-temperature-cure (121 $^\\circ $ C) epoxy prepreg.TenCate Advanced Composites This roughly 60%/40% fiber/resin composite provides low CTE and high modulus.",
"The modulus of the isogrid core is a strong function of the core thickness.",
"FEM studies led to increasing the core thickness as much as possible given the fixed positions of the primary mirror surface and the gondola mount surface.",
"At its thickest point, roughly half the radius from the optical axis, the core is roughly 30 cm thick.",
"At its thinnest, around the central hole and at outer edge of the mirror, the core is only 6 cm thick.",
"This tapered shape maximizes the strength at the areas of highest bending stress, while reducing mass where the stresses are low.",
"The optical surface of the primary mirror was made by laying up multiple layers of prepreg onto a positive graphite mold.",
"The mold was rough-machined from graphite using a computer numerically controlled (CNC) mill and then hand polished to the final figure.",
"After the front facesheet was laid up on the mold it was cured in an autoclave while vacuum pressed against the mold.",
"After curing the facesheet was pulled off the mold and inspected.",
"Once inspected the optical surface was placed back on the mold, and the isogrid was built up in small modular sections.",
"The assembly flow for the primary mirror is shown in Fig.",
"REF .",
"Figure: Primary Mirror Manufacturing Process"
],
[
"Secondary Mirror ",
"The comparatively small size of the secondary mirror ($$ 573 mm) greatly reduced the complexity of the engineering and manufacturing.",
"Lightweight $\\sim $ 0.5 m infrared and submillimeter optics are widely manufactured, and as such we could rely on existing capabilities of different vendors throughout the process.",
"Corning NetOptix Inc.69 Island Street, Keene, NH 03431 developed a stiff, aggressively light-weighted design based on preliminary designs by Alliance Spacesystems, and provided two rough-machined mirror blanks.",
"After surveying the blanks at UPenn, the one with the lower degree of surface figure error was diamond-turned by NiPro Optics Inc.7 Marconi, Irvine, CA 92618.",
"The progression of the diamond-turning is shown in Figure REF .",
"Figure: Images of the secondary mirror at NiPro Optics Inc. taken before (left), during (center), and after (right) diamond turning.",
"After the final machining pass (right) the mirror is nearly optical quality, and produces a clear image of the tool rest."
],
[
"Metrology",
"Complete characterization of the full BLAST-TNG telescope optical system is only possible during the flight.",
"In-band absorption from water vapor prevents ground-based observations of far-field sources with the flight receiver.",
"To evaluate the focus for the BLAST-Pol experiment, the telescope was refocused to image near-field sources up to a few hundred yards away.",
"While this approach is useful for testing focus routines, the WFE introduced by moving the focus, and the atmospheric absorption make meaningful evaluation of the telescope beam impractical.",
"To characterize the expected telescope performance for BLAST-TNG, we rely mainly on analytical performance models and finite element analysis.",
"Extensive FEM under all loading conditions was performed at Alliance Spacesystems, with optical analysis performed throughout the design process to ensure that the 10 $$ m rms WFE condition was met under all gravitational, thermal, and hygroscopic stresses expected during operation.",
"The accuracy of the final primary mirror surface figure presented the largest uncertainty throughout the design process.",
"Because the surface accuracy and size scale of the primary mirror are unprecedented for a composite mirror, there is little heritage for predicting the surface figure errors of the finished surface, and large-scale deviations from the mold surface.",
"Additionally, limited metrology data was available during the intermediate stages of the manufacture and for the final product.",
"Traditional evaluation techniques for large-aperture infrared and submillimeter telescope optics such as infrared interferometry [21], wavefront sensing [24], and coordinate measuring machine (CMM) profilometry were all cost-prohibitive given the size of the primary mirror.",
"Figure: Surface error of the primary mirror graphite mold after final polishing measured using a commercial laser tracker.",
"Deviations shown are calculated after subtracting the best-fit conic section and tip/tilt.Both the mold and the primary mirror surface were surveyed using a laser trackerFARO Technologies Inc., a technique which has been demonstrated at the <1 $$ m accuracy level [25], and surface data recorded on a $\\sim $ 5 cm square grid.",
"The final survey of the mold is shown in Fig.",
"REF .",
"After removing the best-fit conic section and tip/tilt of the surface, the mold demonstrates a surface error of 8.9 $$ m rms, and 40.6 $$ m peak-to-valley.",
"The optical surface of the primary mirror was also surveyed after initial release from the mold, and twice after the mirror assembly was completed before application of the VDA optical surface.",
"While surface measurements were sufficient for placing an upper limit on the surface figure error, large-spatial-scale surface errors were not repeatably observed between subsequent measurements.",
"We attribute these systematic errors to the relative thermal and mechanical stability of the mirror itself during these measurements, as compared with the mold."
],
[
"Point-Spread Function",
"The available metrology data were incorporated into a model of the system optical response based on the nominal optics design and the measured feedhorn beam pattern.",
"The model uses the Zemax model for the full optical system as the nominal reference prescription, models the feedhorn response as a true Gaussian, and assumes that the primary mirror surface perfectly replicates the mold surface.",
"Although the surface of the mold exhibits a number of high frequency surface errors, the impact to performance is minor due to the wavelength compared to the size of the deformations.",
"We also assume that the secondary mirror is refocused to account for deviations from the nominal primary mirror conic section.",
"Figure: Modeled point-spread function for the BLAST-TNG optical system at 250 and 500 m, based on reference optical prescription (“Nominal Optics\"), the telescope obscuration (“Nominal + Struts\"), and the measured surface error of the primary mirror mold (“Nominal + Struts + Mold\").",
"The model includes the measured response of the Gaussian illumination of the Lyot stop by the feedhorns.The predicted point-spread functions (PSF) of the system at 250 and 500 $$ m are plotted in Fig.",
"REF .",
"Because the Lyot stop underilluminates the primary mirror, the resolution is set by the image of the stop on the primary mirror.",
"The predicted Strehl ratio of the system exceeds the 80% convention for the diffraction limit based on the Maréchal condition across all but the shortest wavelengths.",
"The anticipated WFE of the system is low enough that despite a reduction of power in the main beam, the resolution is not degraded beyond the diffraction limit.",
"A summary of the expected optical performance is given in Table REF .",
"Table: Expected Telescope Performance Summary"
],
[
"Balloon-Borne Telescope Platform",
"The telescope is mounted in an altazimuth configuration on a pointed gondola which is suspended from the balloon flight train.",
"The gondola comprises two major mechanical systems: an outer frame which allows the telescope to rotate in azimuth, and an inner frame that can be precisely pointed in elevation with respect to the outer frame by way of a direct-drive motor.",
"The outer frame is suspended by four steel cables from the a high-torque pivot motor which attaches to the base of the balloon flight train.",
"Primary azimuth pointing is achieved by a high moment of inertia reaction wheel which provides fine control of the scan speed, but saturates if large slews or fast azimuth scans are required.",
"The pivot motor servos off the reaction wheel speed to provide coarse pointing during slews and dump angular momentum from the reaction wheel to the balloon flight train.",
"BLAST-TNG operates primarily in a continuous raster-scan mode, with faster ($\\sim $ 0.5$^{\\circ }$ /s) scans in azimuth and slow (<0.1$^\\circ $ /s) drifts in elevation.",
"Data taken during azimuth turnarounds is discarded due to these vibrationally-induced thermal instabilities.",
"Typical observations will map square or circular regions on the sky of sizes from a $\\sim $ 5-50 square degrees, with the lower limit set by inefficiencies in the observation-to-turnaround time, and the upper limit set by the scan speed and the 1/f knee of the detectors.",
"A suite of pointing sensors are used to continuously measure the attitude, geographic location, and trajectory of the payload and are read in by the flight computers to calculate the real-time telescope pointing solution in right ascension (RA) and declination (DEC).",
"Absolute pointing information of the telescope is primarily provided by two autonomous daytime-operating star cameras, mounted on a carbon fiber truss above the cryostat, pointed parallel to the optical axis.",
"These cameras contain a high-resolution integrating CCD camera controlled by a single-board computer, both mounted in an aluminum pressure vessel.",
"The camera observes a 2$^{\\circ }$ by 2.5$^{\\circ }$ area, and the exposure time, aperture, and focus can be controlled by the single-board computer and stepper motors mounted to the camera lens [26].",
"The control computer runs the attitude-determination program STARS developed for the EBEX experiment [27], [28].",
"However, the typical scan speed for BLAST-TNG is around 0.5 $^\\circ $ /s, meaning that to avoid blurring of the images, the star cameras can only capture images at the turnarounds of the scans, which are a few seconds apart.",
"Two three-axis fiber-optic gyroscope units KVH Industries DSP-1760 mounted on opposite sides of the inner frame are used to precisely measure the angular velocity of the inner frame and interpolate the telescope pointing in between star camera solutions.",
"These sensors are able to compute the solution to <5$^{\\prime }$ rms during flight, and to <5$^{\\prime \\prime }$ rms after post-flight pointing reconstruction [29].",
"Figure: Photograph of the BLAST-TNG gondola, showing the telescope mounting surface on the front of the inner frame, the cryogenic receiver (painted red), and the two autonomous daytime star cameras mounted above the receiver.",
"Each star camera has a long baffle to reject stray light and reflections off of the telescope baffle.",
"For flight, both baffles will be painted white to reduce their thermal emissivity."
],
[
"Mechanical Requirements",
"The balloon-borne platform presents a distinct set of constraints and requirements, combining the gravitational stresses of a ground-based telescope, and the mass constraints and extreme thermal environment of a space telescope.",
"For a given size balloon, the altitude of the balloon during flight is determined by the buoyancy of the helium in the balloon and the mass of the payload.",
"To reach the required altitudes with the 34 million cubic feet balloon used by BLAST-TNG the this places a absolute maximum mass requirement of 3,600 kg.",
"Reducing the mass as much as possible below this maximum reduces both the structural demands on the gondola and suspension elements and the torque output from the pointing motors.",
"For BLAST-TNG an upper limit of 3,200 kg was set on the total payload mass, though we expect the actual mass of the payload to be closer to 2,700 kg.",
"The main mechanical requirement of the inner frame is that it maintain precise alignment between the telescope, the receiver, and the star cameras.",
"Any relative motion of the telescope with respect to the receiver will cause spurious blurring and streaking of the images.",
"Equally important is that the telescope beam not move with respect to the star cameras, as this would cause an elevation-dependent systematic pointing offset in the maps and jitter during elevation turnarounds.",
"The absolute offset angle between the star camera beam and the telescope beam is not critical, as long as they are roughly aligned such that the star cameras have a clear view of the sky at all elevation angles.",
"This means the star cameras must be aligned to the telescope bore sight to within a few degrees.",
"Misalignment between the telescope and the receiver shifts the center of the beam on the focal planes, leading to under-illumination of the pixels at the edge of the array.",
"To achieve the necessary rigidity of the inner frame, we required that the relative pointing misalignment between the telescope and the receiver due to deformation of the mounting structure be less than half the FWHM of the beam at the smallest wavelength of observation over the full range of observed elevations between 20 and 60 degrees."
],
[
"Telescope Mount Design",
"To keep the BLAST-TNG inner frame stiff and compact, while separating the bending stresses from the telescope optics, the frame is composed of two joined sections: a ring of alumnum c-channel which attaches around the waist of the cryostat, and a welded monocoque structure of thin aluminum sheets which forms a broad flat surface for the telescope to mount at the front of the frame.",
"A cut-out in this front plate allows the telescope beam to pass through the inner frame, and affords access to the cryostat front window.",
"The cryostat readout electronics are mounted securely to a welded rack hanging below the rear section of the inner frame.",
"Aluminum 6061-T6 was selected for the frame material because it is easily machined, welded, relatively inexpensive, and there are many readily available standard structural beams and other elements that can be incorporated into the design.",
"Because aluminum is also a good thermal conductor, the frame can be used as a heat sink for the flight electronics.",
"The monocoque structure contains four internal ribs, thin vertical sheets of metal, in addition to thick sidewalls which make it very stiff along the direction of the optical axis to minimize sag from the weight of the telescope.",
"The location of the internal webbing structure is shown in Fig.",
"REF .",
"To reduce mass, most panels include oval cut-outs where their presence would not unacceptably reduce the structural strength or interfere with the mounting locations of the mirror, motors, or electronics.",
"A series of 13 mm-thick pads were welded to the front facesheet for mounting the telescope.",
"Before joining the front and rear frame sections, these pads were machined flat and coplanar after they were welded on in a machine large enough to accommodate the full monocoque assembly.",
"Only after surfacing all these mounts were the bolt holes tapped for the telescope mount.",
"This process ensured that the telescope would mount on a flat surface which would not stress the optics."
],
[
"Modeling Pointing Performance",
"In order to size the structural members of the inner frame, a detailed finite element model (FEM) was developed and a series of trade studies performed to achieve the desired stiffness of the frame with as low a mass as possible.",
"A number of parameters were varied in these studies, including the monocoque web thickness, the height and web thickness of the rear c-channel, the thickness of the top (cryostat mount) and bottom rear plates, the cryostat support gussets, geometry of the front and rear sections, and mesh size.",
"To quickly evaluate the results of the FEM simulations of the inner frame assembly without running a full opto-mechanical analysis of the telescope and mount, the pointing offset between the receiver and the telescope optics were estimated for each simulation.",
"The x, y, and z coordinates of of five finite elements were tracked before and after running the simulations.",
"Two elements at the front and back of the cryostat top plate were used to define the cryostat boresight vector, $\\vec{V}_{cryo}$ .",
"Three elements on the telescope optics bench, at the rear of the three secondary mirror struts, were used to define a plane perpendicular to the optical axis of the telescope.",
"The normal vector to this optics bench plane, $\\vec{N}_{OB}$ points along the telescope optical axis.",
"The pointing offset, $theta$ , was defined as angle between these two vectors: $\\cos (\\theta ) = \\vec{N}_{OB} \\bullet \\vec{V}_{cryo} $ .",
"Because this bending-induced pointing offset changes with elevation, the FEM simulation was run at 20$^{\\circ }$ and 60$^{\\circ }$ , the upper and lower elevation limits during flight, and the differential pointing offset, $\\Delta \\theta $ , between these angles was calculated.",
"Figure: Results of finite element modeling of bending of the inner frame under 1-g gravitaional loading at different elevation angles.",
"The deformation scale is exaggerated by several hundred times in order to make the direction of the bending more clear.",
"The largest displacement in the model is the cryostat, and is only 200 m. Figures A and C show the calculated bending in z-direction of the front facesheet of the inner frame monocoque.",
"The six rectangles around the perimeter are the telescope mount points.",
"The four vertical dashed lines show the location of the internal ribs of the monocoque.",
"As expected most of the bending occurs between the ribs.",
"Figures B and C show the total displacement from bending of the inner frame, cryostat, and simplified telescope.Because the simulations were meant to evaluate the stiffness of the inner frame itself, not all elements in the assembly were modeled.",
"Only the exterior of the cryostat was included, and the internal components were not modeled.",
"The internal optical components have been modeled separately [30].",
"The star cameras and their mount were not included in these simulations, as we required the star camera mount be stiff enough that any bending would be subdominant to the bending of the inner frame.",
"The aluminum flexures that attach the telescope optics bench to the front of the inner frame were modeled as perfectly rigid elements as the details of their design were still being developed at the manufacturer.",
"This assumption means that the simulations likely over-estimate the bending of the front facesheet of the inner frame.",
"The Fig.",
"REF shows the deformation of the front facesheet of the monocoque from the simulations at the upper and lower limits of the elevation range.",
"The magnitude of the deformation across the full face is less than 10 $$ m. The mean offset in z was calculated for each of the six telescope mount points at each elevation, and these offsets were put into the detailed telescope FEM model developed at Alliance Spacesystems.",
"The displacements of the mount points had no observed effect on the primary mirror bending or WFE.",
"The results of the FEM simulations of the as-built inner frame are shown in Fig.",
"REF .",
"At an elevation of 30$^{\\circ }$ , the pointing offset is 28$^{\\prime \\prime }$ , and at an elevation of 60$^{\\circ }$ , the pointing offset is 19$^{\\prime \\prime }$ , giving a differential pointing offset over range of observing elevations of 9$^{\\prime \\prime }$ .",
"This is less than half the beam FWHM at the shortest observed wavelength.",
"This offset is of roughly the same order as the anticipated differential pointing offset due to bending of the telescope optics alone.",
"Even if the bending of the telescope optics causes the beam to move in the same direction as the bending of the inner frame, we expect that the elevation-dependent pointing offset from bending/motion of the telescope optics, inner frame, and cryostat to be less than the size of the beam at 250 $$ m."
],
[
"Thermal Environment",
"The BLAST-TNG payload thermal environment is controlled by extensive baffling and Sun shields.",
"The thermal environment of the telescope and the payload must be carefully controlled to avoid direct solar illumination of the optical system, and to ensure all components operate within allowable temperature ranges.",
"The flight trajectory and the conditions during launch, ascent, and descent are largely unpredictable, and can vary widely between different launches.",
"As such, it is necessary to design the gondola platform to handle a wide range conditions and stresses.",
"The Sun shield design follows the approach from the BLASTPol experiment, detailed in Soler et.",
"al., 2014 [31].",
"A 6.5 m long aluminized mylar baffle surrounds the telescope optics.",
"The baffle is formed around a truss of carbon fiber tubes CST Composites bonded to aluminum fittings, incorporating design elements from the X-Calibur gondola [32].",
"An outer Sun shield built from welded aluminum pipeGSM Industrial Inc. encloses the entire payload and acts as a ground screen protecting the entire instrument from both direct and reflected solar illumination.",
"A detailed ThermalDesktop Cullimore and Ring Inc. model was used to determine the placement of the reflective panels and predict the operating temperatures of the major systems and electronic components.",
"Figure: Render of the BLAST-TNG payload, with critical components labeled, including the telescope and pointed gondola platform.",
"The cryogenic receiver, shown in red, supports the two boresight star cameras.",
"The large asymmetric Sun shields enclose the payload, protecting the components from solar illumination and Earth-shine, and allow the gondola to point to within 35 ∘ ^{\\circ } in azimuth from the Sun on the starboard side.The visibility of different regions of the sky throughout the flight is set by the geometry of the telescope baffle.",
"The telescope pointing scheduler is constrained by limits determined from the thermal model based on the geometry of the baffle and the position of the Sun.",
"At every elevation angle, the allowed azimuth angles are determined by ensuring that there is no direct thermal illumination of the telescope primary or secondary mirrors.",
"Because the balloon may in latitude from its launch location depending on stratospheric wind currents, these constraints are calculated at extreme cases of ±10$^{\\circ }$ latitude drifts, and recalculated at different dates throughout the anticipated flight.",
"To allow for pointing towards certain high-priority targets, the baffle is designed to allow pointing to within approximately 35$^{\\circ }$ of the sun at most elevation angles throughout the flight."
],
[
"Conclusion",
"The BLAST-TNG experiment features one of the most technologically ambitious telescopes ever flown on a balloon experiment.",
"With a 2.5 m diameter carbon fiber composite primary mirror, BLAST-TNG will be able to map the submillimeter polarized thermal emission from interstellar dust at sub-arcminute resolution, probing the magnetic field structure of molecular clouds and the diffuse interstellar medium on previously unexplored angular scales.",
"The telescope has been completed and will be integrated with the balloon gondola and cryogenic receiver during pre-flight systems integration at the NASA Columbia Scientific Ballooning Facility in Palestine, TX, in preparation for a planned 28 day stratospheric balloon flight from McMurdo Station, Antarctica during the winter of 2018/2019.",
"The BLAST-TNG collaboration acknowledges the support of NASA under award numbers NNX13AE50G and 80NSSC18K0481, and the NNX13CM03C.",
"Detector development is supported in part by NASA through NNH13ZDA001N-APRA.",
"J.D.S.",
"acknowledges the support from the European Research Council (ERC) under the Horizon 2020 Framework Program via the Consolidator Grant CSF-648505.",
"S.G. is supported through a NASA Earth and Space Science Fellowship (NESSF) NNX16AO91H.",
"The BLAST-TNG telescope is supported in part through the NASA SBIR/STTR office and developed at Alliance Spacesystems.",
"The collaboration also acknowledges the extensive machining, design, and fabrication efforts of Jeffrey Hancock and Harold Borders at the University of Pennsylvania and Matthew Underhill at Arizona State University.",
"Paul Dowkontt at Washington University in St. Louis provided design assistance for the sun shield fitting design.",
"The BLAST-TNG team also recognizes the contribution of undergraduate and post-baccalaureate interns to the gondola development, especially Mark Giovinazzi, Erin Healy, Gregory Kofman, Aaron Mathews, Timothy McSorely, Michael Plumb, Steven Russel, and Nathan Schor."
]
] | 1808.08597 |
[
[
"Deep Probabilistic Logic: A Unifying Framework for Indirect Supervision"
],
[
"Abstract Deep learning has emerged as a versatile tool for a wide range of NLP tasks, due to its superior capacity in representation learning.",
"But its applicability is limited by the reliance on annotated examples, which are difficult to produce at scale.",
"Indirect supervision has emerged as a promising direction to address this bottleneck, either by introducing labeling functions to automatically generate noisy examples from unlabeled text, or by imposing constraints over interdependent label decisions.",
"A plethora of methods have been proposed, each with respective strengths and limitations.",
"Probabilistic logic offers a unifying language to represent indirect supervision, but end-to-end modeling with probabilistic logic is often infeasible due to intractable inference and learning.",
"In this paper, we propose deep probabilistic logic (DPL) as a general framework for indirect supervision, by composing probabilistic logic with deep learning.",
"DPL models label decisions as latent variables, represents prior knowledge on their relations using weighted first-order logical formulas, and alternates between learning a deep neural network for the end task and refining uncertain formula weights for indirect supervision, using variational EM.",
"This framework subsumes prior indirect supervision methods as special cases, and enables novel combination via infusion of rich domain and linguistic knowledge.",
"Experiments on biomedical machine reading demonstrate the promise of this approach."
],
[
"Introduction",
"Deep learning has proven successful in a wide range of NLP tasks [2], [3], [6], [12], [46].",
"The versatility stems from its capacity of learning a compact representation of complex input patterns [11].",
"However, success of deep learning is bounded by its reliance on labeled examples, which are expensive and time-consuming to produce.",
"Indirect supervision has emerged as a promising direction for breaching the annotation bottleneck.",
"A powerful paradigm is joint inference [5], [36], [9], [10], which leverages linguistic and domain knowledge to impose constraints over interdependent label decisions.",
"More recently, another powerful paradigm, often loosely called weak supervision, has gained in popularity.",
"The key idea is to introduce labeling functions to automatically generate (noisy) training examples from unlabeled text.",
"Distant supervision is a prominent example that used existing knowledge bases for this purpose [7], [29].",
"Data programming went further by soliciting labeling functions from domain experts [42], [1].",
"Indirect-supervision methods have achieved remarkable successes in a number of NLP tasks, but they also exhibit serious limitations.",
"Distant supervision often produces incorrect labels, whereas labeling functions from data programming vary in quality and coverage, and may contradict with each other on individual instances.",
"Joint inference incurs greater modeling complexity and often requires specialized learning and inference procedures.",
"Since these methods draw on diverse and often orthogonal sources of indirect supervision, combining them may help address their limitations and amplify their strengths.",
"Probabilistic logic offers an expressive language for such an integration, and is well suited for resolving noisy and contradictory information [44].",
"Unfortunately, probabilistic logic generally incurs intractable learning and inference, often rendering end-to-end modeling infeasible.",
"In this paper, we propose deep probabilistic logic (DPL) as a unifying framework for indirect supervision (Figure REF ).",
"Specifically, we made four contributions.",
"First, we introduce a modular design to compose probabilistic logic with deep learning, with a supervision module that represents indirect supervision using probabilistic logic, and a prediction module that performs the end task using a deep neural network.",
"Label decisions are modeled as latent variables and serve as the interface between the two modules.",
"Second, we show that all popular forms of indirect supervision can be represented in DPL by generalizing virtual evidence [45], [33].",
"Consequently, these diverse methods can be easily combined within a single framework for mutual amplification.",
"Third, we show that our problem formulation yields a well-defined learning objective (maximizing conditional likelihood of virtual evidence).",
"We proposed a modular learning approach by decomposing the optimization over the supervision and prediction modules, using variational EM, which enables us to apply state-of-the-art methods for probabilistic logic and deep learning.",
"Figure: Example of cross-sentence relation extraction for precision cancer treatment.Finally, we applied DPL to biomedical machine reading [41], [34].",
"Biomedicine offers a particularly attractive application domain for exploring indirect supervision.",
"Biomedical literature grows by over one million each yearhttp://ncbi.nlm.nih.gov/pubmed, making it imperative to develop machine reading methods for automating knowledge curation (Figure REF ).",
"While crowd sourcing is hardly applicable, there are rich domain knowledge and structured resources to exploit for indirect supervision.",
"Using cross-sentence relation extraction and entity linking as case studies, we show that distant supervision, data programming, and joint inference can be seamlessly combined in DPL to substantially improve machine reading accuracy, without requiring any manually labeled examples.The DPL code and datasets will be made available at http://hanover.azurewebsites.net."
],
[
"Distant supervision",
"This paradigm was first introduced for binary relation extraction [7], [29].",
"In its simplest form, distant supervision generates a positive example if an entity pair with a known relation co-occurs in a sentence, and samples negative examples from co-occurring entity pairs not known to have the given relation.",
"It has recently been extended to cross-sentence relation extraction [41], [34].",
"In principle, one simply looks beyond single sentences for co-occurring entity pairs.",
"However, this can introduce many false positives and prior work used a small sliding window and filtering (minimal-span) to mitigate training noise.",
"Even so, accuracy is relatively low.",
"Both quirkpoon2017 and pengal17 used ontology-based string matching for entity linking, which also incurs many false positives, as biomedical entities are highly ambiguous (e.g., PDF and AAAS are gene names).",
"Distant supervision for entity linking is relatively underexplored, and prior work generally focuses on Freebase entities, where links to the corresponding Wikipedia articles are available for learning [16]."
],
[
"Data Programming",
"Instead of annotated examples, domain experts are asked to produce labeling functions, each of which assigns a label to an instance if the input satisfies certain conditions, often specified by simple rules [42].",
"This paradigm is useful for semantic tasks, as high-precision text-based rules are often easy to come by.",
"However, there is no guarantee on broad coverage, and labeling functions are still noisy and may contradict with each other.",
"The common denoising strategy assumes that labeling functions make random mistakes, and focuses on estimating their accuracy and correlation [42], [1].",
"A more sophisticated strategy also models instance-level labels and uses instance embedding to estimate instance-level weight for each labeling function [25]."
],
[
"Joint Inference",
"Distant supervision and data programming focus on infusing weak supervision on individual labels.",
"Additionally, there is rich linguistic and domain knowledge that does not specify values for individual labels, but imposes hard or soft constraints on their joint distribution.",
"For example, if two mentions are coreferent, they should agree on entity properties [36].",
"There is a rich literature on joint inference for NLP applications.",
"Notable methodologies include constraint-driven learning [5], general expectation [9], posterior regularization [10], and probabilistic logic [36].",
"Constraints can be imposed on relational instances or on model expectations.",
"Learning and inference are often tailor-made for each approach, including beam search, primal-dual optimization, weighted satisfiability solvers, etc.",
"Recently, joint inference has also been used in denoising distant supervision.",
"Instead of labeling all co-occurrences of an entity pair with a known relation as positive examples, one only assumes that at least one instance is positive [13], [24]."
],
[
"Probabilistic Logic",
"Probabilistic logic combines logic's expressive power with graphical model's capability in handling uncertainty.",
"A representative example is Markov logic [44], which define a probability distribution using weighted first-order logical formulas as templates for a Markov model.",
"Probabilistic logic has been applied to incorporating indirect supervision for various NLP tasks [35], [36], [38], but its expressive power comes at a price: learning and inference are generally intractable, and end-to-end modeling often requires heavy approximation [20].",
"In DPL, we limit the use of probabilistic logic to modeling indirect supervision in the supervision module, leaving end-to-end modeling to deep neural network in the prediction module.",
"This alleviates the computational challenges in probabilistic logic, while leveraging the strength of deep learning in distilling complex patterns from high-dimension data."
],
[
"Knowledge-Rich Deep Learning",
"Infusing knowledge in neural network training is a long-standing challenge in deep learning [47].",
"hu2016harnessing,hu2016deep first used logical rules to help train a convolutional neural network for sentiment analysis.",
"DPL draws inspiration from their approach, but is more general and theoretically well-founded.",
"hu2016harnessing,hu2016deep focused on supervised learning and the logical rules were introduced to augment labeled examples via posterior regularization [10].",
"DPL can incorporate both direct and indirect supervision, including posterior regularization and other forms of indirect supervision.",
"Like DPL, hu2016deep also refined uncertain weights of logical rules, but they did it in a heuristic way by appealing to symmetry with standard posterior regularization.",
"We provide a novel problem formulation using generalized virtual evidence, which shows that their heuristics is a special case of variational EM and opens up opportunities for other optimization strategies.",
"Deep generative models also combine deep learning with probabilistic models, but focus on uncovering latent factors to support generative modeling and semi-supervised learning [22], [21].",
"Knowledge infusion is limited to introducing structures among the latent variables (e.g., Markov chain) [17].",
"In DPL, we focus on learning a discriminative model for predicting the latent labels, using a probabilistic model defined by probabilistic logic to inject indirect supervision."
],
[
"Deep Probabilistic Logic",
"In this section, we introduce deep probabilistic logic (DPL) as a unifying framework for indirect supervision.",
"Label decisions are modeled as latent variables.",
"Indirect supervision is represented as generalized virtual evidence, and learning maximizes the conditional likelihood of virtual evidence given input.",
"We first review the idea of virtual evidence and show how it can be generalized to represent any form of indirect supervision.",
"We then formulate the learning objective and show how it can be optimized using variational EM.",
"Given a prediction task, let $\\mathcal {X}$ denote the set of possible inputs and $\\mathcal {Y}$ the set of possible outputs.",
"The goal is to train a prediction module $\\Psi (x,y)$ that scores output $y$ given input $x$ .",
"Without loss of generality, we assume that $\\Psi (x,y)$ defines the conditional probability $P(y|x)$ using a deep neural network with a softmax layer at the top.",
"Let $X=(X_1,\\cdots ,X_N)$ denote a sequence of inputs and $Y=(Y_1,\\cdots ,Y_N)$ the corresponding outputs.",
"We consider the setting where $Y$ are unobserved, and $\\Psi (x,y)$ is learned using indirect supervision."
],
[
"Virtual evidence",
"Pearl [33] first introduced the notion of virtual evidence, which has been used to incorporate label preference in semi-supervised learning [43], [45], [23] and grounded learning [32].",
"Suppose we have a prior belief on the value of $y$ , it can be represented by introducing a binary variable $v$ as a dependent of $y$ such that $P(v=1|y=l)$ is proportional to the prior belief of $y=l$ .",
"$v=1$ is thus an observed evidence that imposes soft constraints over $y$ .",
"Direct supervision (i.e., observed label) for $y$ is a special case when the belief is concentrated on a specific value $y=l^*$ (i.e., $P(v=1|y=l)=0$ for any $l\\ne l^*$ ).",
"The virtual evidence $v$ can be viewed as a reified variable for a potential function $\\Phi (y)\\propto P(v=1|y)$ .",
"This enables us to generalize virtual evidence to arbitrary potential functions $\\Phi (X,Y)$ over the inputs and outputs.",
"In the rest of the paper, we will simply refer to the potential functions as virtual evidences, without introducing the reified variables explicitly."
],
[
"DPL",
"Let $K=(\\Phi _1,\\cdots ,\\Phi _V)$ be a set of virtual evidence derived from prior knowledge.",
"DPL comprises of a supervision module over K and a prediction module over all input-output pairs (Figure REF ), and defines a probability distribution: $P(K,Y|X)\\propto \\prod _v~\\Phi _{v}(X, Y)\\cdot \\prod _i~\\Psi (X_i, Y_i)$ Without loss of generality, we assume that virtual evidences are log-linear factors, which can be compactly represented by weighted first-order logical formulas [44].",
"Namely, $\\Phi _v(X,Y)=\\exp (w_v\\cdot f_v(X,Y))$ , where $f_v(X,Y)$ is a binary feature represented by a first-order logical formula.",
"A hard constraint is the special case when $w_v=\\infty $ (in practice, it suffices to set it to a large number, e.g., 10).",
"In prior use of virtual evidence, $w_v$ 's are generally pre-determined from prior knowledge.",
"However, this may be suboptimal.",
"Therefore, we consider a general Bayesian learning setting where each $w_v$ is drawn from a pre-specified prior distribution $w_v\\sim P(w_v|\\alpha _v)$ .",
"Fixed $w_v$ amounts to the special case when the prior is concentrated on the preset value.",
"For uncertain $w_v$ 's, we can compute their maximum a posteriori (MAP) estimates and/or quantify the uncertainty."
],
[
"Distant supervision",
"Virtual evidence for distant supervision is similar to that for direct supervision.",
"For example, for relation extraction, distant supervision from a knowledge base of known relations will set $f_{KB}(X_i,Y_i)=\\mathbb {I}[\\text{\\tt In-KB}(X_i,r) \\wedge Y_i=r]$ , where $\\text{\\tt In-KB}(X_i,r)$ is true iff the entity tuple in $X_i$ is known to have relation $r$ in the KB."
],
[
"Data programming",
"Virtual evidence for data programming is similar to that for distant supervision: $f_{L}(X_i,Y_i)=\\mathbb {I}[L(X_i) = Y_i]$ , where $L(X_i)$ is a labeling function provided by domain experts.",
"Labeling functions are usually high-precision rules, but errors are still common, and different functions may assign conflicting labels to an instance.",
"Existing denoising strategy assumes that each function makes random errors independently, and resolves the conflicts by weighted votes [42].",
"In DPL, this can be done by simply treating error probabilities as uncertain parameters and inferring them during learning."
],
[
"Joint inference",
"Constraints on instances or model expectations can be imposed by introducing the corresponding virtual evidence [10] (Proposition 2.1).",
"The weights can be set heuristically [5], [26], [36] or iteratively via primal-dual methods [10].",
"In addition to instance-level constraints, DPL can incorporate arbitrary high-order soft and hard constraints that capture the interdependencies among multiple instances.",
"For example, identical mentions in proximity probably refer to the same entity, which is useful for resolving ambiguous mentions by leveraging their unambiguous coreferences (e.g., an acronym in apposition of the full name).",
"This can be represented by the virtual evidence $f_{\\tt Joint}(X_i,Y_i,X_j,Y_j)=\\mathbb {I}[{\\tt Coref}(X_i,X_j) \\wedge Y_i=Y_j]$ , where ${\\tt Coref}(X_i,X_j)$ is true iff $X_i$ and $X_j$ are coreferences.",
"Similarly, the common denoising strategy for distant supervision replaces the mention-level constraints with type-level constraints [13].",
"Suppose that $X_E\\subset X$ contains all $X_i$ 's with co-occurring entity tuple $E$ .",
"The new constraints simply impose that, for each $E$ with known relation $r\\in KB$ , $Y_i=r$ for at least one $X_i\\in X_E$ .",
"This can be represented by a high-order factor on $(X_i,Y_i: X_i\\in X_E)$ .",
"[t] DPL Learning Input: Virtual evidences $K=\\Phi _{1:V}$ , deep neural network $\\Psi $ , inputs $X=(X_1,\\cdots ,X_N)$ , unobserved outputs $Y=(Y_1,\\cdots ,Y_N)$ .",
"Output: Learned prediction module $\\Psi ^*$ Initialize: $\\Phi ^0 \\sim \\text{priors}$ , $\\Psi ^0 \\sim \\text{uniform}$ .",
"$t=1:T$ $q^t(Y) \\leftarrow &\\arg \\min _{q}~D_{KL}(\\prod _i~q_i(Y_i)~||~&\\\\&\\prod _v~\\Phi ^{t-1}_v(X,Y)\\cdot \\prod _i~\\Psi ^{t-1}(X_i,Y_i)) &\\\\\\Phi ^t \\leftarrow &\\arg \\min _{\\Phi }~D_{KL}(q^t(Y)~||~ \\prod _v~\\Phi _v(X,Y)) &\\\\\\Psi ^t \\leftarrow & \\arg \\min _{\\Psi }~D_{KL}(q^t(Y)~||~\\prod _i~\\Psi (X_i,Y_i)) &$ $\\Psi ^*=\\Psi ^T$ ."
],
[
"Parameter learning",
"Learning in DPL maximizes the conditional likelihood of virtual evidences $P(K|X)$ .",
"We can directly optimize this objective by summing out latent $Y$ to compute the gradient and run backpropagation.",
"In this paper, however, we opted for a modular approach using variational EM.",
"See Algorithm REF .",
"In the E-step, we compute a variational approximation $q(Y)=\\prod _i~q_i(Y_i)$ by minimizing its KL divergence with $P(Y|K,X)$ , which amounts to computing marginal probabilities $q_i(Y_i)=P(Y_i|K,X)=\\sum _{Y_{-i}}~P(Y_i, Y_{-i}|K,X)$ , with current parameters $\\Phi , \\Psi $ .",
"This is a standard probabilistic inference problem.",
"Exact inference is generally intractable, but there are a plethora of approximate inference methods that can efficiently produce an estimate.",
"We use loopy belief propagation [31] in this paper, by conducting message passing in $P(K,Y|X)$ iteratively.",
"Note that this inference problem is considerably simpler than end-to-end inference with probabilistic logic, since the bulk of the computation is encapsulated by $\\Psi $ .",
"Inference with high-order factors of large size can be challenging, but there is a rich body of literature for handling such structured factors in a principled way.",
"In particular, in distant supervision denoising, we alter the message passing schedule so that each at-least-one factor will compute messages to its variables jointly by renormalizing their current marginal probabilities with noisy-or [18], which is essentially a soft version of dual decomposition [4].",
"In the M-step, we treat the variational approximation $q_i(Y_i)$ as probabilistic labels, and use them to optimize $\\Phi $ and $\\Psi $ via standard supervised learning, which is equivalent to minimizing the KL divergence between the probabilistic labels and the conditional likelihood of $Y$ given $X$ under the supervision module ($\\Phi $ ) and prediction module ($\\Psi $ ), respectively.",
"For the prediction module, this optimization reduces to standard deep learning.",
"Likewise, for the supervision module, this optimization reduces to standard parameter learning for log-linear models (i.e., learning all $w_v$ 's that are not fixed).",
"Given the probabilistic labels, it is a convex optimization problem with a unique global optimum.",
"Here, we simply use gradient descent, with the partial derivative for $w_v$ being $\\mathbb {E}_{\\Phi (Y,X)}~[f_v(X,Y)] - \\mathbb {E}_{q(Y)}~[f_v(X,Y)]$ .",
"For a tied weight, the partial derivative will sum over all features that originate from the same template.",
"The second expectation can be done by simple counting.",
"The first expectation, on the other hand, requires probabilistic inference in the graphical model.",
"But it can be computed using belief propagation, similar to the E-step, except that the messages are limited to factors within the supervision module (i.e., messages from $\\Psi $ are not longer included).",
"Convergence is usually fast, upon which the marginal for each $Y_i$ is available, and $\\mathbb {E}_{\\Phi (Y,X)}~[f_v(X,Y)]$ is simply the fraction of $Y$ that renders $f_v(X,Y)$ to be true.",
"Again, this parameter learning problem is much simpler than end-to-end learning with probabilistic logic, as it focuses on refining uncertain weights for indirect supervision, rather than learning complex input patterns for label prediction (handled in deep learning).",
"Figure: Example of DPL combining various indirect supervision using probabilistic logic.",
"The prediction module is omitted to avoid clutter."
],
[
"Example",
"Figure REF shows a toy example on how DPL combines various indirect supervision for predicting drug-gene interaction (e.g., gefitinib can be used to treat tumors with EGFR mutations).",
"Indirect supervision is modeled by probabilistic logic, which defines a joint probability distribution over latent labeling decisions for drug-gene mention pairs in unlabeled text.",
"Here, distant supervision prefers classifying mention pairs of known relations, whereas the data programming formula opposes classifying instances resembling citations, and the joint inference formula ensures that at least one mention pair of a known relation is classified as positive.",
"Formula weight signifies the confidence in the indirect supervision, and can be refined iteratively along with the prediction module."
],
[
"Handling label imbalance",
"One challenge for distant supervision is that negative examples are often much more numerous.",
"A common strategy is to subsample negative examples to attain a balanced dataset.",
"In preliminary experiments, we found that this was often suboptimal, as many informative negative examples were excluded from training.",
"Instead, we restored the balance by up-weighting positive examples.",
"In DPL, an additional challenge is that the labels are probabilistic and change over iterations.",
"In this paper, we simply used hard EM, with binary labels set using 0.5 as the probability threshold, and the up-weighting coefficient recalculated after each E-step."
],
[
"Biomedical Machine Reading",
"There is a long-standing interest in biomedical machine reading (e.g., morgan2008overview, kim2009overview), but prior studies focused on supervised approaches.",
"The advent of big biomedical data creates additional urgency for developing scalable approaches that can generalize to new reading tasks.",
"For example, genome sequencing cost has been dropping faster than Moore's Law, yet oncologists can only evaluate tumor sequences for a tiny fraction of patients, due to the bottleneck in assimilating relevant knowledge from publications.",
"Recently, pengal17 formulated precision oncology machine reading as cross-sentence relation extraction (Figure REF ) and developed the state-of-the-art system using distant supervision.",
"While promising, their results still leave much room to improve.",
"Moreover, they used heuristics to heavily filter entity candidates, with significant recall loss.",
"In this section, we use cross-sentence relation extraction as a case study for combining indirect supervision using deep probabilistic logic (DPL).",
"First, we show that DPL can substantially improve machine reading accuracy in a head-to-head comparison with pengal17, using the same entity linking method.",
"Next, we apply DPL to entity linking itself and attain similar improvement.",
"Finally, we consider further improving the recall by removing the entity filter.",
"By applying DPL to joint entity linking and relation extraction, we more than doubled the recall in relation extraction while attaining comparable precision as pengal17 with heavy entity filtering."
],
[
"Evaluation",
"Comparing indirect supervision methods is challenging as there is often no annotated test set for evaluating precision and recall.",
"In such cases, we resort to the standard strategy used in prior work by reporting sample precision (estimated proportion of correct system extractions) and absolute recall (estimated number of correct system extractions).",
"Absolute recall is proportional to recall and can be used to compare different systems (modulo estimation errors)."
],
[
"Datasets",
"We used the same unlabeled text as pengal17, which consists of about one million full text articles in PubMed Central (PMC)www.ncbi.nlm.nih.gov/pmc.",
"Tokenization, part-of-speech tagging, and syntactic parsing were conducted using SPLAT [40], and Stanford dependencies [28] were obtained using Stanford CoreNLP [27].",
"For entity ontologies, we used DrugBankwww.drugbank.ca and Human Gene Ontology (HUGO)www.genenames.org.",
"DrugBank contains 8257 drugs; we used the subset of 599 cancer drugs.",
"HUGO contains 37661 genes.",
"For knowledge bases, we used the Gene Drug Knowledge Database (GDKD) [8] and the Clinical Interpretations of Variants In Cancer (CIVIC)civic.genome.wustl.edu.",
"Together, they contain 231 drug-gene-mutation triples, with 76 drugs, 35 genes and 123 mutations."
],
[
"Cross-sentence relation extraction",
"Let $e_1,\\cdots ,e_m$ be entity mentions in text $T$ .",
"Relation extraction can be formulated as classifying whether a relation $R$ holds for $e_1,\\cdots ,e_m$ in $T$ .",
"To enable a head-to-head comparison, we used the same cross-sentence setting as pengal17, where $T$ spans up to three consecutive sentences and $R$ represents the ternary interaction over drugs, genes, and mutations (whether the drug is relevant for treating tumors with the given gene mutation)."
],
[
"Entity linking",
"In this subsection, we used the entity linker from Literome [37] to identify drug, gene, and mutation mentions, as in pengal17.",
"This entity linker first identifies candidate mentions by matching entity names or synonyms in domain ontologies, then applies heuristics to filter candidates.",
"The heuristics are designed to enhance precision, at the expense of recall.",
"For example, one heuristics would filter candidates of length less than four, which eliminates key cancer genes such as ER or AKT."
],
[
"Prediction module",
"We used the same graph LSTM as in pengal17 to enable head-to-head comparison on indirect supervision strategies.",
"Briefly, a graph LSTM generalizes a linear-chain LSTM by incorporating arbitrary long-ranged dependencies, such as syntactic dependencies, discourse relations, coreference, and connections between roots of adjacent sentences.",
"A word might have precedents other than the prior word, and its LSTM unit is expanded to include a forget gate for each precedent.",
"See pengal17 for details.",
"Table: DPL combines three indirect supervision strategies for cross-sentence relation extraction"
],
[
"Supervision module",
"We used DPL to combine three indirect supervision strategies for cross-sentence relation extraction (Table REF ).",
"For distant supervision, we used GDKD and CIVIC as in pengal17.",
"For data programming, we introduced labeling functions that aim to correct entity and relation errors.",
"Finally, we incorporated joint inference among all co-occurring instances of an entity tuple with the known relation by imposing the at-least-one constraint (i.e., the relation holds for at least one of the instances).",
"For development, we sampled 250 positive extractions from DPL using only distant supervision [34] and excluded them from future training and evaluation."
],
[
"Experiment results",
"We compared DPL with the state-of-the-art system of pengal17.",
"We also conducted ablation study to evaluate the impact of indirect-supervision strategies.",
"For a fair comparison, we used the same probability threshold in all cases (an instance is classified as positive if the normalized probability score is at least 0.5).",
"For each system, sample precision was estimated by sampling 100 positive extractions and manually determining the proportion of correct extractions by an author knowledgeable about this domain.",
"Absolute recall is estimated by multiplying sample precision with the number of positive extractions.",
"Table: Comparison of sample precision and absolute recall (all instances and unique entity tuples) in test extraction on PMC.DPL + 𝙴𝙼𝙱\\tt EMB is our full system using PubMed-trained word embedding, whereas DPL uses the original Wikipedia-trained word embedding in pengal17.",
"Ablation: DS (distant supervision), DP (data programming), JI (joint inference).Table: Comparison of sample precision and absolute recall (all instances and unique entity tuples) in test extraction on PMC.",
"Both use same indirect supervision and Wikipedia-trained word embedding.Table REF shows the results.",
"DPL substantially outperformed pengal17, improving sample precision by ten absolute points and raising absolute recall by 25%.",
"Combining disparate indirect supervision strategies is key to this performance gain, as evident from the ablation results.",
"While distant supervision remained the most potent source of indirect supervision, data programming and joint inference each contributed significantly.",
"Replacing out-of-domain (Wikipedia) word embedding with in-domain (PubMed) word embedding [39] also led to a small gain.",
"pengal17 only compared graph LSTM and linear-chain LSTM in automatic evaluation, where distant-supervision labels were treated as ground truth.",
"They found significant but relatively small gains by graph LSTM.",
"We conducted additional manual evaluation comparing the two in DPL.",
"Surprisingly, we found rather large performance difference, with graph LSTM outperforming linear-chain LSTM by 13 absolute points in precision and raising absolute recall by over 20% (Table REF ).",
"This suggests that pengal17 might have underestimated the performance gain by graph LSTM using automatic evaluation.",
"Table: DPL combines three indirect supervision strategies for entity linking."
],
[
"Entity linking",
"Let $m$ be a mention in text and $e$ be an entity in an ontology.",
"The goal of entity linking is to predict $\\tt Link(m,e)$ , which is true iff $m$ refers to $e$ , for every candidate mention-entity pair $m,e$ .",
"We focus on genes in this paper, as they are particularly noisy."
],
[
"Prediction module",
"We used BiLSTM with attention over the ten-word windows before and after a mention.",
"The embedding layer is initialized by word2vec embedding trained on PubMed abstracts and full text [39].",
"The word embedding dimension was 200.",
"We used 5 epochs for training, with Adam as the optimizer.",
"We set learning rate to 0.001, and batch size to 64."
],
[
"Supervision module",
"As in relation extraction, we combined three indirect supervision strategies using DPL (Table REF ).",
"For distant supervision, we obtained all mention-gene candidates by matching PMC text against the HUGO lexicon.",
"We then sampled a subset of 200,000 candidate instances as positive examples.",
"We sampled a similar number of noun phrases as negative examples.",
"For data programming, we introduced labeling functions that used mention characteristics (longer names are less ambiguous) or syntactic context (genes are more likely to be direct objects and nouns).",
"For joint inference, we leverage linguistic phenomena related to coreference (identical, appositive, or synonymous mentions nearby are likely coreferent).",
"Table: Comparison of gene entity linking results on a balanced test set.",
"The string-matching baseline has low precision.",
"By combining indirect supervision strategies, DPL substantially improved precision while retaining reasonably high recall.Table: Comparison of gene entity linking results on BioCreative II test set.",
"GNormPlus is the state-of-the-art system trained on thousands of labeled examples.",
"DPL used only indirect supervision."
],
[
"Experiment results",
"For evaluation, we annotated a larger set of sample gene-mention candidates and then subsampled a balanced test set of 550 instances (half are true gene mentions, half not).",
"These instances were excluded from training and development.",
"Table REF compares system performance on this test set.",
"The string-matching baseline has a very low precision, as gene mentions are highly ambiguous, which explains why pengal17 resorted to heavy filtering.",
"By combining indirect supervision strategies, DPL improved precision by over 50 absolute points, while retaining a reasonably high recall (86%).",
"All indirect supervision strategies contributed significantly, as the ablation tests show.",
"We also evaluated DPL on BioCreative II, a shared task on gene entity linking [30].",
"We compared DPL with GNormPlus [48], the state-of-the-art supervised system trained on thousands of labeled examples in BioCreative II training set.",
"Despite using zero manually labeled examples, DPL attained comparable F1 and recall (Table REF ).",
"The difference is mainly in precision, which indicates opportunities for more indirect supervision."
],
[
"Joint entity and relation extraction",
"An important use case for machine reading is to improve knowledge curation efficiency by offering extraction results as candidates for curators to vet.",
"The key to practical adoption is attaining high recall with reasonable precision [34].",
"The entity filter used in pengal17 is not ideal in this aspect, as it substantially reduced recall.",
"In this subsection, we consider replacing the entity filter by the DPL entity linker Table REF .",
"Specifically, we added one labeling function to check if the entity linker returns a normalized probability score above $p_{\\tt TRN}$ for gene mentions, and filtered test instances if the gene mention score is lower than $p_{\\tt TST}$ .",
"We set $p_{\\tt TRN}=0.6$ and $p_{\\tt TST}=0.3$ from preliminary experiments.",
"The labeling function discouraged learning from noisy mentions, and the test-time filter skips an instance if the gene is likely wrong.",
"Not surprisingly, without entity filtering, pengal17 suffered large precision loss.",
"All DPL versions substantially improved accuracy, with significantly more gains using the DPL entity linker.",
"Table: Comparison of sample precision and absolute recall (all instances and unique entity tuples) when all gene mention candidates are considered.",
"pengal17 used distant supervision only.",
"RE: DPL relation extraction.",
"EL: using DPL entity linking in RE training (TRN) and/or test (TST).Table: Error analysis for DPL relation extraction."
],
[
"Scalability",
"DPL is efficient to train, taking around 3.5 hours for relation extraction and 2.5 hours for entity linking in our PubMed-scale experiments, with 25 CPU cores (for probabilistic logic) and one GPU (for LSTM).",
"For relation extraction, the graphical model of probabilistic logic contains around 7,000 variables and 70,000 factors.",
"At test time, it is just an LSTM, which predicted each instance in less than a second.",
"In general, DPL learning scales linearly in the number of training instances.",
"For distant supervision and data programming, DPL scales linearly in the number of known facts and labeling functions.",
"As discussed in Section 3, joint inference with high-order factors is more challenging, but can be efficiently approximated.",
"For inference in probabilistic logic, we found that loopy belief propagation worked reasonably well, converging after 2-4 iterations.",
"Overall, we ran variational EM for three iterations, using ten epochs of deep learning in each M-step.",
"We found these worked well in preliminary experiments and used the same setting in all final experiments.",
"Figure: Example of relation-extraction errors corrected by DPL with additional indirect supervision."
],
[
"Accuracy",
"To understand more about DPL's performance gain over distant supervision, we manually inspected some relation-extraction errors fixed by DPL after training with additional indirect supervision.",
"Figure REF shows two such examples.",
"While some data programming functions were introduced to prevent errors stemming from citations or flattened tables, none were directly applicable to these examples.",
"This shows that DPL can generalize beyond the original indirect supervision.",
"While the results are promising, there is still much to improve.",
"Table REF shows estimated precision errors for relation extraction by DPL.",
"(Some instances have multiple errors.)",
"Entity linking can incorporate more indirect supervision.",
"Joint entity linking and relation extraction can be improved by feeding back extraction results to linking.",
"Improvement is also sorely needed in classifying mutations and gene-mutation associations.",
"The prediction module can also be improved, e.g., by adding attention to graph LSTM.",
"DPL offers a flexible framework for exploring all these directions."
],
[
"Conclusion",
"We introduce DPL as a unifying framework for indirect supervision, by composing probabilistic logic with deep learning.",
"Experiments on biomedical machine reading show that this enables novel combination of disparate indirect supervision methodologies, resulting in substantial gain in accuracy.",
"Future directions include: combining DPL with deep generative models; exploring alternative optimization strategies; applications to other domains."
],
[
"Acknowledgements",
"We thank David McAllester, Chris Quirk, and Scott Yih for useful discussions, and the three anonymous reviewers for helpful comments."
]
] | 1808.08485 |
[
[
"Some inequalities for interpolational operator means"
],
[
"Abstract Using the properties of geometric mean, we shall show for any $0\\le \\alpha ,\\beta \\le 1$, \\[f\\left( A{{\\nabla }_{\\alpha }}B \\right)\\le f\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right){{\\nabla }_{\\beta }}A \\right){{\\sharp}_{\\alpha }}f\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right){{\\nabla }_{\\beta }}B \\right)\\le f\\left( A \\right){{\\sharp}_{\\alpha }}f\\left( B \\right)\\] whenever $f$ is a non-negative operator log-convex function, $A,B\\in \\mathcal{B}\\left( \\mathcal{H} \\right)$ are positive operators, and $0\\le \\alpha ,\\beta \\le 1$.",
"As an application of this operator mean inequality, we present several refinements of the Aujla subadditive inequality for operator monotone decreasing functions.",
"Also, in a similar way, we consider some inequalities of Ando's type.",
"Among other things, it is shown that if $\\Phi $ is a positive linear map, then \\[\\Phi \\left( A{{\\sharp}_{\\alpha }}B \\right)\\le \\Phi \\left( \\left( A{{\\sharp}_{\\alpha }}B \\right){{\\sharp}_{\\beta }}A \\right){{\\sharp}_{\\alpha }}\\Phi \\left( \\left( A{{\\sharp}_{\\alpha }}B \\right){{\\sharp}_{\\beta }}B \\right)\\le \\Phi \\left( A \\right){{\\sharp}_{\\alpha }}\\Phi \\left( B \\right).\\]"
],
[
"Introduction and Preliminaries",
"We denote the set of all bounded linear operators on a Hilbert space $\\mathcal {H}$ by $\\mathcal {B}\\left( \\mathcal {H} \\right)$ .",
"An operator $A\\in \\mathcal {B}\\left( \\mathcal {H} \\right)$ is said to be positive (denoted by $A\\ge 0$ ) if $\\left\\langle Ax,x \\right\\rangle \\ge 0$ for all $x\\in \\mathcal {H}$ .",
"If a positive operator is invertible, it is said to be strictly positive and we write $A>0.$ The axiomatic theory for connections and means for pairs of positive matrices have been studied by Kubo and Ando [10].",
"A binary operation $\\sigma $ defined on the cone of strictly positive operators is called an operator mean if for $A,B>0,$ (i) $I\\sigma I=I$ , where $I$ is the identity operator; (ii) ${{C}^{*}}\\left( A\\sigma B \\right)C\\le \\left( {{C}^{*}}AC \\right)\\sigma \\left( {{C}^{*}}BC \\right)$ , $\\forall C\\in \\mathcal {B}(\\mathcal {H})$ ; (iii) $A_{n}\\downarrow A$ and $B_{n}\\downarrow B$ imply $A_{n}\\sigma B_{n}\\downarrow A\\sigma B$ , where ${{A}_{n}}\\downarrow A$ means that ${{A}_{1}}\\ge {{A}_{2}}\\ldots $ and ${{A}_{n}}\\rightarrow A$ as $n\\rightarrow \\infty $ in the strong operator topology; (iv) $A\\le B\\quad \\& \\quad C\\le D\\quad \\text{ }\\Rightarrow \\quad \\text{ }A\\sigma C\\le B\\sigma D, \\forall C,D>0.",
"$ For a symmetric operator mean $\\sigma $ (in the sense that $A\\sigma B=B\\sigma A$ ), a parametrized operator mean ${{\\sigma }_{\\alpha }}$ ($\\alpha \\in \\left[ 0,1 \\right]$ ) is called an interpolational path for $\\sigma $ (or Uhlmann's interpolation for $\\sigma $ ) if it satisfies (c1) $A{{\\sigma }_{0}}B=A$ (here we recall the convention ${{T}^{0}}=I$ for any positive operator $T$ ), $A{{\\sigma }_{1}}B=B$ , and $A{{\\sigma }_{\\frac{1}{2}}}B=A\\sigma B$ ; (c2) $\\left( A{{\\sigma }_{\\alpha }}B \\right)\\sigma \\left( A{{\\sigma }_{\\beta }}B \\right)=A{{\\sigma }_{\\frac{\\alpha +\\beta }{2}}}B$ for all $\\alpha ,\\beta \\in \\left[ 0,1 \\right]$ ; (c3) the map $\\alpha \\in \\left[ 0,1 \\right]\\mapsto A{{\\sigma }_{\\alpha }}B$ is norm continuous for each $A$ and $B$ .",
"It is straightforward to see that the set of all $\\gamma \\in \\left[ 0,1 \\right]$ satisfying $\\left( A{{\\sigma }_{\\alpha }}B \\right){{\\sigma }_{\\gamma }}\\left( A{{\\sigma }_{\\beta }}B \\right)=A{{\\sigma }_{\\left( 1-\\gamma \\right)\\alpha +\\gamma \\beta }}B$ for all $\\alpha ,\\beta $ is a convex subset of $\\left[ 0,1 \\right]$ including 0 and 1.",
"Therefore (REF ) is valid for all $\\alpha ,\\beta ,\\gamma \\in \\left[ 0,1 \\right]$ (see [7]).",
"Typical interpolational means are so-called power means $A{{m}_{\\upsilon }}B={{A}^{\\frac{1}{2}}}{{\\left( \\frac{1}{2}\\left( I+{{\\left( {{A}^{-\\frac{1}{2}}}B{{A}^{-\\frac{1}{2}}} \\right)}^{\\upsilon }} \\right) \\right)}^{\\frac{1}{\\upsilon }}}{{A}^{\\frac{1}{2}}},\\quad \\text{ }-1\\le \\upsilon \\le 1$ and their interpolational paths are [8], $A{{m}_{\\upsilon ,\\alpha }}B={{A}^{\\frac{1}{2}}}{{\\left( \\left( 1-\\alpha \\right)I+\\alpha {{\\left( {{A}^{-\\frac{1}{2}}}B{{A}^{-\\frac{1}{2}}} \\right)}^{\\upsilon }} \\right)}^{\\frac{1}{\\upsilon }}}{{A}^{\\frac{1}{2}}},\\quad \\text{ }0\\le \\alpha \\le 1.$ In particular, we have $A{{m}_{1,\\alpha }}B=A{{\\nabla }_{\\alpha }}B=\\left( 1-\\alpha \\right)A+\\alpha B,$ $A{{m}_{0,\\alpha }}B=A{{\\sharp }_{\\alpha }}B={{A}^{\\frac{1}{2}}}{{\\left( {{A}^{-\\frac{1}{2}}}B{{A}^{-\\frac{1}{2}}} \\right)}^{\\alpha }}{{A}^{\\frac{1}{2}}},$ $A{{m}_{-1,\\alpha }}B=A{{!",
"}_{\\alpha }}B={{\\left( {{A}^{-1}}{{\\nabla }_{\\alpha }}B \\right)}^{-1}}.$ They are called the weighted arithmetic, weighted geometric, and weighted harmonic interpolations respectively.",
"It is well-known that $A{{!",
"}_{\\alpha }}B\\le A{{\\sharp }_{\\alpha }}B\\le A{{\\nabla }_{\\alpha }}B,\\quad \\text{ }0\\le \\alpha \\le 1$ In [5], Aujla et al.",
"introduced the notion of operator log-convex functions in the following way: A continuous real function $f:\\left( 0,\\infty \\right)\\rightarrow \\left( 0,\\infty \\right)$ is called operator log-convex if $f\\left( A{{\\nabla }_{\\alpha }}B \\right)\\le f\\left( A \\right){{\\sharp }_{\\alpha }}f\\left( B \\right),\\quad \\text{ }0\\le \\alpha \\le 1$ for all positive operators $A$ and $B$ .",
"After that, Ando and Hiai [2] gave the following characterization of operator monotone decreasing functions: Let $f$ be a continuous non-negative function on $\\left( 0,\\infty \\right)$ .",
"Then the following conditions are equivalent: (a) $f$ is operator monotone decreasing; (b) $f$ is operator log-convex; (c) $f\\left( A\\nabla B \\right)\\le f\\left( A \\right)\\sigma f\\left( B \\right)$ for all positive operators $A$ , $B$ and for all symmetric operator means $\\sigma $ .",
"In Theorem REF below, we provide a more precise estimate than (REF ) for operator log-convex functions.",
"As a by-product, we improve both inequalities in (REF ).",
"Additionally, we give refinement and two reverse inequalities for the triangle inequality.",
"Our main application of Theorem REF is a subadditive behavior of operator monotone decreasing functions.",
"Recall that a concave function (not necessarily operator concave) $f:(0,\\infty )\\rightarrow [0,\\infty )$ enjoys the subadditive inequality $f(a+b)\\le f(a)+f(b), a,b>0.$ Operator concave functions (equivalently, operator monotone) do not enjoy the same subadditive behavior.",
"However, in [3] it was shown that an operator concave function $f:(0,\\infty )\\rightarrow (0,\\infty )$ satisfies the norm version of (REF ) as follows $|||f(A+B)|||\\le |||f(A)+f(B)|||,$ for positive definite matrices $A,B$ and any unitraily invariant norm $|||\\;\\;|||$ .",
"Later, the authors in [6] showed that (REF ) is still valid for concave functions $f:(0,\\infty )\\rightarrow (0,\\infty )$ (not necessarily operator concave).",
"We emphasize that (REF ) does not hold without the norm.",
"In [4], it is shown that an operator monotone decreasing function $f:(0,\\infty )\\rightarrow (0,\\infty )$ satisfies the subadditive inequality $f(A+B)\\le f(A)\\nabla f(B),$ for the positive matrices $A,B.$ In Corollary REF , we present multiple refinements of (REF ).",
"The celebrated Ando's inequality asserts that if $\\Phi $ is a positive linear map and $A,B\\in \\mathcal {B}\\left( \\mathcal {H} \\right)$ are positive operators, then $\\Phi \\left( A{{\\sharp }_{\\alpha }}B \\right)\\le \\Phi \\left( A \\right){{\\sharp }_{\\alpha }}\\Phi \\left( B \\right),\\quad \\text{ }0\\le \\alpha \\le 1.$ Recall that, a linear map $\\Phi $ is positive if $\\Phi \\left( A \\right)$ is positive whenever $A$ is positive.",
"We improve and extend this result to Uhlmann's interpolation ${{\\sigma }_{\\alpha \\beta }}$ ($0\\le \\alpha ,\\beta \\le 1$ ).",
"Precisely speaking, we prove that $\\begin{aligned}\\Phi \\left( A{{\\sigma }_{\\alpha \\beta }}B \\right)&\\le \\Phi \\left( \\left( A{{\\sigma }_{\\alpha }}B \\right){{\\sigma }_{\\beta }}\\left( A{{\\sigma }_{0}}B \\right) \\right){{\\sigma }_{\\alpha }}\\Phi \\left( \\left( A{{\\sigma }_{\\alpha }}B \\right){{\\sigma }_{\\beta }}\\left( A{{\\sigma }_{1}}B \\right) \\right) \\\\& \\le \\Phi \\left( A \\right){{\\sigma }_{\\alpha \\beta }}\\Phi \\left( B \\right).\\end{aligned}$ This result is included in Section ."
],
[
"On the operator log-convexity",
"Our first main result in this paper reads as follows.",
"Theorem 2.1 Let $A,B\\in \\mathcal {B}\\left( \\mathcal {H} \\right)$ be positive operators and $0\\le \\alpha \\le 1$ .",
"If $f$ is a non-negative operator monotone decreasing function, then $f\\left( A{{\\nabla }_{\\alpha }}B \\right)\\le f\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right){{\\nabla }_{\\beta }}A \\right){{\\sharp }_{\\alpha }}f\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right){{\\nabla }_{\\beta }}B \\right)\\le f\\left( A \\right){{\\sharp }_{\\alpha }}f\\left( B \\right)$ for any $0\\le \\beta \\le 1$ .",
"Assume $f$ is operator monotone decreasing.",
"We start with the useful identity $A{{\\nabla }_{\\alpha }}B=\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right){{\\nabla }_{\\beta }} A\\right){{\\nabla }_{\\alpha }}\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right) {{\\nabla }_{\\beta }}B\\right),$ which follows from (REF ) with $A=A\\nabla _0B$ and $B=A\\nabla _1B$ .",
"Then we have $f\\left( A{{\\nabla }_{\\alpha }}B \\right)&=f\\left( \\left( \\left( A{{\\nabla }_{\\alpha }}B \\right) {{\\nabla }_{\\beta }}A\\right){{\\nabla }_{\\alpha }}\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right) {{\\nabla }_{\\beta }}B\\right) \\right) \\nonumber \\\\& \\le f\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right) {{\\nabla }_{\\beta }}A\\right){{\\sharp }_{\\alpha }}f\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right){{\\nabla }_{\\beta }}B \\right) \\\\& \\le \\left( f\\left( A{{\\nabla }_{\\alpha }}B \\right){{\\sharp }_{\\beta }}f(A) \\right){{\\sharp }_{\\alpha }}\\left( f\\left( A{{\\nabla }_{\\alpha }}B \\right){{\\sharp }_{\\beta }}f(B) \\right) \\\\& \\le \\left(\\left( f\\left( A \\right){{\\sharp }_{\\alpha }}f\\left( B \\right) \\right) {{\\sharp }_{\\beta }}f(A)\\right){{\\sharp }_{\\alpha }}\\left( \\left( f\\left( A \\right){{\\sharp }_{\\alpha }}f\\left( B \\right) \\right) {{\\sharp }_{\\beta }}f(B)\\right) \\\\& =\\left(\\left( f\\left( A \\right){{\\sharp }_{\\alpha }}f\\left( B \\right) \\right) {{\\sharp }_{\\beta }}\\left(f(A)\\sharp _0f(B)\\right)\\right){{\\sharp }_{\\alpha }}\\left( \\left( f\\left( A \\right){{\\sharp }_{\\alpha }}f\\left( B \\right) \\right) {{\\sharp }_{\\beta }}\\left(f(A)\\sharp _1f(B)\\right)\\right) \\\\& =f\\left( A \\right){{\\sharp }_{(1-\\beta )\\alpha +\\beta \\alpha }}f\\left( B \\right) \\\\& =f\\left( A \\right){{\\sharp }_{\\alpha }}f\\left( B \\right) \\nonumber $ where the inequalities (REF ), () and () follow directly from the log-convexity assumption on $f$ together with (REF ), the equalities () and () are obtained from the property (c1) and (REF ), respectively.",
"This completes the proof.",
"As promised in the introduction, we present the following refinement of Aujla inequality (REF ), as a main application of Theorem REF .",
"Corollary 2.1 Let $A,B\\in \\mathcal {B}\\left( \\mathcal {H} \\right)$ be positive operators.",
"If $f$ is a non-negative operator monotone decreasing function, then $f(A+B)&\\le f(3A\\nabla B)\\sharp f(A\\nabla 3B)\\\\&\\le f(2A)\\sharp f(2B)\\\\&\\le f(2A)\\nabla f(2B)\\\\&\\le f(A)\\nabla f(B).$ In Theorem REF , let $\\alpha =\\beta =\\frac{1}{2}$ and replace $(A,B)$ by $(2A,2B).$ This implies the first and second inequalities immediately.",
"The third inequality follows from the second inequality in (REF ), while the last inequality follows properties of operator means and the fact that $f$ is operator monotone decreasing.",
"Remark 2.1 Let $A,B\\in \\mathcal {B}\\left( \\mathcal {H} \\right)$ be positive operators and $0\\le \\alpha \\le 1$ .",
"If $f$ is a function satisfying $f\\left( A{{\\nabla }_{\\alpha }}B \\right)\\le f\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right) {{\\nabla }_{\\beta }} A\\right){{\\sharp }_{\\alpha }}f\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right) {{\\nabla }_{\\beta }}B\\right),$ for $0\\le \\beta \\le 1,$ then $f$ is operator monotone decreasing.",
"This follows by taking $\\beta =1$ in (REF ) and equivalence between (a) and (b) above.",
"Corollary 2.2 Let $A,B\\in \\mathcal {B}\\left( \\mathcal {H} \\right)$ be positive operators.",
"If $g$ is a non-negative operator monotone increasing, then $g\\left( A{{\\nabla }_{\\alpha }}B \\right)\\ge g\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right){{\\nabla }_{\\beta }}A \\right){{\\sharp }_{\\alpha }}g\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right){{\\nabla }_{\\beta }}B \\right)\\ge g\\left( A \\right){{\\sharp }_{\\alpha }}g\\left( B \\right)$ for any $0\\le \\alpha ,\\beta \\le 1$ .",
"It was shown in [2] that operator monotonicity of $g$ is equivalent to operator log-concavity ( $g\\left( A{{\\nabla }_{\\alpha }}B \\right)\\ge g\\left( A \\right){{\\sharp }_{\\alpha }}g\\left( B \\right)$ ).",
"The proof goes in a similar way to the proof of Theorem REF .",
"Remark 2.2 In [2], we have for non-negative operator monotone decreasing function $f$ , any operator mean $\\sigma $ and $A,B>0,$ $f(A\\nabla _{\\alpha }B)\\le f(A)!_{\\alpha }f(B)\\le f(A) \\sigma f(B),\\; 0\\le \\alpha \\le 1.$ Better estimates than (REF ) may be obtained as follows, where $0\\le \\alpha , \\beta \\le 1,$ $f\\left( A{{\\nabla }_{\\alpha }}B \\right)&=f\\left( \\left( \\left( A{{\\nabla }_{\\alpha }}B \\right) {{\\nabla }_{\\beta }}A\\right){{\\nabla }_{\\alpha }}\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right) {{\\nabla }_{\\beta }}B\\right) \\right) \\nonumber \\\\& \\le f\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right) {{\\nabla }_{\\beta }}A\\right){{!",
"}_{\\alpha }}f\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right){{\\nabla }_{\\beta }}B \\right) \\nonumber \\\\& \\le \\left( f\\left( A{{\\nabla }_{\\alpha }}B \\right){{!",
"}_{\\beta }}f(A) \\right){{!",
"}_{\\alpha }}\\left( f\\left( A{{\\nabla }_{\\alpha }}B \\right){{!",
"}_{\\beta }}f(B) \\right) \\nonumber \\\\& \\le \\left(\\left( f\\left( A \\right){{!",
"}_{\\alpha }}f\\left( B \\right) \\right) {{!",
"}_{\\beta }}f(A)\\right){{!",
"}_{\\alpha }}\\left( \\left( f\\left( A \\right){{!",
"}_{\\alpha }}f\\left( B \\right) \\right) {{!",
"}_{\\beta }}f(B)\\right) \\nonumber \\\\& =\\left(\\left( f\\left( A \\right){{!",
"}_{\\alpha }}f\\left( B \\right) \\right) {{!",
"}_{\\beta }}\\left(f(A)!_0f(B)\\right)\\right){{!",
"}_{\\alpha }}\\left( \\left( f\\left( A \\right){{!",
"}_{\\alpha }}f\\left( B \\right) \\right) {{!",
"}_{\\beta }}\\left(f(A)!_1f(B)\\right)\\right) \\nonumber \\\\& =f\\left( A \\right){{!",
"}_{(1-\\beta )\\alpha +\\beta \\alpha }}f\\left( B \\right) \\nonumber \\\\& =f\\left( A \\right){{!",
"}_{\\alpha }}f\\left( B \\right) \\nonumber \\\\& \\le f\\left( A \\right){{\\sigma }}f\\left( B \\right) \\nonumber $ In the following we improve the well-known weighted operator arithmetic-geometric-harmonic mean inequalities (REF ).",
"Theorem 2.2 Let $A,B\\in \\mathcal {B}\\left( \\mathcal {H} \\right)$ be positive operators.",
"Then $\\begin{aligned}A{{!",
"}_{\\alpha }}B&\\le \\left( \\left( A{{\\sharp }_{\\alpha }}B \\right){{\\sharp }_{\\beta }}A \\right){{!",
"}_{\\alpha }}\\left( \\left( A{{\\sharp }_{\\alpha }}B \\right) {{\\sharp }_{\\beta }}B\\right) \\\\& \\le A{{\\sharp }_{\\alpha }}B \\\\& \\le \\left( \\left( A{{\\sharp }_{\\alpha }}B \\right){{\\sharp }_{\\beta }}A \\right){{\\nabla }_{\\alpha }}\\left( \\left( A{{\\sharp }_{\\alpha }}B \\right) {{\\sharp }_{\\beta }}B\\right) \\\\& \\le A{{\\nabla }_{\\alpha }}B\\end{aligned}$ for $0\\le \\alpha ,\\beta \\le 1$ .",
"It follows from the proof of Theorem REF that $A{{\\sharp }_{\\alpha }}B=\\left( \\left( A{{\\sharp }_{\\alpha }}B \\right){{\\sharp }_{\\beta }}A \\right){{\\sharp }_{\\alpha }}\\left( \\left( A{{\\sharp }_{\\alpha }}B \\right){{\\sharp }_{\\beta }}B \\right),\\quad \\text{ }0\\le \\alpha ,\\beta \\le 1.$ Thus, we have $A{{\\sharp }_{\\alpha }}B&=\\left( \\left( A{{\\sharp }_{\\alpha }}B \\right){{\\sharp }_{\\beta }}A \\right){{\\sharp }_{\\alpha }}\\left( \\left( A{{\\sharp }_{\\alpha }}B \\right){{\\sharp }_{\\beta }}B \\right) \\nonumber \\\\& \\le \\left( \\left( A{{\\sharp }_{\\alpha }}B \\right) {{\\sharp }_{\\beta }}A\\right){{\\nabla }_{\\alpha }}\\left( \\left( A{{\\sharp }_{\\alpha }}B \\right) {{\\sharp }_{\\beta }}B\\right) \\\\& \\le \\left( \\left( A{{\\nabla }_{\\alpha }}B \\right){{\\nabla }_{\\beta }} A \\right) \\nabla _{\\alpha }\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right){{\\nabla }_{\\beta }}B \\right) \\\\& =\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right){{\\nabla }_{\\beta }} \\left(A\\nabla _0B\\right) \\right) \\nabla _{\\alpha }\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right){{\\nabla }_{\\beta }}\\left(A\\nabla _1B\\right) \\right) \\nonumber \\\\& =A{{\\nabla }_{\\alpha }}B $ where in the inequalities (REF ) and () we used the weighted operator arithmetic-geometric mean inequality and the equality () follows from (REF ).",
"This proves the third and fourth inequalities.",
"As for the first and second inequalities, replace $A$ and $B$ by $A^{-1}$ and $B^{-1}$ , respectively, in the third and fourth inequalities $A{{\\sharp }_{\\alpha }}B\\le \\left( \\left( A{{\\sharp }_{\\alpha }}B \\right){{\\sharp }_{\\beta }}A \\right){{\\nabla }_{\\alpha }}\\left( \\left( A{{\\sharp }_{\\alpha }}B \\right){{\\sharp }_{\\beta }}B \\right)\\le A{{\\nabla }_{\\alpha }}B$ which we have just shown.",
"Then take the inverse to obtain the required results (thanks to the identity $A^{-1}\\sharp _{\\alpha }B^{-1}=(A\\sharp _{\\alpha }B)^{-1}$ ).",
"This completes the proof.",
"Remark 2.3 We notice that similar inequalities maybe obtained for any symmetric mean $\\sigma $ , as follows.",
"First, observe that if $\\sigma ,\\tau $ are two symmetric means such that $\\sigma \\le \\tau $ , then the set $T=\\lbrace t:0\\le t\\le 1\\;{\\text{and}}\\;\\sigma _t\\le \\tau _t\\rbrace $ is convex.",
"Indeed, assume $t_1,t_2\\in T$ .",
"Then for the positive operators $A,B$ , we have $A\\sigma _{\\frac{t_1+t_2}{2}}B&=(A\\sigma _{t_1}B)\\sigma (A\\sigma _{t_2}B)\\\\&\\le (A\\tau _{t_1}B)\\tau (A\\tau _{t_2}B)\\\\&=A\\tau _{\\frac{t_1+t_2}{2}}B,$ where we have used the assumptions $\\sigma \\le \\tau $ and $t_1,t_2\\in T.$ This proves that $T$ is convex, and hence $T=[0,1]$ since $0,1\\in T$ , trivially.",
"Thus, we have shown that if $\\sigma \\le \\tau $ then $\\sigma _{\\alpha }\\le \\tau _{\\alpha },$ for all $0\\le \\alpha \\le 1.$ Now noting that $A\\sigma _{\\alpha } B= \\left((A\\sigma _{\\alpha }B)\\sigma _{\\beta }A\\right)\\sigma _{\\alpha }\\left((A\\sigma _{\\alpha }B)\\sigma _{\\beta }B\\right),$ and proceeding as in Theorem REF , we obtain $f\\left( A{{\\nabla }_{\\alpha }}B \\right)\\le f\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right){{\\nabla }_{\\beta }}A \\right){{\\sigma }_{\\alpha }}f\\left( \\left( A{{\\nabla }_{\\alpha }}B \\right){{\\nabla }_{\\beta }}B \\right)\\le f\\left( A \\right){{\\sigma }_{\\alpha }}f\\left( B \\right)$ for any $0\\le \\beta \\le 1$ and the operator log-convex function $f$ .",
"This provides a more precise estimate than $(c)$ above.",
"On the other hand, proceeding as in Theorem REF , we obtain $A\\sigma _{\\alpha } B\\le \\left((A\\sigma _{\\alpha }B)\\sigma _{\\beta }A\\right)\\nabla _{\\alpha }\\left((A\\sigma _{\\alpha }B)\\sigma _{\\beta }B\\right)\\le A\\nabla _{\\alpha }B,$ observing that $\\sigma _{\\alpha }\\le \\nabla _{\\alpha }.$ This provides a refinement of the latter basic inequality.",
"Taking into account (REF ), it follows that $A+B=\\alpha A+\\left( 1-\\alpha \\right)\\left( A\\nabla B \\right)+\\alpha B+\\left( 1-\\alpha \\right)\\left( A\\nabla B \\right).$ As a consequence of this inequality, we have the following refinement of the well-known triangle inequality $\\left\\Vert A+B \\right\\Vert \\le \\left\\Vert A \\right\\Vert +\\left\\Vert B \\right\\Vert .$ Corollary 2.3 Let $A,B\\in \\mathcal {B}\\left( \\mathcal {H} \\right)$ .",
"Then, for $\\alpha \\in \\mathbb {R}$ , $\\left\\Vert A+B \\right\\Vert \\le \\left\\Vert \\alpha A+\\left( 1-\\alpha \\right)\\left( A\\nabla B \\right) \\right\\Vert +\\left\\Vert \\alpha B+\\left( 1-\\alpha \\right)\\left( A\\nabla B \\right) \\right\\Vert \\le \\left\\Vert A \\right\\Vert +\\left\\Vert B \\right\\Vert .$ Remark 2.4 Using Corollary REF , we obtain the reverse triangle inequalities $\\left\\Vert A \\right\\Vert -\\left\\Vert B \\right\\Vert \\le \\frac{1}{2}\\left(\\left\\Vert A \\nabla _{-\\alpha }(2B)\\right\\Vert +\\left\\Vert A \\nabla _{\\alpha }(2B)\\right\\Vert -2\\left\\Vert B \\right\\Vert \\right)\\le \\left\\Vert A-B \\right\\Vert $ and $\\left\\Vert B \\right\\Vert -\\left\\Vert A \\right\\Vert \\le \\frac{1}{2}\\left(\\left\\Vert B \\nabla _{-\\alpha }(2A)\\right\\Vert +\\left\\Vert B \\nabla _{\\alpha }(2A)\\right\\Vert -2\\left\\Vert A \\right\\Vert \\right)\\le \\left\\Vert A-B \\right\\Vert ,$ where $\\alpha \\in \\mathbb {R}.$"
],
[
"A glimpse at the Ando's inequality",
"In this section, we present some versions and improvements of Ando's inequality (REF ).",
"Theorem 3.1 Let $A,B\\in \\mathcal {B}\\left( \\mathcal {H} \\right)$ be positive operators and $\\Phi $ be a positive linear map.",
"Then for any $0\\le \\alpha ,\\beta \\le 1$ , $\\Phi \\left( A{{\\sharp }_{\\alpha }}B \\right)\\le \\Phi \\left( \\left( A{{\\sharp }_{\\alpha }}B \\right){{\\sharp }_{\\beta }}A \\right){{\\sharp }_{\\alpha }}\\Phi \\left( \\left( A{{\\sharp }_{\\alpha }}B \\right){{\\sharp }_{\\beta }}B \\right)\\le \\Phi \\left( A \\right){{\\sharp }_{\\alpha }}\\Phi \\left( B \\right).$ In particular, $\\begin{aligned}\\sum \\limits _{j=1}^{m}{{{A}_{j}}{{\\sharp }_{\\alpha }}{{B}_{j}}}&\\le \\left( \\sum \\limits _{j=1}^{m}{\\left( {{A}_{j}}{{\\sharp }_{\\alpha }}{{B}_{j}} \\right){{\\sharp }_{\\beta }} {{A}_{j}} } \\right){{\\sharp }_{\\alpha }}\\left( \\sum \\limits _{j=1}^{m}{ \\left( {{A}_{j}}{{\\sharp }_{\\alpha }}{{B}_{j}} \\right){{\\sharp }_{\\beta }} {{B}_{j}} } \\right) \\\\& \\le \\left( \\sum \\limits _{j=1}^{m}{{{A}_{j}}} \\right){{\\sharp }_{\\alpha }}\\left( \\sum \\limits _{j=1}^{m}{{{B}_{j}}} \\right).\\end{aligned}$ We omit the proof of (REF ) because it is proved in a way similar to that of (REF ) in Theorem REF .",
"Now, if in (REF ) we take $\\Phi :{{M}_{nk}}\\left( \\mathbb {C} \\right)\\rightarrow {{M}_{k}}\\left( \\mathbb {C} \\right)$ defined by $\\Phi \\left( \\left( \\begin{matrix}{{X}_{1,1}} & {} & {} \\\\{} & \\ddots & {} \\\\{} & {} & {{X}_{n,n}} \\\\\\end{matrix} \\right) \\right)={{X}_{1,1}}+\\ldots +{{X}_{n,n}}$ and apply $\\Phi $ to $A={\\text{diag}}\\left( {{A}_{1}},\\ldots ,{{A}_{n}} \\right)$ and $B={\\text{diag}}\\left( {{B}_{1}},\\ldots ,{{B}_{n}} \\right)$ , we get (REF ).",
"In the following, we present a more general form of (REF ) will be shown.",
"Theorem 3.2 Let $A,B\\in \\mathcal {B}\\left( \\mathcal {H} \\right)$ be positive operators and $\\Phi $ be any positive linear map.",
"Then we have the following inequalities for Uhlmann's interpolation ${{\\sigma }_{\\alpha \\beta }}$ and $0\\le \\alpha ,\\beta \\le 1$ , $\\begin{aligned}\\Phi \\left( A{{\\sigma }_{\\alpha \\beta }}B \\right)&\\le \\Phi \\left( \\left( A{{\\sigma }_{\\alpha }}B \\right){{\\sigma }_{\\beta }}\\left( A{{\\sigma }_{0}}B \\right) \\right){{\\sigma }_{\\alpha }}\\Phi \\left( \\left( A{{\\sigma }_{\\alpha }}B \\right){{\\sigma }_{\\beta }}\\left( A{{\\sigma }_{1}}B \\right) \\right) \\\\& \\le \\Phi \\left( A \\right){{\\sigma }_{\\alpha \\beta }}\\Phi \\left( B \\right).\\end{aligned}$ Thanks to (REF ), we obviously have $\\begin{aligned}&\\left( \\left( A{{\\sigma }_{\\alpha }}B \\right){{\\sigma }_{\\beta }}\\left( A{{\\sigma }_{0}}B \\right) \\right){{\\sigma }_{\\alpha }}\\left( \\left( A{{\\sigma }_{\\alpha }}B \\right){{\\sigma }_{\\beta }}\\left( A{{\\sigma }_{1}}B \\right) \\right)\\\\& =\\left( A{{\\sigma }_{\\alpha \\left( 1-\\beta \\right)}}B \\right){{\\sigma }_{\\alpha }}\\left( A{{\\sigma }_{\\alpha \\left( 1-\\beta \\right)+\\beta }}B \\right) \\\\&= A{{\\sigma }_{\\alpha \\beta }}B.\\end{aligned}$ Now, the desired result follows directly from the above identities.",
"Remark 3.1 From simple calculations, we have the following inequalities for positive operators $A,B\\in \\mathcal {B}\\left( \\mathcal {H} \\right)$ , any positive linear map $\\Phi $ and $0\\le \\alpha ,\\beta ,\\gamma ,\\delta \\le 1$ , $\\begin{aligned}\\Phi \\left( A{{\\sigma }_{\\alpha \\left( 1-\\beta \\right)+\\beta \\left( \\left( 1-\\alpha \\right)\\gamma +\\alpha \\delta \\right)}}B \\right)&\\le \\Phi \\left( \\left( A{{\\sigma }_{\\alpha }}B \\right){{\\sigma }_{\\beta }}\\left( \\left( A{{\\sigma }_{\\gamma }}B \\right) \\right) \\right){{\\sigma }_{\\alpha }}\\Phi \\left( \\left( A{{\\sigma }_{\\alpha }}B \\right){{\\sigma }_{\\beta }}\\left( \\left( A{{\\sigma }_{\\delta }}B \\right) \\right) \\right) \\\\& \\le \\Phi \\left( A \\right){{\\sigma }_{\\alpha \\left( 1-\\beta \\right)+\\beta \\left( \\left( 1-\\alpha \\right)\\gamma +\\alpha \\delta \\right)}}\\Phi \\left( B \\right).\\end{aligned}$ Apparently, (REF ) reduces to (REF ) when $\\gamma =0$ and $\\delta =1$ .",
"(H. R. Moradi) Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran.",
"E-mail address: hrmoradi@mshdiau.ac.ir (S. Furuichi) Department of Information Science, College of Humanities and Sciences, Nihon University, 3-25-40, Sakurajyousui, Setagaya-ku, Tokyo, 156-8550, Japan.",
"E-mail address: furuichi@chs.nihon-u.ac.jp (M. Sababheh) Department of Basic Sciences, Princess Sumaya University for Technology, Amman 11941, Jordan.",
"E-mail address: sababheh@yahoo.com; sababheh@psut.edu.jo"
]
] | 1808.08342 |
[
[
"On the joint distribution of the marginals of multipartite random\n quantum states"
],
[
"Abstract We study the joint distribution of the set of all marginals of a random Wishart matrix acting on a tensor product Hilbert space.",
"We compute the limiting free mixed cumulants of the marginals, and we show that in the balanced asymptotical regime, the marginals are asymptotically free.",
"We connect the matrix integrals relevant to the study of operators on tensor product spaces with the corresponding classes of combinatorial maps, for which we develop the combinatorial machinery necessary for the asymptotic study.",
"Finally, we present some applications to the theory of random quantum states in quantum information theory."
],
[
"Introduction",
"The Wishart ensemble was historically the first probability distribution on matrices which was studied [46].",
"The main motivation for Wishart was statistics; later, Wigner [45] modelled complex, analytically intractable Hamiltonians in nuclear physics with Hermitian random matrices, and the field of Random Matrix Theory [29], [1] was born.",
"Nowadays, random matrices play a significant role in many sub-fields of mathematics, such as operator algebras, combinatorics and algebraic geometry, integrable systems and partial differential equations, as well as in other disciplines such as theoretical physics or telecommunication.",
"In this paper, we are motivated by a recent application of random matrices to Quantum Information Theory [32].",
"The mathematical formalism of Quantum Information Theory is constructed upon the central notion of quantum states, also known in the physical literature as density matrices.",
"These are positive semidefinite $d \\times d$ complex matrices, normalized to have unit trace; here, $d$ is the number of degrees of freedom of the quantum system under consideration.",
"Density matrices model “open quantum systems”, that is quantum systems which interact with an environment (which, most of the times, is too complicated to be taken under consideration).",
"Isolated systems, (which are called “closed”) are modeled traditionally by unit vectors in $\\mathbb {C}^d$ , which we choose to identify with the rank-one projections on the corresponding vector space; these projections are the extremal points of the convex set of density matrices.",
"One might want to study random quantum states for several different reasons.",
"Foremost, we would like to understand what are the typical mathematical (or physical, or even information-theoretical) properties of a typical state, where typical should be understood as randomly distributed with respect to some natural (or physically relevant) probability distribution.",
"Another reason one would like to understand random density matrices comes from the empirical observation that, in situations where explicit examples satisfying some desired properties are hard to come by, one should simply pick the sought-for object at random; in many cases, with large probability, the random sample will have the desired properties.",
"Random quantum states (and random quantum channels) have been a valuable source of (counter-)examples in Quantum Information Theory (see, e.g.",
"the recent review paper [16]).",
"There is a large literature on random density matrices and their applications to quantum information theory, most of it focusing on spectral properties of one random matrix.",
"In particular, the focus was on random states of single quantum systems and bipartite quantum systems (mostly related to the study of entanglement).",
"In this work, we tackle a fundamentally different question: Given a multipartite random density operator, what is the joint probability distribution of its different marginals?",
"Recall that, for a quantum state of a multi-partite system, the marginal of a subset of systems is the partial trace with respect to the complementary set of systems.",
"We shall study the question above in different settings, of increasing generality, first with just 4-partite states, and then for quantum states with arbitrarily many subsystems.",
"Depending on the relative sizes of the subsystems and on their rates of asymptotic growth, we shall exhibit two types of behavior.",
"In one situation, where the growth rates of the system dimensions are the same, we prove that the whole set of (balanced) marginals are asymptotically free, meaning, in broad terms, that they behave like independent random matrices (although they might share one or more subsystems).",
"In a different regime, where the dimensions of some of the subsystems are being kept fixed, we do not have asymptotic freeness, but we provide exact formulas for the limiting joint free cumulants, in terms of the types of marginals involved.",
"We state next, informally, two of the main results of this paper, corresponding to the asymptotic regimes described above; for the more general and precise results, we refer the reader to Theorem REF and, respectively, Theorem REF .",
"Theorem Let $\\rho ^{(N)} \\in \\mathcal {M}_{N^{2r}}(\\mathbb {C})$ be a random pure state of $2r$ -partite quantum system, where each subsystem is $N$ -dimensional.",
"Then, the $\\binom{2r}{r}$ marginals $\\rho _S^{(N)}$ , with $S \\subseteq \\lbrace 1,2, \\ldots , 2r\\rbrace $ a set of cardinality $r$ , are asymptotically free: their (rescaled) joint distribution converges in moments to that of $\\binom{2r}{r}$ free Marčenko-Pastur elements.",
"Equivalently, their limiting joint distribution is the same as that of $\\binom{2r}{r}$ independent copies of, say, $\\rho _{\\lbrace 1,2,\\ldots , r\\rbrace }^{(N)}$ .",
"In the unbalanced case, we have, informally, for 4-partite systems, the following result.",
"Theorem Let $\\rho _N \\in \\mathcal {M}_{Nm^2M}(\\mathbb {C})$ be a random pure state of 4-partite quantum system $\\mathcal {H}_{ABCD}$ , where $\\dim \\mathcal {H}_A= N$ , $\\dim \\mathcal {H}_B = \\dim \\mathcal {H}_C= m$ and $\\dim \\mathcal {H}_D= M \\sim cN$ .",
"Assuming $m,c$ are fixed constants, the (rescaled) joint distribution of the marginals $(\\rho _{AB}, \\rho _{BC})$ converges, in moments, as $N \\rightarrow \\infty $ , to a pair of non-commutative random variables $(x_{AB},x_{AC})$ having the following free cumulants: $\\kappa (x_{f(1)}, x_{f(2)}, \\ldots , x_{f(p)}) = cm^{-\\mathsf {alt}(f)},$ where $f \\in \\lbrace AB, AC\\rbrace ^p$ is an arbitrary word in the letters $AB, AC$ , and $\\mathsf {alt}(f)$ is the number of different consecutive values of $f$ , counted cyclically: $\\mathsf {alt}(f) := |\\lbrace a \\, : \\, f(a) \\ne f(a+1)\\rbrace |,$ where $f(p+1):=f(1)$ .",
"The contribution of our paper is threefold.",
"First, we introduce, in full detail, the notions of combinatorial maps relevant for the random matrix computations we perform, and we extend them to the case of matrices having a tensor product structure.",
"Our presentation starts at a basic level, gradually adding layers of complexity, and can be used by readers with a quantum information background as an introduction to the subject.",
"We develop the necessary combinatorial techniques to deal with the types of maps appearing in our study (combinatorial maps with vertices of two colors and edges of $2r$ colors).",
"Secondly, we contribute to the theory of random matrices and free probability by computing the limiting distribution of a family of random matrices obtained as marginals of a unique random object.",
"Although, globally, the random matrix model is standard (Wishart matrices), taking (intersecting) marginals (i.e.",
"partial traces) and considering their joint distribution is new; for this reason, in order to emphasize the importance of the tensor product structure of the Hilbert space, we shall call the random matrices we study Wishart tensors.",
"We prove asymptotic freeness in the balanced case, in a very general setting, and obtain the limiting free cumulants in the unbalanced setting; the explicit form of the free cumulants (see the second informal theorem above) is very interesting, involving a parameter counting the number of different consecutive letters appearing in the respective word.",
"Thirdly, from the point of view of quantum information theory, our study shows that the marginals of a random pure quantum state behave independently in the balanced case and in the large $N$ limit: the moment statistics of the whole set of marginals are the same as if the marginals were independent.",
"The situation is different in the unbalanced case: there is a strong correlation between, say, the marginals $\\rho _{AB}$ and $\\rho _{AC}$ of a pure random 4-partite quantum state $\\psi _{ABCD}$ when $\\dim \\mathcal {H}_A \\gg \\dim \\mathcal {H}_B = \\dim \\mathcal {H}_C$ .",
"Since the main focus of our paper is on random quantum states over Hilbert spaces with a tensor product structure, let us give now a brief survey of the literature on the subject, emphasizing the point of contact with our work.",
"Physicists started working on ensembles of quantum states in the early '90s, when Page computed the average entropy of entanglement of a bipartite random pure state [37].",
"The study of probability measures induced by metrics in the one-party case was initiated by Hall in [22] and developed by Sommers and Życzkowski in [49], [50], [41], [42]; see also [36] for the limiting eigenvalue distribution of the Bures ensemble.",
"In the multi-partite case, the study of random tensors in quantum information theory was initiated [2], where superpositions of random product states were investigated.",
"Later, specific models of randomness were studied in [38] in the case of random matrix product states and in [17], [18] for random graph states.",
"In [10], Christandl, Doran, Kousidis, and Walter studied the joint distribution of all the single particle marginals of a multi-partite quantum state in a very general setting, allowing for different distributions of the global state, and making use of the Duistermaat-Heckman measures from Lie theory.",
"The current work is, to our knowledge, the first instance where the question of the joint distribution of the possibly overlapping marginals of a random quantum state is considered; we do so in the simplest framework, that of the Wishart ensemble.",
"In the framework of quantum information theory, this corresponds to considering a random pure quantum state on a multipartite Hilbert space, and tracing out some of the subsystems to obtain the marginals.",
"The paper is organized as follows.",
"In Section we recall some well-known results about Wishart matrices and random density matrices, which can be seen as marginals of bipartite Wishart tensors.",
"This simple situation is also the occasion to introduce the machinery of combinatorial maps.",
"We provide two proofs of the classical Marčenko-Pastur theorem, one using the language of permutations and their metric properties, and another one using combinatorial maps; the reader can see from this example how the two approaches mirror each other.",
"In Section , we study in full detail the case of 4-partite Wishart tensors.",
"We consider two different asymptotical regimes: a balanced regime, where the dimension of all the spaces are equal, and an unbalanced regime, where two of the four spaces have fixed dimension.",
"We compute the limiting joint distribution of the two 2-marginals in both regimes: in the balanced case, we show that the marginals are asymptotically free, while in the unbalanced case we compute the (non-trivial) limiting mixed free cumulants.",
"Finally, in Section , we study the general multipartite case.",
"In the balanced case, we show again that the marginals are asymptotically free; in the other asymptotic regimes, we only have partial results: we list the different regimes that take place, but leave their detailed description for future work.",
"Acknowledgments.",
"L.L.",
"is a JSPS International Research Fellow.",
"The work of S.D.",
"was partially supported by the Australian Research Council grant DP170102028.",
"I.N.",
"'s research has been supported by the ANR projects StoQ (grant number ANR-14-CE25-0003-01) and NEXT (grant number ANR-10-LABX-0037-NEXT), and by the PHC Sakura program (grant number 38615VA).",
"I.N.",
"also acknowledges the hospitality of the Technische Universität München.",
"The authors would like to thank the Institut Henri Poincaré in Paris for its hospitality and for hosting the trimester on “Analysis in Quantum Information Theory”, during which part of this work was undertaken.",
"S.D.",
"and I.N.",
"would also like to thank the organizers of the “QUATR-17” conference in Skoltech/Moscow, and especially Leonid Chekhov, for bringing together researchers in random tensor theory and quantum information theory."
],
[
"The limiting eigenvalue distribution of random density matrices",
"In this section, we discuss the different ensembles of random density matrices from the literature, focusing on the induced ensemble, which will be the one we shall study in the later sections.",
"We also compute the limiting eigenvalue distribution of (rescaled) random density matrices, using two different formalisms: an algebraic one, emphasizing the role of permutations, and a combinatorial one, featuring the theory of combinatorial maps.",
"Although the two proofs given will be equivalent, we present both in full detail in order to introduce the main objects and to prepare the reader for the more complicated situations discussed in the later sections."
],
[
"Random density matrices",
"To start, let us fix some notation.",
"Density matrices with $N$ degrees of freedom are represented by unit trace, positive semidefinite $N \\times N$ matrices: $\\mathcal {M}_N^{1,+} := \\lbrace \\rho \\in \\mathcal {M}_N(\\mathbb {C}) \\, : \\, \\rho \\ge 0 \\text{ and } \\operatorname{Tr} \\rho = 1\\rbrace .$ This is a convex body, whose extreme points are rank one projections which we identify (up to a phase) with vectors $x \\in \\mathbb {C}^N$ , $\\Vert x\\Vert =1$ , called pure states.",
"We consider first the canonical distribution on pure states, that is the Lebesgue measure on the unit sphere of $\\mathbb {C}^N$ .",
"Integrating polynomials in the state's coordinates with respect to this measure is quite straightforward, see e.g. [20].",
"The idea is to relate the spherical integral to a Gaussian one with the help of a change of variable to the polar coordinates, and then use Wick's (or Isserlis' [26]) formula to evaluate the Gaussian integral; we recall this result next.",
"Proposition 2.1 Let $X_1, \\ldots , X_k$ be a $k$ -tuple of random variables having a joint (complex) Gaussian distribution.",
"If $k$ is odd, then $\\mathbb {E}[X_1 \\cdots X_k] = 0$ .",
"If $k=2l$ is even, then $\\mathbb {E}[X_1 \\cdots X_k] = \\sum _{\\begin{array}{c}p=\\lbrace \\lbrace i_1,j_1\\rbrace ,\\ldots , \\lbrace i_l,j_l\\rbrace \\rbrace \\\\ \\text{pairing of }\\lbrace 1,\\ldots ,k\\rbrace \\end{array}} \\quad \\prod _{s=1}^l \\mathbb {E}[X_{i_s} X_{j_s}].$ Let us move now to ensembles on the whole set of $N \\times N$ density matrices, $\\mathcal {M}_N^{1,+}$ .",
"Again, there is a natural candidate here, the normalization of the Lebesgue measure on the ambient space.",
"It turns out however, that this measure is just a specialization of a 1-parameter family of probability distributions, called the induced measures.",
"Introduced by Życzkowski and Sommers in [49], these measures have the advantage of being interesting and natural both from the physical and the mathematical perspectives.",
"Let us start with some motivating consideration from quantum physics.",
"Assume the physical system we are interested in (which has $N$ degrees of freedom) is not isolated, but coupled to an environment, having $M$ degrees of freedom.",
"In most physical applications, the environment is big and inaccessible, so we choose not to model it; in other words, if $\\Psi \\in \\mathbb {C}^N \\otimes \\mathbb {C}^M$ is the (pure) quantum state describing jointly the system and the environment, we only have access to the state of the system $\\rho = [\\operatorname{id}_N \\otimes \\operatorname{Tr}_M](\\Psi \\Psi ^*) \\in \\mathcal {M}_N^{1,+}.$ In the equation above, we assume that the vector $\\Psi $ is normalized, $\\Vert \\Psi \\Vert =1$ .",
"The main idea of [49] is to consider $\\Psi $ uniformly distributed on the unit sphere of $\\mathbb {C}^N \\otimes \\mathbb {C}^M$ ; this leads to the following definition.",
"Definition 2.2 The induced measure of parameters $(N,M)$ is the image measure of the uniform probability distribution on the unit sphere of $\\mathbb {C}^N \\otimes \\mathbb {C}^M$ through the map $\\mathbb {C}^N \\otimes \\mathbb {C}^M & \\rightarrow \\mathcal {M}_N^{1,+}\\\\\\Psi & \\mapsto [\\operatorname{id}_N \\otimes \\operatorname{Tr}_M](\\Psi \\Psi ^*),$ where $\\Psi \\Psi ^*$ denotes the self-adjoint rank one projection on $\\mathbb {C} \\Psi $ .",
"Importantly, the uniform (Lebesgue) probability measure on the convex body $\\mathcal {M}_N^{1,+}$ is exactly the induced measure with parameters $(N,N)$ [49].",
"A detailed mathematical analysis of the induced measures defined above was performed in [33], where it was emphasized that quantum states distributed along the induced measures are just normalized Wishart random matrices.",
"To make this observation more precise, let us briefly remind the reader the definition of the Wishart ensemble (we refer the reader to [23], [7], [25] for detailed treatments of the Wishart ensemble from a random matrix theory perspective, and to the excellent book [3] for a quantum information theory point of view).",
"Let $X \\in \\mathcal {M}_{N \\times M}(\\mathbb {C})$ be a Ginibre random matrix, that is a matrix having i.i.d.",
"complex standard Gaussian entries (no symmetry is assumed here).",
"For completeness, we rewrite formally what we mean by standard i.i.d.",
"complex Gaussian entries.",
"The entries of $X$ form a set of $NM$ complex numbers $x_{i,j}$ each with density $\\frac{1}{2i\\pi }e^{-|x_{i,j}|^2} \\mathrm {d}\\bar{x}_{i,j}\\mathrm {d}x_{i,j},$ which rewrites in terms of the real part $r_{i,j}$ and imaginary part $s_{i,j}$ of $x_{i,j}$ as $\\frac{1}{\\pi }e^{-(r_{i,j}^2+s_{i,j}^2)}\\mathrm {d}r_{i,j}\\mathrm {d}s_{i,j}.$ A Wishart matrix of parameters $(N,M)$ is defined as $W = XX^*$ .",
"The relation to random density matrices from the induced ensemble has been made mathematically rigorous in [33].",
"Proposition 2.3 Let $W$ be a random Wishart matrix of parameters $(N,M)$ .",
"Then, $\\rho = \\frac{W}{\\operatorname{Tr}W}$ is a random density matrix distributed along the induced measure from Definition REF of parameter $(N,M)$ .",
"Although the normalization by the trace is a highly non-trivial (and non-linear) operation, in practice, for large random matrices, it does not pose technical difficulties.",
"There are two reasons for this: first, in equation (REF ), the trace $\\operatorname{Tr} W$ and the normalized density matrix $\\rho $ are independent random variables; this fact is similar to the result in classical probability which states that the norm and the direction of a (standard) Gaussian vector are independent random variables.",
"The second reason which allows us to deal in a simple manner with the trace normalization is that the trace of a Wishart random matrix is a chi-squared random variable and thus concentrates very well around its average $\\mathbb {E} \\operatorname{Tr} W = NM$ (see [3]) $\\forall t>0, \\qquad \\mathbb {P}\\left[ |\\operatorname{Tr} W - NM| > tNM \\right] \\le 2 \\exp \\left(- \\frac{t^2NM}{2+4t/3} \\right).$"
],
[
"The limiting eigenvalue distribution",
"We compute in this section the eigenvalue distribution of Wishart matrices, and thus of random density matrices, in the large $N$ limit.",
"We shall work with the simpler model of Wishart matrices, and then translate the results to quantum states in Corollary REF .",
"We shall present two proofs of the well-known convergence to the Marčenko-Pastur distribution, one using permutations and the other one using combinatorial maps; this will be the occasion to introduce these two proof techniques and to familiarize the reader with the main objects appearing in the respective theories.",
"In the limit of large matrix dimension ($N \\rightarrow \\infty $ ), the behavior of Wishart matrices depends on the asymptotic ratio $M/N$ .",
"The most common situation is when $M/N \\rightarrow c \\in (0,\\infty )$ , in which case the matrix converges to the well-known Marčenko-Pastur distribution [30].",
"Although this result is straightforward and very well-known, we provide a self-contained proof in order to compare the approach of this section with the one in the next section.",
"Proposition 2.4 Let $W_N$ be a sequence of random Wishart matrices of parameters $(N,M_N)$ , where $M_N$ is an integer sequence with the property that $M_N \\sim cN$ as $N \\rightarrow \\infty $ , where $c \\in (0,\\infty )$ is a constant.",
"The sequence $N^{-1}W_N$ converges, in moments, towards the Marčenko-Pastur distribution $\\forall p \\ge 1, \\qquad \\lim _{N \\rightarrow \\infty } \\mathbb {E} \\frac{1}{N} \\operatorname{Tr}\\left(\\frac{W_N}{N}\\right)^p = \\int x^p \\mathrm {d}\\mathrm {MP}_c(x),$ where $\\mathrm {d}\\mathrm {MP}_c=\\max (1-c,0)\\delta _0+\\frac{\\sqrt{(b-x)(x-a)}}{2\\pi x} \\; \\mathbf {1}_{[a,b]}(x) \\, \\mathrm {d}x,$ with $a = (1-\\sqrt{c})^2$ and $b=(1+\\sqrt{c})^2$ .",
"We plot the density of the Marčenko-Pastur distribution, along with Monte-Carlo simulations in Figure REF ; for other regimes, see [3].",
"The mass term for $c<1$ is easily explained by the fact that in this case $M<N$ and the rank of the $N\\times N$ matrix $W$ is $M<N$ thus $W$ shall have $N-M\\sim _{N\\rightarrow \\infty }N(1-c)$ null eigenvalues.",
"Figure: The density of the Marčenko-Pastur distribution MP c \\mathrm {MP}_c for c=1c=1 (left) and c=5c=5 (right).We shall use the method of moments, i.e.",
"compute Gaussian expectations of the form $\\mathbb {E} \\operatorname{Tr}(W^{p})= \\int _{(N,M)} \\operatorname{Tr}\\bigl ((XX^*)^{p}\\bigr ) \\mathrm {d}\\mu (X),$ with $\\mathrm {d}\\mu (X)=\\frac{1}{(2i\\pi )^{NM}}e^{-\\operatorname{Tr}(XX^*)}\\mathrm {d}X^*\\mathrm {d}X, \\ \\textrm { and } \\ \\mathrm {d}X^*\\mathrm {d}X=\\prod _{\\begin{array}{c}1\\le i\\le N \\\\ 1\\le j\\le M\\end{array}}\\mathrm {d}\\bar{x}_{i,j}\\mathrm {d}x_{i,j}.$ The proof consists of three steps.",
"First, we show the exact formula $\\mathbb {E} N^{-1} \\operatorname{Tr}\\left[(N^{-1}W_N)^p\\right] = \\sum _{\\alpha \\in \\mathcal {S}_p} N^{\\#(\\gamma \\alpha )} M^{\\#\\alpha },$ where $\\gamma \\in \\mathcal {S}_p$ is the full cycle permutation $\\gamma = (1, 2, 3, \\ldots p)$ and $\\#\\alpha $ denotes the number of cycles of the permutation $\\alpha $ .",
"Note that we dropped the dependence on $N$ of the parameter $M$ , in order to keep the notation light; the reader should keep in mind that $M = M_N$ is a function of $N$ which grows as $M \\sim cN$ .",
"The second step, Lemma REF , will consist in analyzing the dominating terms in (REF ).",
"We will show that the surviving permutations are in bijection with non-crossing partitions, recovering the moments of the Marčenko-Pastur distribution $\\lim _{N \\rightarrow \\infty } \\mathbb {E} N^{-1} \\operatorname{Tr}\\left[(N^{-1}W_N)^p\\right] = \\sum _{\\alpha \\in \\mathrm {NC}(p)} c^{\\#\\alpha }.$ Using Voiculescu's $R$ -transform, we compute in a third step the Cauchy transform of the probability measure having the moments above, and then, by Stieltjes inversion, we recover the exact expression of the density (REF ).",
"First step.",
"To show (REF ), we need to perform the integration on the left-hand-side with the help of the Wick formula from Proposition REF .",
"We are going to use a graphical reading of the Wick formula introduced in [15].",
"In this framework, matrices (and more generally, tensors) are represented by boxes having decorations corresponding to the vector spaces the matrix is acting on.",
"The decorations have two attributes: shape, distinguishing vector spaces of various dimensions, and shading, distinguishing primal (filled symbols) from dual (empty symbols) spaces.",
"We depict in Figure REF (from left to right) the diagram of a Wishart matrix $W=XX^*$ , then, in the center, the same diagram, with $X^*$ replaced by the transpose of $\\bar{X}$ , and then the diagram for the second moment $\\operatorname{Tr}(W^2)$ .",
"Figure: Diagrams for Wishart matrices.",
"On the left, the diagram for W=XX * W=XX^*, with X∈ℳ N×M X \\in \\mathcal {M}_{N \\times M}.",
"The Hilbert space ℂ N \\mathbb {C}^N is depicted by round decorations, while ℂ M \\mathbb {C}^M is depicted by square decorations.",
"The tensor contraction between the two square decorations corresponds to the matrix product X·X * X \\cdot X^*.",
"In the center panel, we have the same diagram, after replacing X * X^* by (X ¯) ⊤ (\\bar{X})^\\top ; notice that taking the transposition amounts to inverting the shading of the decorations of the X ¯\\bar{X} box.",
"In the last panel, we depict the scalar Tr(W 2 )=Tr(XX * XX * )\\operatorname{Tr}(W^2) = \\operatorname{Tr}(XX^*XX^*).In order to establish the formula (REF ), we need to apply the Wick formula to the quantity $\\operatorname{Tr}(W^p)$ , for an arbitrary $p \\ge 1$ .",
"In [15] (see also [16]), it has been shown that computing a Gaussian expectation can be done in a graphical way, as follows.",
"Given a diagram containing boxes $X$ and $\\bar{X}$ corresponding to random matrices (or tensors) with i.i.d.",
"standard complex Gaussian entries, if the number of $X$ -boxes is different than the number of $\\bar{X}$ -boxes, the expectation (over the randomness in $X$ ) is zero.",
"If one has, say, $p$ $X$ -boxes and $p$ $\\bar{X}$ -boxes, the expectation of the diagram is a sum indexed by permutations $\\alpha \\in \\mathcal {S}_p$ , where the terms are obtained by deleting the $X$ and the $\\bar{X}$ boxes, and connecting the corresponding attached decorations with the permutation $\\alpha $ : the decorations of the $i$ -th $\\bar{X}$ -box are to be connected to the corresponding decorations of the $\\alpha (i)$ -th $X$ box.",
"We would like to warn the reader at this point that the above convention is opposite from the one used in [15], where $\\alpha $ was connecting $X$ -boxes to $\\bar{X}$ -boxes.",
"It turns out that the current convention makes the connection between the Wick graphical calculus and the theory of combinatorial maps more transparent, justifying our choice.",
"As an example, in our moment problem, the two diagrams appearing when computing $\\mathbb {E} \\operatorname{Tr}(XX^*XX^*)$ are depicted in Figure REF .",
"Notice that the new diagrams are entirely made out of loops, so their (scalar) values are given by $N^aM^b$ , where $a$ (resp.",
"$b$ ) is the number of loops corresponding to $\\mathbb {C}^N$ , that is to round decorations (resp.",
"$\\mathbb {C}^M$ , i.e.",
"square decorations).",
"Figure: The two diagrams corresponding to the graphical application of the Wick formula for 𝔼Tr(W 2 )\\mathbb {E} \\operatorname{Tr}(W^2), see Figure , rightmost panels.",
"On the left, the diagram corresponding to the identity permutation (the blue wires connect the decorations of each XX-box to the corresponding ones of the X ¯\\bar{X}-box directly following it).",
"On the right is the diagram corresponding to the permutation α=(12)\\alpha = (12).",
"The two diagrams contain only loops, so their values are NM 2 NM^2,respectively N 2 MN^2M.Moving to the general case of an arbitrary $p$ , we write $\\mathbb {E} \\operatorname{Tr}(W^p) = \\sum _{\\alpha \\in \\mathcal {S}_p} \\mathcal {D}_\\alpha $ , where $\\mathcal {D}_\\alpha $ is the diagram obtained by deleting the $p$ $X$ - and $\\bar{X}$ -boxes and by connecting the corresponding decorations according to the permutation $\\alpha $ .",
"It is clear that $\\mathcal {D}_\\alpha $ consists only of loops corresponding to the Hilbert spaces $\\mathbb {C}^N$ and $\\mathbb {C}^M$ ; it follows that in order to evaluate such a diagram, one has to count the number of loops of each type.",
"Let us start by counting the loops attached to square decorations, each giving a contribution of $M$ .",
"Note that in the original diagram (before taking the expectation), the square decoration of the $i$ -th $X$ -box is connected to the square decoration of the $\\bar{X}$ -box belonging to the same, $i$ -th, group.",
"It is then easy to see that each distinct cycle of the permutation $\\alpha $ gives rise to a loop, then the number of $M$ -loops is $\\#\\alpha $ , thus giving a total contribution of $M^{\\#\\alpha }$ .",
"The same reasoning can be applied when counting the contribution of loops attached to round decorations (which correspond to the Hilbert space $\\mathbb {C}^N$ ), with one difference: in the initial wiring of the diagram (before taking the expectation), the $i$ -th $X$ -box is connected to the $(i-1)$ -th $\\bar{X}$ -box (where the subtraction operation is understood cyclically, modulo $p$ ).",
"In other words, the initial wiring is given by the full-cycle permutation $\\gamma $ , with $\\gamma (i) = i+1$ ; note that we are numbering the boxes $1, 2, \\ldots , p$ from right to left.",
"A similar combinatorial argument shows that the number of loops is, in this case, $\\#(\\gamma \\alpha )$ , for a final contribution of $N^{\\#(\\gamma \\alpha )}$ .",
"We conclude that $\\mathcal {D}_\\alpha = N^{\\#(\\gamma \\alpha )}M^{\\# \\alpha }$ , proving (REF ).",
"Second step.",
"We now move to the second step of the proof, which is computing the limit $N \\rightarrow \\infty $ of the moment formula (REF ).",
"Lemma 2.5 The asymptotic moments of the (normalized) random matrices $W_N$ are given by a sum over non-crossing partitions $\\lim _{N \\rightarrow \\infty }\\mathbb {E} N^{-1} \\operatorname{Tr}\\left[(N{^{-1}}W_N)^p\\right] =\\sum _{\\tilde{\\alpha }\\in \\mathrm {NC}(p)}c^{\\#\\tilde{\\alpha }}.$ Proof of Lemma REF .",
"Since we need to find the dominating terms in the sum, we have to maximize the function $\\mathcal {S}_p \\ni \\alpha \\mapsto \\#\\alpha + \\#(\\gamma \\alpha )$ .",
"The following lemma contains the key combinatorial insight which allows us to perform this task.",
"The result below is contained in [6] (see also [35] for a textbook presentation).",
"Lemma 2.6 For a permutation $\\alpha \\in \\mathcal {S}_p$ , let $|\\alpha |$ denote the minimum number of transpositions that multiply to $\\alpha $ ; $|\\alpha |$ is called the length of the permutation $\\alpha $ and satisfies the relations $ |\\alpha | + \\#\\alpha = p \\qquad \\qquad |\\alpha | = |\\alpha ^{-1}| \\qquad \\qquad |\\alpha \\beta | = |\\beta \\alpha |$ for all permutations $\\alpha ,\\beta \\in \\mathcal {S}_p$ .",
"The mapping $d(\\alpha , \\beta ) := |\\alpha ^{-1} \\beta |$ defines a distance on $\\mathcal {S}_p$ .",
"For two fixed permutations $\\alpha , \\beta \\in \\mathcal {S}_p$ , the permutations $\\chi \\in \\mathcal {S}_p$ saturating the triangle inequality $d(\\alpha , \\chi ) + d(\\chi , \\beta ) \\ge d(\\alpha , \\beta )$ are called geodesic; we write $\\alpha - \\chi - \\beta $ .",
"The set of geodesic permutations between the identity permutation $\\mathrm {id}$ and the full cycle permutation $\\gamma \\in \\mathcal {S}_p$ is in bijection with the set of non-crossing partitions $\\mathrm {NC}(p)$ : a partition $\\tilde{\\alpha }\\in \\mathrm {NC}(p)$ encodes the cycle structure of $\\chi $ , and the elements inside a given cycle of $\\chi $ have the same cyclic ordering as in $\\gamma $ .",
"Using the lemma above and the asymptotic relation $M_N \\sim cN$ , the exponent of $N$ in the general term of (REF ) can be bounded as follows $\\nonumber \\#\\alpha + \\#(\\gamma \\alpha ) &= p-|\\alpha | + p-|\\gamma \\alpha |\\\\&= 2p-(|\\alpha | + |\\alpha ^{-1}\\gamma ^{-1}|)\\\\\\nonumber &\\le 2p - |\\gamma ^{-1}| = p+1,$ where we have used the triangle inequality.",
"The permutations $\\alpha $ saturating this inequality are precisely the geodesic ones, i.e.",
"the ones satisfying $\\mathrm {id} - \\alpha - \\gamma $ .",
"These are in bijection with non-crossing partitions $\\tilde{\\alpha }\\in \\mathrm {NC}(p)$ , and one has $\\# \\alpha = \\# \\tilde{\\alpha }$ , where the $\\#$ notation denotes at the same time the number of cycles of a permutation $\\alpha $ and the number of blocks of the corresponding non-crossing partition $\\tilde{\\alpha }$ .",
"We have shown $\\mathbb {E} \\operatorname{Tr}(W^p) \\sim N^{p+1}\\sum _{\\tilde{\\alpha }\\in \\mathrm {NC}(p)}c^{\\#\\tilde{\\alpha }}.$ Third step.",
"The general term in the sum in the right hand side of (REF ) is a multiplicative function over the blocks of non-crossing partition $\\tilde{\\alpha }$ : the contribution of each cycle is $c$ , independently of the length of the block.",
"We have thus identified the free cumulants of the limiting distribution of the random matrices $W_N$ : $\\kappa _n = c$ , for all $n \\ge 1$ (we refer the reader to [35] for the definition and the basic properties of free cumulants).",
"In order to obtain the density of the probability distribution having the moments above, we use Voiculescu's $\\mathcal {R}$ -transform machinery.",
"We have $\\mathcal {R}(z) = \\sum _{n=0}^\\infty \\kappa _{n+1} z^n = \\sum _{n=0}^\\infty c z^n = \\frac{c}{1-z}.$ The Cauchy transform and the $\\mathcal {R}$ -transform are related by the implicit equation $G(\\mathcal {R}(z) + 1/z) = z$ , which we solve for $G$ , obtaining $G(w) = \\frac{1-c+w-\\sqrt{(w-a)(w-b)}}{2w}.$ Above, we have chosen the solution of the second degree equation in $G$ such that $G(w) \\sim w^{-1}$ as $w \\rightarrow \\infty $ , and we have set $a = (1-\\sqrt{c})^2$ and $b = (1+\\sqrt{c} )^2$ .",
"Note that the function $G$ can have poles only at $w=0$ , with residue $\\max (1-c,0)$ , explaining the atom at 0, when $0<c<1$ .",
"The expression for the density is obtained using the Stieltjes inversion formula $\\frac{\\mathrm {d} \\mathrm {MP}_c}{\\mathrm {d}x} = - \\frac{1}{\\pi }\\lim _{\\varepsilon \\rightarrow 0} \\Im G(x+i \\varepsilon ).",
"$ For random density matrices, one has to simply take into account the trace normalization: if $\\rho _N$ is a sequence of random density matrices from the induced ensemble with parameters $(N, M_N)$ , we can write $\\rho _N = W_N / \\operatorname{Tr} W_N$ , for $W_N$ a sequence of random Wishart matrices of parameters $(N, M_N)$ .",
"We have then the following corollary.",
"Corollary 2.7 Let $\\rho _N \\in \\mathcal {M}_N^{1,+}$ be a sequence of random density matrices from the induced ensemble of parameters $(N,M_N)$ , where $M_N$ is an integer sequence with the property that $M_N \\sim cN$ as $N \\rightarrow \\infty $ , with $c \\in (0,\\infty )$ a constant.",
"The sequence $cN \\rho _N$ converges, in moments, towards the Marčenko-Pastur distribution $\\mathrm {MP}_c$ .",
"Write $cN \\rho _N = cN \\frac{W_N}{\\operatorname{Tr}W_N} = \\frac{W_N}{N} \\cdot \\frac{cN^2}{\\operatorname{Tr}W_N}.$ From equation (REF ), it follows that $cN^2/(\\operatorname{Tr}W_N)$ converges, almost surely, to 1.",
"Together with Proposition REF , this proves the claim.",
"Note that one can prove much stronger statements of convergence than the ones we cited; importantly, one can show that the largest eigenvalue of (properly normalized) Wishart and random density matrices converges, almost surely, towards the right edge of the support of the limiting Marčenko-Pastur distribution [8].",
"Remark 2.8 Note that if $W \\in \\mathcal {M}_{N_1N_2}(\\mathbb {C})$ is a Wishart tensor of parameters $(N_1N_2,M)$ , then $W^{\\prime } = \\operatorname{Tr}_2(W)$ is a Wishart matrix of parameters $(N_1, N_2M)$ .",
"Indeed, if $W = XX^*$ , with $X \\in \\mathcal {M}_{N_1N_2 \\times M}(\\mathbb {C})$ a Gaussian matrix, then $W^{\\prime } = YY^*$ , where $Y \\in \\mathcal {M}_{N_1 \\times N_2M}(\\mathbb {C})$ is the matrix obtained by “reshaping” $X$ into a matrix of appropriate dimensions.",
"This equivalence comes from the fact that both the partial trace and the matrix multiplication correspond to tensor contractions.",
"In the random density matrix picture, we have, for a random vector $\\Psi \\in \\mathbb {C}^{N_1} \\otimes \\mathbb {C}^{N_2} \\otimes \\mathbb {C}^{M}$ , $\\rho _1 = [\\operatorname{id}_{N_1} \\otimes \\operatorname{Tr}_{N_2}](\\rho _{23}) = [\\operatorname{id}_{N_1} \\otimes \\operatorname{Tr}_{N_2}\\otimes \\operatorname{Tr}_{M}](\\Psi \\Psi ^*).$"
],
[
"A combinatorial map version of the proof",
"In this subsection we re-prove the formulas (REF ) and (REF ) using combinatorial map methods instead of the results on distances between permutations (Lemma REF ).",
"Let us first provide a few definitions.",
"Definition 2.9 A connected labeled bicolored combinatorial map, or simply a bicolored map, is a triplet $\\mathcal {M}=(E, \\sigma _\\circ , \\sigma _\\bullet )$ where $E$ is a set of edges labeled from 1 to $p$ $\\sigma _\\circ $ and $\\sigma _\\bullet $ are permutations on $E$ , the group $\\langle \\sigma _\\circ ,\\sigma _\\bullet \\rangle $ generated by $ \\sigma _\\circ $ and $\\sigma _\\bullet $ acts transitively on $E$ .",
"We call white $vertices$ (resp.",
"black $vertices$ , resp.",
"$faces$ ) the elements of the unique decomposition of $\\sigma _\\circ $ (resp.",
"$\\sigma _\\bullet $ , resp.",
"$\\sigma _\\bullet \\sigma _\\circ $ ) in disjoint cycles.",
"A bicolored map $\\mathcal {M}=(E, \\sigma _\\circ , \\sigma _\\bullet )$ satisfies the following classical result: $\\#\\sigma _\\circ + \\#\\sigma _\\bullet + \\#(\\sigma _\\bullet \\sigma _\\circ ) - p = 2 - 2g(\\mathcal {M}),$ where $g(\\mathcal {M})$ is a non-negative integer.",
"A map can also be seen as a graph drawn on a two-dimensional surface, up to continuous deformations of the edges: each disjoint cycle of $\\sigma _\\circ $ (resp.",
"$\\sigma _\\bullet $ ) is a white (resp.",
"black) point (vertex) of that surface, each element of $E$ is an arc between a black and a white vertex, and the ordering of the cycles of $ \\sigma _\\circ $ , and $\\sigma _\\bullet $ define an ordering of the half-edges incident to the vertices.",
"The graph is said to be cellularly embedded (or simply embedded, in the context of this paper) on that surface if the connected components of the complement of the graph on that surface are homeomorphic to discs, in which case these components correspond to the faces, the disjoint cycles of $\\sigma _\\bullet \\sigma _\\circ $ .",
"The genus (number of holes) of a surface on which the underlying graph of a map can be embedded is precisely the integer $g(\\mathcal {M})$ in (REF ).",
"A genus 0 map can therefore be drawn on the plane without crossing.",
"Finally, the transitivity condition forces the combinatorial map to be connected.",
"Figure: A trace of pp matrices can be represented as a vertex of valency pp with a cyclic ordering of incident half-edges.With the notations of the proof of Prop.",
"REF , we define a bicolored map on $E=\\lbrace 1,\\ldots ,p\\rbrace $ by setting $\\sigma _\\bullet := \\gamma $ , and $\\sigma _\\circ := \\alpha $ .",
"As illustrated in Fig.",
"REF , in this representation, a single black vertex of valency $p$ is associated to $\\operatorname{Tr}(W^p)$ , each matrix $W$ corresponding to an edge incident to it.",
"The trace induces a cyclic counter-clockwise ordering of these half-edges around the vertexAs will appear in the following, the moments are expressed in terms of labeled maps.",
"This is because the trace corresponding to the only black vertex does not come with a factor $1/p!$ .",
"(the matrices are labeled growingly from right to left, and the corresponding half-edges are labeled growingly when going counter-clockwise around the black vertex).",
"The free index corresponding to the matrix $X$ (resp $X^\\ast $ ) is associated to the left (resp.",
"right) side of the half-edge $ XX^*$ .",
"Each Wick pairing induces a permutation $\\sigma _\\circ $ which defines the white vertices.",
"The maps corresponding to the examples of Figure REF are shown in Figure REF .",
"Figure: Combinatorial maps for the example of Figure .Notation 2.10 Throughout the text, we will denote $\\mathbb {M}_{p}$ the set of labeled connected bicolored maps with a single black vertex and $p$ edges, and $\\mathbb {M}^g_{p}$ its subset of maps having genus $g$ .",
"The two elements of $\\mathbb {M}_{2} = \\mathbb {M}^0_{2}$ are shown in Fig.",
"REF , and an example of map in $\\mathbb {M}^0_{7}$ is shown on the left of Fig.",
"REF .",
"Denoting $F(\\mathcal {M})$ the number of faces of a map $\\mathcal {M}$ , and $V(\\mathcal {M})$ its total number of vertices, we can therefore restate the key relation (REF ) as follows: $\\mathbb {E} \\operatorname{Tr}\\bigl (W^{p}\\bigr ) = \\sum _{\\mathcal {M}\\in \\mathbb {M}_{p}} N^{F(\\mathcal {M})} M^ {V(\\mathcal {M}) - 1}.$ The genus (REF ) of a connected map in $\\mathbb {M}_{p}$ writes $2-2g(\\mathcal {M})= V(\\mathcal {M}) + F(\\mathcal {M}) -p ,$ so that, if we assume $M\\sim cN$ , $\\mathbb {E} \\operatorname{Tr}\\bigl (W^{p}\\bigr ) = (1+o(1))\\sum _{\\mathcal {M}\\in \\mathbb {M}_{p}} N^{p + 1 -2g(\\mathcal {M})} c^ {V(\\mathcal {M}) - 1}.$ Since $g\\ge 0$ , the bound (REF ) on the number of cycles $\\#\\alpha + \\#(\\gamma \\alpha )$ is now obtained using the positivity of the genus instead of the distances between permutations, and the leading terms in $N$ correspond to planar maps.",
"Therefore, with the Notation REF , $\\lim _{N\\rightarrow + \\infty }\\frac{1}{N^{p+1}} \\mathbb {E} \\operatorname{Tr}\\bigl (W^{p}\\bigr ) = \\sum _{\\mathcal {M}\\in \\mathbb {M}_{p}^0}c^ {V(\\mathcal {M}) - 1}.$ It is easily seen by studying the permutations $\\sigma _\\bullet = \\gamma $ and $\\sigma _\\circ = \\alpha $ , that elements of $\\mathbb {M}_{p}^0$ with $V(\\mathcal {M})$ vertices correspond to non-crossing partitions with $V(\\mathcal {M})-1$ disjoint cycles.",
"Another simple way of seeing this is to notice that maps in $\\mathbb {M}_{p}^0$ with $V(\\mathcal {M})$ vertices and $F(\\mathcal {M})$ faces are in bijection with bicolored labeled plane trees with $p$ edges, $V(\\mathcal {M})-1$ white vertices, and $F(\\mathcal {M})$ black vertices (plane trees are all bicolored), as explained below.",
"Such trees are themselves in bijection with non-crossing partitions with $V(\\mathcal {M})-1$ disjoint cycles throughout a dual mapping.",
"Starting from a map in $\\mathbb {M}_{p}^0$ , a vertex is added in each face and an edge is added between each corner of each white vertex and the newly added vertices (the ordering of edges around vertices is given by the clockwise ordering of appearance of corners around faces)Note that by linking the newly added vertices to the corners around the black vertex instead of the white vertices, we have a duality between elements of $\\mathbb {M}_{p}^0$ , see Def.",
"REF ..",
"This procedure is illustrated in Fig.",
"REF .",
"It is a particular case of Tutte's bijection for bicolored maps [43] and is known to be bijective.",
"Figure: Bijection between elements of 𝕄 p 0 \\mathbb {M}_p^0 and labeled plane trees.The map obtained from $M\\in \\mathbb {M}_{p}^0$ is connected, has $V(M) - 1 +F(M) = p+1$ vertices, and its number of edges is the number of corners of $M$ around white vertices, which is $p$ , and is therefore a tree.",
"In the case where $c=1$ , $\\lim _{N\\rightarrow + \\infty }\\frac{1}{N^{p+1}} \\mathbb {E} \\operatorname{Tr}\\bigl (W^{p}\\bigr )$ is just the cardinal of $\\mathbb {M}_{p}^0$ , namely the number of plane trees with $p$ edges, which is well-known [24] to be the Catalan number $C_{p}= \\frac{1}{p+1}\\binom{2p}{p} $ , $\\lim _{N\\rightarrow + \\infty }\\frac{1}{N^{p+1}} \\mathbb {E} \\operatorname{Tr}\\bigl (W_{c=1}^{p}\\bigr )= C_{p}.$ At finite $N$ and for $c=1$ , the contribution of order $N^{p+1-2g}$ is given by the number of maps in $\\mathbb {M}_{p}$ whose genus is $g$ .",
"The very same transformation which mapped maps in $\\mathbb {M}_{p}^0$ to plane trees can be applied.",
"It maps elements of $\\mathbb {M}_{p}^g$ to bicolored maps of genus $g$ with a single face.",
"Maps with a single face are called unicellular maps [12].",
"Conversely, the inverse transformation maps bicolored unicellular maps of genus $g$ to genus $g$ bicolored maps with a single black vertex.",
"Unicellular maps of genus $g$ are the object of many studies, and they can be counted exactly using various recursive formulas (see for instance the Harer-Zagier formula [48], Lehman-Walsh formula [47], Goupil-Schaeffer formula [21], and Chapuy's formula [13], [11], [9]).",
"Using such results, (REF ) can be expressed exactly for any $p$ ."
],
[
"Free cumulants and maps",
"We elaborate here on the relation between moments and free cumulants in the case of a Wishart distribution.",
"As was pointed out in the proof of Proposition REF (third step), the free cumulants of the limiting distribution are $\\kappa _n=c$ for all $n\\ge 1$ .",
"We consider again the expectation value of the traces of powers of the matrix $W$ $\\mathbb {E}\\operatorname{Tr}(W^p)&=\\sum _{i_1,\\ldots ,i_p=1}^N\\mathbb {E}(W_{i_1i_p}\\ldots W_{i_3i_2}W_{i_2i_1}),\\\\&=\\sum _{i_1,\\ldots ,i_p=1}^N\\sum _{\\mathcal {J}\\vdash \\lbrace 1,\\ldots ,p\\rbrace }\\prod _{J_r \\in \\mathcal {J}}\\mathcal {K}(W_{J_r}), $ where the second sum is taken over all partitions $\\mathcal {J}$ of $\\lbrace 1,\\ldots ,p\\rbrace $ , and we define $\\mathcal {K}(W_{J_r})=\\mathcal {K}(\\lbrace W_{i_{k+1}i_k}\\rbrace _{\\begin{array}{c}k\\in J_r \\\\ k\\in \\mathbb {Z}_p\\end{array}})$ as the classical cumulants of the family of random variables $\\lbrace W_{i_{k+1}i_k}\\rbrace _{\\begin{array}{c}k\\in J_r \\\\ k\\in \\mathbb {Z}_p\\end{array}}$ .",
"The claim () can be obtained from the definition of the generating function of classical cumulants.",
"Depending on conventions, this cumulant generating function can be defined either as the logarithm of the characteristic function or of the moment generating function of the corresponding probability density.",
"For reference on classical cumulants, see for instance [35], [34] or [28].",
"As moments, these classical cumulants can be represented graphically.",
"In the mathematical-physics literature, the graphical objects representing them are often described as connected combinatorial maps (or ribbon graphs) with half-edges or connected opened combinatorial maps, see Remark REF for more details.",
"In our case, as is shown in the next paragraphs, they will be represented as usual combinatorial maps with one white vertexThe fact that there is only one white vertex is the direct translation of the “connected” conditions introduced in the mathematical-physics literature.",
"and one black vertex.",
"The fact that combinatorial maps with one white vertex and one black vertex represent classical cumulants is the translation of the decomposition of classical cumulants in terms of classical moments for some random variables $X_1,\\ldots , X_s$ , which is the cumulant-moment formula in classical probability [35], making use of the Möbius function on the lattice of partitions: $\\mathcal {K}(X_{\\vert J_r \\vert },\\ldots ,X_1 )= \\sum _{\\mathcal {I}\\vdash \\lbrace 1,\\dots , {\\vert J_r \\vert }\\rbrace }(-1)^{|\\mathcal {I}|-1} (|\\mathcal {I}|-1)!",
"\\prod _{I_m\\in \\mathcal {I}}\\mathbb {E}\\left(\\prod _{k\\in I_m}X_k\\right),$ for the random variables $X_k = W_{i_{k+1}i_k}$ .",
"The moments $\\mathbb {E}\\left(\\prod _{k\\in I_m}X_k\\right)$ are then computed using Wick's theorem (Prop.",
"REF ).",
"As detailed previously, these Wick pairings can be represented graphically, either using $X-$ and $X^*-$ boxes, or using the combinatorial map representation.",
"In this last representation, identifying $W_{i_{k+1}i_k}$ with the integer $k\\in I_m$ , the Wick pairings induce a permutation $\\alpha _m$ on the set $I_m$ (see the proof of Prop.",
"REF ).",
"Each cycle of $\\alpha _m$ contributes with $\\lfloor N c \\rfloor $ .",
"As before, we can define a bicolored map in $\\mathbb {M}_{\\vert I_m \\vert }$ by setting $\\sigma _\\circ = \\alpha _m$ , and $\\sigma _\\bullet = \\gamma _m$ , where $\\gamma _m$ is the cycle $(x_1, \\ldots , x_{\\vert I_m \\vert })$ , in which $x_1<x_2< \\ldots <x_{\\vert I_m \\vert }$ are all the elements of $I_m$ .",
"The difference with what has been done before, where we were considering the expectation of a trace, is that here, the indices $\\lbrace i_k, i_{k+1}\\rbrace $ are not summed.",
"This means that the faces of the map $(I_m, \\alpha _m, \\gamma _m)$ (the disjoint cycles of $\\gamma _m\\alpha _m$ ) do not factorize in the expression of $\\mathbb {E}\\left(\\prod _{k\\in I_m}X_k\\right)$ , and instead we have a Kronecker delta for each corner of each white vertex: $\\mathbb {E}\\left(\\prod _{k\\in I_m}W_{i_{k+1}i_k}\\right)= \\sum _{\\alpha _m \\in \\mathcal {S}_{\\vert I_m \\vert }}\\lfloor Nc \\rfloor ^{(\\#\\alpha _m)} \\prod _{k\\in I_m} \\delta _{i_k,i_{\\alpha _m(k) + 1}}.$ Furthermore, because of the factors $(-1)^{|\\mathcal {I}|-1} (|\\mathcal {I}|-1)!$ in the expression of the cumulants (REF ), and because the products of Kronecker deltas factorize, the terms corresponding to maps which have more than one white vertex will be canceled by products of terms corresponding to maps with fewer white vertices.",
"The only non-vanishing terms are those for which $\\mathcal {I}$ is reduced to $\\lbrace 1,\\ldots , \\vert J_r\\vert \\rbrace $ , and the cumulants can therefore be expressed in terms of cycles of $J_r$ of length $\\vert J_r\\vert $ , or equivalently using a graphical expansion over bicolored maps with two vertices, $\\mathcal {K}(W_{J_r})= \\lfloor Nc \\rfloor \\sum _{ \\begin{array}{c}{\\mathcal {M}\\in \\mathbb {M}_{\\vert J_r \\vert }}\\\\{V(\\mathcal {M}) = 2}\\end{array} }\\ \\prod _{k\\in J_r} \\delta _{i_k,i_{\\alpha _\\mathcal {M}(k) + 1}},$ where we have denoted $\\alpha _\\mathcal {M}$ the cycle $\\sigma _\\circ $ for the map $\\mathcal {M}$ .",
"We do not prove these statements here, however we give what we hope is an illuminating example in the present case.",
"Consider the cumulant of three matrix elements ${W_{i_6i_5}, W_{i_4i_3}, W_{i_2i_1}}$ , $ \\mathcal {K}(W_{i_6i_5}, W_{i_4i_3}, W_{i_2i_1})=\\mathbb {E}(W_{i_6i_5}W_{i_4i_3}W_{i_2i_1})-\\mathbb {E}(W_{i_6i_5}W_{i_4i_3})\\mathbb {E}(W_{i_2i_1}) \\\\-\\mathbb {E}(W_{i_6i_5})\\mathbb {E}(W_{i_4i_3}W_{i_2i_1}) - \\mathbb {E}(W_{i_6i_5}W_{i_2i_1})\\mathbb {E}(W_{i_4i_3})+2\\mathbb {E}(W_{i_6i_5})\\mathbb {E}(W_{i_4i_3})\\mathbb {E}(W_{i_2i_1}).$ Figure: Graphical expansion of the cumulant 𝒦(W i 6 i 5 ,W i 4 i 3 ,W i 2 i 1 )\\mathcal {K}(W_{i_6i_5}, W_{i_4i_3}, W_{i_2i_1}) from its moments decomposition and the Wick theorem.",
"Identical terms are shown in the same color.The graphical representation of the Wick pairings of equation (REF ) is shown in Fig.",
"REF .",
"The dashed edges indicate which $X,X^*$ are paired together.",
"In order to simplify (and compactify) the drawings, the vertices of empty types represent the $X$ -boxes, while the filled vertices represent the $X^*$ -boxes.",
"In this representation, it is the disconnected parts of the graphical expansion of $\\mathbb {E}(W_{i_6i_5}W_{i_4i_3}W_{i_2i_1})$ which are suppressed by the remaining terms of the right hand side of (REF ), that are all disconnected in nature.",
"The weights associated to the only remaining terms are $\\lfloor Nc \\rfloor \\delta _{i_1i_6}\\delta _{i_2i_3}\\delta _{i_4i_5}$ and $\\lfloor Nc \\rfloor \\delta _{i_1i_4}\\delta _{i_2i_5}\\delta _{i_3i_6}$ .",
"In Fig.",
"REF , we represent the same expansion using the representation in terms of combinatorial maps.",
"Figure: This is the translation of the previous Figure in the language of combinatorial maps.",
"Every corner on a white vertex carries a δ i k ,i k ' \\delta _{i_k, i_{k^{\\prime }}}, where i k i_k and i k ' i_{k^{\\prime }} are the incident indices on the drawing.",
"The only graphs which do not cancel are the ones corresponding to ⌊Nc⌋δ i 1 i 6 δ i 2 i 3 δ i 4 i 5 \\lfloor Nc \\rfloor \\delta _{i_1i_6}\\delta _{i_2i_3}\\delta _{i_4i_5} and ⌊Nc⌋δ i 1 i 4 δ i 2 i 5 δ i 3 i 6 \\lfloor Nc \\rfloor \\delta _{i_1i_4}\\delta _{i_2i_5}\\delta _{i_3i_6} (shown in black).Using the expression (REF ) in the classical moment-cumulant formula (REF ) above, we have $\\mathbb {E}\\operatorname{Tr}(W^p)&=\\sum _{i_1,\\ldots ,i_p=1}^N\\sum _{\\mathcal {J}\\vdash \\lbrace 1,\\ldots ,p\\rbrace }\\prod _{J_r\\in \\mathcal {J}} \\lfloor Nc \\rfloor \\sum _{ \\begin{array}{c}{\\mathcal {M}\\in \\mathbb {M}_{\\vert J_r \\vert }}\\\\{V(\\mathcal {M}) = 2}\\end{array} }\\ \\prod _{k\\in J_r} \\delta _{i_k,i_{\\alpha _\\mathcal {M}(k) + 1}}\\\\&= \\sum _{\\mathcal {J}\\vdash \\lbrace 1,\\ldots ,p\\rbrace }\\lfloor N c\\rfloor ^{|\\mathcal {J}|}\\sum _{i_1,\\ldots ,i_p=1}^N \\prod _{J_r\\in \\mathcal {J}} \\sum _{ \\begin{array}{c}{\\mathcal {M}\\in \\mathbb {M}_{\\vert J_r \\vert }}\\\\{V(\\mathcal {M}) = 2}\\end{array} }\\ \\prod _{k\\in J_r} \\delta _{i_k,i_{\\alpha _\\mathcal {M}(k) + 1}}.$ We can rewrite the terms on the second line in terms of bicolored maps in $\\mathbb {M}_p$ .",
"The partition $\\mathcal {J}$ divides the edges incident to its black vertex in groups $J_r$ which are all linked to the same white vertex, one for each group.",
"For each one of these groups, a sum is taken over all possible maps with two vertices, of the product of Kronecker deltas over the corners around the white vertex.",
"Any choice of $|\\mathcal {J}|$ two-vertices maps provides a different map in $\\mathbb {M}_p$ .",
"Denoting $\\mathbb {M}_p[\\mathcal {J}]$ the subset of $\\mathbb {M}_p$ of maps for which for every $J_r\\in \\mathcal {J}$ , $(\\alpha _\\mathcal {M})_{|J_r}$ is a unique cycle of length $|J_r|$ , we can therefore rewrite this sum as $\\mathbb {E}\\operatorname{Tr}(W^p) = \\sum _{\\mathcal {J}\\vdash \\lbrace 1,\\ldots ,p\\rbrace }\\lfloor N c\\rfloor ^{|\\mathcal {J}|}\\sum _{i_1,\\ldots ,i_p=1}^N \\, \\sum _{ \\mathcal {M}\\in \\mathbb {M}_{p}[\\mathcal {J}]} \\, \\prod _{k\\in \\lbrace 1,\\ldots , p\\rbrace } \\delta _{i_k,i_{\\alpha _\\mathcal {M}(k) + 1}}.$ Now, we have precisely one index per corner around the black vertex, and we still have one Kronecker delta per corner around a white vertex, so the sum over $i_1,\\ldots , i_p$ yields a factor $N^{F(\\mathcal {M})}$ , so that $\\mathbb {E}\\operatorname{Tr}(W^p) = \\sum _{\\mathcal {J}\\vdash \\lbrace 1,\\ldots ,p\\rbrace }\\lfloor N c\\rfloor ^{|\\mathcal {J}|} \\sum _{ \\mathcal {M}\\in \\mathbb {M}_{p}[\\mathcal {J}]} N^{F(\\mathcal {M})} = \\sum _{ \\mathcal {M}\\in \\mathbb {M}_p} \\lfloor N c\\rfloor ^{V(\\mathcal {M})-1} N^{F(\\mathcal {M})},$ and we indeed recover (REF ), and therefore the results of Lemma REF which state that the leading terms correspond to the planar maps.",
"However, we can now pull back the planarity result to the level of the cumulants.",
"From the calculation above, we see that the cumulants are the two-vertices maps which are then re-arranged (their black vertices are crushed into a single one, and the edges are naturally ordered around this single vertex as they are labeled from 1 to $p$ ).",
"In order for the resulting map to contribute to the large $N$ limit, it has to be planar, thus the collection of two-vertices maps we start from all have to be planar.",
"Indeed it is a classical result that the genus of a map is greater or equal to that of a submap (see e.g. [31]).",
"We reprove it in our case in Prop.",
"REF .",
"If all the two-vertices maps are planar, in addition, the partition $\\mathcal {J}$ has to be non-crossing for the map to be planar.",
"Proposition 2.11 Consider a map $\\mathcal {M}\\in \\mathbb {M}_p$ , and the submap $\\mathcal {M}^{\\prime }$ obtained by keeping the black vertex, a single white vertex, and all the edges between them.",
"If $\\mathcal {M}^{\\prime }$ is non-planar, then $\\mathcal {M}$ is also non-planar.",
"Suppose that $g(\\mathcal {M}^{\\prime })>0$ , and consider the set of edges of the combinatorial map that do not belong to $\\mathcal {M}^{\\prime }$ .",
"Among these edges, we first remove all the ones adjacent to univalent white vertices, and we remove the isolated white vertices created in this process.",
"The obtained map has the same genus as the initial map.",
"For each of the remaining edge we proceed as follows.",
"If the edge is adjacent to only one face and to a white vertex which is not univalent, then the map has a non zero genus and the proposition is true.",
"If the edge is adjacent to two different faces, withdrawing it does not change the genus.",
"We do so, and then repeat this process.",
"At each step, we may create univalent white vertices, which we remove, together with the isolated white vertices thus created.",
"Either we find during the process an edge which is not a bridge and is adjacent to only one face (and the proposition is proved), or we have erased all the edges that did not belong to $\\mathcal {M}^{\\prime }$ , without changing the genus of the map.",
"We are thus left with a map with non vanishing genus $g(\\mathcal {M})=g(\\mathcal {M}^{\\prime })>0$ .",
"We can now claim that the free cumulant of order $p$ is represented by the unique planar map with one white vertex, one black vertex and $p$ edges.",
"The interest of this remark lies in the fact that it gives a heuristic for reading freeness directly out of the combinatorial maps expression of moments (intuition which will be used in Sections REF and REF ).",
"Indeed looking at moments of alternating products of different random matrices, one expects to express these moments using colored combinatorial mapsSee later sections.",
"(where the colors label different types of edges indexing Wick pairing between different matrices).",
"If one believesThis can actually be shown, however this lies out of the main scope of this paper.",
"that white vertices with adjacent edges of different colors correspond to mixed cumulants, it is easy to guess if in the large $N$ limit the different matrices converge to different free elements, just by confronting the weights of the white vertices with only one adjacent color to those of white vertices with strictly more than one adjacent color.",
"Remark 2.12 We notice here that the following expectation values of products of matrix elements such that $i_k\\ne i_q, \\ \\forall k\\ne q$ (indices are fixed and not summed) $\\mathbb {E}(W_{i_1i_p}\\ldots W_{i_3i_2}W_{i_2i_1})=M=\\lfloor Nc \\rfloor ,$ thus are equal up to a rescaling in $N$ to the free cumulants $\\kappa _n=c$ .",
"A similar fact has been used in the study of Hermitian (formal) matrix integrals and simple combinatorial maps in [5].",
"In this reference a combinatorial maps interpretation of the relation between moments and classical cumulants is given by bijective means valid at all orders in the large $N$ asymptotic expansion.",
"In particular the bijective method is valid at the first order of the expansion and thus can be interpreted in the context of free probability and the corresponding free cumulants.",
"Remark 2.13 The cumulants can also be expressed in terms of the white vertices with incident-pending half-edges, whose both sides are labeled by pairs $(i_k,i_{k+1})$ , while the half-edges themselves are labeled by the first index of the pair $k\\in I_r$ .",
"In this representation, the moments are obtained by gluing the cumulants around the black vertex while respecting the natural order of natural integers, and the free cumulants are the connected terms, for which the indices of the half-edges satisfy a non-crossing condition."
],
[
"The four-partite case: joint distribution of two marginals",
"In this section, we describe the joint distribution of two marginals, $\\rho _{AB}$ and $\\rho _{AC}$ , of a random pure quantum state $\\rho _{ABCD} = |\\psi \\rangle \\langle \\psi |_{ABCD}$ .",
"We prove all results both using metric properties of the permutation groups and the combinatorial maps approach with the underlying discrete geometry, in parallel.",
"As before, we state our results for Wishart tensors; the corresponding results for random density matrices can be obtained by renormalizing the Wishart matrices (see Remark REF , as well as the very last paragraph of this section).",
"Let $X = X_{ABCD} \\in \\mathcal {H}_A \\otimes \\mathcal {H}_B \\otimes \\mathcal {H}_C \\otimes \\mathcal {H}_D$ be a random Gaussian tensor, i.e.",
"a random tensor with i.i.d.",
"standard complex Gaussian entries.",
"We are going to study here the joint distribution of the two marginals $W_{AB} := [\\operatorname{id} \\otimes \\operatorname{id} \\otimes \\operatorname{Tr} \\otimes \\operatorname{Tr}](XX^*) \\qquad \\text{ and } \\qquad W_{AC}:=[\\operatorname{id} \\otimes \\operatorname{Tr} \\otimes \\operatorname{id} \\otimes \\operatorname{Tr}](XX^*),$ in some asymptotic regime, where the dimension of some of the Hilbert spaces grow to infinity.",
"In order to study mixed moments of the two random matrices $W_{AB}$ and $W_{AC}$ , we are going to assume from now on that $\\dim \\mathcal {H}_B = \\dim \\mathcal {H}_C.$ We consider the following two asymptotic regimes: the balanced regime: $\\dim \\mathcal {H}_B = \\dim \\mathcal {H}_C = N \\rightarrow \\infty $ , $\\dim \\mathcal {H}_D / \\dim \\mathcal {H}_A \\rightarrow c$ , for some constant $c \\in (0,\\infty )$ ; the unbalanced regime: $\\dim \\mathcal {H}_B = \\dim \\mathcal {H}_C = m$ , $\\dim H_A \\sim N$ , $\\dim H_D \\sim cN$ , $N \\rightarrow \\infty $ , for some constants $m \\ge 1$ and $c \\in (0, \\infty )$ .",
"As special cases of the balanced asymptotic regime, we shall emphasize the following two sub-cases (see Remark REF ): $\\dim \\mathcal {H}_B = \\dim \\mathcal {H}_C = N$ , $\\dim H_A \\sim c_1N$ , $\\dim H_D \\sim c_4N$ , $N \\rightarrow \\infty $ , for some constants $c_{1,4} \\in (0, \\infty )$ ; $\\dim \\mathcal {H}_B = \\dim \\mathcal {H}_C = N$ , $\\dim H_A = k$ , $\\dim H_D = l$ , $N \\rightarrow \\infty $ , for some integer constants $k,l \\ge 1$ .",
"The balanced regime will be studied in Section REF , while the unbalanced regime will be studied in Section REF .",
"We first compute the exact combinatorial expression for the moments."
],
[
"Exact expression for the moments",
"In this section, we prove a non-asymptotical result, describing the mixed moments of the marginals $W_{AB}$ , $W_{AC}$ as a combinatorial sum.",
"This result is the starting point for the asymptotic computations in Sections REF and REF .",
"In order to describe the mixed moments of the two marginals, we write $W_f := \\prod _{1 \\le a \\le p}^{\\longleftarrow } W_{f(a)},$ for a word $f \\in \\lbrace AB,AC\\rbrace ^p$ of length $p$ in the letters $(AB), (AC)$ .",
"Here again, the matrices are labeled from right to left.",
"We also denote $N_X:=\\dim \\mathcal {H}_X$ , for $X=A,B,C,D$ ; note that we have $N_B=N_C =:N_{B,C}$ .",
"Theorem 3.1 The mixed moments of the two marginals $W_{AB},W_{AC}$ defined in (REF ) are given by the following exact formula: $\\mathbb {E} \\operatorname{Tr} W_f = \\sum _{\\alpha \\in \\mathcal {S}_p} N_A^{\\#(\\gamma \\alpha )} N_D^{\\#\\alpha } N_{B,C}^{L(f,\\alpha )},$ where $\\gamma = (1 2 3 \\ldots p )$ is the full cycle permutation and $L(f,\\alpha )$ is a combinatorial function described in equations (REF )-().",
"The proof is a standard application of the graphical Wick formalism from [15], similar to the proof of the moment formula in Proposition REF .",
"First, consider the diagram associated to the trace of the operator $W_f$ (see Figure REF for a simple example): each of the $p$ $X$ (or $\\bar{X}$ ) boxes has 4 decorations, corresponding to the 4 Hilbert spaces $\\mathcal {H}_{A,B,C,D}$ .",
"The sum over permutations $\\alpha \\in \\mathcal {S}_p$ comes from the application of the Wick formula, and the different factors in the general term count the different types of loops, as follows: There are $\\#(\\gamma \\alpha )$ loops associated to the Hilbert space $\\mathbb {C}^{N_A}$ , since the initial wiring of the boxes is given by the cyclic permutation $\\gamma $ , corresponding to matrix multiplication.",
"The total number of loops is the number of cycles of the permutation obtained as the product of the permutation describing the initial wiring (here, $\\gamma $ , connecting $X$ boxes to $\\bar{X}$ boxes) and the permutation coming from the graphical Wick formula ($\\alpha $ , connecting $\\bar{X}$ boxes to $X$ boxes).",
"There are $\\#\\alpha $ loops associated to the Hilbert space $\\mathbb {C}^{N_D}$ , since the initial wiring of the boxes is given by the identity, corresponding to the partial trace over the fourth tensor factor (which appears in all the $W_{f(i)}$ ).",
"We denote by $L(f,\\alpha )$ the number of loops corresponding to the second ($B$ ) and the third ($C$ ) tensor factors.",
"This quantity is less obvious to evaluate, since it depends in a non-trivial way on both the specific word $f$ and on the permutation $\\alpha $ .",
"These loops contribute each a factor of $N_{B,C}$ .",
"Figure: The diagram of the random variable Tr(W AB W AC )\\operatorname{Tr}(W_{AB}W_{AC}).",
"Each tensor box has 4 decorations corresponding to ℋ A \\mathcal {H}_A (round shape), ℋ B ,ℋ C \\mathcal {H}_B, \\mathcal {H}_C (square shapes), and ℋ D \\mathcal {H}_D (diamond shape), displayed in this order from top to bottom.We discuss next the exponent $L(f,\\alpha )$ , counting the number of loops associated to systems $B$ and $C$ .",
"As in the other cases, we have $L(f,\\alpha ) = \\#(\\hat{\\gamma }_f\\, \\hat{\\alpha }),$ where $\\hat{\\gamma }_f, \\hat{\\alpha }\\in \\mathcal {S}_{2p}$ encode, respectively, the initial and the Wick wirings of the $B,C$ decorations.",
"In order to define properly these permutations, let us relabel the $2p$ decorations of the $X$ boxes by $\\bigl \\lbrace (1,B), (2,B), \\ldots , (p, B), (1,C), (2,C), \\ldots , (p, C)\\bigr \\rbrace $ to keep track of the system they refer to (as usual, the numbering is increasing from right to left).",
"Then, using cyclical notation (i.e.",
"$p+1 \\equiv 1$ ), we have $ \\hat{\\gamma }_f(a,B)& = {\\left\\lbrace \\begin{array}{ll}(a,B) & \\qquad \\text{ if } f(a) = AC \\\\(a+1,B) & \\qquad \\text{ if } f(a) = AB \\text{ and } f(a+1) = AB \\\\(a+1, C) & \\qquad \\text{ if } f(a) = AB \\text{ and } f(a+1) = AC\\end{array}\\right.}",
"\\\\ \\hat{\\gamma }_f(a,C)& = {\\left\\lbrace \\begin{array}{ll}(a,C) & \\qquad \\text{ if } f(a) = AB \\\\(a+1, B) & \\qquad \\text{ if } f(a) = AC \\text{ and } f(a+1) = AB \\\\(a+1, C) & \\qquad \\text{ if } f(a) = AC \\text{ and } f(a+1) = AC\\end{array}\\right.}",
"\\\\ \\hat{\\alpha }(i, Z)& = (\\alpha (i), Z), \\quad \\text{ for } Z=B,C.$ As a direct application of (REF ), we have, for the example in Fig.",
"REF , $\\mathbb {E} \\operatorname{Tr}(W_{AB}W_{AC}) = N_A N_{B,C}^3 N_D^2 + N_A^2N_{B,C}N_D,$ where the two terms in the right hand side correspond respectively to the identity permutation and to the transposition $(12)$ .",
"We shall now present a different point of view on Theorem REF , using combinatorial maps.",
"First let us introduces some important (and somehow non-standard) notation for integer intervals: for integers $a,b$ , we denote $\\llbracket a, b\\rrbracket := \\lbrace a, a+1, \\ldots , b-1, b\\rbrace .$ Orbits.",
"Before going to the combinatorial maps version of this result, we provide a more formal definition of the objects we count, in terms of the permutation.",
"If $I\\subset \\lbrace A,B,C,D\\rbrace $ , we denote $[p]^I := \\bigsqcup _{i\\in I}\\bigl \\lbrace (a,i)\\bigr \\rbrace _{a\\in \\llbracket 1,p\\rrbracket },$ so that for instance, $[p]^{B,C} = \\bigl \\lbrace (a,B) \\bigr \\rbrace _{a\\in \\llbracket 1,p\\rrbracket } \\sqcup \\bigl \\lbrace (a,C) \\bigr \\rbrace _{a\\in \\llbracket 1,p\\rrbracket }$ , where $a$ runs through the set of edgesThe letter $a$ stands for arêtes, which means edges in French..",
"In some sense, $[p]^I$ is the set of all potential colored edges with color set $I$ that can be obtained from the $p$ edges of a map.",
"Definition 3.2 (Orbits) We call orbits for $f$ and $\\alpha $ , the cycles respectively induced by $\\gamma \\alpha $ , $\\hat{\\gamma }_f\\, \\hat{\\alpha }$ , and $\\alpha $ on the sets $[p]^A$ , $[p]^{B,C}$ , and $[p]^D$ .Where we canonically identified $[p]^A$ and $[p]^D$ with $\\llbracket 1, p\\rrbracket $ .",
"Of course, these are in bijection with the loops in the box representation.",
"Note that the orbits involving the colors $A$ and $D$ are already understood from Section (corresponding to the bipartite quantum system case), and only the orbits involving the colors $B$ and $C$ present a new behavior.",
"Combinatorial maps.",
"As just mentioned, as far as the colors $A$ and $D$ are concerned, the behavior is unchanged from Section .",
"So, naturally, if we define the map $\\mathcal {M}= (\\gamma , \\alpha )\\in \\mathbb {M}_p$ , the orbits corresponding to the color $A$ are simply the faces of $\\mathcal {M}$ , while there is one orbit of color $D$ per white vertex of the map.",
"Colors $A$ and $D$ therefore contribute with a factor $N_A^{F(\\mathcal {M})}N_D^{V(\\mathcal {M}) - 1}$ .",
"We now focus on the orbits involving colors $B$ and $C$ .",
"We can provide each edge $a\\in \\llbracket 1,p\\rrbracket $ of the map with a color $j(a)\\in \\lbrace B,C\\rbrace $ : the non traced color in $\\lbrace B,C\\rbrace $ for the matrix number $a$ of the word $f$ (i.e.",
"the color different from $A$ in $f(a)$ ).",
"For instance, the moment $\\mathbb {E}(\\operatorname{Tr}W_{AB}W_{AC})$ of the trace of Fig.",
"REF involves the two maps in $\\mathbb {M}_2$ (Fig.",
"REF ), but in addition, one edge carries the color $B$ , the other the color $C$ .",
"We consider the set of the colored edges, $[p]^{B,C}_f = \\bigl \\lbrace \\bigl (a,j(a)\\bigl ) \\bigr \\rbrace _{a\\in \\llbracket 1,p\\rrbracket }.$ It is the subset of $[p]^{B,C}$ corresponding to the colors which are not traced.",
"Some orbits contain at least one element of $[p]^{B,C}_f$ , or equivalently, follow at least one edge of the map $\\mathcal {M}$ .",
"In the map language, when such an orbit reaches a corner on the black vertex after having followed a colored edge $\\bigl (a, j(a)\\bigr )$ , $a\\in \\llbracket 1,p\\rrbracket $ and $j(a)\\in \\lbrace B, C\\rbrace $ , it will go to the colored edge $\\bigl (a+1, j(a+1)\\bigr )$ .",
"Apart from the information on the color, the orbit locally behaves, around the black vertex, as the face of the map $\\mathcal {M}$ , which goes from $a$ to $a+1$ .",
"However, the orbits and the faces of the map do not behave alike around white vertices.",
"Indeed, arriving from a colored edge $(a, j(a))$ at a white vertex $v_\\circ $ , the orbit goes to the first edge around $v_\\circ $ counterclockwise that also carries the color $j(a)$ , namely $\\alpha ^q(a),\\quad \\text{where} \\quad q=\\min \\bigl \\lbrace s\\in \\mathbb {N}^\\ast \\mid j(a) =j( \\alpha ^s(a))\\bigr \\rbrace .$ If the integer $q$ is greater than one, the behavior differs from that of the face, which goes to $\\alpha (a)$ .",
"A simple way of solving this issue is to locally split the white vertex into two vertices, one for each color $B$ and $C$ , as illustrated in Fig.",
"REF .",
"Figure: Local duplication of white vertices to obtain the map ℳ f \\mathcal {M}_f.Note that this operation has no effect if all the edges incident to the white vertex have the same color.",
"Otherwise, duplicating as in Fig.",
"REF a white vertex $v_\\circ $ with both incident colors $B$ and $C$ , we obtain a new map in $\\mathbb {M}_p$ with one additional white vertex, and the number of orbits remains unchanged.",
"However, now, around the two vertices of color $B$ and $C$ resulting from splitting $v_\\circ $ , the orbits and the faces behave alike.",
"Performing this operation on all the white vertices, we obtain a map $\\mathcal {M}_f \\in \\mathbb {M}_p$ , with the same black vertex $\\gamma $ as $\\mathcal {M}$ , but with two permutations $\\alpha _B$ and $\\alpha _C$ , which are simply the restrictions of $\\alpha $ to the two sets of edges that respectively carry the colors $B$ and $C$ .",
"This defines a tricolored map $\\mathcal {M}_{f} = (\\sigma _\\bullet , \\alpha _B, \\alpha _C)$ , where $\\sigma _\\bullet =\\gamma $ , and such that edges only link white vertices and edges of color $B$ or $C$ (i.e.",
"it is also a bicolored map $(\\sigma _\\bullet , \\alpha _B \\alpha _C)$ in the sense of Def.",
"REF ).",
"Because if $j(a)$ is e.g.",
"$B$ , $\\alpha _B(a)= \\alpha ^q(a)$ defined in (REF ), the faces of $\\mathcal {M}_f$ are precisely the orbits, and therefore the colors $B$ and $C$ contribute with a factor $N_{B,C}^{F(\\mathcal {M}_f)}$ .",
"And lastly, some orbits do not follow any edge of the map.",
"They correspond to the restriction of $\\hat{\\gamma }_f \\hat{\\alpha }$ to the set $[p]^{B,C} \\setminus [p]^{B,C}_f,$ and in the map language, to white vertices of $\\mathcal {M}$ whose incident edges all share the same color.",
"Such white vertices are those left invariant by the operation of Fig.",
"REF , and therefore their number is just $2V(\\mathcal {M})-V(\\mathcal {M}_f)-1.$ Indeed, $2(V(\\mathcal {M})-1)-(V(\\mathcal {M}_f)-1)$ counts one for each white vertex with only adjacent edges of a single color, while it counts zero for each white vertex with both adjacent colors.",
"We have therefore the following formulation of Thm.",
"REF .",
"Theorem REF in the maps formulation For $f$ a word of length $p$ in the alphabet $\\lbrace AB,AC\\rbrace $ , the mixed moments of the two marginals $W_{AB}, W_{AC}$ are expressed exactly using a sum over combinatorial maps $\\mathbb {E}\\operatorname{Tr}(W_f)= \\sum _{\\mathcal {M}\\in \\mathbb {M}_p} N_A^{F(\\mathcal {M})}N_D^{V(\\mathcal {M})-1}N_{B,C}^{L(f,\\, \\mathcal {M})},$ where $L(f, \\mathcal {M})$ coincides with $L(f, \\alpha )$ and can be expressed as $L(f,\\mathcal {M}) = F(\\mathcal {M}_f)+ 2V(\\mathcal {M})-V(\\mathcal {M}_f)-1.$ A bound on the exponent $L$ .",
"We establish now some key results that will be used in the following sections, regarding the range of the functional $L(f, \\cdot )$ .",
"Note that the other exponents in (REF ) or (REF ) have been dealt with in Lemma REF in terms of permutations, where it has been shown that $|\\alpha | + |\\alpha ^{-1}\\gamma ^{-1}| \\ge |\\gamma ^{-1}| = p-1,$ with equality if the permutation $\\alpha $ is geodesic with respect to $\\gamma ^{-1}$, i.e.",
"it is associated to a non-crossing partition (the cycle structure of $\\alpha $ is non-crossing and inside each cycle the elements have the same cyclic ordering as $\\gamma ^{-1}$ , or in terms of maps, using the genus (REF ).",
"The next results are a generalization of the above fact, which corresponds to the situation where $f$ is constant.",
"Proposition 3.3 For $f$ a word of length $p$ in the alphabet $\\lbrace AB,AC\\rbrace $ and $\\alpha \\in \\mathcal {S}_p$ , we have the bound $L(f,\\mathcal {M}) \\le p+1,$ with equality iff the map $\\mathcal {M}$ is planar, and all the edges of $\\mathcal {M}$ incident to a common white vertex have the same color.",
"In terms of the permutation $\\alpha $ , $L(f, \\alpha ) = p+1$ with equality iff $\\alpha $ is geodesic and $\\alpha \\le \\ker f$ (for the usual partial order on partitions), where $\\ker f$ is the partition having (at most) two blocks, $f^{-1}(\\lbrace AB\\rbrace )$ and $f^{-1}(\\lbrace AC\\rbrace )$ .",
"Proof in the map language.",
"The relation of the Euler characteristic for the map $\\mathcal {M}_f$ writes $2-2g(\\mathcal {M}_f)=V(\\mathcal {M}_f) - p + F(\\mathcal {M}_f).$ we can therefore rewrite (REF ) as $L(f,\\mathcal {M}) = p + 1 -2g(\\mathcal {M}_f)-2\\Delta _f(\\mathcal {M}),$ where we have denoted $\\Delta _f(\\mathcal {M})$ the number of white vertices reached by both colors $B$ and $C$ , $\\Delta _f(\\mathcal {M})=V(\\mathcal {M}_f)-V(\\mathcal {M}).$ In particular, this vanishes iff in $\\mathcal {M}$ , all the edges incident to a common white vertex have the same color, and is positive otherwise.",
"Therefore, $L(f,\\mathcal {M}) = p + 1$ iff $\\Delta _f(\\mathcal {M})= g(\\mathcal {M}_f)=0$ .",
"Moreover, if $\\Delta _f(\\mathcal {M})=0$ , the maps $\\mathcal {M}_f$ and $\\mathcal {M}$ coincide, so that $L(f,\\mathcal {M}) = p + 1$ iff $\\Delta _f(\\mathcal {M})= g(\\mathcal {M})=0$ .",
"$\\Box $ Note that, using the again the genus formula for $\\mathcal {M}$ , we obtain the following exact expression for the moments $\\mathbb {E}\\operatorname{Tr}(W_f)= \\sum _{\\mathcal {M}\\in \\mathbb {M}_p} N_A^{p+2-2g(\\mathcal {M})-V(\\mathcal {M})}N_D^{V(\\mathcal {M})-1}N_{B,C}^{p+1-2g(\\mathcal {M}_f)-2\\Delta _f(\\mathcal {M})}.$ Proof in terms of permutations.",
"Let us first restate the proposition in the form of a lemma.",
"To recover Prop.",
"REF , it suffices to use the fact that $\\#(\\hat{\\gamma }_f \\hat{\\alpha }) + |\\hat{\\gamma }_f \\hat{\\alpha }|= 2p.$ Lemma 3.4 For any $f \\in \\lbrace AB, AC\\rbrace ^p$ and $\\alpha \\in \\mathcal {S}_p$ , we have $|\\hat{\\gamma }_f \\hat{\\alpha }| \\ge p-1,$ with equality iff $\\alpha $ is geodesic and $\\alpha \\le \\ker f$ (for the usual partial order on partitions), where $\\ker f$ is the partition having (at most) two blocks, $f^{-1}(\\lbrace AB\\rbrace )$ and $f^{-1}(\\lbrace AC\\rbrace )$ .",
"Let us start by re-writing the permutation $\\hat{\\gamma }_f \\in \\mathcal {S}_{2p}$ (we are using the labeling of (REF ), $[p]^{B,C} = \\lbrace (1,B), \\ldots , (p, B),(1,C), \\ldots , (p, C)\\rbrace $ ) $\\hat{\\gamma }_f = \\delta _C \\left( \\gamma ^B \\oplus \\mathrm {id}^C \\right) \\delta _C,$ where $\\gamma ^B$ (resp.",
"$\\mathrm {id}^C$ ) denotes the permutation $\\gamma $ (resp.",
"$\\mathrm {id}$ ) acting on $[p]^B=\\lbrace (1,B), \\ldots , (p, B)\\rbrace $ (resp.",
"on $[p]^C=\\lbrace (1,C), \\ldots , (p, C)\\rbrace $ ), and $\\delta _C$ is a product of transpositions, $\\mathcal {S}_{2p} \\ni \\delta _C := \\prod _{a \\in f^{-1}(\\lbrace AC\\rbrace )} \\bigl ((a,B) (a,C)\\bigr ).$ Indeed, one can check formula (REF ) by comparing it with equations (REF ) and ().",
"We now apply [14] to the permutations $\\hat{\\gamma }_f,\\hat{\\alpha }\\in \\mathcal {S}_{2p}$ : $|\\hat{\\gamma }_f\\hat{\\alpha }| + |\\hat{\\gamma }_f| + |\\hat{\\alpha }| \\ge 2|\\hat{\\gamma }_f \\vee \\hat{\\alpha }|,$ where the join operation $\\vee $ on the right hand side should be understood as acting on the partitions induced by the cycles of the two permutations (the join of two partitions is their least upper bound).",
"The notation $| \\cdot |$ is extended to partitions as $|\\pi | := 2p - \\#\\pi $ , for a partition $\\pi $ of $[2p]$ , where $\\#\\pi $ denotes the number of blocks of $\\pi $ .",
"Since $\\hat{\\gamma }_f$ has one cycle of length $p$ and $p$ fixed points, and $\\hat{\\alpha }= \\alpha ^B \\oplus \\alpha ^C$ (using the same notation as above), the inequality becomes $|\\hat{\\gamma }_f\\hat{\\alpha }| &\\ge 4p - 2\\#(\\hat{\\gamma }_f \\vee \\hat{\\alpha }) - (p-1) - (2p - 2\\#\\alpha )\\\\&= p+1 + 2(\\#\\alpha - \\#(\\hat{\\gamma }_f \\vee \\hat{\\alpha })).$ We claim that $\\#(\\hat{\\gamma }_f \\vee \\hat{\\alpha }) = 1 + \\mathsf {pure}(\\alpha , f)$ , where $\\mathsf {pure}(\\alpha ,f)$ denotes the number of cycles of $\\alpha $ on which the restriction of $f$ is constant, i.e.",
"in the map language, the white vertices whose incident edges all carry the same colors, so that $\\mathsf {pure}(\\alpha ,f)$ is given by (REF ).",
"Indeed, the blocks of the partition $\\hat{\\gamma }_f \\vee \\hat{\\alpha }$ correspond to the blocks of $\\alpha ^B$ and $\\alpha ^C$ merged by the block of length $p$ of $\\hat{\\gamma }_f$ , together with each block of $\\alpha ^{B,C}$ matching only fixed points of $\\hat{\\gamma }_f$ .",
"It is now clear that the latter are exactly blocks of $\\alpha ^B$ on which $f \\equiv AC$ or blocks of $\\alpha ^C$ on which $f \\equiv AB$ ; we conclude that there are exactly $\\mathsf {pure}(\\alpha ,f)$ of those, proving the claim.",
"Hence, $|\\hat{\\gamma }_f\\hat{\\alpha }| \\ge p-1 + 2(\\#\\alpha - \\mathsf {pure}(\\alpha ,f)) \\ge p-1,$ proving the inequality.",
"Let us now characterize the permutations $\\alpha $ which saturate the inequality (assuming $f$ is fixed).",
"First, the last inequality above should be an equality, so $\\alpha = \\mathsf {pure}(\\alpha ,f)$ ; in other words, $f$ should be constant on the blocks of $\\alpha $ , which is the condition $\\alpha \\le \\ker f$ appearing in the statement of the lemma.",
"Moreover, we can easily see that this condition is equivalent to the fact that the permutations $\\hat{\\alpha }$ and $\\delta _C$ commute.",
"Indeed, by direct computation, we can see that, for any $a \\in \\llbracket 1, p\\rrbracket $ and any $Z=B,C$ , we have $[\\delta _C \\hat{\\alpha }\\delta _C](a,Z) = {\\left\\lbrace \\begin{array}{ll}(\\alpha (a),Z) &\\quad \\text{ if } f(a) = f(\\alpha (a))\\\\(\\alpha (a),{\\bar{Z}}) &\\quad \\text{ if } f(a) \\ne f(\\alpha (a)),\\end{array}\\right.",
"}$ where we denote by $\\bar{Z}$ the complement of $Z$ in $\\lbrace B,C\\rbrace $ .",
"Hence, $\\delta _C \\hat{\\alpha }\\delta _C = \\hat{\\alpha }$ iff $f$ is constant on the cycles of $\\alpha $ , which is the claimed statement.",
"Using (REF ), we have $|\\hat{\\gamma }_f\\hat{\\alpha }| = |\\delta _C \\left( \\gamma ^B \\oplus \\mathrm {id}^C \\right) \\delta _C \\hat{\\alpha }| = |\\left( \\gamma ^B \\oplus \\mathrm {id}^C \\right) \\hat{\\alpha }| = |(\\gamma \\alpha )^B \\oplus \\alpha ^C| = |\\alpha | + |\\alpha \\gamma |\\ge p-1,$ with equality iff $\\alpha $ is geodesic, proving the other equality condition in the statement and finishing the proof.",
"Reformulation of the exponent $L$ .",
"In this paragraph, we give a more intuitive expression for the exponent $L$ .",
"We define $\\mathsf {alt}(f,\\alpha )$ as the total number of changes of colors around the cycles of $\\alpha $ $\\mathsf {alt}(f,\\alpha ) = \\bigl \\vert \\lbrace a \\in \\llbracket 1, p \\rrbracket \\, : \\, f(a) \\ne f(\\alpha (a))\\rbrace \\bigr \\vert .$ In the map language, $\\mathsf {alt}(f,\\mathcal {M})$ is the total number of corners around white vertices of $\\mathcal {M}$ whose incident edges $a$ and $\\alpha (a)$ have different colors $j(a) \\ne j(\\alpha (a))$ .",
"Proposition 3.5 For any $f \\in \\lbrace AB, AC\\rbrace ^p$ and $\\alpha \\in \\mathcal {S}_p$ , where $\\alpha $ is geodesic (or equivalently, where $\\mathcal {M}=(\\gamma , \\alpha )$ is planar), we have $L(f,\\alpha ) = p+1 - \\mathsf {alt}(f,\\alpha ).$ We also provide two proofs, first in terms of permutations, and then in terms of maps.",
"In the following, for $\\mathcal {M}=(\\gamma , \\alpha )$ , we identify $L(f,\\alpha )$ and $L(f,\\mathcal {M})$ , as well as $\\mathsf {alt}(f,\\alpha )$ and $\\mathsf {alt}(f,\\mathcal {M})$ .",
"We stress that (REF ) is true only for a geodesic permutation $\\alpha $ , or equivalently a planar map $\\mathcal {M}$ .",
"In general, for non-planar maps, $2(g(\\mathcal {M}_f) + \\Delta _f(\\mathcal {M})) \\ne \\mathsf {alt}(f,\\mathcal {M})$ .",
"For instance, if $\\Delta _f(\\mathcal {M}) = 0$ , $\\mathcal {M}_f = \\mathcal {M}$ , so that $2(g(\\mathcal {M}_f) + \\Delta _f(\\mathcal {M}))= 2g(\\mathcal {M})$ , but $\\mathsf {alt}(f,\\mathcal {M}) = 0$ .",
"Proof in terms of permutations.",
"Let us assume that $\\alpha $ is geodesic (w.r.t.",
"$\\gamma ^{-1}$ , i.e.",
"it satisfies $|\\alpha |+|\\alpha \\gamma | = |\\gamma ^{-1}| = p-1$ ) and prove $|\\hat{\\gamma }_f \\hat{\\alpha }| = p-1 + \\mathsf {alt}(f,\\alpha ),$ which proves the proposition using (REF ).",
"We write, as before, $|\\hat{\\gamma }_f\\hat{\\alpha }| &= |\\delta _C \\left( \\gamma ^B \\oplus \\mathrm {id}^C \\right) \\delta _C \\hat{\\alpha }|\\\\&= |\\delta _C \\hat{\\alpha }\\delta _C ( \\gamma ^B \\oplus \\mathrm {id}^C )| \\\\&\\le |\\delta _C \\hat{\\alpha }\\delta _C \\hat{\\alpha }^{-1}| + | \\hat{\\alpha }( \\gamma ^B \\oplus \\mathrm {id}^C )| \\\\&= |\\alpha | + |\\alpha \\gamma | + |\\delta _C \\hat{\\alpha }\\delta _C \\hat{\\alpha }^{-1}| \\\\&= p-1 + \\mathsf {alt}(f,\\alpha ),$ where we have used the triangle inequality and the fact that $\\delta _C \\hat{\\alpha }\\delta _C \\hat{\\alpha }^{-1} = \\prod _{a \\, : \\, f(a) \\ne f(\\alpha ^{-1}(a))} \\bigl ((a,B) (a,C)\\bigr ),$ which follows easily from (REF ).",
"To conclude, we need to show that the triangle inequality used above is saturated, which is equivalent to the three permutations $\\delta _C \\hat{\\alpha }\\delta _C - \\hat{\\alpha }- (\\gamma ^B)^{-1} \\oplus \\mathrm {id}^C$ lying on a geodesic.",
"Since $\\delta _C \\hat{\\alpha }\\delta _C \\hat{\\alpha }^{-1}$ is a product of disjoint transpositions, this is in turn equivalent to the fact that, for all $a$ such that $f(a) \\ne f(\\alpha ^{-1}(a))$ , the elements $(a,B)$ and $(a,C)$ are contained in the same cycle of the permutation $\\varepsilon :=\\delta _C \\hat{\\alpha }\\delta _C(\\gamma ^B \\oplus \\mathrm {id}^C)$ .",
"This permutation acts in the following way: $(a,B) &\\mapsto {\\left\\lbrace \\begin{array}{ll}(\\alpha (a+1), B) &\\quad \\text{ if } \\qquad f(\\alpha (a+1)) = f(a+1) \\\\(\\alpha (a+1),C) &\\quad \\text{ if } \\qquad f(\\alpha (a+1)) \\ne f(a+1)\\end{array}\\right.",
"}\\\\(a,C) &\\mapsto {\\left\\lbrace \\begin{array}{ll}(\\alpha (a),C) &\\quad \\text{ if } \\qquad f(\\alpha (a)) = f(a) \\\\(\\alpha (a), B) &\\quad \\text{ if } \\qquad f(\\alpha (a)) \\ne f(a).\\end{array}\\right.",
"}$ In other words, on the $B$ -level, $[p]^B$ , $\\varepsilon $ acts as $\\alpha \\gamma $ , which is the permutation associated to the non-crossing partition $\\alpha ^{\\mathrm {Kr}}$ (see [35]), while on the $C$ -level, $[p]^C$ , $\\varepsilon $ acts as $\\alpha $ .",
"Using the fact that $\\alpha $ is non-crossing and the definition of $\\alpha ^{Kr}$ (see [35], and note that in our notation, we also have $\\bar{a} > a$ ), we can easily see that $(a,B)$ and $(a,C)$ belong to the same cycle of $\\varepsilon $ , whenever $f(a) \\ne f(\\alpha ^{-1}(a))$ ; we refer the reader to Figure REF for a graphical illustration of this fact.",
"$\\Box $ Figure: Diagram showing that whenever f(a)≠f(α -1 (a))f(a) \\ne f(\\alpha ^{-1}(a)), (a,B)(a,B) and (a,C)(a,C) belong to the same cycle of ε=δ C α ^δ C (γ B ⊕ id C )\\varepsilon =\\delta _C \\hat{\\alpha }\\delta _C(\\gamma ^B \\oplus \\mathrm {id}^C).",
"Since the partition α Kr \\alpha ^{\\operatorname{Kr}} is non-crossing (blue partition), the element (a,C)(a,C) (in red) must “escape” the interval [a,α(a)][a,\\alpha (a)] through (a,B)(a,B) (in blue).Proof in the map language.",
"We assume that the map $\\mathcal {M}\\in \\mathbb {M}_p$ is planar, and prove by induction that $\\mathsf {alt}(f, \\mathcal {M}) = 2g(\\mathcal {M}_f)+2\\Delta _f(\\mathcal {M}).$ If $v_\\circ $ is a white vertex and $v_\\bullet $ is the only black vertex, we denote $\\mathcal {M}^{v_\\circ }$ the submap obtained from $\\mathcal {M}$ by keeping only $v_\\circ $ and $v_\\bullet $ and the edges linking them.",
"We can apply the operation of Fig.",
"REF to the white vertex of $\\mathcal {M}^{v_\\circ }$ , thus obtaining a map $\\mathcal {M}_f^{v_\\circ }$ .",
"Because the map $\\mathcal {M}$ is planar, the map $\\mathcal {M}_f$ can be constructed by recursively inserting the maps $\\mathcal {M}_f^{v_\\circ }$ in the corners in the appropriate way (inverse operation of that shown in Fig.",
"REF ), where $v_\\circ $ spans the white vertices of $\\mathcal {M}$ .",
"Therefore, any operation on the edges of a given $\\mathcal {M}_f^{v_\\circ }$ does not affect the faces incident to other $\\mathcal {M}_f^{v^{\\prime }_\\circ }$ , and the genus of $\\mathcal {M}_f$ is the sum of the genera of all the $\\mathcal {M}_f^{v_\\circ }$ $g(\\mathcal {M}_f) = \\sum _{\\begin{array}{c}{v_\\circ \\in \\mathcal {M}}\\\\{\\text{white vertex}}\\end{array}} g(\\mathcal {M}_f^{v_\\circ }).$ Note that this is not true for a non-planar map $\\mathcal {M}$ .",
"We can therefore study what happens locally, for a single $\\mathcal {M}^{v_\\circ }$ .",
"The second step is to notice that if two consecutive edges around a $v_\\circ $ have the same color, then removing one or the other will not affect the genus $g(\\mathcal {M}_f^{v_\\circ })$ .",
"We can therefore consider that there are no two consecutive edges of the same color.",
"The case with 3 edges of each color is shown in Fig.",
"REF a) below.",
"Figure: a) A six edges ℳ v ∘ \\mathcal {M}^{v_\\circ }.",
"b-d) Recursive operations on ℳ f v ∘ \\mathcal {M}_f^{v_\\circ }.Firstly, in the case where $\\mathcal {M}^{v_\\circ }$ is made of only two edges of two different colors, we indeed have $2=\\mathsf {alt}(f, \\mathcal {M}^{v_\\circ }) = 2g(\\mathcal {M}_f^{v_\\circ })+2\\Delta _f(\\mathcal {M}^{v_\\circ })=2\\cdot 0+2\\cdot (3-2)=2$ .",
"If now there are more than two edges, it is easy to see that a single face visits all the edges twice in $\\mathcal {M}_f^{v_\\circ }$ (see the dotted face in Fig.",
"REF b)).",
"Furthermore, these edges are not bridges, and therefore, deleting any edge in $\\mathcal {M}_f^{v_\\circ }$ , the genus decreases by one (there is one more face and one less edge), and the number of corners incident to edges of different colors around $v_\\circ $ in $\\mathcal {M}^{v_\\circ }$ decreases by two.",
"In the resulting map (Fig.",
"REF c)), two edges have two incident faces.",
"Deleting one of them, the genus does not vary, nor the number of corners incident to edges of different colors, and we recover a map with less edges and with the same property (Fig.",
"REF d)).",
"By induction we deduce that the number of corners incident to edges of different colors around $v_\\circ $ in $\\mathcal {M}^{v_\\circ }$ is twice the genus of $\\mathcal {M}_f^{v_\\circ }$ plus two.",
"Summing over white vertices (REF ), we obtain the sought relation $\\mathsf {alt}(f, \\mathcal {M}) = 2g(\\mathcal {M}_f)+2\\Delta _f(\\mathcal {M})$ .",
"$\\Box $"
],
[
"The balanced asymptotical regime",
"We focus in this section on the asymptotic regime where $N:=\\dim \\mathcal {H}_B = \\dim \\mathcal {H}_C \\rightarrow \\infty $ , which we call balanced; for the case when the size of the $B,C$ subsystems stays bounded, see Section REF .",
"In this regime, where $\\mathcal {H}_{B,C}$ grow, it turns out that the asymptotic behavior of $\\mathcal {H}_{A,D}$ is not so important, as long as the ratio $\\dim \\mathcal {H}_D / \\dim \\mathcal {H}_A$ converges to a positive constant as $N \\rightarrow \\infty $ .",
"We present next the main result of this section, and discuss several particular asymptotic scenarios later.",
"Theorem 3.6 Let $X_N \\in \\mathbb {C}^{N_A} \\otimes \\mathbb {C}^N \\otimes \\mathbb {C}^{N} \\otimes \\mathbb {C}^{N_D}$ be a sequence of random Gaussian tensors, where $N_{A,D}$ are arbitrary functions of $N$ satisfying $N_D \\sim cN_A$ as $N \\rightarrow \\infty $ , for some constant $c \\in (0,\\infty )$ .",
"Then, the normalized marginals $(N_A^{-1}N^{-1}W_{AB}^{(N)}, N_A^{-1}N^{-1}W_{AC}^{(N)})$ defined in (REF ) converge in distribution, as $N \\rightarrow \\infty $ , to a pair of identically distributed and free elements $(x_{AB},x_{AC})$ , where $x_{AB}$ and $x_{AC}$ have a $\\mathrm {MP}_{c}$ distribution.",
"Equivalently, for any word in the two marginals $f \\in \\lbrace AB, AC\\rbrace ^p$ , we have $\\lim _{N \\rightarrow \\infty } \\mathbb {E} (N_A N)^{-p-1} \\operatorname{Tr} \\prod _{1 \\le i \\le p}^{\\longleftarrow } W_{f(i)}^{(N)} = \\sum _{\\alpha \\in \\mathrm {NC}(p), \\, \\alpha \\le \\ker f} c^{\\#\\alpha },$ where $\\ker f$ is the partition having two blocks corresponding to the occurrences of $AB$ (resp.",
"$AC$ ) in the word $f$ .",
"This rewrites in terms of planar bicolored maps with one white vertex as $\\lim _{N \\rightarrow \\infty } \\mathbb {E} (N_A N)^{-p-1} \\operatorname{Tr} \\prod _{1 \\le i \\le p}^{\\longleftarrow } W_{f(i)}^{(N)} = \\sum _{\\mathcal {M}\\in \\mathbb {M}_p^{0}, \\, \\Delta _f(\\mathcal {M}) =0} c^{V(\\mathcal {M}) - 1},$ where the sum is restricted to maps whose white vertices only have incident edges of the same color, and we recall that $\\mathbb {M}_p^{0}$ is the subset of the elements of $\\mathbb {M}_p$ of vanishing genus.",
"We first prove formula (REF ).",
"Starting from the exact moment formula of Theorem REF , let us analyze the contribution of $N$ , through its exponent $L(f,\\alpha )$ .",
"From Proposition REF , $L(f,\\alpha ) \\le p+1,$ with equality iff $\\alpha $ is geodesic and $\\alpha \\le \\ker f$ .",
"Hence, we have $\\mathbb {E} \\operatorname{Tr} W^{(N)}_f &= (1+o(1))N^{p+1} \\sum _{\\alpha \\in \\operatorname{NC}(p),\\, \\alpha \\le \\ker f} N_A^{\\#(\\gamma \\alpha )}N_D^{\\#\\alpha } \\\\&= (1+o(1))(N_AN)^{p+1} \\sum _{\\alpha \\in \\operatorname{NC}(p),\\, \\alpha \\le \\ker f} \\left( \\frac{N_D}{N_A}\\right)^{\\#\\alpha } \\\\&= (1+o(1))(N_AN)^{p+1} \\sum _{\\alpha \\in \\operatorname{NC}(p),\\, \\alpha \\le \\ker f} c^{\\#\\alpha },$ proving the claimed formula.",
"Above, we have used the key fact that, for the surviving $\\alpha $ terms (i.e.",
"the permutations which are geodesic w.r.t.",
"$\\gamma ^{-1}$ ), $\\#(\\gamma \\alpha ) = p+1 - \\#\\alpha $ .",
"We now show how the moment formula (REF ) implies the main claim.",
"Using the moment-cumulant formula [35], one can read the asymptotic free cumulants directly off the moment formula: $\\lim _{N \\rightarrow \\infty } \\mathbb {E} (N_A N)^{-p-1} \\operatorname{Tr} \\prod _{1 \\le i \\le p}^{\\longleftarrow } W_{f(i)}^{(N)} = \\sum _{\\alpha \\in \\mathrm {NC}(p)} \\quad \\prod _{b \\text{ block of }\\alpha } c\\mathbf {1}_{f \\text{ is constant on } b} .$ Hence, mixed cumulants vanish (implying freeness, see [35]), and the distribution of the limiting variables $x_{AB}$ , $x_{AC}$ is Marčenko-Pastur of parameter $c$ , ending the proof.",
"Remark 3.7 As a special case of the result above, one can consider the case where all the Hilbert spaces have, up to constants, the same dimension $\\dim \\mathcal {H}_A = \\lfloor c_1N\\rfloor $ , $\\dim \\mathcal {H}_D = \\lfloor c_4N \\rfloor $ and $\\dim \\mathcal {H}_{B} = \\dim \\mathcal {H}_C = N$ .",
"Then, the normalized marginals $(c_1N^2)^{-1} (W_{AB},W_{AC})$ converge in moments, as $N \\rightarrow \\infty $ , towards two free elements having $\\mathrm {MP}_{c_4/c_1}$ distribution.",
"The multi-partite equivalent of this result will be considered in Section REF .",
"Similarly, when the subsystems $A$ and $D$ have fixed dimension $\\dim \\mathcal {H}_A = k$ , $\\dim \\mathcal {H}_D = l$ and $\\dim \\mathcal {H}_{B} = \\dim \\mathcal {H}_C = N$ the normalized marginals $(kN)^{-1} (W_{AB},W_{AC})$ converge in moments, as $N \\rightarrow \\infty $ , towards two free elements having $\\mathrm {MP}_{l/k}$ distribution.",
"Remark 3.8 One can interpret the asymptotical freeness of the two marginals $W_{AB}$ and $W_{AC}$ in the following way: the two marginals behave as if they come from independent random tensors $X$ and $Y$ : $W_{AB} = [\\operatorname{id} \\otimes \\operatorname{id} \\otimes \\operatorname{Tr} \\otimes \\operatorname{Tr}](XX^*) \\quad \\text{ and } \\qquad \\tilde{W}_{AC}=[\\operatorname{id} \\otimes \\operatorname{Tr} \\otimes \\operatorname{id} \\otimes \\operatorname{Tr}](YY^*).$ Indeed, for the random matrices above, the conclusion of the theorem above follows from the very general asymptotic freeness results of Voiculescu: the random matrices $W_{AB}$ and $\\tilde{W}_{AC}$ are independent and unitarily invariant, and they converge to Marčenko-Pastur elements.",
"One can understand this parallel using the fact that, in the asymptotical regime under consideration here, the amount of fresh randomness (the Hilbert spaces $\\mathcal {H}_{B,C}$ ) is growing.",
"This situation is to be contrasted with the behavior of the marginals in the fixed $B,C$ regime discussed in Section REF .",
"Remark 3.9 The remark above has interesting applications to quantum information theory.",
"As the quantum marginals $\\rho _{AB}$ and $\\rho _{AC}$ are rescaled versions of the Wishart matrices $W_{AB,AC}$ , the same asymptotic freeness result holds, with a different scaling (precisely, it is the random matrices $N_D N \\rho _{AB}, N_D N\\rho _{AC}$ which are asymptotically free).",
"Thus, the previous remark implies that, in the large $N$ limit, the quantum marginals $\\rho _{AB}$ and $\\rho _{AC}$ “forget” that they are marginals of the same quantum state $|\\psi \\rangle _{ABCD}$ and behave like independent random density matrices from the induced ensemble with parameter $c = \\lim N_D/N_A$ .",
"In particular, these marginals become uncorrelated asymptotically, the intuition for this fact being that the amount of “fresh randomness” from the systems $\\mathcal {H}_B$ and $\\mathcal {H}_C$ , which have dimension growing to infinity, is enough to erase the correlations from system $\\mathcal {H}_A$ , and this independently on the ratio $\\dim \\mathcal {H}_A$ vs. $\\dim \\mathcal {H}_{B,C}$ .",
"As an application of Theorem REF , let us consider the product of the two marginals $W_{AB}$ and $W_{AC}$ , or, to be exact, its self-adjoint version $P:=W_{AB}^{1/2} W_{AC} W_{AB}^{1/2}$ .",
"Applying Theorem REF , the random matrix $P$ converges in moments to the element $x_{AB}^{1/2}x_{AC}x_{AB}^{1/2}$ , where $x_{AB}$ , $x_{AC}$ are two free elements having distribution $\\mathrm {MP}_{c}$ .",
"In free probability theory, the (self-adjoint) multiplication operation of free elements is known as the free multiplicative convolution (denoted by $\\boxtimes $ ), see [35].",
"In our case, we are interested in the probability measure $\\mathrm {MP}_{c} \\boxtimes \\mathrm {MP}_{c} = \\mathrm {MP}_{c}^{\\boxtimes 2}$ .",
"Exact formulas for the densities of the above distributions have been computed in [39] in the case $c=1$ and in [19] in the general case.",
"We compare Monte Carlo simulations to the exact densities in Figure REF .",
"Figure: The density of the free multiplicative square of the Marčenko-Pastur distribution MP c \\mathrm {MP}_c for c=1c=1 (left) and c=2c=2 (right) versus Monte-Carlo simulations.",
"The supports of the two probability measures are [0,27/4][0, 27/4] (left) and [(71-1717)/8,(71+1717)/8]≈[0.11,17.64][(71-17 \\sqrt{17})/8,(71+17 \\sqrt{17})/8] \\approx [0.11, 17.64] (right).We discuss next another formulation of the asymptotic moment formula (REF ), useful in practice when one has to evaluate specific mixed moments in the $AB$ /$AC$ marginals.",
"Proposition 3.10 In the same setting as Theorem REF with $c=1$ , the asymptotical mixed moments of the marginals $W_{AB}$ , $W_{AC}$ are indexed by arbitrary words $f \\in \\lbrace AB, AC\\rbrace ^p$ , or, equivalently, by two integer vectors $r,s$ : $\\prod _{1 \\le i \\le p}^{\\longleftarrow } W_{f(i)} = W_{AB}^{r_m} W_{AC}^{s_m} \\cdots W_{AB}^{r_2} W_{AC}^{s_2} W_{AB}^{r_1} W_{AC}^{s_1}.$ Asymptotically, we have $\\lim _{N \\rightarrow \\infty }(N_A N)^{-p-1} \\mathbb {E} \\operatorname{Tr}W_{AB}^{r_m} W_{AC}^{s_m}\\cdots W_{AB}^{r_1} W_{AC}^{s_1} = \\sum _{\\sigma , \\pi \\in \\mathrm {NC}(m),\\, \\sigma \\le \\pi }\\mathrm {Cat}_\\sigma (r) \\mathrm {Cat}_{\\pi ^\\mathrm {Kr}}(s) \\operatorname{Mob}(\\sigma , \\pi ),$ where $\\operatorname{Cat}_\\sigma $ is the multiplicative extension of Catalan numbers $\\operatorname{Cat}_\\sigma (r):=\\prod _{b \\text{ cycle of } \\sigma } \\mathrm {Cat}_{\\sum _{i \\in b}r_i},$ $\\mathrm {Kr}$ denotes the Kreweras complementation (see [27] or [35]) and $\\operatorname{Mob}$ is the Möbius function on the non-crossing partition lattice (see [35]), defined for $\\sigma \\le \\pi $ by $\\operatorname{Mob}(\\sigma ,\\pi ) := \\operatorname{Mob}(\\sigma ^{-1}\\pi ) = \\prod _{b \\text{ cycle of }\\sigma ^{-1}\\pi } (-1)^{|b|-1} \\mathrm {Cat}_{|b|-1}.$ The result follows from Theorem REF and from the formula for the moments of a product of free random variables [35]: denoting by $x_{AB}$ and $x_{AC}$ the limits in distribution of $W_{AB}$ , resp.",
"$W_{AC}$ and by $\\operatorname{tr}$ the expectation in the non-commutative probability space where $x_{AB}$ and $x_{AC}$ live, we have $\\operatorname{tr}(x_{AB}^{r_{q}} x_{AC}^{s_q}\\cdots x_{AB}^{r_1} x_{AC}^{s_1}) &= \\sum _{\\sigma \\in \\operatorname{NC}(q)} \\operatorname{tr}_\\sigma (x_{AB}^{r_q}, \\ldots , x_{AB}^{r_1}) \\kappa _{\\sigma ^{\\mathrm {Kr}}}(x_{AC}^{s_q}, \\ldots , x_{AC}^{s_1}) \\\\&= \\sum _{\\sigma \\in \\operatorname{NC}(q)} \\operatorname{Cat}_\\sigma (r) \\sum _{\\pi ^{\\prime } \\le \\sigma ^{\\mathrm {Kr}}}\\operatorname{tr}_{\\pi ^{\\prime }}(x_{AC}^{s_m}, \\ldots , x_{AC}^{s_1}) \\operatorname{Mob}(\\pi ^{\\prime },\\sigma ^{\\mathrm {Kr}})\\\\&= \\sum _{\\sigma \\le \\pi } \\operatorname{Cat}_\\sigma (r) \\operatorname{Cat}_{\\pi ^{\\mathrm {Kr}}}(s)\\operatorname{Mob}(\\sigma ,\\pi ),$ where we have used $\\pi ^{\\prime } = \\pi ^{\\mathrm {Kr}}$ .",
"As an application of the proposition above, we give below the explicit moments in the two simplest cases.",
"When $q=1$ , writing $p = r_1 + s_1$ , we have just one term in the sum (corresponding to $\\sigma = \\pi = \\includegraphics [scale=.7, trim=0 5px 0 0]{part-1}$ ) $\\lim _{N \\rightarrow \\infty } (c_1N^2)^{-p-1} \\mathbb {E} \\operatorname{Tr}W_{AB}^{r_1} W_{AC}^{s_1} = \\mathrm {Cat}_{r_1}\\mathrm {Cat}_{s_1}.$ When $q=2$ , write similarly $p=r_1+s_1+r_2+s_2$ ; this time, the sum contains three terms, corresponding respectively to $\\sigma = \\pi = \\includegraphics [scale=.7, trim=0 5px 0 0]{part-12}$ , $\\lbrace \\sigma =\\includegraphics [scale=.7, trim=0 5px 0 0]{part-12}, \\pi = \\includegraphics [scale=.7, trim=0 5px 0 0]{part-11}\\rbrace $ , and $\\sigma = \\pi = \\includegraphics [scale=.7, trim=0 5px 0 0]{part-11}$ : $\\lim _{N \\rightarrow \\infty } (c_1N^2)^{-p-1} \\mathbb {E} \\operatorname{Tr}W_{AB}^{r_2} W_{AC}^{s_2}W_{AB}^{r_1} W_{AC}^{s_1} &= \\mathrm {Cat}_{r_1}\\mathrm {Cat}_{r_2}\\mathrm {Cat}_{s_1+s_2} \\\\&- \\mathrm {Cat}_{r_1}\\mathrm {Cat}_{r_2}\\mathrm {Cat}_{s_1}\\mathrm {Cat}_{s_2}\\\\&+\\mathrm {Cat}_{r_1+r_2}\\mathrm {Cat}_{s_1}\\mathrm {Cat}_{s_2}.$ Remark 3.11 The number of terms in equation (REF ) is given by the number of pairs $(\\sigma , \\pi )\\in \\mathrm {NC}(m)^2$ such that $\\sigma \\le \\pi $ .",
"The set of all such pairs is known as the set of 2-chains (or intervals) in the lattice of non-crossing partitions and has been enumerated by Kreweras in [27]: their number is given by the Fuss-Catalan numbers of order 2 (sequence A001764 in [40]) $\\mathrm {FC}_2(q):= \\frac{1}{2q+1} \\binom{3q}{q}.$ Remarkably, these numbers are also the $q$ -th moments of the free multiplicative square of the Marčenko-Pastur distribution of parameter 1: $\\forall q \\ge 1, \\qquad \\int x^q \\mathrm {d} \\mathrm {MP}_1^{\\boxtimes 2}(x) = \\frac{1}{2q+1} \\binom{3q}{q}.$ Bijection with trees.",
"At the end of Section (Fig.",
"REF ), we described a bijective mapping between planar bicolored maps in $\\mathbb {M}_p^{0}$ .",
"Naturally, this bijection still applies in the present context, as the sum in (REF ) still involves elements of $\\mathbb {M}_p^{0}$ .",
"However we now have additional coloring information, which will translate into a coloring of the trees.",
"The edges carry a color in $\\lbrace B,C\\rbrace $ , but because the maps involved in (REF ) satisfy $\\Delta _f(\\mathcal {M}) =0$ , all the edges incident to the same white vertex share the same color.",
"We color each white vertex with the color of the incident edges.",
"We add a vertex in every face, and new edges linking it to the corners of the face, so that the result remains planar, and delete the initial edges and the black vertex.",
"A tree is always bicolored, but here we see that the vertices of the corresponding tree are partitioned into two sets: white vertices, and vertices carrying a color $B$ or $C$ , such that edges may only link white vertices to colored vertices.",
"Figure: Bijection between tricolored maps with one black vertex, and tricolored trees in 𝕋 f \\mathbb {T}_{f}.Furthermore, the labeling of the edges now translates into a labeling of the colored corners of the tree: starting on corner 1 and following the face around the tree counter-clockwisely, we alternatively encounter corners incident to white vertices, and corners incident to colored vertices, labeled with $p$ , then $p-1$ , ...until returning to corner 1 (the ordering is reversed with respect to the initial bicolored map, because of the dual bijective mapping).",
"All the labeled colored trees are not images of the colored bicolored maps: the corner corresponding to the edge number $a$ should be incident to a vertex of color $j(a)$ .",
"Therefore, starting from corner 1 and going around the tree in the clockwise direction, we should encounter corners incident to vertices of color $j(1)$ , then $j(2)$ , then $j(3)$ , and so on.",
"We denote $\\mathbb {T}_{f}$ the set of tricolored trees with white vertices, and vertices of color $B$ or $C$ , and with a such that the colors of the colored vertices encountered in the counterclockwise face are $(j(1), j(p), j(p-1),\\ldots j(1))$ , which is also the reverse word $f$ (adding color $A$ on every vertex).",
"In the case where $c=1$ , the quantity $\\lim _{N \\rightarrow \\infty } \\mathbb {E} (N_A N)^{-p-1} \\operatorname{Tr}W_{AB}^{r_q} W_{AC}^{s_q}\\cdots W_{AB}^{r_1} W_{AC}^{s_1}$ of Prop.",
"REF therefore counts the number of tricolored trees in $\\mathbb {T}_{p}$ .",
"With the notations of Prop.",
"REF , we therefore have: $\\vert \\mathbb {T}_{f} \\vert = \\sum _{\\sigma , \\pi \\in \\mathrm {NC}(q),\\, \\sigma \\le \\pi }\\mathrm {Cat}_\\sigma (r) \\mathrm {Cat}_{\\pi ^\\mathrm {Kr}}(s) \\operatorname{Mob}(\\sigma , \\pi ).$ In particular, this is always a non-negative quantity.",
"Remark that because of the tree structure, we can easily find recursive relations for the quantity $\\vert \\mathbb {T}_{f} \\vert $ .Note that these relations can also be found from the Schwinger-Dyson equations applied to the matrix formulation of the moments.",
"In order to write the recursive relations, we rather denoteFor practical reasons, the labeling is reversed with respect to our usual convention.",
"$\\mathcal {W}_{u_1,\\ldots , u_q}^{d_1,\\ldots , d_q} = \\vert \\mathbb {T}_{f} \\vert = \\lim _{N \\rightarrow \\infty } (N_A N)^{-p-1} \\mathbb {E} \\operatorname{Tr}W_{AB}^{u_1} W_{AC}^{d_1}\\cdots W_{AB}^{u_q} W_{AC}^{d_q}.$ We find: $&\\mathcal {W}_{u_1,\\ldots , u_q}^{d_1,\\ldots , d_q + 1}=\\sum _{k=1}^{q-1}\\sum _{s=0}^{d_k-1}\\mathcal {W}_{u_1,\\ldots , u_k}^{d_1, \\ldots ,s}\\mathcal {W}_{u_{k+1},\\ldots , u_q}^{d_{k+1}, \\ldots ,d_q+d_k-s} + \\sum _{s=0}^{d_q-1}\\mathcal {W}_{u_1,\\ldots , u_q}^{d_1, \\ldots ,s} C_{d_q - s}\\\\&\\mathcal {W}_{u_1,\\ldots , u_q + 1}^{d_1,\\ldots , d_q }=\\sum _{k=1}^{q}\\sum _{s=0}^{u_k-1}\\mathcal {W}_{u_1,\\ldots , u_{k-1},s}^{d_1,\\ldots , d_{k-1}, d_q}\\mathcal {W}_{u_q+u_k - s,u_{k+1},\\ldots , u_{q-1}}^{d_k, d_{k+1},\\ldots ,d_{q-1}} + \\sum _{s=0}^{u_q-1}\\mathcal {W}_{u_1,\\ldots , s}^{d_1, \\ldots ,d_q} C_{u_q - s}.$ However, solving these relations directly is a difficult task, and the solution is found considerably more easily using the techniques of the proof of Prop.",
"REF ."
],
[
"Comparing with the Fuss-Catalan matrix model",
"In this subsection, we would like to compare the matrix model discussed above in the balanced regime where $c_1=c_4=1$ (i.e.",
"$\\mathcal {H}_A = \\mathcal {H}_B = \\mathcal {H}_C = \\mathcal {H}_D = \\mathbb {C}^N$ ) with another one having the same asymptotic moments (the Fuss-Catalan numbers), the so-called free Bessel laws of parameter 2 from [4].",
"More precisely, the latter matrix model is given by $Q = X_1X_2X_2^*X_1^*$ , where $X_{1,2}$ are i.i.d.",
"$N \\times N$ complex Gaussian random matrices.",
"The exact moments of the random matrix $Q$ are given by $\\mathbb {E} \\operatorname{Tr} Q^p = \\sum _{\\alpha _1, \\alpha _2 \\in \\mathcal {S}_p} N^{\\#(\\gamma \\alpha _1\\alpha _2) + \\#\\alpha _1 + \\#\\alpha _2}.$ Indeed, in our usual representation in maps, the sum is taken over bicolored maps with one black vertex, such that white vertices only have incident edges of the same color.",
"In that sense, white vertices inherit the color 1 or 2 of their incident edges.",
"If $\\alpha _1$ (resp.",
"$\\alpha _2$ ) is the permutation whose cycles encode the vertices of color 1 (resp.",
"2), then we have one orbit for each white vertex (so $\\#\\alpha _1 + \\#\\alpha _2$ ), and one for each face of the map (so $\\#(\\gamma \\alpha _1\\alpha _2)$ ).",
"We wish to compare the expression (REF ) with the exact moments (REF ) of the model studied in Section REF , which we report here for $N_A=N_B=N_C=N_D$ , and for $P=W_{AB}^{1/2}W_{AC}W_{AB}^{1/2}$ : denoting $f_0$ the length $2p$ word $AB, AC, AB, AC, \\ldots $ (there are no two consecutive edges of the same color), $\\mathbb {E} \\operatorname{Tr} P^p = N^{2(2p+1)}\\sum _{\\mathcal {M}\\in \\mathbb {M}_{2p}} N^{-2g(\\mathcal {M})-2g(\\mathcal {M}_{f_0}) -2\\Delta _{f_0}(\\mathcal {M})}.$ Using that the Euler characteristics of the map writes $F(\\mathcal {M}) + V(\\mathcal {M}) = 2p+2 - 2g(\\mathcal {M})$ , we notice that (REF ) is very similar to (REF ), provided that we impose the condition $\\Delta _{f_0}(\\mathcal {M}) = 0.$ In that case, $\\mathcal {M}=\\mathcal {M}_{f_0}$ , so that $g(\\mathcal {M}_{f_0})=g(\\mathcal {M})$ .",
"We rewrite $\\mathbb {E} \\operatorname{Tr} Q^p = N^{2p+1}\\sum _{ \\begin{array}{c}{\\mathcal {M}\\in \\mathbb {M}_{2p}}\\\\{\\Delta _{f_0}(\\mathcal {M}) = 0}\\end{array}} N^{-2g(\\mathcal {M})}.$ As a consequence, the asymptotic moments of the two matrix models are identical, but the lower orders are different: $\\mathbb {E} N^{-2}\\operatorname{Tr}(N^{-4} P) &= 1 + N^{-2} \\\\\\mathbb {E} N^{-2}\\operatorname{Tr}(N^{-4} P)^2 &= 3+ 8N^{-2} + 8N^{-4} + 5N^{-6}\\\\\\mathbb {E} N^{-2}\\operatorname{Tr}(N^{-4} P)^3 &= 12+ 54N^{-2} + 135N^{-4} + 278N^{-6}+170 N^{-8} + 71 N^{-10},\\\\[+1ex]\\mathbb {E} N^{-1}\\operatorname{Tr}(N^{-2}Q) &= 1 \\\\\\mathbb {E} N^{-1}\\operatorname{Tr}(N^{-2}Q)^2 &= 3+N^{-2} \\\\\\mathbb {E} N^{-1}\\operatorname{Tr}(N^{-2}Q)^3 &= 12 + 21N^{-2} + 3 N^{-4}.$ Figure: Maps contributing to 𝔼N -2 Tr(N -4 P) 2 \\mathbb {E} N^{-2}\\operatorname{Tr}(N^{-4} P)^2.",
"We have denoted g f 0 =g(ℳ f 0 )g_{f_0}=g(\\mathcal {M}_{f_0}).",
"The edges labeled 1 is always the upper left one (blue).",
"The only maps contributing to 𝔼N -1 Tr(N -2 Q) 2 \\mathbb {E} N^{-1}\\operatorname{Tr}(N^{-2}Q)^2 are those with Δ f 0 =0\\Delta _{f_0}=0, however in that case, the contribution in NN is only given by the genus gg of the map.The coefficients of the lower orders in $N$ of $P$ are bigger than the ones of $Q$ .",
"This originates from the fact that in the $Q$ case, the lower orders come from maps in $\\mathbb {M}_{2p}$ with $\\Delta _{f_0}(\\mathcal {M}) =0$ with non-trivial genus, while in the case of $P$ , lower orders are obtained either from higher genus combinatorial mapsIt is also important to notice that in the $P$ case the genus appears in two ways in the exponent of $N$ - once for the combinatorial map $\\mathcal {M}$ and once for the combinatorial map $\\mathcal {M}_{f_0}$ .",
"This has to be taken into account when comparing the $Q$ case against the $P$ case.",
"or combinatorial maps containing both white vertices adjacent to both type $AB$ and type $AC$ edges (i.e.",
"with non-vanishing $\\Delta _{f_0}$ ).",
"In Figure REF , we show all the maps contributing to $\\mathbb {E} N^{-2}\\operatorname{Tr}(N^{-4} P)^2$ .",
"The values of $g$ , $\\Delta _{f_0}$ and $g_{f_0} = g(\\mathcal {M}_{f_0})$ are shown.",
"For instance the maps in the upper right box of Fig.",
"REF are planar but contain one white vertex adjacent to one edge of type $AB$ and one edge of type $AC$ , and therefore contribute to $\\mathbb {E} N^{-2}\\operatorname{Tr}(N^{-4} P)^2$ at order $N^{-2}$ .",
"The maps contributing to $\\mathbb {E} N^{-1}\\operatorname{Tr}(N^{-2}Q)^2$ are those for which $\\Delta _{f_0} = 0$ .",
"Their contribution to $\\mathbb {E} N^{-1}\\operatorname{Tr}(N^{-2}Q)^2$ is $N^{-2g}$ , but their contribution to $\\mathbb {E} N^{-2}\\operatorname{Tr}(N^{-4} P)^2$ is $N^{-4g}$ , because $g(\\mathcal {M}_{f_0})=g(\\mathcal {M})$ .",
"Keeping in mind the aim of comparing the two matrix models mixed moments with the moments of $P$ we start by defining the following operation on combinatorial maps Definition 3.12 Let $\\mathcal {M}_1,\\mathcal {M}_2 \\in \\mathbb {M}(p) \\times \\mathbb {M}(p^{\\prime })$ be two combinatorial maps with respectively $p$ and $p^{\\prime }$ edges.",
"Both maps have a labeling of the edges.",
"We define the gluing convolution as $\\odot : \\ &\\mathbb {M}(p)\\times \\mathbb {M}(p^{\\prime })\\rightarrow \\mathbb {M}(p+p^{\\prime })\\\\&(\\mathcal {M}_1, \\mathcal {M}_2)\\longmapsto \\mathcal {M}$ where $\\mathcal {M}$ is the empty map if $p\\ne p^{\\prime }$ and is otherwise obtained from $\\mathcal {M}_1$ and $\\mathcal {M}_2$ by stacking their black vertices one onto the other, in such way that the edges of $\\mathcal {M}_2$ slip into the corners of the black vertex of $\\mathcal {M}_1$ and edges of $\\mathcal {M}_1$ and $\\mathcal {M}_2$ alternate around the black vertex of the newly created map $\\mathcal {M}$ .",
"In order to select a unique way to perform this operation, we ask that the edge with label $i$ in $\\mathcal {M}_1$ is followed by the edge labeled $i$ in $\\mathcal {M}_2$ when following the ordering of the edges around the black vertex of $\\mathcal {M}$ .",
"Graphically one obtains the local construction shown on Fig.",
"REF .",
"Figure: Gluing convolution of two maps.",
"The gluing convolution is made locally around the black vertex.",
"The rest of the maps stays untouched.We also define the splitting of a vertex.",
"Definition 3.13 A vertex-splitting is a local move on a vertex with at least two corners of a combinatorial map.",
"It is performed by choosing two corners of the considered vertex and splitting the vertex along a straight line between these two corners.",
"Figure: A vertex-splitting move applied to a white vertex.Notice that the vertex splitting move can be understood in the language of permutations.",
"In this language, a $p$ -valent white vertex is a cycle of length $p$ , $(\\alpha _c)$ of $\\alpha =\\prod _{c\\in \\alpha } (\\alpha _c)$ .",
"Such a cycle writes $(\\alpha _c)=(a_1 a_2 \\ldots a_p)$ .",
"Choosing two corners of a white vertex amounts to picking two elements $a_k, a_{k^{\\prime }}$ in $(\\alpha _c)$ such that $a_{k^{\\prime }}\\ne a_k$ .",
"This corresponds to the choice of the two corners of the white vertex located between edges $a_k,a_{k+1}$ and $a_{k^{\\prime }}, a_{k^{\\prime }+1}$ .",
"These elements appear in the cycle $(\\alpha _c)=(a_1 a_2 \\ldots a_k a_{k+1}\\ldots a_{k^{\\prime }} a_{k^{\\prime }+1} a_p)$ .",
"The splitting of a white vertex is just the composition of $(\\alpha _c)$ with the transposition $(a_k a_{k^{\\prime }})$ , as $(\\alpha _c)(a_k a_{k^{\\prime }})=(\\alpha _{c^{\\prime }})(\\alpha _{c^{\\prime \\prime }})$ where $(\\alpha _{c^{\\prime }})=(a_1 a_2\\ldots a_{k}a_{k^{\\prime }+1}\\ldots a_p)$ and $(\\alpha _{c^{\\prime \\prime }})=(a_{k+1}\\ldots a_{k^{\\prime }})$ forms the two new white vertices.",
"We consider the partial order on $\\mathbb {M}^0_p$ defined as follows, for $\\mathcal {M},\\mathcal {M}^{\\prime } \\in \\mathbb {M}^0_p$ two planar combinatorial maps, we say that $\\mathcal {M}\\le \\mathcal {M}^{\\prime }$ if and only if there exists a finite sequence of maps $\\lbrace \\mathcal {M}_i\\rbrace _{i=0}^F$ such that $\\mathcal {M}_0=\\mathcal {M}^{\\prime }$ , $\\mathcal {M}_F=\\mathcal {M}$ , and $\\mathcal {M}_{i+1}$ can be obtained from $\\mathcal {M}_i$ by applying a vertex-splitting move on one white vertex of $\\mathcal {M}_i$ .",
"We now define the Tutte dual of a map.",
"It is a particular case of one of the Tutte bijections for bicolored maps [43], as was the bijection presented in Section , Fig.",
"REF .",
"Though it can be defined for more general sets of maps we assume $\\mathcal {M}\\in \\mathbb {M}_p$ , Definition 3.14 The Tutte dual $\\mathcal {T}(\\mathcal {M})$ of $\\mathcal {M}\\in \\mathbb {M}_p$ is obtained as follows.",
"In every face of $\\mathcal {M}$ we apply the following rules We draw a white vertex inside the face under consideration.",
"We draw a new edge between each corner adjacent to the black vertex inside this face and the newly created white vertex inside this face.",
"Once we have followed this procedure for every face of $\\mathcal {M}$ , we erase all initial edges and white vertices.",
"We notice that the Tutte dual preserves the genus of the map, that is to say that $g(\\mathcal {M})=g(\\mathcal {T}(\\mathcal {M}))$ .",
"Remark that the only difference with the bijection presented in Section , Fig.",
"REF , is that the new edges are added between the new vertices and the black vertex, instead of the white vertices.",
"A consequence of this choice is that after the bijection of Fig.",
"REF , one no longer has elements of $\\mathbb {M}_p$ , while we do in the present case, thus the name “dual\".",
"We report the reader to [43] for the general bijection encoding both cases.",
"We then have the following proposition Proposition 3.15 Let $\\mathcal {M}_1, \\mathcal {M}_2$ be two planar maps in $\\mathbb {M}^0_p$ (i.e.",
"$g(\\mathcal {M}_1)=g(\\mathcal {M}_2)=0$ ).",
"Then $g(\\mathcal {M}_1\\odot \\mathcal {M}_2)=0$ if and only if $\\mathcal {M}_2\\le \\mathcal {T}(\\mathcal {M}_1)$ .",
"First notice that by construction of $\\mathcal {T}(\\mathcal {M}_1)$ , $g(\\mathcal {M}_1\\odot \\mathcal {M}_2)=0$ if $\\mathcal {M}_2=\\mathcal {T}(M_1)$ .",
"Then when performing a vertex-splitting move on a white vertex of $\\mathcal {M}_2$ in $\\mathcal {M}_1\\odot \\mathcal {M}_2$ the number of vertices is raised by one $V\\rightarrow V^{\\prime }=V+1$ , the number of edges stays the same, and the number of faces is decreased by one $F\\rightarrow F^{\\prime }=F-1$ as a consequence the genus stays constant under such a move.",
"Thus we have $\\mathcal {M}_2\\le \\mathcal {T}(\\mathcal {M}_1)\\Rightarrow g(\\mathcal {M}_1 \\odot \\mathcal {M}_2)=0$ .",
"Assume now that $\\mathcal {M}_2\\lnot \\le \\mathcal {T}(M_1)$ .",
"Therefore there exists a vertex in $\\mathcal {M}_2$ adjacent to two edges of $\\mathcal {M}_2$ in $\\mathcal {M}_1\\odot \\mathcal {M}_2$ whose starting points are at corners that belong to two different faces of $\\mathcal {M}_1$ .",
"Thus edges of $\\mathcal {M}_1$ and $\\mathcal {M}_2$ cross in $\\mathcal {M}_1 \\odot \\mathcal {M}_2$ , which is equivalent to saying that $g(\\mathcal {M}_1 \\odot \\mathcal {M}_2)\\ne 0$ .",
"Thus we have $\\mathcal {M}_2 \\le \\mathcal {T}(\\mathcal {M}_1) \\Leftarrow g(\\mathcal {M}_1\\odot \\mathcal {M}_2)$ .",
"If we come back to the definition of the Tutte dual we notice that it is is equivalent to the Kreweras complementation.",
"This similarity is pictured on the Fig.",
"REF , where the graphical representation of non-crossing partition has been borrowed from [35].",
"Note however that the Tutte dual is defined for every maps not just the planar ones (the latter corresponding to non-crossing partitions).",
"Figure: Above: Tutte duality.",
"The initial map ℳ\\mathcal {M} on the left, its Tutte dual 𝒯(ℳ)\\mathcal {T}(\\mathcal {M}) on the right.",
"The middle map represents the intermediate steps of the construction.",
"The labeling of the edges of 𝒯(ℳ)\\mathcal {T}(\\mathcal {M}) is inherited from the map ℳ\\mathcal {M}.Below: Kreweras complementation map.We can write $\\mathbb {E} \\operatorname{Tr} P^p &=N^{4p+2}\\sum _{\\mathcal {M} \\in \\mathbb {M}_{2p}} N^{-2g(\\mathcal {M})-2(g(\\mathcal {M}_{f_0}) + \\Delta _{f_0}(\\mathcal {M}))}\\\\\\mathbb {E} \\operatorname{Tr} Q^p &= N^{2p+1}\\sum _{\\mathcal {M}_1, \\mathcal {M}_2 \\in \\mathbb {M}(p)} N^{-2g(\\mathcal {M}_1 \\odot \\mathcal {M}_2)} \\\\&= N^{2p+1}\\sum _{\\mathcal {M} \\in \\mathbb {M}_{2p} \\, : \\, \\Delta _{f_0}(\\mathcal {M}) = 0} N^{-2g(\\mathcal {M})},$ The proof of the equations (), () is straightforward.",
"From () we see the connection with the formula (REF ), while from () we see why the moments of $P$ are larger than those of $Q$ : the terms in the sum for the moments of $Q$ () are a subset of the terms in (REF ).",
"Indeed, we only have $\\left.\\mathbb {M}_{2p}\\right|_{\\Delta _{f_0}=0} =\\mathbb {M}_p\\odot \\mathbb {M}_p$ , and this equality does not hold in the planar case, $\\left.\\mathbb {M}^0_{2p}\\right|_{\\Delta _{f_0}=0} \\subsetneq \\mathbb {M}^0_p\\odot \\mathbb {M}^0_p$ The extension of the $\\odot $ operation to sets of maps is understood in the straightforward way as the set of all maps with $2p$ edges that can be obtained by making the gluing convolution of two maps with $p$ edges..",
"In the limit $N\\rightarrow \\infty $ , thanks to Proposition REF , the equation () rewrites $\\lim _{N\\rightarrow \\infty }\\frac{1}{N^{2p+1}}\\mathbb {E} \\operatorname{Tr} Q^p &= \\sum _{\\mathcal {M}_1\\in \\mathbb {M}_p} \\Biggl ( \\sum _{\\mathcal {M}_2 \\le \\mathcal {T}(\\mathcal {M}_1)} 1\\Biggr ),$ which mimics, in the language of maps, the expression obtained in [35] with cumulants equal to one in the language of permutations.",
"This translates the fact that in the large $N$ limit the moments of $Q$ are the moments of the multiplicative convolution of two Marčenko-Pastur laws."
],
[
"The unbalanced asymptotical regime",
"In this section, we study the asymptotical regime where the Hilbert spaces $\\mathcal {H}_B$ and $\\mathcal {H}_C$ have fixed dimension, while the dimensions of $\\mathcal {H}_A$ and $\\mathcal {H}_D$ grow to infinity.",
"To be more precise, we assume in this section that $\\dim \\mathcal {H}_B = \\dim \\mathcal {H}_C = m$ , for some positive integer constant $m$ ; $\\dim \\mathcal {H}_A = N$ , $\\dim \\mathcal {H}_D = \\lfloor cN \\rfloor $ , for some constant $c \\in (0,\\infty )$ , where $N \\rightarrow \\infty $ .",
"Contrary to the results proven in Section REF , in this setting, the random matrices $W_{AB}$ and $W_{AC}$ are no longer asymptotically free.",
"One can understand this fact, stated precisely in the theorem below, by noticing that the “shared randomness” between the two random matrices ($\\dim \\mathcal {H}_A = N \\rightarrow \\infty $ ) is much larger than the “fresh randomness” ($\\dim \\mathcal {H}_B = \\dim \\mathcal {H}_C = m$ , fixed).",
"Theorem 3.16 In the asymptotical regime described above, the pairs of random matrices $\\left((mN)^{-1}W_{AB}, (mN)^{-1}W_{AC}\\right)$ converge in distribution, as $N \\rightarrow \\infty $ , to a pair of non-commutative random variables $(x_{AB},x_{AC})$ having the following free cumulants: $\\kappa (x_{f(1)}, x_{f(2)}, \\ldots , x_{f(p)}) = cm^{-\\mathsf {alt}(f)},$ where $f \\in \\lbrace AB, AC\\rbrace ^p$ is an arbitrary word in the letters $AB, AC$ , and $\\mathsf {alt}(f)$ is the number of different consecutive values of $f$ , counted cyclically: $\\mathsf {alt}(f) := |\\lbrace a \\, : \\, f(a) \\ne f(a+1)\\rbrace |,$ where $f(p+1):=f(1)$ .",
"Equivalently, for any word in the two marginals $f \\in \\lbrace AB, AC\\rbrace ^p$ , we have $\\mathbb {E} \\operatorname{Tr} W_f = (1+o(1)) (mN)^{p+1} \\sum _{\\alpha \\in \\mathrm {NC}(p)} c^{\\#\\alpha } m^{-\\mathsf {alt}(f,\\alpha )},$ where $\\mathsf {alt}(f,\\alpha )$ has been defined in (REF ) as $ \\mathsf {alt}(f,\\alpha ) = \\bigl \\vert \\lbrace a \\in \\llbracket 1, p \\rrbracket \\, : \\, f(a) \\ne f(\\alpha (a))\\rbrace \\bigr \\vert .",
"$ We begin by analyzing the exact moment formula of Theorem REF : $\\mathbb {E} \\operatorname{Tr} W_f = \\sum _{\\alpha \\in \\mathcal {S}_p} N^{\\#(\\gamma \\alpha )} \\lfloor cN \\rfloor ^{\\#\\alpha } m^{L(f,\\alpha )}.$ Note that since $m$ is fixed, the dominating terms correspond to permutations $\\alpha $ maximizing the exponent $\\#\\alpha + \\#(\\alpha \\gamma )$ ; these permutations have been shown before to be exactly the non-crossing ones, so we have $\\mathbb {E} \\operatorname{Tr} W_f = (1+o(1)) N^{p+1} \\sum _{\\alpha \\in \\mathrm {NC}(p)} c^{\\#\\alpha } m^{L(f,\\alpha )}.$ For $\\alpha $ non-crossing, we know from Proposition REF that $L(f,\\alpha ) = p+1 - \\mathsf {alt}(f,\\alpha )$ , and the conclusion concerning the cumulants follows from Speicher's moment-cumulant formula.",
"Remark 3.17 In the degenerate case $m=1$ (i.e.",
"there are no $B$ and $C$ systems), one has that $W_{AB} = W_{AC}=W_A$ is a Wishart matrix of parameters $(N, \\lfloor cN \\rfloor )$ , and one recovers the result from the classical Marčenko-Pastur setting, Proposition REF : the free cumulants of $x_{AB} = x_{AC}$ are all equal to $c$ (see equation (REF ) and the comments following it).",
"Remark 3.18 If, after taking the limit $N \\rightarrow \\infty $ in the theorem above, one takes the limit $m \\rightarrow \\infty $ , we recover the result of Theorem REF .",
"Indeed, the only free cumulants surviving are the ones with $\\mathsf {alt}(f)=0$ (i.e.",
"mixed cumulants vanish), proving that $x_{AB}$ and $x_{AC}$ are asymptotically free.",
"Remark 3.19 Elaborating on the preceding remarks, we notice that the eigenvalues distribution of the matrix $P=W_{AB}^{1/2}W_{AC}W_{AB}^{1/2}$ for $m=1$ is the distribution of the square of the eigenvalues of a Wishart matrix, the corresponding density, that we denote $\\mathrm {d}\\rho _{m=1,c}(x)$ writes $\\mathrm {d}\\rho _{m=1,c}(x)=\\mathrm {d}\\textrm {MP}_c(\\sqrt{x})$ , while if $m\\rightarrow \\infty $ we end up with $\\mathrm {d}\\rho _{m=\\infty ,c}(x)= \\mathrm {d}\\textrm {MP}_c^{\\boxtimes 2}(x)$ .",
"This fact does not depend on the rate at which $m$ is sent to infinity with respect to the $N$ limit.",
"Thus the parameter $m$ allows one to canonically interpolate between the two different distributions $\\mathrm {d}\\textrm {MP}_c(\\sqrt{x})$ and $\\mathrm {d}\\textrm {MP}_c^{\\boxtimes 2}(x)$ , the former being the distribution of the square of eigenvalues of a Wishart random matrix, while the latter is the free multiplicative convolution of two Marčenko-Pastur distribution.",
"Notice also, that one can also consider the (non-selfadjoint) matrix $\\tilde{P}=W_{AB}W_{AC}$ which has the same moments and eigenvalues to reach the same conclusion.",
"Finally, we plot Monte-Carlo simulations of the eigenvalues of $P$ versus the square (resp.",
"the free multiplicative square) of a Marčenko-Pastur distribution in Figure REF .",
"Figure: Plots of Monte-Carlo simulations of the eigenvalues of P=W AB 1/2 W AC W AB 1/2 P=W_{AB}^{1/2}W_{AC}W_{AB}^{1/2} (yellow histogram) versus the square of a Marčenko-Pastur distribution (red curve) and the free multiplicative square of the same Marčenko-Pastur distribution (blue curve).",
"On the top row, we have N=600N=600, c=1c=1 and m=1,2,5m=1,2,5, while on the bottom row we have N=600N=600, c=5c=5, m=1,2,5m=1,2,5.We record below some low mixed moments in the variables $x_{AB}$ , $x_{AC}$ (we denote by $\\operatorname{tr}$ the expectation in the non-commutative probability space where these variables live): $\\operatorname{tr}(x_{AB}x_{AC}) &= c^2+\\frac{c}{m^2}\\\\\\operatorname{tr}(x_{AB}x_{AB}x_{AC}) &= c^3+c^2+\\frac{2 c^2}{m^2}+\\frac{c}{m^2}\\\\\\operatorname{tr}(x_{AB}x_{AB}x_{AC}x_{AC}) &= c^4+2 c^3+c^2+\\frac{4 c^3}{m^2}+\\frac{4 c^2}{m^2}+\\frac{c}{m^2}+\\frac{c^2}{m^4}\\\\\\operatorname{tr}(x_{AB}x_{AC}x_{AB}x_{AC}) &= c^4+2 c^3+\\frac{4 c^3}{m^2}+\\frac{4 c^2}{m^2}+\\frac{2 c^2}{m^4}+\\frac{c}{m^4}.$ Let us now explore the consequences of the formulas for the mixed free cumulants from this section to quantum information theory.",
"As in Section REF , the two quantum marginals $\\rho _{AB}$ and $\\rho _{AC}$ , when properly rescaled, converge in moments, jointly, to a pair of non-commutative random variables having free cumulants as in (REF ): $N_DN(\\rho _{AB},\\rho _{AC}) \\rightarrow (x_{AB},x_{AC})$ .",
"This allows one to compute the asymptotic value of any correlation function involving $\\rho _{AB}$ and $\\rho _{AC}$ .",
"For example, the rescaled overlap between the matrices converges to $\\lim _{N \\rightarrow \\infty } N_A N \\langle \\rho _{AB} , \\rho _{AC} \\rangle = \\frac{1}{c^2} \\kappa (x_{AB}, x_{AC}) = 1+ \\frac{1}{cm^2}.$ The computation above should be compared with the similar limit from the balanced regime of Section REF , where the two marginals were uncorrelated: $\\lim _{N \\rightarrow \\infty } N_A N \\langle \\rho _{AB} , \\rho _{AC} \\rangle =0$ ."
],
[
"The general multipartite case",
"In this section we consider the more general situation of a random Wishart tensor defined on a Hilbert space which is factorized in an arbitrary number of factors.",
"The section consists of three parts: we first derive the general, non-asymptotic mixed moment formula, and then consider two asymptotic regimes: the balanced regime, where all tensor factors have the same dimension, and the unbalanced regime, where some of the tensor factors (the ones corresponding to the “moving legs”) are being kept fixed.",
"In the balanced case, we prove that the marginals are asymptotically free (Proposition REF in the Wishart setting and Theorem REF in quantum information language), while in the unbalanced case, we show in an example that it is not possible to factorize the expression of the mixed cumulant functions over the cycles of the non-crossing partitions.",
"Indeed they depend more finely on the structure of the non-crossing partitions.",
"This implies that we cannot give an expression for the mixed free cumulants.",
"However, we expect that this situation can be dealt with in the framework of free probability with amalgamation.",
"Such results will be presented in a following paper.",
"We consider complex tensor $X$ of size $N_1\\times N_2\\times \\cdots \\times N_n$ , an un-normalized quantum state in the said Hilbert space $\\mathbb {C}^{N_1} \\otimes \\cdots \\otimes \\mathbb {C}^{N_2}$ .",
"The density matrix of the corresponding pure state is $X\\otimes X^*$ , the (un-normalized) unit rank projection on the space $\\mathbb {C} X$ .",
"For a given set $I\\subset \\lbrace 1,\\ldots , n\\rbrace $ , we denote $\\widehat{I} = \\lbrace 1,\\ldots ,n\\rbrace \\setminus I $ , and define the reduced density matrix as the tensor $X._{\\widehat{I}}\\bar{X}$ obtained by summing, for each $i\\in \\widehat{I}$ , the index of position $i$ of $X$ with the index of position $i$ of $\\bar{X}$ , $X._{\\widehat{I}}\\bar{X} =[\\mathrm {id}_I \\otimes \\mathrm {Tr}_{\\widehat{I}}](XX^*)$ .",
"This partial contraction of two tensors can also be understood as the matrix $[X._{\\widehat{I}}\\bar{X}]$ , whose first (resp.",
"second) sub-index of position $j\\in I$ is the free-index of position $j$ of $X$ (resp.",
"$\\bar{X}$ ).",
"For instance, for $n=4$ , choosing $\\widehat{I}=\\lbrace 3,4\\rbrace $ , $[X\\cdot _{ 3,4}\\bar{X}]_{i_1,i_2\\, ;\\, i^{\\prime }_1,i^{\\prime }_2} = \\sum _{i_3=1}^{N_3}\\sum _{i_4=1}^{N_4}X_{i_1,i_2,i_3,i_4} \\bar{X}_{i^{\\prime }_1,i^{\\prime }_2, i_3,i_4}.$ There is a canonical one-to-one correspondence which maps $I$ to $\\lbrace 1,\\ldots ,\\vert I \\vert \\rbrace $ while preserving the ordering of natural integers.",
"We denote $\\mathcal {S}_{\\vert I \\vert }$ the set of permutations of $\\vert I \\vert $ elements.",
"In the following, we implicitly make use of these canonical bijections when saying that a permutation $\\sigma \\in \\mathcal {S}_{\\vert I \\vert }$ acts on $I$ and has $I^{\\prime }$ of same cardinality as an image.",
"For instance if $I=\\lbrace A,B,C\\rbrace $ and $I^{\\prime }=\\lbrace A,C,E\\rbrace $ , the identity $\\operatorname{id}:I\\rightarrow I^{\\prime }$ is understood as the map $A\\rightarrow A, B\\rightarrow C, C\\rightarrow E $ .",
"Given $\\mathcal {I}\\in \\mathbb {N}$ , a permutation $\\sigma \\in \\mathcal {S}_\\mathcal {I}$ , and two matrices $P$ and $Q$ , whose two indices have $\\mathcal {I}$ sub-indices, we define the product $P\\cdot _\\sigma Q$ as the twisted contraction $\\bigl (P\\cdot _\\sigma Q \\bigr )_{i_1,\\cdots ,i_\\mathcal {I}\\, ;\\, i^{\\prime }_1,\\cdots , i^{\\prime }_\\mathcal {I}} = \\sum _{\\begin{array}{c}{j_1, \\cdots , j_\\mathcal {I}}\\\\{j^{\\prime }_1, \\cdots , j^{\\prime }_\\mathcal {I}}\\end{array}} \\prod _{b=1}^\\mathcal {I}\\delta _{j^{\\prime }_b}^{j_{\\sigma (b)}} P_{i_1,\\cdots ,i_\\mathcal {I}\\, ;\\, j_1,\\cdots j_\\mathcal {I}}Q_{j^{\\prime }_1,\\cdots , j^{\\prime }_{\\mathcal {I}}\\, ;\\, i^{\\prime }_1,\\cdots ,i^{\\prime }_\\mathcal {I}}.$ We define the associated trace $\\operatorname{Tr}\\cdot _{\\sigma }$ accordingly.",
"We are interested in computing expectations of the form $\\mathbb {E}\\operatorname{Tr}\\cdot _{\\sigma _p} [X._{\\widehat{I}_p}\\bar{X}]\\cdot _{\\sigma _{p-1}}\\,\\ldots \\, \\cdot _{\\sigma _3} [X._{\\widehat{I}_3}\\bar{X}] \\cdot _{\\sigma _2} [X._{\\widehat{I}_2}\\bar{X}] \\cdot _{\\sigma _1}[X._{\\widehat{I}_1}\\bar{X}],$ for some integer $p$ , some non-necessarily distinct sets $I_a$ which all have the same number of elements $\\vert I_a\\vert = \\mathcal {I}$ , and some permutations $\\sigma _a\\in \\mathcal {S}_\\mathcal {I}$ , (with our convention, $\\sigma _a:I_a \\rightarrow I_{a+1}$ , and $I_{p+1}=I_1$ ).",
"We denote $W_{I} = [X._{\\widehat{I}}\\bar{X}]$ , so that the objects under focus are rewritten as $\\mathbb {E}\\operatorname{Tr}_{\\mathbf {\\sigma }} W_{\\mathbf {f}} (\\lbrace N_i\\rbrace ) =\\mathbb {E}\\operatorname{Tr}\\cdot _{\\sigma _p} W_{I_p}\\cdot _{\\sigma _{p-1}}\\,\\ldots \\, \\cdot _{\\sigma _3} W_{I_3} \\cdot _{\\sigma _2} W_{I_2} \\cdot _{\\sigma _1}W_{I_1},$ where we respectively denoted $\\mathbf {f}$ and $\\mathbf {\\sigma }$ the ordered lists $\\mathbf {f}=[I_1,\\ldots , I_p]$ of $p$ subsets of $\\llbracket 1,n \\rrbracket $ , and $\\mathbf {\\sigma }=[\\sigma _1,\\ldots , \\sigma _p]$ of $p$ permutations in $\\mathcal {S}_\\mathcal {I}$ .",
"Note that depending on the $\\lbrace I_a\\rbrace $ and the $\\lbrace \\sigma _a\\rbrace $ , all choices are not possible for $\\lbrace N_1, \\ldots , N_n\\rbrace $ .",
"In general the above defined object is always meaningful when $N_1=\\cdots =N_n$ , but interesting cases can be considered for specific $\\lbrace I_a\\rbrace $ and $\\lbrace \\sigma _a\\rbrace $ .",
"To highlight this, we separate for a given $\\mathbf {f}$ , the colors which are traced for every $W_{I_a} = [X._{\\widehat{I}_a}\\bar{X}]$ , which, without loss of generality, we can suppose to be the colors from $r$ to $n$ , $\\lbrace r,\\ldots , n\\rbrace = \\bigcap _{a=1}^p \\widehat{I}_a.$ We also suppose that the colors from 1 to $l<r$ are the common fixed points of all the $\\lbrace \\sigma _a\\rbrace $ , $\\forall i \\in \\llbracket 1, l\\rrbracket ,\\ \\forall a \\in \\llbracket 1, p\\rrbracket ,\\quad \\sigma _a(i) = i,$ where we use $\\llbracket \\cdot , \\cdot \\rrbracket $ to denote integer intervals, see (REF ).",
"For each $a\\in \\llbracket 1,p\\rrbracket $ , among the colors $\\llbracket l+1 , r-1 \\rrbracket $ , the colors $\\widehat{I}_a \\setminus \\llbracket r, n\\rrbracket = \\widehat{I}_a \\cap \\llbracket l+1,r-1\\rrbracket $ are traced, and $\\sigma _a$ only acts non-trivially on the colors $J_a = I_a \\setminus \\llbracket 1, l\\rrbracket = I_a \\cap \\llbracket l+1,r-1\\rrbracket ,$ whose cardinal we denote $k = \\mathcal {I}- l.$ Note that the sets $J_a$ generalize the color $j(a)$ of Section REF .",
"In the 4-partite case $ABCD$ we considered in Section REF , we had $l=1$ (which corresponded to color $A$ ), $r=4$ (which corresponded to color $D$ ), and $k=1$ .",
"The only non-trivial action of the permutations $\\sigma $ was either $J_a = \\lbrace 2\\rbrace \\equiv \\lbrace B\\rbrace $ or $J_a = \\lbrace 3\\rbrace \\equiv \\lbrace C\\rbrace $ , color which was denoted by $j(a)$ .",
"The notations in the general case are illustrated in Fig.",
"REF .",
"Figure: A typical sample of an expectation ().",
"The colors linked by red edges belong to the corresponding J a J_as, while the blue edges link colors in the I a ∖〚1,l〛I_a \\setminus \\llbracket 1, l\\rrbracket .Before moving on to the moment computation, let us point out that the data defining the moment $(\\mathbf {\\sigma },\\mathbf {f})$ could be replaced by a single list of ordered subsets of $\\llbracket 1, n \\rrbracket $ ."
],
[
"Exact expression for the moments",
"Proposition 4.1 We suppose that $N_{l+1} = \\cdots = N_{r-1} = N_J$We put independent $N_i$ 's for the colors $i\\in \\llbracket 1,n\\rrbracket $ which are fixed points of all the initial wirings or which are always traced, and put a common $N_J$ for the others.",
"We stress however that more general cases might possibly be considered, if the colors can be separated into two sets if $\\llbracket l+1,r-1\\rrbracket = K_1\\sqcup K_2$ , such that the support of any orbit is either included in $K_1$ , or in $K_2$ .",
"We may then choose a different $N_{J_1}$ and $N_{J_2}$ for colors in $K_1$ and $K_2$ .. Then, with the previous notations, $\\mathbb {E}\\operatorname{Tr}_{\\mathbf {\\sigma }} W_{\\mathbf {f}} (\\lbrace N_i\\rbrace ) = \\sum _{\\alpha \\in \\mathcal {S}_p}\\, \\prod _{i=1}^l N_i^{\\#(\\gamma \\alpha )}\\, \\prod _{j=r}^n N_j^{\\#\\alpha }\\, N_J^{L(\\mathbf {f}, \\mathbf {\\sigma }, \\alpha )},$ in which $\\gamma =(12 \\cdots p)$ , and $L(\\mathbf {f}, \\mathbf {\\sigma }, \\alpha )$ is a combinatorial function defined in (REF ).",
"As in Section REF , the moments will be expressed as a sum over Wick wirings $\\alpha \\in \\mathcal {S}_p$ – or equivalently over maps in $\\mathbb {M}_p$ – of some weight.",
"Each loop in the box representation contributes to this weight with a factor $N_i$ .",
"We want to describe these loops as orbits as was done in Def.",
"REF , i.e.",
"in terms of the cycles of some permutations acting on the product of $r-l-1$ copies of $\\llbracket 1, p\\rrbracket $ , of the form $[p]^i=\\lbrace (1,i), \\ldots , (p,i)\\rbrace $ for each color $i$ in $\\llbracket l+1, r-1\\rrbracket $ .",
"As detailed several times along this paper, to every permutation $\\alpha \\in \\mathcal {S}_p$ corresponds a combinatorial map $\\mathcal {M}$ with one black vertex and $\\#\\alpha $ white vertices, and whose $p$ labeled edges are disposed from 1 to $p$ counterclockwise around the black vertex and correspond to matrices $W_{I_a} = [X._{\\widehat{I}_a}\\bar{X}]$ .",
"The edge labeled $a$ therefore carries the set $I_a$ of $\\mathcal {I}$ colors, $l$ of which belong to every edge.",
"Let us take a closer look at the orbits in the map formulation.",
"As was previously the case in Section REF , an orbit which has color $i$ , when it arrives on a white vertex, leaves this vertex on the next edge also carrying the color $i$ , counterclockwise.",
"When an orbit of color $i$ arrives on a black vertex from the edge labeled $a$ , it goes to the following edge around that vertex counterclockwise, labeled $a+1$ , but changes color to $\\sigma _a(i)$ .",
"For the colors 1 to $l$ , these are actually the usual faces of the map, $F(\\mathcal {M})$ , as was the case for color $A$ in Section REF .",
"The colors $\\llbracket 1,l \\rrbracket $ therefore contribute with a factor $\\prod _{i=1}^l N_i^{F(\\mathcal {M})} = \\prod _{i=1}^l N_i^{\\#(\\gamma \\alpha )}$ .",
"We may therefore as well forget these colors, and label the edges with the sets $J_a$ instead of $I_a$ (as was done for $j(a)$ in Sec.",
"REF ).",
"The colors which belong to the sets $\\widehat{I}_a$ will be taken care of further, and we now focus on the remaining colors, which belong to at least one set $J_a$ .",
"In this general case, their behavior might differ from the particular case previously treated in Section REF .",
"Indeed, for colors in $\\llbracket r+1, l-1 \\rrbracket $ , a given edge might appear several times on the same orbit (at most $k$ times).",
"Therefore the orbits cannot in general be defined as the cycles of a permutation of the $p$ edges, or equivalently the faces of a combinatorial map without colors, which was the key point in Section REF .",
"In order to bypass this difficulty, we define a labeling for $pk$ copies of the edges, one per each couple $(a,i)$ , where $a\\in \\llbracket 1, p\\rrbracket $ labels the edge, and $i \\in J_a$ is a color which is neither traced, nor a fixed point of all the $\\sigma _b$ .",
"On these $pk$ elements, we define the following permutation $\\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }} : (a, i) \\mapsto (a+1, \\sigma _a(i)).$ We label each cycle of the permutation starting from the smallest color of the edge of smallest label in the cycle.",
"For instance, if $J_1=\\lbrace B,C\\rbrace $ , $J_2=\\lbrace B,D\\rbrace $ , $J_3=\\lbrace C,D\\rbrace $ , $\\sigma _1=\\mathrm {id}$ , $\\sigma _2=\\mathrm {id}$ , and $\\sigma _3$ is the transposition in $\\mathcal {S}_2$ , then $\\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }}=\\Bigl ( (1,B), (2,B), (3, C), (1, C), (2,D), (3,D) \\Bigr ),$ and if $J_1=\\lbrace B,C\\rbrace $ , $J_2=\\lbrace B,D\\rbrace $ , $J_3=\\lbrace B,C\\rbrace $ , $J_4=\\lbrace C,D\\rbrace $ , $\\sigma _1=\\mathrm {id}$ , $\\sigma _2=\\mathrm {id}$ , and $\\sigma _3$ and $\\sigma _4$ are the transposition in $\\mathcal {S}_2$ , then $\\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }}=\\Bigl ( (1,B), (2,B), (3, B), (4, D)\\Bigr )\\Bigl ((1,C), (2,D), (3,C) , (4,C) \\Bigr ).$ For colors in $\\llbracket l+1, r-1 \\rrbracket $ , the behavior of the faces around white vertices is similar to Section REF , and the duplication operation of Figure REF generalizes locally: performing the following duplication operation (illustrated for $\\mathcal {I}=3$ ) on every white vertex does not change locally the incident external orbit, and removes the color conditions on the white vertices.",
"Figure: Local duplication of white vertices.This would precisely create one copy of an edge for each one of the $kp$ couples $(a,i)$ .",
"More precisely, if $i\\in J_a$ , denoting $\\alpha _i(a)$ the first edge following $a$ around the white endpoint of $a$ and containing color $i$ , $\\alpha _i(a) = \\alpha ^q(a),\\quad \\text{where} \\quad q=\\min \\lbrace s\\in \\mathbb {N}^\\ast \\mid i \\in J_{\\alpha ^s(a)}\\rbrace ,$ ($q$ depends on $\\alpha $ , $i$ , and $a$ ) then the permutation defining the resulting white vertices is $\\alpha _{\\mathbf {f}} : (a, i) \\mapsto (\\alpha _i(a), i).$ We can now define a (non-necessarily connected) combinatorial map $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }} = (\\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }}, \\alpha _{\\mathbf {f}})$ from the two permutations $\\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }}$ and $\\alpha _{\\mathbf {f}}$ , acting on $\\llbracket 1, p\\rrbracket \\times \\llbracket 1, k\\rrbracket $ .",
"See the examples in Fig.",
"REF and REF .",
"Figure: The map ℳ 𝐟,σ \\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }} for an example in the case ().",
"The white vertex in the original map ℳ\\mathcal {M} is tripled into 3 white vertices having half-edges of the same color attached.",
"The order of the half-edges around the black vertex is given by the permutation Γ 𝐟,σ \\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }}.Figure: The maps ℳ 𝐟,σ \\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }} for two examples in the case ().Among the orbits which act on colors in $\\llbracket l+1, n \\rrbracket $ , one has to take into account the orbits which are entirely included in the white vertices of $\\mathcal {M}$ .",
"The number of such orbits contained in a given vertex is $l$ , plus the number of colors in $\\llbracket l+1, r-1\\rrbracket $ which do not appear on any edge.",
"The total number of the latter can be expressed as $(r-l-1)\\bigl (V(\\mathcal {M}) - 1\\bigr ) - \\bigl ( V(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) - V_\\bullet \\bigr (\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }})),$ where we have denoted the number of black vertices of $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ as $V_\\bullet \\bigr (\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) =V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma }) = \\#\\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }}.$ This is an integer between 1 and $k$ which does not depend on $\\alpha $.",
"These orbits therefore contribute with a factor $ \\prod _{i=r}^n N_i^{V(\\mathcal {M}) - 1} N_J^{(r-l-1)(V(\\mathcal {M}) - 1) - ( V(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) - V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma }))},$ where we recall thatNotice also that if one wants to make explicit the analogy between $\\mathcal {M}$ and $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ in this expression, one can rewrite it as $\\prod _{i=r}^n N_i^{V(\\mathcal {M}) - 1} N_J^{(r-l-1)(V(\\mathcal {M}) - V_\\bullet (\\mathcal {M})) - ( V(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) - V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma }))}$ or $\\prod _{i=r}^n N_i^{V(\\mathcal {M}) - 1} N_J^{(r-l-1)(V(\\mathcal {M}) - \\#\\gamma ) - ( V(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) - \\#\\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }})}$ $V(\\mathcal {M}) - 1 = \\#\\alpha $ .",
"The remaining orbits run along at least one edge.",
"Because the permutations $\\Gamma _{\\mathbf {f},\\mathbf {\\sigma }}$ and $\\alpha _{\\mathbf {f}}$ encode precisely the way they behave locally around the black and the white vertices respectively, their total number is precisely given by the number of faces of the map $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ , $F(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) = \\#(\\Gamma _{\\mathbf {f},\\mathbf {\\sigma }}\\, \\alpha _{\\mathbf {f}}),$ hence contributing with a factor $N_J^{F(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) }$ .",
"Putting all of this together, we have shown that $\\mathbb {E}\\operatorname{Tr}_{\\mathbf {\\sigma }} W_{\\mathbf {f}} (\\lbrace N_i\\rbrace ) = \\sum _{\\mathcal {M}\\in \\mathbb {M}_p}\\, \\prod _{i=1}^l N_i^{F(\\mathcal {M})}\\, \\prod _{j=r}^n N_j^{V(\\mathcal {M})-1}\\, N_J^{L(\\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})},$ where we have defined $L(\\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M}) := F(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) + (r-l-1)\\bigl (V(\\mathcal {M}) - 1\\bigr ) - \\bigl ( V(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) - V_\\bullet \\bigr (\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }})).$ Identifying $L(\\mathbf {f}, \\mathbf {\\sigma }, \\alpha ) = L(\\mathbf {f}, \\mathbf {\\sigma }, (\\gamma , \\alpha ) )$ , this concludes the proof.",
"In the above proof we defined the companion map $\\mathcal {M}_{\\mathbf {f},\\mathbf {\\sigma }}$ to $\\mathcal {M}$ .",
"This companion map is a generalization of the previous companion map $\\mathcal {M}_f$ introduced in section REF .",
"Since this is an important object, we recall its construction in the following definition Definition 4.2 For all integer $p$ , we associate to the triplet $(\\mathcal {M},\\mathbf {f}, \\mathbf {\\sigma })$ , with $\\mathcal {M}$ a combinatorial map in $\\mathbb {M}_p$ , $\\mathbf {f}$ the list of colors of the edges of $\\mathcal {M}$ and $\\mathbf {\\sigma }$ the list of permutations labeling the corners of the black vertex of $\\mathcal {M}$ , a combinatorial map $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ called unfolded map from the following data: The set $E_{\\mathbf {f}}:=\\bigl \\lbrace (a,i)\\bigr \\rbrace _{a \\in \\llbracket 1,p \\rrbracket , \\, i\\in J_a}$ is the set of edges.",
"The permutations $\\alpha _{\\mathbf {f}}$ and $\\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }}$ , where $\\alpha _{\\mathbf {f}}$ is the permutation that defines the white vertices after local duplication, and $\\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }}$ defines the black vertices of $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ .",
"Denoting $\\alpha _i(a) = \\alpha ^q(a),\\quad \\text{where} \\quad q=\\min \\lbrace s\\in \\mathbb {N}^\\ast \\mid i \\in J_{\\alpha ^s(a)}\\rbrace , \\nonumber $ the permutation $\\alpha _{\\mathbf {f}}$ is defined as $\\alpha _{\\mathbf {f}}:& \\ E_{\\mathbf {f}} \\rightarrow E_{\\mathbf {f}} \\nonumber \\\\& (a,i) \\mapsto (\\alpha _i(a),i), \\nonumber $ and $\\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }}$ is defined as $\\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }}:& \\ E_{\\mathbf {f}} \\rightarrow E_{\\mathbf {f}} \\nonumber \\\\&(a,i)\\mapsto (a+1,\\sigma _a(i)).",
"\\nonumber $ If we compare with Definition REF then we notice that we do not require that the group $\\langle \\alpha _{\\mathbf {f}}, \\Gamma _{\\mathbf {f},\\mathbf {\\sigma }} \\rangle $ acts transitively on $E_{\\mathbf {f}}$ , this is because $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ can be disconnected.",
"Remark also that each white vertex of $\\mathcal {M}$ is duplicated into $k=|J_a|$ white vertices of $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ if all the incident edges are labeled by the same set $J_a$ ; and into strictly more than $k$ white vertices if at most $k-1$ colors are common to all the edges.",
"We therefore define the quantity $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M}) := V(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) - V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma }) - k(V(\\mathcal {M}) - 1),$ which vanishes if and only if all the edges incident to a given white vertex of $\\mathcal {M}$ share the same color set $J_a$ , and is positive otherwise.",
"Moreover, if the number of black vertices is $V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma })$ , the number of connected components $K$ of a given map $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ is an integer in $\\lbrace 1, \\ldots , V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma })\\rbrace $ , so that we define $\\Sigma (\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) = V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma }) - K(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}),$ which is an integer between 0 and $V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma })-1$ .",
"Theorem 4.3 With the previous notations, the mixed moments of the marginals $\\lbrace W_{I_a}\\rbrace $ are expressed exactly using a sum over combinatorial maps $\\mathbb {E}\\operatorname{Tr}_{\\mathbf {\\sigma }} W_{\\mathbf {f}} = \\sum _{\\mathcal {M}\\in \\mathbb {M}_p}\\, \\prod _{i=1}^l N_i^{2+p -V(\\mathcal {M}) - 2g(\\mathcal {M})} \\prod _{j=r}^n N_j^{V(\\mathcal {M})-1} N_J^{kp + V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma }) + (r-l-2k-1)(V(\\mathcal {M})-1)-\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})},$ in which we have denoted $\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M}) = 2\\bigl (g(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }})+ \\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M}) + \\Sigma (\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }})\\bigr ),$ where $g(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }})\\ge 0$ is the genus of the map $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ , and $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M})\\ge 0$ and $\\Sigma (\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }})\\ge 0$ have been defined above.",
"Before starting with the proof, notice that the quantity $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M})$ seems to penalize the mixed cumulants in the large $N_J$ regime, if we trust the heuristic relating monochromatic white vertices (i.e.",
"white vertices incident to edges of the same color) to non-mixed cumulants.",
"This is an indication for freeness at large $N_J$ .",
"The only thing we need to prove is that $L(\\mathbf {f}, \\mathbf {\\sigma }, \\alpha ) = kp + V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma }) + (r-l-2k-1)(V(\\mathcal {M})-1)-\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})$ , which follows from the Euler characteristics of the map $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ , $2K(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) - 2g(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) = F(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) - kp + V(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}).$ We conclude using Proposition REF and the definitions of $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M})$ and $\\Sigma (\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }})$ .",
"Remark 4.4 In the case where $k=0$ , the result above degenerates, and we recover the classical Marčenko-Pastur result from Proposition REF (see also equation (REF ) for the combinatorial map approach).",
"Indeed, in this case $r=l+1$ , and the factor with $N_J$ is trivially equal to 1; moreover, the quantities $\\mathbf {\\sigma }$ and $\\mathbf {f}$ are trivial and do not play any role.",
"The result reads $\\mathbb {E}\\operatorname{Tr}W_n = \\sum _{\\mathcal {M}\\in \\mathbb {M}_p}\\, \\prod _{i=1}^l N_i^{2+p -V(\\mathcal {M}) - 2g(\\mathcal {M})} \\prod _{j=r}^n N_j^{V(\\mathcal {M})-1}.$ If we denote $N_{tot}:=\\prod _{i=1}^l N_i$ and consider the scaling where $N^{\\prime }_{tot}:= \\prod _{j=r}^n N_j \\sim c N_{tot}$ , we obtain $\\mathbb {E}\\operatorname{Tr}W_n = (1+o(1)) N_{tot}^{1+p}\\sum _{\\mathcal {M}\\in \\mathbb {M}_p} N_{tot}^{-2g(\\mathcal {M})}c^{V(\\mathcal {M})-1}.$ Assuming moreover $N_{tot} \\rightarrow \\infty $ , the limit selects planar maps, and we obtain (REF ): $\\lim _{N_{tot} \\rightarrow \\infty }\\mathbb {E}\\frac{1}{N_{tot}}\\operatorname{Tr}\\left(\\frac{W_n}{N_{tot}}\\right)^p = \\sum _{\\mathcal {M}\\in \\mathbb {M}_p^0}c^{V(\\mathcal {M})-1}.$ Remark 4.5 The first non-trivial case corresponds to $k=1$ , where just one of the $n$ legs of the tensors can “move around”.",
"In this case, it is clear that the permutations $\\mathbf {\\sigma }$ do not play any role, and $\\mathbf {f}$ is just a function from $\\llbracket 1, p\\rrbracket $ to the set of tensor legs, selecting for each tensor the leg which “moves around”.",
"We have $\\mathbf {f}:\\llbracket 1, p\\rrbracket \\rightarrow \\llbracket r \\rrbracket ]$ , and, since $\\mathbf {\\sigma }$ is trivial, $V_\\bullet (\\mathbf {f})=1$ .",
"Assuming, to keep things simple, that all the vector spaces have dimension $N$ , the result of Theorem REF reads $\\mathbb {E}\\operatorname{Tr}W_\\mathbf {f}&= \\sum _{\\mathcal {M}\\in \\mathbb {M}_p} N^{(r-1)[2+p-V(\\mathcal {M})-2g(\\mathcal {M})]}N^{V(\\mathcal {M})-1}N^{p + V_\\bullet (\\mathbf {f}) + (r-2)(V(\\mathcal {M})-1)-\\tilde{L} ( \\mathbf {f},\\mathcal {M})}\\\\&= N^{r(p+1)}\\sum _{\\mathcal {M}\\in \\mathbb {M}_p} N^{-2(r-1)g(\\mathcal {M}) - \\tilde{L}(\\mathbf {f},\\mathcal {M})}.$ In order to find the dominating contributions, we have to identify the maps $\\mathcal {M}\\in \\mathbb {M}_p$ which cancel both $g(\\mathcal {M})$ and $\\tilde{L}(\\mathbf {f},\\mathcal {M})$ .",
"The first condition corresponds to $\\mathcal {M}$ being planar, while the second one is equivalent to the cancellation of the three quantities $g(\\mathcal {M}_{\\mathbf {f}})$ ,$\\Delta _{\\mathbf {f}}(\\mathcal {M})$ , $\\Sigma (\\mathcal {M}_{\\mathbf {f}})$ from (REF ).",
"We focus on the $\\Delta $ quantity: it follows from (REF ) that this quantity is zero iff all the edges incident to a white vertex in $\\mathcal {M}$ have the same color given by $f$ ; in other words, the partition introduced by the white vertices of $\\mathcal {M}$ on $\\llbracket 1, p\\rrbracket $ has to be smaller than the partition $\\mathbf {f}$ introduces on the same set.",
"We claim now that the conditions $g(\\mathcal {M}) = 0$ and $\\Delta _{\\mathbf {f}}(\\mathcal {M})=0$ imply that the other two quantities appearing with a negative sign in the exponent of $N$ cancel.",
"Indeed, $\\Sigma (\\mathcal {M}_{\\mathbf {f}})=0$ , since $V_\\bullet (\\mathbf {f})=1$ .",
"Since $\\Delta _{\\mathbf {f}}(\\mathcal {M})=0$ , it also follows that $\\mathcal {M}_{\\mathbf {f}}=\\mathcal {M}$ , and thus $g(\\mathcal {M}_{\\mathbf {f}})=0$ .",
"To summarize, we reach the same conclusion as in Theorem REF , see equation (REF ): $\\lim _{N \\rightarrow \\infty } N^{-r(p+1)} \\mathbb {E}\\operatorname{Tr}W_\\mathbf {f}= |\\mathbb {M}_p^{0,\\mathbf {f}}|,$ where $\\mathbb {M}_p^{0,\\mathbf {f}} := \\lbrace \\mathcal {M}\\in \\mathbb {M}_p \\, : \\, &\\mathcal {M}\\text{ is planar and $\\mathbf {f}$ assigns the same color to the edges}\\\\& \\text{incident to the white vertices of $\\mathcal {M}$}\\rbrace .$"
],
[
"The balanced asymptotical regime",
"As a first step, we focus on the limit where $N_1, \\ldots , N_n$ all grow to infinity, while the ratio between $N_1\\ldots N_l$ and $N_r \\ldots N_n$ converges to a fixed constant at infinity.",
"More precisely, we can consider for instance $\\forall i \\in \\llbracket 1,l \\rrbracket , N_i=N_J=N$ while $\\forall j \\in [\\![r,n]\\!",
"], \\ \\textrm {lim}_{N\\rightarrow \\infty } N_j/N = c^{\\frac{1}{n-r+1}}$ or pick $j \\in [\\![r,n]\\!",
"]$ such that $N_j \\sim cN$ at infinity, while $N_i=N$ for $i\\ne j$ .",
"In that case, Theorem REF writes $\\mathbb {E}\\operatorname{Tr}_{\\mathbf {\\sigma }} W_{\\mathbf {f}} = N^{(k+l)(p-2) + l + n + V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma })}\\, \\sum _{\\mathcal {M}\\in \\mathbb {M}_p}\\, h(N)^{V(\\mathcal {M})-1} N^{ - 2lg(\\mathcal {M}) -\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M}) -(2k+2l - n)(V(\\mathcal {M}) - 2)},$ where $h(N)$ is an arbitrary function such that $h(N) \\rightarrow c$ when $N\\rightarrow + \\infty $ .",
"Notice that the factor $N^{V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma })}$ is out of the sum since, as already emphasized earlier, it only depends on $\\mathbf {f},\\mathbf {\\sigma }$ .",
"The terms $g(\\mathcal {M})$ , $\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})$ and $V(\\mathcal {M}) - 2$ are all non-negative, so we have three cases, depending on the sign of $2k+2l - n$ .",
"If $2k+2l - n<0$ , the maps $\\mathcal {M}$ which survive in the large $N$ limit are those which satisfy $g(\\mathcal {M})=0\\ (\\text{if } l\\ne 0),\\quad \\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})=0,\\quad \\text{and}\\quad V(\\mathcal {M}) = p + 1.$ Since there is a unique solution to this system, given by the only tree in $\\mathbb {M}_p$ , in that case, in the large $N$ limit, $\\lim _{N\\rightarrow \\infty } \\frac{1}{N^{(n-k-l)p + l + V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma }) }}\\, \\mathbb {E}\\operatorname{Tr}_{\\mathbf {\\sigma }} W_{\\mathbf {f}} = c^{p}.$ This case is therefore trivial, and we do not consider it.",
"In the following, we first consider the case $2k+2l - n=0$ in Subsection REF , and then the case $2k+2l - n>0$ in Subsection REF ."
],
[
"In this section we study the case where $2k+2l - n$ vanishes, or equivalently $\\mathcal {I}= l + k = \\frac{n}{2},$ the number of colors $n$ is even, and for each edge, precisely half the colors are traced.",
"The equation (REF ) rewrites as $\\mathbb {E}\\operatorname{Tr}_{\\mathbf {\\sigma }} W_{\\mathbf {f}} = N^{(k+l)p + l + V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma }) }\\sum _{\\mathcal {M}\\in \\mathbb {M}_p}\\, h(N)^{V(\\mathcal {M})-1} N^{ - 2lg(\\mathcal {M}) -\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})},$ where here $ h(N)$ is an arbitrary function whose limit at infinity is $ c$ .",
"Large $N$ limit, case $l\\ne 0$ .",
"In the large $N$ limit, the sum restricts to planar maps, which are such that edges incident to a given white vertex have the same set of colors $I_a$ , and for which the map $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ is planar and has as many connected components as black vertices.",
"We define the set of “free\" maps $\\mathbb {M}_{\\mathbf {f}, \\mathbf {\\sigma }}^{(0) \\text{free}} := \\bigl \\lbrace \\mathcal {M}\\in \\mathbb {M}_p \\mid g(\\mathcal {M}) = 0, g(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) = 0, \\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M})=0, \\text{ and }\\Sigma (\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) = 0\\bigr \\rbrace .$ The following result is just a consequence of Theorem REF .",
"Proposition 4.6 In the large $N_1,\\ldots , N_n$ regime, if $ l + k = \\frac{n}{2}$ and with the previous notations, $\\lim _{N\\rightarrow \\infty } \\frac{1}{N^{\\frac{n}{2} p + l + V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma }) }}\\, \\mathbb {E}\\operatorname{Tr}_{\\mathbf {\\sigma }} W_{\\mathbf {f}} = \\sum _{\\mathcal {M}\\in \\mathbb {M}_{\\mathbf {f}, \\mathbf {\\sigma }}^{(0) \\text{free}}} c^{\\, V(\\mathcal {M}) - 1}.$ Note that since $V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma })$ might be smaller than $k$ , the moments corresponding to $\\mathbf {\\sigma }$ and $\\mathbf {f}$ might have a weaker scaling in $N$ than the usual $N^{\\frac{n}{2} (p + 1)}$ .",
"General case including $l=0$ .",
"In the case where $l=0$ , the map $\\mathcal {M}$ is a priori non necessarily planar in the large $N$ limit.",
"In this paragraph, we show that maps with vanishing $\\tilde{L}$ are in fact necessarily planar.",
"As a consequence, the system defining the “free\" maps in (REF ) can in fact be reduced to $\\mathbb {M}_{\\mathbf {f}, \\mathbf {\\sigma }}^{(0) \\text{free}} = \\bigl \\lbrace \\mathcal {M}\\in \\mathbb {M}_p \\mid g(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) = 0, \\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M})=0, \\text{ and }\\Sigma (\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) = 0\\bigr \\rbrace ,$ thus extending the preceding result to the case $l=0$ .",
"Indeed, we have the following result Proposition 4.7 A map $\\mathcal {M}\\in \\mathbb {M}_p$ in a triplet $(\\mathcal {M}, \\mathbf {f}, \\mathbf {\\sigma })$ which satisfies $g(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) = 0 \\quad \\text{and} \\quad \\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M})=0,$ is necessarily planar: $g(\\mathcal {M}) = 0$ .",
"Relying on the following Lemma REF , the proposition is proven easily by noticing that, if $g(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) = 0$ , the genus $g(\\mathcal {M})$ is a non-negative integer which is smaller than one.",
"$\\Box $ Lemma 4.8 If $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M})=0$ , we have the following bound for the number of faces of the combinatorial map $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ , $F(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }})\\le k F(\\mathcal {M}), \\quad \\text{ and } \\quad g(\\mathcal {M}) \\le \\frac{g(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }})}{k} + \\frac{k-1 }{k}.$ Because $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M})=0$ , the orbits, though different from the faces, cover precisely the faces of $\\mathcal {M}$ .",
"The difference is that if a face starts on an edge $a\\in \\llbracket 1,p \\rrbracket $ with a color $i \\in J_a$ , after going around $\\mathcal {M}$ , it might come back for the first time on the same side of the edge $a$ for a color $j\\ne i$ .",
"In that case, it would follow the face of $\\mathcal {M}$ again a certain number of times, until it reaches again the same side of the edge $a$ , for the same color $i$ .",
"An orbit, which is projected to a face of $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ , is therefore a multiple of a face of $\\mathcal {M}$ .",
"Each side of each edge $a$ is used exactly once per color in $J_a$ .",
"If all the orbits go exactly once around the map, then each face of $\\mathcal {M}$ corresponds to $k$ orbits (and thus faces of $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ ), as they never pass twice on the same side of an edge, and an edge has $k$ colors which are not traced.",
"If not, there are less orbits, so that the first inequality is proven.",
"To prove the second inequality, we consider the Euler characteristics of $\\mathcal {M}$ , $2g(\\mathcal {M}) = 2 - V(\\mathcal {M}) + p - F(\\mathcal {M})$ .",
"Since $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M})=0$ , we replace $V(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) - V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma })= k(V(\\mathcal {M}) - 1)$ , $2kg(\\mathcal {M}) &= 2k - kV(\\mathcal {M}) + kp - kF(\\mathcal {M}) \\\\& = k - \\Bigl (V(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) - V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma })\\Bigr )+ kp - kF(\\mathcal {M}).$ Now using the Euler characteristics of $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ , $ kp - V(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) = 2g(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) + F(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) - 2K(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }})$ , we find that $2kg(\\mathcal {M}) = k + 2g(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) + F(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) - 2K(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) + V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma })- kF(\\mathcal {M}).$ Using the first inequality of (REF ), we are left with $2kg(\\mathcal {M}) \\le k + 2g(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) - 2K(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) + V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma }).$ We conclude using that $K(\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) \\ge 1$ and $V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma }) \\le k$ .",
"Remark 4.9 A consequence of Proposition REF is that keeping the $N_i$ finite for $i\\in \\llbracket 1,l\\rrbracket $ does not lead to more interesting behavior.",
"Indeed, a map with $\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})=0$ also has vanishing genus, so that the large $N$ limit also selects all the maps in $\\mathbb {M}_{\\mathbf {f}, \\mathbf {\\sigma }}^{(0) \\text{free}}$ , and we recover the results of the present case.",
"Application to the asymptotic freeness of permuted marginals of random Gaussian tensors.",
"We now consider an application of Theorem REF to a new situation, where we consider the set of all possible marginals of the tensor $X\\otimes X^*$ , with all possible permutations acting on half of the tensor legs.",
"More precisely, let $n$ be even, and put, for an $(n/2)$ -subset $I$ of $\\llbracket 1,n\\rrbracket $ , $W_I = [\\mathrm {id}_{I} \\otimes \\mathrm {Tr}_{\\llbracket 1,n\\rrbracket \\setminus I}](XX^*)$ , where $X \\in (\\mathbb {C}^N)^{\\otimes n}$ is a Gaussian tensor; note that we assume here, for the sake of simplicity, that all the Hilbert space dimensions are $N$ .",
"The general case of different Hilbert space dimensions, say $N_i \\sim c_i N$ for some constants $c_i >0$ and $N \\rightarrow \\infty $ , can be easily obtained.",
"Let us introduce, for every permutation $\\pi \\in \\mathcal {S}_{n/2}$ , $W_{I,\\pi } = P_\\pi W_I P_\\pi ^{-1} \\in \\mathcal {M}_{N^{n/2}}(\\mathbb {C})$ the matrix obtained by permuting the tensor legs of $W_I$ according to the permutation $\\pi $ .",
"For example, in the case $n=4$ , there are 12 possible matrices $W_{I,\\pi }$ .",
"We present in Figure REF two examples: $W_{\\lbrace 1,2\\rbrace ,(1)(2)} = W_{\\lbrace 1,2\\rbrace }$ and $W_{\\lbrace 1,3\\rbrace ,(12)} = F W_{\\lbrace 1,3\\rbrace } F$ , where $F \\in \\mathcal {U}_{d^2}$ is the flip operator, acting on simple tensors by $F(x \\otimes y) = y \\otimes x$ .",
"Figure: Diagrams for the permuted marginals W {1,2},(1)(2) W_{\\lbrace 1,2\\rbrace ,(1)(2)} and W {1,3},(12) W_{\\lbrace 1,3\\rbrace ,(12)}.This is therefore a particular case of the large $N_1,\\ldots , N_n$ regime, for specific permutations $\\sigma _a\\in \\mathbf {\\sigma }$ , which factorize as $\\forall a\\in \\llbracket 1,p\\rrbracket , \\qquad \\sigma _a = \\pi _{a+1}^{-1}\\pi _a.$ Note that the permutations $\\sigma _a$ might share some fixed points (recall that we denote by $l$ the number of these common fixed points, and we also put $k=n/2-l$ ).",
"Proposition REF applies, and the maps that contribute to the large $N$ limit are a subset of $\\mathbb {M}_{\\mathbf {f}, \\mathbf {\\sigma }}^{(0) \\text{free}}$ : in particular, they are planar, and the edges that are attached to the same white vertices all have the same set of colors $J_a$ .",
"Moreover, the hypothesis (REF ) has stronger consequences.",
"Indeed, we have the following lemmas.",
"Lemma 4.10 If the permutations in $\\mathbf {\\sigma }$ factorize as in (REF ), for any $\\mathcal {M}\\in \\mathbb {M}_p$ , the map $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ has precisely $k$ black vertices.",
"As a consequence, if $\\Sigma (\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }})=0$ , the maps $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ have $k$ connected components.",
"In the combinatorial maps setting, the picture is the following: when arriving to an edge, the color of an orbit changes by multiplying with $\\pi _a^{-1}$ , and when leaving the edge, one multiplies with the inverse permutation $\\pi _a$ .",
"Formally, since the black vertices of $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ correspond to the cycles of the permutation $\\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }}$ from (REF ), we show that $\\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }}$ has precisely $k$ cycles.",
"Indeed, we have $(a,i) \\xrightarrow{} (a+1, \\pi _{a+1}^{-1}\\pi _a(i)) \\xrightarrow{} (a+2, \\underbrace{\\pi _{a+2}^{-1}\\pi _{a+1}\\pi _{a+1}^{-1}\\pi _a(i)}_{\\pi _{a+2}^{-1}\\pi _a(i)})\\xrightarrow{} \\cdots \\xrightarrow{} (\\underbrace{a+p}_a, \\underbrace{\\pi _{a+p}^{-1}\\pi _a(i)}_i),$ for all $i \\in J_a$ , proving the claim (the addition on the first coordinate is done modulo $p$ ).",
"Lemma 4.11 If $\\mathcal {M}$ is such that $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M})=\\Sigma (\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }})=0$ , and the permutations in $\\mathbf {\\sigma }$ factorize as in (REF ), the edges incident to a given white vertex are all labeled by the same permutation $\\pi $ .",
"Consider two edges $a$ and $b$ incident to the same white vertex in $\\mathcal {M}$ .",
"As $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M})=0$ , they have the same set of colors $J_a=J_b$ .",
"Take a color $i\\in J_a$ .",
"In the unfolded map $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ , the white vertex corresponding to the color $i \\in J_a$ has attached to it, among others, the edges $(a,i)$ and $(b,i)$ .",
"The edge $(a,i)$ has its other end attached to the black vertex corresponding to the cycle containing $(a,i)$ in the permutation $\\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }}$ .",
"In order for the map $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ to have exactly $k$ connected components (see Lemma REF ), the edge $(b,i)$ must be connected to the same black vertex (otherwise, the aforementioned white vertex would be connected to two different black vertices, decreasing the number of connected components).",
"However, from the proof of Lemma REF , we see that the black vertex has attached to it the following edges: $\\lbrace (c,\\pi _c^{-1}\\pi _a(i))\\rbrace _{c \\in \\llbracket 1,p\\rrbracket }$ .",
"Hence, we must have $\\pi _b^{-1}\\pi _a(i) = i$ for all $i$ , which is the claim.",
"Lemma 4.12 If $\\mathcal {M}$ is such that $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M})=\\Sigma (\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }})=0$ , and the permutations in $\\mathbf {\\sigma }$ factorize as in (REF ), the unfolded map $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ consists of $k$ copies of the original map $\\mathcal {M}$ (when discarding the edge coloring).",
"From the proof of Lemma REF , $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ has $k$ connected components, and each one of them has one black vertex with $p$ incident edges, ordered from 1 to $p$ .",
"Discarding the edge coloring, the permutation defining the black vertex of each one of the connected components is therefore $\\gamma $ .",
"These $p$ edges are attached to white vertices in $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ .",
"Now since all the edges $a$ incident to some white vertex $v$ of $\\mathcal {M}$ have the same sets $J_a$ , each white vertex $v$ of $\\mathcal {M}$ corresponds to $k=|J_a|$ white vertices $\\lbrace v^i\\rbrace _{i \\in J_a}$ in the unfolded map $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ , with the same edges $(a, \\alpha (a), \\alpha ^2(a), \\cdots )$ attached to it, in the same order.",
"Therefore, discarding the edge coloring, the permutation defining the white vertices of a given connected component of $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ is just $\\alpha $ .",
"This proves that each connected component of $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ is isomorphic to $\\mathcal {M}$ .",
"We arrive now at one of the main results of this paper, the asymptotic freeness of all the (permuted) balanced marginals of a random multipartite quantum state.",
"Theorem 4.13 In the asymptotic setting described above, the family of random matrices $(W_{f,\\pi })_{f \\in \\binom{\\llbracket 1,n \\rrbracket }{n/2},\\, \\pi \\in \\mathcal {S}_{n/2}}$ converges, as $N \\rightarrow \\infty $ , to a family of $\\binom{n}{n/2}(n/2)!$ free random variables, each having a Marčenko-Pastur distribution of parameter 1.",
"At the level of moments, this reads $\\lim _{N \\rightarrow \\infty } \\mathbb {E}N^{-n(p+1)/2} \\operatorname{Tr}W_{f,\\pi } = \\sum _{\\mathcal {M}\\in \\mathbb {M}^{bal}_{f,\\pi }} 1,$ where $f$ is a word of $p$ $(n/2)$ -subsets of $\\llbracket 1,n \\rrbracket $ , $\\pi $ is a word of $p$ permutations and $\\mathbb {M}^{bal}_{f,\\pi } := \\lbrace \\mathcal {M}\\in \\mathbb {M}_p \\, : \\, \\mathcal {M}\\text{ is planar and $f,\\pi $ are constant on the white vertices of $\\mathcal {M}$}\\rbrace .$ Remark 4.14 The above result can be easily generalized to the case where the dimensions of the Hilbert spaces are not identical: one has to replace the “1” in the right hand side of (REF ) with the appropriate product of constants $c_i$ , where the dimensions of the Hilbert spaces scale as $N_i \\sim c_i N$ , with $N \\rightarrow \\infty $ .",
"The statement about freeness follows easily from the moment formula, using the correspondence between planar maps and non-crossing partitions and the fact that a map $\\mathcal {M}$ belongs to $\\mathbb {M}^{bal}_{f,\\pi }$ iff its corresponding non-crossing partition $\\alpha $ satisfies $\\alpha \\le (\\ker f \\wedge \\ker {\\pi })$ (see also the end of the proof of Theorem REF ).",
"We now show the moment formula (REF ).",
"Note that we are actually in the setting of Theorem REF and of Proposition REF , with the permutations $\\mathbf {\\sigma }$ being given by $\\sigma _a = \\pi _{a+1}^{-1}\\pi _a$ , $\\forall a \\in \\llbracket 1, p\\rrbracket $ .",
"Applying now Proposition REF (remember that all Hilbert space dimensions are $N$ in our current setting), we get $\\lim _{N \\rightarrow \\infty } \\mathbb {E}N^{-n(p +1)/2 } \\operatorname{Tr}W_{f,\\pi } = \\sum _{\\mathcal {M}\\in \\mathbb {M}_{\\mathbf {f}, \\mathbf {\\sigma }}^{(0) \\text{free}}} 1,$ where we recall that (see (REF )) $\\mathbb {M}_{\\mathbf {f}, \\mathbf {\\sigma }}^{(0) \\text{free}} = \\lbrace \\mathcal {M}\\, : \\, \\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M}) = 0\\rbrace $ , the functional $\\tilde{L}$ being defined in (REF ).",
"From Lemma REF , we get that for all maps $\\mathcal {M}$ , $V_\\bullet (f,\\sigma ) = k$ .",
"It is now enough to show that $\\mathbb {M}_{\\mathbf {f}, \\mathbf {\\sigma }}^{(0) \\text{free}} = \\mathbb {M}^{bal}_{f,\\pi }$ .",
"Let us start with the inclusion $\\mathbb {M}_{\\mathbf {f}, \\mathbf {\\sigma }}^{(0) \\text{free}} \\subseteq \\mathbb {M}^{bal}_{f,\\pi }$ .",
"First, from $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M}) = 0$ , we get that $f$ is constant on the white vertices of $\\mathcal {M}$ , see (REF ).",
"The fact that $\\pi $ is also constant on the white vertices of $\\mathcal {M}$ follows form Lemma REF , while the planarity of $\\mathcal {M}$ follows from Proposition REF .",
"For the other inclusion, consider a map $\\mathcal {M}\\in \\mathbb {M}^{bal}_{f,\\pi }$ .",
"Since $f$ is constant on the white vertices of $\\mathcal {M}$ , we have $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}(\\mathcal {M}) = 0$ .",
"Also, recall from the proof of Lemma REF that, for any edge $a$ and color $i \\in J_a$ , the black vertex to which $(a,i)$ is connected in $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ has the following incident edges: $\\lbrace (c,\\pi _c^{-1}\\pi _a(i))\\rbrace _{c \\in \\llbracket 1, p\\rrbracket }$ .",
"In particular, using the fact that $\\pi $ is constant on the white vertices of $\\mathcal {M}$ , all edges $(b,i)$ incident to the same white vertex as $(a,i)$ in $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ are connected to the same black vertex, and thus $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ has as many connected components as black vertices (i.e.",
"$k$ ), so $\\Sigma (\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}) = 0$ .",
"Finally, from Lemma REF , the map $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ consists of $k$ copies of $\\mathcal {M}$ , so that if $\\mathcal {M}$ is planar, $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ is too.",
"This concludes the proof.",
"Remark 4.15 The result above has a simple interpretation in the case of two marginals $W_{f,\\mathrm {id}}$ and $W_{\\widehat{f}, \\mathrm {id}}$ , for complementary sets $f, \\widehat{f}$ : $f \\sqcup \\widehat{f} = \\llbracket 1,n \\rrbracket $ .",
"Indeed, for all $N$ , these two marginals have exactly the same eigenvalues, and their eigenvector unitary operators are independent (this follows from the fact that they act on non-overlapping tensor factors).",
"Free independence is in this case a consequence of Voiculescu's classical result [44].",
"The discussion above generalizes easily to an arbitrary number of non-overlapping marginals.",
"Case where less than half the colors are traced Proposition 4.16 In the large $N$ regime, if $ l + k > \\frac{n}{2}$ and with the previous notations, $\\lim _{N\\rightarrow \\infty } \\frac{1}{N^{n+(k+l)(p-2) + l + V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma })-\\mu }}\\, \\mathbb {E}\\operatorname{Tr}_{\\mathbf {\\sigma }} W_{\\mathbf {f}} = \\sum _{\\mathcal {M}\\in \\mathbb {M}_{\\mathbf {f}, \\mathbf {\\sigma }}^{>}}\\, c^{\\, V(\\mathcal {M}) - 1},$ where $\\mu =\\min \\lbrace 2lg(\\mathcal {M}) + \\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M}) + (2k+2l - n)(V(\\mathcal {M}) - 2) \\, : \\, \\mathcal {M}\\in \\mathbb {M}_p\\rbrace \\ge 0$ and $ \\mathbb {M}_{\\mathbf {f}, \\mathbf {\\sigma }}^{>} := \\lbrace \\mathcal {M}\\in \\mathbb {M}_p \\, : \\, 2lg(\\mathcal {M}) + \\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M}) + (2k+2l - n)(V(\\mathcal {M}) - 2) = \\mu \\rbrace .$ It follows from (REF ), but let us detail the computations starting from Theorem REF .",
"Under the usual balanced scaling, the exponent of $N$ from Theorem REF can be bounded as follows $l(2+p-&V(\\mathcal {M})-2g(\\mathcal {M})) + (n-r+1)(V(\\mathcal {M})-1)+kp+V_\\bullet (f,\\sigma )+\\\\&\\qquad \\qquad \\qquad \\qquad +(r-l-2k-1)(V(\\mathcal {M})-1) - \\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M}) \\\\&= (n-2l-2k)(V(\\mathcal {M})-1) + (k+l)p+l-2lg(\\mathcal {M})+ V_\\bullet (f,\\sigma )-\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})\\\\&= n+(k+l)(p-2)+l+ V_\\bullet (f,\\sigma ) -\\\\&\\qquad \\qquad \\qquad \\qquad - [2lg(\\mathcal {M}) + \\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M}) + (2k+2l - n)(V(\\mathcal {M}) - 2)]\\\\&\\le n+(k+l)(p-2)+l+ V_\\bullet (f,\\sigma )-\\mu .$ The fact that $\\mu \\ge 0$ follows from the hypothesis $n-2l-2k<0$ , and from the bounds $V(\\mathcal {M})\\ge 2$ , $g(\\mathcal {M})\\ge 0$ , $ \\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})\\ge 0$ .",
"If all the edges have the same set of colors, the minimum $\\mu $ is achieved by the unique planar map in $\\mathbb {M}_p$ having two vertices (one black and one white).",
"However we stress that in the general case, this map does not have a vanishing $ \\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})\\ge 0$ , and therefore the minimization problem seems highly non-trivial.",
"Indeed, depending on the values taken by the coefficient of $V(\\mathcal {M})-2$ , two maps with different $g$ , $\\tilde{L}$ , and $V$ might have the same minimal value of $2lg + \\tilde{L} + (2k+2l - n)(V - 2)$ .",
"We illustrate this with a very simple example in the case $l=1$ (color $A$ ), $k=2$ (colors $B,C$ ), and $n$ is kept as a parameter.",
"We take $\\sigma _a=\\pi _{a+1}^{-1}\\pi _a$ , $\\pi _1=(12)$ (the transposition) and $\\pi _2=(1)(2)$ (the identity), so that both $\\sigma _1$ and $\\sigma _2$ are transpositions.",
"In that case, $\\includegraphics [width=0.55]{Ex422.pdf}$ if $n=3$ , $2k+2l - n = 3$ , and the only map in $\\mathbb {M}_{\\mathbf {f}, \\mathbf {\\sigma }}^{>}$ is that with two vertices, if $n=5,6$ , $2k+2l - n = 1,0$ and the only map in $\\mathbb {M}_{\\mathbf {f}, \\mathbf {\\sigma }}^{>}$ is the tree, if $n=4$ , $2k+2l - n = 2$ and the two maps belong to $\\mathbb {M}_{\\mathbf {f}, \\mathbf {\\sigma }}^{>}$ .",
"The unbalanced asymptotical regime We consider in this section the asymptotical regime where the dimensions of the Hilbert spaces that “move around” are fixed; in this respect, the setting here generalized the 4-partite situation considered in Section REF .",
"In this case, we fix $N_J = m$ , and set $\\forall i \\in \\llbracket 1,l \\rrbracket , N_i=N$ and $\\forall j \\in \\llbracket r,n\\rrbracket , N_j \\sim c^{\\frac{1}{n-r+1}}N$ as $N\\rightarrow \\infty $ .",
"Theorem REF writes $\\mathbb {E}\\operatorname{Tr}_{\\mathbf {\\sigma }} W_{\\mathbf {f}} &= N^{pl + n-r+1}\\sum _{\\mathcal {M}\\in \\mathbb {M}_p} h(N)^{V(\\mathcal {M})-1} N^{ - 2lg(\\mathcal {M}) - (l+r-n-1)(V(\\mathcal {M})-2)} \\\\\\nonumber &\\hspace{113.81102pt}\\times m^{kp + V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma }) + (r-l-2k-1)(V(\\mathcal {M})-1) -\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})},$ where $h$ is an arbitrary function whose limit at infinity is $c$ .",
"Again we have different behaviors depending on the factor $l+r-n-1 = 2l+k-n$ in front of $V(\\mathcal {M})-2$ : If $l+r-n-1<0$ , the large $N$ limit is trivial, the only surviving map being the one with $p+1$ vertices (1 black and $p$ white).",
"This situation was detailed in the beginning of Sec.",
"REF .",
"If $l+r-n-1>0$ , then $l >0$ , and the large $N$ limit selects the only planar map with two vertices.",
"The limit is also trivial.",
"Besides the two trivial cases above, the only remaining situation is $l+r-n-1=0$ (or, equivalently $2l+k=n$ ) and we therefore assume it in the following.",
"In this case, Theorem REF writes $\\mathbb {E}\\operatorname{Tr}_{\\mathbf {\\sigma }} W_{\\mathbf {f}} = N^{l(p+1)}\\sum _{\\mathcal {M}\\in \\mathbb {M}_p}\\, h(N)^{V(\\mathcal {M})-1} N^{ - 2lg(\\mathcal {M})} m^{kp + V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma }) + (r-l-2k-1)(V(\\mathcal {M})-1) -\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})},$ where $(r-l-2k-1)(V(\\mathcal {M})-1) -\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})$ is a generalization of $``\\mathsf {alt}\"$ which we defined in Def.",
"REF , Section for $n=4$ .",
"We recall that $\\mathbb {M}_p^{(0)}$ is the subset of planar elements of $\\mathbb {M}_p$ .",
"Proposition 4.17 In this regime, with the previous notations, $\\lim _{N\\rightarrow \\infty } \\frac{1}{N^{l(p+1)}}\\, \\mathbb {E}\\operatorname{Tr}_{\\mathbf {\\sigma }} W_{\\mathbf {f}} = \\sum _{\\mathcal {M}\\in \\mathbb {M}_p^{(0)}}\\, c^{\\, V(\\mathcal {M}) - 1} m^{kp + V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma }) + (r-l-2k-1)(V(\\mathcal {M})-1) -\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})}.$ Note that if for $a\\in \\llbracket 1, p \\rrbracket $ we assume $V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma })=k$ , and we define $\\tilde{W}_{f(a)} = \\frac{1}{N^l m^k} W_{f(a)}$ , as well as $\\tilde{c} = cm^{r-l-2k-1}$ , this rewrites as $\\lim _{N\\rightarrow \\infty } \\frac{1}{N^lm^k}\\, \\mathbb {E}\\operatorname{Tr}_{\\mathbf {\\sigma }} \\tilde{W}_{\\mathbf {f}} = \\sum _{\\mathcal {M}\\in \\mathbb {M}_p^{(0)}}\\, \\tilde{c}^{V(\\mathcal {M}) - 1} m^{ -\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})}.$ An example for $n=5$ .",
"We now focus on the quantity $\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})$ , in the following interesting case of “ABCDE\": $n=5$ , $l=1$ (color $A$ ), $r=5$ (color $E$ is always traced), and $k=2$ (the edges carry two colors among $\\lbrace B,C,D\\rbrace $ ); the colors $A$ and $E$ correspond to spaces of dimension growing to infinity, while the colors $B,C,D$ correspond to $\\mathbb {C}^m$ .",
"As $l+r - n - 1=0$ , Prop.",
"REF and Eq.",
"(REF ) apply.",
"The study of the properties of $\\tilde{L}$ in this case shows that the corresponding cumulant functions do not factorize over the cycles of non-crossing partitions, but depend more subtly on these non-crossing partitions.",
"In what follows, we study the special case of a map with two vertices (one black, one white), and we show, by the means of an example, that in the general case the factorization property does not hold.",
"To start, we recall the expression of $\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})$ $\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M})=2(g(\\mathcal {M}_{\\mathbf {f},\\mathbf {\\sigma }})+\\Delta _{\\mathbf {f},\\mathbf {\\sigma }}(\\mathcal {M})+\\Sigma (\\mathcal {M}_{\\mathbf {f},\\mathbf {\\sigma }})).$ Any permutation $\\sigma \\in \\mathcal {S}_2$In the following we omit color $A$ from the support and image of the $\\sigma _a$ 's as it is a common fixed point.",
"defines a partial permutation on $\\llbracket l+1, r-1\\rrbracket = \\llbracket 2, 4\\rrbracket $ , which we complete canonically to have a permutation $\\Phi (\\sigma )$ of $\\llbracket 2, 4\\rrbracket $ .",
"Indeed, there is only one missing color in the support and image of each permutation $\\sigma _a$ , so we just pair them (e.g.",
"if $\\sigma $ is $B\\rightarrow B, C\\rightarrow D$ , then $\\Phi (\\sigma )$ is $B\\rightarrow B, C\\rightarrow D, D\\rightarrow C$ , and if $\\sigma $ is $B\\rightarrow C, C\\rightarrow D$ , then $\\Phi (\\sigma )$ is $B\\rightarrow C, C\\rightarrow D, D\\rightarrow B$ ).",
"We recall that $\\vert \\tilde{\\sigma }\\vert $ is the length of $\\tilde{\\sigma }$ , i.e.",
"its number of inversions.",
"Proposition 4.18 In the $n=5$ example described above, if $g(\\mathcal {M})=0$ , $V_\\bullet (\\mathbf {f}, \\mathbf {\\sigma })=k=2$ , and $V(\\mathcal {M})=2$ , then $\\tilde{L} ( \\mathbf {f}, \\mathbf {\\sigma }, \\mathcal {M}) = \\sum _{a=1}^p\\, \\vert \\Phi (\\sigma _a)\\vert $ To begin, note that the white vertex of $\\mathcal {M}$ can be duplicated into 2 or 3 white vertices in the unfolded map $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ , depending on whether the edges share the same set of colors or not; hence, $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}$ is either 0 or 1.",
"We first assume that $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}=0$ .",
"This means that e.g.",
"$B$ is not in any of the color sets $J_a$ , the $\\sigma _a$ are either the identity, either the transposition, on $\\lbrace C,D\\rbrace $ , and that $\\Phi (\\sigma _a)(B) = B$ for all $a$ .",
"We can assume that there are only transpositions in the corners, because when one has the identity permutation, one can collapse the two neighboring edges into a single one without changing the genus of $\\mathcal {M}$ or $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ , nor $\\Sigma (\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }})$ .",
"Here one can make a simple recurrence, which is quite similar to the one appearing in the proof of Proposition REF in terms of maps: pick any $a$ , and replace $\\sigma _a$ and $\\sigma _{a+1}$ by two identities.",
"The number of inversions decreases by two.",
"In $\\mathcal {M}_{\\mathbf {f}, \\mathbf {\\sigma }}$ , this amounts to “flipping\" the two edges, as shown in the figure below (the half-edges incident to the white vertices remain the same, but the endpoints incident to the black vertices are exchangedIndeed changing $\\sigma _a$ and $\\sigma _{a+1}$ changes the cycles of $\\Gamma _{\\mathbf {f}, \\mathbf {\\sigma }}$ while keeping $\\hat{\\alpha }$ invariant).",
"The flip clearly decreases the genus or the number of connected components.",
"To see this, note that we can decompose the flip in edge-deletions and creations.",
"The edges have (the same) two faces incident each.",
"Delete one of them, and the other gets a single face visiting both sides.",
"Delete the second edge: if it is a bridge the number of connected components increases (and $\\Sigma _{\\mathbf {f}, \\mathbf {\\sigma }}$ goes from 1 to 0) and the genus is unchanged, so $\\tilde{L}$ decreases by two.",
"If not, $\\Sigma _{\\mathbf {f}, \\mathbf {\\sigma }}$ is unchanged, and the genus decreases by one, so $\\tilde{L}$ decreases by two, and we get the formula, after creating two new edges to complete the flip (creating the two new edges does not change the number of connected components nor the genus).",
"Figure: NO_CAPTIONWe now assume that $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}=1$ .",
"The strategy is to pick some $a$ such that $\\sigma _{a+1}$ contains $B$ in its image (we can always find one, since $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}\\ne 0$ ), and replace $\\sigma _a$ and $\\sigma _{a+1}$ so that $B$ is a fixed point of $\\sigma _{a+1}$ .",
"Then do the same for $\\sigma _{a-1}$ and $\\sigma _{a}$ , and so on.",
"At each step, one should be careful that the variation in the number of inversions is the same as $\\tilde{L}$ .",
"In the end, $B$ is a common fixed point of all the $\\sigma _a$ , so we are in the $\\Delta _{\\mathbf {f}, \\mathbf {\\sigma }}=0$ case treated above, and we can conclude.",
"There are two cases, depending whether $B$ is in the pre-image of $\\sigma _{a+1}$ or not: If $B$ is in the pre-image of $\\sigma _{a+1}$ (and is not a fixed point of $\\sigma _{a+1}$ , in which case we go to the next step right away), we exchange it with the other color in the pre-image of $\\sigma _{a+1}$ , both in $\\sigma _a$ and $\\sigma _{a+1}$ : $\\begin{tabular}{rcccl} &\\sigma _a & &\\sigma _{a+1}& \\\\ c_a&\\mapsto & c_{a+1} &\\mapsto & B \\\\c^{\\prime }_a&\\mapsto & B &\\mapsto & c^{\\prime }_{a+2}\\end{tabular}\\raisebox {-1.5ex}{\\quad \\leadsto \\quad }\\begin{tabular}{rcccl} & & && \\\\ c_a&\\mapsto & B &\\mapsto & B \\\\c^{\\prime }_a&\\mapsto & c_{a+1} &\\mapsto & c^{\\prime }_{a+2}\\end{tabular}.$ There are two sub-cases: either the number of inversions decreases by two, in which case we verify that $\\tilde{L}$ decreases by two (6 possibilities), or the number of inversions is constant, in which case we verify that $\\tilde{L}$ is also constant (4 possibilities).",
"If $B$ is not in the pre-image of $\\sigma _{a+1}$ , we just exchange the pre-image of $B$ with $B$ , and leave the other color in the pre-image of $\\sigma _{a+1}$ untouched.",
"Again, either the number of inversions decreases by two, in which case we verify that $\\tilde{L}$ decreases by two (12 possibilities), or the number of inversions is constant, in which case we verify that $\\tilde{L}$ is also constant (8 possibilities).",
"This concludes the proof Figure: For π 1 =(1)(2)\\pi _1=(1)(2), π 2 =(12)\\pi _2=(12), π 3 =(1)(2)\\pi _3=(1)(2), π 4 =(12)\\pi _4=(12), one has L ˜(𝐟,σ,ℳ)=2\\tilde{L}(\\mathbf {f},\\mathbf {\\sigma }, \\mathcal {M})=2, while the value of L ˜\\tilde{L} for both submaps with one white vertex is 2, so the sum over the one white vertex submaps of L ˜\\tilde{L} does not match L ˜(𝐟,σ,ℳ)\\tilde{L}(\\mathbf {f},\\mathbf {\\sigma }, \\mathcal {M}).",
"However, if π 1 =(1)(2)\\pi _1=(1)(2), π 2 =(1)(2)\\pi _2=(1)(2), π 3 =(1)(2)\\pi _3=(1)(2), π 4 =(1)(2)\\pi _4=(1)(2), then L ˜(𝐟,σ,ℳ)=0\\tilde{L}(\\mathbf {f},\\mathbf {\\sigma }, \\mathcal {M})=0 and the one white vertex submaps also have L ˜=0\\tilde{L}=0.",
"This behavior indicates that there is no factorization over the cycles of α\\alpha .When trying to generalize the formula of the result above to $V(\\mathcal {M})>2$ , we face the following difficulty: around a white vertex, there is at least one corner for which we do not have a $\\sigma $ (the white corner which does not face a black corner).",
"On the example of Fig.",
"REF , it is easily seen that the value of $\\tilde{L}$ for the full map is not given as the sum of the value $\\tilde{L}$ for the two submaps with one white vertex.",
"This indicates that there is no factorization of $\\tilde{L}$ over the cycles of $\\alpha $ .",
"It is also possible to display a formula for the case of maps whose edges are only of color $CD$ .",
"This formula shows that there is no factorization over the cycles of $\\alpha $ .",
"Thus it is not possible to write the free cumulants in the context of (scalar) free probability.",
"It seems that the right framework needed to tackle this unbalanced scaling is the one of free probability with amalgamation over an algebra $\\mathcal {B}$ that is contained in $\\mathbb {1}_A\\otimes \\mathcal {M}_m(\\mathbb {C})\\otimes \\mathcal {M}_m(\\mathbb {C})\\supset \\mathcal {B}$ .",
"In order to limit the length of this paper, we postpone the exploration and the exposition of such results to future work."
]
] | 1808.08554 |
[
[
"Nonlocal Entanglement of 1D Thermal States Induced by Fermion Exchange\n Statistics"
],
[
"Abstract When two identical fermions exchange their positions, their wave function gains phase factor $-1$.",
"We show that this distance-independent effect can induce nonlocal entanglement in one-dimensional (1D) electron systems having Majorana fermions at the ends.",
"It occurs in the system bulk and has nontrivial temperature dependence.",
"In a system having a single Majorana at each end, the nonlocal entanglement has a Bell-state form at zero temperature and decays as temperature increases, vanishing suddenly at certain finite temperature.",
"In a system having two Majoranas at each end, it is in a cluster-state form and its nonlocality is more noticeable at finite temperature.",
"By contrast, thermal states of corresponding 1D spins do not have nonlocal entanglement."
],
[
"Nonlocal Entanglement of 1D Thermal States Induced by Fermion Exchange Statistics YeJe Park Jeongmin Shim S.-S. B. Lee [Present address: Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience, Ludwig-Maximilians-Universität, Theresienstraße 37, D-80333 München, Germany] H.-S. Sim[]hssim@kaist.ac.kr Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea When two identical fermions exchange their positions, their wave function gains phase factor $-1$ .",
"We show that this distance-independent effect can induce nonlocal entanglement in one-dimensional (1D) electron systems having Majorana fermions at the ends.",
"It occurs in the system bulk and has nontrivial temperature dependence.",
"In a system having a single Majorana at each end, the nonlocal entanglement has a Bell-state form at zero temperature and decays as temperature increases, vanishing suddenly at certain finite temperature.",
"In a system having two Majoranas at each end, it is in a cluster-state form and its nonlocality is more noticeable at finite temperature.",
"By contrast, thermal states of corresponding 1D spins do not have nonlocal entanglement.",
"Topological phases of matter are notions of zero-temperature ground states.",
"They have interesting properties such as ground-state degeneracy, edge states, and unusual excitations.",
"For example, topological phases of 1D electron systems have Majorana zero modes localized at system ends [1], [2], [3], [4].",
"The phases are identified by entanglement spectrum [2], [3], entanglement entropy [5], or a nonlocal string order parameter [6], and classified by symmetries and the number of the zero modes [2], [3], [4].",
"The non-Abelian statistics of Majorana fermions is a key to topological quantum computing [7], [8].",
"A natural question is how quantum properties of topological phases are thermally suppressed [9], [10], [11], [12], [13], [14], [15], [16], [17].",
"It involves a number of largely unexplored points.",
"First of all, tools identifying topological phases of pure states are not readily applicable to thermal states; e.g.",
"for mixed states entanglement spectrum [19], [20], [18] is ill-defined and entanglement entropy overestimates entanglement.",
"Secondly, pure ground states have definite quantum numbers in presence of symmetries, such as number parity of 1D electrons, and they can be thermally mixed.",
"Consequences of such mixing need to be studied.",
"Lastly, topological phases of 1D electron pure states can be classified by means of the 1D spins obtained by Jordan-Wigner transformation (JWT) [21].",
"It is interesting to see whether this fermion-spin correspondence is applicable to thermal states.",
"We will address the above points for 1D electrons.",
"In this Letter, we study thermal states of 1D electron systems having Majorana zero modes, using two mixed-state entanglement measures [22], the entanglement of formation [23] and logarithmic negativity [24], [25], [26].",
"We find that the thermal states can have nonlocal and length-independent entanglement of occupation number in the bulk.",
"It has nontrivial temperature ($T$ ) dependence, depending on the number of the zero modes.",
"In a system having one zero mode at each end, nonlocal entanglement occurs in a Bell-state form at $T=0$ , decreases at larger $T$ , then vanishes at certain finite $T$ .",
"In a system of two zero modes at each end, it occurs in a cluster-state form and its nonlocality is noticeable in a finite-$T$ window.",
"The nonlocal entanglement results from the non-Abelian statistics of Majorana fermions and the fermion exchange statistics.",
"By contrast, the 1D spins, obtained by JWT of the 1D electrons, do not have any nonlocal entanglement.",
"Figure: (a) Kitaev chain having one Majorana zero mode γ 1,4 \\gamma _{1,4} at each end.",
"(b) It is partitioned into the left (B) and right (A and C) by dashed imaginary cuts.",
"There are Majorana fermions γ 2,2 ¯,3 ¯,3 \\gamma _{2,\\bar{2},\\bar{3},3} at the cuts.",
"Majorana fusion pairs (arrows) γ 2 +iγ 2 ¯ \\gamma _2 + i \\gamma _{\\bar{2}} and γ 3 ¯ +iγ 3 \\gamma _{\\bar{3}} + i \\gamma _3 annihilate the ground states of the chain.",
"A fusion pair γ 2 ¯ +iγ 3 ¯ \\gamma _{\\bar{2}} + i \\gamma _{\\bar{3}} is localized within the left, while fusion pairs (c) γ 1 +iγ 2 \\gamma _1 + i \\gamma _2, γ 3 +iγ 4 \\gamma _3 + i \\gamma _4 or (d) γ 2 +iγ 3 \\gamma _2 + i \\gamma _3, γ 1 +iγ 4 \\gamma _1 + i \\gamma _4 in the right.Nonlocal entanglement in Kitaev chain.— We first consider a Kitaev chain [1] of spinless electrons.",
"It represents a topological phase (BDI class) of 1D fermions protected by time-reversal and parity symmetries [2], [3], [4].",
"It has one Majorana zero mode $\\gamma _{a=1,4}$ at each end [Fig.",
"REF (a)].",
"We study entanglement between the left B and right AC of imaginary cuts through physical bonds connecting electron sites [Fig.",
"REF (b)], partitioning the chain into regions A, B, C. We use fermion operators $f_{ab} \\equiv (\\gamma _a + i \\gamma _{b}) / \\sqrt{2}$ and their occupation numbers $\\hat{n}_{ab} = f^\\dagger _{ab} f_{ab}$ , formed by Majoranas at the ends $\\gamma _{a=1,4}$ and the cuts $\\gamma _{a=2,\\bar{2},\\bar{3},3}$ .",
"The system Hamiltonian is $\\hat{H}_\\textrm {I} = - \\sum _{j=1}^{N-1} [\\frac{t}{2} (c_j^{\\dagger } c_{j+1} + c_{j+1}^{\\dagger } c_j) + \\frac{\\Delta }{2} (c_j c_{j+1} + c^\\dagger _{j+1} c^\\dagger _j )] +\\sum _{j=1}^N \\mu c_j^{\\dagger } c_j$ with hopping energy $t$ , pairing $\\Delta $ , chemical potential $\\mu $ , and electron creation $c_j^\\dagger $ at site $j =1,\\cdots ,N$ .",
"We choose [27] $t = \\Delta >0$ and $\\mu = 0$ for simplicity.",
"Then the chain has two degenerate ground states, $|0 \\rangle _\\textrm {I} = |0_{14} 0_{2\\bar{2}} 0_{\\bar{3}3} \\cdots \\rangle $ (of even number parity) and $|1 \\rangle _\\textrm {I} = f^\\dagger _{14} |0 \\rangle _\\textrm {I}$ (odd).",
"At equilibrium, it is described by the thermal state $\\rho _\\textrm {I} (T) = e^{- \\beta \\hat{H}_\\textrm {I}}/\\textrm {Tr} e^{- \\beta \\hat{H}_\\textrm {I}}$ with $\\beta = k_B T$ and Boltzmann constant $k_B$ .",
"We analyze entanglement between B and AC in $\\rho _\\textrm {I} (T=0) = (|0 \\rangle \\langle 0 |_\\textrm {I} + |1 \\rangle \\langle 1 |_\\textrm {I})/2$ at $T=0$ .",
"For this, we write [28] the ground states using operators $f_{12}$ , $f_{\\bar{2}\\bar{3}}$ , $f_{34}$ localized within A, B, C, respectively [Fig.",
"REF (c)], |0I = 12 ( 1 + f12f34+ f12f23+ f2 3f34)|012 023 034 , |1I = 12( f12+ f34+ f23+ f12f23f34) |012 023 034 .",
"To see the entanglement, we need to map [29], [30] them into qubits since fermion states lack tensor product structure.",
"Before mapping, we reorder operators such that those belonging to a subsystem are grouped together; $|0 \\rangle _\\textrm {I} = \\frac{1}{2} (1 + f_{12}^\\dagger f_{34}^\\dagger - f_{\\bar{2}\\bar{3}}^\\dagger f_{12}^\\dagger + f_{\\bar{2} \\bar{3}}^\\dagger f_{34}^\\dagger )|0_{\\bar{2}\\bar{3}} 0_{12} 0_{34} \\cdots \\rangle $ , $|1 \\rangle _\\textrm {I} = \\frac{1}{2} (f_{12}^\\dagger + f_{34}^\\dagger + f_{\\bar{2}\\bar{3}}^\\dagger - f_{\\bar{2}\\bar{3}}^\\dagger f_{12}^\\dagger f_{34}^\\dagger ) |0_{\\bar{2}\\bar{3}} 0_{12} 0_{34} \\cdots \\rangle $ .",
"Here, $f_{12}^\\dagger $ and $f_{34}^\\dagger $ belonging to AC are collected to the right of $f_{\\bar{2}\\bar{3}}^\\dagger $ belonging to B, generating the fermion exchange sign $-1$ in some coefficients.",
"We call this ordering as “subsystem operator grouping”.",
"Then occupation number states are mapped onto qubit states, | 0I |0Iq= 12(| 023 q (|012q |034 q +|112 q |134 q) - |123 q (|112 q |034 q -|012q | 134 q)), |1I |1Iq= 12 ( | 023q (|112 q | 034 q +|012 q |134 q) + |123q (|012 q | 034 q -|112 q |134 q)).",
"This shows that $\\rho _\\textrm {I} (T=0)$ has maximal entanglement between B ($\\bar{2} \\bar{3}$ ) and AC (12,34).",
"This entanglement can be alternatively seen writing the states using $\\lbrace f_{23}, f_{14}\\rbrace $ instead of $\\lbrace f_{12}, f_{34}\\rbrace $ .",
"The result [28] is $|0 \\rangle _\\textrm {I} = \\frac{1}{\\sqrt{2}}(1 + i f^\\dagger _{\\bar{2} \\bar{3}} f^\\dagger _{23}) |0_{\\bar{2} \\bar{3}} 0_{23} 0_{14} \\cdots \\rangle $ , $|1 \\rangle _\\textrm {I} = \\frac{1}{\\sqrt{2}} (1 + i f^\\dagger _{\\bar{2} \\bar{3}} f^\\dagger _{23}) f^\\dagger _{14} |0_{\\bar{2} \\bar{3}} 0_{23} 0_{14} \\cdots \\rangle $ .",
"We map these onto qubits, $|0\\rangle _\\textrm {I} \\mapsto |\\text{Bell}\\rangle ^q |0_{14}\\rangle ^q$ and $|1\\rangle _\\textrm {I} \\mapsto |\\text{Bell}\\rangle ^q |1_{14}\\rangle ^q$ .",
"The map shows a Bell state entangling B ($\\bar{2}\\bar{3}$ ) and AC (23) |Bellq = 12 (|023q |023q + i |123 q |123q).",
"Their mixture also has the Bell state, $\\rho _\\textrm {I}(T=0) \\mapsto \\rho _\\textrm {I}^q(T=0) = |\\text{Bell}\\rangle \\langle \\text{Bell}|^q \\otimes (|0_{14} \\rangle \\langle 0_{14}|^q + |1_{14} \\rangle \\langle 1_{14}|^q)/2$ .",
"The end qubit $|n_{14} \\rangle ^q$ does not affect the entanglement.",
"The Bell state $|\\text{Bell} \\rangle ^q$ is nonlocal as $f^\\dagger _{\\bar{2} \\bar{3}}$ and $f^\\dagger _{23}$ create nonlocal fermions.",
"It originates [31] from entanglement generation $|0_{2\\bar{2}} 0_{\\bar{3}3} \\rangle \\rightarrow |0_{\\bar{2}\\bar{3}} 0_{23} \\rangle + i |1_{\\bar{2} \\bar{3}} 1_{23} \\rangle $ by changing non-Abelian Majorana fusion pairs from $\\gamma _2 + i\\gamma _{\\bar{2}}$ , $\\gamma _{\\bar{3}} + i \\gamma _3$ [Fig.",
"REF (b)] to $\\gamma _{\\bar{2}} + i \\gamma _{\\bar{3}}$ , $\\gamma _2 + i \\gamma _3$ [Fig.",
"REF (d)].",
"It occurs in the chain bulk and is independent of lengths of A, B, C. It is unaffected by parity mixing as it occurs in both $|0\\rangle _\\textrm {I}$ and $|1\\rangle _\\textrm {I}$ .",
"It is robust against quasiparticle poisoning [32], [33], [34] due to protection by the gap $\\Delta $ .",
"This can be seen as a bulk-edge correspondence of the Kitaev chain; the entanglement occurs in the bulk for other values of $\\mu $ , $t$ , and $\\Delta $ [27] for which end Majoranas appear.",
"We emphasize the importance of the fermion exchange sign in the mapping (REF ).",
"As the entanglement is physical, it should be invariant under basis change between $\\lbrace f_{12}, f_{34} \\rbrace $ and $\\lbrace f_{14}, f_{23} \\rbrace $ .",
"However, if we were to map $|n=0,1\\rangle _\\textrm {I}$ into qubits without the subsystem operator grouping, we would obtain the states of Eq.",
"(REF ) but with positive signs replacing the negative signs, | 0sI = 12(| 023 s (|012s |034 s +|112 s |134 s) + |123 s (|112 s |034 s +|012s | 134 s)), |1sI = 12 ( | 023s (|112 s | 034 s +|012 s |134 s) + |123s (|012 s | 034 s +|112 s |134 s)), where superscripts $s$ are for the discussion about spins below.",
"Although $|n \\rangle ^s_\\textrm {I}$ 's have entanglement between B and AC similar to $|n \\rangle _\\textrm {I}^q$ in Eq.",
"(REF ), their equal mixture $\\rho _\\textrm {I}^s(T=0)=(|0 \\rangle \\langle 0|^s + |1 \\rangle \\langle 1 |^s)/2$ has no entanglement in contradiction with the presence of $|\\textrm {Bell}\\rangle ^q$ in $\\rho _\\textrm {I}(T=0)$ in Eq.",
"(REF ).",
"This demonstrates that the fermion statistics has to be taken into account in order to correctly quantify entanglement in fermion mixed states.",
"We notice that $\\rho _\\textrm {I} (T=0)$ provides a good example resolving an issue [35], [36] of fermion entanglement whether particular orderings of fermions have to be chosen when one maps fermion occupations onto qubits.",
"To see whether the fermion-spin correspondence [21], [2] is valid for the mixed state $\\rho _\\textrm {I} (T=0)$ , we compare the Kitaev chain with the spin chain obtained by JWT [28] of $\\hat{H}_\\textrm {I}$ .",
"The ground states $|n \\rangle ^s_\\textrm {I}$ of the spin chain turn out to be the same as those in Eq.",
"(REF ) obtained from the electron states $|n \\rangle _\\textrm {I}$ by mapping into qubits without taking the fermion exchange sign into account.",
"$|n \\rangle ^s_\\textrm {I}$ 's are eigenstates of JWT of the parity operator $\\hat{P} = \\prod _{j} e^{i \\pi c^\\dagger _j c_j}$ with eigenvalue $(-1)^n$ so that the parity symmetry is mapped by JWT [21].",
"In Eq.",
"(REF ) $|n_{ab}=0,1 \\rangle ^s$ is a state of region R ($= \\textrm {A}$ for $ab=12$ , $\\textrm {B}$ for $\\bar{2}\\bar{3}$ , $\\textrm {C}$ for 34) of the spin chain [28].",
"The pure ground states in Eq.",
"(REF ) have the same entanglement as those in Eq.",
"(REF ), as expected from the correspondence.",
"However, $\\rho _\\textrm {I}^s(T=0)$ has no entanglement, showing that the correspondence breaks down for mixed states.",
"Nonlocal entanglement at $T>0$ .— We quantify entanglement between B and AC in the thermal state $\\rho _\\textrm {I} (T)$ by two bipartite mixed-state measures, logarithmic negativity $\\mathcal {LN}$ [24], [25], [26] and the entanglement of formation $\\mathcal {F}$ [23].",
"$\\mathcal {LN}$ is easier to compute than other measures.",
"We compute $\\mathcal {LN} (\\rho _\\textrm {I}) \\equiv \\log _2 \\textrm {Tr} \\left|(\\rho _\\textrm {I}^q)^{T_\\textrm {B}}\\right|$ .",
"$(\\rho _\\textrm {I}^q)^{T_\\textrm {B}}$ is the partial transpose with respect to B of $\\rho ^q_\\textrm {I}$ obtained by mapping $\\rho _\\textrm {I}$ onto qubits with the subsystem operator grouping.",
"$\\textrm {Tr} |\\cdot |$ is the trace norm.",
"$\\mathcal {F}$ is a mixed-state generalization of entanglement entropy $\\mathcal {S}$ , as $\\mathcal {F}= \\mathcal {S}$ for pure states.",
"We compute $ \\mathcal {F} (\\rho _\\textrm {I}) \\equiv \\inf _{\\rho ^q_\\textrm {I} = \\sum _i w_k |\\psi _k \\rangle \\langle \\psi _k |} [ \\sum _k w_k \\, \\mathcal {S} (|\\psi _k \\rangle ) ]$ , exploring all possible decompositions of $\\rho ^q_\\textrm {I}$ into normalized pure states $|\\psi _k \\rangle $ with weight $w_k$ and finding the optimal decomposition for which $\\sum _k w_k \\, \\mathcal {S} (|\\psi _k \\rangle )$ is the lowest.",
"$\\mathcal {S} (|\\psi _k \\rangle ) \\equiv - \\textrm {Tr} ( \\rho ^\\textrm {B}_k \\log _2 \\rho ^\\textrm {B}_k )$ is entanglement entropy between B and AC in $|\\psi _k \\rangle $ and $\\rho ^\\textrm {B}_k = \\textrm {Tr}_\\textrm {B} |\\psi _k \\rangle \\langle \\psi _k|$ .",
"$\\mathcal {LN}$ and $\\mathcal {F}$ have been used for studying many-body states [37], [38], [39], [40], [41], [42], [43].",
"Figure: Temperature dependence of nonlocal entanglement between the regions B and AC of Kitaev chain.",
"It is quantified by ℒ𝒩\\mathcal {LN} and ℱ\\mathcal {F}.",
"It is maximal at T=0T=0 and survives up to T SD T_\\textrm {SD}.",
"It decays not exponentially nor algebraically near T SD T_\\textrm {SD}.We can write $\\rho _\\textrm {I} (T) = \\rho _\\textrm {I,cuts} (T) \\rho ^{\\prime }_\\textrm {I} (T)$ .",
"The first factor $\\rho _\\textrm {I,cuts} \\propto e^{-\\beta \\hat{H}_\\textrm {I,cuts}}$ accounts for thermal excitations at the cuts, where $\\hat{H}_\\textrm {I,cuts} = \\Delta (f_{2 \\bar{2}}^\\dagger f_{2 \\bar{2}} + f_{\\bar{3} 3}^\\dagger f_{\\bar{3} 3})$ .",
"$\\rho _\\textrm {I,cuts}$ can have entanglement between B and AC since each excitation at the cuts is mapped onto one of the four Bell type states: $|\\text{Bell}\\rangle ^q$ , $|0_{\\bar{2}\\bar{3}}\\rangle ^q|1_{23}\\rangle ^q - i |1_{\\bar{2}\\bar{3}}\\rangle ^q |0_{23} \\rangle ^q$ (excited by $f_{2\\bar{2}}^\\dagger $ ), $|0_{\\bar{2}\\bar{3}}\\rangle ^q |1_{23}\\rangle ^q + i |1_{\\bar{2}\\bar{3}}\\rangle ^q |0_{23} \\rangle ^q$ (by $f_{\\bar{3}3}^\\dagger $ ), and $|0_{\\bar{2}\\bar{3}}\\rangle ^q |0_{23}\\rangle ^q - i |1_{\\bar{2}\\bar{3}}\\rangle ^q|1_{23} \\rangle ^q$ (by $f_{2\\bar{2}}^\\dagger f_{\\bar{3}3}^\\dagger $ ).",
"These Bell states originate from entanglement generation by changing non-Abelian fusion pairs from Fig.",
"REF (b) to (d).",
"The second factor $\\rho ^{\\prime }_\\textrm {I}$ accounts for excitations localized in one of B and AC, and does not contribute to the entanglement.",
"Hence, $\\mathcal {LN}(\\rho _\\textrm {I}) = \\mathcal {LN}(\\rho _\\textrm {I,cuts})$ and $\\mathcal {F}(\\rho _\\textrm {I}) = \\mathcal {F}(\\rho _\\textrm {I,cuts})$ .",
"$\\mathcal {LN}$ and $\\mathcal {F}$ are computable since $\\rho _\\textrm {I, cuts}$ is mapped onto a two-qubit state.",
"Both show qualitatively the same $T$ dependence in Fig.",
"REF .",
"At $T = 0$ , $\\mathcal {LN}=\\mathcal {F}=1$ because of $|\\textrm {Bell} \\rangle ^q$ .",
"This value is related to the quantum dimension $\\sqrt{2}$ of Majoranas; among four ($= \\sqrt{2}^4$ ) electron-number states formed by the four cut Majoranas $\\gamma _{2,\\bar{2},\\bar{3},3}$ , only two even-parity states $|0_{\\bar{2}\\bar{3}} 0_{23} \\rangle $ and $|1_{\\bar{2}\\bar{3}} 1_{23} \\rangle $ are allowed at $T=0$ due to superconductivity, resulting in $\\ln _2 2 = 1$ .",
"As $T$ increases, the four Bell states become more mixed, so $\\mathcal {LN}$ and $\\mathcal {F}$ decrease, vanishing at $T_\\textrm {SD}$ , kB TSD = - (2-1) 1.135 .",
"Interestingly, $d\\mathcal {LN}/dT$ and $d\\mathcal {F}/dT$ are discontinuous at $T_\\textrm {SD}$ , contrary to exponential or algebraic decay; $\\mathcal {LN} \\propto \\delta T$ and $\\mathcal {F} \\propto (\\delta T)^2\\log \\delta T$ at $T = T_\\textrm {SD} - \\delta T$ with $\\delta T \\ll \\Delta $ , and $\\mathcal {LN}, \\mathcal {F} = 0$ for $T>T_\\textrm {SD}$ [28].",
"This behavior may be called “entanglement sudden death” [44].",
"It results from thermal mixing of states of different parity in $\\rho _\\textrm {I}(T)$ ; it does not occur for a mixture of states of even parity [28].",
"The overall $T$ dependence is independent of cut positions.",
"By contrast, for the spin chain, $\\mathcal {LN} = \\mathcal {F} = 0$ at any $T$ as at $T=0$ .",
"This indicates that the nonvanishing entanglement in the Kitaev chain is a consequence of the fermion statistics absent in spins.",
"Figure: (a) A chain having two Majorana zero modes γ a=1,1 ' ,4 ' ,4 \\gamma _{a=1,1^{\\prime },4^{\\prime },4} at each end.",
"(b)It is partitioned as in Fig. .",
"Four Majoranas γ a=2 ' ,2,2 ¯,2 ¯ ' ,3 ¯ ' ,3 ¯,3,3 ' \\gamma _{a=2^{\\prime },2, \\bar{2}, \\bar{2}^{\\prime }, \\bar{3}^{\\prime }, \\bar{3} ,3, 3^{\\prime }} are revealed at each cut.",
"The arrows indicate the fusion pairs annihilating |0〉 II |0 \\rangle _\\textrm {II}.",
"(c) The Majoranas are fused into fermions localized in B or AC.Chain with two Majoranas at an end.— We next study a chain having two Majorana zero modes $\\gamma _{a=1,1^{\\prime },4^{\\prime },4}$ at each end [Fig.",
"REF (a)].",
"It represents a topological phase (D class) of 1D fermions distinct from the Kitaev chain $\\hat{H}_\\textrm {I}$ [2], [3], [4].",
"Its Hamiltonian is $\\hat{H}_\\textrm {II} = -\\frac{\\Delta }{2}\\sum _{j=1}^{N-2} (c_j + c_j^\\dagger )(c_{j+2} -c_{j+2}^\\dagger )$ .",
"Because of $\\gamma _{a=1,1^{\\prime },4^{\\prime },4}$ , it has four degenerate ground states $|n \\rangle _\\textrm {II}$ = $(f_{11^{\\prime }}^\\dagger )^{n_{11^{\\prime }}} (f_{44^{\\prime }}^\\dagger )^{n_{44^{\\prime }}}|0_{11^{\\prime }} 0_{2 \\bar{2}^{\\prime }} 0_{2^{\\prime } \\bar{2}} 0_{\\bar{3}^{\\prime } 3} 0_{\\bar{3} 3^{\\prime }} 0_{44^{\\prime }}\\cdots \\rangle $ where $n = n_{11^{\\prime }} + 2n_{44^{\\prime }}$ and $n_{11^{\\prime }},n_{44^{\\prime }} = 0,1$ .",
"Changing Majorana fusion pairs from Fig.",
"REF (b) to (c), we rewrite $|n\\rangle _\\textrm {II}$ using operators local with respect to regions A, B, C [28] |n II = 12 (f11')n11' (f44')n44' (1 + f22'f22')(1 + f33'f3 3') |011' 022' 022' 033' 033' 044' .",
"Mapping it onto qubits, we first observe entanglement between A and BC (AB and C), and find that the result has a factor of a Bell state $|\\textrm {Bell}_2\\rangle ^q$ ( $|\\textrm {Bell}_3\\rangle ^q$ ) localized at the cut between A and BC (AB and C) |Bella=2,3 q = 12(|0aa' q |0aa' q + |1aa' q |1aa' q) that entangles qubits $n_{22^{\\prime }}$ and $n_{\\bar{2}\\bar{2}^{\\prime }}$ ($n_{\\bar{3}\\bar{3}^{\\prime }}$ and $n_{33^{\\prime }}$ ).",
"Entanglement between B and AC is, however, nonlocal due to the fermion statistics.",
"To see this, we need to map the ground states into qubits after the subsystem operator grouping such that operators $f_{\\bar{2} \\bar{2}^{\\prime }}^\\dagger $ , $f_{\\bar{3} \\bar{3}^{\\prime }}^\\dagger $ of B are collected to the left of those of AC: $|n \\rangle _\\textrm {II} = \\frac{1}{2} (1 - f_{\\bar{2}\\bar{2}^{\\prime }}^\\dagger f_{22^{\\prime }}^\\dagger + f_{\\bar{3}\\bar{3}^{\\prime }}^\\dagger f_{3 3^{\\prime }}^\\dagger +f_{\\bar{2}\\bar{2}^{\\prime }}^\\dagger f_{\\bar{3}\\bar{3}^{\\prime }}^\\dagger f_{22^{\\prime }}^\\dagger f_{3 3^{\\prime }}^\\dagger )(f_{11^{\\prime }}^\\dagger )^{n_{11^{\\prime }}} (f_{44^{\\prime }}^\\dagger )^{n_{44^{\\prime }}} | 0_{22^{\\prime }} 0_{\\bar{2}\\bar{2}^{\\prime }} 0_{\\bar{3}\\bar{3}^{\\prime }} 0_{33^{\\prime }} \\cdots \\rangle $ , |nII |nIIq = |CLq |n11'q |n44'q, |CLq = 12 [|02 2' q |022' q ( |033' q |033' q +|133'q |133' q) - |12 2'q |122'q ( |033'q |033' q -|133'q |133' q)].",
"$|n\\rangle _\\textrm {II}$ has a factor of a four-qubit cluster state [45], [46] $|\\textrm {CL}\\rangle ^q$ .",
"$|\\textrm {CL}\\rangle ^q$ is nonlocal as it cannot be written as a product of locally entangled states $|\\textrm {Bell}_2 \\rangle ^q \\otimes |\\textrm {Bell}_3 \\rangle ^q$ .",
"It is independent of cut positions.",
"Because the end qubits $|n_{11^{\\prime }} \\rangle ^q \\otimes |n_{44^{\\prime }} \\rangle ^q$ are decoupled [Eq.",
"(REF )], the equal mixture of ground states $\\rho _\\textrm {II} (T=0)$ also has the cluster-state entanglement.",
"Since pure excited states have similar cluster-state entanglement, the thermal state $\\rho _\\textrm {II} (T) = e^{- \\beta \\hat{H}_\\textrm {II}}/\\textrm {Tr} e^{- \\beta \\hat{H}_\\textrm {II}}$ can have the nonlocal and length-independent entanglement between B and AC.",
"By contrast, for the spin chain obtained by JWT [28] of $\\hat{H}_\\textrm {II}$ , entanglement between B and AC is local.",
"The equal mixture of ground states $\\rho _\\textrm {II}^s(T=0)$ has entanglement $|\\textrm {Bell}_2 \\rangle ^s \\otimes |\\textrm {Bell}_3 \\rangle ^s$ between B and AC, where $|\\textrm {Bell}_2 \\rangle ^s$ ($|\\textrm {Bell}_3 \\rangle ^s$ ) is the Bell state in Eq.",
"(REF ) localized at the cut between A and BC (AB and C).",
"Similarly, the entanglement is local in all the excited states.",
"Consequently, the thermal state $\\rho _\\textrm {II}^s(T)$ of the spin chain is decomposed as $\\rho _\\textrm {II}^s(T) = \\rho _\\textrm {II,cut2}^s(T)\\otimes \\rho _\\textrm {II,cut3}^s(T) \\otimes {\\rho _\\textrm {II}^s}^{\\prime }(T)$ , where $\\rho _\\textrm {II,cut2}^s$ ($\\rho _\\textrm {II,cut3}^s$ ) can have local entanglement at the cut between A and BC (AB and C) and ${\\rho _\\textrm {II}^s}^{\\prime }$ has no entanglement.",
"Since $\\rho ^s_\\textrm {II}(T)$ is identical to the result of mapping $\\rho _\\textrm {II}(T)$ into qubits without the subsystem operator grouping, we see that the nonlocal entanglement in the electron state $\\rho _\\textrm {II}(T)$ derives from the fermion exchange statistics.",
"We compute $\\mathcal {LN}$ for the electron state $\\rho _\\textrm {II}(T)$ and the spin $\\rho ^s_\\textrm {II}(T)$ in Fig.",
"REF ; $\\mathcal {F}$ is hard to compute for four-qubit mixed states [47].",
"For electrons, $\\mathcal {LN}(\\rho _\\textrm {II})= \\mathcal {LN}(\\rho _\\textrm {II,cuts})$ where $\\rho _\\textrm {II,cuts} \\propto e^{-\\beta \\hat{H}_{\\textrm {II,cuts}}}$ and $\\hat{H}_\\textrm {II,cuts} = \\Delta (f_{22^{\\prime }} f_{\\bar{2} \\bar{2}^{\\prime }}+f_{\\bar{3} \\bar{3}^{\\prime }} f_{33^{\\prime }}+\\textrm {H.c})$ .",
"In contrast, for spins, the decomposition of $\\rho ^s_\\textrm {II}$ above implies $\\mathcal {LN}(\\rho ^s_\\textrm {II}) = \\mathcal {LN}(\\rho ^s_\\textrm {II,cut2})+ \\mathcal {LN}(\\rho ^s_\\textrm {II,cut3}) = 2 \\mathcal {LN}(\\rho ^s_\\textrm {II,cut2})$ .",
"For both $\\rho _\\textrm {II}(T=0)$ and $\\rho ^s_\\textrm {II}(T=0)$ , $\\mathcal {LN}$ equals 2 and decreases at larger $T$ .",
"$\\mathcal {LN}$ does not distinguish between the nonlocal entanglement of $\\rho _\\textrm {II}(T=0)$ and the local one of $\\rho ^s_\\textrm {II}(T=0)$ .",
"However, it distinguishes at finite $T$ ($ > T_{\\text{SB}}^{\\text{NL}}$ ), showing a larger value for $\\rho _\\textrm {II}(T)$ .",
"The excess $\\mathcal {LN}(\\rho _\\textrm {II}(T)) - \\mathcal {LN}(\\rho ^s_\\textrm {II}(T))$ is a marker of the breakdown of the fermion-spin correspondence, i.e., the nonlocality of entanglement between B and AC in $\\rho _\\textrm {II}(T)$ .",
"Interestingly, the excess starts to appear at $T = T_\\textrm {SB}^\\textrm {NL}$ , increases with $T$ till $T^\\textrm {L}_\\textrm {SD}$ (at which the local entanglement of $\\rho ^s_\\textrm {II}$ vanishes), then decreases with $T$ , vanishing at $T^\\textrm {NL}_\\textrm {SD}$ .",
"$d \\mathcal {LN} / dT$ is discontinuous at $T_\\textrm {SB}^\\textrm {NL}$ , $T^\\textrm {L}_\\textrm {SD}$ , $T^\\textrm {NL}_\\textrm {SD}$ , which are related to entanglement sudden birth and death [44].",
"It is nontrivial that the quantum nonlocality induced by the fermion statistics is more visible at finite $T$ .",
"This may be because the marker is based on a bipartite entanglement measure rather than a multipartite (4-qubit) one.",
"Figure: Temperature dependence of entanglement (ℒ𝒩\\mathcal {LN}) between B and AC for the cases of the electron chain of H ^ II \\hat{H}_\\textrm {II} and the corresponding spin chain.The electrons have ℒ𝒩\\mathcal {LN} greater than or equal to the spins.",
"The excess indicates nonlocality of the entanglement in the electrons as the entanglement is local in the spins.",
"The nonlocality is visible in T∈(T SB NL ,T SD NL )T \\in (T_\\textrm {SB}^\\textrm {NL}, T^\\textrm {NL}_\\textrm {SD}).Inset: dℒ𝒩/dTd \\mathcal {LN} / dT is discontinuous at T SB NL T_\\textrm {SB}^\\textrm {NL}, T SD L T_\\textrm {SD}^\\textrm {L}, T SD NL T^\\textrm {NL}_\\textrm {SD}.Conclusion.— We find nonlocal entanglement in 1D electrons having end Majoranas.",
"It is independent of subsystem lengths, occurs in the bulk with protection by energy gap, and survives up to certain temperature.",
"Its form (Bell state vs. cluster state) and its finite-temperature behavior depend on the number of the end Majoranas.",
"This is an entanglement version of bulk-edge correspondence for the 1D electrons.",
"The nonlocal entanglement will exist universally in general 1D fermions belonging to BDI and D classes, provided that their subsystems A, B, C are longer than Majorana localizaion length $\\xi $ .",
"($\\xi $ was zero in our models.)",
"It is because the nonlocal entanglement originates from Majorana non-Abelian fusion and the fermion statistics, not depending on system specifics.",
"It will be interesting to rigorously confirm this universality.",
"Our results illustrate a number of interesting aspects of fermion entanglement.",
"First, the fermion statistics plays a crucial role in entanglement, especially for mixed states: It can generate quantum nonlocality.",
"Due to this, the fermion-spin correspondence established [21], [6] for pure states breaks down: The thermal state $\\rho _\\textrm {I}$ of electrons ($\\rho _\\textrm {I}^s$ of spins) has nonlocal (no) entanglement.",
"For the thermal state $\\rho _\\textrm {II}$ of electrons ($\\rho _\\textrm {II}^s$ of spins), the amount of entanglement between B and AC can differ from (always equals) the sum of local entanglement at the two cuts.",
"Even for pure states, the form of entanglement can differ between the two connected by JWT; e.g., cluster in $|n \\rangle _\\textrm {II}$ vs. $\\textrm {Bell}\\otimes \\textrm {Bell}$ in $|n \\rangle _\\textrm {II}^s$ .",
"Second, bipartite mixed-state entanglement measures unveil nontrivial temperature dependence of the nonlocal entanglement.",
"Their sudden birth and death provide a characteristic example of finite-temperature behavior of topological phases, which is in stark contrast with usual quantum-to-classical crossover.",
"Third, the electron thermal states are mixtures of states of different parity.",
"For the case where the parity mixing is not allowed for some purposes, e.g., where only states of even parity are thermally mixed, our results of $\\mathcal {LN}$ and $\\mathcal {F}$ provide a lower bound of the entanglement [30], [28].",
"Hence for that case, nonlocal entanglement is also generated by the fermion statistics.",
"Fourth, our finding will be useful for generalizing and characterizing the notion of topological phases to thermal states, overcoming difficulties faced by tools for pure states such as ground-state degeneracy, bulk-edge correspondence, topological entanglement entropy [48], [49].",
"We thank E. Berg, R. Fazio, P. Fendley, F. Pollmann, and S. Ryu for valuable discussions, and the support by Korea NRF (Grant Nos.",
"2015R1A2A1A15051869 and 2016R1A5A1008184)."
]
] | 1808.08505 |
[
[
"Preflight Characterization of the BLAST-TNG Receiver and Detector Arrays"
],
[
"Abstract The Next Generation Balloon-borne Large Aperture Submillimeter Telescope (BLAST-TNG) is a submillimeter mapping experiment planned for a 28 day long-duration balloon (LDB) flight from McMurdo Station, Antarctica during the 2018-2019 season.",
"BLAST-TNG will detect submillimeter polarized interstellar dust emission, tracing magnetic fields in galactic molecular clouds.",
"BLAST-TNG will be the first polarimeter with the sensitivity and resolution to probe the $\\sim$0.1 parsec-scale features that are critical to understanding the origin of structures in the interstellar medium.",
"BLAST-TNG features three detector arrays operating at wavelengths of 250, 350, and 500 $\\mu$m (1200, 857, and 600 GHz) comprised of 918, 469, and 272 dual-polarization pixels, respectively.",
"Each pixel is made up of two crossed microwave kinetic inductance detectors (MKIDs).",
"These arrays are cooled to 275 mK in a cryogenic receiver.",
"Each MKID has a different resonant frequency, allowing hundreds of resonators to be read out on a single transmission line.",
"This inherent ability to be frequency-domain multiplexed simplifies the cryogenic readout hardware, but requires careful optical testing to map out the physical location of each resonator on the focal plane.",
"Receiver-level optical testing was carried out using both a cryogenic source mounted to a movable xy-stage with a shutter, and a beam-filling, heated blackbody source able to provide a 10-50 $^\\circ$C temperature chop.",
"The focal plane array noise properties, responsivity, polarization efficiency, instrumental polarization were measured.",
"We present the preflight characterization of the BLAST-TNG cryogenic system and array-level optical testing of the MKID detector arrays in the flight receiver."
],
[
"INTRODUCTION",
"The Next Generation Balloon-borne Large Aperture Submillimeter Telescope (BLAST-TNG) is a submillimeter mapping experiment which features three microwave kinetic inductance detector (MKID) arrays operating over 30% bandwidths centered at 250, 350, and 500 $$ m (1200, 857, and 600 GHz).",
"These highly-multiplexed, high-sensitivity arrays, featuring 918, 469, and 272 dual-polarization pixels, for a total of 3,318 detectors, are coupled to a 2.5 m diameter primary mirror and a cryogenic optical system providing diffraction-limited resolution of 30$^{\\prime \\prime }$ , 41$^{\\prime \\prime }$ , and 50$^{\\prime \\prime }$ respectively.",
"The arrays are cooled to $\\sim $ 275 mK in a liquid-helium-cooled cryogenic receiver which will enable observations over the course of a 28-day stratospheric balloon flight from McMurdo Station in Antarctica as part of NASA's long-duration-balloon (LDB) program, planned for the 2018/2019 winter campaign.",
"BLAST-TNG is the successor to the BLASTPol and BLAST balloon-borne experiments which flew five times between 2005 and 2012[1], [2].",
"Achieving diffraction-limited, sub-arcminute resolution and telescope pointing accuracy is one of the highest priorities for the success of the BLAST-TNG mission.",
"Although the science goals of BLAST-TNG are similar to the 2012 BLASTPol mission, most of the major instrument systems have been rebuilt and improved since the last flight.",
"A new 2.5 m aperture Cassegrain telescope, featuring a lightweight composite carbon fiber reinforced polymer (CFRP) primary mirror designed and built by Alliance Spacesystems,4398 Corporate Center Dr, Los Alamitos, CA 90720 will enable an increase in resolution to 30$^{\\prime \\prime }$ at 250 $$ m, from BLASTPol's 2.5$^{\\prime }$ at the same band.",
"With improved detector sensitivity and a increase in detector count by a factor of 12, we expect BLAST-TNG will have more than six times the mapping speed of BLASTPol.",
"The new cryostat has demonstrated a 28 day hold-time, enabling observations of many more targets at greater depth than were possible during the $\\sim $ 13 day BLASTPol flight in 2012.",
"The primary science goal of BLAST-TNG is to map the polarized thermal emission from galactic interstellar dust around star-forming regions and in the diffuse interstellar medium (ISM).",
"These maps will yield $\\sim $ 250,000 polarization vectors on the sky, allowing us to explore correlations between the magnetic field dispersion, polarization fraction, cloud temperature, and column density.",
"Quantifying the relationships between these variables over a large sample of clouds will yield testable relationships which can be fed back into numerical simulations.",
"The Planck satellite has observed strong correlations between the orientation of Galactic magnetic fields and large-scale ISM structures [3], as well as the interior of giant molecular clouds (GMCs)[4].",
"While BLASTPol was able to observe the magnetic fields within GMCs at higher resolution than Planck [2], [5], BLAST-TNG will be the first experiment to probe the fields within the characteristic filamentary structures within GMCs observed by Herschel [6].",
"Combining the BLAST-TNG data with molecular cloud simulations, [7] and numerical models of dust emission [8] and grain properties, [9] will give unprecedented insight into the interplay between the gravitational, turbulent and magnetic field contributions to star and cloud formation, as well as the physics of grain alignment and mass flow within the interstellar medium.",
"Polarized dust emission is also the dominant foreground for observations of the cosmic microwave background (CMB).",
"Characterization of these foregrounds is one of the most important requirements in the search for the gravitational wave signature of cosmic inflation [10].",
"While the power spectrum from polarized dust foregrounds is thought to be lowest at small angular scales, there is limited high-resolution observational data of the diffuse ISM [4], [11].",
"BLAST-TNG will be able to make the deepest maps to date of the dust emission in the types of dark, diffuse regions of the sky favored by state of the art CMB polarization experiments.",
"BLAST-TNG will probe angular scales not well-characterized to date, and explore correlations between diffuse dust emission and structures in the cold neutral medium [12] at submillimeter wavelengths where the intensity of the thermal dust signal dominates.",
"With its high pixel count and photon-noise-limited detectors, BLAST-TNG will produce maps of diffuse ISM with higher fidelity than the highest frequency Planck polarization maps at 353 GHz."
],
[
"Cryogenic Receiver",
"Success of BLAST-TNG's observational goals relies on making sensitive and stable observations of a large sample of molecular clouds, and a varied sample of regions of the ISM with different densities and radiative environments.",
"This is enabled primarily through by the high-sensitivity MKID detector arrays [13], [14], careful control of polarization systematics, and crucially, a cryogenic receiver that which operates autonomously for the full 28 day flight."
],
[
"Cold Optics",
"The cryogenic receiver encloses a series of cold re-imaging optics arranged in a modified Offner relay configuration.",
"A similar configuration was flown in the BLAST/BLASTPol optics box.",
"The design features a cold Lyot stop at the image of the primary mirror which significantly reduces the optical loading on the detectors from stray light.",
"The optics bench, shown in Fig.",
"REF , cools the 4 K reimaging optics, and supports the band-defining filters which split the telescope beam to the three focal plane arrays.",
"Details of the optical design can be found in Lourie, et.",
"al[15], and in refs.",
"tyrblasttngspie,bradblastspie.",
"Figure: Photograph of the BLAST-TNG 4 K reimaging optics with critical components labeled."
],
[
"Cryostat Design",
"To simplify the mechanical design and minimize the number of pressure vessels, the cryostat is based on liquid helium-only system.",
"A 250-L liquid helium tank cools the optics and cold electronics to 4 K, and backs the operation of the sub-Kelvin refrigeration system, described in further detail in Section REF , and in Galitzki, et.",
"al.",
"[16].",
"The 4 K cold plate is integrated into the tank, and forms the lower cap of the liquid helium dewar as shown in Fig.",
"REF .",
"The cold plate has a domed center to maximize structural rigidity while reducing mass.",
"The optics bench is bolted to the thick rim of the tank and located with precision alignment features machined around the perimeter.",
"By mounting the optics to the perimeter, the optics are isolated from pressure-induced bowing at the center of plate.",
"The 4 K optics cavity is enclosed by an 1100-series aluminum shroud.",
"The interior of this shroud is coated with an absorptive coating made from Stycast 2850-MT/Cat 23LVHenkel Adhesives, North America mixed with 10% by mass of powdered charcoalGeneral Pencil Company, Inc. Redwood City, CA to absorb stray in-band light as well infrared radiation to prevent indirect loading of the sub-Kelvin components[18].",
"All housekeeping electronics are accessed via the top lid of the cryostat.",
"A series of six pass-through pipes are welded into the liquid tank to allow wiring, coaxial connections from the detector focal plane readout, and axles from the cryogenic actuators to be passed directly from the cold-plate to the top of the cryostat.",
"These pass-throughs can be accessed by removing the top lids of the vacuum shell and vapor-cooled shields, so that making changes to the wiring harnesses does not require disassembly or removal of all of the cryostat shells.",
"Figure: Cross-section render of the BLAST-TNG cryostat with critical components labeled.Two intermediate thermal shrouds made of 1100-series aluminum enclose the 4 K volume to reduce the conductive and radiative loading on the liquid helium bath.",
"As the liquid helium boils, the vapor is forced through two copper spiral heat exchangers [19], one bolted to each of these intermediate vapor-cooled stages (VCS), cooling the first intermediate stage (VCS1) to 40 K and the second stage (VCS2) to 140 K. The entrance and exit apertures of the spiral heat exchangers sit within the helium fill port, which is the only port attached to the helium tank.",
"Two spring-loaded PTFE plugs seal the fill port at each VCS stage, allowing vapor pressure in the tank to build to 25 mbar above atmospheric pressure and force the cold helium gas from the cryogen boil-off through the heat exchangers[19].",
"These plugs are removed during cryogen transfers.",
"A TAVCOTAVCO Sales & Service Company, Inc. Gilbert, AZ 1-atm absolute pressure valve regulates the pressure at the outlet of the VCS2 heat exchanger.",
"The 4, 40, 140, and 300 K stages are separated by G10 fiberglass cylinders with wall thicknesses of 0.5, 1.0, and 1.6 mm respectively.",
"The G10 cylinders are assembled from epoxy-coated woven fiberglass, and are assembled such that the warp of the G10 fibers is oriented circumferentially around the cylinders which reduces the effective thermal conductivity of the supports [20].",
"The VCS stages are a highly coupled system, and the cooling power of the heat exchangers is proportional to the boiloff rate of the helium bath, providing negative thermal feedback to the 4 K stage.",
"The VCS typically reach ±5 K of their equilibrium temperatures within 48 hours of the initial liquid helium transfer and reach equilibrium temperatures stable to <1 K within 4 days.",
"If not properly controlled, infrared loading through the cryostat window can dominate the loading at each stage.",
"A series of metal-mesh band-defining and thermal/infrared blocking filters, and low-pass band-defining filters reflects infrared light back out of the cryostat, while passing submillimeter wavelengths through to the cold optics and FPAs.",
"The filter arrangement at each temperature stage has been adjusted between “light runs\" with the window installed, in order to optimize the infrared rejection and maximize in-band transmission.",
"The radiative load on the 4 K stage is particularly sensitive to the filter arrangement at the two VCS.",
"The low-pass filters reject out-of-band submillimeter light, but absorb infrared wavelengths, which must be rejected earlier in the filter stack.",
"Where allowed by space constraints, the filters are tipped at opposing angles to reduce Fabry-Perot resonances and multiple reflections."
],
[
"Sub-Kelvin Refrigeration System",
"The BLAST-TNG focal plane arrays are cooled to $\\sim $ 275 mK via a closed-cycle $^3$ He sorption refrigerator, backed by a $\\sim $ 1 K superfluid, pumped $^4$ He volume (the “pumped pot\"), which draws liquid helium from the main liquid helium tank.",
"The $^3$ He refrigerator is a copy of that flown on the BLASTPol and BLAST experiments[1], and built for the MUSTANG instrument on the Green Bank Telescope[21].",
"Figure: Photograph of the BLAST-TNG cold plate, showing the cold optics box covered by the magnetic shielding, and the sub-Kelvin refrigeration components.The geometry of the BLAST-TNG superfluid system is designed to minimize the consumption of liquid helium during operation, and run as cold as possible to reduce loading on the FPAs.",
"The flow of liquid helium into the pumped pot is regulated by a 0.25 mm diameter rate-limiting capillary, and can be turned on and off by a cryogenic valveSwagelok Solon, OH.",
"This is contrasted by similar systems in the BLASTPol and SPIDER cryostats in which the pumped-superfluid pot is continually filled by a smaller capillary [1], [22].",
"While introducing a cryogenic valve increases the complexity and risk of in-flight valve failure, it greatly reduces the consumption of liquid helium over the course of the flight and the larger capillary size reduces the risk clogs or ice-plugs.",
"Additionally, stopping the flow of liquid helium into the pot lowers the base temperature of the system, since the pot will operate at a lower vapor pressure for a given pumping speed.",
"During flight, the pot is pumped to the ambient pressure at $\\sim $ 35 km altitude through a 19 mm diameter pump tube.",
"We expect in-flight operation below 1.3 K, an improvement from the 1.8 K BLASTPol system [23].",
"The 4 K valve is actuated via a G10 fiberglass shaft, through a ferrofluidic feed-throughFerroTec Corporation, Santa Clara, CA, driven by a geared stepper motor mounted outside the vacuum shell on the top of the cryostat.",
"The diameter of the capillary was chosen such that the 200 mL pumped-pot can be completely filled in less than half an hour.",
"While filling, some amount of helium entering the pot will be pumped directly through the pot and into the pump tube rather than collecting.",
"By measuring the flow rate of helium gas through the pump with the 4 K valve open and closed, we find the flow rate of liquid helium through the capillary to be 13.3 mL/min, which collects in the pot at a rate of 8.3 mL/min, a filling efficiency of $\\sim $ 60%.",
"With the valve open, the pot temperature rises above 2 K. The capillary diameter is such that if the pot valve were to be stuck in its open position due to a mechanical failure, the flight maximum flight time would be reduced to 13 days before the full helium tank is depleted.",
"The helium consumption of the pumped-pot during operation can be quantified by calculating an average equivalent thermal load which would consume the same volume of liquid helium over the course of the flight.",
"The pot is sized such that the $^3$ He refrigerator can be recycled without refilling, and in practice must be refilled every $\\sim $ 3 days.",
"By turning off the flow of helium into the pumped-pot when it is full, we reduce the equivalent thermal load compared to BLASTPol by nearly 85%, from 23 mW to 3.5 mW, even while increasing the capillary diameter from 0.038 to 0.25 mm.",
"The cryostat and the sub-Kelvin system must be able to operate entirely autonomously.",
"While commanding and communications from the ground will be available during the flight, the telemetry bandwidth is limited and equipment failures could cause contact to be lost completely.",
"Housekeeping thermometry is continuously read out via a combination of custom thermometry bias/demodulation electronics and commercial off-the shelf data acquisition hardwareLabJack Corporation, Lakewood, CO. Any time the array temperatures exceed threshold values, the flight computer triggers the pot valve to open and recycle the $^3$ He sorption refrigerator.",
"The thermal loading is low enough that lab testing indicates that the sorption refrigerator will have to be cycled only once every every 4-5 days.",
"The cryostat, designed at the University of Pennsylvania[24] and built by Precision Cryogenic SystemsPrecision Cryogenic Systems, Indianapolis, IN, was first delivered in October, 2015.",
"Preliminary testing indicated the presence of excess loading on each of the thermal stages.",
"During dark tests with the windows covered at each thermal stage, VCS1/2 ran at 65 K and 165 K respectively, and the loading on the liquid helium bath was $\\sim $ 40% larger than modeled, corresponding to a shortened 22.5 day hold time.",
"The excess loading was attributed to un-modeled radiative loads from light leaks between thermal stages and inadequate multilayer insulation (MLI) around feedthroughs and fixtures [16].",
"In June, 2017 the cryostat experienced a catastrophic cryogenic failure during a pre-cooling procedure with liquid nitrogen, when an ice plug on the fill port caused an over-pressurization of the liquid cryogen tank rupturing the tank welds.",
"The rupture caused liquid nitrogen to spill into the cavity between the tank and VCS1 where it flash-boiled.",
"The ensuing pressure wave destroyed most of the VCS and 4 K shrouds, the magnetic shielding around the optics box, the focal plane mounts, the plumbing for the sub-Kelvin refrigeration system and the housekeeping wiring.",
"Crucially, however, the cold optics, the cold plate, $^3$ He refrigerator, the heat exchangers, and the vacuum vessel were determined to be undamaged.",
"The cryostat was rebuilt and assembled, and underwent its first dark test in December, 2017.",
"The rebuilt BLAST-TNG cryogenic receiver performance has been validated during extensive laboratory testing and has benefited from key redesigns from its initial conception.",
"Rather than hand-cutting and wrapping single layers of polyester-fiber-backed aluminized mylar to form MLI blankets, custom-designed laser-cut 10-layer blankets of Coolcat 2 NWRUAG Space GmbH, Vienna, Austria were purchased.",
"Layering these blankets provided 10, 20, and 30 layers of aluminized mylar at the around the 4 K, 40 K, and 140 K stages.",
"The laser-cut slits for the various housekeeping components, along with the addition of metallic baffles around the motor axle shafts significantly reduced the light leaks between stages.",
"Dark testing indicates that the excess loading at each stage has been reduced, and the performance matches the modeled 28 day hold time."
],
[
"MKID Detector Arrays",
"The BLAST-TNG detectors are based on arrays of Microwave Kinetic Inductance Detectors (MKIDS).",
"MKIDs are superconducting, lumped element inductive/capacitive (LC) circuits with a resonant frequency and quality factor that is sensitive to changes in incident radiation.",
"Absorbed radiation with enough energy to break Cooper pairs in the circuit causes a change in the kinetic inductance of the device, changing its impedance and causing a shift in the resonant frequency[25].",
"MKIDs can be highly multiplexed, and arrays can be formed by capacitively coupling multiple resonators to a single transmission line.",
"BLAST-TNG achieves multiplexing factors up to $\\sim $ 1000 (see Table REF ) using a `tile-and-trim' approach in which arrays of resonators with identical inductive absorbers and capacitive elements of multilayer TiN/Ti/TiN films are laid out on a silicon substrate, before trimming the capacitors with a deep-reactive-ion-etch to uniquely tune the resonant frequency of each resonator [26].",
"Fabrication errors and wafer non-uniformity can cause displacement of the resonances from their designed frequencies, leading to an ambiguity in the mapping between resonant frequency and physical location on the array.",
"Collisions or overlap between resonances can lead to unusable detectors and reduced yield.",
"The physical location of each resonator was mapped at NIST-Boulder using an custom array of optical light-emiting diodes (LEDs) designed for each FPA [27], [28].",
"The three BLAST-TNG detector arrays are shown in Fig.",
"REF .",
"The 350 and 500 $$ m arrays are both read out on a single transmission line, while the 250 $$ m array is split into three identical rhombus-shaped subarrays, each with its own transmission line.",
"By using the long, thin inductive element of the MKID as the absorber itself, each resonator is sensitive to single linear polarization.",
"Dual-polarization pixels are formed by coupling two single-polarization-sensitive detectors to a single feedhorn, in a crossoverless configuration which achieves less than 3% cross-polar coupling [29].",
"Single-pixel testing with a temperature-controlled blackbody load indicate that the detectors are photon-noise-limited for at their 275 mK operating temperature and 15 pW expected optical loading [30], [26].",
"Table: BLAST-TNG Focal Plane ArraysThe BLAST-TNG detector arrays are read out using a highly multiplexed, field-programmable gate array (FPGA)-based digital spectrometer.",
"This readout is the first of its kind to have been developed for the second generation Reconfigurable Open Architecture Computing Hardware (ROACH-2) board developed by the CASPER collaboration[31].",
"Each readout module includes a ROACH-2 board, MUSIC [32] digital-to-analog and analog-to-digital converter boards (DAC/ADC) and a set of analog radio-frequency (RF) components, which are housed in a custom enclosure.",
"Using firmware designed for BLAST-TNG, a single board is capable of simultaneous readout of over 1000 detectors at a rate of 488 Hz, over 512 MHz of RF bandwidth.",
"Each ROACH-2 module generates a baseband carrier waveform containing the resonant frequencies of each detector.",
"The carrier signal, which is multiplexed on a single coaxial cable, is upconverted to RF and passed to the detector array, where its phase is modulated by the sky signal.",
"The carriers are then amplified by a $\\sim $ 4 K SiGe cryogenic low-noise amplifier (developed at Arizona State University), converted to baseband, and looped back into the ROACH-2, where they are digitized, demodulated, and stored to disk.",
"Five readout modules (three for the 250 $$ m array, one each for the 350 and 500$$ m arrays) are mounted in an enclosure mounted on the balloon gondola frame.",
"Details of the readout and pre-flight demonstration are presented in Gordon et al.",
"2017 [33].",
"Figure: Photographs of each of the three focal plane arrays before mounting in their carriers.",
"The photolithographed MKIDs can be seen, along with the meandering transmission line on which they are read out."
],
[
"Detector Integration",
"The first detector tests in the rebuilt BLAST-TNG receiver began in February, 2018.",
"The 350 $$ m array was installed first and run completely in the dark to characterize the non-optical thermal loading on the array.",
"Each array is mounted on a rigid carbon fiber mount which mounts to the 4 K optics box, and supports the array off of a 1.4 K intercept stage.",
"These mounts conduct less than 1 $$ W per array of thermal power to $^3$ He refrigerator[17].",
"During dark tests, the array operated successfully at 275 mK.",
"The first cryogenic run with the windows and filters installed was conducted in April, 2018 with both the 250 and 350 $$ m FPAs installed.",
"The optical loading on the detectors did not affect the operating temperature, allowing for preliminary optical testing.",
"As of early May 2018, all three MKID FPAs are mounted in the receiver in flight configuration.",
"Initial results indicate that all arrays are fully operational, with high detector yield and expected sensitivity levels.",
"Vector network analyzer (VNA) sweeps for each array taken in the flight receiver are shown in Fig.",
"REF .",
"Figure: VNA sweeps for each of the arrays taken in the BLAST-TNG flight receiver.",
"Data has been high-pass filtered to remove ripple and slope from cable attenuation.",
"The sweep shown from the 250 m array is for one of the three identical subarrays of MKIDs that make up the full FPA.Understanding the response of the receiver to polarized light is critical to characterizing the instrument and analyzing the maps made during the flight.",
"The detector cross-polar coupling has been measured at NIST to be less than 3% [29].",
"The polarization efficiency and cross-polar response of the receiver is measured by observing the response to a chopped thermal source which provides a near-square-wave chop between 300 K and 315 K. The source fills the receiver beam and provides a uniform signal across most of the array.",
"To ensure the detectors operate when viewing a 300 K blackbody we place a 4 K 4% neutral density filter (NDF) at the entrance to the optics box.",
"A metal-mesh polarizing grid provided by Cardiff University is placed in front of the cryostat and mounted at a 45$^{\\circ }$ angle to the optical axis.",
"The angled grid passes linearly-polarized light parallel to the grid orientation, and reflects perpendicularly-polarized light to an absorber outside the cryostat.",
"Detector time streams are recorded using the flight ROACH-2 readout hardware during chops and the angle of the polarizer grid is stepped to measure the response as a function of polarization angle.",
"Detector time streams measured during preliminary measurements are shown in Fig.",
"REF for representative X and Y-polarized resonators on the 250 $$ m array.",
"Initial results, without the HWP installed, indicate a maximum cross-polar coupling of 4-6% across all three arrays.",
"The degree of instrumentally-induced polarization signal in the receiver is measured by using the 4 K half wave plate (HWP) in the receiver cold optics.",
"Instrumental polarization is characterized by repeating the same observations of the chopped thermal source with the external grid at a fixed angle while stepping the HWP.",
"Repeating these measurements at different grid angles allows polarization effects inherent to the receiver design to be identified and accounted for during data analysis.",
"Initial polarization characterization was done with the HWP removed from the system, while improvements were made to the heat-strapping of the HWP rotator.",
"The HWP has been reinstalled in the system, and will undergo full characterization during June, 2018.",
"Figure: Preliminary polarization response demonstration of the 250 m FPA, showing response to a 300 K/77 K chop viewed through a polarizer grid at different angles.",
"Upper plot shows response of an X-Pol resonator and the lower plot shows a Y-Pol.",
"Response is in arbitrary uncalibrated units read in using the ROACH electronics in flight configuration.",
"Chop time streams are arbitrarily offset along the y-axis for better visibility.",
"Polarizer grid absolute angles are arbitrary and are not referenced to the detector antenna axis."
],
[
"Noise",
"The BLAST-TNG detector arrays have been demonstrated to be photon-noise-limited at the expected in-flight optical loading [30], and a number of improvements have been made to the receiver RF system to maintain this low-noise performance.",
"In order to minimize the conductive thermal loading on the liquid helium bath and the two VCS, we use thin (0.86 mm diameter) stainless steel coaxial cable COAX CO. LTD., Kanagawa, Japan between 4 K and 300 K. These cables have sufficiently low thermal conductivity, but have relatively high signal attenuation, contributing to a round-trip signal attenuation of $\\sim $ -30 dBm.",
"Maintaining high signal-to-noise detector operation requires sufficient cold amplification, which is achieved with 4 K SiGe low-noise amplifiers designed by Arizona State University.",
"Increasing the bias power of the amplifiers increases their gain, but also increases the load on the liquid helium bath and reduces the hold time of the cryostat.",
"By cascading two SiGe amplifiers we achieve sufficient gain and demonstrated that the comparing the white noise level (away from the MKID resonant frequencies) of the cold amplifiers exceeds that of the warm readout electronics.",
"Initial tests of the cascaded amplifier scheme with the 250 $$ m array indicate that the detector noise level exceeds that of the full readout chain.",
"By comparing the white noise level on and off resonance we have demonstrated that the system is detector-noise limited across most of the readout band.",
"Though work is still ongoing to optimize the readout tone power for each detector, we expect the receiver and readout architecture to achieve photon-noise-limited performance in flight.",
"Figure: Response of three MKIDs on the 250m array to different beam-filling optical loads, measured through the BLAST-TNG cryogenic receiver with a vector network analyzer.",
"The three curves show a different optical load: a metal plate covering the window of the receiver (blue), a 300 K blackbody (orange), and a 300 K blackbody viewed through a 2.85 % neutral density filter (NDF).",
"The resonator towards the center of the figure is a dark pixel (non-optically-coupled), while the resonators on either side of the figure are feedhorn-coupled to the cold optics.",
"As the optical power is increased, the resonant frequencies and quality factors of the optically-coupled resonators decreases.",
"The dark pixels show no change with the optical load.",
"There is a clear difference in the quality factor and resonator depth between the dark and light pixels, due to the optical loading.",
"We expect the optical loading in flight to be significantly reduced compared to these laboratory tests."
],
[
"Responsivity",
"The sensitivity of the receiver is determined by measuring the detector response through the full cold optical system to known optical loads.",
"For these measurements to be accurate, the optical load must fill the entire beam of the receiver.",
"The same chopped optical source used in the polarization measurements was stepped in 5 K steps from 300 K to 330 K. At each temperature, the resonant frequencies of the MKIDs were determined using the ROACH-2 readout in vector network analyzer (VNA) mode, and the response to a 1 Hz chop between the room temperature and hot blackbody loads was recorded.",
"Additional data was recorded viewing a 300 K blackbody through a room-temperature 2.85% neutral density filter (NDF), and viewing a 77 K liquid nitrogen source.",
"The response of approximately a dozen dark (not-feedhorn-coupled) MKIDs on each array can put limits on the cross-talk between pixels.",
"During these laboratory tests, a 4.0% NDF was placed at the entrance to the 4 K optics box to reduce the optical load and ensure the detectors operated when viewing the 300 K thermal load.",
"The response of several resonators is shown in Fig.REF .",
"Full characterization of the response of each array is still ongoing."
],
[
"Spectral Response",
"The bandpass of each FPA will be measured with a Fourier transform spectrometer (FTS).",
"An FTS, designed to operate at the BLAST-TNG wavebands, has been built at the University of Pennsylvania in collaboration with Cardiff University, and is currently being tested using the BLAST-TNG receiver.",
"These measurements are ongoing, and we expect to finalize the bandpass measurements before summer 2018."
],
[
"Conclusion",
"BLAST-TNG will be the most sensitive submillimeter polarimeter to date, and will build on the observing techniques developed for BLAST-Pol to make deeper maps of more science targets at higher resolution.",
"The cryogenic receiver, MKID arrays, and ROACH-2-based readout are operating in flight configuration.",
"Array level optical characterization data is still being analyzed, but initial results indicate the detector arrays in the flight receiver will maintain the photon-noise-limited performance demonstrated during single-pixel testing.",
"The receiver will be integrated with the balloon gondola and telescope during pre-flight systems integration at the NASA Columbia Scientific Ballooning Facility in Palestine, TX, in preparation for a planned 28 day stratospheric balloon flight from McMurdo Station, Antarctica during the winter of 2018/2019.",
"The BLAST-TNG collaboration acknowledges the support of NASA under award numbers NNX13AE50G and 80NSSC18K0481, and the NNX13CM03C.",
"Detector development is supported in part by NASA through NNH13ZDA001N-APRA.",
"J.D.S.",
"acknowledges the support from the European Research Council (ERC) under the Horizon 2020 Framework Program via the Consolidator Grant CSF-648505.",
"S.G. is supported through a NASA Earth and Space Science Fellowship (NESSF) NNX16AO91H.",
"The BLAST-TNG telescope is supported in part through the NASA SBIR/STTR office and developed at Alliance Spacesystems.",
"The BLAST-TNG collaboration would like to acknowledge the Xilinx University Program for their generous donation of five Virtex-6 FPGAs for use in our ROACH-2 readout electronics.",
"The collaboration also acknowledges the extensive machining, design, and fabrication efforts of Jeffrey Hancock and Harold Borders at the University of Pennsylvania and Matthew Underhill at Arizona State University, as well as Richard Gummer and the team at Precision Cryogenics Inc, without whose dedication the development and subsequent rebuild of the receiver would not have been possible.",
"The BLAST-TNG team also recognizes the contribution of undergraduate and post-baccalaureate interns to the receiver development, especially Mark Giovinazzi, Erin Healy, Gregory Kofman, Ariana Martino, Aaron Mathews, Timothy McSorely, Michael Plumb, and Nathan Schor, and Stephen Russell who built the calibration chopper used in the receiver testing."
]
] | 1808.08489 |
[
[
"A Perspective on Unique Information: Directionality, Intuitions, and\n Secret Key Agreement"
],
[
"Abstract Recently, the partial information decomposition emerged as a promising framework for identifying the meaningful components of the information contained in a joint distribution.",
"Its adoption and practical application, however, have been stymied by the lack of a generally-accepted method of quantifying its components.",
"Here, we briefly discuss the bivariate (two-source) partial information decomposition and two implicitly directional interpretations used to intuitively motivate alternative component definitions.",
"Drawing parallels with secret key agreement rates from information-theoretic cryptography, we demonstrate that these intuitions are mutually incompatible and suggest that this underlies the persistence of competing definitions and interpretations.",
"Having highlighted this hitherto unacknowledged issue, we outline several possible solutions."
],
[
"Introduction",
"Consider a joint distribution over “source” variables $X_{0}$ and $X_{1}$ and “target” $Y$ .",
"Such distributions arise in many settings: sensory integration, logical computing, neural coding, functional network inference, and many others.",
"One promising approach to understanding how the information shared between $X_{0}$ , $X_{1}$ , and $Y$ is organized is the partial information decomposition (PID) [1].",
"This decomposition seeks to quantify how much of the information shared between $X_{0}$ , $X_{1}$ , and $Y$ is done so redundantly, how much is uniquely attributable to $X_{0}$ , how much is uniquely attributable to $X_{1}$ , and finally how much arises synergistically by considering both $X_{0}$ and $X_{1}$ together.",
"Unfortunately, the lack of a commonly accepted method of quantifying these components has hindered PID's adoption.",
"In point of fact, several proposed axioms are not mutually consistent.",
"And, to date, there is little agreement as to which should hold.",
"Here, we take a step toward rectifying these issues by bringing to light a potentially fundamental inconsistency in the intuitions commonly and often implicitly brought to bear upon information decomposition.",
"We make the intuitions quantitative by appealing to information-theoretic cryptography.",
"Taken together, our observations suggest that the context in which PID is applied should determine how its components are quantified.",
"Our development proceeds as follows.",
"sec:pid briefly describes the two-source PID.",
"sec:intuitions calls out the two distinct intuitions often used in interpreting PID.",
"sec:distribution introduces a prototype distribution that highlights the issues and we interpret it through the lenses of the two intuitions.",
"sec:secrets defines secret key agreement rates and computes them for the prototype distribution.",
"sec:natural then discusses how the two intuitions relate to secret key agreement rates and identifies when the latter result in viable decompositions.",
"Finally, sec:conclusion summarizes our findings and speculates as to how future developments can bring consistency to PID."
],
[
"Partial Information Decomposition",
"Two-source PID seeks to decompose the mutual information $I{X_{0} X_{1} : Y}$ between “sources” $X_{0}$ and $X_{1}$ and a “target” $Y$ into four nonnegative components.",
"The components identify information that is redundant, uniquely associated with $X_{0}$ , uniquely associated with $X_{1}$ , and synergistic: I$X_{0}$ $X_{1}$ : $Y$ = $\\operatorname{I_{\\partial }}\\left[ X_{0} \\cdot X_{1} \\rightarrow Y \\right]$ redundant + $\\operatorname{I_{\\partial }}\\left[ X_{0} \\rightarrow Y \\setminus X_{1} \\right]$ unique from $X_{0}$ + $\\operatorname{I_{\\partial }}\\left[ X_{1} \\rightarrow Y \\setminus X_{0} \\right]$ unique from $X_{1}$ + $\\operatorname{I_{\\partial }}\\left[ X_{0} X_{1} \\rightarrow Y \\right]$ .",
"synergistic Furthermore, the mutual information between $X_{0}$ and $Y$ is decomposed into two components: I$X_{0}$ : $Y$ = $\\operatorname{I_{\\partial }}\\left[ X_{0} \\cdot X_{1} \\rightarrow Y \\right]$ redundant + $\\operatorname{I_{\\partial }}\\left[ X_{0} \\rightarrow Y \\setminus X_{1} \\right]$ .",
"unique from $X_{0}$ And, similarly: I$X_{1}$ : $Y$ = $\\operatorname{I_{\\partial }}\\left[ X_{0} \\cdot X_{1} \\rightarrow Y \\right]$ redundant + $\\operatorname{I_{\\partial }}\\left[ X_{1} \\rightarrow Y \\setminus X_{0} \\right]$ .",
"unique from $X_{1}$ In this way, PID relates the four component informations.",
"However, it does not uniquely determine how to quantify them.",
"To do this, a definition must be supplied for one of them and then the others follow.",
"This allows for a range of choices.",
"In the case that one wishes to directly quantify the unique informations $\\operatorname{I_{\\partial }}\\left[ X_{0} \\rightarrow Y \\setminus X_{1} \\right]$ and $\\operatorname{I_{\\partial }}\\left[ X_{1} \\rightarrow Y \\setminus X_{0} \\right]$ , a consistency relation must hold when they are computed independently: $\\operatorname{I_{\\partial }}\\left[ X_{0} \\rightarrow Y \\setminus X_{1} \\right]$ + I$X_{1}$ : $Y$ = $\\operatorname{I_{\\partial }}\\left[ X_{1} \\rightarrow Y \\setminus X_{0} \\right]$ + I$X_{0}$ : $Y$ ."
],
[
"The Camel and the Elephant",
"There are two common ways of thinking about PID.",
"These approaches differ only in the (implied) directionality of cause and effect—a property unspecified by PID.",
"In the first approach, one thinks of $X_{0}$ and $X_{1}$ as “inputs” that, when combined, produce $Y$ , a “output”.",
"While seemingly helpful labels, their use already imports an unwarranted semantics to the relationship between the three random variables.",
"In this, it inadvertently begs the main issue we wish to raise here, while at the same time illustrating the issue.",
"When taking this view of PID, one generally asks questions such as “How much information in $X_{0}$ is uniquely conveyed to $Y$ ?”.",
"From this vantage, considering the role of the individual channels $X_{0}\\rightarrow Y $ and $X_{1} \\rightarrow Y $ might or might not help develop intuition.",
"Recalling the aphorism “a camel is a horse designed by committee”, we call this the camel intuition as particular input events $X_{0}$ and $X_{1}$ come together to describe an output $Y$ .",
"In the second approach, one considers $X_{0}$ and $X_{1}$ as “noisy observations” or “representations” of a single underlying object $Y$ .",
"When taking this view, one might ask a question such as “How much information in $Y$ is uniquely captured by $X_{0}$ ?”.",
"Under this, the individual channels $Y \\rightarrow X_{0} $ and $Y \\rightarrow X_{1} $ take on primary importance.",
"After the parable of the blind men describing an elephant, we call this the elephant intuition since particular objects $Y$ may be described by various, possibly partial, representations, $X_{0}$ and $X_{1}$ ."
],
[
"The Pointwise Unique Distribution",
"The pointwise unique distribution [2] is given by the events and probabilities displayed in tab:dist: at any time exactly one of $X_{0}$ or $X_{1}$ is a `1' or `2' and matches $Y$ , while the other is `0'.",
"Let's now interpret this distribution by adopting the camel and elephant intuitions in turn.",
"We will see that they provide contradictory interpretations of the relationships between the variables.",
"Table: The pointwise unique distribution.Adopting the camel intuition, we consider the ways in which $X_{0}$ influences $Y$ .",
"It is easy to see that half of the time (tab:dist's 1 and 3 rows) $X_{0}$ is unable to say anything about the state of $Y$ .",
"The other half of the time (the 2 and 4 rows) $X_{0}$ and $Y$ are perfectly correlated, while $X_{1}$ is ignorant as to their state.",
"Analogously, this is true when considering how $X_{1}$ influences $Y$ .",
"In this way, we interpret the distribution's PID as consisting entirely of unique informations.",
"The camel intuition is summarized in tab:intuitions.",
"When adopting the elephant intuition, however, a strikingly different picture emerges.",
"Taking the viewpoint of $Y$ , both single channel distributions $p(X_{0} |Y)$ and $p(X_{1} |Y)$ are identical.",
"So, any information shared with one must be redundantly shared with the other.",
"These channels do not allow one to determine the states of either $X_{0}$ or $X_{1}$ .",
"What is learned, however, is that exactly one of them matches $Y$ , while the other is `0'.",
"Furthermore, removing the remaining uncertainty in the values of $X_{0}$ and $X_{1}$ requires observing one of them—a synergistic effect.",
"The resulting elephant analysis is also summarized in tab:intuitions.",
"Table: Camel and elephant intuitions applied to tab:dist's pointwise unique distribution.The camel intuition takes the view that X 0 X_{0} and X 1 X_{1} supply YY with unique informations, though only one of them at a time.The elephant intuition takes the view that YY provides both X 0 X_{0} and X 1 X_{1} with the same information, but it gets erased on the way to exactly one of them.In short, the two directional PID interpretations lead to contradictory quantifications.",
"From the viewpoint of camels, elephant approaches create redundancy where there is none.",
"From the vantage of elephants, camels draw distinctions where none exist.",
"This has been discussed by Ref.",
"[3] regarding whether or not unique information should depend on I$X_{0}$ : $X_{1}$ .",
"From the camel's point of view, ignoring this as a constraint may “artificially correlate” $X_{0}$ and $X_{1}$ and thereby inflate redundancy.",
"This viewpoint can be more directly illustrated by considering the intermediate distribution from which $\\operatorname{I_{\\textrm {BROJA}}}\\!\\!$ [4]—an elephant—computes unique information for the pointwise unique distribution: Table: NO_CAPTIONFrom the elephant's view, I$X_{0}$ : $X_{1}$ is irrelevant."
],
[
"Secret Key Agreement",
"Secret key agreement is a fundamental concept within information-theoretic cryptography [5].",
"The central idea is that if three parties, Alice, Bob, and Eve, observe some joint probability distribution $ABE \\sim p(a, b, e)$ where Alice has access only to $a$ , Bob $b$ , and Eve $e$ , is it possible for Alice and Bob to agree upon a secret key of which Eve has no knowledge.",
"The degree to which they may generate such a secret key immediately depends upon the structure of the joint distribution $ABE$ .",
"It also depends upon whether Alice and Bob are allowed to publicly communicate.",
"Concretely, consider Alice, Bob, and Eve each receiving $n$ independent, identically distributed samples from $ABE$ —Alice receiving $A^n$ , Bob $B^n$ , and Eve $E^n$ .",
"A secret key agreement scheme consists of functions $f$ and $g$ , as well as a protocol for public communication ($h$ ) allowing either Alice, Bob, neither, or both to communicate.",
"In the case of a single party being permitted to communicate—say, Alice—she constructs $C = h(A^n)$ and then broadcasts it to all parties.",
"In the case that both parties are permitted communication, they take turns constructing and broadcasting messages of the form $C_i = h_i(A^n, C_{[0 \\ldots i-1]})$ (Alice) and $C_i = h_i(B^n, C_{[0 \\ldots i-1]})$ (Bob) [6].",
"Formally, a secret key agreement scheme is considered $R$ -achievable if for all $\\epsilon > 0$ : KA (1)= f(An, C) KB (2)= g(Bn, C) p(KA = KB = K) (3) 1 - IK : C En (4) 1n K̋ (5) R - where $(1)$ and $(2)$ denote the method by which Alice and Bob construct their keys $K_A$ and $K_B$ , respectively, $(3)$ states that their keys must agree with arbitrarily high probability, $(4)$ states that the information about the key which Eve—armed with both her private information $E^n$ as well as the public communication $C$ —be arbitrarily small, and $(5)$ states that the key consists of approximately $R$ bits per sample.",
"The greatest rate $R$ such that an achievable scheme exists is known as the secret key agreement rate.",
"Notational variations indicate which parties are permitted to communicate.",
"In the case that Alice and Bob are not allowed to communicate, their rate of secret key agreement is denoted $\\operatorname{S}(A : B ~||~ E)$ .",
"When only Alice is allowed to communicate their secret key agreement rate is $\\operatorname{S}(A \\rightarrow B ~||~ E)$ .",
"And, similarly, if only Bob is permitted to communicate.",
"When both Alice and Bob are allowed to communicate, their secret key agreement rate is denoted $\\operatorname{S}(A \\leftrightarrow B ~||~ E)$ .",
"In this, we modified the standard notation for secret key agreement rates to emphasize which party or parties communicate.",
"In the case of no communication, $\\operatorname{S}(A : B ~||~ E)$ is given by [7]: $\\operatorname{S}(A : B ~||~ E)$ = where $X Y$ denotes the Gács-Körner common random variable [8].",
"It is worth noting that this quantity does not vary continuously with the distribution and generically vanishes.",
"In the case of one-way communication, $\\operatorname{S}(A \\rightarrow B ~||~ E)$ is given by [9]: $\\operatorname{S}(A \\rightarrow B ~||~ E)$ = { IB : K | C - IE : K | C } where the maximum is taken over all variables $C$ and $K$ , such that the following Markov condition holds: $C K A BE$ .",
"It suffices to consider $K$ and $C$ such that $|K| \\le |A|$ and $|C| \\le |A|^2$ .",
"There are no such solutions for $\\operatorname{S}(A \\leftrightarrow B ~||~ E)$ , however both upper- and lower-bounds are known [6].",
"Let us now consider the pointwise unique distribution of tab:dist and the ability of $X_{0}$ and $Y$ to agree upon a secret key while $X_{1}$ eavesdrops.",
"Secret key agreement rates have been associated with unique informations before.",
"An upper bound on $\\operatorname{S}(A \\leftrightarrow B ~||~ E)$ —the intrinsic mutual information [10]—is known to not satisfy the consistency condition eq:consistency [11].",
"More recently, the relationship between a particular method of quantifying unique information and one-way secret key agreement has been considered [12].",
"This can be interpreted four different ways.",
"First, neither $X_{0}$ nor $Y$ may be allowed to communicate.",
"Second, only $Y$ can communicate.",
"Third, only $X_{0}$ is permitted to communicate.",
"Finally, both $X_{0}$ and $Y$ may be allowed to communicate.",
"Note that the eavesdropper $X_{1}$ is not allowed to communicate in any secret sharing schemes here.",
"Looking at this distribution, a general strategy becomes clear: both $X_{0}$ and $Y$ need some scheme to determine when they agree (the 2 and 4 rows).",
"Broadly, the only way in which both $X_{0}$ and $Y$ can come to understand if they match or not is if $X_{0}$ is permitted to broadcast whether she observed a 0 or not.",
"Therefore, in the instances where $X_{0}$ is not communicating there is no ability to agree upon a key: $\\operatorname{S}(X_{0} : Y ~||~ X_{1}) = \\operatorname{S}(Y \\rightarrow X_{0} ~||~ X_{1}) = {0}{}$ .",
"However, when $X_{0}$ is allowed communication a key can be agreed upon: $\\operatorname{S}(X_{0} \\rightarrow Y ~||~ X_{1}) = \\operatorname{S}(X_{0} \\leftrightarrow Y ~||~ X_{1}) = {1/2}{}$ .",
"It is known that $\\operatorname{S}(X_{0} \\leftrightarrow Y ~||~ X_{1}) = {1/2}{}$ due to the convergence of upper and lower bounds in this instance.",
"These rates are summarized in tab:skars.",
"Table: The variety of secret sharing schemes and their rates for thepointwise unique distribution of tab:dist."
],
[
"Directionality, Naturalness, and Consistency",
"We are now in a position to integrate the two intuitions with the results of secret key agreement rates.",
"The camel intuition, with the channels $X_{0} \\rightarrow Y $ and $X_{1} \\rightarrow Y $ taking center stage, most closely aligns with the one-way secret key agreement rates $\\operatorname{S}(X_{0} \\rightarrow Y ~||~ X_{1})$ and $\\operatorname{S}(X_{1} \\rightarrow Y ~||~ X_{0})$ .",
"This also agrees with sec:distribution's quantification (compare tab:intuitions,tab:skars): $\\operatorname{I_{\\partial }}\\left[ X_{0} \\rightarrow Y \\setminus X_{1} \\right]$ = $\\operatorname{S}(X_{0} \\rightarrow Y ~||~ X_{1})$ and $\\operatorname{I_{\\partial }}\\left[ X_{1} \\rightarrow Y \\setminus X_{0} \\right]$ = $\\operatorname{S}(X_{1} \\rightarrow Y ~||~ X_{0})$ .",
"The elephant intuition, with its focus on the channels $Y \\rightarrow X_{0} $ and $Y \\rightarrow X_{1} $ is more naturally aligned with the one-way secret key agreement rates $\\operatorname{S}(Y \\rightarrow X_{0} ~||~ X_{1})$ and $\\operatorname{S}(Y \\rightarrow X_{0} ~||~ X_{1})$ .",
"This again accords with sec:distribution's quantification: $\\operatorname{I_{\\partial }}\\left[ X_{0} \\rightarrow Y \\setminus X_{1} \\right]$ = $\\operatorname{S}(Y \\rightarrow X_{0} ~||~ X_{1})$ and $\\operatorname{I_{\\partial }}\\left[ X_{1} \\rightarrow Y \\setminus X_{0} \\right]$ = $\\operatorname{S}(Y \\rightarrow X_{1} ~||~ X_{0})$ .",
"There are, however, difficulties with these approaches.",
"The first difficulty concerns the camel intuition.",
"If the one-way secret key agreement rates $\\operatorname{S}(X_{0} \\rightarrow Y ~||~ X_{1})$ and $\\operatorname{S}(X_{1} \\rightarrow Y ~||~ X_{0})$ are used to quantify the unique informations $\\operatorname{I_{\\partial }}\\left[ X_{0} \\rightarrow Y \\setminus X_{1} \\right]$ and $\\operatorname{I_{\\partial }}\\left[ X_{1} \\rightarrow Y \\setminus X_{0} \\right]$ , respectively, the consistency relation given by eq:consistency is not necessarily satisfied.",
"Importantly, though, if $\\operatorname{S}(Y \\rightarrow X_{0} ~||~ X_{1})$ and $\\operatorname{S}(Y \\rightarrow X_{1} ~||~ X_{0})$ are used, the resulting PID is always consistent.",
"One concludes that the elephant intuition is the more natural of the two when using one-way secret key agreement rates to quantify unique informations.",
"There is another difficulty.",
"PID is defined to be agnostic to directionality.",
"Furthermore, only one of the myriad proposed PID axioms is contingent on any inherent directionality—the Blackwell Property [13] and it is an elephant.",
"In this sense, neither the camel nor the elephant intuitions are consistent with PID.",
"Again relating to secret key agreement, this implies that unique informations should more closely align with either the pair $\\operatorname{S}(X_{0} : Y ~||~ X_{1})$ and $\\operatorname{S}(X_{1} : Y ~||~ X_{0})$ or with the pair $\\operatorname{S}(X_{0} \\leftrightarrow Y ~||~ X_{1})$ and $\\operatorname{S}(X_{1} \\leftrightarrow Y ~||~ X_{0})$ ; neither of which adopt any sort of directionality.",
"Both approaches bring their own further difficulties.",
"On the one hand, the no-communication secret key agreement rate is not continuous in the space of distributions, whereas PID is generally considered to vary continuously.",
"On the other hand, the two-way secret key agreement rate $\\operatorname{S}(X_{0} \\leftrightarrow Y ~||~ X_{1})$ has no known closed-form solution, only upper and lower bounds, and so it cannot be practically computed.",
"Furthermore and perhaps more fundamentally, whether or not the two-way secret key agreement rate results in a consistent decomposition is not known.",
"That said, our extensive searches of examples for which the upper and lower bounds converge are encouraging—they have not resulted in any violations of eq:consistency."
],
[
"Conclusion",
"At present, a primary barrier for PID's general adoption as a useful and possibly a central tool in analyzing how complex systems store and process information is an agreement on a method to quantify its component informations.",
"Here, we posited that one reason for disagreement stems from conflicting intuitions regarding the decomposition's operational behavior.",
"This suggests several possibilities.",
"The first is that PID is inherently context-dependent and quantification depends on a notion of directionality.",
"In this case, the elephant intuition is apparently more natural, as adopting closely related notions from cryptography results in a consistent PID.",
"If context demands the camel intuition, though, either a noncryptographic method of quantifying unique information is needed or consistency must be enforced by augmenting the secret key agreement rate.",
"The second possibility suggested by our observations is that intuitions which project a directionality on the decomposition are inherently flawed and that any correct quantification must be independent of direction.",
"Interestingly, cryptographic notions may still play a role here.",
"Though, since there is as yet no known way to compute the two-way secret key agreement rate, its application remains open.",
"A final possibility is that associating secret key agreement rates with unique information is fundamentally flawed and that, ultimately, PID quantifies unique information as something distinct from the ability to agree upon a secret key.",
"Given that one of the main factors driving PID's creation was the need for interpretability, ensuring that the intuitions brought to bear are consistent with the quantitative values is of the utmost importance.",
"We described three quantitative regimes, each corresponding to a specific directionality or the lack thereof.",
"While it is possible that each can play a distinct role in the understanding of complex systems, our hope is that a single method will emerge as the most useful and accepted approach to understanding the organization of information within a joint probability distribution."
],
[
"Acknowledgments",
"All calculations herein were performed using the dit Python package [14].",
"We thank P. Banerjee, E. Olbrich, and D. Feldspar for many helpful discussions.",
"As a faculty member, JPC thanks the Santa Fe Institute and the Telluride Science Research Center for their hospitality during visits.",
"This material is based upon work supported by, or in part by, Foundational Questions Institute grant FQXi-RFP-1609, the U.S. Army Research Laboratory and the U.S. Army Research Office under contracts W911NF-13-1-0390 and W911NF-13-1-0340 and grant W911NF-18-1-0028, and via Intel Corporation support of CSC as an Intel Parallel Computing Center."
]
] | 1808.08606 |
[
[
"Deep Learning: Computational Aspects"
],
[
"Abstract In this article we review computational aspects of Deep Learning (DL).",
"Deep learning uses network architectures consisting of hierarchical layers of latent variables to construct predictors for high-dimensional input-output models.",
"Training a deep learning architecture is computationally intensive, and efficient linear algebra libraries is the key for training and inference.",
"Stochastic gradient descent (SGD) optimization and batch sampling are used to learn from massive data sets."
],
[
"Introduction",
"Deep learning (DL) is a form of machine learning that uses hierarchical layers of abstraction to model complex structures.",
"DL requires efficient training strategies and these are at the heart of today's successful applications which range from natural language processing to engineering and financial analysis.",
"While deep learning has been available for several decades there were only a few practical applications until the early 2010s when the field has changed for several reasons.",
"The renaissance is due to a number of factors, in particular Hardware and software for accelerated computing (GPUs and specialized linear algebra libraries) Increased size of datasets (Massive Data) Efficient algorithms, such as stochastic gradient descent (SGD).",
"The goal of our article is to provide the reader with an overview of computational aspects underlying the algorithms and hardware, which allow modern deep learning models to be implemented at scale.",
"Many of the leading Internet companies employ DL at scale [21].",
"The most impressive accomplishment of DL is its many successful applications in research and business.",
"These applications include algorithms such as Google Neural Machine Translation [64] closes the gap with humans in accuracy of the translation by 55-85% (estimated by people on a 6-point scale).",
"One of the keys to the success of the model is the use of Google's huge dataset.",
"Chat Bots which predict natural language response have been available for many years.",
"Deep learning networks can significantly improve the performance of chatbots [23].",
"Nowadays they provide help systems for companies and home assistants such as Amazon's Alexa and Google home.",
"Voice Generation was taken to the next level by DL based solutions.",
"Google WaveNet, developed by DeepMind [38], generates speech from text and reduces the gap between the state-of-the art and human-level performance by over 50% for both US English and Mandarin Chinese.",
"Google Maps were improved after DL was developed to analyze more than 80 billion Street View images and to extract names of roads and businesses [63].",
"Companies like Google Calico and Google Brain Health develop DL for health care diagnostics.",
"Adversarial Auto-encoder model found new molecules to fight cancer.",
"Identification and generation of new compounds was based on available biochemical data [26].",
"Convolutional Neural Nets (CNNs) have been developed to detect pneumonia from chest X-rays with better accuracy than practicing radiologists [44].",
"Another CNN model is capable of identifying skin cancer from biopsy-labeled test images [14].",
"[51] has discovered two new planets using DL and massive data from NASA’s Kepler Space Telescope.",
"In more traditional engineering, science applications, such as spatio-temporal and financial analysis DL showed superior performance compared to traditional statistical learning techniques [41], [10], [22], [54], [16], [15] In this paper we discuss computationally intensive algorithms for training deep learning models.",
"The main advantage of DL models is the ability to learn complex relationships in high dimensions.",
"However, a large number of samples is required to make useful predictions.",
"Ability to train DL models using large number (millions) of high-dimensional inputs is the key to the current success of those models.",
"A specialized software that utilizes accelerated hardware computing architectures, such as Graphical Processing Units (GPUs) or Tensor Processing Units (TPU) is typically used to train DL models.",
"We start in Section by reviewing deep learning models.",
"In Section we review most commonly used optimization algorithm and highlight that the core operation of those are matrix-matrix multiplications.",
"Is Section we review techniques to accelerate linear algebra operations (e.g.",
"matrix-matrix multiplications).",
"In Section we review specialized processors that are currently used for training large scale models and the software libraries that support those architectures are discussed in Section .",
"Finally, we conclude with Section where we suggest some further readings."
],
[
"Deep Learning",
"Simply put, DL constructs an input-output map.",
"Let $Y$ represent an output (or response) to a task which we aim to solve based on the information in a given high dimensional input matrix, denoted by $X$ .",
"An input-output mapping is denoted by $Y = F(X)$ where $X=(X_1,\\ldots ,X_p)$ is a vector of predictors.",
"[42] view the theoretical roots of DL in Kolmogorov's representation of a multivariate response surface as a superposition of univariate activation functions applied to an affine transformation of the input variable [29], [30].",
"An affine transformation of a vector is a weighted sum of its elements (linear transformation) plus an offset constant (bias).",
"Our Bayesian perspective on DL leads to new avenues of research including faster stochastic algorithms, hyper-parameter tuning, construction of good predictors, and model interpretation.",
"On the theoretical side, DL exploits Kolmogorov's “universal basis”.",
"The fact that DL forms a universal `basis', which we recognize in this formulation, dates back to Poincare and Hilbert.",
"By construction, deep learning models are very flexible and gradient information can be efficiently calculated for a variety of architectures.",
"On the empirical side, the advances in DL are due to a number of factors, in particular: New activation functions, e.g.",
"rectified linear unit ($\\text{ReLU}(x) = \\max (0,x)$ ).",
"Dropout as a variable selection technique and use of multiple layers Computationally efficient routines to train and evaluate the models as well as accelerated computing on graphics processing unit (GPU) and tensor processing unit (TPU).",
"Computational software such as TensorFlow or PyTorch.",
"Similar to a classic basis decomposition, the deep approach uses univariate activation functions to decompose a high dimensional $X$ .",
"Let $ Z^{(l)} $ denote the $l$ th layer, and so $ X = Z^{(0)}$ .",
"The final output $Y$ can be numeric or categorical.",
"The explicit structure of a deep prediction rule is then $Z^{(1)} & = f^{(1)} \\left( W^{(0)} X + b^{(0)} \\right) \\nonumber \\\\Z^{(2)} & = f^{(2)} \\left( W^{(1)} Z^{(1)} + b^{(1)} \\right) \\\\\\ldots & \\nonumber \\\\Z^{(L)} & = f^{(L)} \\left( W^{(L-1)} Z^{(L-1)} + b^{(L-1)} \\right)\\nonumber \\\\\\hat{Y} (X \\mid W,b) & = f^{L+1}(W^{(L)} Z^{(L)} + b^{(L)}) \\nonumber \\,.$ Here $W = (W^{(0)},\\dots ,W^{(L)})$ , are weight matrices and $b = (b^{(0)},\\dots ,b^{(L)})$ are threshold or activation levels.",
"Designing a good predictor depends crucially on the choice of univariate activation functions $ f^{(l)}$ .",
"The $Z^{(l)}$ are hidden features which the algorithm will extract.",
"For a regression problem, we use an identity function for the last layer $f^{L+1} = I$ and for a classification problem, we use a logistic function $f^{L+1}(x) = 1/(1+e^{-x})$ .",
"For a more extended overview of deep learning models, see [43], [31], [18], [49].",
"From a practical perspective, given a large enough data set of “test cases\", we can empirically learn an optimal predictor.",
"From a statistical point of view, Equation (REF ) can be viewed as a hierarchical generalized linear model with a simple GLM being a specific case when $L=0$ .",
"When $L>0$ , it is practically impossible to interpret neither parameters of the model $(W,b)$ nor the outputs of hidden layers $Z^{(l)}$ .",
"Thus, while adding hidden layers allows for learning more complex relations in the data, it prevents us from explaining the prediction rule.",
"Explainability of deep learning models is one of the hurdles that prevents those from being used in heavily regulated industries, such as finance or insurance.",
"It is possible to calculate derivative of the output $\\hat{Y}(X)$ with respect to any of the inputs $X_i$ and thus derive a measure of sensitivity or importance of each of the inputs.",
"The key constraint of this approach is that it is limited to be applied to one input predictor at a time.",
"On the other hand, the main advantage of deep learning model is the ability to capture interactions among the inputs.",
"Thus, the sensitivity analysis is very limiting.",
"Ability to explain predictions of deep learning models is an open area of research.",
"An extension to sensitivity based analysis of deep learning models was proposed by [52] who use not only derivatives of the model output but also of the output of each of the hidden layers and derive metrics that allow to explain the interactions among inputs.",
"[56] use a modification of the sensitivity approach called integrated gradients to extract explanations of the prediction rules.",
"[45] proposed to fit an interpretable model locally around a specific value of the input vector.",
"[24] propose a method to explain deep learning predictions for different parts of the input space (population of samples).",
"[35] demonstrate empirical performance of several techniques for deep learning model interoperability."
],
[
"A Probabilistic View of DL",
"Probabilistically the output $Y$ can be viewed as a random variable being generated by a probability model $p\\left(Y\\mid \\hat{Y}(X\\mid W,b)\\right)$ , where $\\hat{Y}(X\\mid W,b)$ is a prediction by a deep learning models with weights $W = (W^{(0)},\\dots ,W^{(L)})$ , and $b = (b^{(0)},\\dots ,b^{(L)})$ .",
"Then, given parameters $(W,b)$ , the negative log-likelihood defines a loss $\\mathcal {L} $ as $\\mathcal {L}(Y, \\hat{Y} ) = - \\log p\\left(Y\\mid \\hat{Y}(X\\mid W,b)\\right).$ Given a training sample $D = \\lbrace (X_i,Y_i)\\rbrace _{i=1}^n$ , the $L_2$ -norm, $\\mathcal {L}( Y_i, \\hat{Y}( X_i)) = 1/n\\sum _{i=1}^n(Y_i - \\hat{Y}( X_i))^2_2$ is traditional least squares, and negative cross-entropy loss is $\\mathcal {L}( Y_i, \\hat{Y}( X_i)) = -\\sum _{i=1}^n Y_i \\log \\hat{Y} ( X_i )$ for multi-class logistic classification.",
"There is a bias-variance trade-off, which is controlled by adding a regularization term and optimizing the regularized loss $\\mathcal {L}_{\\lambda }(Y, \\hat{Y} ) = - \\log p\\left(Y\\mid \\hat{Y}(X\\mid W,b)\\right)- \\log p( W, b \\mid \\lambda ).$ The regularization term is a negative log-prior distribution over parameters, namely $- \\log p( W, b \\mid \\lambda ) & = \\lambda \\phi (W,b),\\\\p( W, b \\mid \\lambda ) & \\propto \\exp ( - \\lambda \\phi (W,b)).$ Deep predictors are regularized maximum a posteriori (MAP) estimators, where $-p( W, b | D ) & \\propto -p\\left(Y\\mid \\hat{Y}(X\\mid W,b)\\right) p( W, b\\mid \\lambda ) \\\\& \\propto \\exp \\left( - \\log p\\left(Y\\mid \\hat{Y}(X\\mid W,b)\\right) - \\log p( W, b) \\right).$ Training requires the solution of a highly nonlinear optimization problem $\\operatornamewithlimits{arg\\,max}_{W,b} \\; \\log p( W, b \\mid D).$ This problem is solved using Stochastic Gradient Descent (SGD) which iteratively updates the parameters $(W,b)$ by taking a step in the direction negative to the gradient.",
"The key property is that $ \\nabla _{W,b} \\log p( W, b \\mid D) $ is computationally inexpensive to evaluate using back-propagation algorithms that implemented using modern matrix computation libraries for various hardware architectures.",
"It makes a fast implementation on large datasets possible.",
"TensorFlow and TPUs provide a state-of-the-art framework for a plethora of deep learning architectures.",
"From a statistical perspective, one caveat is that the posterior is highly multi-modal and providing good hyper-parameter (e.g.",
"number of layers and neurons per layer) tuning can be expensive.",
"This is clearly a fruitful area of research for state-of-the-art stochastic Bayesian MCMC algorithms to provide more efficient algorithms.",
"For shallow architectures, the alternating direction method of multipliers (ADMM) provides an efficient optimization solution.",
"For more details on probabilistic and Bayesian perspective on deep learning, see [42]."
],
[
"Optimization Algorithms",
"We now discuss two types of algorithms for training learning models.",
"First, we discuss stochastic gradient descent, which is a very general algorithm that efficiently works for large scale datasets and has been used for many deep learning applications.",
"Second, we discuss specialized statistical learning algorithms, which are tailored for certain types of traditional statistical models."
],
[
"Stochastic Gradient Descent",
"Stochastic gradient descent (SGD) is a default gold standard for minimizing the a function $f(W,b)$ (maximizing the likelihood) to find the deep learning weights and offsets.",
"SGD simply minimizes the function by taking a negative step along an estimate $g^k$ of the gradient $\\nabla f(W^k, b^k)$ at iteration $k$ .",
"The gradients are available via the chain rule applied to the superposition of semi-affine functions.",
"The approximate gradient is estimated by calculating $g^k = \\frac{1}{|E_k|} \\sum _{i \\in E_k} \\nabla \\mathcal {L}( Y_i ,\\hat{Y}( X_i \\mid W^{k},b^k)),$ where $E_k \\subset \\lbrace 1,\\ldots ,T \\rbrace $ and $|E_k|$ is the number of elements in $E_k$ .",
"When $|E_k| >1$ the algorithm is called batch SGD and simply SGD otherwise.",
"Typically, the subset $E$ is chosen by going cyclically and picking consecutive elements of $\\lbrace 1,\\ldots ,T \\rbrace $ , $E_{k+1} = [E_k \\mod {T}]+1$ .",
"The direction $g^k$ is calculated using a chain rule (a.k.a.",
"back-propagation) providing an unbiased estimator of the gradient computed using the entire sample $\\nabla f(W^k,b^k)$ .",
"Specifically, this leads to $\\mathrm {E}(g^k) = \\mathrm {E}\\left(\\nabla f(W^k, b^k)\\right).$ At each iteration, SGD updates the solution $(W,b)^{k+1} = (W,b)^k - t_k g^k.$ Deep learning algorithms use a step size $t_k$ (a.k.a learning rate) that is either kept constant or a simple step size reduction strategy, such as $t_k = a\\exp (-kt)$ is used.",
"The hyper parameters of reduction schedule are usually found empirically from numerical experiments and observations of the loss function progression.",
"One caveat of SGD is that the descent in $f$ is not guaranteed, or it can be very slow at every iteration.",
"Stochastic Bayesian approaches ought to alleviate these issues.",
"For example, [60] provide a scalable MCMC algorithm that can be used to train multi-modal loss function that arise when training deep learning architectures.",
"The variance of the gradient estimate $g^k$ can also be near zero, as the iterates converge to a solution.",
"To tackle those problems a coordinate descent (CD) and momentum-based modifications can be applied.",
"Alternative directions method of multipliers (ADMM) can also provide a natural alternative, and leads to non-linear alternating updates, see [7].",
"The CD evaluates a single component $E_k$ of the gradient $\\nabla f$ at the current point and then updates the $E_k$ th component of the variable vector in the negative gradient direction.",
"The momentum-based versions of SGD, or so-called accelerated algorithms were originally proposed by [36].",
"For a more recent discussion, see [37].",
"The momentum term adds memory to the search process by combining new gradient information with the previous search directions.",
"Empirically momentum-based methods have been shown to have better convergence for DL networks [57].",
"The gradient only influences changes in the velocity of the update which then updates the variable $v^{k+1} = & \\mu v^k - t_k g((W,b)^k)\\\\(W,b)^{k+1} = & (W,b)^k +v^k$ The hyper-parameter $\\mu $ controls the dumping effect on the rate of update of the variables.",
"The physical analogy is the reduction in kinetic energy that allows to “slow down\" the movements at the minima.",
"This parameter can also be chosen empirically using cross-validation.",
"Nesterov's momentum method (a.k.a.",
"Nesterov acceleration) calculates the gradient at the point predicted by the momentum.",
"One can view this as a one-step look-ahead strategy with updating scheme $v^{k+1} = & \\mu v^k - t_k g((W,b)^k +v^k)\\\\(W,b)^{k+1} = & (W,b)^k +v^k.$ Another popular modification [65], adaptively scales each of the learning parameter at each iteration $c^{k+1} = & c^k + g((W,b)^k)^2\\\\(W,b)^{k+1} = & (W,b)^k - t_k g(W,b)^k)/(\\sqrt{c^{k+1}} - a),$ where $a$ is typically a small number, e.g.",
"$a = 10^{-6}$ that prevents dividing by zero.",
"This method is called AdaGrad.",
"PRMSprop takes the AdaGrad idea further and places more weight on recent values of gradient squared to scale the update direction, i.e.",
"we have $c^{k+1} = dc^k + (1-d)g((W,b)^k)^2.$ The Adam method [27] combines both PRMSprop and momentum methods and leads to the following update equations $v^{k+1} = & \\mu v^k - (1-\\mu )t_k g((W,b)^k +v^k)\\\\c^{k+1} = & dc^k + (1-d)g((W,b)^k)^2\\\\(W,b)^{k+1} = & (W,b)^k - t_k v^{k+1}/(\\sqrt{c^{k+1}} - a).$ Initial guess in model weights and choice of optimization algorithms parameters plays crucial role in rate of convergence of the SGD and its variants [57].",
"Second order methods solve the optimization problem by solving a system of nonlinear equations $\\nabla f(W,b) = 0$ by applying the Newton's method $(W,b)^+ = (W,b) - \\lbrace \\nabla ^2f(W,b) \\rbrace ^{-1}\\nabla f(W,b).$ Here SGD simply approximates $\\nabla ^2f(W,b)$ by $1/t$ .",
"The advantages of a second order method include much faster convergence rates and insensitivity to the conditioning of the problem.",
"An ill-conditioned problem is the one that has “flat directions\" in which function changes very slowly and it makes SGD rates low.",
"In practice, second order methods are rarely used for deep learning applications [9].",
"The major disadvantage is their inability to train models using batches of data as SGD does.",
"Second order methods require the inverse Hessian matrix, which in turn requires the entire data set to be calculated.",
"Since a typical DL model relies on large scale data sets, second order methods become memory and computationally prohibitive at even modest-sized training data sets."
],
[
"Automatic Differentiation (AD)",
"To calculate the value of the gradient vector, at each step of the optimization process, deep learning libraries require calculations of derivatives.",
"In general, there are three different ways to calculate those derivatives.",
"First, is numerical differentiation, when a gradient is approximated by a finite difference $f^{\\prime }(x) = (f(x+h)-f(x))/h$ and requires two function evaluations.",
"However, the numerical differentiation is not backward stable [19], meaning that for a small perturbation in input value $x$ , the calculated derivative is not the correct one.",
"Second, is a symbolic differentiation which has been used in symbolic computational frameworks such as Mathematica or Maple for decades.",
"Symbolic differentiation uses a tree form representation of a function and applies chain rule to the tree to calculate the symbolic derivative of a given function.",
"Figure REF shows a tree representation of of composition of affine and sigmoid functions.",
"Figure: Tree form representation of composition of affine and sigmoid functions: 1 e -b-wx +1\\frac{1}{e^{-b-wx}+1}The advantage of symbolic calculations is that analytical representation of derivative is available for further analysis.",
"For example, when derivative calculation is in an intermediate step of the analysis.",
"Third way to calculate a derivate is to use automatic differentiation (AD).",
"Similar to symbolic differentiations AD recursively applies the chain rule and calculates the exact value of derivative and thus avoids the problem of numerical instability.",
"The difference between AD and symbolic differentiation is that AD provides the value of derivative evaluated at a specific point rather than an analytical representation of the derivative.",
"AD does not require analytical specification and can be applied to a function defined by a sequence of algebraic manipulations, logical and transient functions applied to input variables and specified in a computer code.",
"AD can differentiate complex functions which involve IF statements and loops, and AD can be implemented using either forward or backward mode.",
"Consider an example of calculating a derivative of the following function with respect to x.",
"3mm0mmtrueflexiblegraydkgreenmauvetrue def sigmoid(x,b,w): \tv1 = w*x; \tv2 = v1 + b \tv3 = 1/(1+exp(-v2)) In the forward mode an auxiliary variable, called a dual number, will be added to each line of the code to track the value of the derivative associated with this line.",
"In our example, if we set x=2, w=3, b=5, we get the calculations given in Table REF .",
"Table: Forward AD algorithmVariables dv1,dv2,dv3 in Table REF correspond to partial (local) derivatives of each intermediate variables v1,v2,v3 with respect to $x$ , and are called dual variables.",
"Tracking for dual variables can either be implemented using source code modification tools that add new code for calculating the dual numbers or via operator overloading.",
"The reverse AD also applies chain rule recursively but starts from the outer function, as shown in Table REF .",
"Table: Reverse AD algorithmFor DL, derivatives are calculated by applying reverse AD algorithm to a model which is defined as a superposition of functions.",
"A model is defined either using a general purpose language as it is done in PyTorch or through a sequence of function calls defined by framework libraries (e.g.",
"in TensorFlow).",
"Forward AD algorithms calculate the derivative with respect to a single input variable, but reverse AD produces derivatives with respect to all intermediate variables.",
"For models with a large number of parameters, it is much more computationally feasible to perform the reverse AD.",
"In the context of neural networks the reverse AD algorithms is called back-propagation and was popularized in AI by [46].",
"According to [49] the first version of what we call today back-propagation was published in 1970 in a master's thesis [32] and was closely related to the work of [39].",
"However, similar techniques rooted in Pontryagin's maximization principle [2] were discussed in the context of multi-stage control problems [5], [6].",
"[11] applies back-propagation to calculate first order derivative of a return function to numerically solve a variational problem.",
"Later [12] used back-propagation to derive an efficient algorithm to solve a minimization problem.",
"The first neural network specific version of back-propagation was proposed in [61] and an efficient back-propagation algo ritm was discussed in [62].",
"Modern deep learning frameworks fully automate the process of finding derivatives using AD algorithms.",
"For example, PyTorch relies on autograd library which automatically finds gradient using back-propagation algorithm.",
"Here is a small code example using autograd library in Python.",
"3mm0mmtrueflexiblegraydkgreenmauvetrue # Thinly wrapped numpy import autograd.numpy as np # Basically everything you need from autograd import grad # Define a function like normal with Python and Numpy def tanh(x): \ty = np.exp(-x) \treturn (1.0 - y) / (1.0 + y) # Create a function to compute the gradient grad_tanh = grad(tanh) # Evaluate the gradient at x = 1.0 print(grad_tanh(1.0))"
],
[
"Architecture Optimization",
"Currently, there is no automated way to find a good deep learning architecture.",
"An architecture is defined by number of hidden layers, a number of neuron on each layer, parameters that define weight sharing layers, such as convolution layers or recurrent layers.",
"All of those parameters that defined an architecture belong to the set of hyperparameters.",
"Another group of hyperparameters specify the settings for stochastic gradient descent algorithms, e.g.",
"learning rate, momentum, etc.",
"It is not uncommon to use hand-tuning to find a deep learning architecture, when a modeler hand-picks several candidates and choses the one that performs the best on out-of-sample data.",
"It is usually done iteratively and might take weeks or months.",
"An easiest automated way to find an optimal set of hyperparameters is grid search, when space of hyperparameters is discretized using a grid and a model is estimated for each node of the grid.",
"This approach is used, for example, to find an optimal penalty weight for a LASSO model.",
"However, this approach is not feasible, when number of hyperparameters is large.",
"A random search rather samples from the grid randomly.",
"This, does not guarantee the optimal architecture will be identified but works rather well in practice [1].",
"Figure REF shows an example of randomly chosen grid points, while searching for an optimal number of neurons on the first hidden layer $n_1$ and the best learning rate $\\alpha $ .",
"Figure: Random grid search for hyperparametersBayesian optimization [55], [53] for hyperparameters search is more sample efficient, i.e.",
"requires less model evaluations to find the best candidate.",
"Bayesian methods rely on approximating the relations between hyper-parameters and model performance using a Gaussian Process surrogate model [34].",
"Gaussian process surrogates have the attractive property that the posterior distribution over function value at any point follows a Gaussian distribution.",
"The stochastic nature of the surrogate allows to quantify uncertainty over the function values and to explore the hyper-parameter space using approaches that can alternate exploration (searching input regions associated with high uncertainty levels of output values) and exploitations (searching regions of local minimal).",
"However, sequential nature of the search process prevents from distributed parallel evaluations of models and is usually less preferred compared to random search when a large number of compute nodes is available.",
"One can run several instances of Bayesian search in parallel using different initial values [50].",
"For example, Google's default architecture search algorithm [17] uses batched Bayesian optimization with Matérn kernel Gaussian process.",
"However, this approach, empirically is less efficient compared to random search.",
"Techniques to speed up Bayesian search include early stopping [20] and using a fraction of the data to evaluate models [47].",
"Genetic-like algorithms provide advantage of sample efficiency and of parallel computing.",
"Recently, [25] proposed a population based training approach, that evaluates multiple models in parallel and then generates new model candidates by modifying architectures of the models that performed best thus far."
],
[
"Scalable Linear Algebra",
"The key computational routine required to evaluate a DL model specified by Equation (REF ) is matrix-matrix multiplication.",
"In the context of DL models weights $W$ and inputs and outputs of each layer $X,Z^{(1)},\\dots ,Z^{(L)},\\hat{Y}$ are called tensors.",
"For example, in image processing input $X$ is a three dimensional tensor, which is made up by three matrices that correspond to red, green and blue color channels.",
"Thus, one of the key operations while training DL or calculating a prediction is a matrix-matrix multiplication, with matrix-vector, dot product or saxpy $a x + y$ (scalar $a$ times $x$ plus $y$ ) being a special cases.",
"Naive implementation of matrix-matrix multiplication would invoke a loop over the elements of the input matrices, which is inefficient.",
"We can parallelize the operations, even on a single processor.",
"Concurrency arises from performing the same operations on different pieces of data is is performed using Single Instruction Multiple Data (SIMD) instructions.",
"SIMD performs multiple independent algebraic operations in one clock cycle.",
"It is achieved by dividing each algebraic operation into multiple simpler ones with separate hardware in the processor for each of the simple operations.",
"The calculations are performed in a pipeline fashion, a.k.a conveyor belt.",
"For example, an addition operation can have the following components Find the location of inputs Copy data to register Align the exponents; the addition .3e-1+.6e-2 becomes .3e-1+.06e-1 Execute the addition of the mantissas Normalize the result.",
"When performed one at a time as in a loop, each addition takes 5 cycles.",
"However, when pipelined, we can do it in 1 cycle.",
"Modern processes might have more than 20 components for addition or multiplication operations [13].",
"GPU computing takes it further by using a set of threads for the same instruction (on different data elements), NVIDIA calls it SIMT (Single Instruction Multiple Threads).",
"Vectorized operations that rely on SIMD or SIMT replace naive loop implementation for calculating the matrix-matrix multiplication.",
"A vector processor comes with a repertoire of vector instructions, such as vector add, vector multiply, vector scale, dot product, and saxpy.",
"These operations take place in vector registers with input and output handled by vector load and vector store instructions.",
"For example, vectorization can speedup vector dot product calculations by two orders of magnitude, as shown in code: 3mm0mmtrueflexiblegraydkgreenmauvetrue N = int(1e6) a = np.random.rand(N) b = np.random.rand(N) tic = time.time() c = np.dot(a,b) toc = time.time() print(\"Vec dot time: \" + str((toc-tic)*1000) + \" ms\") c = 0 tic = time.time() for i in range(N): \t c+=a[i]*b[i] toc = time.time() print(\"Loop dot time: \" + str((toc-tic)*1000) + \" ms\") Vec dot time: \t 1.372 ms Loop dot time: 435.639 ms When calculations are performed on vectors of different dimensions, modern numerical libraries, such as Python's numpy perform those using broadcasting.",
"It involves “broadcasting” the smaller array across the larger array so that they have the same shape.",
"The vectorized operation is performed on the broadcasted and the other vector.",
"Broadcasting does not make copies of data and usually leads to efficient algorithm implementations.",
"For example, to perform b+z, where b is a scalar and z is an $n$ -vector, the numeric library will create a vector of length $n$ b_broadcast = (b,b,...,b) and then will compute (b,b,...,b) + z.",
"Further, matrix operations implemented by a linear algebra library take into account the memory hierarchy and aim at maximizing the use of the fastest cache memory which is co-located with the processor on the same board [13].",
"In summary, a modeler should avoid loops in their model implementations and always look for ways to vectorize the code.",
"Another way a modern DL framework speed up calculations is by using quantization [58], which simply replaces floating point calculations with 8-bit integers calculations.",
"Quantization allows to train larger models (less memory is required to store the model) and faster model evaluations, since cache can be used more efficiently.",
"In some case you'll have a dedicated hardware that can accelerate 8-bit calculations too.",
"Quantization also allows to evaluate large scale models on embedded and mobile devices, and enables what is called edge computing, when data is analyzed locally instead of being shipped to a remote server.",
"Edge computing is essential for Internet of Things (IoT) and robotics systems"
],
[
"Hardware Architectures",
"Usage of efficient hardware architectures is an important ingredient in today's success of DL models.",
"Design and optimization of DL hardware systems is currently an active area of research in industry and academia.",
"We currently see an “arms race” among large companies such as Google and Nvidia and small startups to produce the most economically and energy efficient deep learning systems."
],
[
"GPU Computing",
"In the last 20 years, the video gaming industry drove forward huge advances in Graphical Processing Unit (GPU), which is a special purpose processor for calculations required for graphics processing.",
"Since operations required for graphics heavily rely on linear algebra, and GPUs have become widely used for non-graphics processing, specifically for training deep learning models.",
"GPUs rely on data parallelism, when the body of a loop is executed for all elements in a vector: 3mm0mmtrueflexiblegraydkgreenmauvetrue for i in range(10000): a[i] = 2*b[i] Our data is divided among multiple processing units available, and each processor executes the same statement a = 2*b on its local data in parallel.",
"In graphics processing usually the same operation is independently applied to each pixel of an image, thus GPUs are strongly based on data parallelism.",
"The major drawback of GPU computing is the requirement to copy data from CPU to GPU memory which incurs a long latency.",
"Throughput computing, processing large amounts of data at high rates, plays a central role in GPU architectures.",
"High throughput is enabled by a large number of threads and ability to switch fast between them.",
"Modern GPUs would typically have several thousand cores, compare it to the latest Intel i9-family processors that have up to 18 cores.",
"Further, most recent GPUs from NVIDIA would include up to a thousand of so-called tensor cores, that can perform multiply-accumulate operation on a small matrix in one clock cycle.",
"Development of GPU code requires skills and knowledge typically not available to modelers.",
"Fortunately, most deep learning modelers do not need to program GPUs directly and use software libraries that have implementations of the most widely used operations, such as matrix-matrix multiplications.",
"Currently, Nvidia dominates the market for GPUs, with the next closest competitor being AMD.",
"Recently, AMD announced the release of a platform called ROCm to provide more support for deep learning.",
"The status of ROCm for major deep learning libraries such as PyTorch, TensorFlow, MxNet, and CNTK is still under development.",
"Let us demonstrate the speed up provided by using GPU using a code example: 3mm0mmtrueflexiblegraydkgreenmauvetrue dtype = torch.FloatTensor N = 50000 x = torch.randn(N,N).type(dtype) y = torch.randn(N,N).type(dtype) start = time.time() z = x*y end = time.time() print(\"CPU Time:\",(end - start)*1000) if torch.cuda.is_available(): start = time.time() x = x.cuda() y = y.cuda() end = time.time() print(\"Copy to GPU Time:\",(end - start)*1000) \t start = time.time() a = x*y end = time.time() print(\"GPU Time:\",(end - start)*1000) start = time.time() a = a.cpu() end = time.time() print(\"Copy from GPU Time:\",(end - start)*1000) CPU Time: 11.6 Copy to GPU Time: 28.9 GPU Time: 0.24 Copy from GPU Time: 33.2 The matrix multiplication operation itself is performed 48 times faster on GPU (11.6 ms vs 0.24 ms).",
"However, copying data from main memory to GPU memory and back adds another 62.1 ms (28.9 + 33.2).",
"Thus, to efficiently use GPU architectures, it is necessary to minimize amount of data transferred between main and GPU memories"
],
[
"Intel Xeon Phi",
"Recently, in response to the dominance of GPU processors in scientific computing and machine learning, Intel has released a co-processor Intel Xeon Phi.",
"As a GPU, Xeon Phi provides a large number of cores and has a considerable latency in starting up.",
"The main difference is that Xeon Phi has general purpose cores, while a set of GPU instructions is limited.",
"An ordinary C code can be executed on a Xeon Phi processor.",
"However, the ease of use of GPU libraries for linear algebra operations make those the default architecture choice."
],
[
"DL Specific Architectures",
"Companies such as Google or Facebook use deep learning models at extreme scales.",
"Recent computational demand for training and deploying deep learning models at those scales fueled development of custom hardware architectures for deep learning.",
"The Intel's Nervana NNP team is focusing on developing a co-processor with fast and reliable bi-directional data transfer.",
"They use a proprietary numeric format called Flexpoint, to increase the throughput.",
"Further, the power consumption is reduced by shrinking circuit size.",
"Google's Tensor Processing Units (TPU) [48] has two processors, each having 128x128 matrix multiply units (MXU).",
"Each MXU can perform multiple matrix operations in one clock cycle.",
"Google uses TPUs for all of its online services such as Search, Street View, Google Photos, and Google Translate.",
"TPU uses Complex Instruction Set Computer (CISC) design style which focuses on implementing instructions for high-level complex tasks such as matrix-matrix multiplication with in one clock cycle.",
"In contrast, a typical general purpose CPU follows a Reduced Instruction Set Computer (RISC) design and implements a large number of small primitive instructions (load, multiply,...) and assumes every operation can be represented as a combination of those simple primitives.",
"There are several other established and startup companies working on developing custom hardware architectures for deep learning computing.",
"Most approaches rely on usage of Field-programmable gate array (FPGA) designs [4].",
"For example, Microsoft's Brainwave [33] hardware, which used FPGA is claimed to address the inflexibility of other computing platforms by providing a design that scales across range of data types.",
"Other processor's inflexibility comes from the fact that a set of specific instructions is available at any given architecture."
],
[
"Software Frameworks",
"Python is by far the most commonly used language for DL.",
"There are a number of deep learning libraries available, with almost every major tech company backing a different library.",
"Widely used deep learning libraries include TensorFlow (Google), PyTorch and Caffe2 (Facebook), MxNet (Amazon), CNTK (Microsoft).",
"All of those frameworks have Python support.",
"For R users Keras library (https://keras.rstudio.com) provides a high-level interface to TensorFlow, and is the most robust option at this point.",
"One of the major differences between different libraries is the use of dynamic vs. static graph computations.",
"Some libraries, such as MxNet and TensorFlow, allow for both.",
"In static setting a model is fully specified before the training process.",
"In dynamic graphs, structure is defined “in-thr-fly” as code gets executed, which is the way our traditional programs are executed.",
"Static graphs provide the opportunity to pre-process the model and to optimize the computations and thus are preferred for large scale models used in production.",
"Dynamic settings provide more flexibility and is typically used during the research and development phase of the model development.",
"Furthermore, dynamic models are easier to debug and easier to learn for those who are familiar with traditional object-oriented programming."
],
[
"PyTorch",
"PyTorch is native Python library rather than a binding to library written in another language.",
"Thus, it provides an intuitive and friendly interface for Python users to build and train deep learning models on CPU and GPU hardware.",
"Pytorch is widely used for research as it provides a way to build models dynamically using native Python functions.",
"On a flexibility-code simplicity scale Pytorch is an attractive option for a researcher who is using Python."
],
[
"TensorFlow",
"TensorFlow is an open source framework written in C++ with interfaces available for many other languages such as Python.",
"Although TensorFlow assumes a steeper learning curve when compared to other DL frameworks, its performance on large scale problems across different hardware architectures and support for many popular machine learning algorithms made it a popular choice among practitioners."
],
[
"Compiler Based Approach",
"Traditional DL systems consists of high-level interface libraries, such as PyTorch or TensorFlow which perform computationally intensive operations by calling functions from libraries optimized for a specific hardware as shown in Figure REF .",
"Currently, hardware manufactures have to develop a software stack (a set of libraries) specific to their processors.",
"Nvidia developed CUDA libraries, Intel has MKL library, Google developed TPU library.",
"The reason why Nvidia and not AMD is the GPU of choice for deep learning models is because of Nvidia's greater level of software support for linear algebra and other DL specific computations.",
"Figure: Deep Learning System HierarchyHowever, usage of vendor-developed libraries can be limiting.",
"Some expressions might require a complex combination of function calls or might be impossible to write using the functions provided by the vendor library.",
"For example, vendor library might not support sparse matrices.",
"A different approach has recently emerged that relies on compiling linear algebra expressions written in special language to a code which is optimized for a given hardware architecture.",
"This approach solves two problems, it allows to perform computations that are not implemented in hardware specific-library, and facilitates support for wider a range of architectures, including mobile ones.",
"Recent examples include tensor comprehensions by Facebook [59], TVM from U Washington [8], TACO [28] and Google's XLA (Accelerated Linear Algebra) compiler.",
"Code below demonstrates impact of Numba, which compiles functions written directly in Python.",
"Numba uses annotations to compile Python code to native machine instructions.",
"When original python code is mostly performing linear algebra operations, the resulting native machine instructions will lead to performance similar to C, C++ and Fortran.",
"Numba generates optimized machine code using the LLVM compiler infrastructure.",
"Numba supports compilation of Python to run on either CPU or GPU hardware and is designed to integrate with the Python scientific software libraries.",
"3mm0mmtrueflexiblegraydkgreenmauvetrue from numba import jit, double import math import numpy as np import time @jit(nopython = True) def mydot(a,b,c): for i in range(N): c+=a[i]*b[i] N = int(1e6); a = np.random.rand(N); b = np.random.rand(N) c = 0 tic = time.time() mydot(a,b,c) toc = time.time() print(\"Numba dot time: \" + str((toc-tic)*1000) + \" ms\") c = 0 tic = time.time() for i in range(N): c+=a[i]*b[i] toc = time.time() print(\"Loop dot time: \" + str((toc-tic)*1000) + \" ms\") Numba dot time: \t138.628959656 ms Loop dot time: \t630.362033844 ms"
],
[
"Concluding Remark",
"The goal of our paper is to provide an overview of computational aspects of DL.",
"To do this, we have discussed the core linear algebra, computational routines required for training, and inference using the DL models as well as the importance of hardware architectures for efficient model training.",
"A brief introduction SGD optimization and its variants, that are typically used to find parameters (weights and biases) of a deep learning model is also provided.",
"For further reading, see [3].",
"Although, DL models have been almost exclusively used for problems of image analysis and natural language processing, more traditional data sets, which arise in finance, science and engineering, such as spatial [41], [10] and temporal [40] data can be efficiently analyzed using deep learning.",
"There are a number of areas of future research for Statisticians.",
"In particular, uncertainty quantification and model selection such as architecture design.",
"To algorithmic improvements and Bayesian deep learning.",
"We hope this review will make DL models accessible for statisticians."
]
] | 1808.08618 |
[
[
"Unsupervised Hypergraph Feature Selection via a Novel Point-Weighting\n Framework and Low-Rank Representation"
],
[
"Abstract Feature selection methods are widely used in order to solve the 'curse of dimensionality' problem.",
"Many proposed feature selection frameworks, treat all data points equally; neglecting their different representation power and importance.",
"In this paper, we propose an unsupervised hypergraph feature selection method via a novel point-weighting framework and low-rank representation that captures the importance of different data points.",
"We introduce a novel soft hypergraph with low complexity to model data.",
"Then, we formulate the feature selection as an optimization problem to preserve local relationships and also global structure of data.",
"Our approach for global structure preservation helps the framework overcome the problem of unavailability of data labels in unsupervised learning.",
"The proposed feature selection method treats with different data points based on their importance in defining data structure and representation power.",
"Moreover, since the robustness of feature selection methods against noise and outlier is of great importance, we adopt low-rank representation in our model.",
"Also, we provide an efficient algorithm to solve the proposed optimization problem.",
"The computational cost of the proposed algorithm is lower than many state-of-the-art methods which is of high importance in feature selection tasks.",
"We conducted comprehensive experiments with various evaluation methods on different benchmark data sets.",
"These experiments indicate significant improvement, compared with state-of-the-art feature selection methods."
],
[
"Introduction",
"Nowadays confronting high dimensional data is expected in practical applications.",
"The complexity of many algorithms in various fields like machine learning and pattern recognition highly depends on the dimensionality of data.",
"Two types of dimensionality reduction algorithms can be used to solve the 'curse of dimensionality' problem: feature extraction and feature selection.",
"Feature extraction methods try to define new features based on primary features and redescribe data based on them.",
"In opposite, feature selection techniques aim to select the most descriptive subset of features.",
"Unlike feature extraction techniques, feature selection algorithms preserve the primary representation of the data [1].",
"Feature selection methods are categorized into three groups: supervised, semi-supervised, and unsupervised.",
"Supervised and semi-supervised methods like generalized fisher score (GFS) [2], hypergraph based information-theoretic feature selection (MII_HG) [3], and hypergraph regularized Lasso (HLasso) [4] need labeled data to perform feature selection.",
"Although these methods can be effective, collecting labels for data can be time-consuming or even impractical in many cases.",
"Unlike supervised and semi-supervised methods, unsupervised methods such as feature selection via joint embedding learning and sparse regression (JELSR) [5] and self-representation hypergraph low-rank feature selection (SHLFS) [6] need no labeled data for feature selection.",
"Hence unsupervised methods are used more widely in many applications.",
"In unsupervised feature selection methods, providing an efficient representation of data is of great importance.",
"Because the structure of data is preserved based on the constructed representation which without having an effective representation, preservation of the defined structure can be ineffective.",
"In the last decade, various algorithms such as JELSR [5], laplacian score (LapScore) [7], similarity preserving feature selection (SPFS) [8], and multi-cluster feature selection (MCFS) [9] used graphs to model data.",
"In graphs each edge can only be connected to at most another edge; So, graph-based methods can be adopted to model only dual relationships among data points.",
"On the other hand, in real applications, different data points can be related to each other as a group and as a result, there exist multiple relationships among them which should be modeled and in the next steps, preserved.",
"To overcome this problem, in the last few years, some algorithms have been using hypergraphs instead to model interconnection among data points; For instance, JHLSR [10], MII_HG [3], and SHLFS [6].",
"In hypergraphs, hyperedges are used instead of conventional edges.",
"Hyperedges can connect any number of vertices.",
"Although hypergraphs improve ordinary graphs in many ways, they have an important limitation: there are only two states for a vertex and a hyperedge: a hyperedge completely consists of that vertex or not, but the vertex may need to have partial participation in that hyperedge; Because importance of other vertices in that hyperedge can be more or less critical.",
"So, a generalized form of hypergraphs is needed in which amount of attendance of a vertex in a hyperedge can adopt continuous values instead of only two states.",
"Feature selection methods usually encounter the problem of inaccurate measurements and outlier in practical applications.",
"So, it is of great importance for a feature selection method to overcome this problem.",
"Proposing a model without handling noise and outlier can lead to focus on wrong data structure and as a result, inefficient feature selection.",
"One way to overcome this problem is using low-rank representation in hypergraph-based feature selection methods proposed in [11], [6].",
"In other words, hypergraph low-rank feature selection (HLR_FS) [11] and self-representation hypergraph low-rank feature selection (SHLFS) [6] use low-rank representation in order to deal with noise and outlier.",
"One common disadvantage of the mentioned methods is behaving different data points equally; While different data points can vary much in importance and role in defining the data structure.",
"Another important point in feature selection methods is computational cost.",
"As it was clarified, it is awaited to meet high dimensional data in the task of feature selection.",
"Therefore, the computational complexity of an algorithm is of high importance.",
"An algorithm can be effective on small-sized data, but it may be infeasible to adopt it on a high dimensional practical data set.",
"In this paper, to overcome the weaknesses of the previously proposed methods, we introduce a novel low-rank feature selection method using soft hypergraphs which captures the importance of different data points.",
"In the proposed point-weighting framework, we treat different data points based on their representation power and role in defining the data structure.",
"We propose using soft hypergraphs in order to resolve the limitation of ordinary hypergraphs.",
"Unlike ordinary hypergraphs, in soft hypergraphs, different vertices can have different participation in a hyperedge which leads to more accurate modeling.",
"Moreover, the proposed approach uses cluster centroids to build the hypergraph which results in reducing the computational cost, and also better handling of noise, outlier, and redundancy.",
"The reason for this can be described as follows: Usually, in real situations, some data points can have similar properties.",
"So in order to preserve data structure, there is no need to use all data points.",
"Points with analogous properties can be grouped as a cluster and a representative data point can be used instead of the cluster to represent the properties of that group.",
"In this way, computational complexity decreases and the effect of noise and outlier reduces; Because the small differences in properties of some data points can be only the result of noise; Moreover, redundant information of almost similar data points will be discarded.",
"The best choice for this representative point is the centroid of that cluster; Since it has a good representation power.",
"We use the constructed hypergraph to preserve local relationships between centroids.",
"To preserve the complete structure of data, the global structure needs to be preserved along with local relationships between centroids.",
"For this purpose, we try to maintain the global structure of data by adding a term which also helps resolve the problem of data labels unavailability in unsupervised learning.",
"So, in brief, in this paper, by weighting points, we focus on data points with high representation power; Especially cluster centroids.",
"We also adopt low-rank representation and propose a novel term for global structure preservation.",
"We have tested our proposed framework using various evaluation methods on different benchmark datasets which indicates significant improvement, compared with state-of-the-art feature selection methods.",
"The innovative aspects of the proposed method can be summarized as follows: i) Introducing a new soft hypergraph to model data for unsupervised feature selection.",
"ii) Proposing a point-weighting framework.",
"iii) Adopting low-rank representation in the point-weighting framework.",
"iv) Introducing a new approach to preserve the global structure of data.",
"v) Providing an efficient algorithm to solve the proposed optimization problem.",
"The notations used in this paper and their definitions are summarized in table REF .",
"Here the organization of the rest of the paper is explained.",
"In Section , we review concepts of hypergraph-based feature selection.",
"In Section , we explain our proposed framework for feature selection.",
"In section , we provide an efficient algorithm to solve the proposed optimization problem.",
"In Section , we calculate the computational cost of our proposed algorithm.",
"Section exhibits our experimental results.",
"Finally, in Section , we conclude our paper.",
"Table: The list of important notations used in this paper and their brief explanation."
],
[
"Hypergraph-based feature selection",
"An important step in feature selection algorithms is modeling data.",
"Graphs can be used in order to model local relationships between data points.",
"But they can model only dual relationships.",
"To overcome this problem, in recent studies, hypergraphs has been used instead [6], [10], [4].",
"In this section, first, we explain the fundamentals of hypergraphs.",
"Afterward, principals of soft hypergraphs are explained.",
"Then, we review hypergraph-based feature selection methods quickly."
],
[
"Hypergraph fundamentals",
"An ordinary graph can be defined as the triplet: G = (V, E, w); Where V is the set of vertices, E is the set of edges, and w is a function which assigns each edge a weight.",
"An edge consists of at most two vertices; So it can only model dual relationships among vertices.",
"As it was explained earlier, in Section , in many applications, there exist multiple relationships among data points.",
"So a conventional graph is not appropriate to thoroughly model relationships among them.",
"To tackle this limitation of conventional graphs, hypergraphs can be used instead.",
"Hypergraphs can be considered as the generalized form of simple graphs.",
"They can be shown using the following notation: $G_H$ = (V, E, w).",
"Similar to graphs, V and w are the set of vertices and the weight function respectively.",
"Although instead of simple edges, hyperedges are adopted in hypergraphs to show interconnection among vertices; They are denoted by E. Hyperedges are the extended mode of traditional edges and can include multiple vertices; As a result, they can model multiple relationships amid vertices.",
"In order to determine the relationship between different vertices and hyperedges, the incidence matrix H is defined as follows: $h_v^{e}={\\left\\lbrace \\begin{array}{ll}1, & \\text{if}\\ vertex~v \\in hyperedge~e \\\\0, & \\text{elsewhere}\\end{array}\\right.",
"}$ Where $h_v^{e}$ is the element in vth row and eth column of matrix H. Akin to conventional graphs, degree is defined for vertices and hyperedges.",
"Degree of a vertex is the total weights of hyperedges which it is comprised in, i.e.",
"$ d(v)=\\sum _{e~=~1}^{|E|} w(e)h_v^{e}$ Where $|E|$ is the number of hyperedges.",
"Degree of a hyperedge is the total amount of contributions of different vertices in it which in conventional hypergraphs is equal to the number of vertices it embraces, i.e.",
"$ \\delta (e)=\\sum _{v~=~1}^{|V|} h_v^{e}$ Where $|V|$ is the number of vertices, $h_{v}^{e}$ is the element in vth row and eth column of matrix H."
],
[
"Soft hypergraph",
"Even though hypergraphs cover the main limitation of ordinary graphs, they have the problem of equal participation of different vertices in their hyperedges.",
"In hyperedges, all vertices are treated equally, but different vertices have different importance and contributions in hyperedges.",
"In addition, there is no need to define the incidence matrix binary, While it can be continuous; At least in our case.",
"This type of hypergraph is called soft hypergraph [12].",
"An example of soft hypergraph is provided in figure REF ; $E=\\lbrace e_1, e_2, e_3\\rbrace $ is the set of hyperedges and $V=\\lbrace v_1, v_2, v_3, v_4, v_5\\rbrace $ is the set of vertices.",
"Hyperedge $e_3$ models multiple relationship between vertices $v_3$ , $v_4$ , and $v_5$ .",
"Since the incidence matrix is not binary, these vertices could have different participation in that hyperedge, i.e.",
"0.18, 0.59, and 0.42 respectively.",
"Figure: An example of soft hypergraph."
],
[
"A review on hypergraph-based feature selection methods",
"In this section, we review some state-of-the-art feature selection methods which use hypergraphs to perform data modeling.",
"We list related algorithms and provide a brief explanation about them.",
"JHLSR [10]: This framework adopts sparse representation [13] to construct its hypergraph.",
"Sparse representation for a data point can be described as the minimal subset of linearly dependent vertices.",
"JHLSR needs one hyperedge for each data point; Moreover, Because of not using soft hypergraph, they have to repeat the process of finding the sparse representation of each data point several times with different sparsification parameters.",
"As a result, the constructed hypergraph becomes very large which increases the computational complexity of the proposed model.",
"As it was explained earlier, encountering noise and outlier is inevitable, but JHLSR does not provide any solution to handle this problem.",
"SHLFS [6]: Some supervised feature selection methods such as [11], [14] try to generate the response vector using the linear combination of features and in the last step, report features with the most contribution in generating the response vector, as output.",
"SHLFS modifies this method to be applicable in unsupervised learning and tries to generate each feature, using the linear combination of other features.",
"SHLFS also adopts low-rank representation to handle noise and outlier.",
"HLasso [4]: In order to construct the hypergraph, HLasso connects every data point to its k nearest neighbors which results in a large hypergraph which imposes much computational cost; Moreover, since it adopts ordinary hypergraphs, instead of soft hypergraphs, the nearest neighbors of each data point are treated equally, while they can have different distances from that point.",
"After modeling data, HLasso uses hypergraph laplacian to preserve the distances of each point from its nearest neighbors.",
"HLasso does not handle noise and outlier.",
"HLR_FS [11]: This method constructs an ordinary hypergraph by connecting data points with the same label.",
"HLR_FS also uses a low-rank constraint to make the model robust against noise and outlier.",
"The above methods overlook the importance of treating different data points based on their representation power and role in defining the data structure.",
"Moreover, none of them adopt soft hypergraphs in order to model data.",
"In this paper, we propose an unsupervised hypergraph feature selection method via a novel point-weighting framework and low-rank representation (referred to as HPWL).",
"Since each centroid is the indicator of its cluster and its corresponding points, it has good representational power.",
"So at the first step, we should find the centroids.",
"In this way, any general clustering method can be adopted.",
"We use the k-means algorithm in our experiments.",
"the k-means algorithm is as follows: [H] Input: X, m. Output: $\\mu $ , $\\zeta $ .",
"Step 1: Initialization: Initialize cluster centers $\\mu _1$ , $\\mu _2$ , ..., $\\mu _m$ randomly; Step 2: Updating clusters and their centers: repeat $\\zeta _i = \\operatornamewithlimits{\\arg \\!\\min }_j{||x_i - \\mu _j||^2}$ , for i = 1, 2, ..., n $\\mu _j = \\frac{\\sum _{i=1}^m 1\\lbrace \\zeta _i=j\\rbrace x_i}{\\sum _{i=1}^m 1\\lbrace \\zeta _i=j\\rbrace }$ , for j = 1, 2, ..., m until convergence; k-means clustering algorithm.",
"Where X is the data matrix, $x_i$ is the ith data point, m is the number of clusters, n is the number of data points, $\\mu _i$ is the center of the ith cluster, and $\\zeta _i$ indicates the cluster number of the ith data point.",
"Using k-means, we determine m clusters.",
"Since in our proposed method each centroid needs to be one of the data points, we consider our ith centroid, i.e.",
"$c_i$ the nearest data point to $\\mu _i$ ; For i = 1, 2, ..., m. We use obtained centroids for hypergraph construction.",
"Treating all data points equally instead of focusing on centroids, results in useless computational complexity and redundant information; Moreover, it makes the algorithm sensitive to noise and outlier which leads to an ineffective selection of features."
],
[
"Soft hypergraph construction",
"We construct our soft hypergraph by focusing on the defined centroids.",
"By using soft hypergraph, we can model and then preserve multiple relationships among different centroids precisely.",
"Each centroid is considered as a vertex in the hypergraph.",
"To generate hyperedges, we connect every centroid to its $l$ nearest centroids.",
"To construct the H matrix, suppose that we are generating the hyperedge corresponding to the centroid i, i.e.",
"$h^i$ .",
"Its jth element is computed as follows: $h_j^i={\\left\\lbrace \\begin{array}{ll}a_j^i, & \\text{if}\\ c_j \\in \\text{\\lbrace $l$ nearest centroids of } c_i\\rbrace \\\\0, & \\text{elsewhere}\\end{array}\\right.",
"}$ Where $h_j^i$ is the element in jth row and ith column of matrix H, $a_j^i = {exp}{\\Big (-\\frac{||c_i - c_j||_{2}^{2}}{\\sigma ^{2}}\\Big )}$ ; $\\sigma $ is average Euclidean distance among centroids and $c_i$ , $c_j$ are two centroids.",
"$H~=~[h^1,~h^2,...~,~h^m]$ where $h^i$ is the ith column of H, i.e.",
"the ith hyperedge which is the hyperedge corresponding to the ith centroid.",
"Our algorithm is capable of differentiating the importance of different centroids.",
"We use equal values to initialize the weights of hyperedges.",
"To select the most informative hyperedges and assign a higher weight to them, hyperedges weights should be updated.",
"So the number of hyperedges directly affects the computational complexity.",
"Hence a large number of hyperedges as used in [10] imposes too much computational complexity.",
"In the proposed method, the number of vertices and the number of hyperedges are equal to the number of centroids which is much lower than the number of data points.",
"So it does not generate too many hyperedges and its computational cost is much lower than other hypergraph-based methods such as [10], [15], [4], [16]."
],
[
"Data embedding",
"To perform the task of feature selection, we use a transformation matrix (T) to project data into a low dimensional space.",
"If $x_i$ , $x_j$ are close to each other, we want $x_i T$ , $x_j T$ to be close to each other too.",
"After finding the optimal T, we use $\\ell _{2}$ norm to find primary features which have the most role in setting up this transformation matrix.",
"As it was explained earlier in Section , real-world data sets generally are corrupted and include noise and outlier; As a result, the rank of matrices is high in practical applications [17], [18]; Moreover, it has been demonstrated that high-dimensional data sets usually have low-dimensional representations, i.e.",
"they can be redescribed in a low-dimensional space [19].",
"In order to remedy the aforementioned corruption and robust selection of features, we can use low-rank representation.",
"Hence, we suppose that the transformation matrix is low-rank and we consider the following low-rank constraint for it: rank(T) $\\le $ r Since features are finally ranked according to matrix T, in order not to use uninformative and redundant features in data embedding, matrix T also needs to be sparse; Moreover, some elements of T may have small values which can be only the effect of errors.",
"Trying to make matrix T sparse, also helps to set the values of these elements to zero."
],
[
"Local relationships between centroids",
"Importance of preservation of local neighborhood relationships among data points has been well discussed in recent works [7], [5].",
"In order to maintain data local relationships, we focus on the centroids and then we try to preserve local neighborhood relationships among them.",
"It has been demonstrated that using Gaussian kernel as similarity function can model local neighborhoods [20].",
"So, we try to preserve the Gaussian similarity between different centroids.",
"In other words, we try to maintain the distances between different centroids.",
"To achieve this goal, we need to minimize the following objective function: $\\Psi ^{(1)}(W,~T)~=~\\frac{1}{2}{{\\sum }_{c_{i},c_{j}\\in V}}\\alpha _{i,j}\\times ||(c_iT)^T~-~(c_jT)^T||_2^2$ where $c_iT$ and $c_jT$ are transformed centroids, $||.||_2$ is $\\ell _2$ norm of a vector, and $\\alpha _{i,j}$ is calculated using the following formula: $\\alpha _{i,j}~=~{\\sum }_{e \\in E}~\\frac{w(e)h(c_i,~e)h(c_j,~e)}{\\delta (e)d(c_i)}$ Where $c_i$ is the ith centroid, $d(c_i)$ is its degree in hypergraph, w(e) is the weight of hyperedge e, and $\\delta (e)$ is its degree.",
"$\\alpha _{i,j}$ determines the importance of closing $c_iT$ and $c_jT$ which is zero in case centroids i and j are not connected to each other; Because h($c_i$ , e)h($c_j$ , e) = 0.",
"In the other case, namely h($c_i$ , e)h($c_j$ , e) $>$ 0, the higher value of this factor means these two centroids are closer which leads to a larger $\\alpha _{i,j}$ ; Moreover, $\\frac{w(e)}{\\delta (e)d(c_i)}$ determines the normalized weight of that hyperedge for $c_i$ .",
"So the larger $\\alpha _{i,j}$ is, the higher the importance of closing $c_iT$ and $c_jT$ is.",
"As it was explained, $\\alpha _{i,j}$ also indicates importance of closing $c_i$ and $c_j$ .",
"So minimizing $\\Psi ^{(1)}$ leads to preservation of distances among different centroids.",
"We have: $\\Psi ^{(1)}(W,~T)~=~\\frac{1}{2}{{\\sum }_{c_{i},c_{j}\\in V}}\\alpha _{i,j}\\times ||(c_iT)^T~-~(c_jT)^T||_2^2~=~tr(T^TC^T\\Delta CT)$ Where C is matrix of centroids and $\\Delta $ is the unnormalized hypergraph laplacian matrix defined by [21]: $\\Delta ~=~I_{|V|}~-~D_v^{-1}HWD_e^{-1}H^T$ $D_v$ , $D_e$ , and W are diagonal matrices of vertices degrees, hyperedges degrees, and hyperedges weights respectively and $I_{|V|}$ is a $|V| \\times |V|$ identity matrix.",
"It is worth noting that generally, minimizing $tr\\big (F(C)^T\\Delta F(C)\\big )$ , where F(C) is a function (in our case CT), leads to closing $f(c_i)$ and $f(c_j)$ where $f(c_i)=F(C)_i$ , and $c_i$ and $c_j$ are in at least one common hyperedge; In addition, the amount of effort to close $f(c_i)$ and $f(c_j)$ is determined by contributions of $c_i$ and $c_j$ in their common hyperedge and the normalized hyperedge weight.",
"This analysis clarifies the importance of using soft hypergraphs where different vertices are eligible to have different participation in a hyperedge because elements of the incidence matrix can adopt continuous values.",
"Considering the low-rank constraint for transformation matrix, we have: $\\begin{aligned}\\min _{W,~T}\\Psi ^{(1)}(W,~T)~=~tr(T^TC^T\\Delta CT)\\\\\\text{s.t.",
"rank(T) $\\le $ r}\\end{aligned}$ Where C is the matrix of centroids and considering low-rank constraint is helpful in handling noise and outlier.",
"We also add $\\ell 2,1$ norm of matrix T as a regularizer to try to make it sparse and as a result, not to use uninformative features in data embedding and reducing the effect of errors: $\\Theta ^{(1)}(T)~=~||T||_{2, 1}$ Although $||T||_{2, 1}$ is convex, its derivative does not exist when at least one of its rows is zero.",
"Hence we adopt the definition in [5]: $\\Theta (T)~=~tr(T^TBT)$ Where B is a $d\\times d$ diagonal matrix which its ith element is calculated as follows: $b_i^i~=~\\frac{1}{2||t_i||_2}$ Where $t_i$ is the ith row of T. Adding this term, changes the optimization problem to: $\\begin{aligned}\\min _{W,~T}\\Psi ^{(1)}(W,~T)~+~\\rho \\Theta (T)=~tr(T^TC^T\\Delta CT)~+~\\rho \\times tr(T^TBT)\\\\\\text{s.t.",
"rank(T) $\\le $ r}\\end{aligned}$ Where $\\rho $ is a regularization parameter.",
"In the next step, in order to control model complexity and setting the weight of uninformative or redundant hyperedges to zero, two constraints are added to the optimization problem (REF ) which results: $\\begin{aligned}\\min _{W,~T}\\Psi (W,~T)~+~\\rho \\Theta (T)~=~tr(T^TC^T\\Delta CT)~+~\\rho \\times tr(T^TBT)~+~\\kappa ||diag(W)||_2^2\\\\s.t.~\\sum _{i~=~1}^mW_i=1,~W_i\\ge 0~for~i=0,1,...,m,~\\text{rank(T) $\\le $ r}\\end{aligned}$ Where diag(W) denotes the main diagonal of W, i.e.",
"the vector of hyperedges weights, $W_i = w_i^i$ which is the weight of the ith hyperedge, $\\kappa $ is a regularization parameter, and $||.||_2$ is $\\ell _{2}$ norm of a vector.",
"Although the low-rank constraint added in Section REF helps our proposed framework become robust to noise and outlier, it makes the optimization problem non-convex and NP-hard [22].",
"An appropriate way to apply this constraint is to produce T by multiplying two low-rank matrices, i.e.",
"T = PQ where P $\\in $ $R^{d\\times r}$ and Q $\\in $ $R^{r\\times k}$ : $\\begin{aligned}\\min _{W,~P,~Q}~[\\Psi (W,~P,~Q)+\\rho \\Theta (P,~Q)]~=~tr(Q^TP^TC^T\\Delta CPQ)~+~\\rho \\times tr(Q^TP^TBPQ)~+\\\\\\kappa ||diag(W)||_2^2~~~s.t.~\\sum _{i~=~1}^mW_i=1,~W_i\\ge 0~for~i=0,1,...,m\\end{aligned}$"
],
[
"Global structure",
"As it was discussed, our proposed framework preserves local relationships between centroids by maintaining the distances among them.",
"But in order to conserve the complete structure of data, the global structure of data should be preserved along with local relationships.",
"Since data labels have a significant role in defining the structure of data, many supervised methods such as [23], [24], [3], [2] focus on them to preserve data structure.",
"But in unsupervised learning, data labels are not available.",
"This motivates us to introduce an alternative way of preserving global structure in our unsupervised framework.",
"It has been demonstrated that data points within the same class have high linear dependency and correlation [13].",
"So, trying to preserve the correlation between data points, can be used as an alternative to focusing on data labels for unsupervised learning.",
"Hence, in the proposed framework, we try to maintain the correlation between data points for global structure preservation; Especially data points with more representation power and role in defining the data structure.",
"Since hypergraph construction and using it have high computational cost, we proposed using cluster centroids in Section REF .",
"However, using only centroids leads to inevitable loss of some information.",
"In order to use all the information with minor computational cost, we use all data points in the equation (REF ).",
"In this way, our goal can be minimizing the following objective function: $\\Upsilon ^{(1)}(T)~=~||(XT)(XT)^T~-~XX^T||_F^2$ Where XT is the transformed data and $||.||_F$ is the Frobenius norm of a matrix.",
"In order to normalize the above correlation matrices, we divide each matrix by the length of its generating vectors, i.e.",
"k and d respectively: $\\Upsilon ^{(2)}(T)~=~||\\frac{XT(XT)^T}{k}~-~\\frac{XX^T}{d}||_F^2$ Which can be reformulated as follows: $\\Upsilon ^{(3)}(T)~=~||XT(XT)^T~-~\\frac{k}{d}~\\times ~XX^T||_F^2$ Where k is the dimensionality of embedding and d is the initial number of features.",
"As in local relationships preservation, more representative data points should be emphasized on, especially the centroids.",
"As a data point gets further from its corresponding centroid, not only its importance and representation power reduces, but it gets more likely an outlier.",
"So it is not of high importance to preserve correlation for these data points.",
"For this purpose, we define diagonal matrix D which its ith element is calculated as follows: $d_i^i={\\left\\lbrace \\begin{array}{ll}1, & \\text{if}\\ x_i \\text{ is one of the centroids}\\\\affinity(x_i, c_{\\zeta _i}), & \\text{elsewhere}\\end{array}\\right.",
"}$ Where $c_{\\zeta _i}$ is the centroid of the cluster which $x_i$ belongs to.",
"So, $affinity(x_i, c_{\\zeta _i})$ is the Gaussian similarity between the data point i and its corresponding centroid.",
"Note that $affinity(x_i, c_{\\zeta _i})~\\le ~1$ .",
"The further a data point gets from its corresponding centroid, the smaller its corresponding value in D will be.",
"To focus on data points with better representation power and neglect outliers we change the equation (REF ) to: $\\Upsilon ^{(4)}(T)~=~||D^\\frac{1}{2}[XT(XT)^T~-~\\frac{k}{d}~\\times ~XX^T]D^\\frac{1}{2}||_F^2$ We are actually multiplying the changes of correlations between different data points by their importance, prior to calculating the Frobenius norm which is equivalent to emphasizing on more representative points, especially the centroids.",
"The problem is that the defined term is not convex with respect to T. So we need to modify its representation: $\\begin{aligned}\\Upsilon ^{(4)}(T)~=~||D^\\frac{1}{2}[XT(XT)^T~-~\\frac{k}{d}~\\times ~XX^T]D^\\frac{1}{2}||_F^2\\\\=~||D^\\frac{1}{2}XT(XT)^TD^\\frac{1}{2}~-~\\frac{k}{d}~\\times ~D^\\frac{1}{2}XX^TD^\\frac{1}{2}||_F^2\\\\=~||D^\\frac{1}{2}XTT^TX^TD^\\frac{1}{2}~-~\\frac{k}{d}~\\times ~D^\\frac{1}{2}XX^TD^\\frac{1}{2}||_F^2\\end{aligned}$ And in the last step, to make it convex, we use an approximation provided in [8] and we have: $\\Upsilon (T)~=~||D^\\frac{1}{2}XT~-~Z_k||_F^2$ Where $Z_k = \\Gamma _k\\Xi _k^{1/2}$ , $\\Xi _k$ is the diagonal matrix of k eigenvalues of $[\\frac{k}{d}~\\times ~D^\\frac{1}{2}XX^TD^\\frac{1}{2}]$ sorted descendingly, and $\\Gamma _k$ is the matrix of their corresponding eigenvectors put together column-wise.",
"So, minimizing $\\Upsilon (T)$ results in preserving the correlation between data points."
],
[
"Final optimization problem",
"To find appropriate solutions for W and T, we should minimize $\\Psi (W, T)$ , $\\Theta (T)$ , and $\\Upsilon (T)$ simultaneously, and T needs to be replaced with PQ.",
"In this way, our optimization problem becomes as follows: $\\begin{aligned}\\min _{W,~P,~Q}~[\\Psi (W,~P,~Q)+\\tau \\Upsilon (P,~Q)+\\rho \\Theta (P,~Q)]~=~tr(Q^TP^TC^T\\Delta CPQ)~+~\\tau ||D^\\frac{1}{2}XPQ~-~Z_k||_F^2\\\\+~\\kappa ||diag(W)||_2^2~+~\\rho \\times tr(Q^TP^TBPQ)~~~s.t.~\\sum _{i~=~1}^mW_i=1,~W_i\\ge 0~for~i=0,1,...,m\\end{aligned}$ The optimization problem (REF ) is the final optimization problem we aim to solve."
],
[
"Optimization",
"Our proposed objective function consists of three factors: W, P, and Q.",
"Even though our optimization problem is generally non-convex and complicated, if we fix two factors and consider the other factor a variable, the problem becomes much simpler.",
"So, we use coordinate descent algorithm to solve it which is summarized in algorithm .",
"Each outer iteration includes three steps as follows: Fix Q, W, optimize P. Fix P, W, find the optimal value of Q.",
"Solve for W by fixing P and Q.",
"[H] Input: X and parameters $\\tau $ , $\\rho $ , $\\kappa $ , and r. Output: features scores.",
"Step 1: Hypergraph construction.",
"1.1 cluster X using k-means algorithm; 1.2 generate matrix H based on the equation (REF ); Step 2: Updating P, Q: repeat 2.1 Update P, Q using the equations (REF ), (REF ) respectively; 2.2 Update B according to the equation (REF ); until convergence; Step 3: Updating W: Update W using the coordinate descent algorithm explained in Section REF ; Step 4: Updating $\\Delta $ : Update $\\Delta $ based on the equation (REF ); Step 5: Checking convergence and calculating features scores if necessary: If P, Q have converged, calculate features scores using $\\ell 2$ norm of T rows, otherwise go to Step 2; novel hypergraph point-weighting framework via low-rank representation (HPWL)"
],
[
"Updating P",
"By considering Q and W as constants, we need to solve the following optimization problem: $\\begin{aligned}\\min _P~tr(Q^TP^TC^T\\Delta CPQ)~+~\\tau ||D^\\frac{1}{2}XPQ~-~Z_k||_F^2\\\\+~\\rho \\times tr(Q^TP^TBPQ)\\end{aligned}$ Since the objective function in (REF ) is convex with respect to P, it suffices to take its derivative with respect to P and set it to zero: $\\begin{aligned}C^T(\\Delta + \\Delta ^T)CPQQ^T~+~2\\tau X^TD^\\frac{1}{2}(D^\\frac{1}{2}XPQ-Z_k)Q^T+~2\\rho BPQQ^T~=~0\\end{aligned}$ Which results: $\\begin{aligned}P~=~\\tau [C^T\\Delta ^\\prime C~+~\\tau X^TDX~+~\\rho B]^{-1}X^TD^\\frac{1}{2}Z_kQ^{-1}\\end{aligned}$ Where $\\Delta ^\\prime = (\\Delta + \\Delta ^T) / 2$ ."
],
[
"Updating Q",
"Similarly, we need to solve the following optimization problem to update Q: $\\begin{aligned}\\min _Q~tr(Q^TP^TC^T\\Delta CPQ)~+~\\tau ||D^\\frac{1}{2}XPQ~-~Z_k||_F^2\\\\+~\\rho \\times tr(Q^TP^TBPQ)\\end{aligned}$ Setting its derivative with respect to Q, to zero results: $\\begin{aligned}P^TC^T(\\Delta + \\Delta ^T)CPQ~+~2\\tau P^TX^TD^\\frac{1}{2}(D^\\frac{1}{2}XPQ-Z_k)+~2\\rho P^TBPQ~=~0\\end{aligned}$ Which leads to: $\\begin{aligned}Q~=~\\tau P^{-1}[C^T\\Delta ^\\prime C~+~\\tau X^TDX~+~\\rho B]^{-1}X^TD^\\frac{1}{2}Z_k\\end{aligned}$ Where $\\Delta ^\\prime = (\\Delta + \\Delta ^T) / 2$ ."
],
[
"Updating W",
"Since P and Q are fixed, in order to update W, we need to solve the following optimization problem: $\\begin{aligned}\\min _W~tr(Q^TP^TC^T\\Delta CPQ)~+~\\kappa ||diag(W)||_2^2~~~s.t.~\\sum _{i~=~1}^mW_i=1,~W_i\\ge 0~for~i=0,1,...,m\\end{aligned}$ Based on the equation (REF ), $\\Delta ~=~I_{|V|}~-~D_v^{-1}HWD_e^{-1}H^T$ .",
"The optimization problem (REF ) can be simplified to: $\\begin{aligned}\\min _W~tr(Q^TP^TC^TD_v^{-1}HWD_e^{-1}H^TCPQ)~+~\\kappa ||diag(W)||_2^2\\\\s.t.~\\sum _{i~=~1}^mW_i=1,~W_i\\ge 0~for~i=0,1,...,m\\end{aligned}$ Let $R=Q^TP^TC^TD_v^{-1}H$ , $S=D_e^{-1}H^TCPQ$ .",
"for simplification.",
"We have: $\\begin{aligned}\\min _W~tr(RWS)~+~\\kappa ||diag(W)||_2^2\\\\s.t.~\\sum _{i~=~1}^mW_i=1,~W_i\\ge 0~for~i=0,1,...,m\\end{aligned}$ We have the following equalities: $\\begin{aligned}tr(RWS)~+~\\kappa ||diag(W)||_2^2~=~tr\\Bigg (\\begin{bmatrix}r^1~r^2~\\hdots ~r^m\\end{bmatrix}\\begin{bmatrix}W_1&&\\\\ &\\ddots &\\\\&&W_m\\end{bmatrix}\\begin{bmatrix}s_1\\\\ s_2\\\\ \\vdots \\\\ s_m\\end{bmatrix}\\Bigg )\\\\+~\\kappa ||diag(W)||_2^2~=~tr\\bigg (\\sum _{i=1}^m W_i r^i s_i\\bigg )+~\\kappa ||diag(W)||_2^2~=~\\sum _{i=1}^m\\bigg [\\sum _{j=1}^k r_j^i s_i^j W_i\\bigg ]\\\\+~\\kappa ||diag(W)||_2^2~=~\\sum _{i=1}^m\\bigg [\\Omega _iW_i~+~\\kappa W_i^2\\bigg ]\\end{aligned}$ Where $\\Omega _i=\\sum _{j=1}^k r_j^i s_i^j$ , $s_i$ is the ith row of S, and $r^i$ is the ith column of R. So the optimization problem (REF ) can be simplified to: $\\begin{aligned}\\min _W~\\sum _{i=1}^m\\bigg [\\Omega _iW_i~+~\\kappa W_i^2\\bigg ]\\\\s.t.~\\sum _{i~=~1}^mW_i=1,~W_i\\ge 0~for~i=0,1,...,m\\end{aligned}$ Where $\\Omega _i=\\sum _{j=1}^k r_j^i s_i^j$ .",
"We use the coordinate descent algorithm to solve the optimization problem (REF ); Because it has the potential to provide a sparse solution, i.e.",
"setting weights of uninformative hyperedges to zero.",
"In each iteration of the coordinate descent algorithm, we select two consecutive hyperedges, like $W_i$ and $W_{i+1}$ , to update their weights.",
"Since $\\sum _{i~=~1}^mW_i=1$ , $(W_i + W_{i+1})$ will not change after each inner iteration.",
"So we have: $\\begin{aligned}W_i^*, W_{i+1}^*~=~\\operatornamewithlimits{\\arg \\!\\min }_{W_i, W_{i+1}}{\\bigg [\\Omega _iW_i~+~\\Omega _{i+1}W_{i+1}~+~\\kappa (W_i^2+W_{i+1}^2)\\bigg ]}\\\\s.t.~W_i, W_{i+1} \\ge 0,~~W_i + W_{i+1} = c\\end{aligned}$ Where $W_i^*$ , $W_{i+1}^*$ are the optimal values of $W_i$ , $W_{i+1}$ respectively.",
"Replacing $W_{i+1}$ with $(c-W_i)$ results in: $\\begin{aligned}W_i^*~=~\\operatornamewithlimits{\\arg \\!\\min }_{W_i}{\\bigg [\\Omega _iW_i~+~\\Omega _{i+1}(c-W_i)~+~\\kappa (W_i^2+(c-W_i)^2)\\bigg ]}\\\\s.t.~0 \\le W_i \\le c\\end{aligned}$ Which can be simplified to: $\\begin{aligned}W_i^*~=~\\operatornamewithlimits{\\arg \\!\\min }_{W_i}{\\bigg [2\\kappa W_i^2 + (\\Omega _i-\\Omega _{i+1}-2\\kappa c)W_i\\bigg ]}\\\\s.t.~0 \\le W_i \\le c\\end{aligned}$ Since the objective function in (REF ) is convex, we have the following equations: ${\\left\\lbrace \\begin{array}{ll}W_i^* = 0,~W_{i+1}^* = c & if~2\\kappa c \\le (\\Omega _i - \\Omega _{i + 1})\\\\W_{i+1}^* = 0,~W_i^* = c & if~2\\kappa c \\le (\\Omega _{i + 1} - \\Omega _i)\\\\W_i^* = \\frac{-\\Omega _i+\\Omega _{i+1}+2\\kappa c}{4\\kappa },~W_{i+1}^*=c - W_i^* & \\text{elsewhere}\\end{array}\\right.",
"}$ Where $W_i^*$ , $W_{i+1}^*$ are the updated values of $W_i$ , $W_{i+1}$ respectively, $\\Omega _i=\\sum _{j=1}^k r_j^i s_i^j$ , and $c = (W_i + W_{i+1})$ .",
"Using this method, the value of the objective function in (REF ) reduces after each inner iteration of updating W. As it can be seen in the equation (REF ), the provided solution by the coordinate descent algorithm is potentially sparse.",
"After finding the optimal values of P and Q, matrix T is constructed using T=PQ.",
"Each row of T corresponds to a feature and score of each feature is calculated based on $\\ell 2$ norm of its corresponding row in T. Finally, top-ranked features are selected."
],
[
"Computational cost",
"The first step of algorithm is hypergraph construction.",
"To construct the hypergraph, data points should be clustered using the k-means method.",
"its computational cost is $\\mathcal {O}(t_0mn)$ ; Where $t_0\\ll n$ , $m\\ll n$ , and n are the number of iterations, clusters, and data points respectively.",
"In step 2, matrices P and Q need to be updated.",
"In Section REF , because only centroids participated in the hypergraph, the incidence matrix H was defined $\\in R^{m \\times m}$ .",
"But, if needed, H can be defined $\\in R^{n \\times m}$ by adding zero valued rows for non-centroid data points, without any change in the hypergraph.",
"Updating P, Q requires calculating the inverse of a $d\\times d$ matrix where d is the number of features.",
"Since in feature selection problems, usually the number of data points is much fewer than the number of features, based on [25], by defining matrix H $\\in R^{n \\times m}$ , the equations (REF ), (REF ) can be reformulated as follows: $P~=~\\frac{\\tau }{\\rho }B^{-1}X^T[\\frac{1}{\\rho }(\\Delta ^\\prime +\\tau D)XB^{-1}X^T+I_n]^{-1}D^\\frac{1}{2}Z_kQ^{-1}$ $Q~=~\\frac{\\tau }{\\rho } P^{-1}B^{-1}X^T[\\frac{1}{\\rho }(\\Delta ^\\prime +\\tau D)XBX^T + I_n]^{-1}D^\\frac{1}{2}Z_k$ By this approach the inverses of $n\\times n$ matrices can be calculated instead which yields the time complexity of $\\mathcal {O}(min(d,n)^3)$ for the second step.",
"The third step consists of m inner iterations and computational cost of each iteration is k; Where k is the dimensionality of embedding.",
"As a result, the time complexity of this step is $\\mathcal {O}(mk)$ .",
"Since in equation (REF ), after each outer iteration, the values of H and $D_e$ does not change, there is no need to compute $\\Delta $ from the ground up.",
"So, the computational complexity of this step is $m^2\\log m$ .",
"Since $m\\ll n$ , in almost all cases its lower than step 2.",
"So The computational cost of each step of algorithm is $max\\lbrace \\mathcal {O}(min(d,n)^3), \\mathcal {O}(mk), \\mathcal {O}(t_0mn)\\rbrace $ , $m,t_0\\ll n$ ; Which is less than several state-of-the-art algorithms such as [10], [7], [5], [9] in most cases."
],
[
"Convergence study",
"In Section REF , it was proved that the value of the objective function decreases after each inner iteration.",
"The proposed objective function has lower bound of zero; Moreover, we have the following proposition: Proposition 1.",
"By using coordinate descent method, value of the objective function in (REF ) reduces after each iteration of step 2 of algorithm .",
"Proof: let $P^{(i)}, Q^{(i)} =\\operatornamewithlimits{\\arg \\!\\min }_{P, Q}{\\Psi (P, Q, W) + \\Upsilon (P, Q) + \\rho \\times tr(Q^TP^TB^{(i-1)}PQ)}$ Where $P^{(i)}, Q^{(i)}$ are the values of P, Q after the ith iteration and $B^{(i-1)}$ is the value of B after iteration (i-1).",
"As a result: $\\begin{aligned}\\Psi (P^{(i)}, Q^{(i)}, W) + \\Upsilon (P^{(i)}, Q^{(i)}) + \\rho \\times tr((Q^{(i)})^T(P^{(i)})^TB^{(i-1)}P^{(i)}Q^{(i)}) \\le \\\\\\Psi (P^{(i-1)}, Q^{(i-1)}, W) + \\Upsilon (P^{(i-1)}, Q^{(i-1)}) + \\rho \\times tr((Q^{(i-1)})^T(P^{(i-1)})^TB^{(i-1)}P^{(i-1)}Q^{(i-1)})\\end{aligned}$ Since based on the definition in (REF ), $b_i^i~=~\\frac{1}{2||t_i||_2}$ , equation (REF ) can be rewritten as follows: $\\begin{aligned}\\Psi (P^{(i)}, Q^{(i)}, W) + \\Upsilon (P^{(i)}, Q^{(i)}) + \\rho \\times \\sum _{j=1}^d\\frac{||t_j^{(i)}||_2^2}{||t_j^{(i-1)}||_2} \\le \\\\\\Psi (P^{(i-1)}, Q^{(i-1)}, W) + \\Upsilon (P^{(i-1)}, Q^{(i-1)}) + \\rho \\times \\sum _{j=1}^d\\frac{||t_j^{(i-1)}||_2^2}{||t_j^{(i-1)}||_2}\\end{aligned}$ Based on a lemma provided in [25], we have: $\\begin{aligned}\\sum _{j=1}^d ||t_j^i||_2 - \\sum _{j=1}^d \\frac{||t_j^i||_2^2}{||t_j^{(i-1)}||_2} \\le \\sum _{j=1}^d ||t_j^{(i-1)}||_2 - \\sum _{j=1}^d \\frac{||t_j^{(i-1)}||_2^2}{||t_j^{(i-1)}||_2}\\end{aligned}$ According to the equations (REF ), (REF ), the following result holds: $\\begin{aligned}\\Psi (P^{(i)}, Q^{(i)}, W) + \\Upsilon (P^{(i)}, Q^{(i)}) + \\rho \\times \\sum _{j=1}^d ||t_j^{(i)}||_2^2 \\le \\\\\\Psi (P^{(i-1)}, Q^{(i-1)}, W) + \\Upsilon (P^{(i-1)}, Q^{(i-1)}) + \\rho \\times \\sum _{j=1}^d ||t_j^{(i-1)}||_2^2\\end{aligned}$ Which is equivalent to: $\\begin{aligned}\\Psi (P^{(i)}, Q^{(i)}, W) + \\Upsilon (P^{(i)}, Q^{(i)}) + \\rho \\times \\Theta (P^{(i)}, Q^{(i)}) \\le \\\\\\Psi (P^{(i-1)}, Q^{(i-1)}, W) + \\Upsilon (P^{(i-1)}, Q^{(i-1)}) + \\rho \\times \\Theta (P^{(i-1)}, Q^{(i-1)})\\end{aligned}$ This completes the proof.",
"Algorithm generally converges in two iterations; See figures REF , REF .",
"This number of iterations is fewer than several other feature selection methods such as [10], [5]."
],
[
"Comparison with other methods",
"The computational cost of some state-of-the-art feature selection methods which mentioned their computational complexity, along with the number of outer iterations which they need to converge, is provided in table REF .",
"As it can be seen, focusing on centroids helps our proposed method need a fewer number of iterations to converge and have lower computational complexity in most cases.",
"Table: Computational cost of state-of-the-art methods beside number of outer iterations each one needs to converge."
],
[
"Parameter determination",
"Our proposed objective function includes three parameters that need to be optimized manually: $\\tau $ , $\\kappa $ , and $\\rho $ .",
"$\\kappa $ and $\\rho $ are regularization parameters which adjust sparsity of W and T. Parameter $\\tau $ is used to set the importance of global structure preservation versus maintaining local neighborhood relationships among different centroids.",
"We tuned these parameters empirically by grid search.",
"In our experiments, we used $m = \\left\\lfloor {\\frac{n}{10}}\\right\\rfloor $ for the number of centroids."
],
[
"Evaluation methods",
"In order to exhibit the effectiveness of our proposed method, we tested it using classification methods.",
"Since our proposed method does not collaborate with any specific classifier, it can be tested using any classification method.",
"We evaluated the proposed method using four general classification methods: K-Nearest Neighbours (KNN), Support Vector Machine with linear kernel (L-SVM), Support Vector Machine with radial basis kernel (RB-SVM), and Naïve Bayes (NB).",
"These classifiers have been used to evaluate feature selection methods widely in recent studies [23], [26], [5], [9], [27], [28]."
],
[
"Data sets",
"Six publicly available benchmark data sets were used in our experiments: Gene-Expression [29], Smoke-Cancer [30], Various-Cancers [31], Burkitt-Lymphoma [31], Mouse-Type [31], and Hepatitis-C [31].",
"The number of classes in these datasets ranges from 2 to 10.",
"These datasets also have a different number of samples: from 187 to 801.",
"Moreover, these data sets are from different areas.",
"The details of these data sets are provided in table REF .",
"Table: Details of benchmark data sets used in our experiments."
],
[
"Comparison methods",
"We compared the accuracy of our proposed method with seven state-of-the-art feature selection frameworks.",
"These methods treat all data points equally and do not focus on more representative points, Also the importance of handling noise, redundancy and outlier is underestimated in them.",
"Here we provide a brief explanation of each one of them: [font=$\\bullet $] JHLSR [10] Uses sparse representation for hypergraph construction.",
"FScore [24] This supervised framework scores the features one by one.",
"This method tries to select the features which have the same value for data points within the same class, and different values for data points from different classes.",
"l21RFS [25] Tries to preserve similarity between data points and measures regression loss by $\\ell _{2, 1}$ norm.",
"This framework is supervised with class labels.",
"TraceRatio [27] The main idea of this framework is similar to FScore.",
"But it evaluates features jointly.",
"We used the supervised form of this method in our experiments.",
"LapScore [7] Selects features which can best preserve local structure of data.",
"JELSR [5] Adopts a graph to model data structure.",
"This method performs feature selection by jointly learning embedding and sparse regression.",
"MCFS [9] First carries out manifold learning and then spectral regression.",
"From the above methods, FScore, l21RFS, and TraceRatio are supervised with class labels and other methods are unsupervised.",
"The differences between the unsupervised methods are provided in table REF .",
"Table: Differences between the proposed method and other state-of-the-art unsupervised feature selection methods.",
"a ^atreating data points differently based on their importance.",
"b ^badopting hypergraph for data representation.",
"c ^cunifying two learning instead of performing them consecutively."
],
[
"Experimental settings",
"Since the optimal number of features is unknown, we performed comprehensive experiments with 10, 20, 30, ..., 200 number of features on each benchmark dataset.",
"As in [10], [5] we randomly select 50% of data as training data and the other 50% as test data.",
"We repeated this procedure five times and reported average and standard deviation of accuracies.",
"Training data is given to feature selection methods as input.",
"When the selection of features is conducted, reported features of test data are selected and along with corresponding labels are passed to a classification algorithm.",
"Finally, classification accuracy is reported."
],
[
"Experiments Results",
"The result of the exhaustive experiments conducted on various data sets prove efficiency of our proposed method.",
"Although our proposed unsupervised framework, has lower computational complexity in most cases, it outperforms state-of-the-art supervised and unsupervised feature selection methods.",
"Average and standard deviation of accuracies of each method on every data set are reported in tables REF , REF , REF , REF .",
"The highest accuracies are boldfaced.",
"We enhance average classification accuracy by 3.79% (JHLSR), 16.88% (FScore), 7.56% (l21RFS), 7.3% (TraceRatio), 8.38% (LapScore), 8.12% (JELSR), and 7.38% (MCFS) by KNN classification method.",
"Using L-SVM classifier, we improve average classification accuracy by 3.47% (JHLSR), 16.75% (FScore), 8.98% (l21RFS), 8.12% (TraceRatio), 9.13% (LapScore), 8.98% (JELSR), and 8.22% (MCFS).",
"By classifying using RB-SVM, we improve average classification accuracy by 3.93% (JHLSR), 18.86% (FScore), 8.1% (l21RFS), 8.15% (TraceRatio), 9.02% (LapScore), 8.93% (JELSR), and 8.29% (MCFS).",
"Using NB classification method, we enhance average classification accuracy by 4.75% (JHLSR), 14.19% (FScore), 7.33% (l21RFS), 7.46% (TraceRatio), 9.37% (LapScore), 8.57% (JELSR), and 7.46% (MCFS).",
"Accuracies of different methods for different number of features is also compared in figures REF , REF , REF , REF .",
"Table: Average KNN classification accuracy of the methods on benchmark data sets.Table: Average L-SVM classification accuracy of the methods on benchmark data sets.Table: Average RB-SVM classification accuracy of the methods on benchmark data sets.Table: Average NB classification accuracy of the methods on benchmark data sets.Figure: KNN classification accuracy vs. number of selected features on benchmark data sets.Figure: Various-CancersFigure: L-SVM classification accuracy vs. number of selected features on benchmark data sets.Figure: Various-CancersFigure: RB-SVM classification accuracy vs. number of selected features on benchmark data sets.Figure: NB classification accuracy vs. number of selected features on benchmark data sets.Figure: Various-CancersFigure: Convergence of matrix T on different data sets."
],
[
"Convergence Analysis",
"Since features are finally selected based on matrix T, we analyze changes of matrix T, along with the value of the objective function through iterations in order to study the convergence of algorithm .",
"Let $err(i) = ||T^{(i)} - T^{(i-1)}||_F^2~/~(d \\times k)$ Where $T^{(i)}$ is the value of T after outer iteration number i and $(d \\times k)$ is the number of elements of T. err(i) calculates the normalized change of T after an iteration.",
"Generally, both T and the objective function converge to a global minimum after two iterations.",
"See figures REF , REF .",
"Figure: Various-CancersFigure: Convergence of the proposed objective function on different data sets."
],
[
"Effect of weighting points",
"To show the importance of weighting points of the proposed framework we conducted some experiments.",
"We implemented the ordinary version of our point-weighting framework.",
"In this new framework, all data points are treated equally.",
"For this purpose, we replaced matrix D with the identity matrix.",
"We also changed the soft hypergraph construction method.",
"The new hypergraph was constructed by connecting each data point to its $l$ nearest neighbors.",
"This soft hypergraph is widely used in recent studies [16], [4].",
"This hypergraph contains n vertices and n hyperedges; While the hypergraph of our proposed method contains only m vertices and m hyperedges; Where $m \\ll n$ .",
"Although our point-weighting framework has much lower computational complexity than the ordinary version, conducted experiments support its effectiveness by achieving higher classification accuracy on different benchmark datasets using different evaluation methods, in most cases.",
"See figures REF , REF , REF , and REF .",
"These experiments exhibit the importance of behaving different data points based on their representation power and role in defining the data structure.",
"Figure: KNN classification accuracies of different methods on benchmark data sets.Figure: L-SVM classification accuracies of different methods on benchmark data sets.Figure: RB-SVM classification accuracies of different methods on benchmark data sets.Figure: NB classification accuracies of different methods on benchmark data sets."
],
[
"Effect of soft hypergraph",
"In soft hypergraphs, different vertices can have different participation in a hyperedge.",
"This participation can be based on their importance and role in that hyperedge.",
"This helps the framework to model data structure more precisely.",
"We carried out some experiments to show the superiority of soft hypergraphs to ordinary hypergraphs.",
"To do so, we converted the soft hypergraph of our framework to an ordinary hypergraph and tested it on some benchmark data sets using diverse evaluation methods.",
"As it can be seen in figures REF , REF , REF , and REF , using soft hypergraph results more classification accuracy in most cases.",
"Figure: KNN classification accuracies of different methods on benchmark data sets.Figure: L-SVM classification accuracies of different methods on benchmark data sets.Figure: RB-SVM classification accuracies of different methods on benchmark data sets.Figure: NB classification accuracies of different methods on benchmark data sets.Figure: KNN classification accuracies of different methods on benchmark data sets."
],
[
"Effect of global structure preservation",
"In the proposed framework, we preserve the global structure of data by maintaining the correlation between data points.",
"Since in unsupervised frameworks data labels are unavailable and data points from the same class generally have high linear dependency and correlation [13], this approach performs as an alternative to employment of data labels in supervised learning for structure preservation; Moreover, it helps to repair the inevitable loss of information caused by our low computational hypergraph which is constructed only based on cluster centroids.",
"In order to test the importance of this term, we performed some experiments on different datasets using various evaluation methods.",
"Although adding global structure preservation term has minor effect in computational cost of the proposed algorithm, as it can be seen in figures REF , REF , REF , and REF , it leads to much more effective selection of features in most cases.",
"Figure: L-SVM classification accuracies of different methods on benchmark data sets.Figure: RB-SVM classification accuracies of different methods on benchmark data sets.Figure: NB classification accuracies of different methods on benchmark data sets."
],
[
"Conclusion",
"In this paper, a novel point-weighting framework for hypergraph feature selection was proposed which adopts low-rank representation to handle noise and outlier, and captures the importance of different data points.",
"Focusing on centroids helps the proposed method to have lower computational complexity with respect to many state-of-the-art feature selection methods, in most cases.",
"We also behave different data points based on their representation power and role in defining the data structure.",
"We conducted experiments to show the effectiveness of the innovative ideas of this paper.",
"Exhaustive experiments exhibit the effectiveness of our proposed method in comparison with other state-of-the-art feature selection methods."
]
] | 1808.08414 |
[
[
"Dehn surgery on the minimally twisted seven-chain link"
],
[
"Abstract We classify all the exceptional Dehn surgeries on the minimally twisted chain links with six and seven components."
],
[
"Introduction",
"Let $M$ be a compact orientable three-manifold with some boundary tori.",
"We say as usual that $M$ is hyperbolic if its interior admits a finite-volume complete hyperbolic metric (which is then unique by Mostow and Prasad's rigidity theorem).",
"Recall that a Dehn filling of $M$ is the operation that consists of attaching solid tori to some (possibly all) of the boundary components of $M$ , a manipulation that is essentially determined by the choice of some slopes in the chosen boundary tori.",
"Figure: A notable sequence of hyperbolic links with i≤7i\\le 7 components.",
"These are the figure-eight knot, the Whitehead link, and some chain links with 3,...,73,\\ldots , 7 components.",
"Those with 5,6,5, 6, and 7 components are minimally twisted.We say that a Dehn filling is hyperbolic if the resulting manifold is still hyperbolic, and exceptional otherwise.",
"The goal of this paper is to make a further step in the classification of all the exceptional fillings in a natural sequence of hyperbolic link complements, initiated in [34] and [35].",
"The sequence is shown in Figure REF .",
"The main result is Theorem REF below, where we exhibit a complete classification of all the exceptional fillings of the last two link complements shown in the figure.",
"Figure REF contains a sequence of notable hyperbolic links.",
"These are the figure-eight knot, the Whitehead link, and some particular chain links with $i=3,\\ldots , 7$ components.",
"Let $M_1,\\ldots , M_7$ be the complements of these links.",
"Each $M_i$ is conjectured [2] to have smallest volume among hyperbolic manifolds with $i$ cusps (this conjecture has been proved in the cases $i=1, 2$ , and 4 by Cao – Meyerhoff [10], Agol [2], and Yoshida [41]).",
"Another important feature of this sequence is that each $M_i$ is obtained as a $(-1)$ -filling of the subsequent one $M_{i+1}$ , as one sees via a blow-down as in Figure REF .",
"See [39], [24], [37] for more information on hyperbolic chain links.",
"Figure: A blow-down.The manifolds $M_1,\\ldots , M_7$ appear naturally in many contexts.",
"The manifold $M_3$ was called the magic manifold by Gordon and Wu in [20] because of its many interesting fillings; it plays a role in the study and/or classification of the closed hyperbolic manifolds of smallest volume [18], of the pseudo-Anosov mapping classes with small dilatation [22], [26], and of the cusped hyperbolic 3-manifold with the largest number of exceptional fillings [31].",
"The manifolds $M_4, M_5$ , and $M_6$ form a superb triple of highly symmetric hyperbolic manifolds.",
"They decompose into regular ideal octahedra, tetrahedra, and octahedra respectively, and they are characterised (together with $M_3$ ) by the configuration of the many thrice-punctured spheres they contain [42].",
"The manifolds $M_5$ and $M_6$ cover two very natural hyperbolic orbifolds, shown in Figure REF .",
"The first is the boundary of the 5-simplex and decomposes into 5 regular ideal tetrahedra.",
"The second is obtained by mirroring an ideal regular octahedron.",
"The manifolds $M_3$ , $M_5$ , and $M_6$ are principal congruence link complements [7], while $M_5$ and $M_6$ are also the smallest hyperbolic 3-manifolds admitting a regular tessellation [21].",
"Figure: The manifolds M 5 M_5 and M 6 M_6 double cover two natural orbifolds.",
"The deck transformation is a π\\pi rotation around the dotted axis.The fillings of $M_4$ were used to classify the four-manifolds with shadow-complexity one [28] and to build knots with long unknotting tunnels [11].",
"It was noted in [40], [14] that many cusped manifolds in the census [9] are obtained by filling $M_5$ .",
"Among these, we find many Berge knots complements [4] and other hyperbolic manifolds with interesting exceptional surgeries [3], [5], [15], [19], [25].",
"The question of classifying all the exceptional fillings of $M_6$ was raised in [16].",
"We answer to this question here.",
"The manifold $M_6$ appears in the construction of hyperbolic four-manifolds with arbitrarily many cusps [29] and of arithmetic link extensions [6] in $S^3$ .",
"Its fundamental group is biorderable [27].",
"By filling $M_6$ we obtain yet more hyperbolic manifolds with interesting exceptional fillings [16], [38].",
"The manifold $M_7$ lacks all the beautiful symmetries of $M_4, M_5$ , and $M_6$ .",
"It is the first and the only non-arithmetic manifold in the sequence $M_1, \\ldots , M_7$ .",
"The hyperbolic Dehn fillings of the figure-eight knot complement $M_1$ were famously described by Thurston in his notes [39].",
"The exceptional fillings of the magic manifold $M_3$ were then classified by Martelli and Petronio in [34].",
"Later on, all the exceptional fillings on $M_5$ were listed by Martelli, Petronio, and Roukema in [35].",
"The main result of this paper is a complete classification of all the exceptional filings of the complement $M_7$ of the minimally twisted chain link with seven components, the last one of the sequence in Figure REF .",
"This of course includes also a classification of all the exceptional fillings of $M_6$ .",
"The author would like to thank Nathan Dunfield and the anonymous referee for providing helpful comments on a first draft of this paper."
],
[
"The general strategy",
"The exceptional Dehn fillings on a multi-cusped hyperbolic manifold $M$ may be infinite in number, but can typically be described using a finite amount of data.",
"For instance, the magic manifold contains infinitely many exceptional fillings, which could be grouped into explicit families and were fully described in few pages in [34].",
"To classify the exceptional fillings of $M_7$ we adopt the same general strategy of [34], [35], that we now outline.",
"More generally, let $M$ be any hyperbolic manifold with some boundary tori.",
"Recall that a filling of $M$ is determined by a set of slopes, one for each filled boundary torus (we are allowed to leave some boundary tori unfilled).",
"Following [35], we say that an exceptional Dehn filling on $M$ is isolated if any proper subset of the chosen slopes produces a hyperbolic Dehn filling.",
"Thurston's Dehn filling Theorem implies the following: Theorem 1.1 Every hyperbolic $M$ has only finitely many isolated exceptional fillings.",
"To classify all the exceptional fillings of $M$ we must fulfill the following tasks: classify all the isolated exceptional fillings of $M$ , recognise the filled manifolds in each case, and if necessary, proceed recursively on each filled manifold.",
"The third point is necessary if the filled exceptional manifold contains some hyperbolic piece in its prime or JSJ decomposition, a case that did not occur in [34] and [35], but that will arise here in this paper.",
"We could achieve all these objectives for the complement $M_7$ of the minimally twisted chain link with seven components.",
"To this purpose we made an essential use of the formidable programs SnapPy [12], Regina [8], and Recognizer [36].",
"Task (1) was fulfilled via find_exceptional_fillings.py, a python script written by the author already used in [35] and publicly available [33] to be performed on any cusped hyperbolic three-manifold.",
"The computer-assisted proof is rigorous thanks to the hikmot libraries [23]."
],
[
"The output",
"When accomplished, the general strategy produces finitely many families of exceptional fillings, but as the number of cusps increases their number explodes, and it soon becomes impossible to write fully comprehensive tables as it was done for the magic manifold in [34].",
"We now describe the outcome of our research.",
"The most concise amount of useful information that we can give is the following.",
"Table: Numbers of isolated exceptional fillings on M i M_i.Theorem 1.2 The number of isolated exceptional fillings of $M_1,\\ldots , M_7$ is shown in Table REF .",
"The complete lists of fillings can be downloaded from [33].",
"The first impression that we get from looking at Table REF is that the number of isolated exceptional fillings of $M_i$ with a fixed number $k$ of slopes is roughly constant as $i=1,\\ldots , 7$ varies, and grows roughly exponentially in $k$ ."
],
[
"Data reduction",
"The manifold $M_7$ has 255,209 isolated exceptional fillings overall, and we would like to describe what these filled manifolds are.",
"We cannot of course describe them all on a single table; instead, we try to reduce the amount of data that is necessary to understand and present them in some reasonable way.",
"It was already remarked in [35] that all the exceptional fillings of $M_5$ can actually be deduced from a very short lists of rules: a collection of 7 basic exceptional fillings plus a list of 5 isometries of $M_5$ and of some of its fillings generate all the exceptional fillings.",
"Everything could be described in [35] in a half-page long theorem.",
"(See Remark REF below for some corrections of the tables in [35].)",
"We would like to find a similar small generating set of rules for the manifolds $M_6$ and $M_7$ .",
"As a first step, we quotient the exceptional fillings of $M_i$ by the action of its isometry group ${\\rm Isom}(M_i)$ .",
"The isometry groups of $M_1,\\ldots , M_7$ have order: $8, \\qquad 8,\\qquad 12,\\qquad 64,\\qquad 240,\\qquad 192,\\qquad 28.$ These are respectively $D_8, \\qquad D_8, \\qquad D_{12}, \\qquad G_{64}, \\qquad S_5 \\times \\mathbb {Z}_{2}, \\qquad D_8 \\times S_4, \\qquad D_{28}.$ Here $D_{2n}$ is the dihedral group of order $2n$ and the symbol $G_{64}$ indicates some non-abelian group of order 64 that does not split as a direct product.",
"The manifolds $M_4, M_5$ , and $M_6$ are arithmetic, decompose into regular ideal tetrahedra or octahedra, and have an extraordinary number of symmetries.",
"On the other hand, the last manifold $M_7$ is not arithmetic [37] and has only few isometries: the 28 symmetries of the chain link that one infers from Figure REF and nothing more than that.",
"Table: Numbers of isolated exceptional fillings on M i M_i up to the action of the symmetry group of M i M_i.The numbers of isolated exceptional fillings on $M_i$ for $i=1,\\ldots , 7$ considered up to the action of ${\\rm Isom}(M_i)$ are listed in Table REF .",
"These numbers are smaller than those of Table REF , but yet too big for our purposes, especially with the least symmetric and largest manifold $M_7$ .",
"We now want to reduce them further.",
"Recall that each $M_{i-1}$ is a filling of $M_i$ .",
"We use the standard meridian/longitude basis to identify the slopes on the boundary tori of $M_i$ with $\\mathbb {Q}\\cup \\lbrace \\infty \\rbrace $ .",
"If $s$ is a set of slopes we denote by $M_i(s)$ the manifold obtained by filling $M_i$ via $s$ .",
"Via a blow-down as in Figure REF we see that $M_{i-1} = M_i(-1)$ for all $i\\ge 2$ .",
"Note that there is no need of specifying which boundary component is filled thanks to the cyclic symmetry of all chain links.",
"We now say that an exceptional filling of $M_i$ factors through $M_{i-1}$ if it contains $-1$ or any slope in the orbit of $-1$ along the action of ${\\rm Isom}(M_i)$ .",
"Since we are classifying the exceptional slopes of $M_i$ inductively on $i$ , it is natural to exclude those that factor through $M_{i-1}$ .",
"The surviving slopes are then collected in Table REF .",
"Table: Numbers of isolated exceptional fillings on M i M_i that do not factor through M i-1 M_{i-1}, up to the action of the symmetry group of M i M_i.The numbers in Table REF are extraordinarily small for $M_1,\\ldots , M_5$ and are quite reasonable also for $M_6$ .",
"We can say informally that every $M_i$ adds a very small number of exceptional fillings to those of $M_{i-1}$ when $i\\le 5$ .",
"Only 6+10+15+3+3+47=84 basic exceptional fillings generate all the exceptional fillings of the manifolds $M_1,\\ldots , M_6$ .",
"These 84 exceptional fillings are described in the tables at the end of the paper.",
"The following theorem summarizes these discoveries.",
"Theorem 1.3 The numbers of isolated exceptional fillings on $M_i$ that do not factor through $M_{i-1}$ , up to the action of the symmetry group of $M_i$ , are listed in Table REF .",
"The fillings of $M_1,\\ldots , M_6$ are described in the Tables REF , REF , REF , REF , REF , REF , REF , REF , and REF .",
"In the tables at the end of the paper we can find the exceptional slopes and a description of the 84 filled exceptional manifolds.",
"Among these, 81 are graph manifolds and 3 are irreducible manifolds whose JSJ contains a hyperbolic piece, the figure-eight complement $M_1$ .",
"The manifold $M_6$ is the first manifold in the list that has some exceptional fillings that are not graph manifolds.",
"A precise description of all the exceptional fillings of $M_6$ is given in Theorem REF ."
],
[
"The manifold $M_7$",
"We are left with the 2,007 exceptional fillings of $M_7$ from Table REF .",
"These are yet too many to be reproduced here.",
"Why does $M_7$ have such a big number of isolated fillings that do not factor through $M_6$ ?",
"This is probably due again to its lack of symmetries: the group ${\\rm Isom}(M_i)$ acts transitively on the cusps of $M_i$ for all $i$ , and the groups ${\\rm Isom}(M_4)$ , ${\\rm Isom}(M_5)$ , and ${\\rm Isom}(M_6)$ have a formidable amount of additional symmetries that send the slope $-1$ on any boundary torus $T$ to the sets of slopes (respectively) $\\left\\lbrace -1, \\frac{1}{2}, \\frac{3}{2}, 3\\right\\rbrace , \\qquad \\left\\lbrace -1, \\frac{1}{2}, 2\\right\\rbrace , \\qquad \\lbrace -1,1\\rbrace $ on any boundary torus.",
"Therefore $M_{i-1}$ is a filling of $M_i$ in multiple ways (respectively: in 16, 15, and 12 different ways), and hence there are many possibilities for an exceptional set of slopes on $M_i$ to factor through $M_{i-1}$ .",
"(We remark that the slopes $\\lbrace 0,1,2,\\infty \\rbrace $ , $\\lbrace 0,1,\\infty \\rbrace $ , and $\\lbrace 0,\\infty \\rbrace $ are exceptional on $M_4$ , $M_5$ , and $M_6$ respectively.)",
"On the other hand ${\\rm Isom}(M_7)$ acts trivially on the slopes of a single boundary torus, so $M_6$ is a filling of $M_7$ in only 7 distinct ways.",
"Since $\\lbrace 0, \\infty \\rbrace $ are exceptional, the first important non-exceptional slopes are $-1$ and 1.",
"It is natural to expect that most exceptional fillings of $M_7$ should contain either $-1$ or 1.",
"This is indeed the case, as Table REF shows quite impressively.",
"We denote by $N_6$ the hyperbolic manifold $N_6=M_7(1)$ .",
"Table: Numbers of isolated exceptional fillings on M 7 M_7 that do not contain the slopes -1-1 and 1, up to the action of the symmetry group of M 7 M_7."
],
[
"Another sequence of links",
"We are still left with the problem of listing all the exceptional fillings of the new manifold $N_6 = M_7(1)$ .",
"By mirroring the blow-down in Figure REF we see that $N_6$ is also the complement of a chain link with six components.",
"It will be convenient to see $N_6$ as the last member of another sequence of chain link complements shown in Figure REF , that parallels somehow that of Figure REF .",
"Figure: Another sequence of hyperbolic chain links.",
"Each link complement is a 1-filling of the subsequent one.For every $i=3,\\ldots , 6$ , let $N_i$ be the complement of the chain link in Figure REF with $i$ components.",
"These are all hyperbolic.",
"Each $N_i$ is a 1-filling of $N_{i+1}$ , and $N_6$ is a 1-filling of $M_7$ .",
"The manifolds $N_3, \\ldots , N_5$ cannot be obtained as a Dehn filling of $M_6$ because they have some interesting exceptional fillings that $M_6$ does not have, as we will see.",
"The volumes of the manifolds $M_i$ and $N_i$ are shown in Table REF .",
"Table: The approximated volumes of the manifolds M i M_i and N i N_i for i≥3i\\ge 3.It might be interesting to compare the numbers of exceptional fillings of the sequence $N_i$ with those of $M_i$ .",
"These are listed in Tables REF and REF .",
"The symmetries of $N_3, \\ldots , N_6$ are only those of the corresponding chain links, so they form a group of order 12, 16, 20, and 24.",
"Table: Numbers of isolated exceptional fillings on N i N_i.Table: Numbers of isolated exceptional fillings on N i N_i, up to the action of the symmetry group of N i N_i."
],
[
"Some notable fillings",
"Recall that our goal is to describe the exceptional fillings of $N_6$ with the minimum amount of information.",
"Using SnapPy we discover some notable fillings of $N_3,\\ldots , N_6$ in Table REF .",
"The table shows that the fillings $\\lbrace -3, -2, -1, 1\\rbrace $ on each $N_3, N_4, N_5$ and the fillings $\\lbrace -2, -1, 1\\rbrace $ on $N_6$ are diffeomorphic to either $N_{i-1}$ or some fillings of $M_3, M_4, M_5, M_6$ .",
"Since we have already examined the exceptional fillings of these manifolds, we disregard them: we say that a filling of $N_3, \\ldots , N_6$ factors if it contains one of these slopes (that is $\\lbrace -3,-2,-1,1\\rbrace $ for $N_3, N_4, N_5$ , and $\\lbrace -2, -1, 1\\rbrace $ for $N_6$ ).",
"We are happy with this definition because the number of isolated exceptional fillings of $N_3,\\dots , N_6$ that do not factor is very small: see Table REF .",
"The following theorem summarizes our discoveries.",
"Theorem 1.4 The numbers of isolated exceptional fillings of $N_i$ that do not factor, up to the action of the symmetry group of $N_i$ , are listed in Table REF .",
"The fillings are described in the Tables REF , REF , REF , and REF .",
"Among these 3+3+5+10 = 21 exceptional fillings, we find 14 graph manifolds and 7 irreducible manifolds whose JSJ contains a hyperbolic piece.",
"The hyperbolic pieces that arise are $M_1, M_2, M_3$ , and $M_4$ .",
"Table: Some fillings of N i N_i are diffeomorphic either to N i-1 N_{i-1} or to some filling of M 3 ,M 4 ,M 5 ,M 6 M_3, M_4, M_5, M_6.",
"For instance from this table we infer that N 4 (-3)=M 5 (3,2 3)N_4(-3) = M_5(3,\\frac{2}{3}) and N 5 (-1)=M 5 (-1)=M 4 N_5(-1) = M_5(-1) = M_4.Table: Numbers of isolated exceptional fillings on N i N_i that do not factor, up to the action of the symmetry group of N i N_i."
],
[
"A final improvement",
"We conclude this discussion by further reducing the numbers of Table REF .",
"Using SnapPy, we note the isometry $M_7(-2,-2) = N_6(-3)$ .",
"Since we have already classified the exceptional fillings of $N_6$ , we may disregard all the fillings containing $(-2,-2)$ .",
"We say that a filling of $M_7$ factors if it contains the slope 1, $-1$ , or the pair $(-2,-2)$ in two consecutive boundary components.",
"The final survivors are counted in Table REF .",
"We will identify them in Tables REF and REF .",
"Table: Numbers of isolated exceptional fillings on M 7 M_7 that do not contain the slopes -1,1-1,1, and (-2,-2)(-2,-2), up to the action of the symmetry group of M 7 M_7.We summarize our discoveries on $M_7$ .",
"Theorem 1.5 The numbers of isolated exceptional fillings of $M_7$ that do not factor, up to the action of the symmetry group of $M_7$ , are listed in Table REF .",
"The fillings are described in the Tables REF and REF .",
"Among the exceptional fillings of $M_7$ we find infinitely many pairwise non-diffeomorphic closed manifolds whose JSJ has a hyperbolic piece.",
"A complete description of all the manifolds that can be obtained as exceptional fillings of $M_7$ is given in Theorem REF .",
"Remark 1.6 The tables in [35] of all the closed isolated exceptional fillings of $M_5$ contain a few mistakes that we correct here.",
"In [35] the fillings $(-1, -2, -1, -3, -2)$ and $(-1, -2, -1, \\frac{1}{2}, \\frac{1}{2})$ should be replaced with the correct ones $(-3, -2, -1, -3, -2)$ and $(-1, 2, -1, \\frac{1}{2}, \\frac{1}{2})$ respectively.",
"On the other hand, the exceptional fillings $(-1, -\\frac{1}{2}, -1, \\frac{1}{2}, \\frac{5}{3})$ , $(-1, -\\frac{1}{3}, -1, \\frac{2}{3}, \\frac{3}{2})$ , $(-1, \\frac{1}{2}, 3, -1, -\\frac{1}{2})$ , and $(-1, \\frac{1}{2}, -1, \\frac{1}{3}, \\frac{3}{2})$ should be removed from [35] because they are actually not isolated.",
"For this reason the wrong numbers 5232 and 52 appeared in [35] instead of the correct ones 4818 and 48 that we display here."
],
[
"The exceptional fillings",
"In the previous section we have reduced all the isolated exceptional fillings of the hyperbolic manifolds $M_1,\\ldots , M_7$ to some, as small as possible, “generating” set.",
"We now describe explicitly these generating exceptional fillings."
],
[
"Notation",
"We use the same notation of [32], [35] for Seifert and graph manifolds, which seems standard.",
"We quickly recall it here.",
"Given a compact surface $\\Sigma $ , possibly with boundary, and some pairs $(p_1,q_1), \\ldots , (p_k,q_k)$ of coprime integers, the notation $X = \\big ( \\Sigma , (p_1,q_1), \\ldots , (p_k,q_k) \\big )$ denotes the 3-manifold $X$ obtained as follows.",
"We remove $k$ open discs from $\\Sigma $ , thus getting a new surface $\\Sigma ^{\\prime }$ .",
"Then we attach $k$ solid tori to the (unique) oriented circle bundle over $\\Sigma ^{\\prime }$ by killing the slopes $(p_1,q_1), \\ldots , (p_k,q_k)$ in any $k$ boundary tori.",
"We use here as a basis a meridian in $\\partial \\Sigma ^{\\prime }$ and a longitude $\\lbrace {\\rm pt}\\rbrace \\times S^1$ , oriented as a positive basis.",
"Note that the case $p_i = 0$ is allowed here.",
"It is a standard fact on Seifert manifolds that if $p_i\\ne 0$ for all $i$ then $X$ is a Seifert manifold, while if $p_i=0$ for some $i$ then $X$ “degenerates” to a connected sum of lens spaces and solid tori.",
"More specifically, if $\\Sigma $ is orientable we have $\\big ( \\Sigma , (0,1), (p_2,q_2), \\ldots , (p_k,q_k) \\big )=L({ p_2},{ q_2}) \\# \\cdots \\# L({ p_k},{ q_k}) \\#_{2g} (S^2 \\times S^1) \\#_b(D^2 \\times S^1)$ where $g$ and $b$ are the genus and the number of boundary components of $\\Sigma $ .",
"Concretely, in most cases the surface $\\Sigma $ will be either $S^2, D, A$ , or $P$ , that is a sphere, a disc, an annulus, or a pair of pants.",
"The manifold $X$ described in this way is naturally equipped with an orientation and a standard meridian/longitude basis on each boundary torus.",
"There is no need to distinguish between the boundary tori of a Seifert manifold since they are all equivalent up to diffeomorphism.",
"With that in mind, given some manifolds $X, Y, Z,$ and matrices $A, B \\in {\\rm GL}(2,\\mathbb {Z})$ we may write $X \\bigcup \\nolimits _{A} Y \\bigcup \\nolimits _B Z, \\qquad X/_A, \\qquad X \\bigcup \\nolimits _A\\nolimits ^B Y$ to denote some graph manifolds that decomposes along tori into the pieces $X,Y,Z$ , glued via the maps $A, B$ .",
"In the second example two distinct boundary components of $X$ are identified via $A$ .",
"In the third, two manifolds $X, Y$ have two pairs of boundary tori glued via $A$ and $B$ .",
"All the matrices here will have $\\det = -1$ .",
"We also use the notation $T_A$ to denote a torus fibration over $S^1$ with monodromy $A$ , and in this case we have $\\det A = 1$ ."
],
[
"Ambiguities",
"The same graph manifold may be described in various different ways and unfortunately in many occasions there is no preferred description.",
"A useful set of moves that modify the notation of a graph manifold was collected in [34].",
"We report here for completeness the ones that are more relevant for us: we will use them at various points.",
"Here are the first ones: $\\big ( \\Sigma , (a,b), \\ldots (y,z) \\big ) & = & \\big ( \\Sigma , (a,-b), \\ldots (y,-z) \\big ) \\\\\\big (\\Sigma ,({ a},{ b}),({c},{ d}),\\ldots \\big )& = & \\big (\\Sigma ,({ a},{ b+ka}),({c},{ d-kc}),\\ldots \\big )\\\\\\big (\\Sigma ,({ a},{ b}),\\ldots \\big )& = & \\big (\\Sigma ,({ a},{ b+ka}),\\ldots \\big )\\quad {\\rm if\\ }\\partial \\Sigma \\ne \\emptyset \\\\\\big (\\Sigma ,({ 1},{ 0}),({a},{ b}),\\ldots \\big )& = & \\big (\\Sigma ,({ a},{ b}),\\ldots \\big )$ In the first move we change the orientation of the manifold.",
"In this paper we have chosen to write a Seifert manifold as standardly as possible, with positive normalized numbers if the manifold has boundary: the unique oriented Seifert manifold fibering over the orbifold $(D,2,2)$ is usually denoted as $\\big (D,({2},{ 1}),({2},{1})\\big )$ , although the notation $\\big (D,({2},{ 1}),({2},{-1})\\big )$ would also be natural since it visibly expresses the fact that the Euler number vanishes; there are two oriented Seifert manifolds fibering over $(D,2,3)$ , and these are $\\big (D,({2},{ 1}),({3},{1})\\big )$ and $\\big (D,({2},{ 1}),({3},{2})\\big )$ .",
"They are orientation-reversingly diffeomorphic.",
"The following moves involve the gluing of two pieces: $X \\bigcup \\nolimits _A Y & = & X \\bigcup \\nolimits _{-A} Y \\\\\\big (\\Sigma ,({ a},{ b}),\\ldots \\big )\\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} m & n \\\\ p & q \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"X & = & \\big (\\Sigma ,({ a},{ b+k a}),\\ldots \\big )\\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} m+k n & n \\\\ p+k q & q \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"X \\\\X\\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} m & n \\\\ p & q \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (\\Sigma ,({ a},{ b}),\\ldots \\big )& = & X\\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} m & n \\\\ p-k m & q-k n \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!",
"}\\big (\\Sigma ,({ a},{ b+ka}),\\ldots \\big ) $ The moves (, ) also apply when two boundary tori of the same block are glued together, but move (REF ) does not!",
"Note that with (REF , , ) it is not possible to change the absolute value $|n|$ of the top-right element $n$ in the gluing matrix.",
"Indeed $|n|$ has an important geometric significance: it is the geometric intersection of the fibers of the two glued Seifert manifolds.",
"There are also some more complicated moves that occur in more sporadic cases.",
"The following reflect the fact that $\\big (D,({2},{ 1}),({2},{1})\\big )$ has an alternative description as the orientable circle bundle $S\\begin{picture}(12,12)\\put (2,0){\\times }\\put (2,4.5){\\sim }\\end{picture}S^1$ over the Möbius strip $S$ .",
"$\\big (D,({2},{ 1}),({2},{1})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} m & n \\\\ p & q \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"X & = &(S\\begin{picture}(12,12)\\put (2,0){\\times }\\put (2,4.5){\\sim }\\end{picture}S^1)\\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} n & n-m \\\\ q & q-p \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"X \\\\X\\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} m & n \\\\ p & q \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({2},{ 1}),({2},{1})\\big ) & = & X \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} m+p & n+q \\\\ -m & -n \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"(S\\begin{picture}(12,12)\\put (2,0){\\times }\\put (2,4.5){\\sim }\\end{picture}S^1)$ Finally, it is sometimes useful to understand how the manifold “degenerates” when we perform a longitudinal filling: $\\big (S^2,({ a},{ b}),({ c},{ d}),({ 0},{ 1})\\big ) & = & L({ a},{ b})\\# L({ c},{ d}) \\\\\\big (D,({0},{ 1}),({a},{b})\\big )\\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} m & n \\\\ p & q \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (\\Sigma ,\\ldots \\big )& = & L({ a},{ b})\\#\\big (\\Sigma ^{\\prime },({ n},{ q}),\\ldots \\big )$ Here $\\Sigma ^{\\prime }$ is $\\Sigma $ with one boundary component capped off.",
"In this paper $S^2 \\times S^1$ is also denoted as the lens space $L(0,1)$ ."
],
[
"Zero and infinity",
"Let $M$ be the complement of any chain link $L\\subset S^3$ .",
"The fillings 0 and $\\infty $ on any boundary component of $M$ are easily understood, and they are always exceptional.",
"The filling $\\infty $ corresponds to the removal of a component from $L$ , so we get the complement of an open chain in $S^3$ as in Figure REF -(1), that is easily identified as the graph manifold $A \\times S^1, P \\times S^1$ , or $P\\times S^1 \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\cdots \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"P \\times S^1$ depending on the number of components of $L$ .",
"Figure: The fillings ∞\\infty and 0 on any chain link.The filling 0 may be modified with a handle slide as shown in Figure REF -(2).",
"The resulting manifold is the complement of another chain link (with two components less) attached to a $P \\times S^1$ .",
"These exceptional fillings 0 and $\\infty $ will appear on all the tables concerning the various chain links studied in this paper."
],
[
"The exceptional fillings",
"We can finally describe the exceptional fillings of $M_6$ and $M_7$ .",
"Let us start with $M_6$ .",
"For completeness, we also review all the exceptional fillings of $M_1,\\ldots , M_5$ , already classified in [34], [35].",
"All the tables are postponed to the end of the paper for the sake of clarity.",
"Theorem 2.1 Every isolated exceptional filling of $M_i$ with $i=1,\\ldots , 6$ that does not factor through $M_{i-1}$ is equivalent, up to the action of ${\\rm Isom}(M_i)$ , to precisely one of those listed in Tables REF , REF , REF , REF , REF , REF , REF and REF .",
"The filled manifold is also shown there.",
"Recall that factoring through $M_{i-1}$ is equivalent to containing $-1$ in some cusp, or any other slope obtained from $-1$ by the action of ${\\rm Isom}(M_i)$ .",
"The tables show the isolated exceptional slopes (one representative for each orbit of ${\\rm Isom}(M_i)$ ), the filled manifold, and its integral first homology group.",
"The notation for the filled manifold sometimes differ from [34], [35] via some of the moves listed in Section REF .",
"We now do the same with $N_3,\\ldots , N_6$ .",
"Recall that “factoring” here means that the filling slopes contain at least one of the numbers $\\lbrace -3,-2,-1,1\\rbrace $ for $N_3, N_4, N_5$ and of $\\lbrace -2,-1,1\\rbrace $ for $N_6$ .",
"Theorem 2.2 Every isolated exceptional filling of $N_3, \\ldots , N_6$ that does not factor is equivalent, up to the action of ${\\rm Isom}(N_i)$ , to precisely one of those listed in Tables REF , REF , REF , and REF .",
"The filled manifold is also listed there.",
"Finally, we turn to $M_7$ .",
"Recall that “factoring” here means that the filling slope contains either $-1$ , 1, or the pair $(-2,-2)$ in two consecutive boundary tori.",
"Theorem 2.3 Every isolated exceptional filling of $M_7$ that does not factor is equivalent, up to the action of ${\\rm Isom}(M_7)$ , to precisely one of those listed in Tables REF and REF .",
"The tables shown so far contain a fair amount of information.",
"From these, we can easily deduce which kinds of non-hyperbolic filling we can obtain from each manifold $M_i$ .",
"We do this in the following sections."
],
[
"The manifold $M_5$",
"The following theorem was already proved in [35].",
"Theorem 2.4 The closed non-hyperbolic fillings of $M_5$ are precisely the manifolds: $\\big (D,({a},{ b}),({c},{d})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({e},{ f}),({g},{h})\\big ),$ $\\big (D,({2},{ 1}),({2},{1})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 1+n & 2+n \\\\ -n & -1-n \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({2},{ 1}),({3},{1})\\big ),$ $\\big (A,({a},{ b})\\big ) \\big /_{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!",
"}, \\qquad \\big (A,({2},{ 1})\\big ) \\big /_{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 1 & 2 \\\\ 0 & -1 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!",
"}$ where $(a,b)$ , $(c,d)$ , $(e,f)$ , $(g,h)$ are arbitrary pairs of coprime integers, and $n\\in \\lbrace 0,1,2,3\\rbrace $ .",
"We note that the first family in the theorem contains many different kinds of manifolds.",
"Proposition 2.5 The manifolds $X$ that may be obtained via the description $X = \\big (D,({a},{ b}),({c},{d})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({e},{ f}),({g},{h})\\big )$ are precisely the following: The lens spaces and connected sums of two lens spaces.",
"The Seifert manifolds fibering over $S^2$ with 3 exceptional fibres.",
"The Seifert manifolds fibering over $\\mathbb {P}^2$ with 2 exceptional fibres.",
"The Seifert manifold $(K,1)$ .",
"The graph manifolds whose JSJ decomposition is as in the description of $X$ .",
"Here $K$ is the Klein bottle and $(K,1)$ is the fibration over $K$ with Euler number 1.",
"We use the moves described in Section REF .",
"If one of $a,c,e,g$ is zero, we get a connected sum of two lens spaces.",
"Now suppose $a,c,e,g \\ne 0$ .",
"If $a=1$ , we get a Dehn filling of $\\big (D,({e},{ f}),({g},{h})\\big )$ , hence either a lens space, a connected sum of two lens spaces, or a Seifert manifold fibering over $S^2$ with 3 exceptional fibres.",
"We are left with the case $|a|, |c|, |e|, |g| \\ge 2$ .",
"In general, we get a graph manifold whose JSJ decomposition is as in the description of $X$ .",
"There is only one exceptional case to consider: if $a=c=2$ , then up to some moves we get $X & = \\big (D,({2},{ 1}),({2},{1})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} k & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({e},{ f}),({g},{h})\\big ) \\\\&= \\big ( S \\begin{picture}(12,12)\\put (2,0){\\times }\\put (2,4.5){\\sim }\\end{picture}S^1 \\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 1 & 1-k \\\\ 0 & -1 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({e},{ f}),({g},{h})\\big )$ for some $k\\in \\mathbb {Z}$ .",
"If $k=1$ the fibers of the two blocks match to give a Seifert manifold with two exceptional fibres over $\\mathbb {RP}^2$ .",
"If $e=g=2$ also the right block has another fibration and we get $X = \\big (S \\begin{picture}(12,12)\\put (2,0){\\times }\\put (2,4.5){\\sim }\\end{picture}S^1) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 1 & 1-k \\\\ l & l-1-kl \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({2},{ 1}),({2},{1})\\big ) =\\big (S \\begin{picture}(12,12)\\put (2,0){\\times }\\put (2,4.5){\\sim }\\end{picture}S^1 \\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 1+l & l-k-kl \\\\ -1 & k-1 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (S \\begin{picture}(12,12)\\put (2,0){\\times }\\put (2,4.5){\\sim }\\end{picture}S^1 \\big )$ for some $k,l\\in \\mathbb {Z}$ .",
"The two fibres match when $l-k-kl=0$ and $k-1=\\pm 1$ .",
"We get two cases $(k,l)=(0,0)$ or $(2,-2)$ and in both cases we get the Seifert manifold $(K,\\pm 1)$ .",
"The generic case (5) consists precisely of all the irreducible 3-manifolds whose JSJ decomposition consists of two Seifert pieces, each fibering over a disc with two cone points, whose fibers meet in the glued torus with geometric intersection one.",
"Theorem REF says that $M_5$ has also some more sporadic exceptional fillings where this geometric intersection is 2, 3, 4, or 5.",
"Among the exceptional fillings of $M_5$ we also find another family that can be analyzed in a similar fashion: Proposition 2.6 The manifolds $X$ that may be obtained via the description $\\big (A,({a},{ b})\\big ) \\big /_{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!",
"}$ are precisely the following: The manifold $S^2\\times S^1$ .",
"The torus bundles of type $T_{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} b & 1 \\\\ -1 & 0 \\\\ \\end{array} \\right)}}.$ The graph manifolds whose JSJ decomposition is as in the description of $X$ .",
"When $a=0$ we get $S^2\\times S^1$ .",
"When $a=1$ we get the torus bundle $T_{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} b & -1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}} = T_{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} b & 1 \\\\ -1 & 0 \\\\ \\end{array} \\right)}}.$ When $|a| \\ge 2$ we get a manifold whose JSJ decomposition is as described.",
"As above, the manifolds that we get in (3) are precisely all the irreducible 3-manifolds whose JSJ decomposition consists of a single piece fibering over an annulus with a cone point, whose fibers meet in the glued torus with geometric intersection one.",
"Theorem REF exhibits also a sporadic example with geometric intersection 2.",
"As we already knew from [35], all the exceptional fillings of $M_5$ are graph manifolds.",
"We now discover here that this is not the case for $M_6$ ."
],
[
"The manifold $M_6$ .",
"We now turn to the exceptional fillings of $M_6$ .",
"Theorem 2.7 The closed non-hyperbolic fillings of $M_6$ are precisely the manifolds: $\\big (D,({a},{ b}),({c},{d})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (A,({e},{ f})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({g},{ h}),({i},{j})\\big ),$ $\\big (A,({a},{ b})\\big )\\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}}^{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}} \\big (A,({c},{ d})\\big ),$ $M_1 \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} -1 & 0 \\\\ 1 & 1 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({2},{ 1}),({2},{1})\\big ), \\qquad M_1 \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} -1 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({2},{ 1}),({2},{1})\\big )$ where $(a,b)$ , $(c,d)$ , $(e,f)$ , $(g,h)$ , $(i,j)$ are arbitrary pairs of coprime integers.",
"The Tables REF , REF , REF and REF show that the exceptional fillings $\\infty , \\quad \\left(-2,-\\dfrac{1}{2},\\cdot ,\\dfrac{1}{2},2\\right),\\quad \\left(-3,-2,-\\dfrac{1}{3},3,2,\\dfrac{1}{3}\\right), \\quad \\left(-3,-\\dfrac{3}{2},-\\dfrac{1}{2},2,2,\\dfrac{1}{3}\\right)$ give rise precisely to all the manifolds listed in the theorem.",
"Conversely, a case by case analysis of the manifolds listed in Theorem REF and Tables REF , REF , REF , and REF shows that all the exceptional fillings of $M_6$ are of one of these types.",
"Here are the details.",
"Using the moves of Section REF we see that $\\big (D,({a},{ b}),({c},{d})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (A,({1},{ f})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({g},{ h}),({i},{j})\\big )= $ $\\big (D,({a},{ b}),({c},{d})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 1 & f \\\\ 0 & -1 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({g},{ h}),({i},{j})\\big ),$ $\\big (A,({a},{ b})\\big )\\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}}^{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}} \\big (A,({1},{ d})\\big )= \\big (A,({a},{ b})\\big ) \\big /_{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 1 & d \\\\ 0 & -1 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!",
"}.$ By applying the moves (, ) we deduce that we can actually obtain in this way all the manifolds of the following two types: $\\big (D,({a},{ b}),({c},{d})\\big ) \\bigcup \\nolimits _B \\big (D,({g},{ h}),({i},{j})\\big ), \\qquad \\big (A,({a},{ b})\\big ) /_B$ where $B$ is any matrix that can be written as $B = \\begin{pmatrix}1 + mf & f \\\\-m-n-mnf & -(1 + nf)\\end{pmatrix}$ for some $m,n,f\\in \\mathbb {Z}$ .",
"In other words, here $B$ is any matrix $B = \\tiny \\left(\\begin{array}{@{}c@{\\ }c@{}} r & f \\\\ s & t \\\\ \\end{array} \\right)$ with $\\det B = -1$ such that $r \\equiv 1 \\mod {f}$ and $t \\equiv -1 \\mod {f}$ .",
"When the manifold is of the first type we can also exchange $B$ with $-B$ using the move (REF ) from Section REF , so we may also get $r \\equiv -1 \\mod {f}$ and $t \\equiv 1 \\mod {f}$ in that case.",
"All the manifolds that arise from Theorem REF and Tables REF , REF , REF and REF are of this kind, except of course the two manifolds whose JSJ decomposition contains a hyperbolic piece.",
"Theorem REF exhibits a couple of important differences between $M_6$ and the manifolds $M_1, \\ldots , M_5$ .",
"The first is that all the graph manifolds come into two big families, and there are no sporadic manifolds outside of these.",
"The second is of course the presence of two irreducible manifolds whose JSJ decomposition contains some hyperbolic piece.",
"The following proposition furnishes some details on the graph manifolds produced by the first family.",
"Proposition 2.8 The manifolds $X$ that may be obtained via the description $X = \\big (D,({a},{ b}),({c},{d})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (A,({e},{ f})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({g},{ h}),({i},{j})\\big )$ are precisely the following: The manifolds that arise in Proposition REF .",
"The connected sums of three lens spaces.",
"The connected sums of a Seifert manifold over $S^2$ with 3 exceptional fibres and a lens space.",
"The Seifert manifolds over $S^2$ with 4 exceptional fibres.",
"The Seifert manifolds over $K$ with 0 or 1 exceptional fibres.",
"The graph manifolds whose JSJ decomposition is $X = \\big (D,({a},{ b}),({c},{d})\\big ) \\bigcup \\nolimits _B \\big (D,({g},{ h}),({i},{j})\\big )$ with $B = \\begin{pmatrix}1 + mf & f \\\\-m-n-mnf & -(1 + nf)\\end{pmatrix}.$ The graph manifolds whose JSJ decomposition is $X = \\big (S,({a},{ b})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({c},{ d}),({e},{f})\\big )$ The graph manifolds whose JSJ decomposition is as in the description of $X$ .",
"Here $K$ and $S$ are the Klein bottle and the Möbius strip.",
"If $a=0$ we get a connected sum $L({ c},{ d}) \\# \\big (S^2, (f,e),(g,h),(i,j)\\big )$ .",
"The second addendum may in turn give rise to a connected sum of two lens spaces.",
"If $e=0$ we get a connected sum of two lens spaces.",
"So we suppose that $a,c,e,g,i \\ne 0$ .",
"If $a=1$ we get a manifold as in Proposition REF .",
"So we suppose $|a|, |c|, |g|, |i| \\ge 2$ .",
"If $e=1$ we get a manifold $\\big (D,({a},{ b}),({c},{d})\\big ) \\bigcup \\nolimits _B \\big (D,({g},{ h}),({i},{j})\\big )$ where $B$ is any matrix that can be written as $B = \\begin{pmatrix}1 + mf & f \\\\-m-n-mnf & - (1 + nf)\\end{pmatrix}.$ See the proof of Theorem REF .",
"If $f=0$ we get (4).",
"If $f\\ne 0$ we get a graph manifold as in (6), except possibly when one (or both) piece is $\\big (D,({2},{ 1}),({2},{1})\\big )$ and the fibrations match: in this way we could obtain a Seifert fibration over $\\mathbb {RP}^2$ with two singular fibres or over $K$ without singular fibres; the former was already obtained in (1) and the latter will be obtained in the next paragraph by other means, so we ignore it.",
"We can now suppose that also $|e| \\ge 2$ .",
"We get a manifold of type (8), except when the left (or the right) block is $\\big (D,({2},{ 1}),({2},{1})\\big )$ and its alternative fibration matches with that of the central block.",
"This may happen and we get a manifold of type (7), unless this happens on both extreme blocks simultaneously and in this case we get (5).",
"We do the same analysis with the second family of graph manifolds.",
"Proposition 2.9 The manifolds $X$ that may be obtained via the description $\\big (A,({a},{ b})\\big )\\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}}^{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}} \\big (A,({c},{ d})\\big )$ are precisely the following: The manifolds $(S^2\\times S^1) \\# L(p,q)$ .",
"The torus bundles $T_C$ with $C = \\begin{pmatrix}-(1 + mf) & -f \\\\-m-n-mnf &- (1 + nf)\\end{pmatrix}.$ The graph manifolds whose JSJ decomposition is $X = \\big (A,({a},{ b})\\big ) /_B$ with $B = \\begin{pmatrix}1 + mf & f \\\\-m-n-mnf &-(1 + nf)\\end{pmatrix}.$ All graph manifolds whose JSJ decomposition is as in the description of $X$ .",
"If $a=0$ we get $(S^2 \\times S^1) \\# L(d,c)$ .",
"So we suppose $a, c \\ne 0$ .",
"If $c=1$ we get (3), unless $a=1$ and in this case we get (2).",
"If $|a|, |c| \\ge 2$ we get (4).",
"We note in particular that we get all the torus bundles with monodromy $\\tiny \\left(\\begin{array}{@{}c@{\\ }c@{}} -1 & 0 \\\\ a & -1 \\\\ \\end{array} \\right)$ .",
"However, we do not get the identity matrix!",
"We deduce the following.",
"Corollary 2.10 Among the six orientable flat 3-manifolds, five can be obtained by Dehn filling $M_6$ , but the 3-torus cannot.",
"The four that fiber over $S^2$ with three exceptional fibers or over $\\mathbb {RP}^2$ with two exceptional fibers can already be obtained from the magic manifold $M_3$ .",
"The one that fibers over $S^2$ with four exceptional fibers (equivalently, over $K$ ) is obtained with $M_6$ .",
"One important novelty in the exceptional fillings of $M_6$ is of course the presence of two sporadic irreducible manifolds whose JSJ decomposition contains a hyperbolic piece.",
"Both exceptional manifolds decompose into the figure-eight knot complement $M_1$ and the Seifert manifold $\\big (D,({2},{ 1}),({2},{1})\\big )$ , which is diffeomorphic to the $I$ -bundle $K\\begin{picture}(12,12)\\put (2,0){\\times }\\put (2,4.5){\\sim }\\end{picture}I$ over the Klein bottle $K$ and to the orientable $S^1$ -bundle $S \\begin{picture}(12,12)\\put (2,0){\\times }\\put (2,4.5){\\sim }\\end{picture}S^1$ over the Möbius stip $S$ .",
"By looking at the gluing matrices, we note that in both cases the meridian of the figure-eight complement (which is also the shortest curve in a flat cusp section) is attached to the fiber of the alternative fibration $S \\begin{picture}(12,12)\\put (2,0){\\times }\\put (2,4.5){\\sim }\\end{picture}S^1$ , that is represented as the slope $(-1, 1)$ in the fibration $\\big (D, (2,1), (2,1) \\big )$ ."
],
[
"The manifold $M_7$ .",
"We now turn to $M_7$ .",
"We recall that the simplest method we found to describe all the exceptional fillings of $M_7$ was to study an alternative sequence of chain links $N_3, \\ldots , N_6$ .",
"We first note the quite surprising fact that $N_3$ contains infinitely many distinct exceptional fillings with hyperbolic pieces.",
"As an aside, this implies that the manifolds $N_3, N_4, N_5$ are not fillings of $M_6$ .",
"We can now list all the exceptional fillings of $M_7$ .",
"Theorem 2.11 The closed non-hyperbolic fillings of $M_7$ are precisely the manifolds: $\\big (D,({a},{ b}),({c},{d})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (A,({e},{ f})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (A,({g},{ h})\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({i},{ j}),({k},{l})\\big ),$ $\\big (A,({a},{ b})\\big )\\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}}^{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}} \\big (A,({c},{ d})\\big ),$ $ M_5\\big ((a,b), (c,d), (e,f), (g,h)\\big ) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({i},{ j}),({k},{l})\\big ), $ $M_2(a,b) \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} -1 & 0 \\\\ 1 & 1 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({2},{ 1}),({2},{1})\\big ),$ $\\big (A,({2},{ 1})\\big ) \\big /_{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} n-1 & n \\\\ 1 & 1 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!",
"}$ where $(a,b)$ , $(c,d)$ , $(e,f)$ , $(g,h)$ , $(i,j)$ , $(k,l)$ are arbitrary pairs of coprime integers and $n\\in \\lbrace 3, 4, 5, 6\\rbrace $ .",
"In the third family we suppose $|i|, |k| \\ge 2$ .",
"We note the reappearance of some sporadic graph manifolds in the list, which were absent in $M_6$ .",
"The four sporadic graph manifolds listed in the last row are not members of the previous families.",
"We have to check that all the exceptional fillings of $M_7$ are of this type, and to this purpose we only need to verify this for the manifolds listed in Tables REF , REF , REF , REF , REF , and REF .",
"Concerning graph manifolds, this is easily settled using the moves described in Section REF when necessary.",
"We are left with the non-graph manifolds with non-trivial JSJ decomposition.",
"The manifolds $N_i(0)$ with $i=3,\\ldots , 6$ are obviously a filling of $N_7(0)$ and hence can be excluded, since their fillings are already contained in the third family.",
"The tables contain three manifolds $X_i = M_2 \\bigcup \\nolimits _{A_i} \\big (D,({2},{ 1}),({2},{1})\\big ),$ where $A_i$ is one of the matrices $A_1= \\begin{pmatrix} -1 & 2 \\\\ 1 & -1 \\end{pmatrix}, \\qquad A_2= \\begin{pmatrix} -1 & 0 \\\\ 1 & 1 \\end{pmatrix}, \\qquad A_3 = \\begin{pmatrix} -1 & 1 \\\\ 1 & 0 \\end{pmatrix}.$ There are also more manifolds where $M_2$ is replaced either by $M_1$ or by $M_2(-2)$ and $A_i$ is still of one of these three types.",
"Since $M_1 = M_2(-1)$ via an isometry that acts as the identity on the other boundary torus, these manifolds are fillings of the $X_i$ and can be ignored.",
"To conclude we need to show that the manifolds $X_1$ and $X_3$ are fillings of $X = M_5 \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"P \\times S^1.$ It will be convenient to use the moves in Section REF and write them as $X_1 & = M_2 \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} -1 & 2 \\\\ 0 & 1 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({2},{ 1}),({2},{-1})\\big ), \\\\X_3 & = M_2 \\bigcup \\nolimits _{{\\tiny {\\left(\\begin{array}{@{}c@{\\ }c@{}} -1 & 1 \\\\ -1 & 2 \\\\ \\end{array} \\right)}}\\phantom{\\Big |}\\!\\!}",
"\\big (D,({2},{ -1}),({2},{-1})\\big ).$ Using SnapPy we find an isometry from $M_2$ to $M_5(-1,-2,-2)$ that acts on a boundary torus as the matrix $B = \\begin{pmatrix}1 & -1 \\\\ 0 & 1 \\end{pmatrix}.$ We also note that there are isometries of $M_5$ that act on the cusps like the matrices $C_1 = \\begin{pmatrix}0 & 1 \\\\-1 & 1\\end{pmatrix}, \\qquad C_2 = \\begin{pmatrix}-1 & 1 \\\\-1 & 0\\end{pmatrix}$ We deduce that both $X_1$ and $X_3$ are Dehn fillings of $X$ because $\\begin{pmatrix} -1 & 2 \\\\ 0 & 1 \\end{pmatrix} =\\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} C_1B,\\qquad \\begin{pmatrix} -1 & 1 \\\\ -1 & 2 \\end{pmatrix} =\\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} C_2B.$ The proof is complete.",
"Corollary 2.12 The 3-torus $S^1\\times S^1 \\times S^1$ is not a filling of $M_7$ .",
"In particular, the Borromean rings complement is not a filling of $M_7$ .",
"As pointed out by the referee, this corollary is actually well-known: since the minimally twisted chain links are strongly invertible, every Dehn surgery on $M_7$ is a double branched cover over some link in $S^3$ .",
"By a result of Fox [17], the 3-torus cannot be realized as a double branched cover on any link in $S^3$ .",
"Therefore it cannot be realized as a Dehn surgery on any minimally twisted chain link."
],
[
"The cusped census",
"Many cusped hyperbolic manifolds of the Callahan – Hildebrand – Weeks census [9] are Dehn fillings of $M_7$ and a classification of their exceptional fillings can be deduced from the theorems stated here.",
"The Callahan – Hildebrand – Weeks census contains all the cusped hyperbolic manifolds $N$ with complexity $c\\le 7$ (the complexity is the minimum number of tetrahedra in a topological ideal triangulation).",
"To see whether $N$ can be obtained as a filling of $M_i$ for some $i\\le 7$ , a very simple and concrete method consists of drilling the shortest simple closed curve found by SnapPy in $N$ multiple times, until we get a manifold with $i$ cusps.",
"This typically leads to some hyperbolic manifold $N_i$ and one checks via SnapPy whether $N_i$ is isometric to $M_i$ or not.",
"Note that as soon as $N_i=M_i$ we also have $N_{i+1} = M_{i+1}$ because each $M_{i+1}$ is obtained by drilling a shortest curve from $M_i$ .",
"If this crude algorithm fails, of course this does not imply that $N$ is not a filling of $M_i$ .",
"Table: Numbers of cusped hyperbolic manifolds of complexity cc that transform into M i M_i after repeatedly drilling the shortest curve found by SnapPy.",
"If i≥4i\\ge 4 we only count those that were not previously transformed into M i-1 M_{i-1}.We display in Table REF the number of manifolds in each complexity $c\\le 5$ , $c=6$ , and $c= 7$ for which this algorithm produces a positive answer for $M_i$ .",
"We could represent all the manifolds of the census as a filling of $M_7$ , except a number of 10, 37, and 151 of them with $c\\le 5, c=6$ , and $c=7$ respectively.",
"Figure: The Borromean rings."
],
[
"The Borromean rings",
"We have discovered at the end of Section REF that the complement $W$ of the Borromean rings (shown in Figure REF ) is not a Dehn filling of $M_7$ .",
"While performing the drilling algorithm, we notice that in each of the 10 remaining manifolds $N$ with $c(N)\\le 5$ of Table REF , the drilled manifold $N_3$ is in fact isometric to the Borromean rings complement $W$ .",
"More generally, among the 10, 47, and 198 remaining manifolds with $c \\le 5$ , $c=6$ , and $c=7$ , for 10, 34, and 92 of them the manifold $N_3$ is isometric to $W$ .",
"We are now led naturally to the study of the exceptional fillings of the Borromean ring complement $W$ .",
"These are easily classified.",
"We note that $W(-1) = M_2$ is the Whitehead link complement, so we consider only the Dehn fillings that do not factor through $M_2$ .",
"Theorem 3.1 The isolated exceptional fillings on the Borromean rings $W$ , considered up to the action of ${\\rm Isom}(W)$ , are listed in Table REF .",
"The exceptional fillings of all the 301 manifolds with $c\\le 5$ can be deduced from the theorems stated here.",
"These 301 manifolds have either one or two cusps.",
"While completing this paper, we have been informed that Dunfield has recently classified and recognised the exceptional fillings of all the one-cusped hyperbolic manifolds in the much wider $c\\le 9$ census.",
"He discovered in particular that there are almost 206,000 exceptional fillings overall [13]."
],
[
"Proofs",
"The proofs of the Theorems REF , REF , REF , REF , and REF follow the strategy outlined in the introduction.",
"To detect the isolated exceptional fillings we use the python script find_exceptional_fillings.py, that can be downloaded from [33] to be used on any multi-cusped hyperbolic manifold.",
"The script was already used in [35], and some of its routines have been rewritten in a more efficient way to increase its speed.",
"We refer to [35] for a detailed tutorial to the script.",
"As explained there, the script produces two finite lists of fillings: a list of “probably isolated exceptional fillings”, and a list of “probably hyperbolic fillings” that typically contains only closed fillings.",
"To conclude rigorously with the proof we need to prove that indeed all the members of the first list are not hyperbolic, and those of the second are hyperbolic (in the unlucky case where the second list contains cusped manifolds, one should run the program again on each).",
"We have been able to accomplish that in all the steps of the proofs.",
"As in [35], we have used Recognizer and Regina for the first task, and another python script for the second: the script search_geometric_solution.py is designed to determine the hyperbolicity of a long list of fillings of the same manifold $M$ , using retriangulations and finite coverings.",
"We could complete all cases using finite covers of order $\\le 11$ .",
"Few cases needed the degree 11 coverings.",
"Following this strategy, all the exceptional fillings of $M_i$ and $N_i$ with $i\\le 5$ can be classified directly.",
"The program needs few seconds for $i\\le 3$ , few minutes for $i=4$ , and few hours for $i=5$ .",
"To attack $M_6$ we use the formidable symmetries of $M_6$ and note that, thanks to Agol and Lackenby's 6 theorem [1], [30], an exceptional filling of $M_6$ factors either through $M_5$ or through $M_6(-2)$ .",
"So to complete the classification for $M_6$ we only needed to analyze the 5-cusped manifold $M_6(-2)$ .",
"Alternatively, we can directly classify all the fillings on $M_6$ in few days of computer time.",
"We used a similar approach for the manifolds $N_3, \\ldots , N_6$ .",
"Classifying directly the exceptional fillings of $M_7$ takes too much computer time, so we attacked it using the 6 theorem again: we only needed to analyse $M_7(\\alpha )$ with $\\alpha \\in \\lbrace -3, -2, -1, -\\frac{1}{2}, 0, \\frac{1}{2}, 1, 2, \\infty \\rbrace $ .",
"Note that, due also to the presence of fewer symmetries, we have to consider many more cases for $M_7$ than for $M_6$ .",
"The slopes 0 and $\\infty $ are exceptional.",
"The slopes $-1$ and 1 have already been analysed and give $M_6$ and $N_6$ .",
"We ran the program on $M_7(\\alpha )$ for each remaining $\\alpha $ , and after a few days the computation was done.",
"We have also used another script to regenerate from the sometimes small amount of information found (the exceptional fillings that do not factor) all the isolated exceptional fillings of $M_6$ and $M_7$ , by acting with the isometry groups of $M_6$ and $M_7$ .",
"Table: The exceptional fillings of the figure-eight knot complement M 1 M_1.",
"We also show their first homology groups.Table: The isolated exceptional fillings of the Whitehead link complement M 2 M_2 that do not factor through M 1 M_1, up to the action of Isom (M 2 ){\\rm Isom}(M_2).",
"We also show their first homology groups.Table: The isolated exceptional fillings of the magic manifold M 3 M_3 that do not factor through M 2 M_2, up to the action of Isom (M 3 ){\\rm Isom}(M_3).",
"We also show their first homology groups.Table: The isolated exceptional fillings of M 4 M_4 that do not factor through M 3 M_3, up to the action of Isom (M 4 ){\\rm Isom}(M_4).",
"We also show their first homology groups.Table: The isolated exceptional fillings of M 5 M_5 that do not factor through M 4 M_4, up to the action of Isom (M 5 ){\\rm Isom}(M_5).",
"We also show their first homology groups.Table: The non-closed isolated exceptional fillings of M 6 M_6 that do not factor through M 5 M_5, up to the action of Isom (M 6 ){\\rm Isom}(M_6).",
"The dot indicates a cusp that is not filled: it appears when the fillings are not along consecutive components on the link.",
"We also show the first homology groups.Table: The closed isolated exceptional fillings of M 6 M_6 that do not factor through M 5 M_5, up to the action of Isom (M 6 ){\\rm Isom}(M_6).",
"We also show their first homology groups.",
"(Part I.",
")Table: The closed isolated exceptional fillings of M 6 M_6 that do not factor through M 5 M_5, up to the action of Isom (M 6 ){\\rm Isom}(M_6).",
"We also show their first homology groups.",
"(Part II.",
")Table: The closed isolated exceptional fillings of M 6 M_6 that do not factor through M 5 M_5, up to the action of Isom (M 6 ){\\rm Isom}(M_6).",
"We also show their first homology groups.",
"(Part III.",
")Table: The isolated exceptional fillings of N 3 N_3 that do not factor, up to the action of Isom (N 3 ){\\rm Isom}(N_3).",
"We also show their first homology groups.Table: The isolated exceptional fillings of N 4 N_4 that do not factor, up to the action of Isom (N 4 ){\\rm Isom}(N_4).",
"We also show their first homology groups.Table: The isolated exceptional fillings of N 5 N_5 that do not factor, up to the action of Isom (N 5 ){\\rm Isom}(N_5).",
"We also show their first homology groups.Table: The isolated exceptional fillings of N 6 N_6 that do not factor, up to the action of Isom (N 6 ){\\rm Isom}(N_6).",
"We also show their first homology groups.Table: The isolated exceptional fillings of the Borromean rings complement WW that do not factor through M 1 M_1, up to the action of Isom (W){\\rm Isom}(W).",
"We also show their first homology groups.Table: The non-closed isolated exceptional fillings of M 7 M_7 that do not factor, up to the action of Isom (M 7 ){\\rm Isom}(M_7).",
"We also show their first homology groups.Table: The closed isolated exceptional fillings of M 7 M_7 that do not factor, up to the action of Isom (M 7 ){\\rm Isom}(M_7).",
"We also show their first homology groups."
]
] | 1808.08430 |
[
[
"Database-Agnostic Workload Management"
],
[
"Abstract We present a system to support generalized SQL workload analysis and management for multi-tenant and multi-database platforms.",
"Workload analysis applications are becoming more sophisticated to support database administration, model user behavior, audit security, and route queries, but the methods rely on specialized feature engineering, and therefore must be carefully implemented and reimplemented for each SQL dialect, database system, and application.",
"Meanwhile, the size and complexity of workloads are increasing as systems centralize in the cloud.",
"We model workload analysis and management tasks as variations on query labeling, and propose a system design that can support general query labeling routines across multiple applications and database backends.",
"The design relies on the use of learned vector embeddings for SQL queries as a replacement for application-specific syntactic features, reducing custom code and allowing the use of off-the-shelf machine learning algorithms for labeling.",
"The key hypothesis, for which we provide evidence in this paper, is that these learned features can outperform conventional feature engineering on representative machine learning tasks.",
"We present the design of a database-agnostic workload management and analytics service, describe potential applications, and show that separating workload representation from labeling tasks affords new capabilities and can outperform existing solutions for representative tasks, including workload sampling for index recommendation and user labeling for security audits."
],
[
"Introduction",
"Extracting patterns from a SQL query workload has enabled a number of important features in database systems, including workload compression [3], index recommendation [2], modeling user and application behavior [31], [9], [35], query recommendation [1], predicting cache performance [29], [5], and designing benchmarks [35].",
"These techniques can be used as part of a more comprehensive approach to automate database administration .",
"However, the diversity of applications have led to a diversity of solutions, each relying on specialized feature engineering.",
"For example, workload summarization for index recommendation uses the structure of join and group by operators as features [3], query recommendation may pre-process a query into fragments before making recommendations [13], and security audits may require user-defined functions to enforce particular policies [32].",
"In fact, the features and the algorithms to extract them tend to be the significant contributions in the papers in this space.",
"But the state of the art in a variety of applications is to learn features automatically.",
"For instance, Natural Language Processing applications previously relied on parsing and labeling sentences as a pre-processing step, but now use learned vector representations almost exclusively [6], .",
"This approach not only obviates the need for manual feature engineering and pre-processing, but also has the potential to significantly outperform more specialized methods.",
"We see three trends motivating an analogous role for generalized workload representations.",
"First, workload heterogeneity is increasing, making it difficult to maintain SQL parsers and feature extraction routines.",
"The number of SQL-like languages is increasing, with inconsistent support and syntax for even relatively common features such as outer joins.",
"Second, workload scale is increasing.",
"Cloud-hosted, multi-tenant database services including Redshift [8], Snowflake [4], BigQuery [23] and more receive millions of queries daily from thousands of customers using hundreds of schemas; relying on brittle parsers (or worse, manual inspection) to identify query patterns that influence administration decisions is no longer tenable.",
"Third, new use cases for centralized workload management are emerging.",
"For example, SQL debugging [7], database forensics [27], and data use management [32] motivate a more automated analysis of user behavior patterns, and cloud-hosted multi-tenant systems motivate a more automated approach to query routing and resource allocation.",
"In this work, we propose Querc, a database-agnostic systems for mining and managing large-scale and heterogeneous workloads.",
"We model workload management and analysis as a set of query labeling tasks.",
"For instance, workload sampling can be reduced to labeling each query as present or absent in the sample, error prediction involves labeling each query with an error type, query routing involves labeling each query with a cluster resource to which the query should be routed, and so on.",
"Because our framework depends only on the query text (along with typical metadata such as arrival timestamp and userid issuing the query), it can be used with any DBMS and any SQL dialect.",
"In fact, as we will show, features learned with a workload against a particular schema and SQL dialect can be effective even when used with a different schema and SQL dialect.",
"The weakness of this approach is that it requires enormous amounts of data to be effective.",
"But as database products migrate to the cloud, service providers have access to workloads from a large number of customers, potentially even across different database products.",
"Since the input is just the query text, these diverse workloads can be processed as one very large dataset.",
"But the resulting vectors can still be used to train models to support specific applications, as we will show on two representative tasks: workload summarization for index selection and user prediction for security audits and routing."
],
[
"System Architecture",
"Figure REF illustrates the architecture of Querc.",
"There are three applications, X, Y, Z.",
"Each application has its own database, DB(X), DB(Y), and DB(Z), though these may be logical instances in the same physical multi-tenant service.",
"In this example, DB(X) and DB(Y) are tenants in the same service.",
"Each application is also associated with a separate stream of queries (at left), where query(X,t) indicates a batch of queries arriving for application X at time instant $t$ .",
"Each application is associated with one Qworker, but each Qworker operates multiple classifiers.",
"Qworkers may not be entirely stateless, as some labeling tasks process a small window of queries.",
"However, the state is assumed to be small such that the Qworkers do not need their own local storage and can be load balanced and parallelized in typical ways.",
"Each classifier is a pre-trained (embedder, labeler) pair.",
"The same trained embedder may be used across multiple applications.",
"This split design is critical, because we want to learn features using a very large, combined workload, but an individual classifier may perform better when trained on an application-specific workload.",
"In this example, application X and application Y both share the same embedder, EmbedderA, trained on the combined X and Y workloads, written EmbedderA(X,Y).",
"This log sharing between customers may not always be permitted by customers for security reasons, and in this example, application Z uses only its own data.",
"But there is some incentive for customers to pool their data as the additional signal can potentially improve accuracy, and some cloud providers support features to allow data sharing between customers.",
"The Labeler passes the query on to the database, but also transmits the query back to a central training module (“Training, Evaluation, and Offline Labeling” in Figure REF ).",
"The training module manages training sets, including the (parallel) execution of training and evaluation routines, then deploys trained models back to Qworkers.",
"There is significant ongoing research in the database, systems, and ML communities on runtime architectures for training and deploying models (e.g., [21]); we do not discuss them further since our requirements are relatively modest.",
"Since Querc is specialized for query workload analytics rather than general machine learning, one data model can be shared among most applications.",
"The only messages passed between components are labeled queries.",
"A labeled query is a tuple $(Q, c_1,c_2,c_3,\\dots )$ where $c_i$ is a label.",
"This simple model captures situations where a query arrives already equipped with a timestamp, a userid, an IP address, etc., but also captures more verbose query logs that are returned from the database.",
"The training module also records the queries with their predicted labels for retraining, evaluation, and to support offline analysis tasks.",
"Offline tasks are those that do not require or do not allow processing each query separately, and can be implemented as typical batch jobs.",
"For example, query clustering is important for workload summarization [16], but does not require real-time labeling of individual queries.",
"Training data is collected periodically from the databases in the form of query logs.",
"These logs are (batched) sequences of labeled queries, but with additional labels to be used for training, such as runtime, memory usage, error codes, security flags, resource IDs.",
"We do not specify the mechanism by which these logs are transmitted from the database to Querc, since most systems have robust means of exporting logs in appropriate forms.",
"In some applications, Querc may not be in the critical path for query execution to avoid any performance overhead or reduce dependencies.",
"In these cases, queries will be forked to Querc.",
"No change to the architecture is required in this case; queries come in, and labeled queries are collected in the training module.",
"The query is simply not forwarded to the database.",
"Figure: System architecture.",
"Queries arrive for three different applications XX, YY, and ZZ and are processed by one or more (embedder, labeler) pair before being sent on to the database, centralized for offline labeling tasks, or both.This architecture is not designed for continuous learning, as the training is handled separately from real time query labeling.",
"Not all algorithms can support fully continuous learning, and an important design goal is to support simple machine learning algorithms as labelers.",
"Model training is therefore assumed to occur infrequently as a batch job."
],
[
"Learning Vector Representations",
"There are multiple choices for embedders; we describe two initial models we evaluate in this paper: Context prediction models: Mikolov et al.",
"[25], [24], proposed learning a vector representation for words by predicting the next word in a context, and then deriving a vector representation for larger semantic units (sentences, paragraphs, documents) by adding a vector representing the paragraph to each context as an additional “word.\"",
"The learned vector for this virtual context word is used as a representation for the entire paragraph.",
"This \"Doc2Vec\" method has been shown to capture semantic relationships that work well for, say, sentiment classification and clustering tasks [14], .",
"This approach can be applied directly for learning representations of SQL queries: We can use fixed-size context windows to learn a representation for each token in the query, and include an identifier to learn a representation of entire query.",
"LSTM AutoEncoders: The paragraph vector approach in the previous section is viable, but it requires a hyper-parameter for the context size.",
"There is no obvious way to determine a context size for queries, for two reasons: First, there may be semantic relationships between distant tokens in the query.",
"Second, the length of queries vary widely in ad hoc workloads [12], [10].",
"To avoid setting a context size, we can use Long Short-Term Memory (LSTM) networks [36], which are modified Recurrent Neural Networks (RNN) that can automatically learn how much context to remember and how much of it to forget, thereby removing the dependency on a fixed context size.",
"LSTMs have successfully been used in sentence classification, semantic similarity between sentences and sentiment analysis .",
"We use a standard LSTM encoder decoder network [37], [20] with architecture as illustrated in Figure REF .",
"An LSTM autoencoder is trained by sequentially feeding words from the query to the network one word at a time, and then attempting to reproduce the input.",
"The LSTM network not only learns the encoding for the samples, but also the relevant context window associated with the samples.",
"The final output of the encoder network gives us an encoding for the query.",
"Once this network has been trained, an embedded representation for a query can be computed by passing the query to the encoder network, completing a forward pass, and using the hidden state of the final encoder LSTM cell as the learned vector representation.",
"Figure: The LSTM Autoencoder network architecture learns to generate the input token in the decoding phase.",
"Once trained, the encoder can be used to output a vector representation for the text of a query.There are multiple prior approaches in the NLP literature that compare the efficacy of these models and their relative performance , [22], .",
"For this paper, we consider context-based models (i.e., doc2vec) and LSTM AutoEncoders."
],
[
"Applications",
"The applications supported by this system reduce to query labeling, and general workflow consists of two machine learning models: a representation learner (an embedder) and a classifier.",
"We split the task into two parts to allow the same representation to be used for multiple applications.",
"Workload summarization for index recommendation: The goal [3], [16] is to find a representative sample of the workload as input to further database administration, tuning, and testing tasks [3], [33].",
"In particular, workload summarization aids index recommendation, since the recommendation process is typically quadratic in the size of the workload [3].",
"While index recommendation systems are well-studied and ship with most production databases [3], [2], the quality of the representative sample determines the overall quality of the final recommendations.",
"In Section , we show that a simple sampling procedure using learned features delivers a significant runtime improvement over the built-in sampling procedure in the SQL Server database system.",
"Enforcing query routing policies: Query Routing in a distributed database involves identifying the cluster resources on which to execute the incoming query.",
"The policies that govern these routing decisions may involve customer SLAs, security considerations (e.g., certain applications must use a physically distinct cluster from other applications), auditing requirements (e.g., queries from certain accounts or those accessing certain tables must be logged for auditing purposes).",
"Even in modern cloud-hosted database products such as Snowflake [4] and BigQuery [23], these policies tend to be manually encoded, and management of these policies as they evolve, while maintaining multiple heterogeneous clusters used by thousands of customers, is increasingly perceived as untenable.",
"Under the hypothesis that queries that follow a particular policy tend to have similar features, Querc can help identify policy misconfiguration by detecting when a predicted routing decision differs from the assigned routing decision.",
"Error prediction: Particular syntax patterns in the workload may be associated with resource errors or bugs in the database system.",
"In a multi-tenant, multi-database, and high-volume scenario, identification of the syntactic patterns that tend to trigger errors, either manually or with scripts, becomes untenable: there may be hundreds of error codes, each with hundreds of subtle patterns that tend to trigger them, across hundreds of tenant schemas.",
"Using learned features, a classifier to predict errors from syntax is trivial to engineer.",
"This prediction allows the query to be routed to a different runtime environment that is instrumented, equipped with more more memory per node, or running a more stable version of the database engine.",
"We consider this application in a tech report companion to this paper [11].",
"Resource allocation: The structure of the query is not sufficient to accurately predict its runtime or memory footprint, but it can provide a hint that can be used for load balancing, scheduling, and as an input for optimization.",
"If we can coarsely categorize queries as memory-intensive, long-running, etc.",
"with some degree of accuracy, these labels can be used as a simple, database-agnostic way to speculatively allocate resources.",
"Training data is readily available from the query logs themselves.",
"We consider this application in a tech report companion to this paper [11].",
"Query recommendation: The query recommendation problem can be modeled as a prediction of the next query the user will submit to the database based on the recent history of queries [1].",
"This prediction is then shown to the user though an appropriate client application to assist in query authoring.",
"Our framework can generate features that can be used to train query recommendation models.",
"We consider this application in a tech report companion to this paper [11].",
"Security auditing: To the extent that users' individual workloads tend to follow predictable patterns, an anomalous query may be a sign that a user's account has been compromised.",
"By formulating a prediction problem that tries to guess the user that submitted the query from the syntax alone, we can identify anomalous queries for security audits.",
"In our framework, the labeler is a simple classifier $V \\rightarrow user$ .",
"Figure: Workload runtime using indexes recommended under various time budgets.",
"For most time budgets, the workload summaries improve runtimes, even when the embedders were trained on an unrelated workload (lstmSnowflake and doc2vecSnowflake)."
],
[
"Experiments",
"We consider two applications: Workload summarization for index selection, and labeling tasks for security audits and query routing."
],
[
"Workload Summaries for Index Selection",
"The workload summarization task (with respect to index recommendation) is to find a subset $Q_{sub}$ of a given query workload $Q$ , such that the set of indexes recommended based on $Q_{sub}$ is similar to the the set of indexes recommended for the overall workload $Q$ .",
"Previous solutions are primarily variants of the approach of Chaudhuri et al.",
"[3], which uses K-medioids to cluster the queries and selects a witness query from each cluster.",
"However, the authors emphasize that a custom distance function should be developed for specific workloads; our hypothesis is that generic representation learning approaches obviate the need for these custom distance functions.",
"In the Querc framework, this task is offline and does not require real-time labeling of queries.",
"Instead, we perform the task as an offline unsupervised learning task.",
"In our approach, we assign each query to a vector (using a suitably trained embedder), then simply use K-means to find $K$ query clusters and pick the nearest query to the centroid in each cluster as the representative subset.",
"To determine $K$ , we use an intentionally simple method (the “elbow method\" [15]) which runs the K-means algorithm in a loop with increasing $K$ till the rate of change of the sum of squared distances from centroids plateaus.",
"Although better methods exist, we highlight the effect of the learned vectors rather than the choice of $K$ .",
"Figure: Runtime for each query under no indexes and under indexes recommended with a three-minute time budget.",
"For a few specific queries, the presence of a recommended index results in significantly worse performance.Setup: Following the evaluation strategy of Chaudhuri et al.",
"[3], we first run the index selection tool on the entire workload $Q$ , create the recommended indexes, and measure the runtime $t_{orig}$ for the original workload.",
"We then run use the workload summarization algorithm to produce a reduced set of queries $\\mathcal {Q}_{sub}$ , re-run the index selection tool, create the recommended indexes, and again measure the runtime $t_{sub}$ of the entire original workload.",
"We use SQL Server 2016 and the Database Engine Tuning Advisor, which performs its own summarization on the input according to the documentation.",
"We use an $m4.large$ AWS EC2 instance as the server.",
"We use TPC-H with scale factor 1 as the workload for comparison with previous results and to interpret the recommended indexes, but we also show how the method performs when trained on a more complex Snowflake workload.",
"We pass the summarized workload to the tuning advisor, along with a time budget (a parameter supported by the tuning advisor).",
"Each experiment involves clearing caches, generating indexes, applying the indexes, and running the full workload.",
"We report the time running the workload; the time budget specifies the time limit under which the advisor must return a set of recommendations.",
"Results: Figure REF shows the results.",
"The x-axis is the time budget, and the y-axis is the runtime for the entire workload after building the recommended indexes.",
"For time budgets less than 3 minutes, the advisor does not produce any index recommendations for any method, and the runtime is constant at 1200 seconds.",
"As we relax the time budget, different sets of indexes are recommended, each associated with a separate runtime.",
"The full workload (blue line) varies dramatically with the time budget, and surprisingly it gets worse before it gets better.",
"For the summarized workloads, the workload is small enough that the runtimes are constant: Once three minutes have elapsed, the advisor has found the “optimal\" set of indexes, and allowing more time does not change the result.",
"We evaluate four trained embedders: two methods on two workloads.",
"The two methods are Doc2Vec and the LSTMAutoencoder, and the two workloads are TPC-H itself, and a separate workload of 500,000 queries from the Snowflake service.",
"When training the embedder on TPC-H (doc2VecTPCH and lstmTPCH), the advisor finds close-to-optimal indexes in about three minutes as opposed to the six minutes the advisor requires on the full workload.",
"Table: Query Labeling resultsSurprisingly, under tight time budgets, the index recommendations made by the native system can actually hurt performance relative to having no indexes at all.",
"The reason is that the optimizer chooses a bad plan for a few particular queries, but the effect is enough to hurt the overall runtime.",
"Figure REF shows the sequence of queries in the workload on the x-axis and the runtime for each query on the y-axis under no indexes and the low-quality indexes found at the three-minute time budget.",
"All instances of TPC-H query 18 (queries 640-680 in Figure REF ) take much longer than they would take when run without these indexes, because the optimizer finds a bad plan.",
"Transfer Learning: Figure REF also illustrates the capacity for transfer learning using Querc: When training the embedder on the snowflake dataset — a completely unrelated workload to TPC-H workload in the SQL Server dialect — the summarized workload still outperforms native SQL Server for most time budgets.",
"This transfer learning effect allows us to bootstrap new applications without waiting for a representative workload to accumulate, and to avoid having to repeatedly re-implement brittle parsers and feature extractors for each new dialect of SQL we encounter."
],
[
"Labeling for Security Audits",
"We consider the conditions under which the learned features from query syntax are sufficient to predict username and customer account, where each customer has many users.",
"When the predicted username differs from the actual username, we can potentially flag the query for an audit.",
"Predicting username can help flag queries for security audits, account and cluster labels can identify misrouted queries.",
"labels from query syntax using the two embedding methods described in Section over the Snowflake dataset.",
"Setup: We use embedders pre-trained on 500000 Snowflake queries.",
"The experiment itself is run on another dataset of 200000 Snowflake queries labeled with username, account_id and cluster_name for the cluster that ran the query.",
"Next we train classifiers (randomized decision trees) for username and customer account.",
"Results: Table REF shows the results for the labeling experiments.",
"The numbers denote the 10-fold cross validation score on the respective task.",
"We find that LSTM based embedders beats Doc2Vec on all tasks.",
"The LSTM method achieves near perfect accuracy when predicting the customer account, which is because it automatically incorporates signal from the schema, and different customers use primarily different schemas (there are instances of shared schemas, but that is the less common case).",
"The method was completely generic and knows nothing about schemas or queries.",
"For user prediction, the task is more difficult, and the overall accuracy is lower at 55%.",
"Upon further analysis we found that the user labeling task has $> 95\\%$ accuracies for a majority of accounts (Table REF ).",
"The accounts that had poor accuracies for user labeling had one distinctive property: multiple users running the exact same query, making the users nearly indistinguishable.",
"In the sample of workload that we were working with, there were two accounts that had a number of repetitive queries by different users (for instance, $69\\%$ percent of the 74000 queries in an account had more than one user label), and these two accounts also covered around $65\\%$ of the total queries, bringing down the overall accuracy of classifiers.",
"Table: Top accounts with user prediction accuracy."
],
[
"Future Work",
"Other methods: There are a variety of other methods for learning representations of text that we do not evaluate in this paper.",
"Our goal is not to identify the best possible representation learning approach but rather to show that these methods can compete with and outperform classical approaches that rely on task-specific heuristics and feature engineering (extracting JOIN clauses, counting the number of attributes, etc.",
"), and to organize the methods into a coherent system architecture.",
"Alternative methods can be roughly categorized into non-neural-network based methods and neural-network-based methods.",
"The non-neural-network-based methods, including non-negative matrix factorization (NMF), bag-of-words representations, and LDA [22] have been shown to be less effective than neural-network-based-methods in a variety of contexts [25], [19].",
"Apart from the methods considered in this paper, there are more recent neural-network-based methods using Convolutional Neural Networks (CNNs) adapted for text data.",
"However, Yin et al.",
"[34] showed that RNN based methods (e.g., LSTMs) perform well and are robust in a broad range of tasks when compared to CNNs.",
"However, we plan to extend the current work to include a rigorous comparison of the techniques not covered in this paper.",
"Publish pre-trained models: The results in Section demonstrate that the proposed framework in this paper has potential to use pre-trained models on generic workloads to aid analytics for previously unseen query.",
"In future work, we will build this framework as a service which is accessible by third parties.",
"Given the workloads that we have access to from Snowflake [4], such a service could be really beneficial for researchers who do not have access to massive query workloads."
],
[
"Conclusions",
"We presented the architecture for Querc, a database-agnostic workload analytics service that captures the structural and schema patterns present in the query workload automatically, largely eliminating the need for the specialized syntactic feature engineering that has motivated a number of papers in the literature.",
"The proposed architecture provides a new way of organizing a variety of database administration and user productivity tasks, and provides a mechanism by which to automatically adapt database operations to specific query workloads.",
"Our evaluation of this architecture showed that our general framework outperformed or was competitive with previous approaches that required specialized feature engineering, and also admitted simpler classification algorithms because the inputs are numeric vectors with well-behaved algebraic properties rather than result of arbitrary user-defined functions for which few properties can be assumed.",
"The use of transfer learning in Querc allows workload analytics to be SQL dialect independent and enables the capability to bootstrap new analytics tasks and avoid re-implementing brittle codes paths."
]
] | 1808.08355 |
[
[
"A Parametric Framework for Reversible Pi-Calculi"
],
[
"Abstract This paper presents a study of causality in a reversible, concurrent setting.",
"There exist various notions of causality in pi-calculus, which differ in the treatment of parallel extrusions of the same name.",
"In this paper we present a uniform framework for reversible pi-calculi that is parametric with respect to a data structure that stores information about an extrusion of a name.",
"Different data structures yield different approaches to the parallel extrusion problem.",
"We map three well-known causal semantics into our framework.",
"We show that the (parametric) reversibility induced by our framework is causally-consistent and prove a causal correspondence between an appropriate instance of the framework and Boreale and Sangiorgi's causal semantics."
],
[
"Introduction",
"Starting from the 1970s [5] reversible computing has attracted interest in different fields, from thermodynamical physics [3], to systems biology [14], [28], system debugging [30], [16] and quantum computing [17].",
"Of particular interest is its application to the study of programming abstractions for reliable systems: most fault-tolerant schemes exploiting system recovery techniques [2] rely on some form of undo.",
"Examples of how reversibility can be used to model transactions exist in CCS [13] and higher-order $\\pi $ -calculus [19].",
"A reversible system is able to execute both in the forward (normal) direction and in the backward one.",
"In a sequential setting, there is just one order of reversing a computation: one has just to undo the computation by starting from the last action.",
"In a concurrent system there is no clear notion of last action.",
"A good approximation of what is the last action in a concurrent system is given by causally-consistent reversibility, introduced by Danos and Krivine for reversible CCS [12].",
"Causally-consistent reversibility relates causality and reversibility of a concurrent system in the following way: an action can be reversed, and hence considered as a last one, provided all its consequences have been reversed.",
"In CCS [25], there exists just one notion of causality: so-called structural causality, which is induced by the prefixing `.'",
"operator and by synchronisations.",
"As a consequence, there is only one way of reversing a CCS trace, and from an abstract point of view there exists only one reversible CCS.",
"Evidence for this has been given in [22], where an equivalence is shown between the two methods for reversing CCS (namely RCCS [12] and CCSK [27]).",
"When moving to more expressive calculi with name creation and value passing like the $\\pi $ -calculus, matters are more complex.",
"As in CCS, structural causality in the $\\pi $ -calculus is determined by the nesting of the prefixes; for example, in process $\\overline{{b}}{a}.\\overline{{c}}{e}$ the output on channel $c$ structurally depends on the output on $b$ .",
"Extruding (or opening) a name generates an object dependency; for example, in process $\\nu {a} \\;(\\overline{{b}}{a} \\;|\\;a(z)) $ the input action on $a$ depends on the output on $b$ .",
"In the case of parallel extrusions of the same name, for example $\\nu {a} \\;(\\overline{{b}}{a} \\;|\\;\\overline{{c}}{a} \\;|\\;a(z))$ , there exist different interpretations of which extrusion will cause the action $a(z)$ .",
"In what follows, we consider three approaches.",
"The classical and the most used approach to causality in the $\\pi $ -calculus is the one where the order of extrusions matters and the first one of them is the cause of the action $a(z)$ .",
"Some of the causal semantics representing this idea are [15], [6], [8] and all of them are defined for standard (forward-only) $\\pi $ -calculus.",
"In [15] the authors claim that, after abstracting away from the technique used to record causal dependences, the final order between the actions in their semantics coincides with the ones introduced in [6], [8].",
"Hence we group these semantics together as a single approach to causality.",
"Secondly, in [9], action $a(z)$ in the example above depends on one of the extruders, but there is no need to keep track of which one exactly.",
"This causal semantics is defined for the forward-only $\\pi $ -calculus.",
"Finally, the first compositional causal semantics for the reversible $\\pi $ -calculus is introduced in [10].",
"In the above example, parallel extrusions are concurrent and the action $a(z)$ will record dependence on one of them (exactly which one is decided by the context).",
"This causal semantics enjoys certain correctness properties which are not satisfied by other semantics.",
"Here we present a framework for reversible $\\pi $ -calculus that is parametric with respect to the data structure that stores information about an extrusion of a name.",
"Different data structures will lead to different approaches to the parallel extrusion problem, including the three described above.",
"Our framework allows us to add reversibility to semantics where it was not previously defined.",
"By varying the parameters, different orderings of the causally-consistent backward steps are allowed.",
"Our intention is to develop a causal behavioural theory for the framework, in order to better understand different interpretations of reversibility in the $\\pi $ -calculus, and to use this understanding for causal analysis of concurrent programs.",
"A preliminary discussion of the framework appeared in [23], where some initial ideas were given.",
"Moreover in [23] it was argued that it was necessary to modify the semantics of [6] in order to add information about silent actions.",
"In this work we fully develop the idea behind the framework and leave the semantics of [6] unchanged, apart from using a late semantics, rather than early as originally given.",
"Contributions.",
"We present a framework for reversible $\\pi $ -calculus which is parametric in the bookkeeping data structure used to keep track of object dependency.",
"As reversing technique, we will extend the one introduced by CCSK [27], which is limited to calculi defined with GSOS [1] inference rules (e.g., CCS, CSP), to work with more expressive calculi featuring name passing and binders.",
"This choice allows us to have a compositional semantics which does not rely on any congruence rule (in particular the splitting rule used by [10]).",
"Depending on the bookkeeping data structure used to instantiate the framework, we can obtain different causal semantics (i.e., [10], [6], [9]).",
"We then show that our framework enjoys the standard properties for a reversible calculus, namely the loop lemma and causal consistency, regardless of the notion of causality which is used.",
"We prove causal correspondence between the causal semantics introduced in [6] and the matching instance of our framework.",
"The rest of the paper is as follows: syntax and operational semantics of the framework are given in Section .",
"In Section , we show how by using different data structures we can encompass different causal semantics.",
"The main results are given in Section , and Section concludes the paper.",
"Proofs are omitted for space reasons; they can be found in the extended version [24]."
],
[
"The Framework",
"We present the syntax and operational semantics of our parametric framework, after an informal introduction."
],
[
"Informal presentation",
"In [27] a general technique to reverse any CCS-like calculus is given.",
"The key ideas are to use communication keys to identify events, and to make static all the operators of the calculus, since dynamic operators such as choice and prefix are forgetful operators.",
"For example, if we take a CCS process $a.P \\;|\\;\\overline{a}.Q$ a possible computation is: $\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad a.P \\;|\\;\\overline{a}.Q \\xrightarrow{} a[i].P \\;|\\;\\overline{a}[i].Q$ As one can see, prefixes are not destroyed but decorated with a communication key.",
"The obtained process acts like $P \\;|\\;Q$ , since decorated prefixes are just used for backward steps.",
"We bring this idea to the $\\pi $ -calculus.",
"For example by lifting this process into $\\pi $ -calculus we have something like $\\qquad \\qquad \\qquad \\qquad \\quad \\qquad a(x).P \\;|\\;\\overline{{a}}{b}.Q \\xrightarrow{} a(x)[i].P\\lbrace {b^i}/{x}\\rbrace \\;|\\;\\overline{a}b[i].Q$ In the substitution $\\lbrace {b^i}/{x}\\rbrace $ , name $b$ is decorated with the key $i$ to record that it was substituted for variable $x$ in the synchronisation identified by the communication key $i$ .",
"The key $i$ is also recorded in the memories.",
"By choosing to adapt the ideas of [27] to work with the $\\pi $ -calculus, we avoid using the splitting rule of $R\\pi $ [10].",
"In $R\\pi $ each process is monitored by a memory, $m \\triangleright P$ , which is in charge of recording all past events of the process.",
"In this way, the past of the process is not recorded directly in the process.",
"One drawback of this approach is that one needs to resort to a splitting rule of the form $\\qquad \\qquad \\qquad \\qquad \\qquad \\quad \\quad m \\triangleright (P\\;|\\;Q) \\equiv \\langle \\uparrow \\rangle \\cdot m \\triangleright P \\;|\\;\\langle \\uparrow \\rangle \\cdot m \\triangleright Q$ to let both $P$ and $Q$ execute.",
"This rule is not associative and moreover, as shown in [20], introduces some undesired non-determinism, since equivalent processes performing the same action may become non-equivalent processes.",
"The framework has to remember extrusions, and in particular who was the extruder of a certain name, and what is the contextual cause for an action.",
"For example in $\\qquad \\qquad \\qquad \\nu a \\, (\\overline{{b}}{a} \\;|\\;a(x).P) \\xrightarrow{} \\nu a_{\\lbrace i\\rbrace }(\\overline{{b}}{a}[i] \\;|\\;a(x).P)\\xrightarrow{}\\nu a_{\\lbrace i\\rbrace }(\\overline{{b}}{a}[i] \\;|\\;a(x)[j,i].P )$ we have that after the extrusion, the restriction $\\nu a$ does not disappear as in standard $\\pi $ -calculus, but remains where it was, becoming the memory $\\nu a_{\\lbrace i\\rbrace }$ (introduced in [10]).",
"This memory records the fact that name $a$ was extruded because of transition $i$ .",
"Moreover, since it is no longer a restriction but just a decoration, the following transition using name $a$ can take place.",
"Transition $j$ uses $i$ as its contextual cause, indicating that the input action can happen on $a$ because it was extruded by $i$ , and this is recorded in the process $a(x)[j,i].P$ ."
],
[
"Syntax",
"We assume the existence of the following denumerable infinite mutually disjoint sets: the set $\\mathcal {N}$ of names, the set $\\mathcal {K}$ of keys, and the set $\\mathcal {V}$ of variables.",
"Moreover, we let $\\mathcal {K}_{*} = \\mathcal {K}\\cup \\lbrace *\\rbrace $ where $*$ is a special key.",
"We let $a,b,c$ range over $\\mathcal {N}$ ; $x,y$ range over $\\mathcal {V}$ and $i,j,k$ range over $\\mathcal {K}$ .",
"The syntax of the framework is depicted in Figure REF .",
"Processes, given by the $P, Q$ productions, are the standard processes of the $\\pi $ -calculus [29]: $\\mathbf {0}$ represents the idle process; $\\overline{{b}}{c}.P$ is the output-prefixed process indicating the act of sending name $c$ over channel $b$ ; $b(x).P$ is the input-prefixed process indicating the act of receiving a value (which will be bound to the variable $x$ ) on channel $b$ .",
"Process $P \\;|\\;Q$ represents the parallel composition of two processes, while $\\nu a (P)$ represents the fact that name $a$ is restricted in $P$ .",
"Figure: Syntax.Reversibility is defined on the top of the $\\pi $ -calculus.",
"Unlike in the standard $\\pi $ -calculus, executed actions are not discarded.",
"Each of them, followed by the memory, becomes a part of the process that we shall call the history.",
"Reversible processes are given by $X,Y$ productions.",
"A reversible process ${P}$ is a standard $\\pi $ -calculus process $P$ where channels are decorated with instantiators.",
"As we shall see later on, instantiators are used to keep track of substitutions.",
"In a prefix of the form $\\overline{{b}}{a}$ or $b(x)$ we say that name $b$ is used in subject position, while name $a$ and variable $x$ are in object position.",
"We shall use operators $sub(\\cdot )$ and $obj(\\cdot )$ to get respectively the subject and the object of a prefix.",
"The prefix $\\overline{{b}}^j{a^{j_1}}[{i,K}].X$ represents a past output recording the fact that in the past the process $X$ performed an output identified by key $i$ and that its contextual cause set was $K\\subseteq \\mathcal {K}_{*}$ .",
"Prefix $b^{j}(x)[{i,K}].X$ represents a past input recording the fact that the input was identified by key $i$ and its contextual cause set was $K$ .",
"If it is not relevant whether the prefix in the process is an input or an output, we shall denote it with $\\alpha $ ($\\alpha =\\overline{{b}}^j{a^{j_1}}$ or $\\alpha =b^j (x)$ ).",
"Following [10] the restriction operator $\\nu a_{\\Delta }$ is decorated with the memory $\\Delta $ which keeps track of the extruders of a name $a$ .",
"As we shall see later on, we shall abstract away from the form of $\\Delta $ , as different data structures lead to different notions of causality.",
"When $\\mathtt {empty}(\\Delta )=true$ , the data structure is initialised and $\\nu a_{\\Delta }$ will act as the usual restriction operator $\\nu a$ of the $\\pi $ -calculus.",
"The set of reversible processes is denoted with $\\mathcal {X}$ .",
"To simplify manipulation with reversible processes, we shall define history and general context.",
"History context represents the reversible process $X$ made of executed prefixes.",
"For example, we can express the process $X=\\overline{{b}}^{*}{a^*}[{i,K}].\\overline{{c}}^{*}{a^*}[{i^{\\prime },K^{\\prime }}].",
"{P}$ as $X=\\mathtt {H}[{P}]$ with $\\mathtt {H}[\\bullet ] = \\overline{{b}}^{*}{a^*}[{i,K}].\\overline{{c}}^{*}{a^*}[{i^{\\prime },K^{\\prime }}].\\bullet $ .",
"General context is defined on the top of the history context by adding parallel and restriction operators on it.",
"For example, the process $Z\\;|\\;Y\\;|\\;X$ can be written as $C[X]$ if the only relevant element is $X$ .",
"Formally: Definition 1 (History and General context) History contexts $\\mathtt {H}$ and general contexts $C$ are reversible processes with a hole $\\bullet $ , defined by the following grammar: $\\qquad \\qquad \\qquad \\qquad \\mathtt {H}::=\\bullet \\;|\\;\\alpha [{i,K}].\\bullet \\qquad C::=\\mathtt {H}[\\bullet ]\\;\\;|\\;\\: X\\;|\\;\\bullet \\;\\;|\\;\\nu a_{\\Delta } (\\bullet )$ Free names and free variables.",
"Notions of free names and free variables in our framework are standard.",
"It suffices to note that constructs with binders are of the forms: $\\nu {a}_\\Delta (X)$ when $\\mathtt {empty}(\\Delta )$ holds, which binds the name $a$ with scope $X$ ; and $ b(x).P $ , which binds the variable $x$ with scope $P$ .",
"We denote with $\\mathtt {fn}(P)$ and $\\mathtt {fn}(X)$ the set of free names of $P$ and of $X$ respectively.",
"Remark 1 Annotation $b^*$ to a name $b$ , used either in the subject or in the object position, indicates that name $b$ has no instantiators.",
"Since the framework will be parametric in the data structure $\\Delta $ , we specify it as an interface (in the style of a Java interface) by defining the operations that it has to offer.",
"Definition 2 $\\Delta $ is a data structure with the following defined operations: $(i)$ $\\mathtt {init}: \\Delta \\rightarrow \\Delta \\, \\text{ initialises the data structure}$ $(ii)$ $\\mathtt {empty}: \\Delta \\rightarrow \\mathtt {bool}\\, \\text{ predicate telling whether $\\Delta $ is empty}$ $(iii)$ $+ : \\Delta \\times \\mathcal {K}\\rightarrow \\Delta \\, \\text{ operation adding a key to $\\Delta $} $ $(iv)$ $\\#i : \\Delta \\times \\mathcal {K}\\rightarrow \\Delta \\, \\text{ operation removing a key from$\\Delta $}$ $(v)$ $\\in : \\Delta \\times \\mathcal {K}\\rightarrow \\mathtt {bool}\\, \\text{ predicate telling whether a key belongs to $\\Delta $}$ We now define three instances of $\\Delta $ : sets, sets indexed with an element and sets indexed with a set.",
"As we shall see, these three instances will give rise to three different notions of object causality."
],
[
"$\\Gamma $ is a set containing keys (i.e.",
"$\\Gamma \\subseteq \\mathcal {K}$ ).",
"The intuition of $\\nu a_\\Gamma $ is that any of the elements contained in $\\Gamma $ can be a contextual cause for $a$ (i.e., the reason why $a$ is known to the context).",
"Definition 3 (Operations on a set) The operations on a set $\\Gamma $ are defined as: $(i)$ $\\mathtt {init}(\\Gamma )=\\emptyset $ $(ii)$ $\\mathtt {empty}(\\Gamma )=true$ , when $\\Gamma =\\emptyset $ $(iii)$ $+$ is the classical addition of elements to a set $(iv)$ $\\#i$ is defined as the identity, that is $\\Delta _{_{\\#i}}=\\Gamma _{_{\\#i}}=\\Gamma $ .",
"$(v)$ $i\\in \\Gamma $ the key $i$ belongs to the set $\\Gamma $ $\\Gamma _w$ is an indexed set containing keys and $w$ is the key of the action which extruded a name $a$ .",
"In this case the contextual cause for name $a$ can be just $w$ .",
"If there is no cause, then we shall set $w=*$ .",
"Definition 4 (Operations on an indexed set) The operations on an indexed set $\\Gamma _w$ are defined as: $(i)$ $\\mathtt {init}(\\Gamma _w)=\\emptyset _{*}$ $(ii)$ $\\mathtt {empty}(\\Gamma _w)=true$ , when $\\Gamma =\\emptyset \\;\\wedge \\; w=*$ $(iii)$ operation $+$ is defined as: $\\Gamma _w + i={\\left\\lbrace \\begin{array}{ll}(\\Gamma \\cup \\lbrace i\\rbrace )_{i}, \\text{ when } w=*\\\\(\\Gamma \\cup \\lbrace i\\rbrace )_{w}, \\text{ when } w\\ne *\\end{array}\\right.",
"}$ $(iv)$ operation $\\#i$ is defined inductively as: $& (X\\;|\\;Y)_{\\#i}= X_{\\#i}\\;|\\;Y_{\\#i}&&(\\mathtt {H}[X])_{\\#i}=\\mathtt {H}[X_{\\#i}] \\qquad \\qquad ({P})_{\\#i}={P}\\\\& (\\nu a_{\\Gamma _{i}} X)_{\\#i}= \\nu a_{\\Gamma _{*}} X_{\\#i}&&(\\nu a_{\\Gamma _{w}} X)_{\\#i}= \\nu a_{\\Gamma _{w}} X_{\\#i}$ $(v)$ $i\\in \\Gamma _w$ the key $i$ belongs to the set $\\Gamma $ , regardless of $w$ (e.g.",
"$i\\in \\lbrace i\\rbrace _{*}$ ) $\\Gamma _\\Omega $ is a set containing keys indexed with a set $\\Omega \\in \\mathcal {K}_{*}$ .",
"Extruders of name $a$ which are not part of the communication, will be saved in the set $\\Omega $ .",
"In this case the contextual cause for name $a$ is a set $\\Omega $ .",
"If there is no cause, then we shall set $\\Omega =\\lbrace *\\rbrace $ .",
"Definition 5 (Operations on a set indexed with a set) The operations on a set indexed with a set $\\Gamma _{\\Omega }$ are defined as: $(i)$ $\\mathtt {init}(\\Gamma _{\\Omega })=\\emptyset _{\\lbrace *\\rbrace }$ $(ii)$ $\\mathtt {empty}(\\Gamma _{\\Omega })=true$ , when $\\Gamma =\\emptyset \\;\\wedge \\; \\Omega =\\lbrace *\\rbrace $ $(iii)$ operation $+$ is defined as: $(\\Gamma _{\\Omega }) + i=(\\Gamma \\cup \\lbrace i\\rbrace )_{(\\Omega \\cup \\lbrace i\\rbrace )}$ $(iv)$ operation $_{\\#i}$ is defined inductively as: $\\qquad (X\\;|\\;Y)_{\\#i}= X_{\\#i}\\;|\\;Y_{\\#i}\\qquad (\\nu a_{\\Gamma _{\\Omega }} X)_{\\#i}= \\nu a_{\\Gamma _{\\Omega \\setminus \\lbrace i\\rbrace }} X_{\\#i}\\qquad (\\mathtt {H}(X))_{\\#i}=\\mathtt {H}[X_{\\#i}] \\qquad ({P})_{\\#i}={P}$ $(v)$ $i\\in \\Gamma _{\\Omega }$ the key $i$ belongs to the set $\\Gamma $ , regardless $\\Omega $ (e.g.",
"$i\\in \\lbrace i\\rbrace _{\\lbrace *\\rbrace }$ )"
],
[
"Operational Semantics",
"The grammar of the labels generated by the framework is: $\\mu ::=({i},{K},{j}):{\\pi }\\qquad \\qquad \\pi ::=\\overline{{b}}{c}\\;|\\;b(x)\\;|\\;\\overline{{b}}\\langle \\nu {c}_{\\Delta }\\rangle \\;|\\;\\tau $ where $i$ is the key, and $K\\subseteq \\mathcal {K}_{*}$ , $j\\in \\mathcal {K}_{*}$ are the set of contextual causes and an instantiator of $i$ , respectively.",
"If there is no action which caused and/or instantiated $i$ , we denote this with $K=\\lbrace *\\rbrace $ , $j=*$ , respectively.",
"The set $\\mathcal {L}$ of all possible labels generated by the framework is defined as $\\mathcal {L}= \\mathcal {K}\\times \\mathcal {K}_{*} \\times \\mathcal {K}_{*} \\times \\mathcal {A}$ , where $\\mathcal {A}$ is a set of actions ranged over by $\\pi $ .",
"We extend $sub(\\cdot )$ and $obj(\\cdot )$ to apply also to labels.",
"The operational semantics of the reversible framework is given in terms of a labelled transition system (LTS) $(\\mathcal {X},\\mathcal {L},\\xrightarrow{})$ , where $\\mathcal {X}$ is the set of reversible processes; $\\xrightarrow{}= \\xrightarrow{}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow \\cup \\mathrel {\\begin{tikzpicture}[baseline= {( (current bounding box.south) + (0,-0.5ex) )}]\\node [inner sep=.5ex] (1) {\\scriptstyle \\quad };[draw,<-,decorate,decoration={zigzag,amplitude=1.2pt,segment length=1.5mm,pre=lineto,pre length=6pt}](1.south east) -- (1.south west);\\end{tikzpicture}}$ where $\\xrightarrow{}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow $ is the least transition relation induced by the rules in Figures REF and REF ; and $\\mathrel {\\begin{tikzpicture}[baseline= {( (current bounding box.south) + (0,-0.5ex) )}]\\node [inner sep=.5ex] (2) {\\scriptstyle \\quad };[draw,<-,decorate,decoration={zigzag,amplitude=1.2pt,segment length=1.5mm,pre=lineto,pre length=6pt}](2.south east) -- (2.south west);\\end{tikzpicture}}$ is the least transition relation induced by the rules in Figure REF .",
"Definition 6 (Process keys) The set of communication keys of a process $X$ , written $\\mathtt {key}(X)$ , is inductively defined as follows: $&\\mathtt {key}(X\\;|\\;Y) = \\mathtt {key}(X) \\cup \\mathtt {key}(Y)&& \\mathtt {key}(\\alpha [{i,K}].X) = \\lbrace i\\rbrace \\cup \\mathtt {key}(X) \\\\& \\mathtt {key}(\\nu a_{\\Delta } (X)) = \\mathtt {key}(X) && \\mathtt {key}({P}) = \\emptyset &$ Definition 7 A key $i$ is fresh in a process $X$ , written $\\mathtt {fresh}(i,X)$ if $i \\notin \\mathtt {key}(X)$ .",
"The forward rules of a framework are divided into two groups, depending on whether they are parametric with respect to $\\Delta $ or they are common to all the instances of the framework.",
"Figure: Rules that are common to all instances of the framework.Common rules are given in Figure REF .",
"Rules $\\textsc {Out1}$ and $\\textsc {In1}$ generate a fresh new key $i$ which is bound to the action.",
"Rules $\\textsc {Out2}$ and $\\textsc {In2}$ inductively allow a prefixed process $\\mathtt {H}[X]$ to execute if X can execute.",
"Condition $i\\notin Y$ in rule $\\textsc {Par}$ ensures that action keys are unique.",
"Rule $\\textsc {Res}$ is defined in the usual way.",
"Two processes can synchronise through the rule $\\textsc {Com}$ if the additional condition is satisfied ($K=_{*}j$ means $*\\in K$ or $j=*$ or $K=j$ ).",
"After the communication, necessary substitution is applied to the rest of the input process.",
"In the process $Y^{\\prime }\\lbrace {a^{i}}/{x}\\rbrace $ every occurrence of variable $x\\in \\mathtt {fn}(Y^{\\prime })$ is substituted with the name $a^{i}$ , that is, the name $a$ decorated with the key $i$ of the action which was executed.",
"In the further actions of a process $Y^{\\prime }\\lbrace {a^{i}}/{x}\\rbrace $ , the key $i$ will be called the instantiator.",
"The instantiators are used just to keep track of the substitution, not to define a name.",
"For example, the two processes $\\overline{{b}}^j{a^{*}}.",
"{P}$ and $b^{j^{\\prime }}(x).",
"{P}^{\\prime }$ can communicate, even if the instantiators of the name $b$ are not the same.",
"Let us note that we use a late semantics, since substitution happens in the rule $\\textsc {Com}$ .",
"In order to understand how the basic rules work let us consider the following example.",
"Example 1 Let $X=\\overline{{b}}^{*}{a^{*}}.\\mathbf {0}\\;|\\;b^{*}(x).\\overline{{x}}{c^{*}}$ .",
"There are two possibilities for the process $X$ : process $X$ can preform an output and an input action on the channel $b$ while synchronising with environment: $\\overline{{b}}^{*}{a^{*}}.\\mathbf {0}\\;|\\;b^{*}(x).\\overline{{x}}{c^{*}}\\xrightarrow{}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow \\overline{{b}}^{*}{a^{*}}[{i,*}].\\mathbf {0}\\;|\\;b^{*}(x).\\overline{{x}}{c^{*}}\\xrightarrow{}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow \\overline{{b}}^{*}{a^{*}}[{i,*}].\\mathbf {0}\\;|\\;b^{*}(x)[{i^{\\prime },*}].\\overline{{x}}{c^{*}}=Y_1$ As we can notice, the output action $\\overline{{b}}{a}$ is identified by key $i$ , while the input action is identified by key $i^{\\prime }$ .",
"The synchronisation can happen inside of the process $X$ : $\\overline{{b}}^{*}{a^{*}}.\\mathbf {0}\\;|\\;b^{*}(x).\\overline{{x}}{c^{*}}\\xrightarrow{}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow \\overline{{b}}^{*}{a^{*}}[{i,*}].\\mathbf {0}\\;|\\;b^{*}(x)[{i,*}].\\overline{{a}}^{i}{c^{*}}=Y_2$ We can notice that $\\tau $ action is identified with key $i$ and during the synchronisation variable $x$ is substituted with a received name $a$ decorated with the key $i$ of the executed action.",
"In this way we keep track of the substitution of a name.",
"We now define the operation $X_{[{K^{\\prime }}/{K}]\\&i}$ , which updates the contextual cause $K$ of an action identified by $i$ with the new cause $K^{\\prime }$ .",
"Contextual cause update will be used in the the parametric rules of Figure REF ($\\textsc {Open}$ and $\\textsc {Cause Ref}$ ).",
"Formally: Definition 8 (Contextual Cause Update) The contextual cause update of a process, written $X_{[{K^{\\prime }}/{K}]\\&i}$ is defined as follows: $& (X\\;|\\;Y)_{[{K^{\\prime }}/{K}]\\&i}= X_{[{K^{\\prime }}/{K}]\\&i} \\;|\\;Y_{[{K^{\\prime }}/{K}]\\&i}&& \\mathtt {H}[\\alpha [i,K].X]_{[{K^{\\prime }}/{K}]\\&i}=\\mathtt {H}[\\alpha [i,K^{\\prime }].",
"X] &\\\\& (\\nu a_\\Delta (X))_{[{K^{\\prime }}/{K}]\\&i}= \\nu a_\\Delta (X)_{[{K^{\\prime }}/{K}]\\&i}&& \\mathtt {H}[\\alpha [j,K].X]_{[{K^{\\prime }}/{K}]\\&i}=\\mathtt {H}[\\alpha [j,K].",
"X] &$ Figure: Parametric rulesParametric rules are given in Figure REF .",
"Depending on the underlying causal semantics the way a contextual cause is chosen differs.",
"This is why we need to define two predicates: $\\mathtt {Cause}(\\cdot )$ and $\\mathtt {Update}(\\cdot )$ .",
"When instantiating $\\Delta $ with a specific data structure, different implementations of such predicates are needed.",
"We shall define them precisely when discussing various causal semantics in the Section .",
"Every time a label produced by a process $X$ passes the restriction $\\nu a_{\\Delta }$ it needs to check if it is necessary to modify the contextual cause.",
"Depending on whether name $a$ is in the subject or in the object position in the label of an action, rules $\\textsc {Cause Ref}$ or $\\textsc {Open}$ can be used, respectively.",
"Rule $\\textsc {Cause Ref}$ is used when the subject of a label is an already extruded name and a predicate $\\mathtt {Cause}(\\Delta ,K,K^{\\prime })$ tells whether contextual cause $K$ has to be substituted with $K^{\\prime }$ .",
"Rule $\\textsc {Open}$ deals with the scope extrusion of a restricted name.",
"If the restricted name $a$ is used as object of a label with key $i$ we have to record that $i$ is one of the potential extruders of $a$ .",
"Naturally, if $\\mathtt {empty}(\\Delta )=true$ then the first extruder initialises the data structure.",
"Also in this case it might happen that we have to update the contextual cause of the label $i$ .",
"This is why predicate $\\mathtt {Update}(\\Delta ,K,K^{\\prime })$ is used.",
"Two processes can synchronise through the rule $\\textsc {Close}$ satisfying the additional condition.",
"In some semantics, silent actions do not bring the causal information on what is a reason to introduce the operator $_{\\#i}$ , where every time when an extruded name is closed over the context, the key of the closing action is deleted from indexes of $\\Delta _h$ in all restrictions $\\nu a_{\\Delta _h}\\in X^{\\prime }$ .",
"Figure: Backward rules.Backward rules are symmetric to the forward ones; they are presented in Figure REF .",
"The predicates are not necessary for the backward transitions as they are invariant in the history of processes but we keep them to simplify the proofs.",
"In order to better understand the backward rules, we shall consider the following example.",
"Example 2 Let us consider the following processes from Example REF : $Y_1=\\overline{{b}}^{*}{a^{*}}[{i,*}].\\mathbf {0}\\;|\\;b^{*}(x)[{i^{\\prime },*}].\\overline{{x}}{c^{*}}$ ; Process $Y_1$ can perform backward actions on the channel $b$ (an input action identified with key $i^{\\prime }$ and an output action identified with key $i$ ) in any order.",
"For example, let us reverse first the input and then the output action: $Y_1=\\overline{{b}}^{*}{a^{*}}[{i,*}].\\mathbf {0}\\;|\\;b^{*}(x)[{i^{\\prime },*}].\\overline{{x}}{c^{*}}\\mathrel {\\begin{tikzpicture}[baseline= {( (current bounding box.south) + (0,-0.5ex) )}]\\node [inner sep=.5ex] (23) {\\scriptstyle ({i^{\\prime }},{*},{*}):{b(x)}};[draw,<-,decorate,decoration={zigzag,amplitude=1.2pt,segment length=1.5mm,pre=lineto,pre length=6pt}](23.south east) -- (23.south west);\\end{tikzpicture}}\\overline{{b}}^{*}{a^{*}}[{i,*}].\\mathbf {0}\\;|\\;b^{*}(x).\\overline{{x}}{c^{*}}\\mathrel {\\begin{tikzpicture}[baseline= {( (current bounding box.south) + (0,-0.5ex) )}]\\node [inner sep=.5ex] (24) {\\scriptstyle ({i},{*},{*}):{\\overline{{b}}{a}}};[draw,<-,decorate,decoration={zigzag,amplitude=1.2pt,segment length=1.5mm,pre=lineto,pre length=6pt}](24.south east) -- (24.south west);\\end{tikzpicture}}\\overline{{b}}^{*}{a^{*}}.\\mathbf {0}\\;|\\;b^{*}(x).\\overline{{x}}{c^{*}}=X$ We notice that all the necessary elements to reverse the action $b(x)$ are saved in the history part of the process $Y_1$ .",
"$Y_2=\\overline{{b}}^{*}{a^{*}}[{i,*}].\\mathbf {0}\\;|\\;b^{*}(x)[{i,*}].\\overline{{a}}^{i}{c^{*}}$ ; Process $Y_2$ can reverse the communication which happened on the channel $b$ , between its subprocesses.",
"Due to the side condition of the rule $\\textsc {Par$$}$ , it is impossible to reverse an input or an output action separately: $\\overline{{b}}^{*}{a^{*}}[{i,*}].\\mathbf {0}\\;|\\;b^{*}(x)[{i,*}].\\overline{{a}}^{i}{c^{*}}\\lnot \\mathrel {\\begin{tikzpicture}[baseline= {( (current bounding box.south) + (0,-0.5ex) )}]\\node [inner sep=.5ex] (25) {\\scriptstyle ({i},{*},{*}):{\\overline{{b}}{a}}};[draw,<-,decorate,decoration={zigzag,amplitude=1.2pt,segment length=1.5mm,pre=lineto,pre length=6pt}](25.south east) -- (25.south west);\\end{tikzpicture}}\\overline{{b}}^{*}{a^{*}}.\\mathbf {0}\\;|\\;b^{*}(x)[{i,*}].\\overline{{a}}^{i}{c^{*}}$ The backward action above cannot be executed as the key $i$ belongs to the process in parallel ($i\\in \\mathtt {key}(b^{*}(x)[{i,*}].\\overline{{a}}^{i}{c^{*}})$ ).",
"The only possible backward step is: $\\overline{{b}}^{*}{a^{*}}[{i,*}].\\mathbf {0}\\;|\\;b^{*}(x)[{i,*}].\\overline{{a}}^{i}{c^{*}} \\mathrel {\\begin{tikzpicture}[baseline= {( (current bounding box.south) + (0,-0.5ex) )}]\\node [inner sep=.5ex] (26) {\\scriptstyle ({i},{*},{*}):{\\tau }};[draw,<-,decorate,decoration={zigzag,amplitude=1.2pt,segment length=1.5mm,pre=lineto,pre length=6pt}](26.south east) -- (26.south west);\\end{tikzpicture}}\\overline{{b}}^{*}{a^{*}}.\\mathbf {0}\\;|\\;b^{*}(x).\\overline{{x}}{c^{*}}=X$ Remark 2 The choice operator $(+)$ , can be easily added to the framework by following the approach of [27] and by making the operator static.",
"We now review three notions of causal semantics for $\\pi $ -calculus and show how to map them into our framework by giving the definitions for the side conditions in the rules in Figure REF ."
],
[
"Cristescu et al [10] introduce a compositional semantics for the reversible $\\pi $ -calculus.",
"Information about the past actions is kept in a memory added to every process.",
"A term of the form $m\\triangleright P$ represents a reversible process, where memory $m$ is a stack of events and $P$ is the process itself.",
"A memory contains two types of events, one which keeps track of the past action, $\\langle i,k,\\pi \\rangle $ , where elements of a triple are the key, the contextual cause and the executed action, respectively; and one which keeps track of the position of the process in the parallel composition, $\\langle \\uparrow \\rangle $ .",
"Before executing in parallel, a process splits by duplicating its memory and adding event $\\langle \\uparrow \\rangle $ on the top of each copy.",
"This is achievable with specially defined structural congruence rules.",
"The use in [10] of indexed restriction $\\nu a_{\\Gamma }$ was the inspiration for our parametric indexed restriction $\\nu a_{\\Delta }$ .",
"A key point of the semantics of [10] is that it enjoys certain correctness properties: one of which is that two visible transitions are causally related iff for all contexts the corresponding silent transitions are.",
"Since an action can be caused only through the subject of a label we have that contextual cause $K$ will be a singleton.",
"We shall consider one relation between the prefixes into the history.",
"In this way, while changing the cause with the rule $\\textsc {Cause ref}$ , the condition $\\mathtt {Cause}(\\cdot )$ needs to keep track of the instantiation of the cause.",
"Definition 9 (Instantiation relation) Two keys $i_1$ and $i_2$ such that $i_1 , i_2\\in \\mathtt {key}(X)$ and $X=C[b^*(x)[i_1,K_1].Y]$ with $Y=C^{\\prime }[\\alpha ^{j_2}[i_2,K_2].Z]$ , are in instantiation relation, $i_1\\rightsquigarrow _{X}i_2 $ , if $j_2=i_1$ .",
"If $i_1\\rightsquigarrow _{X}i_2 $ holds, we will write $K_1\\rightsquigarrow _{X}K_2 $ .",
"To obtain $R\\pi $ causality in our framework, we need to instantiate the rules of Figure REF with the following predicates.",
"Definition 10 ($R\\pi $ causality) If data structure $\\Delta $ is instantiated with a set $\\Gamma $ , the predicates from Figure REF are defined as: 1.",
"$\\mathtt {Cause}(\\Delta ,K,K^{\\prime })=\\mathtt {Cause}(\\Gamma ,K,K^{\\prime })$ stands for $K=K^{\\prime }$ or $\\exists K^{\\prime }\\in \\Gamma \\; K\\rightsquigarrow _{X}K^{\\prime }$ ; 2.",
"$\\mathtt {Update}(\\Delta ,K,K^{\\prime })=\\mathtt {Update}(\\Gamma ,K,K^{\\prime })$ stands for $K^{\\prime }=K$ .",
"The predicates defined above coincide with the conditions of the semantics introduced in [10].",
"In the following example we shall give the intuition of the $R\\pi $ causality using our framework.",
"Example 3 Let us consider the process $X=\\nu a_\\emptyset (\\overline{{b}}^{*}{a^*}\\;|\\;\\overline{{c}}^{*}{a^*}\\;|\\;a^{*}(x))$ .",
"By applying rule $\\textsc {Open}$ twice and executing concurrently two extrusions on names $b$ and $c$ , we obtain a process: $\\nu a_{\\lbrace i,h\\rbrace } (\\overline{{b}}^{*}{a^*}[{i,*}]\\;|\\;\\overline{{c}}^{*}{a^*}[{h,*}]\\;|\\;a^{*}(x))$ The rule $\\textsc {Cause Ref}$ is used for the execution of the third action.",
"By definition of the predicate $\\mathtt {Cause}(\\cdot )$ , the action $a(x)$ can choose its cause from a set $\\lbrace i,h\\rbrace $ .",
"By choosing $h$ for example, and executing the input action, we get the process: $\\nu a_{\\lbrace i,h\\rbrace } (\\overline{{b}}^{*}{a}[{i,*}]\\;|\\;\\overline{{c}}^{*}{a}[{h,*}]\\;|\\;a^{*}(x)[{l,h}])$ In the memory $[{l,h}]$ we can see that the action identified with key $l$ needs to be reversed before the action with key $h$ .",
"Process $\\overline{{b}}^{*}{a}[{i,*}]$ can execute a backward step at any time with the rule $\\textsc {Open}^\\bullet $ .",
"A compositional causal semantics for standard (i.e., forward only) $\\pi $ -calculus was introduced by Boreale and Sangiorgi [6].",
"Later on, Degano and Priami in [15] introduced a causal semantics for $\\pi $ based on localities.",
"While using different approaches to keep track of the dependences in $\\pi $ -calculus, these two approaches impose the same order of the forward actions (as claimed in [15]).",
"Hence, from the reversible point of view we can take it that the causality notions of these two semantics coincide.",
"In what follows we shall concentrate on the Boreale-Sangiorgi causal semantics.",
"To show the correspondence between the mentioned semantics and our framework, we shall consider it in a late (rather than early, as originally given) version.",
"The precise definition is given in [24].",
"The authors distinguish between two types of causality: subject and the object.",
"To capture the first one, they introduce a causal term $\\mathtt {K}::A$ , where $\\mathtt {K}$ is a set of causes recording that every action performed by $A$ depends on $\\mathtt {K}$ .",
"The object causality is defined on the run (trace) of a process in such a way that once a bound name been extruded, it causes all the subsequent actions using that name in any position of the label.",
"Since an action can be caused through the subject and object position of a label, the contextual cause is a set $K\\subseteq \\mathcal {K}_{*}$ .",
"For example, let us consider a process $\\nu a (\\nu b (\\overline{{c}}{b}\\;|\\;\\overline{{d}}{a}\\;|\\;\\overline{{b}}{a}))$ with a trace $\\xrightarrow{}\\xrightarrow{}\\xrightarrow{}$ .",
"The action $\\overline{{b}}{a}$ depends on the first action because with it name $b$ was extruded and on the second action because with it name $a$ was extruded.",
"It is important to remark that a silent action does not exhibit causes.",
"To capture Boreale-Sangiorgi late semantics we need to give definitions for the predicates in Figure REF .",
"Definition 11 (Boreale-Sangiorgi causal semantics) If an indexed set $\\Gamma _w$ is chosen as a data structure for a memory $\\Delta $ , the predicates from Figure REF are defined as: 1.",
"$\\mathtt {Cause}(\\Delta ,K,K^{\\prime })=\\mathtt {Cause}(\\Gamma _{w},K,K^{\\prime })$ stands for $K^{\\prime }=K\\cup \\lbrace w\\rbrace $ 2.",
"$\\mathtt {Update}(\\Delta ,K,K^{\\prime })=\\mathtt {Update}(\\Gamma _{w},K,K^{\\prime })$ stands for $K^{\\prime }=K\\cup \\lbrace w\\rbrace $ Let us comment on the above definition.",
"After the first extrusion of a name, the cause is fixed and there is no possibility of choosing another cause from the set $\\Gamma $ .",
"To capture this behaviour we use the key of the first extruder, say $w$ , as the index of the set $\\Gamma $ .",
"The following example explains how our framework captures Boreale-Sangiorgi causality.",
"We shall use the same process as in Example REF .",
"Example 4 Consider the process $X=\\nu a_{\\emptyset _*} (\\overline{{b}}^{*}{a^*}\\;|\\;\\overline{{c}}^{*}{a^*}\\;|\\;a^{*}(x))$ .",
"By applying rule $\\textsc {Open}$ and executing the first extrusion on name $b$ , we obtain the process: $\\nu a_{\\lbrace i\\rbrace _{i}} (\\overline{{b}}^{*}{a^*}[{i,*}]\\;|\\;\\overline{{c}}^{*}{a^*}\\;|\\;a^{*}(x))$ In the memory ${\\lbrace i\\rbrace _{i}}$ the index $i$ indicates that name $a$ was extruded with the action $i$ .",
"On the process $ \\overline{{c}}^{*}{a^*}$ , rule $\\textsc {Open}$ can be applied.",
"By definition of the predicate $\\mathtt {Update}(\\cdot )$ , the output action is forced to add $w=i$ in its cause set.",
"Similar for the process $a^{*}(x)$ , by applying the rule $\\textsc {Cause Ref}$ and definition of the predicate $\\mathtt {Cause}(\\cdot )$ .",
"After two executions, we obtain the process: $\\nu a_{\\lbrace i,h\\rbrace _{i}} (\\overline{{b}}^{*}{a^*}[{i,*}]\\;|\\;\\overline{{c}}^{*}{a^*}[{h,\\lbrace i,*\\rbrace }]\\;|\\;a^{*}(x)[{l,\\lbrace i,*\\rbrace }])$ In the memories $[{h,\\lbrace i,*\\rbrace }]$ and $[{l,\\lbrace i,*\\rbrace }]$ we see that both executed actions are caused by action $i$ and this is why it needs to be reversed last.",
"The second and the third action can be reversed in any order.",
"The authors introduced a compositional event structure semantics for the forward $\\pi $ -calculus [9].",
"They represent a process as a pair $(E,\\mathtt {X})$ , where $E$ is a prime event structure and $\\mathtt {X}$ is a set of bound names.",
"Disjunctive objective causality is represented in such a way that an action with extruded name in the subject position can happen if at least one extrusion of that name has been executed before.",
"In the case of parallel extrusions of the same name, an action can be caused by any of them, but it is not necessary to remember which one.",
"Consequently, events do not have a unique causal history.",
"As discussed in [11] this type of disjunctive causality cannot be expressed when we consider processes with a contexts.",
"To adapt this notion of causality to reversible settings we need to keep track of causes; otherwise by going backwards we could reach an undefined state (where the extruder of a name is reversed, but not the action using that name in the subject position).",
"We consider two possibilities for keeping track of causes: the first one is by choosing one of the possible extruders and the second one is recording all of them.",
"In the first case, we would obtain a notion of causality similar to the one introduced in [10].",
"In the following we shall concentrate on the second option.",
"The idea is that, since we do not know which extruder really caused the action on an extruded name, we shall record the whole set of extruders that happened previously.",
"In the framework, the set of executed extruders is set $\\Omega $ .",
"The extrusions which are part of synchronisations will be deleted from $\\Omega $ with the operation $\\#$ .",
"The predicates from the rules of Figure REF are defined as follows: Definition 12 (Disjunctive causality) If an indexed set $\\Gamma _{\\Omega }$ is chosen as a data structure for a memory $\\Delta $ , the predicates are defined as: 1.",
"$\\mathtt {Cause}(\\Delta ,K,K^{\\prime })=\\mathtt {Cause}(\\Gamma _{\\Omega },K,K^{\\prime })$ stands for $K^{\\prime }=K\\cup \\Omega $ 2.",
"$\\mathtt {Update}(\\Delta ,K,K^{\\prime })=\\mathtt {Update}(\\Gamma _{\\Omega },K,K^{\\prime })$ stands for $K^{\\prime }=K$ In the following example we shall give the intuition of how our framework captures the defined notion of causality.",
"Example 5 Let us consider the process $X=\\nu a_{\\emptyset _{\\lbrace *\\rbrace }} (\\overline{{b}}^{*}{a^*}\\;|\\;\\overline{{c}}^{*}{a^*}\\;|\\;a^{*}(x))$ .",
"By applying a rule $\\textsc {Open}$ twice and executing concurrently two extrusions on names $b$ and $c$ , we obtain a process: $\\nu a_{\\lbrace i,h\\rbrace _{\\lbrace *,i,h\\rbrace }} (\\overline{{b}}^{*}{a^*}[{i,*}]\\;|\\;\\overline{{c}}^{*}{a^*}[{h,*}]\\;|\\;a^{*}(x))$ By definition of the predicate $\\mathtt {Cause}(\\cdot )$ , the third action will take the whole index set $\\lbrace *,i,h\\rbrace $ as a set $K$ and we get the process: $\\nu a_{\\lbrace i,h\\rbrace _{\\lbrace *,i,h\\rbrace }} (\\overline{{b}}^{*}{a}[{i,*}]\\;|\\;\\overline{{c}}^{*}{a}[{h,*}]\\;|\\;a^{*}(x)[{l,\\lbrace *,i,h\\rbrace }])$ In the memory $[{l,\\lbrace *,i,h\\rbrace }]$ we see that the first action to be reversed is the action with key $l$ ; the other two actions can be reversed in any order."
],
[
"Properties",
"In this section we shall show some properties of our framework.",
"First we shall show that the framework is a conservative extension of standard $\\pi $ -calculus and that it enjoys causal consistency, a fundamental property for reversible calculi.",
"After that, we shall prove causal correspondence between the causality induced by Boreale-Sangiorgi semantics and causality in the framework when $\\Delta =\\Gamma _w$ .",
"Definition 13 (Initial and Reachable process) A reversible process X is initial if it is derived from a $\\pi $ -calculus process $P$ where all the restricting operators are initialised and in every prefix, names are decorated with a $*$ .",
"A reversible process is reachable if it can be derived from an initial process by using the rules in Figures REF , REF and REF ."
],
[
"Correspondence with the $\\pi $ -calculus",
"We now show that our framework is a conservative extension of the $\\pi $ -calculus.",
"To do so, we first define an erasing function $ that given a reversible process $ X$, by deleting all the past information, generates a $$ process.",
"Then we shall show that there is a \\textit {forward} operational correspondence between a reversible process $ X$ and $ X)$.",
"Let $ P$ be the set of $$-calculus processes; then we have:$ Definition 14 (Erasing function) The function $ \\mathcal {\\mathcal {X}}\\rightarrow \\mathcal {P}$ that maps reversible processes to the $\\pi $ -calculus, is inductively defined as follows: $&X\\;|\\;Y)=X)\\;|\\;Y)\\quad &&\\mathtt {H}[X])=X)&&\\mathbf {0})=\\mathbf {0}\\\\&\\nu a_{\\Delta } (X))= X)\\quad \\quad \\;\\text{if } \\mathtt {empty}(\\Delta )=false \\quad &&b^{j}(x).{P})=b(x).",
"{P})\\\\&\\nu a_{\\Delta } (X))= \\nu a \\;X)\\quad \\text{if } \\mathtt {empty}(\\Delta )=true\\quad &&\\overline{{b}}^j{a^{j^{\\prime }}}.{P})=\\overline{{b}}{a}.",
"{P})$ The erasing function can be extended to labels as: $&({i},{K},{j}):{\\pi })=\\pi )\\quad &&\\overline{{b}}{a})=\\overline{{b}}{a}\\\\&\\overline{{b}}\\langle \\nu {a}_{\\Delta }\\rangle )=\\overline{{b}}\\langle \\nu {a}_{}\\rangle \\quad \\text{when }\\mathtt {empty}(\\Delta )=true\\quad &&b(x))=b(x)\\\\&\\overline{{b}}\\langle \\nu {a}_{\\Delta }\\rangle )=\\overline{{b}}{a}\\qquad \\;\\text{when }\\mathtt {empty}(\\Delta )=false\\quad &&\\tau )=\\tau $ As expected the erasing function discards the past prefixes and name restriction operators that are non-empty.",
"Moreover, it deletes all the information about the instantiators.",
"Every forward move of a reversible process $X$ can be matched in the $\\pi $ -calculus.",
"To this end we use $\\xrightarrow{}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow _\\pi $ to indicate the transition semantics of the $\\pi $ -calculus.",
"Lemma 1 If there is a transition $X\\xrightarrow{}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow Y$ then $X)\\xrightarrow{}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow _{\\pi }Y)$ .",
"We can state the converse of Lemma REF as follows: Lemma 2 If there is a transition $P\\xrightarrow{}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow _{\\pi }Q$ then for all reachable $X$ such that $X)=P$ , there is a transition $X\\xrightarrow{}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow Y$ with $Y)=Q$ .",
"Corollary 1 The relation given by $(X,X))$ , for all reachable processes $X$ , is a strong bisimulation."
],
[
"The main properties of the framework",
"We now prove some properties of our framework which are typical of a reversible process calculus [12], [27], [20], [10].",
"Most of the terminology and the proof schemas are adapted from [12], [10] with more complex arguments due to the generality of our framework.",
"The first important property is the so-called Loop Lemma, stating that any reduction step can be undone.",
"Formally: Lemma 3 (Loop Lemma) For every reachable process $X$ and forward transition $t:X\\xrightarrow{}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow Y$ there exists a backward transition $t^{\\prime }:Y\\mathrel {\\begin{tikzpicture}[baseline= {( (current bounding box.south) + (0,-0.5ex) )}]\\node [inner sep=.5ex] (27) {\\scriptstyle \\;\\mu };[draw,<-,decorate,decoration={zigzag,amplitude=1.2pt,segment length=1.5mm,pre=lineto,pre length=6pt}](27.south east) -- (27.south west);\\end{tikzpicture}}X$ , and conversely.",
"Before stating our main theorems we need to define the causality relation.",
"It is defined on the general framework and it is interpreted as the union of structural and object causality.",
"Definition 15 (Structural cause on the keys) For every two keys $i_1$ and $i_2$ such that $i_1 , i_2\\in \\mathtt {key}(X)$ , we let $i_1 \\sqsubset _{X} i_2 $ if $X=C[\\alpha [i_1,K_1].Y]$ and $i_2\\in \\mathtt {key}(Y)$ .",
"Once having defined structural causal relation on keys, we can extend it to transitions.",
"Definition 16 (Structural causality) Transition $t_1 : X\\xrightarrow{} X^{\\prime }$ is a structural cause of transition $t_2 : X^{\\prime \\prime }\\xrightarrow{} X^{\\prime \\prime \\prime }$ , written $t_1\\sqsubset t_2$ , if $i_1 \\sqsubset _{X^{\\prime \\prime \\prime }} i_2 $ or $i_2 \\sqsubset _{X} i_1 $ .",
"Structural causality, denoted with $\\sqsubseteq $ , is the reflexive and transitive closure of $\\sqsubset $ .",
"Object causality is defined directly on the transitions and to keep track of it we use the contextual cause $K$ .",
"Definition 17 (Reverse transition) The reverse transition of a transition $t: X\\xrightarrow{}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow Y$ , written $t^\\bullet $ , is the transition with the same label and the opposite direction $t^\\bullet : Y\\mathrel {\\begin{tikzpicture}[baseline= {( (current bounding box.south) + (0,-0.5ex) )}]\\node [inner sep=.5ex] (28) {\\scriptstyle \\mu };[draw,<-,decorate,decoration={zigzag,amplitude=1.2pt,segment length=1.5mm,pre=lineto,pre length=6pt}](28.south east) -- (28.south west);\\end{tikzpicture}}X$ , and vice versa.",
"Thus $(t^\\bullet )^\\bullet = t$ .",
"Definition 18 (Object causality) Transition $t_1 : X\\xrightarrow{} X^{\\prime }$ is an object cause of transition $t_2 : X^{\\prime }\\xrightarrow{} X^{\\prime \\prime }$ , written $t_1<t_2$ , if $i_1\\in K_2 $ or $i_2\\in K_1 $ (for the backward transition) and $t_1\\ne t_2^\\bullet $ .",
"Object causality, denoted with $\\ll $ , is the reflexive and transitive closure of $<$ .",
"Example 6 Consider a process $X=\\nu a_{\\Delta } (\\overline{{b}}^{*}{a^*}\\;|\\;\\overline{{c}}^{*}{a^*}\\;|\\;a^*(z))$ and the case when $\\Delta =\\emptyset _*$ , as in Example REF .",
"The executed actions would be $\\xrightarrow{}\\xrightarrow{}\\xrightarrow{}$ .",
"We can notice that $K_2=\\lbrace i_1\\rbrace $ and $K_3=\\lbrace i_1\\rbrace $ , indicating that the second and the third action are caused by the first one.",
"By choosing a different data structure we can obtain different causal order, as mentioned in Example REF and Example REF .",
"Definition 19 (Causality relation and concurrency) The causality relation $\\prec $ is the reflexive and transitive closure of structural and object cause: $\\prec = (\\sqsubseteq \\cup \\ll )^{*}$ .",
"Two transitions are concurrent if they are not causally related.",
"Concurrent transitions can be permuted, and the commutation of transitions is preserved up to label equivalence.",
"Definition 20 (Label equivalence) Label equivalence, $=_{\\lambda }$ , is the least equivalence relation satisfying: $({i},{K},{j}):{\\overline{{b}}\\langle \\nu {a}_{\\Delta }\\rangle }=_{\\lambda }({i},{K},{j}):{\\overline{{b}}\\langle \\nu {a}_{\\Delta ^{\\prime }}\\rangle }$ for all $i, j, K, a ,b$ and $\\Delta , \\Delta ^{\\prime } \\subseteq \\mathcal {K}$ .",
"(Having an indexed set $\\Gamma _w$ for $\\Delta $ we disregard index $w$ , and observe $\\Gamma \\subseteq \\mathcal {K}$ .)",
"Lemma 4 (Square Lemma) If $t_1 : X\\xrightarrow{} Y$ and $t_2 : Y\\xrightarrow{} Z$ are two concurrent transitions, there exist $t^{\\prime }_2 : X\\xrightarrow{} Y_1$ and $t^{\\prime }_1 : Y_1\\xrightarrow{} Z$ where $\\mu _i=_{\\lambda }\\mu ^{\\prime }_i$ .",
"We shall follow the standard notation and say that $t_2$ is a residual of $t^{\\prime }_2$ after $t_1$ , denoted with $t_2=t^{\\prime }_2/t_1$ .",
"Two transitions are coinitial if they have the same source, cofinal if they have the same target, and composable if the target of one is the source of the other.",
"A sequence of pairwise composable transitions is called a trace, written as $t_1; t_2$ .",
"We denote with $\\epsilon $ the empty trace.",
"Notions of target, source, composability and reverse extend naturally to traces.",
"With the next theorem we prove that reversibility in our framework is causally consistent.",
"Definition 21 (Equivalence up-to permutation) Equivalence up-to permutation, $\\sim $ , is the least equivalence relation on the traces, satisfying: $t_1;(t_2/t_1)\\sim t_2;(t_1/t_2)\\qquad t;t^\\bullet \\sim \\epsilon $ Equivalence up-to permutation introduced in [12] is an adaptation of equivalence between traces introduced in [21], [7] that additionally erases from a trace, transitions triggered in both directions.",
"It just states that concurrent actions can be swapped and that a trace made by a transition followed by its inverse is equivalent to the empty trace.",
"Theorem 1 Two traces are coinitial and cofinal if and only if they are equivalent up-to permutation."
],
[
"Correspondence with Boreale and Sangiorgi's semantics",
"We prove causal correspondence between Boreale and Sangiorgi's late semantics (rather than early, as originally given) and the framework when memory $\\Delta $ is instantiated with $\\Gamma _w$ .",
"The precise definitions and the proofs are given in [24]; here we shall give just a brief presentation of the idea.",
"To compare semantics, we observe traces (runs) of the processes.",
"Labels in the framework will bring additional information about the multiset of the structural causes ($K_{F}$ ) of the executed action and a trace in the framework will have the following form: $X_1\\xrightarrow[K_{F1}]{\\mu _1}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow X_2\\ldots X_{n-1}\\xrightarrow[K_{Fn}]{\\mu _n}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow X_{n}$ .",
"To simplify notation, we shall write the transition $A\\xrightarrow[K;k]{\\pi }A_2$ from Boreale and Sangiorgi's semantics as $A\\xrightarrow[K_{B}]{\\zeta }A_2$ , where $\\zeta =k: \\pi $ .",
"Focusing on the structural correspondence, the main difference is in the silent actions.",
"In the framework, silent actions are identified with unique keys, while in Boreale and Sangiorgi's semantics, they just merge the cause sets of the actions participating in the communication.",
"Hence, we need to provide a connection between sets of structural causes in these two semantics.",
"Let us briefly explain our method; more details can be found in [24].",
"Suppose that we have two transition $t$ and $t^{\\prime }$ , where $t: X\\xrightarrow[K_{F}]{({i},{K},{j}):{\\pi }}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow X^{\\prime }$ and $t^{\\prime }: A\\xrightarrow[K_B]{i:\\pi }A^{\\prime }$ where the continuation of the processes $X$ and $A$ is $\\pi .P$By abuse the notation, we shall write $\\pi $ for the prefixes and the labels of the actions in both semantics, since they are essentially the same.",
"We can represent the dependences between the keys in the history of the process $X$ (all executed actions in $X$ ) with a directed graph, in the following way: keys of executed actions will be represented as vertices of a graph (actions which are part of a communication and have the same key, will be represented by two vertices with the same name); order between keys will be represented with directed edges where between the same vertices we shall have edges in both directions.",
"Let as denote this graph $G=(V,E)$ , where $V$ is a multiset of vertices and $E$ set of edges.",
"To show exact correspondence between cause sets $K_{F}$ and $K_B$ we need to take the history part of the process $X$ involved in the execution of an action $\\pi $ .",
"We can do it by taking all the paths in $G$ in which the target vertex will be key $i$ of the action $\\pi $ and we shall obtain the graph $G(i)=(V(i),E(i))$ .",
"The multiset $V(i)$ contains all the keys which cause the action $\\pi $ including $i$ and we can conclude that $K_F=V(i)\\setminus \\lbrace i\\rbrace $ .",
"By removing all bidirectional edges from the graph $G(i)$ and replacing vertices that they connect with vertex renamed to $\\tau _l$ when $l=1,2,\\ldots n$ , we shall obtain the graph $G^{\\prime }=(V^{\\prime },E^{\\prime })$ .",
"(Renaming is applied also on the other edges containing removed vertices.",
"The operation of bidirectional edge contraction is precisely defined in [24].)",
"The set $V^{\\prime }$ differs from $V(i)$ in having $\\tau _l$ vertices instead of the pairs of the vertices with the same name (originally belonging to silent moves in the framework).",
"Hence, we can conclude that $K_B=V^{\\prime }\\setminus (\\lbrace i\\rbrace \\cup \\tau _l)$ .",
"We shall call the algorithm explained above `Removing Keys from a Set', denoted as $\\mathtt {Rem}$ .",
"We shall write $\\mathtt {Rem}(K_{F})=K_B$ , meaning that $K_B$ can be obtained by applying algorithm $\\mathtt {Rem}$ to $K_F$ .",
"Before stating the theorem, we shall give a definition of the erasing function $\\lambda $ and the function $\\gamma $ that maps labels from the framework into labels from Boreale and Sangiorgi's semantics: Definition 22 The function $\\gamma $ that maps label from the framework with a label from Boreale and Sangiorgi's semantics, is inductively defined as follows: $&\\gamma (({i},{K},{j}):{\\pi })=i:\\gamma (\\pi )\\quad \\quad \\text{when }\\pi \\ne \\tau \\quad &&\\gamma (({i},{*},{*}):{\\tau })=\\tau \\\\&\\gamma (\\overline{{b}}\\langle \\nu {a}_{\\Delta }\\rangle )=\\overline{{b}}\\langle \\nu {a}_{}\\rangle \\quad \\text{when }\\mathtt {empty}(\\Delta )=true\\quad &&\\gamma (b(c))=b(c)\\\\&\\gamma (\\overline{{b}}\\langle \\nu {a}_{\\Delta }\\rangle )=\\overline{{b}}{a}\\qquad \\;\\text{when }\\mathtt {empty}(\\Delta )=false\\quad &&\\gamma (\\overline{{b}}{a})=\\overline{{b}}{a}$ Definition 23 The erasing function $\\lambda $ that maps causal processes from Boreale and Sangiorgi's semantics to the $\\pi $ -calculus is inductively defined as follows: $&\\lambda (A\\;|\\;A^{\\prime })=\\lambda (A)\\;|\\;\\lambda (A^{\\prime })&&\\lambda (\\mathtt {K}::A)=\\lambda (A)&&\\lambda (\\nu a (A))=\\nu a (\\lambda (A))&&\\lambda (P)=P&$ Now we have all necessary definitions to state the lemma about structural correspondence between two causal semantics.",
"Lemma 5 (Structural correspondence) Starting from initial $\\pi $ -calculus process $P$ , where $P=A_1=X_1$ , we have: if $P\\xrightarrow[K_{B1}]{\\zeta _1}A_2\\ldots A_{n}\\xrightarrow[K_{Bn}]{\\zeta _{n}}A_{n+1}$ then there exists a trace $P\\xrightarrow[K_{F1}]{\\mu _1}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow X_2\\ldots X_{n}\\xrightarrow[K_{Fn}]{\\mu _{n}}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow X_{n+1}$ and $K_{Fi}$ , such that for all $i$ , $\\lambda (A_{i})=X_{i})$ , $\\zeta _i=\\gamma (\\mu _i)$ and $\\mathtt {Rem}(K_{Fi})=K_{Bi}$ , for $i=1,...,n$ .",
"if $P\\xrightarrow[K_{F1}]{\\mu _1}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow X_2\\ldots X_{n}\\xrightarrow[K_{Fn}]{\\mu _n}\\mathrel {\\hspace{-7.77771pt}}\\rightarrow X_{n+1}$ then there exists a trace $P\\xrightarrow[K_{B1}]{\\zeta _1}A_2\\ldots A_{n}\\xrightarrow[K_{Bn}]{\\zeta _{n}}A_{n+1}$ where for all $i$ , $\\lambda (A_{i})=X_{i})$ , $\\zeta _i=\\gamma (\\mu _i)$ and $\\mathtt {Rem}(K_{Fi})=K_{Bi}$ , for $i=1,...,n$ .",
"Considering the object dependence we have that the first action which extrudes a bound name will cause all future actions using that name in any position of the label.",
"The main difference is that object dependence induced by input action in Boreale and Sangiorgi's semantics is subject dependence as well.",
"The next theorem demonstrates causal correspondence between causality in the framework when memory $\\Delta $ is instantiated with $\\Gamma _w$ and Boreale and Sangiorgi's late causal semantics.",
"Theorem 2 (Causal correspondence) The reflexive and transitive closure of the causality introduced in [6] coincides with the causality of the framework when $\\Delta =\\Gamma _w$ ."
],
[
"Conclusions",
"In a concurrent setting, causally-consistent reversibility relates causality and reversibility.",
"Several works [10], [15], [6], [9], [8] have addressed causal semantics for $\\pi $ -calculus, differing on how object causality is modelled.",
"Starting from this observation, we have devised a framework for reversible $\\pi $ -calculus which abstracts away from the underlying data structure used to record causes and consequences of an extrusion, and hence from the object causality.",
"Depending on the underlying data structure, we can obtain different causal semantics.",
"We have shown how three different semantics [10], [6], [9] can be derived, and we have proved causal correspondence with the semantics introduced in [6].",
"Our framework enjoys typical properties for reversible process algebra, such as loop lemma and causal consistence.",
"As a future work we plan to prove causal correspondence with the semantics [10], [9] and to continue working towards a more parametric framework and to compare it with [26], [18].",
"Moreover it would be interesting to implement our framework in the psi-calculi framework [4], and to develop further the behavioural theory of our framework."
],
[
"Acknowledgments",
"We are grateful to the EXPRESS/SOS reviewers for their useful remarks and suggestions which led to substantial improvements."
]
] | 1808.08655 |
[
[
"Non-linear Waves and Instabilities Leading to Secondary Reconnection in\n Reconnection Outflows"
],
[
"Abstract Reconnection outflows are regions of intense recent scrutiny, from in situ observations and from simulations.",
"These regions are host to a variety of instabilities and intense energy exchanges, often even superior to the main reconnection site.",
"We report here a number of results drawn from investigation of simulations.",
"First, the outflows are observed to become unstable to drift instabilities.",
"Second, these instabilities lead to the formation of secondary reconnection sites.",
"Third, the secondary processes are responsible for large energy exchanges and particle energization.",
"Finally, the particle distribution function are modified to become non-Maxwellian and include multiple interpenetrating populations."
],
[
"Introduction",
"The research of the last two decades has shown that kinetic reconnection is a fast process that develops on Alfvén time-scales [5].",
"This result is a spectacular success for kinetic modelling [3], now confirmed in situ by the Magnetospheric Multiscale Mission [7].",
"However, fast kinetic reconnection is not the solution to all problems in reconnection: fast kinetic reconnection has thus far been observed and modelled only in localised regions.",
"Instead, in many astrophysical and laboratory systems, large amounts of energy are converted over large domains.",
"How can we bring fast kinetic reconnection to large scales?",
"A possible scenario to reach large energy conversion rates on system scales is to imagine a situation where the initiation of reconnection is followed by a chain reaction of more and more secondary reconnection sites [6], [34], [24], [48].",
"Under these conditions, reconnection tends to become chaotic with many reconnection sites being spawned by instability and reabsorbed by island coalescence, leading to fast reconnection [2], [47], [42], [21].",
"Three dimensional reconnection is accompanied by many more instabilities than just the formation of secondary islands in the primary reconnection site seen in two dimensional reconnection: the reconnection inflow[9] and the reconnection outflow [31] host instabilities that lead to secondary reconnection.",
"The first mechanism is primarily present in reconnection separatrices in the case of strong guide fields [32], while the latter is present at all guide fields [30].",
"Outflows from reconnection are rich in free energy that can drive instabilities.",
"Among the possibilities we consider here: Velocity shears around the outflow jet that can drive Kelvin-Helmholtz instability [33].",
"Density and temperature gradients at the front formed by the outflowing jet interacting with the ambient plasma leads to drift-type instabilities [11].",
"Unfavourable curvature of field lines between the separatrices in the outflow region can lead to interchange (Rayleigh-Taylor-type) instabilities [38], [18], [26].",
"Flux ropes in the outflows may be kink unstable [23], [46].",
"Additional instabilities are caused by phase-space features such as anisotropies leading to whistler waves and beams leading to streaming instabilities [16].",
"All these instabilities can cause strong deformation of the flow, leading possibly to turbulence [41], energy exchange [29] and secondary reconnection [31].",
"The 3D scenario for large scale turbulence is than one where reconnection might lead to a chain-reaction type of sequence of events.",
"Reconnection is initiated at one location but the instabilities associated with the flows and the other sources of free energy induced by reconnection lead to the formation of secondary reconnection sites.",
"While not yet observed in simulation, this scenario on large scales (not yet accessible to simulation) can then progress in successive generation of tertiary and further reconnection sites, filling macroscopic domains.",
"Below, we organize our material as follows.",
"Section 2 reports the type of simulations we use to analyze the reconnection outflows and the instabilities developing there.",
"Section 3 investigate the fluctuation spectrum produced in the outflow.",
"Section 4 discusses how the fluctuations interact with the particles energizing them.",
"Conclusions and future directions are outlined in Sect.",
"5."
],
[
"Development of outflow instabilities and secondary reconnection",
"In order to study the properties of outflow instabilities and secondary reconnection, we use particle-in-cell numerical simulations.",
"We consider here the same run previously considered in [31].",
"The system is initialized with a Harris equilibrium [19] ${\\bf B} = B_{0x}\\tanh {(y/\\delta )}{\\bf e_x} + B_{0z}{\\bf e_z},\\;\\;\\; n=n_{0b}+\\frac{n_0}{\\cosh ^2(y/\\delta )}.$ uniquely specified by the mass ratio $m_i/m_e= 256$ , the temperature ratio $T_i/T_e= 5$ and $v_{th,e}/c=0.045$ .",
"We set the density of the uniform plasma background to $n_{0b} = n_0/10$ and the value of the guide field to $B_{0z}/B_{0x}=1/10$ .",
"The evolution is followed using the fully electromagnetic and fully kinetic iPic3D code [35] that treats both electrons and ions as particles.",
"Details are provided in [31].",
"We use coordinates where $x$ is along the initial magnetic field, $y$ is along the initial gradients, and $z$ is along the initial current.",
"Open boundary conditions are imposed in the $x$ and $y$ direction and periodicity is imposed along $z$ .",
"We consider a 3D box of shape $[40.0,15.0,10.0]d_i$ , where $d_i$ is the ion inertial length, which is resolved by a cartesian grid of $[512,192,128]$ cells, each one populated with 125 particles.",
"The spatial resolution is $\\Delta x = 1.25d_e$ , where $d_e$ is the electron inertial length, and the time step is $\\Delta t=\\pi /10 \\omega _{ce}^{-1}$ , where $\\omega _{ce}$ is the electron gyro-frequency.",
"Reconnection is initialised in the centre with an initial x-shaped perturbation that leads to the formation of a central x-line.",
"A reconnection site develops with plasma accelerated towards the reconnection region and expelled out of it.",
"The electron flow pattern in the fully developed non-linear stage is shown in Figure REF .",
"The electrons are first attracted toward the central x-line where the z-directed reconnection electric field accelerates them to high speed.",
"The Lorentz force then deflects the particles towards the outflow.",
"In this region, the system presents a remarkable invariance along $z$ , resembling the same physics of two dimensional fast kinetic reconnection.",
"In the outflow, however, the electron flow pattern becomes distorted and meanders about, eventually passing downstream away of the reconnection region.",
"In this region the electron flow becomes more turbulent.",
"Figure: Visualization of the electron flow around a reconnection site.",
"We report streamlines of the first order moment of the electron distribution (the electron flow velocity) coloured by the intensity of the local electron speed (normalized to the speed of light).The region of electron meandering corresponds to the front formed by the interaction of the outflowing plasma with the surrounding media.",
"At the front, an effect similar to that of a snowplow pushes the plasma outward.",
"A form forms where at least three of the mechanisms mentioned above are present: the field lines wrap around the front gaining unfavourbale curvature that can lead to interchange-type instabilities, the density gradient is unstable to drift modes and the distribution function becomes severely non-maxwellian leading to microinstabilities.",
"Figure REF shows the state of the front after the instability starts to develop.",
"The density (panel a) becomes rippled by a mode that presents a strong perturbation of the $E_z$ (panel b).",
"When the mode structure of these fluctuations is Fourier analysed, the resulting spectrum in $k_z$ is reported in panel c. The observed features are characteristic of a drift mode in the lower-hybrid range.",
"Figure: Early stages of the instability at the front, at time ω ci -1 =15\\omega _{ci}^{-1}=15.",
"The panels show from top to bottom: ion density (a), zz-component of the electric field (b) and the Fourier spectrum in k z k_z of the perturbation of the electric field E z E_z.Figure: Signal from a virtual probe embedded in the simulation at x/d i =7.54x/d_i=7.54, y/d i =7.54y/d_i=7.54, z/d i =5.04z/d_i=5.04.",
"The top panel shows the spectrogram of the E z E_z signal measured.",
"To guide the eye the local lower hybrid frequency is indicated by a white line.",
"The bottom panel shows the magnetic field intensity measured by the virtual probes at the different times.The identification of the instability as having primarily the nature of a lower-hybrid drift instability (LHDI) is confirmed by the temporal spectrum measured at a fixed point reached by the front [11], [10].",
"A spectrogram, obtained with standard windowing methods similar to those used in on-board real space probes, is reported: the observed frequency spectrum is reported at different times.",
"The lower panel reports the corresponding observed local magnetic field intensity.",
"When the front arrives, an intense signal in the lower hybrid range is measured.",
"As the evolution is continued, the ripples in the front become more intense and start to interact leading to conditions where magnetic field of opposite polarity is brought in contact promoting secondary reconnection.",
"Figure REF shows the front at two consecutive times: at later times, the \"fingers\" formed in the front tend to interact and coalesce [49].",
"Figure: Density at the front for two different times.",
"[31] analysed several indicators to detect positively secondary reconnection sites: direct analysis of field line connectivity, energy conversion in the electron frame ($\\mathbf {J}\\cdot (\\mathbf {E}+\\mathbf {v}_{e} \\times \\mathbf {B})$ ), electron agyrotropy, slippage ($\\mathbf {v}_{e\\perp } -\\mathbf {E}\\times \\mathbf {B}/B^2$ ), topological measure of field line breakage [20], [5] ($\\mathbf {b}\\times \\nabla \\times (E_{||}\\mathbf {b})$ ), where $\\mathbf {b}$ is the unit vector along $\\mathbf {B}$ and $E_{||}$ is the parallel component of the non ideal part of Ohm's law [5], normalized as $eE_{||}/m_i c\\omega _{pi}$ .",
"A specific orientation of the magnetic field which allows for the field annihilation and energy release, is an important indicator of magnetic reconnection.",
"In the classical two dimensional picture magnetic field lines of opposite direction approach each other and form an X-point, which, extended to 3D, becomes an X-line denoted by the strong Z-aligned current in our simulation.",
"This sort of magnetic reconnection, however, does not require the field to become exactly zero (hence, no magnetic nulls are formed) on the reconnection site.",
"We use the technique based on the topological degree method [17] as described in [39] to locate and classify magnetic nulls.",
"Indeed, in the simulation reported here no magnetic nulls are present in the central current sheet as summarized in Figure REF .",
"However, the diffusion region around the X line is characterized by strong electron agyrotropy $A=\\left(P_{e,\\perp 1}-P_{e,\\perp 2}\\right)/P_{e,\\perp 1}-\\left(P_{e,\\perp 2}\\right)$ that is shown with volume rendering.",
"No strong energy conversion is associated with the `main' reconnection X-line.",
"In contrast, in the reconnection outflow a number of magnetic nulls form which are depicted by colour spheres in Figure REF .",
"The colour denotes magnetic null's topological type: A and B (red and orange) are the three-dimensional extensions of the X-points called radial nulls; while As and Bs (light blue and blue) represent magnetic islands or magnetic flux ropes.",
"Both radial and spiral nulls are present in the outflows, however the number of spiral ones is larger.",
"Magnetic field lines in the vicinity of the null points in the left outflow are shown with the corresponding colours.",
"A pair of spiral nulls is formed in a swirl of the light blue magnetic field lines, probably, driven by a shear instability.",
"This null pair is embedded in the region of strong energy conversion (see Figure REF ).",
"Other nulls in this outflow are on the interfaces of magnetic fields of different polarities characterized by complex field patterns resembling an X pattern (orange) and merging into flux ropes (purple and pink).",
"Recent observations [15] provide a strong evidence that intermittent energy conversion in the reconnection outflows is associated with the spiral magnetic nulls and twisted magnetic fields.",
"Figure: Combination of different measures at the same time: a vertical cut of the electron current intensity (grayscale); false-colour volume rendering of agyrotropy; magnetic null points (colour spheres) coloured according to their topological type.",
"Selected field lines reconnected once at the primary site are shown in green, while secondary reconnected lines near the nulls are shown in purple, pink, orange and light blue.The picture provides an indication of the scenario described in the introduction: the initial reconnection site located at the centre of the box and forming a X-line produces two outflows that become unstable and produce in turn secondary reconnection sites.",
"In the process the plasma becomes effectively turbulent and a large fraction of the energy is converted to particle heat at these unstable fronts, rather than at the central X-line."
],
[
"Development of intermittent turbulence in reconnection outflows",
"Above we have observed how reconnection tends to become visually turbulent.",
"But is turbulence real?",
"In a recent paper [41], the properties of electric and magnetic fluctuations that are produced by magnetic reconnection have been analysed.",
"Because of the inhomogeneous background it is important to first establish the anisotropy level and in general the 3D properties of turbulence.",
"Analysis of the autocorrelation function of the magnetic field fluctuations have shown that the turbulence that develops in the reconnection jets is anisotropic.",
"In particular, magnetic vortexes are elongated in the direction of the background magnetic field, namely $x$ , with a second smaller anisotropy in the $yz$ plane.",
"The second anisotropy becomes negligible for smaller scales and isotropy is recovered in the $(k_y,k_z)$ plane for $k_{yz} > 1.5$ , with $k_{yz}=\\sqrt{k_y^2+k_z^2}$ .",
"This allows to reduce the 3D spatial spectra to 1D isotropic spectra computed in $(k_y,k_z)$ plane and integrated in $k_x$ .",
"The results of this computation shows magnetic and electric spectra with a clear power law in the sub-ion range $1.5 < k_{yz}d_i < 15$ .",
"As observed in space plasmas [14] the magnetic and electric spectra departs from each other at around $kd_i \\sim 1$ , the electric one proceeding with a spectral slope of $\\sim 1$ and the magnetic one with a slope of $-8/3$ .",
"Recently [36], following simple dimensional arguments, have interpreted this phenomenon as due to the dominance of the Hall-effect at small scales.",
"It is worth remarking how this interpretation still holds in such an anisotropic and inhomogeneous system, where spectra need to be carefully extracted removing large-scale background profiles and border effects.",
"Turbulence is responsible for the transfer of energy from fields to particles.",
"In this work we show that this energy exchange do not take place homogeneously in the reconnection events but is located in small regions in the reconnection outflows where the energy transfer is very intense.",
"In order to quantify the energy exchange we introduce the two dissipation proxies $D_l= {\\bf J} \\cdot {\\bf E}$ and $D_i = {\\bf J} \\cdot ({\\bf E} + {\\bf v}_i \\times {\\bf B})$ [53], where ${\\bf J}$ is the total current, ${\\bf E}$ is the electric field, ${\\bf v}_i$ is the ion fluid velocity, and ${\\bf B}$ is the magnetic field.",
"In panel (a) of Figure REF the Probability Density Functions (PDFs) of $\\delta D_l = D_l - \\langle D_l \\rangle _{x,y,z} $ and $\\delta D_i = D_i - \\langle D_i \\rangle _{x,y,z} $ are plotted, where $\\langle \\rangle _{x,y,z}$ means average along the three axes.",
"The two PDFs are compared with the normalized Gaussian distribution (plotted in dashed-red line).",
"They strongly depart from Gaussian distributions, presenting instead high tails up to several standard deviations $\\sigma $ .",
"In panel (b), the average $D_i$ conditioned to a threshold current density is shown.",
"The plot is constructed as follows: a threshold in the current density magnitude is considered and the average of $D_i$ is computed using all those points in the domain where the value of the current is bigger than the fixed threshold.",
"This average is then normalized to the average of $D_i$ on all points, which gives by definition $\\langle D_i | J=0 \\rangle /\\langle D_i \\rangle = 1$ .",
"The black points in the plots represent the result of such computation for different values of the threshold.",
"The blue curve represents the filling factors, i.e.",
"the fraction of points used for computing the average with respect to the total number of points in the sample.",
"The average of $D_i$ strongly increases when higher threshold are considered up to $J/J_{rms} = 10 $ .",
"Our results confirm that the exchange of energy is local, with larger values of $D_i$ localized in very small volume filling structures.",
"This evidence and the presence of non-Gaussian PDFs of dissipation proxies suggest that magnetic reconnection produces small scales current sheets which are site of strong events of energy exchange between fields and particles.",
"Concisely stated, all these statistics indicate that dissipation in a reconnection event is intermittent.",
"A similar conclusion was reached by [50] who examined the electron frame dissipation surrogate conditioned on magnitude of current density.",
"Figure REF -REF shows the statistics of dissipation proxies presented in Figure REF computed in sub-boxes located in the two reconnection outflows (see Figure REF ).",
"Non-Gaussian statistics and increasing conditioned average of dissipation proxies indicate that intermittent turbulence is at play in both reconnection outflows.",
"Figure: Energy exchange D l =𝐉·𝐄D_l = {\\bf J} \\cdot {\\bf E} in the xyxy plane averaged in the zz direction (a), and in the yzyz plane at x=33d i x = 33\\, d_i (b),x=36d i x = 36\\, d_i (c), x=39d i x = 39\\, d_i (d).",
"The x-line is located at x=20d i x =20 \\, d_i.",
"The three boxes in panel (a) are the onesused for the statical analysis presented in Figure -.Figure: PDFs of D l D_l and D i D_i (a).",
"Red dashed linesrepresent the normalized Gaussian curve.",
"Mean D i D_i conditioned on local current densitythresholds and (right axis) fraction FF of the full box data used to compute the averages (b).Figure: PDFs of D l D_l and D i D_i in BOX 1 BOX_1 (c), BOX 2 BOX_2 (e).Conditioned average of D i D_i and filling factors FF in BOX 1 BOX_1 (d), BOX 2 BOX_2 (f) (left outflow).Figure: PDFs of D l D_l and D i D_i in BOX 3 BOX_3 (c), BOX 4 BOX_4 (e).Conditioned average of D i D_i and filling factors FF in BOX 3 BOX_3 (d), BOX 4 BOX_4 (f) (right outflow)."
],
[
"Energy Exchanges in Reconnection outflows ",
"As shown in Figure REF , the region of the outflow is characterised by intense energy exchange ($\\mathbf {J}\\cdot \\mathbf {E}$ ).",
"Recently the energy budget has been analysed in detail [29] and a large fraction of the energy is deposited as particle energization, while a significant fraction is also transported by the Poynting flux.",
"Figure REF reports the ion temperature at the end of the run.",
"Ions are generally not magnetized in the reconnection region and projecting the pressure tensor in the parallel and perpendicular direction relative to the magnetic field is not productive.",
"Ion energization in reconnection outflows and in reconnection fronts has been analysed in theory and in simulation [1], [40], [4], [25].",
"Complex processes are at play, requiring a full analysis of the phase space and of single particle trajectories to detect with accuracy the specific mechanisms accelerating the particles [13].",
"Figure REF reports the three different kinetic temperatures obtained from the pressure tensor: $T_{i} = P_{ii}/\\rho _{i}$ , for $i= x,\\, y,\\, z$ .",
"The primary region of reconnection tends to heat the ions primarily in the $y$ direction.",
"This effect is due to the mixing of the two populations of ions coming from above and below from the inflow towards the reconnection region.",
"In the outflow, instead, the plasma outflowing along the x-direction mixes with the plasma in the medium causing apparent heating in the x-direction [1].",
"Heating in the $z$ direction is present both in the region of primary reconnection, where it is due to the acceleration of non-magnetised ions in the reconnection region due to the reconnection electric field [37], [12], and in the region of the outflows, where it is a consequence of the instabilities in the outflows.",
"These effects however should not be interpreted as heating in the meaning of increasing thermodynamic temperature.",
"The plasma is far from maxwellian and what appears as heating in the kinetic temperature (i.e.",
"the second order moment of the distribution) is in reality the presence of multiple interpenetrating populations.",
"Figure: Ion temperature in the outflow: from top to bottom: T x T_x (a), T y T_y (b) and T z T_z (c).",
"The left half of the domain is reported at the final time along with selected field lines.Figure REF shows a volume rendering of the full 3D velocity probability distribution for the ions.",
"The distribution is anisotropic and contains multiple populations.",
"When the second order moment is taken to measure a kinetic temperature, the result can be misleading because multiple beams, each with its own temperature, appear as a single plasma with a combined temperature much higher than that of the beams.",
"However this is not a process of heating but one of bulk acceleration of ion populations.",
"In a recent study, each ion component has been tracked back in time to its origin [13].",
"Each component originates from different regions and their trajectories brought them to the same location but with different speeds.",
"Figure: Volume rendering of the ion velocity probability distribution f i (v x ,v y ,v z )f_i(v_x,v_y,v_z) at the position x/d i =10.32x/d_i=10.32, y/d i =7.5y/d_i=7.5, z/d i =5z/d_i=5, obtained averaging over particles contained within a box centred at that location and with side 0.5d i 0.5d_i.Similarly, Figure REF show the parallel and perpendicular electron temperature.",
"The electrons are mostly magnetised and it is more convenient to report the electron temperatures in magnetic coordinates rather than along geometrical axes.",
"The region of primary reconnection causes parallel heating [44].",
"The cause is the reconnection electric field that accelerates the electrons along the $z$ direction [52], [12]: in this region the guide field is the only field present and the acceleration is parallel.",
"The region of secondary instabilities in the outflow shows strong parallel and perpendicular energisation caused by the conversion of electromagnetic energy (i.e.",
"$\\mathbf {J}_{e}\\cdot \\mathbf {E}$ ) [28], [29].",
"Figure: Electron parallel (a) and perpendicular (b) temperature in the outflow.",
"The left half of the domain is reported at the final time along with selected field lines.The electron distribution is typically far smoother than the ion distribution due to the higher thermal speed.",
"However, in the region of the secondary front instability even the electron distribution becomes complex.",
"Figure REF shows a volume rendering of the full 3D velocity probability distribution for the electrons computed as described above for the ions.",
"On a large scale the distribution is bi-maxwellian with different parallel and perpendicular temperatures.",
"Within it we can observe multiple electron populations caused by electron acceleration by the electric field, directed primarily in the directions $z$ (reconnection electric field ) and $y$ (Hall electric field) [51].",
"Acceleration is not present in the $x$ direction where in fact there are no strong macroscopic electric fields.",
"The magnetic field lines show a chaotic behaviour: this condition makes it possible for particles moving along chaotic filed lines to access new acceleration regions of space [8], possibly encountering multiple reconnection sites and increasing their energy in steps.",
"Figure: Volume rendering of the electron velocity probability distribution f e (v x ,v y ,v z )f_e(v_x,v_y,v_z) at the position x/d i =10.32x/d_i=10.32, y/d i =7.5y/d_i=7.5, z/d i =5z/d_i=5, obtained averaging over particles contained within a box centred at that location and with side 0.5d i 0.5d_i."
],
[
"Conclusions and Future Directions",
"The analysis of reconnection outflows in the present case of a weak guide field (1/10 of the main reconnecting field) show the development of an instability in the lower hybrid regime.",
"In the present case, the instability has at least two components.",
"The first, is due to the presence of density gradients formed in the pileup region where the outflow meets the ambient plasma.",
"The second is the pre-existing velocity shears due to the differential velocity between the Harris plasma and the ambient plasma [22], [27], [43], [45].",
"The first instability leads to a Rayleigh-Taylor-type interchange instability in the lower hybrid range, while the latter leads to a kinking of the current layer.",
"Both instabilities feed the onset of a turbulent cascade with the presence of coherent structures and intermittency.",
"The outflows becomes host to secondary reconnection sites where the magnetic field topology becomes chaotic [31].",
"We investigate here the effect of these processes on the energization of particles.",
"The ions and the electrons are energized not only in the primary reconnection site but also, and in some cases predominantly, in the reconnection outflows.",
"Particle energization can be linked to the electric fields operating on the particles.",
"Electric fields do not heat particles in the statistical meaning of increasing their thermal spread, rather they coherently energise all particles, creating beams.",
"Beams originating from different regions interact and interpenetrate creating distribution functions with multiple populations.",
"The end result is that the second order moment of the distribution is increased but the process cannot be interpreted as heating proper but rather as the presence of very non Maxwellian distributions with multiple beams."
],
[
"Acknowledgments",
"The present work was supported by the Onderzoekfonds KU Leuven (Research Fund KU Leuven, GOA scheme and Space Weaves RUN project), by NASA's grant NNX08AO84G and by the US Air Force EOARD Award No.",
"FA9550-14-1-0375.",
"This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No.",
"DE-AC02-05CH11231.",
"Additional computing has been provided by NASA NAS and NCCS High Performance Computing, by the Flemish Supercomputing Center (VSC) and by PRACE Tier-0 allocations."
]
] | 1808.08612 |
[
[
"On the emergence of large and complex memory effects in nonequilibrium\n fluids"
],
[
"Abstract Control of cooling and heating processes is essential in many industrial and biological processes.",
"In fact, the time evolution of an observable quantity may differ according to the previous history of the system.",
"For example, a system that is being subject to cooling and then, at a given time $t_{w}$ for which the instantaneous temperature is $T(t_w)=T_{\\mathrm{st}}$, is suddenly put in contact with a temperature source at $T_{\\mathrm{st}}$ may continue cooling down temporarily or, on the contrary, undergo a temperature rebound.",
"According to current knowledge, there can be only one \"spurious\" and small peak/low.",
"However, our results prove that, under certain conditions, more than one extremum may appear.",
"Specifically, we have observed regions with two extrema and a critical point with three extrema.",
"We have also detected cases where extraordinarily large extrema are observed, as large as the order of magnitude of the stationary value of the variable of interest.",
"We show this by studying the thermal evolution of a low density set of macroscopic particles that do not preserve kinetic energy upon collision, i.e., a granular gas.",
"We describe the mechanism that signals in this system the emergence of these complex and large memory effects, and explain why similar observations can be expected in a variety of systems."
],
[
"Introduction",
"Experimental observations reveal that the response to an excitation of complex condensed matter systems may depend on the entire system's history, and not just on the instantaneous value of the state variables [1], [2], [3], [4], [5], [6], [7], [8].",
"This is usually called memory effect.",
"Memory effects signal the breakdown of the thermodynamic (or hydrodynamic or macroscopic, depending on the physical context) description.",
"Some typical memory effects include shape memory in polymers [4], aging and rejuvenation in spin glasses [9], active matter [10], and polymers [11], and the counterintuitive Mpemba effect [12], [13], [14].",
"One of the most relevant memory effects related to thermal processes was originally observed by Kovacs and collaborators [1] in a polymer system, which was subject to quenching to a low temperature $T_{1}$ from an equilibrium state at temperature $T_{0}>T_1$ .",
"After a long enough waiting time $t_{w}$ , but still relaxing towards equilibrium at $T_{1}$ , the temperature was suddenly increased back to an intermediate value $T_{\\mathrm {st}}$ , $T_{1}< T_{\\mathrm {st}} < T_{0}$ , such that the instantaneous value of the volume $\\mathcal {V}(t=t_w)$ equalled the equilibrium value $\\mathcal {V}_{\\mathrm {st}}$ corresponding to $T_{\\mathrm {st}}$ .",
"Subsequently, the volume $\\mathcal {V}(t)$ did not remain flat but followed a nonmonotonic evolution.",
"This nonmonotonic behavior, denominated later as Kovacs hump, consists in reaching one maximum before returning to its equilibrium value $\\mathcal {V}_{\\mathrm {st}}$ .",
"We have described above the typical cooling procedure, but also a heating protocol can be considered ($T_{0}<T_{\\mathrm {st}}<T_{1}$ ), for which $\\mathcal {V}(t)$ exhibits a single minimum at $t>t_w$ .",
"Quite recently, Kovacs-like memory effects have been thoroughly investigated in glassy systems [15], [16], granular fluids [17], [18], active matter [19], and disordered mechanical systems [20].",
"The memory effect is typically quite small: the maximum deviation of $\\mathcal {V}(t)$ from the stationary value $\\mathcal {V}_{\\mathrm {st}}$ is several orders of magnitude smaller than $\\mathcal {V}_{\\mathrm {st}}$ [1], [17], [18], [15], [19].",
"One of the main aims of our work is to show that the actual memory effects landscape is in general far more complex than expected.",
"First, we show that several extrema—instead of only one—may appear in a single heating/cooling protocol à la Kovacs, contrary to what has been previously observed [15], [16], [17], [18], [19], [20].",
"Second, very large memory humps, of the order of magnitude of the stationary value of the quantity of interest, can be observed.",
"To the best of our knowledge, both features have not yet been reported in the literature.",
"It must be noted that humps much larger than those predicted by linear response theory have recently been found in a nonlinear active matter model [19], but the relative deviation from the steady state is still of a few hundredths therein.",
"Our results are found in a granular fluid but the mechanism presented for these features is quite general.",
"Thus, giant and complex memory effects—not necessarily of the Kovacs-type—may be expected to appear in many natural and artificial systems.",
"These memory effects have obviously important implications in problems like, for instance, system stabilization."
],
[
"Description of the system and theoretical solution",
"We consider a collection of identical solid spheres at low particle density so that collisions are always instantaneous and binary but inelastic, i.e., energy is not conserved and we deal with a granular gas [21], [22].",
"In this case, particles have homogeneous mass density and we employ the rough hard sphere collisional model with constant coefficients of normal and tangential restitution, $\\alpha $ and $\\beta $ , respectively, which is quite realistic for a variety of materials at low particle density [23].",
"Let us discuss first why the granular gas of rough spheres is a good candidate for eventually finding complex memory effects.",
"Memory effects appear always in complex systems that consist of many structural units, for which a continuum description seems in principle appropriate.",
"Within this kind of description, the instantaneous value of the complete set of macroscopic variables completely characterizes the system's time evolution [24].",
"However, there are states that cannot be completely described only with the system macroscopic variables, and it is precisely for these states where a memory effect can be observed.",
"As a matter of fact, this kind of distinct states for which the macroscopic description fails are theoretically very well understood in the context of the kinetic theory of gases [24].",
"Furthermore, the granular gas of rough spheres can have extremely long relaxation times before it falls into a state where the macroscopic description is valid [25], [26], giving room to the emergence of eventual long lasting memory effects.",
"And, most importantly, in this kind of system there are always two intrinsic, independent, and potentially large temperature scales—the translational and rotational granular temperatures—with a highly nonlinear coupling.",
"All these facts open new spaces in the search of novel important features in complex memory effects, including eventually multiple extrema.",
"To keep things simple, we consider the granular gas to be in a spatially homogeneous state at all times.",
"The translational velocities are denoted by $\\mathbf {v}$ , while the angular (or rotational) velocities are denoted by ${}$ .",
"The system is thermalized by a stochastic but homogeneous volume force [27], [28] characterized by a noise intensity $\\chi _0^2$ (see ).",
"The kinetic description of our system starts from the corresponding Boltzmann–Fokker–Planck equation for the granular gas under this kind of forcing [26] (see ).",
"The exact solution to this kinetic equation can be formally expressed by means of an expansion around a Maxwellian distribution with variances $T_t$ (translational temperature) and $T_r$ (rotational temperature) in the translational and angular velocities, respectively.",
"The total granular temperature is given by $T=(T_t+T_r)/2$ , which is proportional to the mean kinetic (translational plus rotational) energy per particle.",
"By adopting a dimensionless time scale $\\tau $ , proportional to the number of collisions per particle (see ), the evolution equations for the temperatures can be written as $\\frac{\\partial \\ln \\theta (\\tau )}{\\partial \\tau }=\\frac{2}{3}\\left[\\mu _{20}(\\tau )-\\mu _{02}(\\tau )-\\gamma (\\tau )\\right],$ $ \\frac{\\partial \\ln \\gamma (\\tau )}{\\partial \\tau }=\\mu _{20}(\\tau )-\\gamma (\\tau ).$ Above, $\\theta \\equiv T_r/T_t$ is the temperature ratio and $\\gamma \\equiv \\left(\\frac{T_{\\mathrm {noise}}}{T}\\frac{1+\\theta }{2}\\right)^{\\frac{3}{2}}$ is a dimensionless measure of the noise intensity, where $T_{\\mathrm {noise}}\\equiv m(3\\chi _0^2/4\\sqrt{\\pi }n\\sigma ^2)^{\\frac{2}{3}}$ with $n$ being the particle density, and $m$ and $\\sigma $ being the mass and diameter of a sphere, respectively.",
"The reduced collisional moments $\\mu _{20}$ and $\\mu _{02}$ (see for more reference) are functionals of the whole velocity distribution and therefore the above system of equations is not closed.",
"In order to solve it, we use the first Sonine approximation, which refers to the first nontrivial truncation of the aforementioned exact infinite expansion [25].",
"For this, together with Eqs.",
"(REF ) and (REF ), we need to incorporate the evolution equations for the fourth-order cumulants and the initial values of $\\gamma $ , $\\theta $ , and these cumulants (see ).",
"We generate a common initial state for all the temperature evolution curves we subsequently analyze.",
"At an arbitrary time, which we choose to be the time origin $\\tau =0$ , and over an arbitrary previous microscopic state, we apply an instantaneous thermal pulse to the granular gas.",
"In this way, the rotational modes ($T_r$ ) of the granular gas are quenched, whereas the translational modes ($T_t$ ) are subject to a large heating.",
"As a result, most of the initial kinetic energy is in the translational modes, so that the total initial temperature is $T(0)=T_t(0)/2$ and the temperature ratio is $\\theta (0)=0$ .",
"Moreover, all the fourth-order cumulants vanish because the initial distribution that results from the heat pulse is a bi-variate ($T_r, T_t$ ) Maxwellian.",
"By this procedure, the system forgets all the previous thermal history of the system, assuring always the same nonequilibrium initial state.",
"From the initial state we have just characterized, the granular gas is left to cool freely, due to the intrinsically inelastic particle collisions [21], for a waiting time $\\tau _w$ .",
"At $\\tau =\\tau _{w}$ , we suddenly apply the stochastic force, with an intensity such that the corresponding steady temperature $T_{\\mathrm {st}}$ to be reached equals the instantaneous temperature value at the moment of turning the noise on, i.e., $T_{\\mathrm {st}} = T(\\tau _w)$ .",
"If $T(\\tau >\\tau _w)$ further departs from $T_{\\mathrm {st}}$ , then a Kovacs-like memory effect is observed.",
"What we call protocol is the thermal procedure that we have just described.",
"Depending on the waiting time $\\tau _w$ for turning the stochastic heating on, the system spans different classes of temperature evolution curves.",
"This is depicted and explained in Fig.",
"REF .",
"For the sake of simplicity, we investigate the two limiting cases in Fig.",
"REF ; i.e., $\\tau _w=0$ (heating protocol, HP) and $\\tau _w\\rightarrow \\infty $ (cooling protocol, CP).",
"In the CP, since the system is left cooling down for a long time, the system is already in the homogeneous cooling state (HCS) [21] at $\\tau _w$ .",
"In the HCS, the temperature $T(t)$ is the only relevant variable and decays in time following Haff's law [21], whereas the temperature ratio and the fourth-order cumulants are time independent, and their values depend only on the parameters $\\alpha $ and $\\beta $ [25].",
"Therefore, the conditions for this protocol at $\\tau _w$ are $\\gamma (\\tau _w)=\\gamma _{\\mathrm {st}}[(1+\\theta _{\\mathrm {st}})/(1+\\theta _{\\mathrm {HCS}})]^{\\frac{3}{2}}$ , $\\theta (\\tau _w)=\\theta _{\\mathrm {HCS}}$ , and the fourth-order cumulants at $\\tau =\\tau _w$ also equal their HCS values.",
"In the HP, the initial conditions for the Kovacs experiment are different.",
"Since we turn on the stochastic force right after the thermal pulse, the initial conditions are those of a bi-variate Maxwellian.",
"Therefore, we have that $\\gamma (\\tau _w)=\\gamma _{\\mathrm {st}}(1+\\theta _{\\mathrm {st}})^{\\frac{3}{2}}$ , $\\theta (\\tau _w)=0$ , and, in addition, all the cumulants vanish at $\\tau =\\tau _w$ .",
"Figure: Illustration of the protocols considered in this work.The granular gas is prepared in an initial state (τ=0\\tau =0) forwhich all the energy is concentrated in the translational degreesof freedom, as described in the text.",
"In a first stage,0<τ<τ w 0<\\tau <\\tau _{w}, the granular gas freely cools.",
"Then, at thewaiting time τ=τ w \\tau =\\tau _{w}, the noise intensity is suddenlyincreased from zero to a value such that the instantaneoustemperature T(τ w )T(\\tau _{w}) coincides with the corresponding steadytemperature T st T_{\\mathrm {st}}.",
"The curves shown for τ>τ w \\tau >\\tau _wcorrespond to the so-called normal Kovacs response.",
"Timeτ\\tau measures the average number of collisions per particle.Note that, in order to visualize the Kovacs effect, the relativedeviations of T(τ)T(\\tau ) from T st T_{\\mathrm {st}} in the response curves havebeen magnified by a factor r=5r=5 for all the protocols, except forthe transition one, for which r=100r=100.",
"All the curves correspondto normal and tangential restitution coefficients α=0.8\\alpha =0.8 andβ=0\\beta =0, respectively."
],
[
"Results and Discussion",
"Two data sets from molecular dynamics (MD) simulations (see ) of the granular gas for both the HP and the CP, together with their corresponding theoretical predictions, are represented in Fig.",
"REFA, which clearly shows the appearance of very large memory effects.",
"The temperature humps displayed here, of approximately $100\\%$ for the CP and $10\\%$ for the HP, are larger by at least two orders of magnitude than previously observed memory effects in athermal systems, which at most range from a few thousandths to a few hundredths of the stationary value of the relevant variable [17], [19].",
"The theoretical curves displayed in Fig.",
"REFA have been obtained by means of a bi-variate Maxwellian approximation, in which all the cumulants are assumed to be zero (see ).",
"Thus, the essential property driving the giant memory effect here is the existence of two independent temperature scales, translational and rotational, i.e., the breakdown of equipartition as given by the fact that $\\theta \\ne 1$ .",
"This is further illustrated in Fig.",
"REFB, which shows $\\theta (\\tau )$ for the same cases as in Fig.",
"REFA.",
"Again, the agreement between theory and simulation is excellent, even at the level of the two contributions to the total temperature.",
"Note that the relaxation time in the CP case is much longer than in the HP one.",
"Let us denote the earliest minimum and maximum in the temperature evolution as $T_{m}$ and $T_{M}$ , respectively.",
"We also define $\\mathcal {H}_{m}\\equiv T_{m}/T_{\\mathrm {st}}-1<0$ , $\\mathcal {H}_{M}\\equiv T_{M}/T_{\\mathrm {st}}-1>0$ , accordingly.",
"In Fig.",
"REF we present contour plots highlighting the regions with large $|\\mathcal {H}_m|$ (HP normal response, CP anomalous response) and $\\mathcal {H}_M$ (HP anomalous response, CP normal response).",
"We also plot the transition line $\\mathcal {H}_M=|\\mathcal {H}_m|$ from normal to anomalous response.",
"Huge Kovacs humps appear, especially in the normal region for the CP, in which the size of reported humps can be as large as $100\\%$ , relative to the steady temperature.",
"In order to characterize and quantify complexity in the thermal response we define the parameter $\\mathcal {S}$ , $\\mathcal {S} =\\mathrm {sgn} ({\\mathcal {H}_{1}})\\frac{\\min (|\\mathcal {H}_{m}|,\\mathcal {H}_{M})}{\\max (|\\mathcal {H}_{m}|,\\mathcal {H}_{M})},$ where $\\mathcal {H}_{1}$ (equal to either $\\mathcal {H}_{m}$ or $\\mathcal {H}_{M}$ ) is the magnitude of the earliest extremum.",
"Note that $\\mathcal {S}=0$ if there is only one extremum.",
"Thus, $\\mathcal {S}\\ne 0$ is the signature of the emergence of more complex response, i.e., with more than one extremum, in the normal-to-anomalous transition.",
"In the transition region, $|\\mathcal {S}|$ attains its maximum value, $|\\mathcal {S}|=1$ , when both extrema are of the same size and neither dominates.",
"The sign of $\\mathcal {S}\\in [-1,1]$ is equal to that of the earliest extremum, providing further information on the detailed structure of the response.",
"Figure REF represents $\\mathcal {S}$ as a function of the coefficients of restitution $\\alpha , \\beta $ , by solving the system of Eqs.",
"(REF ) and (REF ).",
"Panels A, C, and D correspond to the HP, whereas panels B and E correspond to the CP.",
"We have highlighted in blue (red) regions with $\\mathcal {S}<0$ ($\\mathcal {S}>0$ ), whereas all points with “simple” memory behavior, i.e., $\\mathcal {S}=0$ , remain white.",
"The complex regions are thin but still occupy noticeable sections of the parameter space, especially taking into account that they fall into ranges of experimental values of $\\alpha $ and $\\beta $ commonly present in a variety of materials [23].",
"In Fig.",
"REFA (HP), we clearly observe two zones rich in complex memory effects.",
"In panel C, the first complex zone is zoomed in.",
"This region is attached to the smooth limit, $\\beta \\sim -1$ , and only displays $\\mathcal {S}>0$ for high inelasticities, up to $\\alpha =1/\\sqrt{2}$ .",
"In panel D, the second complex region is zoomed in.",
"Within this region, which is close to the quasielastic limit $\\alpha \\sim 1$ , the system displays both $\\mathcal {S}> 0$ and $\\mathcal {S}< 0$ behavior.",
"In Fig.",
"REFB (CP), only one complex Kovacs region next to the quasielastic limit, inside which $\\mathcal {S}> 0$ , has been identified.",
"Panel E shows a close-up thereof.",
"It is important to mention that we have found that all the details of the complex regions emerge in the theoretical solution only when the cumulants are taken into account.",
"This indicates that the temperatures $T_t$ and $T_r$ do not explain in full detail by themselves the complexity of memory effects found in the rough granular gas.",
"Let us also point out that we have found for the HP a critical narrow region with a discontinuous transition from $\\mathcal {S}> 0$ to $\\mathcal {S}<0$ , which is signaled in panel D in Fig.",
"REF and represented in time evolution curves in Fig.",
"REF .",
"In this critical region, the system displays several different mechanisms for the transition from complex to simple—only one extremum—behavior.",
"The latter can be either the normal behavior of molecular systems [1], [9], [15] (also present in nonequilibrium systems) or the anomalous behavior exclusive of nonequilibrium systems [17], [19].",
"This is appropriately tagged in panels A and B of Fig.",
"REF , in which we have labeled the corresponding normal and anomalous regions.",
"In the narrow critical region, $\\mathcal {S}$ discontinuously jumps from (small) negative to positive values and three consecutive temperature extrema appear before stabilization in the stationary value is attained.",
"Otherwise, $\\mathcal {S}$ has a well-defined sign and the transition from complex to simple is continuous.",
"Figure: Contour plots of large extrema in the Kovacs response.Large minima (|ℋ m ||\\mathcal {H}_{m}|) are represented by bluishcontours and large maxima (ℋ M \\mathcal {H}_{M}) by reddishcontours.",
"Dashed lines indicate the ℋ M =|ℋ m |\\mathcal {H}_M=|\\mathcal {H}_m|transition curves, for which the predominant extremum changes sign,from maximum to minimum and vice versa.",
"Above and below these curveswe find ℋ M >|ℋ m |\\mathcal {H}_M> |\\mathcal {H}_m|(ℋ M <|ℋ m |\\mathcal {H}_M< |\\mathcal {H}_m|) and ℋ M <ℋ m \\mathcal {H}_M<\\mathcal {H}_m(ℋ M >|ℋ m |\\mathcal {H}_M>|\\mathcal {H}_m|) behaviors, respectively, in the HP(CP).",
"(A) Heating protocol (HP).",
"(B) Cooling protocol(CP).Figure: Kovacs complexity (𝒮\\mathcal {S}) phase diagrams.Density plot of 𝒮\\mathcal {S} vs. the α,β\\alpha , \\beta completespace parameter: (A) for the HP; (B) for the CP.",
"(C, D) Insets of complex memory regions next to the smoothand the quasielastic limits, respectively, for the HP.",
"(E)Inset of the complex memory region next to the quasielastic limitfor the CP.",
"In the HP, as seen in panels A and D,three different types of transition exist: 𝒮<0\\mathcal {S}<0(bluish, β<-0.65\\beta <-0.65), 𝒮>0\\mathcal {S}>0 (reddish,β>-0.65\\beta > -0.65), and the intermediate mechanism(𝒮≈0\\mathcal {S}\\approx 0) for β≈-0.65\\beta \\approx -0.65, as depictedbelow in Fig. B.",
"However, in the CP, seepanels B and E, only a 𝒮>0\\mathcal {S}>0-typetransition has been observed.Figure: Mechanisms for the transition from normal to anomalous inthe HP.",
"There are three of these mechanisms, which are shownhere by DSMC simulations (symbols) and our theoretical approach(lines).",
"Specifically, through (A) 𝒮>0\\mathcal {S}>0 atβ=0\\beta = 0, (B) 𝒮≈0\\mathcal {S} \\approx 0 (actually a tripleKovacs hump transition mechanism; see inset, where the linejoining the simulation points is a guide to the eye) atβ=-0.65\\beta = -0.65, and (C) 𝒮<0\\mathcal {S} < 0 atβ=-0.8\\beta = -0.8.",
"In order to assist in locating these transitionsin the parameter plane (α,β)(\\alpha ,\\beta ), their positions have beenannotated in Fig.",
"D.Figure REF displays the evolution curves of the temperature for the three different Kovacs transitions that we have found, in all cases depicted here for the HP: the $\\mathcal {S}>0$ transition in panel A, the $\\mathcal {S} \\approx 0$ transition in panel B (in its inset we show the three consecutive humps), and finally the $\\mathcal {S} < 0$ case in panel C. All theoretical curves are compared against the numerical solution of the kinetic equation, obtained by means of the direct simulation Monte Carlo (DSMC) method (see ).",
"The agreement is in general excellent, which once more shows the accuracy of our theoretical approach.",
"Although the size of the humps in the transition regions appear smaller than those in Fig.",
"REFA with simple memory behavior, yet they are of the same order of magnitude as those previously reported in the smooth granular gas [17]."
],
[
"Conclusions",
"Our work puts forward a general mechanism for the emergence of significantly large memory effects.",
"Enormous humps can be expected if the time evolution of the system under scrutiny is controlled by at least two independent and comparable in magnitude physical variables (here the translational and rotational temperatures) but with only one (here the total temperature) being relevant for the macroscopic or hydrodynamic description.",
"In addition, complex Kovacs response, with more than one extremum, can be expected if the time evolution of the system depends on several additional relevant variables.",
"Here, these additional variables are the fourth-order cumulants, whose sometimes nonmonotonic relaxation [25] probably enhances memory effect complexity.",
"So far, and despite the large number of previous works on analogous phenomena, only one extremum in the Kovacs response has been reported.",
"In thermal systems, in which the usual fluctuation-dissipation theorem holds and the stationary (equilibrium) distribution has the canonical shape, this is consistent with linear response results that predict normal behavior with only one maximum [16].",
"In athermal systems, the Kovacs response also includes anomalous behavior, but once more only one extremum has been observed [17], [19].",
"Therefore, an interesting prospect is elucidating whether or not the nonlinear theoretical framework developed in Ref.",
"[19] allows for complex response with more than one extremum.",
"Memory effects of the size and complexity we have observed here can potentially be present in other athermal or molecular systems.",
"Several variables of comparable magnitude must be coupled in their time evolution in nonlinear form, even if only a subset thereof is relevant in the macroscopic description.",
"This may be relevant, for instance, in active matter systems, where nonlinear effects are important in general [19], [29].",
"We think our results are also especially significant for future experimental work, since we expect these large memory effects to be measurable in granular dynamics experiments; a thermally homogeneous system may be achieved by means of homogeneous turbulent air fluidization [28]."
],
[
"Acknowledgements",
"The authors thank Prof. J. S. Urbach for fruitful discussions.",
"This work has been supported by the Spanish Agencia Estatal de Investigación Grants (partially financed by the ERDF) No.",
"MTM2017-84446-C2-2-R, MTM2014-56948-C2-2-P (A.L.",
"), and No.",
"FIS2016-76359-P (F.V.R.",
"and A.S.), and also by Universidad de Sevilla's VI Plan Propio de Investigación Grant PP2018/494 (A.P.).",
"Use of computing facilities from Extremadura Research Centre for Advanced Technologies (CETA-CIEMAT), funded by the ERDF is also acknowledged."
],
[
"Theory",
"The stochastic force ($\\mathbf {F}^{\\mathrm {wn}}$ ) has the form of a white noise: $\\langle {\\bf F}_i^{\\mathrm {wn}} (t) \\rangle = {\\bf 0}$ , $\\langle {\\bf F}_i ^{\\mathrm {wn}} (t) {\\bf F}_j ^{\\mathrm {wn}} (t^{\\prime }) \\rangle = \\mathsf {I}\\,m^2 \\chi _0^2\\delta _{ij}\\delta (t-t^{\\prime })$ , where indexes $i,j$ refer to particles, $\\mathsf {I}$ is the $3\\times 3$ unit matrix, and $\\chi _0^2$ is the white noise intensity.",
"In homogeneous states, the Boltzmann–Fokker–Planck equation characterizing the evolution of a granular gas submitted to the stochastic external force $\\mathbf {F}^{\\mathrm {wn}}$ is written as [25] $\\left(\\partial _t -\\frac{\\chi _0^2}{2}\\nabla _{\\mathbf {v}}^2\\right) f(\\mathbf {v} , ;t)={J[\\mathbf {v}, |f(t)]}.$ Above, $f(\\mathbf {v},;t)$ is the velocity distribution function ($\\mathbf {v}$ and $$ being the translational and angular velocities, respectively) and $J[\\mathbf {v}, |f]$ is the collision integral in the (inelastic) Boltzmann equation for rough spheres, which accounts for the collision rules [25] $ \\widehat{}\\cdot \\mathbf {u}^{\\prime }=-\\alpha \\,\\widehat{{}}\\cdot \\mathbf {u},\\quad \\widehat{{}}\\times \\mathbf {u}^{\\prime }=-\\beta \\,\\widehat{{}}\\times \\mathbf {u}.$ Here, the primes denote postcollisional values, $\\widehat{{}}$ is the unit collision vector joining the centers of the two colliding spheres (from the center of particle 1 to the center of particle 2) and $\\mathbf {u}=\\mathbf {v}_1-\\mathbf {v}_2-{\\frac{\\sigma }{2}}\\widehat{{}}\\times ({}_1+{}_2)$ is the relative velocity of the spheres at their contact point.",
"The coefficient of normal restitution $\\alpha $ takes values between 0 (completely inelastic collision) and 1 (completely elastic collision), while the coefficient of tangential restitution $\\beta $ takes values between $-1$ (completely smooth collision, unchanged angular velocities) and 1 (completely rough collision) [24].",
"Given any one-particle function $A(\\mathbf {v},)$ , its average is defined as $\\langle A(t)\\rangle =n^{-1}\\int d \\mathbf {v}\\int d \\,A( \\mathbf {v}, )f(\\mathbf {v}, ;t)$ , where the number density is given by $n=\\int d \\mathbf {v}\\int d \\, f( \\mathbf {v}, ;t)$ .",
"The basic physical properties are the translational ($T_t$ ), rotational ($T_r$ ), and total ($T$ ) granular temperatures, i.e., ${T_t={\\frac{m}{3}}\\langle v^2\\rangle ,\\quad T_r={\\frac{I}{3}}\\langle \\omega ^2\\rangle ,} \\quad { T=\\frac{T_t+T_r}{2}=T_t\\frac{1+\\theta }{2}},$ where $I$ is the moment of inertia.",
"We have introduced the temperature ratio $\\theta \\equiv T_r/ T_t$ , which is relevant for the analysis that follows and whose steady-state value is independent of the driving amplitude $\\chi _0^2$ .",
"The evolution equations for $T_t$ , $T_r$ , and $T$ are $\\partial _t T_t-{m\\chi _0^2}=-\\xi _t T_t,\\quad {\\partial _t T_r=-\\xi _r T_r},$ $\\partial _t T-\\frac{m\\chi _0^2}{2}=-\\zeta T.$ The equations for $T_t$ and $T_r$ have been obtained by multiplying both sides of Eq.",
"(REF ) by the translational and rotational kinetic energies, respectively, and integrating over all particle velocity values.",
"The parameters $\\xi _t$ and $\\xi _r$ are $\\xi _t=-{\\frac{m}{3n T_t}}\\int d\\mathbf {v}\\int d {}\\,v^2{J[\\mathbf {v}, {}| f]},$ $\\xi _r=-{\\frac{I}{3n T_r}}\\int d\\mathbf {v}\\int d {}\\,\\omega ^2{J[\\mathbf {v},{}| f]},$ respectively.",
"In general, neither $\\xi _t$ nor $\\xi _r$ does have a definite sign, whereas the cooling rate, $ \\zeta =\\frac{\\xi _t T_t+\\xi _rT_r}{2T}=\\frac{\\xi _t+\\xi _r{\\theta }}{1+\\theta },$ is always positive because energy is dissipated in collisions.",
"To proceed further, it is convenient to go to dimensionless variables.",
"Time is measured in a scale $\\tau $ , $\\tau =\\frac{1}{2}\\int _0^t dt^{\\prime }\\,\\nu (t^{\\prime }),\\quad \\nu (t)=4n\\sigma ^2 \\sqrt{\\pi T_t(t)/m},$ which is roughly the accumulated number of collisions per particle, because $\\nu (t)$ is the collision frequency.",
"Dimensionless velocities are introduced as $\\mathbf {c}(t)\\equiv \\frac{\\mathbf {v}}{\\sqrt{2 T_t(t)/m}}, \\quad \\mathbf {w}(t)\\equiv \\frac{{}}{\\sqrt{2 T_r(t)/I}},$ a reduced velocity distribution function as $\\phi (\\mathbf {c},\\mathbf {w};\\tau )\\equiv \\frac{1}{n}\\left[\\frac{4 T_t(t)T_r(t)}{m I}\\right]^{3/2}f(\\mathbf {v}, {};t),$ and the dimensionless collision kernel as $\\mathcal {J}[\\mathbf {c},\\mathbf {w}|\\phi (\\tau )]=\\frac{2}{n\\nu (t)}\\left[\\frac{4 T_t(t) T_r(t)}{m I}\\right]^{3/2} J[\\mathbf {v}, {}|f(t)].$ In dimensionless variables, the evolution equations for the temperatures can be written as Eqs.",
"(REF ) and (REF ) in the main text.",
"Therein, there appear the reduced collisional moments $\\mu _{20}\\equiv \\mu _{20}^{(0)}$ and $\\mu _{02}\\equiv \\mu _{02}^{(0)}$ , where $\\mu _{pq}^{(r)}(\\tau )\\equiv -\\int d\\mathbf {c}\\int d\\mathbf {w}\\,c^pw^q(\\mathbf {c}\\cdot \\mathbf {w})^r{\\mathcal {J}[\\mathbf {c},\\mathbf {w}|\\phi (\\tau )]}.$ Note that, aside from the nondimensionalizing factors, the production rates $\\xi _t$ and $\\xi _r$ are basically identical to $\\mu _{20}$ and $\\mu _{02}$ , respectively.",
"These are functionals of the whole distribution function and thus the evolution equations for the temperatures are not closed.",
"In order to close the dynamical equations, a formally exact expansion in orthogonal polynomials can be performed [25].",
"For isotropic states, we can expand the velocity distribution around the Maxwellian $\\phi _M(c,w) = \\pi ^{-3}e^{-c^2-w^2}$ , $\\phi (\\mathbf {c},\\mathbf {w};\\tau )=\\phi _M(c,w)\\sum _{j=0}^\\infty \\sum _{k=0}^\\infty \\sum _{\\ell =0}^\\infty a_{jk}^{(\\ell )}(\\tau )\\Psi _{jk}^{(\\ell )}(\\mathbf {c},\\mathbf {w}),$ where $\\Psi _{jk}^{(\\ell )}(\\mathbf {c},\\mathbf {w})$ are certain products of Laguerre and Legendre polynomials.",
"By normalization, $a_{00}^{(0)}=1$ , $a_{10}^{(0)}=a_{01}^{(0)}=0$ , and the lowest nontrivial coefficients are those associated with moments of degree four, namely $a_{20}^{(0)}=\\frac{4}{15}\\langle c^4\\rangle -1,\\quad a_{02}^{(0)}=\\frac{4}{15}\\langle w^4\\rangle -1,$ $a_{11}^{(0)}=\\frac{4}{9}\\langle c^2w^2\\rangle -1,\\quad a_{00}^{(1)}=\\frac{8}{15}\\left[\\langle (\\mathbf {c}\\cdot \\mathbf {w})^2\\rangle -\\frac{1}{3}\\langle c^2w^2\\rangle \\right] ,$ which we call the fourth-order cumulants henceforth.",
"Maxwellian approximation.- The simplest description is obtained by substituting the Maxwellian velocity distribution into the collision integrals (REF ).",
"Equivalently, one may consider that all the nontrivial cumulant vanish in this approach, which yields $ \\mu _{20,M}&=1-\\alpha ^2+\\frac{\\kappa (1+\\beta )}{(1+\\kappa )^2}\\left[2+\\kappa (1-\\beta )-\\theta (1+\\beta )\\right], \\\\\\mu _{02,M}&=\\frac{\\kappa (1+\\beta )}{(1+\\kappa )^2}\\left[2+\\kappa ^{-1}(1-\\beta )-\\theta ^{-1}(1+\\beta )\\right].",
"$ where $\\kappa \\equiv 4I/m\\sigma ^2$ is the dimensionless moment of inertia.",
"Insertion of Eq.",
"(REF ) into the evolution equations (REF ) and (REF ) in the main text gives rise to the Maxwellian approximation.",
"First Sonine approximation.- A more elaborate approximation can be done by incorporating the lowest order cumulants, which we defined in Eqs.",
"(REF ) and (REF ), as the first corrections to the Maxwellian.",
"A closed set of six coupled differential equations can be obtained for $\\theta (\\tau )$ , $\\gamma (\\tau )$ , $a_{20}^{(0)}(\\tau )$ , $a_{02}^{(0)}(\\tau )$ , $a_{11}^{(0)}(\\tau )$ , and $a_{00}^{(1)}(\\tau )$ .",
"To do so, explicit—yet not exact—expressions for the collision integrals $\\mu _{pq}^{(r)}$ with $p+q+2r=2$ and 4 are derived in terms of $\\theta $ and those lowest order cumulants.",
"These rather involved expressions can be found in the Supplemental Material of Ref.",
"[25], and are thus omitted here.",
"The resulting set of six differential equations can be numerically solved with appropriate initial conditions for each physical situation, as discussed in the main text.",
"In this way, we obtain the time evolution of the temperatures in the so-called first Sonine approximation, to which we refer throughout this work."
],
[
"Computer simulations",
"We use in this work data sets obtained from computer simulations from two independent and different methods: direct simulation Monte Carlo (DSMC) method, which obtains an exact numerical solution of the relevant kinetic equation [in our case Eq.",
"(REF )] and molecular dynamics (MD) simulation, which solves particles trajectories.",
"A detailed description of the DSMC method may be found elsewhere [30].",
"In our DSMC simulations, and in order to reduce statistical noise in the temperature time evolution curves, we have used an average of 100 statistical replicas of a system with $2\\times 10^6$ particles.",
"In the MD case, we have simulated 1000 inelastic hard spheres at a density $n\\sigma ^{3}=0.01$ and averaged over 500 trajectories."
]
] | 1808.08401 |
[
[
"Flux Transport Dynamo: From Modelling Irregularities to Making\n Predictions"
],
[
"Abstract The flux transport dynamo, in which the poloidal magnetic field is generated by the Babcock--Leighton mechanism and the meridional circulation plays a crucial role, has emerged as an attractive model for the solar cycle.",
"Based on theoretical calculations done with this model, we argue that the fluctuations in the Babcock--Leighton mechanism and the fluctuations in the meridional circulation are the most likely causes of the irregularities of the solar cycle.",
"With our increased theoretical understanding of how these irregularities arise, it can be possible to predict a future solar cycle by feeding the appropriate observational data in a theoretical dynamo model."
],
[
"Introduction",
"The flux transport dynamo model, which started being developed about a quarter century ago [50], [19], [22], has emerged as an attractive theoretical model for the solar cycle.",
"There are several modern reviews [11], [12], [4], [34] surveying the current status of the field.",
"The present paper is not a comprehensive review, but is based on a talk in a Workshop at the International Space Science Institute (ISSI) highlighting the works done by the author and his coworkers.",
"Readers are assumed to be familiar with the phenomenology of the solar cycle and the basic concepts of MHD (such as flux freezing and magnetic buoyancy).",
"Readers not having this background are advised to look at the earlier reviews by the author [11], [12], which were written for wider readership.",
"The initial effort in this field of flux transport dynamo was directed towards developing periodic models to explain the various periodic aspects of the solar cycle.",
"Once sufficiently sophisticated periodic models were available, the next question was whether these theoretical models can be used to understand how the irregularities of the solar cycle arise.",
"There is also a related question: if we understand what causes the irregularities of the cycle, then will that enable us to predict future cycles?",
"We discuss the basic periodic model of the flux transport dynamo in the next Section.",
"Then § 3 discusses the possible causes of the irregularities of the solar cycle.",
"Afterwards in § 4 we address the question whether we are now in a position to predict future cycles.",
"Finally, in § 5 we summarize the limitations of the 2D kinematic dynamo models and the recent efforts of going beyond such simple models."
],
[
"The Basic Periodic Model",
"One completely non-controversial aspect of solar dynamo models is the generation of the toroidal field from the poloidal field by differential rotation.",
"Since differential rotation has now been mapped by helioseismology, this process can now be included in theoretical dynamo models quite realistically.",
"The toroidal field is primarily produced in the tachocline at the bottom of the convection zone and rises from there due to magnetic buoyancy to create the sunspots.",
"Although some authors have argued that the near-surface shear layer discovered by helioseismology can also be important for the generation of the toroidal field [2], the general view is that magnetic buoyancy would limit the growth of magnetic field in this region of strong super-adiabatic gradient.",
"To this generally accepted view that the toroidal field is primarily produced in the tachocline, the flux transport dynamo model adds the following assumptions.",
"The generation of the poloidal field from the toroidal field takes place due to the Babcock–Leighton mechanism.",
"The meridional circulation of the Sun plays a crucial role in the dynamo process.",
"We now comment on these two assumptions.",
"Bipolar sunspots on the solar surface appear with a tilt statistically increasing with latitude, in accordance with the so-called Joy's law.",
"This tilt is produced by the Coriolis force acting on the rising flux tubes [21].",
"[1] and [36] suggested that the poloidal field of the Sun is produced from the decay of such tilted bipolar sunspot pairs.",
"There is now enough evidence from observations of the solar surface that the poloidal field does get built up in this way.",
"The meridional circulation is observed to be poleward at the solar surface and advects the poloidal field generated there, in conformity with observational data of surface magnetic fields.",
"The material which is advected to the poles has to flow back equatorward through deeper layers within the solar convection zone.",
"Since this circulation is driven by the turbulent stresses in the convection zone, we expect the meridional circulation not to penetrate much below the bottom of the convection zone, although a slight penetration helps in explaining several aspects of observational data [42], [3].",
"The early models of the flux transport dynamo assumed the return flow of the meridional circulation to take place at the bottom of the convection zone, where the toroidal field generated by the differential rotation is advected equatorward with this flow, giving a natural explanation of the butterfly diagram representing the equatorward shift of the sunspot belt [19].",
"Such dynamo models have been remarkably successful in explaining many aspects of the observational data [8].",
"Figure: A complicated meridional circulation used by in a dynamocalculation—red corresponding to streamlines of clockwise circulation and blueto anti-clockwise circulation.",
"Note that the flow near the bottom at low latitudesis equatorward.",
"The butterfly diagram obtained with this circulation is solar-like(sunspot activity drifting to lower latitudes with time).While we still do not have unambiguous measurements of the return flow of the meridional circulation, some groups claim to have found evidence for the return flow well above the bottom of the convection zone [24], [54], [45].",
"However, [44] argue that the available helioseismology data still cannot rule out a one-cell meridional circulation spanning the whole of the convection zone in each hemisphere.",
"[27] showed that, even with a meridional circulation much more complicated than the one-cell pattern assumed in the earlier flux transport dynamo papers, it is still possible to match the relevant observational data as long as there is an equatorward flow at the bottom of the convection zone (see Figure 1)."
],
[
"The Origin of the Irregularities of the Solar Cycle",
"The earliest attempts of explaining irregularities of the solar cycle were by regarding them as nonlinear chaos arising out of the nonlinearities of the dynamo equations [51].",
"[5] argued that the Gnevyshev-Ohl rule in solar cycles (i.e.",
"the tendency of alternate cycles to lie above and below the running mean of cycle amplitudes) arises out of period doubling due to nonlinearities.",
"However, the simplest kinds of nonlinearities expected in dynamo equations tend to make the cycles more stable rather than producing irregularities and it has been suggested that stochastic fluctuations are more likely to be the primary reason behind producing irregularities [9].",
"The Babcock–Leighton mechanism for the generation of the poloidal field depends on the tilts of bipolar sunspots.",
"While the average tilt is given by Joy's law, we see considerable scatter around this average tilt, presumably caused by the action of turbulence in the convection zone on the rising flux tubes [37].",
"This scatter in the tilt angles is expected to introduce fluctuations in the Babcock–Leighton mechanism [14].",
"By including this fluctuation in the dynamo models, it is possible to explain many aspects of the irregularities of the cycles including the grand minima [17].",
"One other source of irregularities is the fluctuations in the meridional circulation.",
"A faster meridional circulation will make the solar cycles shorter and vice versa.",
"While we have actual data of meridional circulation variations for not more than a couple of decades, we have data for durations of solar cycles for more than a century, providing indications that the meridional circulation had fluctuations in the past with correlation times of the order of 30–40 years [32].",
"When the meridional circulation is slow and the cycles longer, diffusion has more time to act on the magnetic fields, making the cycles weaker.",
"On such ground, we expect longer cycles to be weaker and shorter cycles to be stronger, leading to what is called the Waldmeier effect [32].",
"Also, when the meridional circulation is sufficiently weak, theoretical dynamo models show that even grand minima can be induced [31].",
"To get these results, the correlation time of the meridional circulation fluctuations was taken to be considerably longer than the cycle period.",
"If the correlation time is taken too short, then one may not get these results [40].",
"We also emphasize that the effect of diffusion in making longer cycles weaker is vital for getting these results.",
"We need to take the value of diffusivity sufficiently high such that the diffusion time scale is shorter than or of the order of the cycle period.",
"This is not the case in the model of [20] in which diffusivity is very low.",
"A longer cycle in such a low-diffusivity model tends to be stronger because differential rotation has time to generate more toroidal field during a cycle, giving the opposite of the Waldmeier effect.",
"Differences between high- and low-diffusivity dynamos were studied by [53].",
"Clearly the high-diffusivity model yields results more in conformity with observational data.",
"Figure: According to the calculations of , the poloidal fieldstrength (γ\\gamma is the value of the poloidal field compared to its averagevalue) and the amplitude of the meridional circulation at the surface have tolie in the shaded region at the beginning of a cycle in order to force the dynamointo a grand minimum.They estimated the probability of this to be about 1.3%, corresponding toabout 13 grand minima in 11,000 years.By analyzing the contents of C-14 in old tree trunks and Be-10 in polar ice cores, it has now been possible to reconstruct the history of solar activity over a few millenia [48].",
"It has been estimated that there have been about 27 grand minima in the last 11,000 years [49].",
"Since grand minima can be caused both by fluctuations in the Babcock-Leighton mechanism and fluctuations in the meridional circulation, a full theoretical model of grand minima should combine both types of fluctuations.",
"If, at the beginning of a cycle, the poloidal field is too weak due to the fluctuations in the Babcock–Leighton mechanism or the meridional circulation is too weak, then the Sun may be forced into a grand minimum.",
"Assuming a Gaussian distribution for fluctuations in both the Babcock–Leighton mechanism and the meridional circulation, [18] developed a comprehensive theory of grand minima that agreed remarkably well with the statistical data of grand minima (see Figure 2 and its caption).",
"However, if there are no sunspots during grand minima, then the Babcock–Leighton mechanism which depends on sunspots may not be operational and how the Sun comes out of the grand minima is still rather poorly understood [33], [28].",
"While discussing irregularities of the solar cycle, it may be mentioned that these irregularities are correlated reasonably well in the two hemispheres of the Sun.",
"Strong cycles are usually strong in both the hemispheres and weak cycles are weak in both.",
"This requires a coupling between the two hemispheres, implying that the turbulent diffusion time over the convection zone could not be more than a few years [6], [23]."
],
[
"Predicting solar cycles",
"The first attempts of predicting solar cycles were based on using observational precursors of solar cycles.",
"There is considerable evidence that the polar field at the end of a solar cycle is correlated with the next cycle.",
"Since the polar field at the end of cycle 23 was rather weak, several authors [47], [46] predicted that the next cycle 24 would be weak.",
"The sunspot minimum between the cycles 23 and 24 (around 2005–2008) was the first sunspot minimum when sufficiently sophisticated models of the flux transport dynamo were available.",
"Whether these models could be used to predict the next cycle became an important question.",
"When a kinematic mean field dynamo code is run without introducing any fluctuations, one finds that the code settles down to a periodic solution if the various dynamo parameters are in the correct range.",
"In order to model actual solar cycles, one has to feed some observational data to the theoretical model in an appropriate manner and then run the code for a future cycle to generate a prediction.",
"The crucial issue here is to figure out what kind of observational data to feed into the theoretical model and how.",
"An understanding of what causes the irregularities of the solar cycle is of utmost interest in deciding this.",
"An attempt by [20] produced the prediction that the cycle 24 would be very strong, in contradiction to what was predicted on the basis of the weak polar field at the end of the cycle 23.",
"Figure: The sunspot number in the last few years.",
"The upper star indicatesthe predicted amplitude of cycle 24 according to , while the lower starindicates the predicted amplitude according to .",
"The circle on thehorizontal axis indicates the time when these predictions were made.Assuming that the fluctuation in the Babcock–Leighton mechanism is the main cause of irregularities in the solar cycle, [14] devised a methodology of feeding observational data of the polar magnetic field into the theoretical model to account for the random kick received by the dynamo due to fluctuations in the Babcock–Leighton mechanism.",
"The dynamo model of [14] predicted that the cycle 24 would be weak, in conformity with the weakness of the polar field at the end of cycle 23.",
"[30] explained the physical basis of what causes the correlation between the polar field at the end of a cycle and the strength of the next cycle.",
"Suppose the fluctuations in the Babcock–Leighton mechanism produced a poloidal field stronger than the usual.",
"This strong poloidal field will be advected to the poles to produce a strong polar field at the end of the cycle and, if the turbulent diffusion time across the convection zone is not more than a few years, this poloidal field will also diffuse to the bottom of the convection zone to provide a strong seed for the next cycle, making the next cycle strong.",
"On the other hand, if the poloidal field produced in a cycle is weaker than the average, then we shall get a weak polar field at the end of the cycle and a weak subsequent cycle.",
"This will give rise to a correlation between the polar field at the end of a cycle and the strength of the next cycle.",
"If the diffusion is assumed to be weak—as in the model of [20]—then different regions of the convection zone may not be able to communicate through diffusion in a few years and we shall not get this correlation.",
"The prediction of [14] that the cycle 24 would be weak was a robust prediction in their model because the polar field at the end of a cycle is correlated to the next cycle in their model and they had fed the data of the weak polar field at the end of cycle 23 into their theoretical model in order to generate their prediction.",
"As can be seen in Figure 3, the actual amplitude of cycle 24 turned out to be very close to what was predicted by [14], making this to be the first successful prediction of a solar cycle from a theoretical dynamo model in the history of our subject.",
"As we have pointed out in the previous Section, the fluctuations of the meridional circulation also can cause irregularities in the solar cycle.",
"This was not realized when the various predictions for cycle 24 were made during 2005–2007.",
"It is observationally found that there is a correlation between the decay rate of a cycle and the strength of the next cycle [26].",
"Now, a faster meridional circulation, which would make a cycle shorter, surely will make the decay rate faster and also the cycle stronger, as pointed out already (a slower meridional circulation would do the opposite).",
"If the effect of the fluctuating meridional circulation on the decay rate is immediate, but on the cycle strength is delayed by a few years, then we can explain the observed correlation.",
"This is confirmed by theoretical dynamo calculations [26].",
"This shows that it may be possible to use the decay rate at the end of a cycle to predict the effect of the fluctuating meridional circulation on the next cycle.",
"This issue needs to be looked at carefully."
],
[
"Conclusion",
"We have pointed out that over the years we have acquired an understanding of how the irregularities of the solar cycle arise and that this understanding helps us in predicting future solar activity.",
"Our point of view is that the fluctuations in the Babcock–Leighton mechanism and the fluctuations in the meridional circulation are the two primary sources of irregularities in the solar cycles.",
"These fluctuations have to be modelled realistically and fed into a theoretical dynamo model to generate predictions.",
"It may be noted that we now have a huge amount of data on the magnetic activity of solar-like stars [13].",
"Some solar-like stars display grand minima and we have evidence for the Waldmeier effect in some of them—see the concluding paragraph of [35].",
"This suggests that dynamos similar to the solar dynamo may be operational in the interiors of solar-type stars and the irregularities of their cycles also may be produced the same way as the irregularities of solar cycles.",
"Work on constructing flux transport dynamo models for solar-like stars has just begun [35].",
"Our ability to model stellar dynamos may throw more light on how the solar dynamo works.",
"Figure: A study of magnetic field evolution on the solar surface from the 3D kinematicdynamo model of , showing how the polar field builds up from a singletilted bipolar sunspot pair due to the Babcock–Leighton mechanism.",
"The differentpanels show the distribution of magnetic field at the following epochs after theemergence of the bipolar sunspots: (a) 0.025 yr, (b) 0.25 yr, (c) 1.02 yr,(d) 2.03 yr, (e) 3.05 yr, (f) 4.06 yr.All the theoretical results we discussed are based on axisymmetric 2D kinematic dynamo models.",
"One inherent limitation of such models is that the rise of a magnetic loop due to magnetic buoyancy and the Babcock–Leighton process of generating poloidal flux from it are intrinsically 3D processes and can be included in 2D models only through very crude averaging procedures [41], [40], [16].",
"As we have discussed, the fluctuations in the Babcock–Leighton process play a crucial role in producing the irregularities of the solar cycle.",
"In order to model these fluctuations realistically, it is essential to treat the Babcock–Leighton process itself more realistically than what is possible in 2D models.",
"The next step should be the construction of 3D kinematic dynamo models for which efforts have begun [52], [38], [39], [25].",
"Such 3D kinematic dynamo models can treat the Babcock–Leighton mechanism more realistically (see Figure 4) and should provide a better understanding of how fluctuations in the Babcock–Leighton mechanism cause irregularities in the dynamo.",
"The magnetic field presumably exists in the form of flux tubes throughout the convection zone and one limitation of a mean field model is that flux tubes are not handled properly [10].",
"A 3D kinematic model allows one to model flux tubes in a more realistic fashion.",
"A proper inclusion of flux tubes in a dynamo model is essential for explaining such interesting observations as the predominance of negative helicity in the norther hemisphere [43], which is presumably caused by the wrapping of the poloidal field around the rising flux tubes [10], [7], [29].",
"This process can be modelled in 2D mean field dynamo models only through drastic simplifications [15].",
"It should be possible to model this better through 3D kinematic dynamo models.",
"In other words, after the tremendous advances made by the 2D kinematic flux transport model during the last quarter century, it appears that that 3D kinematic dynamo models are likely to occupy the centre stage in the coming years."
],
[
"Acknowledgements",
"This work is partly supported by DST through a J.C. Bose Fellowship.",
"I thank VarSITI for travel support for attending the workshop at ISSI and thank ISSI for local hospitality during the workshop."
]
] | 1808.08550 |
[
[
"Proton Mass Decomposition from the QCD Energy Momentum Tensor"
],
[
"Abstract We report results on the proton mass decomposition and also on related quark and glue momentum fractions.",
"The results are based on overlap valence fermions on four ensembles of $N_f = 2+1$ DWF configurations with three lattice spacings and three volumes, and several pion masses including the physical pion mass.",
"With fully non-perturbative renormalization (and universal normalization on both quark and gluon), we find that the quark energy and glue field energy contribute 33(4)(4)\\% and 37(5)(4)\\% respectively in the $\\overline{MS}$ scheme at $\\mu = 2$ GeV.",
"A quarter of the trace anomaly gives a 23(1)(1)\\% contribution to the proton mass based on the sum rule, given 9(2)(1)\\% contribution from the $u, d,$ and $s$ quark scalar condensates.",
"The $u,d,s$ and glue momentum fractions in the $\\overline{MS}$ scheme are in good agreement with global analyses at $\\mu = 2$ GeV."
],
[
"Acknowledgments",
"We thank the RBC and UKQCD collaborations for providing us their DWF gauge configurations.",
"YY is supported by the US National Science Foundation under grant PHY 1653405 “CAREER: Constraining Parton Distribution Functions for New-Physics Searches.\"",
"Y.C.",
"and Z.L.",
"acknowledge the support of the National Science Foundation of China under Grants No.",
"11575196, No.",
"11575197, No.",
"11335001.",
"This work is partially supported by DOE grant DE-SC0013065 and by the DOE TMD topical collaboration.",
"This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No.$~$ DE-AC05-00OR22725.",
"This work used Stampede time under the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1053575.",
"We also thank the National Energy Research Scientific Computing Center (NERSC) for providing HPC resources that have contributed to the research results reported within this paper.",
"We acknowledge the facilities of the USQCD Collaboration used for this research in part, which are funded by the Office of Science of the U.S. Department of Energy."
],
[
"The non-perturbative renormalization of the quark and gauge EMT",
"The renormalized momentum fractions $\\langle x \\rangle ^R$ in the $\\overline{\\textrm {MS}}$ scheme at scale $\\mu $ are $\\langle x \\rangle ^R_{u,d,s}=Z^{\\overline{\\textrm {MS}}}_{QQ}(\\mu )\\langle x \\rangle _{u,d,s}+\\delta Z^{\\overline{\\textrm {MS}}}_{QQ}(\\mu )\\sum _{q=u,d,s} \\langle x \\rangle _{q}+Z^{\\overline{\\textrm {MS}}}_{QG}(\\mu )\\langle x \\rangle _g,\\ \\langle x \\rangle ^R_g=Z^{\\overline{\\textrm {MS}}}_{GQ}(\\mu )\\sum _{q=u,d,s} \\langle x \\rangle _{q}+Z^{\\overline{\\textrm {MS}}}_{GG}\\langle x \\rangle _g,$ where $\\langle x \\rangle _{u,d,s,g}$ is the bare momentum fraction under the lattice regularization, and the renormalization constants at $\\overline{\\textrm {MS}}$ scale $\\mu $ are defined through the RI/MOM scheme at scale $\\mu _R$ , $&&\\left(\\begin{array}{cc}Z^{\\overline{\\textrm {MS}}}_{QQ}(\\mu )+N_f\\delta Z^{\\overline{\\textrm {MS}}}_{QQ} (\\mu )&N_fZ^{\\overline{\\textrm {MS}}}_{QG}(\\mu ) \\\\Z^{\\overline{\\textrm {MS}}}_{GQ}(\\mu ) &Z^{\\overline{\\textrm {MS}}}_{GG}(\\mu )\\end{array}\\right)\\equiv \\left\\lbrace \\left[\\left(\\begin{array}{cc}Z_{QQ}(\\mu _R)+N_f\\delta Z_{QQ}&N_fZ_{QG}(\\mu _R)\\\\Z_{GQ}(\\mu _R)&Z_{GG}(\\mu _R)\\end{array}\\right)\\right.\\right.\\nonumber \\\\&&\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\left.\\left.\\left(\\begin{array}{cc}R_{QQ}(\\frac{\\mu }{\\mu _{R}})+{\\cal O}(N_f\\alpha _s^2) &N_fR_{QG}(\\frac{\\mu }{\\mu _{R}})\\\\R_{GQ}(\\frac{\\mu }{\\mu _{R}}) &R_{GG}(\\frac{\\mu }{\\mu _{R}})\\end{array}\\right)\\right]|_{a^2\\mu _R^2\\rightarrow 0}\\right\\rbrace ^{-1}\\\\&&=\\left\\lbrace \\left(\\begin{array}{cc}\\left(Z_{QQ}R_{QQ}\\right)+N_f\\left(\\delta Z_{QQ}R_{QQ}+Z_{QG}R_{GQ}\\right)&N_f\\left((Z_{QQ}+N_f\\delta R_{QQ}) R_{QG}+Z_{QG}R_{GG}\\right)\\\\\\left(Z_{GQ}R_{QQ}+Z_{GG}R_{GQ}\\right) &\\left(N_fZ_{GQ} R_{QG}+Z_{GG}R_{GG}\\right)\\end{array}\\right)(\\mu _R,\\frac{\\mu }{\\mu _R})|_{a^2\\mu _R^2\\rightarrow 0}\\right\\rbrace ^{-1}$ and $Z_{QQ}(\\mu )=\\left[\\left(Z_{QQ}R_{QQ}\\right)(\\mu _R,\\frac{\\mu }{\\mu _R})|_{a^2\\mu _R^2\\rightarrow 0}\\right]^{-1}$ .",
"In the above equations, the RI/MOM renormalization constants $Z$ of $\\overline{T}^{q,g}_{44}$ are defined with the following conditions suggested by Ref.",
"[20] for cases with quark external legs: $Z_{QQ}(\\mu _R)&=&\\frac{V\\textrm {Tr}\\left[\\Gamma ^{q}_{\\mu \\mu }\\bar{S}_q^{-1}(p)\\left\\langle \\sum _w \\gamma _5 S_q^{\\dagger }(p,w) \\gamma _5 \\gamma _{\\mu }\\overleftrightarrow{D}_{\\mu } S_q(p,w) \\right\\rangle \\bar{S}_q^{-1}(p)\\right]}{\\left[-i\\Gamma ^{q}_{\\mu \\mu }(\\gamma _{\\mu }\\tilde{p}_{\\mu }-\\frac{1}{4}\\tilde{p}\\!\\!\\!/)\\right]Z_q}|_{p^2=\\mu _R^2},\\\\\\delta Z_{QQ}(\\mu _R)&=&\\frac{V\\textrm {Tr}\\left[\\Gamma ^{q}_{\\mu \\mu }\\bar{S}_q^{-1}(p)\\left\\langle \\overline{T}^q_{\\mu \\mu } S_q(p) \\right\\rangle \\bar{S}_q^{-1}(p)\\right]}{\\left[-i\\Gamma ^{q}_{\\mu \\mu }(\\gamma _{\\mu }\\tilde{p}_{\\mu }-\\frac{1}{4}\\tilde{p}\\!\\!\\!/)\\right]Z_q}|_{p^2=\\mu _R^2},\\\\Z_{GQ}(\\mu _R)&=&\\frac{V\\textrm {Tr}\\left[\\Gamma ^{q}_{\\mu \\mu }\\bar{S}_q^{-1}(p)\\left\\langle \\overline{T}^g_{\\mu \\mu } S_q(p) \\right\\rangle \\bar{S}_q^{-1}(p)\\right]}{\\left[-i\\Gamma ^{q}_{\\mu \\mu }(\\gamma _{\\mu }\\tilde{p}_{\\mu }-\\frac{1}{4}\\tilde{p}\\!\\!\\!/)\\right]Z_q}|_{p^2=\\mu _R^2},$ where $V$ is the lattice volume, $p$ is the momentum of the external quark/gluon state, $\\tilde{p}_{\\mu }=\\textrm {sin}p_{\\mu }$ , $\\Gamma ^q_{\\mu \\mu }=i\\gamma _{\\mu }\\tilde{p}_{\\mu }-i\\frac{\\tilde{p}^2_{\\mu }}{\\tilde{p}^2}\\tilde{p}\\!\\!\\!/$ as suggested by Ref.",
"[30], the quark propagators are $\\bar{S}_q(p)\\equiv \\langle S_q(p)\\rangle \\equiv \\langle \\sum _{x} e^{ipx}S_q(p,x)\\rangle $ with $S_q(p,x)=\\sum _{y} e^{-ipy}\\psi (x)\\bar{\\psi }(y) $ , and $Z_q$ is defined through the axial-vector vertex correction and Ward identity [31].",
"Note that the index $\\mu $ in Eqs.",
"(REF )-() is not summed while the results with different values of $\\mu $ can be averaged.",
"For the case of gluon external legs, the definitions are the following, as inspired by Refs.",
"[21], [19], $Z_{QG}(\\mu _R)&=&\\xi ^{-1}Z_{b}(\\mu _R, \\overline{T}^q)-(\\xi ^{-1}-1)Z_{a}(\\mu _R, \\overline{T}^q),\\\\Z_{GG}(\\mu _R)&=&\\xi ^{-1}Z_{b}(\\mu _R, \\overline{T}^g)-(\\xi ^{-1}-1)Z_{a}(\\mu _R, \\overline{T}^g),$ where $\\xi \\equiv \\frac{\\sum _{\\mu }p_{\\mu }^4}{(\\sum _{\\mu }p_{\\mu }^2)^2}$ , and $Z_{a/b}$ are defined by $Z_{a}(\\mu _R, \\overline{T})&=&\\frac{p^2\\langle (k_{\\mu }\\overline{T}_{\\mu \\nu }q_{\\nu })\\mathrm {Tr}[A_{\\rho }(p) A_{\\tau }(-p)\\Gamma _{\\rho \\tau }]\\rangle }{2k^2q^2\\langle \\mathrm {Tr}[A_{\\rho }(p) A_{\\tau }(-p)\\Gamma _{\\rho \\tau }]\\rangle }|_{\\tiny {\\begin{array}{c}p^2=\\mu _R^2, k+q=p, k\\cdot q=0, \\\\ \\Gamma _{\\rho \\tau }=\\delta _{\\rho \\tau }-\\frac{k_{\\rho }k_{\\tau }}{k^2}-\\frac{q_{\\rho }q_{\\tau }}{q^2} \\end{array}}}\\\\Z_{b}(\\mu _R, \\overline{T})&=&\\frac{\\langle (p_{\\mu }\\overline{T}_{\\mu \\nu }p_{\\nu }-l_{\\mu }\\overline{T}_{\\mu \\nu }l_{\\nu })\\mathrm {Tr}[A_{\\rho }(p) A_{\\tau }(-p)\\tilde{\\Gamma }_{\\rho \\tau }]\\rangle }{2p^2\\langle \\mathrm {Tr}[A_{\\rho }(p) A_{\\tau }(-p)\\tilde{\\Gamma }_{\\rho \\tau }]\\rangle }|_{\\tiny {\\begin{array}{c}p^2=\\mu _R^2, l^2=p^2, l\\cdot p=0, \\\\ \\tilde{\\Gamma }_{\\rho \\tau }=\\delta _{\\rho \\tau }-\\frac{p_{\\rho }p_{\\tau }}{p^2}-\\frac{l_{\\rho }l_{\\tau }}{l^2} \\end{array}}},$ with all the repeated indices summed.",
"The 3-loop level result of the matching coefficient $R_{QQ}(\\frac{\\mu }{\\mu _{R}})=1+\\frac{g^2}{16\\pi ^2}C_F[\\frac{8}{3}\\textrm {log}(\\mu ^2/\\mu _R^2)+\\frac{31}{9}]+{\\cal O}(\\alpha _s^2)$ has been obtained in Ref.",
"[20], while just the 1-loop level results of the other $R$ 's are available [21]: $R_{QG}&=&-\\frac{g^2}{16\\pi ^2}[\\frac{2}{3}\\textrm {log}(\\mu ^2/\\mu _R^2)+\\frac{4}{9}]+{\\cal O}(\\alpha _s^2),\\ R_{GQ}=-\\frac{g^2C_F}{16\\pi ^2}[\\frac{8}{3}\\textrm {log}(\\mu ^2/\\mu _R^2)+\\frac{22}{9}]+{\\cal O}(\\alpha _s^2), \\nonumber \\\\R_{GG}&=&1+\\frac{g^2N_f}{16\\pi ^2}[\\frac{2}{3}\\textrm {log}(\\mu ^2/\\mu _R^2)+\\frac{10}{9}]-\\frac{g^2N_c}{16\\pi ^2}\\frac{5}{12}+{\\cal O}(\\alpha _s^2).$ Thus we will use the 3-loop matching for the renormalization of the quark momentum fraction and keep all the matching at 1-loop level in the rest of the renormalization calculation.",
"In the practical lattice calculation, all the $Z$ 's suffer from discretization errors and we need to repeat the calculation of $Z$ at different $p^2$ , match them to the $\\overline{\\textrm {MS}}$ scheme at the $\\mu _R$ scale, evolve them from $\\mu _R$ to a fixed scale such as 2 GeV, and then apply the $a^2\\mu _R^2=a^2p^2$ extrapolation to get the final result of $Z$ .",
"To apply this extrapolation properly, we should calculate the vertex corrections in the quark and gluon states with exactly the same momenta, and combine them first.",
"But it is not necessary for most of the cases except for the quark to gluon mixing, as we will discuss case by case.",
"Figure: The quark EMT renormalization constant Z QQ R QQ Z_{QQ}R_{QQ} (left panel) and the disconnected pieces (δZ QQ R QQ \\delta Z_{QQ}R_{QQ} and Z QG R GQ Z_{QG}R_{GQ}, in the right panel) at different lattice spacings at MS ¯\\overline{\\textrm {MS}} scheme 2 GeV, as a function of a 2 p 2 a^2p^2.",
"With the momenta along the body-diagonal direction, the discretization errors are small and the results have a mild a 2 p 2 a^2p^2 dependence."
],
[
"Quark EMT renormalization",
"For the quark external state, we choose 12 momenta with $\\frac{\\sum _{\\mu }p_{\\mu }^4}{(\\sum _{\\mu }p_{\\mu }^2)^2}<0.27$ for 30 configurations and 6 valence quark masses at each lattice spacing, and use Landau gauge fixed momentum volume sources to improve the signal.",
"With the given momenta, we extrapolated the result to the chiral limit and then applied the continuum matching to get the value at $\\overline{\\textrm {MS}}$ 2 GeV.",
"The values of $Z_{QQ}R_{QQ}$ for three lattice spacings are plotted in the left panel of Fig.",
"REF , and the values at $a^2p^2=0$ limit are obtained based on linear extrapolation of the data in the range $a^2p^2\\in [4,8]$ .",
"It turns out that the $a^2p^2$ dependence is small by using $\\tilde{p}_{\\mu }$ in the definition of the tree level operator and projection, when the chosen momenta are along the body-diagonal direction.",
"The calculation on the 48I ensemble is skipped as its lattice spacing is almost the same as that of the 24I ensemble.",
"We followed the same strategy used in Ref.",
"[31], [32] to analyze the systematic uncertainties and the error budgets are collected in Tab.",
"REF .",
"The systematic uncertainty of $m_s^{sea}\\ne 0$ are estimated as $\\sim $ 1% on the 24I and 32I ensembles and $\\sim $ 0.25% on 32ID ensemble, based on our previous study with multiple sea quark masses [31].",
"The values for the off-diagonal parts of the quark EMT cases at $a$ =0.143, 0.111 and 0.083 fm (at $a^2p^2=0$ limit) are slightly different.",
"They are 0.786(2)(9), 0.796(1)(11) and 0.793(1)(11) respectively.",
"In the right panel of Fig.",
"REF , the other two terms needed by the singlet quark EMT renormalization, $\\delta Z_{QQ}R_{QQ}$ and $Z_{QG}R_{GQ}$ for the disconnected contribution, are plotted.",
"Regardless of the simulation details of the $Z_{QG}$ which will be addressed later, the contribution of second term is much smaller than the first term, and thus the $a^2p^2$ extrapolation can be carried out with the $\\delta Z_{QQ}R_{QQ}$ term alone in the $a^2p^2\\in [4,8]$ range, considering the value of the other term as a systematic uncertainty which is $\\sim $ 0.001.",
"Note that the CDER technique [18] is applied to the correlation function of $\\delta Z_{QQ}$ with the cutoff $r_0\\sim $ 1 fm, $\\left\\langle \\textrm {Tr}[\\overline{T}^{q}_{\\mu \\mu }] S(p) \\right\\rangle =\\left\\langle \\int \\textrm {d}^4x \\textrm {d}^{4}y \\textrm {Tr}[\\overline{T}^{q,g}_{\\mu \\mu }](x) S(p,y) \\right\\rangle \\simeq \\left\\langle \\int \\textrm {d}^4x \\int _{r\\le r_0}\\textrm {d}^{4}r \\textrm {Tr}[\\overline{T}^{q}_{\\mu \\mu }](x) S(p,x+r) \\right\\rangle $"
],
[
"Gauge EMT renormalization",
"In our previous investigation of the glue EMT renormalization [19], we chose the momenta with two transverse components to calculate $Z_{GG}$ : $Z_{GG}(\\mu _R)&=&\\frac{p^2\\langle (\\overline{T}^{g}_{\\mu \\mu }-\\overline{T}^{g}_{\\nu \\nu })\\mathrm {Tr}[A_{\\rho }(p) A_{\\rho }(-p)]\\rangle }{2p^2_{\\mu }\\langle \\mathrm {Tr}[A_{\\rho }(p) A_{\\rho }(-p)]\\rangle }|_{\\tiny {\\begin{array}{c}p^2=\\mu _R^2,\\\\ \\rho \\ne \\mu \\ne \\nu , \\\\p_\\rho =0,\\\\p_\\nu =0\\end{array}}},$ But the discretization errors at the range $a^2p^2\\in [4,8]$ used in the quark external legs are too large to carry out a reliable fit, when we want to combine it with $Z_{GQ}$ .",
"Thus in this work, we introduce two improvements to suppress the $a^2p^2$ corrections: 1.",
"The $a^2p^2$ correction from the HYP smearing on the glue operator: We found that most of the $a^2p^2$ correction in $Z_{GG}$ with the HYP smeared gauge EMT can be removed by the following ratio $f(a^2p^2)$ , $\\tilde{Z}_{GG}(a^2p^2)=Z_{GG}(a^2p^2)f(a^2p^2\\rightarrow 0)/f(a^2p^2),\\ f(a^2p^2)=\\frac{\\langle \\mathrm {Tr}[A^{HYP}_{\\rho }(p) A^{HYP}_{\\rho }(-p)]\\rangle }{\\langle \\mathrm {Tr}[A_{\\rho }(p) A_{\\rho }(-p)]\\rangle },$ where $A_{\\rho }^{HYP}=\\sum _{x} e^{ip\\cdot (x+\\frac{1}{2}\\hat{\\rho })}\\left[\\frac{{\\cal U}_\\rho (x)-{\\cal U}^{\\dagger }_\\rho (x)}{2ig_0a}\\right]_\\text{traceless}$ is the HYP-smeared gauge potential defined from the HYP-smeared gauge link ${\\cal U}_\\rho $ , and $f(a^2p^2)$ is the ratio of two propagators $S(p)\\equiv \\langle \\mathrm {Tr}[A_{\\rho }(p) A_{\\rho }(-p)]\\rangle $ with and without HYP smearing.",
"Since the scale dependence of $f(a^2p^2)$ is cancelled, only the $a^2p^2$ discretization errors exist up to the normalization $f(a^2p^2\\rightarrow 0)$ .",
"Note that such a normalization corresponds to the tadpole effect of the HYP smearing which should be included in the renormalization constant $Z_{GG}$ .",
"Since the value of $S(p)$ doesn't exist at $p^2=0$ , we will have to fit $f(a^2p^2)$ with a polynomial of $a^2p^2$ to get the value of $f(a^2p^2\\rightarrow 0)$ .",
"2.",
"We also extend the calculation of the $Z_{GG}$ to the momenta along the body-diagonal direction, by considering the projected renormalization constants defined in Eq. .",
"They can be rewritten into the combination of the renormalization constant of the traceless diagonal gauge EMT $Z_{GG}$ , and the off-diagonal one $Z_{GG}^{off}$ by $Z_{a}(\\mu _R, \\overline{T}_g)=Z^{off}_{GG}(\\mu _R), Z_{b}(\\mu _R, \\overline{T}_g)=Z^{off}_{GG}(\\mu _R)+\\xi (Z_{GG}-Z^{off}_{GG})(\\mu _R).$ Figure: The gauge EMT renormalization constant, with (left panel) and without (right panel) the a 2 p 2 a^2p^2 improvements described in Eq.",
"() and ().",
"The contribution from Z ˜ GG RI R GG \\tilde{Z}^{RI}_{GG}R_{GG} is dominant and that from N f Z GQ R QG N_fZ_{GQ} R_{QG} is negligible.",
"The a 2 p 2 →0a^2p^2\\rightarrow 0 results with and without improvement can provide consistent predictions.With 949/356/290 configurations respectively, the values of $\\tilde{Z}^{RI}_{GG}R_{GG}$ on 32ID/48I/64I ensembles with 1 step of HYP smearing are illustrated in the left panel of Fig.",
"REF .",
"Note that the ensemble 64I, which has almost the same lattice spacing as 32I (0.0837(2) vs. 0.0828(3)) but with a larger volume and physical pion mass, is used for the calculation, as a similar calculation on 32I would require $\\sim $ 5,000 configurations, which are not available, to reach the similar accuracy.",
"After both improvements above are applied, the $a^2p^2$ dependences of the $\\tilde{Z}^{RI}_{GG}R_{GG}$ are mild.",
"The contributions from the other term $N_fZ_{GQ} R_{QG}$ are also illustrated on the same figures.",
"Both the values and their uncertainties ($\\sim $ 0.005) are much smaller than the statistical uncertainty of $\\tilde{Z}^{RI}_{GG}$ and thus can be dropped safely.",
"With linear extrapolation of the data in the range $a^2p^2\\in [2,7]$ and the polynomial fit of the $f(a^2p^2)$ in the same range, we obtain the renormalization constants at $a$ =0.143, 0.114 and 0.084 fm to be 0.79(3)(6), 0.94(3)(3), and 0.91(3)(4) respectively, with the second error determined from varying the starting/ending points of the $a^2p^2$ range by 1.",
"For comparison, we also illustrate the results obtained from the previous strategy used in Ref.",
"[19] in the right panel of Fig.",
"REF , with the fit of the data in the range of $a^2\\hat{p}^2\\equiv (\\textrm {sin}\\frac{pa}{2})^2\\in [1.5, 5]$ .",
"The $a^2p^2\\rightarrow 0$ results of based on the polynomial fit are consistent with the present linear-fit ones."
],
[
"Mixings",
"For the mixing from glue to quark, both the contributions from $Z_{QQ} R_{QG}$ and $Z_{QG}R_{GG}$ are small, as illustrated in the left panel of Fig.",
"REF .",
"The total contribution can be estimated as $-$ 0.010(10), $-$ 0.005(10), $-$ 0.000(10) per flavor at $a$ =0.143, 0.111 and 0.083 fm respectively, when the linear extrapolation is applied in the range $a^2p^2\\in [4,8]$ .",
"The last piece of the jigsaw is the mixing from quark to glue.",
"The CDER technique used for $\\delta Z_{QQ}$ as in Eq.",
"REF can be applied for the calculation here with the gauge EMT operator.",
"As in the right panel of Fig.",
"REF , the contributions from $\\tilde{Z}^{RI}_{GG}R_{GQ}$ are just a few percent, while those from $Z_{GQ}R_{QQ}$ are sizable.",
"Since the $a^2p^2$ dependence of $\\tilde{Z}^{RI}_{GG}$ is small as in the left panel of Fig.",
"REF , we can approximate the $\\tilde{Z}^{RI}_{GG}(a^2p^2)$ by its $a^2p^2$ extrapolated value and assign $\\sim $ 3% systematic uncertainty from the difference between the values at $a^2p^2$ equal to 4 and 8.",
"After combining the correction and applying the linear extrapolation in the range $a^2p^2\\in [4,8]$ , we obtain the total mixing at $a$ =0.143, 0.111 and 0.083 fm as $-$ 0.34(2)(4), $-$ 0.26(2)(3), and $-$ 0.13(2)(1) respectively."
]
] | 1808.08677 |
[
[
"Theoretical approach to the ductile fracture of polycrystalline solids"
],
[
"Abstract It is shown here that fracture after a brief plastic strain, typically of a few percents, is a necessary consequence of the polycrystalline nature of the materials.",
"The polycrystal undergoing plastic deformation is modeled as a flowing continuum of random deformable polyhedra, representing the grains, which fill the space without leaving voids.",
"Adjacent grains slide with a relative velocity proportional to the local shear stress resolved on the plane of the shared grain boundary, when greater than a finite threshold.",
"The polyhedral grains reshape continuously to preserve matter continuity, being the forces causing grain sliding dominant over those reshaping the grains.",
"It has been shown in the past that this model does not conserve volume, causing a monotonic hydrostatic pressure variation with strain.",
"This effect introduces a novel concept in the theory of plasticity because determines that any fine grained polycrystalline material will fail after a finite plastic strain.",
"Here the hydrostatic pressure dependence on strain is explicitly calculated and shown that has a logarithmic divergence which determines the strain to fracture.",
"Comparison of theoretical results with strains to fracture given by mechanical tests of commercial alloys show very good agreement."
],
[
"Introduction",
"Asking why things break when subjected to strong enough forces may sound superfluous because breaking objects is one of the most early experiences of every person.",
"In reality, explaining why solids undergoing plastic deformation are unable of achieving a steady flow regime and collapse past a finite plastic flow, or with almost no flow at all, is a most important scientific and technical problem yet unsolved.",
"In technical grounds the point is quite serious because of the high expenses associated to fatigue and failure of functional articles.",
"As well, the design of machine parts and structures is always restricted by the strength of the materials they will be made of, which puts limits to their efficiency and bounds costs from below.",
"Since the early investigations of Griffith [1], who claimed that the tensile strength of glass is lowered by the presence of very small pre–existent cracks that concentrate stresses when the material is loaded, and Irwin [2] and Orowan [3], who extended the idea to ductile solids, a great amount of effort has been expended in elucidating why solid materials break from the atomic point of view.",
"Nowadays the question has turned to how solids fail, instead of why they break.",
"Certainly, the two issues are closely related and answering the former question may clarify the latter, but not necessarily.",
"Most of the contemporary research on this subject relies on the hypothesis of cracks, and ascribes brittle behavior to the ability of stressed crack tips to propagate conserving their atomically sharp edges.",
"In ductile solids the tip of the crack blunts, broadens and flows, demanding increasing effort to make it progress [4], [5], [6], [7], [8], [9], [10].",
"Unfortunately, the problem of stress induced crack propagation has proven to be exceedingly complex, and neither theory nor computer simulations [11], [12], [13] have produced conclusive answers on the fracture process and the origin of brittle or ductile fracture.",
"The complex evolution of crack growth has been accurately measured [14], [15], [16], confirming atomic scale model predictions [17], [18] that the dynamics of a crack tip is highly unstable, and steady motion in a given direction is in most situations impossible.",
"We show here that the reason why continued deformation inevitably makes solids to break, undergoing either brittle, ductile or superplastic fracture, is much more basic and simpler than how fracture proceeds.",
"Resorting to a very general model for the structure of the solid, we demonstrate in what follows that fine grained polycrystalline materials are not able of a steady flow, no matter the strength of the forces involved, and should collapse after reaching a finite plastic strain.",
"At a scale much larger than the grain size, polycrystalline matter lacks symmetry constrictions and periodicity, and displays same average packing and properties in all directions, and over its whole extention.",
"Despite this, assimilating an even very fine grained polycrystal to an homogeneous and isotropic continuum may lead to gross errors, no matter the scale, when dealing with it as a dynamical medium.",
"The faceted nature of the structural constituents of a polycrystal determines that the force fields governing their plastic flow yield $\\nabla \\cdot \\vec{v}\\ne 0$ , where $\\vec{v}$ is the velocity field of the material continuum.",
"This means that flow makes the specific volume to vary.",
"Grain elasticity in polycrystalline solids allows for some density variation, and hence the medium can flow up to some limit, yielding ductile behaviour.",
"However, the consequent pressure build up influences strongly the ongoing deformation, which cannot be steady, and finally produces fracture.",
"Thus ductility is closely related to compressibility.",
"The model for the plastic flow of a polycrystalline solid has been extensively studied, principally in the context of superplasticity, but is expected to equally hold for normal ductile solids.",
"However, a brief account of its physical basis and the resulting general theoretical scheme is in order here.",
"The plastic deformation of a fine grained polycrystalline solid is modelled as a flowing continuum of random irregular polyhedra of different shapes and sizes, representing grains, which share faces.",
"The model is essentially the same as the one of Ref. [19].",
"Grains can move over long paths by sliding along the shared surfaces, or grain boundaries, accommodating effortlessly their shapes to preserve matter continuity.",
"Certainly, grain shape accommodation demands some effort, but it is assumed much smaller than the one required for grain sliding.",
"In other words, the shear stress between two sliding grains is greater than the critical resolved shear stress (CRSS) demanded by slip deformation of the crystallites.",
"This way, grain boundary sliding is the rate limiting process in the plastic strain.",
"In the present scheme grains always retain their individuality and mass, and are the dynamical entities.",
"The flow is driven by a field of tensor forces between the grains, determined by the stress tensor.",
"Figure: Local reference system (x ' y ' z ' )(x^\\prime y^\\prime z^\\prime ), with the z ' z^\\prime axis normal to the plane of the commonboundary of two adjacent grains.",
"The relative velocity Δv →\\Delta \\vec{v}of the two grains is in the x ' y ' x^{\\prime }y^{\\prime } plane.",
"The axes of the (xyz)(xyz)frame of reference are in the principal directions of the stresstensor.Fig.",
"REF shows a local frame of reference $(x^{\\prime }y^{\\prime }z^{\\prime })$ with the $x^{\\prime }y^{\\prime }$ plane coincident with the boundary between two adjacent grains.",
"The total shear stress in the shared boundary plane then reads $\\tau _{z^{\\prime }}=(\\sigma _{x^{\\prime }z^{\\prime }}^2+\\sigma _{y^{\\prime }z^{\\prime }}^2)^{1/2}$ , where $\\sigma _{i^{\\prime }j^{\\prime }}$ , $i^{\\prime },j^{\\prime }=x^{\\prime },y^{\\prime },z^{\\prime }$ , stands for the components of the stress tensor in this local coordinate system.",
"There is strong evidence that the sliding relative speed $|\\Delta \\vec{v}|$ of two adjacent grains obeys a linear law of the general form $|\\Delta \\vec{v}|=\\mathcal {Q}(\\tau _{z^{\\prime }}-\\tau _c)$ for $\\tau _{z^{\\prime }}>\\tau _c$ in plastic deformation [19], [20], [21], [22], [23], [24].",
"Here $\\mathcal {Q}$ is a proportionality coefficient and $\\tau _c$ is a critical shear stress such that $|\\Delta \\vec{v}|=0$ when $\\tau _{z^{\\prime }}\\le \\tau _c$ .",
"As $\\Delta \\vec{v}$ is parallel to the shear force in the plane of the interface, its components are given by $\\Delta v_{i^{\\prime }}=\\mathcal {Q}(\\tau _{z^{\\prime }}-\\tau _c)(\\sigma _{i^{\\prime }z^{\\prime }}/\\tau _{z^{\\prime }})$ , $i^{\\prime }=x^{\\prime },y^{\\prime }$ , for $\\tau _{z^{\\prime }}\\ge \\tau _c$ .",
"This expresion for $\\Delta \\vec{v}$ has proven to hold with great accuracy for several aluminium, titanium and magnesium alloys [24], [25], [26].",
"Hence the force law at the grain scale reads $\\begin{aligned}&\\Delta v_{i^{\\prime }}={\\left\\lbrace \\begin{array}{ll}\\mathcal {Q}\\,\\left( 1-\\dfrac{\\tau _c}{\\tau _{z^{\\prime }}}\\right) \\sigma _{i^{\\prime }z^{\\prime }},\\,&i^{\\prime }=x^{\\prime }, y^{\\prime }, \\, \\text{if}\\, \\tau _{z^{\\prime }}>\\tau _c \\\\0, &\\text{ otherwise},\\end{array}\\right.}",
"\\\\&\\Delta v_{z^{\\prime }}\\equiv 0.",
"\\\\\\end{aligned}$ The coefficient $\\mathcal {Q}=\\mathcal {Q}(p,T)$ does not depend on the shear stresses and neither on the orientation of the grain boundary, therefore its dependence on the normal stresses is only via the hydrostatic pressure invariant $p=-(\\sigma _{x^{\\prime }x^{\\prime }}+\\sigma _{y^{\\prime }y^{\\prime }}+\\sigma _{z^{\\prime }z^{\\prime }})/3 .$ The next step is to express the force law (REF ) in the frame of reference $(xyz)$ , common to all grain surfaces, instead of the local ones $(x^\\prime y^\\prime z^\\prime )$ .",
"Given the rotation matrix $R(\\theta ,\\phi )=(R_{ij}(\\theta ,\\phi ))$ connecting the two frames one can put the local stress tensor $(\\sigma _{i^\\prime j^\\prime })=R(\\theta ,\\phi )(\\sigma _{ij})R^T(\\theta ,\\phi )$ in terms of the stress tensor $(\\sigma _{i j})$ of the externally applied forces and the Euler angles $(\\theta ,\\phi )$ of the grain boundary plane.",
"The macroscopic force law is obtained from replacing in Eq.",
"(REF ) and averaging over the Euler angles.",
"Invoking also Hooke's law one has that $\\nabla \\cdot \\vec{v}=\\frac{\\dot{V}}{V}=-\\frac{\\dot{p}}{B},$ where $B$ is the bulk elastic modulus, $\\dot{p}$ the pressure variation rate, and $\\dot{V}/V$ the volume variation rate per unit volume.",
"A detailed account of the procedure would be in excess here because can be found in the literature [19], [23], [24], [25], [26]."
],
[
"The equations of motion",
"After a rather tedious set of mathematical steps [19], [24], [25], [26], [27] the procedure outlined above for the special case of an externally applied unidirectional normal stress $\\sigma $ on a polycrystalline solid, isotropic in the scale much larger than the mean grain size $d$ , yields the complete set of macroscopic equations of motion $\\dot{\\varepsilon }=s\\,\\frac{\\tau _c\\mathcal {Q}(p)}{2d}\\left[ \\cot (2\\theta _c)+2\\theta _c-\\frac{\\pi }{2}\\right] ,$ $\\begin{aligned}\\dot{p}= &sB\\frac{\\tau _c\\mathcal {Q}(p)}{2d}\\bigg [ \\frac{1 -\\cos (2\\theta _c)}{\\sin (2\\theta _c)}-2\\theta _c\\bigg (1+\\frac{2}{\\pi \\sin (2\\theta _c)}\\bigg )\\\\&-\\frac{2}{\\pi }\\cos (2\\theta _c)+\\frac{\\pi }{2}\\bigg ],\\end{aligned}$ where $\\dot{\\varepsilon }$ is the strain rate in the direction of the applied stress $\\sigma $ , $s=\\pm 1$ assumes the positive and negative values for tension and compression, respectively, and the auxiliary variable $\\theta _c$ is given by $\\sin (2\\theta _c)=\\frac{4\\tau _c}{3|\\sigma +p|}.$ The properties of the specific material enters the theoretical formulation through the coefficient $\\mathcal {Q}(p,T)$ , governing grain boundary sliding.",
"It has been studied in detail for fine grained polycrystalline solids and has been shown to be of the general form $\\frac{\\mathcal {Q}(p,T)}{4d}=C_0\\frac{\\Omega ^*}{k_BT}\\exp \\left( -\\frac{\\epsilon _0+\\Omega ^*p}{k_BT}\\right),$ where $k_B$ is the Boltzmann constant, $T$ the absolute temperature, the coefficient $C_0$ depends only on the grain size $d$ , the constant $\\epsilon _0$ is the energy necessary for evaporating a crystal vacancy from the grain boundary, and $\\Omega ^*$ is the excitation volume for the same process.",
"Eqs.",
"(REF ), (REF ) and (REF ) show that plastic flow is essentially a time dependent problem.",
"They govern the coupled time evolution of the three variables, $\\sigma $ , $\\varepsilon $ and $p$ , relevant for the cylindrically symmetric deformation of a polycrystalline continuous medium.",
"The actual behaviour of these variables in specific circumstances depends also on the initial conditions and deformation path ($\\sigma =\\text{constant}$ , $\\dot{\\varepsilon }=\\text{constant}$ , or any other imposed condition between the variables and their time derivatives).",
"The observed dependence on history of the plastic properties of ductile solids is usually attributed to structural variations or deformation induced damage.",
"In the present scheme, history enters through the initial condition for the variable $p$ , which is omitted in the traditional theoretical approaches to plasticity.",
"Here, the system described by Eqs.",
"(REF ), (REF ) and (REF ) behaves always the same way, but its evolution depends on the initial conditions for the variables, which include $p$ [27].",
"In opposition to the classical theory of plasticity, it exists a nontrivial transversal stress $\\sigma _\\perp =-(\\sigma +3p)/2$ which is not an independent variable, but evolves in time as dictated by the equations of motion.",
"One can set $\\sigma _\\perp =0$ as a natural initial condition if the material has been previously annealed, but $\\sigma _\\perp $ is expected to take finite values on the subsequent deformation.",
"As the magnitude inside the square brackets in the right hand side of Eq.",
"(REF ) is positive for any $\\theta _c$ , $\\dot{p}$ has the sign of $s$ .",
"The transversal stress $\\sigma _\\perp $ then decreases monotonically to negative values for positive $\\sigma +p$ .",
"Physically, this means that the plastic stretching in one direction is always accompanied by a finite compression in the plane normal to the deformation axis, which increases monotonically with strain.",
"This explains why necking always precedes ductile fracture [27]."
],
[
"The equations for constant strain rate",
"Replacing Eq.",
"(REF ) in the identity $d\\varepsilon =(\\dot{\\varepsilon }/\\dot{p})\\, dp$ one has that $\\begin{aligned}d\\varepsilon =&s\\frac{2\\dot{\\varepsilon }d}{B\\tau _c}\\bigg [\\frac{1-\\cos (2\\theta _c)}{\\sin (2\\theta _c)}-2\\theta _c\\bigg (1+\\frac{2}{\\pi \\sin (2\\theta _c)}\\bigg )\\\\&-\\frac{2}{\\pi }\\cos (2\\theta _c)+\\frac{\\pi }{2}\\bigg ]^{-1}\\frac{dp}{\\mathcal {Q}(p,T)},\\end{aligned}$ where $\\dot{\\varepsilon }$ is considered as a given constant.",
"As long as $\\dot{\\varepsilon }=\\text{constant}$ , Eq.",
"(REF ) shows that $\\theta _c=\\theta _c(p)$ .",
"Combining the derivatives of Eqs.",
"(REF ) and (REF ) with respect to $p$ it can be shown that $\\frac{dp}{\\mathcal {Q}}=-s\\frac{\\tau _ck_BT}{\\Omega ^*\\dot{\\varepsilon }d}\\cot ^2 (2\\theta _c)\\, d\\theta _c .$ Replacing now Eq.",
"(REF ) in (REF ) and integrating, it is finally obtained $\\begin{aligned}\\varepsilon =&-\\frac{2k_BT}{B\\Omega ^*}\\int _{\\theta _0}^{\\theta _c}d\\theta \\,\\cot ^2(2\\theta _c)\\bigg [\\frac{1-\\cos (2\\theta )}{\\sin (2\\theta )}\\\\&-2\\theta \\bigg (1+\\frac{2}{\\pi \\sin (2\\theta )}\\bigg )-\\frac{2}{\\pi }\\cos (2\\theta )+\\frac{\\pi }{2}\\bigg ]^{-1},\\end{aligned}$ where the limits $\\theta _0$ and $\\theta _c$ correspond to the critical angles for the initial and final values of the strain, $\\varepsilon =0$ and $\\varepsilon $ , respectively.",
"This way, $\\varepsilon $ is related with the auxiliary variable $\\theta _c$ by an expression of the form $\\varepsilon =\\frac{k_BT}{B\\Omega ^*}[F(\\theta _c)-F(\\theta _0)]\\quad (\\dot{\\varepsilon }=\\text{ constant}),$ where $F(\\theta )$ is the universal function $\\begin{aligned}F(\\theta )&=-2\\int _{\\pi /8}^{\\theta } d\\theta \\,\\cot ^2(2\\theta _c)\\bigg [\\frac{1-\\cos (2\\theta )}{\\sin (2\\theta )}\\\\&-2\\theta \\bigg (1+\\frac{2}{\\pi \\sin (2\\theta )}\\bigg )-\\frac{2}{\\pi }\\cos (2\\theta )+\\frac{\\pi }{2}\\bigg ]^{-1},\\end{aligned}$ which is monotonically decreasing in its whole range $(0,\\pi /4)$ and has two singularities, at $\\theta =0$ and $\\theta =\\pi /4$ .",
"If the material has been thoroughly annealed prior to the plastic deformation, it holds the initial condition $p=-\\sigma _0/3$ at $\\varepsilon =0$ , where $\\sigma _0$ is the stress at the beginning of the plastic deformation.",
"The magnitude of $\\varepsilon $ is controlled by the adimensional coefficient appearing in Eqs.",
"(REF ) and (REF ), which is a very small quantity.",
"The bulk modulus $B$ for metals is of the order of $10^{11}\\,\\text{Pa}$ .",
"Previous literature on aluminium and titanium alloys shows that $\\Omega ^*$ is $2.6\\times 10^{-27}\\,\\text{m}^3$ for Al–8090 and $5.9\\times 10^{-28}\\,\\text{m}^3$ for titanium Ti–6Al–4V at rather high temperatures [24].",
"Assuming $\\Omega ^*$ does not vary too much with $T$ one can take these figures to estimate that, at $T=300\\,\\text{K}$ , $\\frac{k_BT}{B\\Omega ^*}\\sim 2.3\\times 10^{-5}\\, -\\, 7.0\\times 10^{-5}.$ Because of the small value of the coefficient (REF ), any significant strain $\\varepsilon $ demands that the function $F(\\theta )$ be large, of the order of $10^3$ to have a strain of a few percents.",
"Hence $\\theta $ , or $\\theta _c$ , or both, must be in one of the two asymptotic regions $\\theta \\gtrsim 0$ or $\\theta \\lesssim \\pi /4$ .",
"The threshold stress $\\tau _c$ for grain sliding is generally in the range $0.5-5\\, \\text{MPa}$ , i. e. much smaller than the applied stresses $\\sigma $ that are customary in mechanical tests.",
"Hence the divergence at $\\theta =0$ should be the right one and appreciable strains occur for $\\theta _c(\\varepsilon ,\\dot{\\varepsilon },T)\\approx 0.$ The other pole of function $F(\\theta )$ corresponds to very slow flux, as occurring in superplastic deformation."
],
[
"Theory in the first order in $\\theta _c$",
"Up to the first order in $\\theta $ the expression in between the square brackets in Eqs.",
"(REF ) and (REF ) reduces to $\\begin{aligned}\\bigg [\\frac{1-\\cos (2\\theta )}{\\sin (2\\theta )}-2\\theta \\bigg ( 1&+\\frac{2}{\\pi \\sin (2\\theta )}\\bigg ) \\\\-\\frac{2\\cos (2\\theta )}{\\pi }&+\\frac{\\pi }{2}\\bigg ]\\approx \\frac{\\pi }{2}-\\frac{4}{\\pi }-\\theta .\\end{aligned}$ The constant $\\pi /2-4/\\pi =0.29756$ is not small enough and we can neglect $\\theta $ when compared with it.",
"Thus, with no significant lost of precision the exact equation $\\begin{aligned}&\\frac{dp}{d\\varepsilon }=sB\\frac{\\tau _c\\mathcal {Q}(p,T)}{2\\dot{\\varepsilon }d}\\bigg [ \\frac{1 -\\cos (2\\theta _c)}{\\sin (2\\theta _c)}\\\\&-2\\theta _c\\bigg (1+\\frac{2}{\\pi \\sin (2\\theta _c)}\\bigg )-\\frac{2}{\\pi }\\cos (2\\theta _c)+\\frac{\\pi }{2}\\bigg ],\\end{aligned}$ can be reduced to the much simpler first order differential equation $\\frac{dp}{d\\varepsilon }=s\\left(\\pi -\\frac{8}{\\pi }\\right)\\frac{C_0B\\tau _c\\Omega ^*}{k_BT\\dot{\\varepsilon }}\\exp \\left( -\\frac{\\epsilon _0+\\Omega ^*p}{k_BT}\\right),$ whose solution can be written as $\\begin{aligned}p-p_0=\\frac{k_BT}{\\Omega ^*}\\ln &\\bigg [ 1-C_0\\frac{\\pi ^2-8}{\\pi }\\frac{\\tau _c B}{\\dot{\\varepsilon }}\\bigg (\\frac{\\Omega ^*}{k_B T}\\bigg )^2 \\\\&\\times \\exp \\bigg ( -\\frac{\\epsilon _0+\\Omega ^*p_0}{k_B T}\\bigg )|\\varepsilon |\\bigg ]\\, .\\end{aligned}$ where it was substituted $s\\varepsilon =|\\varepsilon |$ .",
"Eq.",
"(REF ) expresses the main finding of this work: when the modulus $|\\varepsilon |$ of the strain approaches from below the value $\\varepsilon _{\\text{frac}}=\\frac{\\pi \\dot{\\varepsilon }}{(\\pi ^2-8)C_0\\tau _c B}\\bigg (\\frac{k_B T}{\\Omega ^*}\\bigg )^2\\exp \\bigg ( \\frac{\\epsilon _0+\\Omega ^*p_0}{k_B T}\\bigg )$ the hydrostatic pressure $p$ diverges logarithmically.",
"According to the definition (REF ) positive stresses (tension) contribute negatively to the hydrostatic pressure $p$ .",
"If the sample is conveniently annealed prior to the tensile test then $p_0=-\\sigma _0 /3$ , where $\\sigma _0$ is the applied initial tensile stress.",
"As the test proceeds, $p=-(\\sigma +2\\sigma _\\perp )/3$ increases monotonically with $\\varepsilon $ , and the transversal stress $\\sigma _\\perp $ increases from zero to negative (compressive) values.",
"When $\\varepsilon $ approaches the critical value $\\varepsilon _{\\text{frac}}$ the transversal stress $\\sigma _\\perp $ increases very rapidly, producing the characteristic neck and fracture.",
"Therefore, Eq.",
"(REF ) for $\\varepsilon _{\\text{frac}}$ expresses the strain to fracture of the material."
],
[
"Necking and strain to fracture",
"Eq.",
"(REF ) gives the strain to fracture in terms of the constants of the theory.",
"However one can express it in terms of more standard coefficients and easily measurable quantities.",
"Combining Eqs.",
"(REF ), (REF ), and taking into account the asymptotic approximation (REF ) to write $\\cot (2\\theta _0) +2\\theta _0 -\\frac{\\pi }{2}\\approx \\frac{1}{2\\theta _0}\\approx \\frac{\\sigma _0}{2\\tau _c}\\, ,$ Eq.",
"(REF ) can be written as $\\varepsilon _{\\text{frac}}=\\frac{\\pi }{(\\pi ^2-8)}\\frac{k_B T}{B\\Omega ^*}\\frac{\\sigma _0}{\\tau _c}.$ We recall that $\\sigma _0$ is the stress registered when the plastic deformation at the chosen constant strain rate $\\dot{\\varepsilon }$ begins.",
"The bulk modulus $B$ is in tables and the only undetermined parameter is the product $\\Omega ^*\\tau _c$ .",
"However, $\\Omega ^*\\tau _c$ can be determined independently from other features of the plastic deformation of the sample in order to have a parameter free test of Eq.",
"(REF ).",
"To show how well this expression compares with experiment, we include next a study of a representative commercial steel.",
"Figure: Circles represent the stress–strain experimentaldata for a copper–alloyed high–strength interstitial free steel atthe three strain rates shown in the inset .",
"The continuouslines represent the predictions of Eq.",
"() with the parametersoptimizing the fit to the experimental points, shown in Table.Figure REF shows the results of a mechanical test of a copper–alloyed high–strength interstitial free steel at strain rates 1, 20 and $200\\,\\text{s}^{-1}$ [28], together with the fits of Eq.",
"(REF ) with the asymptotic approximation (REF ).",
"The high quality of the agreement between theory and experiment is apparent in the figure, and the very little dispersion of the fitting parameters $\\Omega ^* B$ and $\\tau _c$ shown in Table REF reinforces this perception.",
"The last column of Table REF displays the strain to failure $\\varepsilon _{\\text{frac}}$ for the three strain rates, as given by Eq.",
"(REF ) where the parameters appearing in the left side of Table REF were substituted.",
"The values are very close to those measured in the mechanical testings.",
"Comparisons between predicted strains to fracture with published results of experimental tests for many other commercial alloys exhibit same agreement as the one shown in Fig.",
"REF and Table REF .",
"Table: Values for the parameters giving the fitsof Fig.",
"and calculated strains to failure."
],
[
"Conclusions",
"Although the existence of cracks and imperfections inside a stressed solid may contribute to accelerate fracture, the general cause of ductile fracture is not in them.",
"An ideal fine–grained polycrystalline material, free of voids and cracks, whose grains are prone to slide, readily accommodating each other's shapes, inevitably should fail after a finite plastic strain.",
"The reason is an elementary condition that was advanced some years ago [19] but omitted in other studies: whatever the mechanisms for stress–dependent grain boundary sliding and grain shape accommodation may be, they must be consistent with density conservation to produce a steady flow.",
"However, it is shown here that if the local shear stresses resolved in the planes of grain interfases have a finite threshold for causing grain sliding, density is not conserved in the overall plastic flow.",
"The grains are increasingly compressed as the sample is being stretched, and hence grain sliding can only proceed at the expenses of elastic volume variations of the crystallites.",
"Fracture after a brief plastic strain, typically of a few percents, is a necessary consequence of the polycrystalline nature of the materials.",
"The model gives a simple and precise closed–form equation for the strain to fracture, which is the strain at which the internal hydrostatic pressure diverges."
]
] | 1808.08670 |
[
[
"Electron dynamics method using a locally projected group diabatic Fock\n matrix for molecules and aggregates"
],
[
"Abstract We propose a method using reduced size of Hilbert space to describe an electron dynamics in molecule and aggregate based on our previous theoretical scheme [ T. Yonehara and T. Nakajima, J. Chem.",
"Phys.",
"\\textbf{147}, 074110 (2017) ].",
"The real-time time-dependent density functional theory is combined with newly introduced projected group diabatic Fock matrix.",
"First, this projection method is applied to a test donor--acceptor dimer, namely, a naphthalene--tetracyanoethylene with and without initial local excitations and light fields.",
"Secondly, we calculate an absorption spectrum of five-unit-polythiophene monomer.",
"The importance of feedback of instantaneous density to Fock matrix is also clarified.",
"In all cases, half of the orbitals were safely reduced without loss of accuracy in descriptions of properties.",
"The present scheme provides one possible way to investigate and analyze a complex excited electron dynamics in molecular aggregates within a moderate computational cost."
],
[
"Introduction",
"Recently, the authors proposed a concise method for describing the quantum dynamics of excited electrons traveling over constituent molecules in a molecular aggregate system by utilizing a group diabatic Fock (GDF) matrix [1].",
"The construction scheme for this GDF matrix using atom centered Gaussian basis set was practically demonstrated by Thoss et al.",
"[2] based on the idea of diabatization of block diagonalization of Hamitonian that was originally introduced by Köppel et al.",
"[3] Before that, the applications of this block diabatization concept for a practical calculation can be found in the works on electron conduction problems.",
"[4], [5] This block diabatic scheme for the electron dynamics provides an intuitive understanding of charge and exciton migration as the quantum mechanical transport of electronic energies in a molecular assembly subject to the inherent electronic propensities of the site molecules, starting from any prepared type of initial local excitations considering the light-electron couplings.",
"In the present article, on the basis of the characteristics of a nearly block structured diabatic representation in the GDF method, we introduce a compact representation within an extracted Hilbert subspace.",
"The aim of this paper is two-fold: (1) to introduce a scheme for reducing the computational cost in keeping with an accurate description of the dynamics, and (2) to investigate how many local orbitals are required for a description of dynamics and absorption spectrum toward an interpretation of electron dynamics.",
"Note that the group diabatic representation treated here does not lead by itself to a reduction of computational cost by describing the dynamics with a small subset of the full orbital space.",
"In fact, the GDF matrix has the same dimension as the original Fock matrix before the block diagonalization.",
"(1) and (2) are obtained by using a projection scheme introduced here as an extraction of local group diabatic orbitals associated with a size of the Hilbert space needed for a description of electron dynamics.",
"The reduction of calculation cost is associated with the reduction of the number of extracted local group orbitals as a basis set for constructing matrix representations of electronic operators utilized in a calculation of electron dynamics.",
"This article improves upon our previous work by providing and assessing a procedure for reducing the computational cost of electron dynamics calculation using the GDF method, which is combined with the real-time time-dependent density functional theory (RT-TDDFT) [6], [7], [8], [9] under the adiabatic approximation for electron functionals.",
"The point is that electron migration of chemical interest over a molecular aggregate under moderate light conditions occurs in a sub Hilbert space consisting of a low number of excited states described by molecular orbitals within a relevant but not very large energy range around the Fermi energy of a system.",
"It is instructive to compare other studies with our present method.",
"There are many studies intensively investigating molecular interactions and their effects on excited energy transfer proceeding in excited molecular aggregates.",
"[10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] For example, Futera et al.",
"[23] successfully utilized and assessed the GDF method originally named a projection operator diabatization scheme[2] to evaluate electronic coupling matrix elements with high level ab initio calculations in the electron-transfer process of a molecular-metal/semiconducter interface.",
"However, a study on a systematic variable description of the excited electron dynamics in a bottom-up approach is rare.",
"Compared to previous studies on excited electron transfer in molecular aggregates, the scheme introduced in the present article has the advantages of a compact description of excited electron migration with ab initio electronic structure calculations.",
"In addition our new scheme allows a systematic improvement of the results by enlarging the projection space.",
"The most prominent feature is that our scheme is intended for a real-time dynamics of excited electrons in molecular aggregates in an external field starting from any pattern of initial local excitations prepared as a perturbation for the electronic state.",
"In this article, we detail the procedures for constructing the projected local orbital space within the group diabatic (GD) representation, and then demonstrate numerical applications.",
"We examine the size of a local orbital space related to dynamics in a systematic way by increasing an energy range for projecting a diabatic local orbital space.",
"A naphthalene(NPTL)$-$ tetracyanoethylene(TCNE) dimer is treated as a test donor$-$ acceptor system.",
"Additionally, we also demonstrate a convergence of absorption spectrum with respect to a size of orbital space using a five-unit polythiophene molecule(5UT) as a typical electron donor species in solar cell materials.",
"To ensure the energy balance between local projected orbitals, we employ an energy width parameter for extracting a relevant subspace with the mean value of the highest occupied molecular orbital (HOMO) and lowest occupied molecular orbital (LUMO) energies in the whole system as a reference energy, and we do not use a scheme which requires direct orbital selection.",
"In Sect.",
"II, we explain the theoretical method for describing the electron dynamics based on the GDF representation within projected local diabatic orbitals, which is followed by numerical examples in the section III."
],
[
"Theoretical method",
"In this section, after a brief summary of the electron dynamics method [1] using a GD representation,[1], [11], [12], [13], [10], [17] we introduce a scheme for constructing a concise matrix form with use of projected diabatic local orbitals having a dominant contribution to the underlying dynamics.",
"The determination of diabatic local projection orbitals playing a main role in the present work requires only the parameters of energy ranges for monomers covering the important orbitals around HOMO and LUMO playing in an excited electron dynamics."
],
[
"Overview of the locally projected-space group diabatic representation",
"The transformation from an atomic orbital (AO) representation of physical operators to a GD form consists of the following two transformations being constructed sequentially: (i) a transformation to a representation using the Löwdin orthogonalized atomic basis functions and (ii) a unitary transformation made from the local orbital sets obtained by the diagonalizations of block sub-matrices corresponding to the predetermined monomer groups in the Fock matrix prepared in step (i).",
"The overview of the process of extracting the diabatic local projection orbitals as the primary topic of the present article is as follows: (a) calculate the mean of the HOMO and LUMO orbital energies of the whole system; (b) set an energy range covering the local orbitals for each monomer in which the value obtained in (a) is placed at the middle; (c) obtain the diabatic local projection orbitals of the monomers in the energy range prepared in (b).",
"We refer to the matrix representation within these projection orbitals as the locally projected-space group diabatic Fock (LP-GDF) representation, of which the details are explained in later subsections.",
"The multiplications of matrices associated with the electronic properties and the analyses of the time-dependent electron density using the newly introduced LP-GDF representation are carried out within the projected orbital space.",
"The information related to the size of the orbital space required for a description of the dynamics without loss of accuracy provides us with insight into the sub-Hilbert space relevant to it.",
"In the following subsection, for a self-contained form of the present article, we first summarize the electron dynamics scheme using a GD representation[1] and then proceed to describe how to obtain the diabatic local projection orbitals and how to construct compact matrix representations by using them as a subset of the basis functions.",
"The first step is to prepare a Fock matrix represented by the Löwdin orthogonalized atomic basis function [24] $\\widetilde{F}_{mn} \\equiv \\langle \\widetilde{\\chi }_m | \\widehat{F} | \\widetilde{\\chi }_n \\rangle ,$ where the orthogonalized Löwdin atomic orbitals(AOs) are expressed by $| \\widetilde{\\chi }_n \\rangle = \\sum _j^{\\textrm {AO}} | \\chi _j \\rangle (\\underline{\\underline{S}}^{-1/2})_{jn}$ with $S_{jn} = \\langle \\chi _j | \\chi _n \\rangle $ being the AO overlap matrix element.",
"Here $\\lbrace \\chi _n \\rbrace $ is the original basis set consisting of AOs."
],
[
"Localized orbitals of a subgroup",
"After the classification of $\\lbrace \\widetilde{\\chi }_n \\rbrace $ into subgroups, e.g., monomers, the block structure of the Fock matrix within the Löwdin basis set is determined with its diagonal blocks $\\lbrace \\underline{\\underline{\\widetilde{F}}}_{G_i G_i}\\rbrace $ and off-diagonal ones $\\lbrace \\underline{\\underline{\\widetilde{F}}}_{G_i G_j}\\rbrace _{i\\ne j}$ with i and j ranging from 1 to $N_g$ .",
"$G_i$ denotes the i-th subgroup.",
"$N_g$ is the number of subgroups in the system.",
"Note that we can employ arbitrary divisions of the component atoms in the whole system.",
"The diagonalization of diagonal blocks corresponding to subgroup $G$ , $\\underline{\\underline{\\widetilde{F}}}_{GG}=\\underline{\\underline{D}}_{G}\\,\\underline{\\underline{\\overline{F}}}_{GG}\\,\\underline{\\underline{D}}_{G}^{\\dagger },$ gives rise to the unitary transformation matrix $\\underline{\\underline{D}}_{G}$ , whose column vectors are the linear coefficient vectors of the localized eigenstates expanded in terms of the Löwdin orthogonalized atomic basis functions for the group $G$ .",
"The dagger symbol attached to a matrix indicates its adjoint form.",
"$ \\underline{\\underline{\\overline{F}}}_{GG} $ is the diagonal matrix having the eigen energies $\\lbrace \\epsilon _{j,{G}}\\rbrace _{j = 1 \\sim M_{G} }$ of the corresponding subgroup $G$ as its elements, and $M_{G}$ is the number of local basis functions spanned at group $G$ .",
"Here, $ {G} \\in \\lbrace {G}_1,...,{G}_{N_g} \\rbrace $ .",
"The elements in off-diagonal blocks of the Fock matrix represented by these localized orbitals, associated with different groups, can take non-zero values, which provide a diabatic character in the representation with the use of the collection of these orbital sets.",
"These group localized orbital sets provide a transformation matrix from a Löwdin representation to the GD one to be explained later.",
"Here, we provide a short comment on the ambiguity of group division in a Löwdin representation of Fock operator.",
"In case using diffuse AO orbitals having far larger distribution than a distance between monomers, a center of Löwdin orbital created by the mixture of original AOs can become close to a nuclei belonging to a different group.",
"Though that is one of the possible problem in the group assignment of Löwdin orbitals associated with a natural group sectoring, it does not cause any problem in case with distances of monomers sufficiently larger than AOs and also Löwdin orbitals.",
"[17] In this article, we do not use diffuse orbitals and treat a case with distance of monomers being larger than a spatial range in orbital distribution."
],
[
"Group diabatic Fock matrix",
"The GD representation of the Fock operator $\\underline{\\underline{\\overline{F}}}$ , as one of the main ingredients in the GDF electron dynamics scheme is constructed via the transformation of the Löwdin representation matrix of the Fock operator $ \\underline{\\underline{\\widetilde{F}}} $ .",
"By using the already obtained unitary matrices with the dimensions of the local basis functions associated with groups, $\\lbrace \\underline{\\underline{D}}_{G_i}\\rbrace _{i=1 \\sim N_g}$ , this transformation is expressed by [1], [11], [12], [10] $\\underline{\\underline{\\overline{F}}}_{G_i\\,G_j}=\\underline{\\underline{D}}_{G_i}^{\\dagger }\\,\\underline{\\underline{\\widetilde{F}}}_{G_i\\,G_j}\\,\\underline{\\underline{D}}_{G_j},$ where i and j range from 1 to the number of groups $N_g$ .",
"The assembly of these sub-matrices $ \\left\\lbrace \\underline{\\underline{\\overline{F}}}_{G_i\\,G_j} \\right\\rbrace _{i,j=1\\sim N_g} $ constructs the GD representation which is called GDF matrix and expressed by $ \\underline{\\underline{F}}^{\\textrm {GD}} $ .",
"The physical meaning of the components in this final form is as follows.",
"This sub-matrices $ \\left\\lbrace \\underline{\\underline{\\overline{F}}}_{G_i G_i} \\right\\rbrace _{i=1 \\sim N_g}$ in the diagonal blocks correspond to the local group eigen energies, while $ \\left\\lbrace \\underline{\\underline{\\overline{F}}}_{G_i G_j} \\right\\rbrace _{i \\ne j}$ , placed at the off-diagonal blocks, describes the interactions between different groups.",
"Note that, in this transformation, the information included in the AO, Löwdin, and GDF representations are the same and no approximation is applied."
],
[
"Transformation from the AO representation to the GD representation",
"A matrix representation of any observable operator $\\hat{O}$ in terms of the constructed GD basis set, $\\underline{\\underline{O}}^{\\mathrm {GD}}$ , is related to that of the original AO basis set, $\\underline{\\underline{O}}^{\\textrm {AO}}$ , as $\\underline{\\underline{O}}^{\\mathrm {GD}}=\\underline{\\underline{U}}^{\\dagger } \\,\\underline{\\underline{O}}^{\\textrm {AO}} \\,\\underline{\\underline{U}},$ where $\\underline{\\underline{U}} \\equiv \\underline{\\underline{S}}^{-1/2} \\underline{\\underline{W}}$ .",
"Here, the diagonal block in the sub-transformation matrix $ \\underline{\\underline{W}}$ is given by $\\underline{\\underline{W}}_{G_i G_i} \\equiv \\underline{\\underline{D}}_{G_i}$ for $i$ , while the off-diagonal one is defined as $\\underline{\\underline{W}}_{G_i G_j} \\equiv \\underline{\\underline{0}}$ .",
"The Fock matrix obeys the same transformation rule and is obtained by setting $\\hat{O}=\\hat{F}$ in the above equations, where we know that $ \\underline{\\underline{F}}^{\\textrm {AO}} = \\underline{\\underline{F}} $ and $ \\underline{\\underline{F}}^{\\mathrm {GD}} = \\underline{\\underline{\\overline{F}}} $ .",
"On the other hand, the transformation of the density matrix from the original AO basis set to the GD one is written as $\\underline{\\underline{\\rho }}^\\mathrm {GD}=\\underline{\\underline{U}} \\,\\underline{\\underline{\\rho }}^\\mathrm {AO} \\,\\underline{\\underline{U}}^{\\dagger }.$ Note that the unitarity of $\\underline{\\underline{W}}$ assures total electron conservation with respect to this transformation, $\\textrm {Tr}\\left[ \\underline{\\underline{\\rho }}^{\\mathrm {AO}} \\underline{\\underline{S}} \\right]= \\mathrm {Tr} \\left[ \\underline{\\underline{\\rho }}^\\mathrm {GD} \\right].$"
],
[
"State coupling",
"We can obtain the essential elements needed for the construction of light–electron coupling by setting $ \\hat{O} = \\hat{\\bf r} $ , $ \\partial _{\\bf r} $ in the previous subsection.",
"Here, boldface denotes a vector in a three-dimensional Cartesian space, and ${\\bf r}$ denotes a composite variable of the electron position in three-dimensional space.",
"The first and second operators are responsible for the light–electron coupling in length and velocity forms.",
"[25], [26] Here, we neglect the nonadiabatic coupling and molecular motion to allow us to focus on an examination of the electron dynamics scheme using projected local diabatic group orbitals.",
"Light-electron couplings within GD representation are expressed by $\\underline{\\underline{L}}^\\mathrm {GD}=+ e \\underline{\\underline{{\\bf r}}}^\\mathrm {GD} {\\bf E}$ for the length gauge and $\\underline{\\underline{L}}^\\mathrm {GD}=- i \\hbar \\dfrac{e}{c} {\\bf A}\\underline{\\underline{\\partial _{\\bf r}}}^\\mathrm {GD}$ for the velocity gauge.",
"[25], [26] ${\\bf E}$ and ${\\bf A}$ denote respectively the three-dimensional electric field vector and electromagnetic field vector potential, which generally depend on a point in a three-dimensional space.",
"$\\underline{\\underline{{\\bf r}}}^{\\textrm {G}D}$ and $\\underline{\\underline{\\partial _{\\bf r}}}^{\\textrm {G}D}$ denote the GD presentation matrices respectively for $\\hat{\\bf O}={\\bf r}$ and $\\partial _{\\bf r}$ in Eq.",
"(REF ).",
"In this study, we employ the length gauge.",
"The details on the treatment for these coupling matrices with further approximations applied in dynamics calculation are given in Subsect.",
"REF ."
],
[
"Locally projected space made from subsets of local site orbitals\n",
"This subsection is an essential part of the present article proposing the LP-GDF scheme.",
"Here, we provide the concrete procedures used in it and the corresponding mathematical expressions.",
"Let us consider the diagonal block of the Fock matrix within the GD representation, $\\left\\lbrace \\underline{\\underline{\\widetilde{F}}}_{G_i G_i}\\right\\rbrace $ and the associated local orbital energies $\\lbrace \\epsilon _{j,G_i} \\rbrace _{j=1\\sim N_{G_i}}$ corresponding to the i–th group site.",
"In a situation where the interactions between group monomers are weak compared to those among the atoms in each monomer, we can safely employ the referential local ground state, where all of the electrons assigned to a local site are filled in ascending order from the lowest-energy local orbital.",
"The excited electron dynamics in a molecular aggregate system under moderate sunlight conditions is expected to proceed in low-excited-state manifolds constructed from the local molecular orbitals in a relevant but not very large energy range around the Fermi energy of the system.",
"In a situation involving rather strong light field or molecular interaction, the Hilbert subspace required for a description of the dynamics will become large.",
"It is worthwhile to examine the size of the subspace relevant to the excited electron dynamics by varying the type of molecular interactions such as light-matter coupling and initial local excitations.",
"Thus, in order to examine the size of the orbital space needed for a description of the electron dynamics at a sufficient accuracy, we establish a procedure for projecting a Hilbert subspace around the Fermi energy of the whole molecular aggregate by using a width parameter for the energy range as follows: (i) Set an energy width $\\Delta \\epsilon _i$ covering the sub-space consisting of local orbitals orbitals for the $i$ -th monomer, $G_{i}$ .",
"Here $i$ ranges from 1 to $N_g$ .",
"(ii) Calculate the mean $\\eta $ of the HOMO and LUMO energies of the whole system.",
"(iii) Extract a subset of group diabatic local orbitals in the energy range $ [ \\eta - \\Delta \\epsilon _i/2, \\eta + \\Delta \\epsilon _i/2 ] $ from the whole orbital space.",
"We refer to this as a projection of the diabatic local orbitals.",
"(iv) Construct minor matrices as representations of the electronic operators within the projected orbital basis set obtained in (iii).",
"(v) Follow the same procedure as that in the GDF electron dynamics scheme other than the projection introduced here.",
"This means that time propagation and property analysis are carried out using the obtained small size matrices.",
"The mathematical form of LP-GDF procedure is compactly given in Appendix .",
"In this article, we employ a version of projection independent on the orbitals during a time propagation but constructed from the initial orbitals in dynamics.",
"The orbital projection scheme using an energy width allows for the systematic and natural treatment for general situations involving the unknown energy orders and spatial localities of the GD orbitals in molecular aggregates.",
"Note that to ensure the energy balance between local projection orbitals, we employ an energy width parameter for extracting a relevant subspace with the mean value of the highest occupied molecular orbital (HOMO) and lowest occupied molecular orbital (LUMO) energies in the whole system as a reference energy, and we do not use a scheme which requires direct orbital selection.",
"In the following numerical demonstrations, for focusing on this essential point in the proposed scheme, we employ a parameter for the energy range independent of the monomers, namely, $\\Delta \\epsilon _i$ = $\\Delta \\epsilon $ , which is expressed by $E_{\\rm bw}$ in the figures.",
"The resultant total number of projection orbitals of the monomers is denoted by $N_{\\rm proj}$ throughout the present article."
],
[
"Initial density matrix: local excitation and electron filling",
"The initial density matrices in the GDF representation are prepared so that the diagonal elements in each diagonal block of the corresponding monomer should be occupied up to the number of electrons assigned to this monomer.",
"A simple example can be found in our previous article.",
"[1] Though an off-diagonal filling responsible for the initial coherence is also possible, for simplicity, we consider only the diagonal part in the preparation of the initial state of the density matrix.",
"This issue will be reported in our future article.",
"In the article, we treat the case of the spin-restricted model within the GDF representation as explained below.",
"This is not rigorously the same as that within the canonical (KS or HF) orbital representation.",
"In a restricted case with the same spatial orbitals for different alpha and beta spins in this model, the occupancy of the GDF local orbitals of each monomer is up to half of the number of electrons assigned to this group from the lowest energy orbital.",
"Therefore, in a strict sense, the initial density matrix mentioned above differs from that of the true ground state of the whole system.",
"In fact, this does not cause any problem for examining the migration dynamics of charge and electronic excitations over the constituent monomers in an assembly.",
"[1] Generally, we can make any type of excitation configuration starting from the reference occupations of the GD orbitals.",
"If we want an initial density associated with an excess or a deficiency of electrons in each monomer for treating the case accompanied with a charge moiety, we merely need to set the occupations to the corresponding number of electrons in each monomer.",
"[1] Throughout this article, we treat cases with an overall singlet spin state in a spin-restricted manner.",
"Appendix provides a practical example for preparing an initial density matrix within GD representation."
],
[
"Time propagation of the density matrix in a GD representation",
"In this study, the calculation of the time propagation of an electronic state is performed in terms of the Liouville–von Neumann equation associated with one particle density matrix as follows: $\\dfrac{\\partial }{\\partial t}\\underline{\\underline{\\rho }}^\\mathrm {GD}=-\\dfrac{i}{\\hbar }\\left[\\underline{\\underline{\\mathcal {F}}}^\\mathrm {GD}\\left\\lbrace \\underline{\\underline{\\rho }}^\\mathrm {GD}(t),t \\right\\rbrace ,\\underline{\\underline{\\rho }}^\\mathrm {GD}\\right],$ where $\\underline{\\underline{\\mathcal {F}}}^\\mathrm {GD}\\equiv \\underline{\\underline{F}}^\\mathrm {GD}+\\underline{\\underline{L}}^\\mathrm {GD}.$ $\\underline{\\underline{L}}^\\mathrm {GD}$ is a light–electron coupling matrix.",
"For the length gauge, their corresponding matrix elements have forms of $\\underline{\\underline{L}}^\\mathrm {GD}=+ e \\underline{\\underline{{\\bf r}}}^\\mathrm {GD} {\\bf E}$ , [25], [26] where ${\\bf E}$ is the three-dimensional electric field vector, which generally depend on a point in a three-dimensional space.",
"We used the dipole approximation, namely, long wavelength approximation, [25], [26] since the wavelength of light treated here is sufficiently large compared to the size of the molecular system treated.",
"In the RT-TDDFT, the Fock matrix depends on the time-dependent density matrix.",
"For convenience of a discussion on the numerical demonstration to be presented in the later section, we label the dynamics described by Eq.",
"(REF ) including this dependency as 'RT' meaning 'real time Fock matrix' while the term 'FF' denoting 'frozen Fock matrix' is used to refer to the approximated dynamics with the replacement of the time dependent Fock by that at the initial simulation time.",
"For a technical simplicity and focusing on the projection scheme of electron dynamics, we employ the pure density functional for including an electronic exchange-correlation throughout this article.",
"For obtaining the time dependent electron density matrix, we solved the non-linear Liouville–von Neumann equation associated with the RT-TDDFT and introduced LP-GDF matrix by using the following two types of numerically stable time integrators, namely (1) the predictor-corrector second order Magnus scheme with linear Fock extrapolation (PC2M-LF) [27] and (2) exponential propagation with predictor-corrector SCF scheme using final corrector as a resultant density (EPPC1).",
"[28] PC2M-LF needs one time of update of Fock matrix for each step while an iteration scheme is applied for EPPC1 until a corrector density is converged.",
"Their details are summarized in Appendix .",
"We followed a dynamics with time steps of 8 and 20 attosecond using a PC2M-LF and EPPC1 time integrator, respectively.",
"Hereinafter we abbreviate femtosecond to fs and attosecond to as.",
"In many cases with a moderate dynamics of density the use of former scheme is sufficient, while the latter was needed for a stable calculation of absorption spectrum using short laser pulses.",
"The Fock, electron dipole transition matrices within the AO representation required for the dynamics calculation were evaluated by using the NTChem2013 software package.",
"[29] We should comment on conservations of properties during dynamics.",
"As a special case, we examined a case starting from a locally excited state without external light field.",
"While within the LP-GD scheme a trace of density matrix is perfectly conserved both for RT-TDDFT and Frozen Fock approximation, an expectation value of Fock operator is conserved only for the latter one.",
"Despite this, in a practical sense, this LP-GD RT-TDDFT scheme provides a good result with respect to a convergence to reference data as $\\Delta _\\epsilon $ increases, which will be shown in Sect.",
"with demonstrations of a userbility of the method through calculations of charge migration dynamics and absorption spectrum."
],
[
"Numerical demonstration",
"Here, the size effect of the projected space on the electron dynamics and absorption spectrum are examined by using the LP-GDF electron dynamics method introduced in the previous section.",
"We compare the time-dependent behaviors of the Mulliken charge of electron donor molecules for NPTL–TCNE with different initial excitation and continuum light field by varying $E_{\\rm bw}$ .",
"The convergence of absorption spectrum for 5UT is also examined in the same manner after showing explicitly the importance of the self-consistency between time-dependent density and the time-dependent Fock matrix during electron dynamics.",
"With respect to the NPTL–TCNE, readers can find further information in our previous article.",
"[1] We validate the efficiency of the present projection method by showing a convergence of result with respect to a reduced size of local orbital space.",
"The schematics of the molecular configurations used in this article as shown in Fig.",
"REF .",
"We also provide information related to the dependence of the computational cost in the electron dynamics calculations with and without Fock build on the size of the projection space, which are detailed in Appendix .",
"Figure: Schematics of the geometries of the molecular aggregates treated in this article:(a) NPTL–TCNE dimer (b) 5UTThe centroids of aggregates were set to origin for all the systems.See the supplementary material for detailed information of their Cartesian coordinates."
],
[
"NPTL–TCNE dimer",
"As a first test system, we treat a dimer system consisting of NPTL and TCNE.",
"In this combination of monomers, NPTL serves as an electron donor molecule, while TCNE plays role as an electron acceptor.",
"The molecular geometries and relative orientations of the monomers used here are the same as those in Panel (l) in Fig.",
"3 in our previous article [1] on the original GDF electron dynamics scheme where readers can find further information including the literature of experimental data.",
"The geometry of each monomer was optimized at the DFT/6-31G(d) level with the use of the PBE exchange correlation functional.",
"[30] The Fock matrix associated with initial optimized KS orbitals and its following time-dependent density matrix for the construction of the GDF matrix was also calculated at the same ab initio level.",
"Both molecules have planar geometries in the optimized geometry in their ground electronic states.",
"Here, as shown in Fig.",
"REF , the principal axis of NPTL was set to be parallel to the X axis, while we set TCNE to be parallel to the Y axis.",
"The molecular planes of these flat molecules are parallel to the X–Y plane.",
"They were placed in a parallel orientation with a slide of 1.24 $\\textrm {Å}$ along the Y axis.",
"The distance between molecular planes was fixed at 4 $\\textrm {Å}$ .",
"Although a dimer is treated here, we considered the crystal data[31] reported in the literature with respect to the relative orientation.",
"This selection of molecular configuration yields a non-vanishing overlap between the frontier orbitals, i.e., the HOMO of NPTL and the LUMO of TCNE.",
"[1] See the supplementary material for a further information on the geometrical coordinate of this system as well as GD and canonical orbital energies.",
"Figure: Convergence of the results for the charge separation dynamics in an NPTL–TCNE dimerwith an increase in the energy range determining the projected local orbital space.The energy width used for orbital projection is expressed byE bw E_{\\rm bw} = Δϵ\\Delta \\epsilon in Hartree.The numbers of projected orbitals, N proj N_{\\rm proj}, are 13, 46, 74 and 120respectively for E bw _{bw} = 0.2, 0.5, 1.0 and 1.6.The exact results obtained by using the full 296 orbitals are indicated by solid red line.See the main text for the details of situations with respect to the treatment of Fock matrix,initial local excitation and light field employed in the panels.Fig.",
"REF provides the results of the excited electron dynamics involving initial local excitations and external light fields.",
"The time dependent behaviors of the Mulliken charge of the donor molecule, NPTL, are displayed with the variation of the energy range covering the projection orbital space $\\Delta \\epsilon $ .",
"In the panels of the figure, $\\Delta \\epsilon $ is expressed as $E_{\\textrm {bw}}$ .",
"In panels, we plotted the Mulliken charge of NPTL as an electron donor system for each case corresponding to the vertical axis.",
"The horizontal axis denotes the time in femtoseconds.",
"The results of RT-TDDFT LP-GDF scheme are displayed in panels of (a–d) while corresponding results of frozen Fock approximation are presented in (e–h).",
"(a/e) and (b/f) show the results starting from the initial local excitation respectively in the donor NPTL and acceptor TCNE, while the other moieties are initially in the ground states within the GDF representation.",
"In these four panels, no light field is irradiated to the system.",
"In panels (c/g) and (d/h), a continuum light field is applied, and its field parameters as a wave length and unit vector of polarization are 700 nm corresponding to $\\omega =0.065$ au and $\\left( \\frac{1}{\\sqrt{3}}, \\frac{1}{\\sqrt{3}}, \\frac{1}{\\sqrt{3}} \\right)$ in the XYZ Cartesian coordinate, respectively.",
"The field strength $Es$ used in the cases of panels (c/g) is 0.015 a.u., while a larger strength of 0.02 is applied in (d/h).",
"The function form of light field is given in Appendix .",
"The continuum light field was replaced by a pulse with a sufficiently large time width $t_w=2.4\\times 10^{4}$ fs and peak time $t_c=0$ .",
"As seen in the RT cases (a) and (b) accompanied with initial local excitations without light fields show the fast convergence with respect to the increase of $E_{\\textrm {bw}}$ .",
"In fact, $E_{\\rm bw}=1.0$ associated with 74 projected local orbitals provides a quantitative convergence to the result obtained by the full 296 orbitals.",
"On the contrary, in (c) and (d) from initial local ground states with continuum light field having moderate strength, the quantitative convergence are achieved by $E_{\\textrm {bw}}$ =1.6 providing 120 projected orbitals.",
"As found in the panels (c/g) and (d/h), the FF approximation provides faster convergence compared to RT results, corresponding to the cases involved with light fields.",
"On the other hand, the differences of the converged results of FF from those of RT means the importance of a consideration of feedback from time dependent density matrix to the Fock matrix.",
"In (c/d/g/h), we employed the light stronger than that of usual sun light for a severe assessment of the present scheme in cases accompanied with hard excitations.",
"In fact, in this strength, the non-linear effect about the density matrix result in the difference in convergence rates between frozen Fock type and RT-TDDFT calculations with respect to an increase of number of projected orbitals.",
"Figure: N proj N_{\\rm proj} dependency of max deviations for donnor moleculeduring dynamics from the results by full orbital calculation in Fig.",
".Deviation is expressed as an absolute value but not relative one.The panels of (a), (b), (c) and (d) in this figure correspond to those of(a/e), (b/f), (c/g) and (d/h) in Fig.",
", respectively.“RT” and “FF” in panels correspond to the results obtained by electron dynamics calculationusing RT-TDDFT scheme considering the self-consistency and frozen Fock approximation.To clarify the convergence of time dependent behaviors of charge migration dynamics in Fig.",
"REF with respect to the number of projected orbitals, we displayed the max deviations of the approximation results measured from the reference data obtained by full orbitals in Fig.",
"REF as a function of $N_{\\rm proj}$ .",
"As found in the figure, in all cases we can reduce a half the number of the orbitals keeping with the accuracy matched to the result obtained by using full orbitals.",
"The associated CPU times spent for the calculations and the ratios of them to those obtained by the full orbitals are summarized in Appendix .",
"In the last part of this subsection, as a supplementary information, we provide an evidence of the efficiency of the orbital extraction scheme for the standard RT-TDDFT and Frozen Fock approximation by guiding to the corresponding results.",
"The electron dynamics calculation using the orbital extraction (projection) in standard KS orbital sequence is equivalent to the GDF electron dynamics calculation with $N_g=1$ , namely no group division.",
"This is because a diagonalization of Löwdin AO representation in case of $N_g=1$ gives rise to exactly the same eigen states for the original Kohn-Sham equation at an initial simulation time.",
"In the other words, in the case of $N_g=1$ , total eigen states of Fock operator do not change through a transformation from a representation with original AO to that with Löwdin orthonormalized AOs.",
"We calculated charge migration of this system by treating the whole system as a monomer, namely, by setting $N_g=1$ .",
"The energy threshold value for orbital extraction and number of selected orbitals were varied within LP scheme.",
"The data were compared with those obtained by the methods without any orbital projection nor any group division, namely full orbital type of standard RT-TDDFT or Frozen Fock approach.",
"Because the setting of $N_g=1$ does not allow local excitation of each monomers at initial simulation time in the same sense as $N_g \\ge 2$ , we calculated only the light field cases starting from initial ground state of whole system, namely, no excitation of that.",
"The results and detailed discussion are included in the Supplementary Information.",
"We should also comment that the Frozen Fock approach without GD representation can provide a better convergence as seen in the comparison between the panel (g) in Fig.",
"2 and (c) in Fig.",
"S2 with respect to the $E_{\\rm bw}=0.2$ and full orbital cases.",
"In the same sense, RT-TDDFT without GD approach can be superior in convergence as found in the comparison between the panel (d) in Fig.",
"2 and (b) in Fig.",
"S2 for $E_{\\rm bw}=1.0$ and full orbital cases."
],
[
"5UT",
"In the second demonstration of the present work, we here apply the method to the calculation of absorption spectrum of 5UT.",
"This molecule plays an important role not only as an electron donor but also a hole transfer material in an organic solar cell.",
"[32] The existence of sulfur atoms in $\\pi $ conjugate system gives rise to the stability of molecular structure as well as the increase of charge conduction through the electron overlap contributed by their d-orbitals.",
"[33] Here we do not discuss a chemical aspect of this system, and just focus on the performance of the method by looking the dependency of results on the size of projected local orbital space.",
"The molecular geometry of 5UT treated here was optimized in the PBE/6-31G(d) level calculation, which is commonly used in the electron dynamics calculation.",
"The geometry information is summarized in the supplementary materials.",
"The light pulse was applied and the time-dependent information of induced electronic dipole moments was converted to absorption spectrum, of which details on functional forms are given in Appendix .",
"We used EPPC1 as the time integrator with the time step of 20as.",
"The total time of simulation carried out was 30 fs.",
"The functional form of light field is given in Appendix .",
"The field strength, $E_s$ , central frequency, $\\omega $ , pulse peak time, $t_c$ , pulse width time, $t_w$ , and angular frequency, $\\omega $ , of the applied light pulse were 0.001au, 100(=2.42 fs), 40(=0.97 fs) and 0.0285, respectively.",
"The polarization vector of light field was set to be parallel to the vector of (1,1,1) in the Cartesian coordinate.",
"We set the group number $N_g$ =1.",
"The initial density used in the electron dynamics was set to that of the DFT ground state.",
"Figure: The importance of feedback of time-dependent information to the Fock matix.We presented absorption spectrums of 5UT with the variation of the damping factor,γ\\gamma , and time cut from the initial in the absorption calculation, T CUT T_{\\rm CUT}.See also the supplementary materials for the mathematical form used for the absorption spectrum.Left panels (a–c) and right ones (d–f) correspond to the results of RT and FF, respectively.Note that in all cases here, the full 394 orbitals are used in the dynamics calculationsfor preparing the light pulse induced electronic dipole moments for the spectrum calculations.Triangles denotes the positions of optically allowed excitationsassociated with the absorption energy obtained by linear response TDDFT calculations.The absorption spectrum evaluated from the formula given in Appendix numerically depends on the damping parameter, $\\gamma $ , and the initial cut off time, $T_{\\rm CUT}$ .",
"We checked that the simulation time was long enough for the examined energy range from 0 to 4 eV.",
"In order to select a set of reliable parameters for absorption spectrum, we compared the obtained spectrum by varying $\\gamma $ and $T_{\\rm CUT}$ , which is presented in Fig.",
"REF including the information of positions of optically-allowed excitations in energy space obtained by LR-TDDFT.",
"The transition properties evaluated by LR-TDDFT including excitation energies, oscillator strength, and dipole moments are summarized in the supplementary material.",
"We also compared the results with and without the feedback of time-dependent density to the Fock matrix in order to show that the instantaneous back-reaction of the density to effective Hamiltonian in the simulation is the key factor for the reproduction of the LR-TDDFT results, which are correspondingly labeled with RT and FF defined in the section of theoretical method, Subsection REF .",
"With respect to the parameters, we found the best parameters $\\gamma =0.004$ both for RT and FF cases in the aspects of the balance of proper smoothness and spectrum peak widths.",
"In turn, the best parameters of $T_{\\rm CUT}$ is 5.3 fs and 6.2 for RT and FF based on the observation of the appearance of maximum peaks.",
"Again, we can see that RT calculations presented in the panels (a–c) correctly reproduce the first peak position given by LR-TDDFT while FF ones fail.",
"Then, we chose the RT type calculation for spectrum calculation with $\\gamma =0.004$ and $T_{\\rm CUT}$ =5.3 fs and proceed to check the dependency on the size of projected local orbital space.",
"Figure: Projected size dependency and convergence of absorption spectrum.we compared the results by projected local orbitals, N proj N_{\\rm proj}=64, 89, and 164correspondingly associated with E bw E_{\\rm bw}=0.5, 0.75, and 1.5to that obtained by using N orbs N_{\\rm orbs}=394 full orbitals for 5UT molecule.Here we employed the damping factor of γ=0.004\\gamma =0.004 and T CUT T_{\\rm CUT}=5.3 fsbased on the observation in Fig.",
".Triangles denotes the positions of optically allowed excitations associated with theabsorption energy obtained by linear response TDDFT calculations,which are labeled with the symbol 'LR.OA'.Fig.",
"REF presents the convergence of the absorption spectrum using RT-TDDFT calculation combined with LP-GDF scheme with respect to the size of projected local orbital space.",
"We can observe the monotonic convergence of the results by projection scheme to that obtained by the 394 full orbitals.",
"In fact, the spectrum shapes by $E_{\\rm bw}$ =0.5, 0.75 and 1.5 corresponding to $N_{\\rm proj}$ =64, 89 and 164 projected orbitals monotonically approaches to that of the full orbital calculation.",
"More than half of the total orbitals were reduced in this spectrum calculation.",
"The information of CPU time is also included in the supplementary material.",
"The success in the reduction of diabatic local orbital space required for the sufficient description suggests the efficiency of the introduced projection method for RT-TDDFT."
],
[
"Concluding remarks",
"In this study, we introduced and assessed the electron dynamics method as a combination of the local orbital projection and GDF electron dynamics scheme within a framework of RT-TDDFT.",
"Through the examination of complex charge migration induced by local excitation and light field for the NPTL–TCNE dimer as a test donor-acceptor system, we showed that the present method allows us to investigate the size of the Hilbert subspace associated with the excited electron dynamics of molecular aggregates.",
"In the application of the projection method to absorption spectrum, we numerically demonstrated the reproduction of the result with use of the reduced number of orbitals.",
"We can expect that this method paves a way to a practical investigation of electron dynamics of molecular aggregates with a further combination of the techniques for reducing the cost in Fock build.",
"We conclude by providing a perspective.",
"The molecular motion, non-adiabatic mixing, and transition between electronic states can affect charge and exciton migration dynamics.",
"There, a nuclear quantum effect may play an important role in electron dynamics in molecular aggregate systems having high density of states.",
"The GD representation scheme can afford a realistic and practical model Hamiltonian for treating these issues.",
"In our future work, we will report on a coarse grained quantum dynamics in molecular aggregates having a nano-size for describing a coupled dynamics of charge and exciton migration associated with their birth, extinction and transport."
],
[
"Supplementary material",
"See supplementary material for the issues of (I) validity check of the time interval employed for describing the charge migration dynamics and (II) Efficiency of orbital projection method for standard RT-TDDFT and Frozen Fock method without a group division in charge migration dynamics for NPTL–TCNE system.",
"(III) group diabatic and canonical orbital energies of NPTL–TCNE dimer (IV) optical transition properties of 5UT obtained by LR-TDDFT calculation (V) geometry data of molecular aggregates of a NPTL–TCNE dimer and 5UT.",
"This research was supported by MEXT, Japan, “Next-Generation Supercomputer Project” (the K computer project) and “Priority Issue on Post-K Computer” (Development of new fundamental technologies for high-efficiency energy creation, conversion/storage and use).",
"Some of the computations in the present study were performed using the Research Center for Computational Science, Okazaki, Japan, and also HOKUSAI system in RIKEN, Wako, Japan."
],
[
"Initial density matrix within GD representation associated with local excitations",
"Essential points in preparing initial density matrix are summarized in a following example.",
"Consider the spin-restricted case with three monomers each of which has 2 electrons and 3 orbitals.",
"Here set $N_g=3$ .",
"See the main text about the approximate treatment associated with this term of 'spin-restricted'.",
"If we want to prepare the initial condition such that the first and third monomers are initially excited from local HOMO to LUMO, the initial density matrix in GDF scheme is constructed as follows: [1] set reference density matrix, namely GDF ground state, $\\underline{\\underline{\\rho }}^{\\textrm {GD:ground}}$ and [2] carry out local HOMO LUMO excitation and obtain the aimed GDF density, $\\underline{\\underline{\\rho }}^{\\textrm {GD:1,3-HL}}$ namely, $& \\underline{\\underline{\\rho }}^{\\textrm {GD:ground}} \\equiv \\left(\\begin{array}{ccccccccc}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right) \\\\\\quad & \\Longrightarrow \\\\\\quad & \\underline{\\underline{\\rho }}^{\\textrm {GD:1,3-HL}} \\equiv \\left(\\begin{array}{ccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right).$"
],
[
"Time integrator",
"We detail the time integrators used for solving non-linear Liouville von Neumann equation including time-dependent Hamiltonian depending on the density matrix employed in this study, namely, (1) the predictor-corrector second order Magnus scheme with linear Fock extrapolation (PC2M-LF) and (2) exponential propagation with predictor-corrector SCF scheme using final corrector as a resultant density (EPPC1).",
"Note again that PC2M-LF needs one time of update of Fock matrix for each step while EPPC1 includes an iteration scheme with respect to a convergence of corrector density.",
"See the main text for the original articles of these schemes."
],
[
"PC2M-LF",
"This scheme consists of the following five processes for each time step, $[1] &\\, F_{3} := - \\frac{3}{4}F_{1a} + \\frac{7}{4} F_{1b} \\\\[2] &\\, D_{4} :=e^{- \\frac{i}{\\hbar } \\frac{dt}{2} F_{3} } D_{2} e^{ \\frac{i}{\\hbar } \\frac{dt}{2} F_{3} }\\\\[3] &\\, F_{5} := F[D_{4}] \\\\[4] &\\, D_{6} :=e^{- \\frac{i}{\\hbar } dt F_{5} } D_{2} e^{ \\frac{i}{\\hbar } dt F_{5} } \\\\[5] &\\,( F_{1a}, F_{1b} ) := ( F_{1b}, F_{5} ) \\,\\, \\Rightarrow \\,\\, \\textrm {End of this step}$ If we consider a time propagation from $t$ to $t+dt$ , $F_3$ and $F_5$ respectively denote the Fock matrices at $t+\\frac{1}{4}dt$ and $t+\\frac{1}{2}dt$ while $D_2$ , $D_4$ and $D_6$ correspond to the density matrices at $t$ , $t+\\frac{1}{2}dt$ and $t+dt$ .",
"$F_3$ in the first step [1] is constructed from the Fock matrices, $F_{1a}$ and $F_{1b}$ , respectively denote the Fock matrices in the previous two times respectively at $t-\\frac{1}{2}dt$ and $t-\\frac{3}{2}dt$ , by taking a linear extrapolation of them.",
"It is known that owing to the time dividing points set properly this scheme has the same accuracy of second order Magnus expansion.",
"The time propagation matrix having exponential form is treated exactly using a spectrum representation obtained by the diagonalization of Fock matrix, which is also applied in the EPPC1 scheme explained in the next subsection."
],
[
"EPPC1",
"This integrator involved with a micro iteration consists of the following procedures: $[1] & \\, F_{N} := F[D_{N}] \\\\[2] & \\, D_{N+1}^{\\rm pred} :=e^{- \\frac{i}{\\hbar } dt F_{N} } \\, D_{N} \\, e^{ \\frac{i}{\\hbar } dt F_{N} } \\\\[3] & \\, F^{\\rm pred}_{N+1} := F[D_{N+1}^{\\rm pred}] \\\\[4] & \\, F^{\\rm mid}:= \\frac{1}{2} \\left( F_{N} + F_{N+1}^{\\rm pred} \\right) \\\\[5] & \\, D_{N+1}^{\\rm corr} :=e^{- \\frac{i}{\\hbar } dt F^{\\rm mid} } \\, D_{N} \\, e^{ \\frac{i}{\\hbar } dt F^{\\rm mid} } \\\\[6] & \\,\\,\\left\\lbrace \\begin{array}{l}\\textrm {If} \\,\\, || D_{N+1}^{\\rm corr} - D_{N+1}^{\\rm pred} || < \\epsilon \\,\\,\\, \\\\\\quad \\textrm {then} \\,\\,\\,D_{N+1} := D_{N+1}^{\\rm corr}\\,\\,\\, \\Rightarrow \\,\\,\\textrm {End of this step} \\\\\\textrm {If} \\,\\, || D_{N+1}^{\\rm corr} - D_{N+1}^{\\rm pred} || \\ge \\epsilon \\,\\,\\, \\\\\\quad \\textrm {then} \\,\\,\\,D_{N+1}^{\\rm pred} := D_{N+1}^{\\rm corr}\\,\\,\\, \\Rightarrow \\,\\,\\textrm {Return to [3]}\\end{array}\\right.$ Here, a time propagation is carried out from $t$ to $t+dt$ , which are correspondingly labeled with $N$ and $N+1$ .",
"$F$ and $D$ are the Fock and density matrices.",
"'mid' denotes a mid point of $t$ and $t+dt$ , namely, $t+\\frac{1}{2}dt$ .",
"'pred' and 'corr' are abbreviations of 'predictor' and 'corrector', respectively.",
"The double '$||$ ' symbol means taking the Frobenius norm of a matrix placed between these two symbols.",
"The threshold is expressed by $\\epsilon =n\\alpha \\xi $ with $n$ being the dimension number of a corresponding matrix.",
"$\\alpha $ is set to be a value proportional to the absolute maximum eigen value of the matrix under consideration.",
"Here, we used 1 as $\\alpha $ for simplicity.",
"Thus, $\\xi $ determines the strictness of the self-consistency between the instantaneous density and the Fock matrix."
],
[
"Mathematical form of LP-GDF procedure",
"Here we provide the formal mathematical expression of LP-GDF scheme.",
"At first, one knows that within the GD representation the identity operator is written as $\\hat{1}\\simeq \\sum _{i=1}^{N_g} \\sum _{j=1}^{N_{G_i}}\\mid \\phi _j^{G_i}\\rangle \\langle \\phi _{j}^{G_{i}}\\mid ,$ where $\\left\\lbrace \\mid \\phi _j^{G_i} \\rangle \\equiv \\sum _k\\mid \\widetilde{\\chi }_{k,G_i} \\rangle \\left[ \\underline{\\underline{D}}_{G_i} \\right]_{kj}\\right\\rbrace $ with $ \\mid \\widetilde{\\chi }_{k,G_i} \\rangle $ being the k-th Löwdin orthogonalized atomic orbital basis function spanned in the G$_i$ -th group is the set of GD localized orbitals for the group labeled by $G_i$ and $j$ ranges from 1 to $N_{G_i}$ , which denotes the number of basis functions spanned at the site, $G_i$ .",
"$i$ ranges from 1 to $N_g$ , i.e., the number of monomer group sites.",
"We used the symbol for approximation in the equation because of the practical use of a finite basis set during computation.",
"Note that the GDF orbitals created from Löwdin orthonormal basis remain to be orthogonal under unitary transformations even if the transformations are carried out in each group.",
"In turn, with respect to the structure of matrix representation of the Fock operator, only the GDF orbital pairs between different monomers are Fock non-orthogonal while the pairs within the same group are Fock orthogonal .",
"Next, we introduce projection operators in order to realize steps (i)–(v) in Subsect.REF as follows: $\\hat{P} \\equiv \\sum _{i=1}^{N_g} \\sum _{j \\in \\Omega _i}^{N_{G_i}}\\mid \\phi _j^{G_i}\\rangle \\langle \\phi _j^{G_i}\\mid $ where $\\Omega _i \\equiv \\left\\lbrace j;| \\epsilon _{j,G_i} - \\bar{\\epsilon } |\\le \\Delta \\epsilon _i/2\\right\\rbrace \\quad \\textrm {and}\\quad \\bar{\\epsilon } \\equiv (\\epsilon _{\\textrm {H}}+\\epsilon _{\\textrm {L}})/2.$ Here, $\\epsilon _{j,G_i}$ is the local orbital energy associated with the GD orbital $\\phi _j^{G_i}$ , while $\\epsilon _{\\textrm {H}}$ and $\\epsilon _{\\textrm {L}}$ are the HOMO and LUMO energies of the whole system.",
"By using these mathematical tools within LP-GDF formulation, we approximate a one electron operator $\\hat{O}$ as follows: $\\hat{O}\\approx \\hat{O}_P \\equiv \\hat{P}\\hat{O}\\hat{P}.$"
],
[
"Function form of Laser pulse field",
"The vector potential of light field as a function of time in the long wave length approximation employed takes a form of ${\\bf A}(t) = \\sum _j^{N_\\textrm {p}}{\\bf A}_{j} \\, f(t;t_{c_j},t_{w_j}) \\, \\text{cos}(\\omega _j (t-t_{c_j})+\\delta _j),$ where bold font means the three dimensional vector in the Cartesian coordinate space.",
"Here, the envelope function is defined by $ f(t;t_c,t_w) \\equiv \\textrm {exp} \\left( - \\left( \\dfrac{t-t_c}{t_w} \\right)^2 \\right) $ .",
"The physical meanings of parameters appeared above are as follows; $t$ denotes time, $t_c$ is a field peak time, $t_w$ stands for a typical gaussian decay time, $\\omega $ means a central angular frequency of field, and $\\delta $ is a carrier envelope phase.",
"$N_\\textrm {p}$ denotes a number of pulses.",
"The electric field vector of external light corresponding to ${\\bf A}(t)$ is given by $ {\\bf E}(t) = -\\dfrac{1}{c} \\dfrac{\\partial {\\bf A}(t) }{\\partial t} $ .",
"In the present article, we employed $N_\\textrm {p}=1$ and $\\delta =0$ ."
],
[
"Absorption spectrum",
"The absorption spectrum was evaluated using the following function, $S(\\omega )\\equiv \\dfrac{1}{3}\\textrm {Tr}[\\underline{\\underline{\\widetilde{\\sigma }}}(\\omega )],$ where $\\underline{\\widetilde{\\underline{\\sigma }}}(\\omega )\\equiv \\dfrac{4\\pi \\omega }{c}\\textrm {Im}[\\underline{\\underline{\\widetilde{\\alpha }}}(\\omega )]$ is an absorption cross section with $[\\underline{\\underline{\\widetilde{\\alpha }}}(\\omega )]_{jk}\\equiv \\dfrac{\\widetilde{\\mu }^{\\rm {ind}}_j(\\omega )}{\\widetilde{E}_k(\\omega )}=\\dfrac{ \\int _{T_{\\rm CUT}}^{T_{\\rm FIN}} e^{ i \\omega t } e^{ - \\gamma t} \\mu _{j}(t) }{ \\int _{T_{\\rm CUT}}^{T_{\\rm FIN}} e^{ i \\omega t } E_k(t) }$ .",
"$\\gamma $ is a damping factor.",
"Here, $ \\mu ^{\\rm {ind}}_j(t) \\equiv \\mu _j(t) - \\mu _j(0) $ and ${E_k(t)}$ are induced dipole moment and external electric field, respectively.",
"The symbol of tilde means taking the Fourier transformation.",
"$T_{\\rm FIN}$ is a final simulation time.",
"$T_{\\rm CUT}$ is the initial time used for a transformation from time dependent induced dipole moment of electrons and external field to absorption spectrum."
],
[
"Computational time",
"Here, we provide information about the computational cost of electron dynamics including analysis of the time-dependent properties.",
"Note that the overhead of parallel computation is also included in time.",
"The aim here is to show examples of the performance of the method in cases with use of moderate computer facilities.",
"The specifications of the computer and compiler utilized in this article are as follows: [For NPTL–TCNE] { Intel(R) Xeon(R) Gold 6148 0 @ 2.40GHz with a cache size of 27.5MB, The Intel FORTRAN compiler in Version 18.0.2.199 with level three optimization. }",
"[For 5UT] { Intel(R) Xeon(R) Gold 6152 0 @ 2.10GHz with a cache size of 30.9MB, The Intel FORTRAN compiler in Version 17.0.4.196 with level three optimization. }",
"In all the calculations 40 cpu cores were used in a parallel manner using openMP for matrix-matrix product and MPI for Fock build if needed."
],
[
"NPTL–TCNE",
"Tab.",
"REF and Tab.",
"REF present the comparison of the computation times using different number of projected local orbitals in cases of PC2M-LF and EPPC1 methods, respectively.",
"The system treated is the NPTL–TCNE dimer.",
"A ratio to the time spent in cases with full orbitals expresses an acceleration in computation by using projected space.",
"The tables include the comparisons between RT and FF.",
"Note that the time intervals of one time step employed here are different between these two integrators.",
"We checked the reproduction of the same result by using these two integrators, which is presented in the supplementary material.",
"In Tab.",
"REF for PC2M-LF, though we can see that a moderate acceleration is attained with a help of projected space and acceleration in FF is superior to RT one including Fock build, the use of RT recommended for the reliable calculation and provides still moderate acceleration by projections of local orbital space.",
"In fact, the computation time in RT with $E_{\\rm bw}=1.6$ corresponding to $N_{\\rm proj}=120$ which provide converged results is reduced to one fourth of that spent in the cases using the full orbitals.",
"As seen in Tab.",
"REF , the acceleration in the calculation by projection in case of EPPC1 including micro iterations accompanied with Fock build for a convergence of corrector density is less than those of PC2M-LF in Tab.",
"REF .",
"Despite of this, the case of $E_{\\rm bw}=1.6$ with $N_{\\rm proj}=120$ need approximately the half of the computational time in the full orbital case.",
"Table: CPU time spent for the dynamics calculations plus the time-dependent analysisfor NPTL–TCNE with PC2M-LF scheme with 1875 time steps using 8 as time interval.The unit of time is second.Numeric value in [] denotes the ratio to the cost in case with use of full orbitalswhile that in {}\\lbrace \\rbrace mean the number of Fock build per one time step.Note the single Fock build per one time step in the PC2M-LF scheme.The spent time needed in the cases of Fig.",
"are the samein each group of the panels (a-d) for RT and (e-h) for FF, namely,in the above RT and FF columns correspond to the group of (a-d) and (e-h) in Fig.",
".See the text for the meanings of E bw =ΔϵE_{\\rm bw}=\\Delta \\epsilon , N proj N_{\\rm proj}, RT and FF.The wall time means the wall clock time associated withopenMP for dynamics and MPI for Fock build.Table: CPU time spent for the dynamics calculations plus the time-dependent analysisfor NPTL–TCNE with EPPC1 scheme with 750 time steps using 20 as time interval.The unit of time is second.Numeric value in [] denotes the ratio to the cost in case with use of full orbitalswhile that presented in {}\\lbrace \\rbrace correspond to the average number of cycles inmicro-iterations needed for the density convergence with ξ=10 -8 \\xi =10^{-8}.See the text for the meanings of E bw =ΔϵE_{\\rm bw}=\\Delta \\epsilon , N proj N_{\\rm proj}, RT and FF.",
"(a-d) and (e-h) correspond to RT and FF, respectively,and all the symbols are the same as Fig.",
"in the main text.The definition of the wall time is the same as that in Tab.",
"."
],
[
"5UT",
"Tab.",
"REF presents the comparison of the computation times using different number of projected local orbitals in cases of EPPC1 methods for 5UT system.",
"The way of presentation is the same as the table for NPTL–TCNE.",
"Here also we show the comparisons between RT and FF.",
"As seen in Tab.",
"REF , the case of $E_{\\rm bw}=1.5$ with $N_{\\rm proj}=164$ associated with the converged spectrum presented in Fig.",
"5 in the main text needs merely less than half of the computational time in the case with the full 394 orbitals.",
"Table: CPU time spent for the dynamics calculations plus the time-dependent analysisfor 5UT with EPPC1 scheme with 1500 time steps using 20as time steps.The unit of time is second.Numeric value in [] denotes the ratio to the cost in case with use of full orbitalswhile that presented in {}\\lbrace \\rbrace correspond to the average number of cycles inmicro-iterations needed for the density convergence with ξ=10 -8 \\xi =10^{-8}.See the text for the meanings of E bw =ΔϵE_{\\rm bw}=\\Delta \\epsilon and N proj N_{\\rm proj}.The wall time means the wall clock time, and the values in parenthesesdenote the CPU efficiency in percent associated openMP for dynamics and MPI for Fock build.Supplementary material for “Electron dynamics method using a locally projected group diabatic Fock matrix for molecule and aggregate”"
],
[
"Convergence of charge migration dynamics with respect\nto the time increment using different integrator for NPTL–TCNE system.",
"Fig.",
"SREF shows the convergence of charge migration dynamics for NPTL–TCNE system with respect to time increments.",
"The panels correspond to those in Fig.",
"2 in the main article.",
"For a stringent check, we used different integrators, namely, PC2M-LF and EPPC1, with the time increments of 8 and 20 as in each step, respectively.",
"The perfect agreement with each other means the convergence of the result.",
"Figure: Check of the convergence of charge migration dynamics with respectto the time increment using different integrator for NPTL–TCNE system.See the main text in this supplementary material about the panels."
],
[
"\nEfficiency of orbital projection method for standard RT-TDDFT and Frozen Fock method\nwithout a group division in charge migration dynamics for NPTL–TCNE system.",
"We demonstrate an efficiency of the present local projection (LP) scheme for a standard RT-TDDFT.",
"Note that the standard RT-TDDFT is equivalent to the case with $N_g$ =1 and a ground state of whole system as an initial state within LP-GDF electron dynamics scheme.",
"This is also verified for the calculation of absorption spectrum for 5UT with $N_g$ =1, which is presented in Fig.",
"5 in the main text.",
"Here, for charge migration dynamics in NPTL–TCNE system, we show the efficiency of LP scheme for the standard RT-TDDFT without any group division in the Fock matrix.",
"We also show the results for the Frozen Fock approach.",
"In this context, panels (a/b) and (c/d) in Fig.",
"SREF with $N_g$ =1 in LP-GDF electron dynamics correspond to (c/d) and (g/h) of Fig.",
"2 in the main text with $N_g$ =2, respectively.",
"The same properties are plotted.",
"Time integrator employed here is EPPC1.",
"Strictly speaking, the initial electronic condition is different for both schemes since the former standard case starts from the initial density matrix made from the ground state of whole system while the initial density matrix was prepared by using the combination of local ground states of monomers in the latter LP-GDF case.",
"Despite this, the time dependent behavior is similar for both cases, which inversely supports validity of group division in Fig.",
"2.",
"Fig.",
"SREF is the maximum error analysis for the LP results compared from the full orbital calculations.",
"Panels (A) and (B) correspond to the cases of (a/b) and (c/d) in Fig.",
"SREF The plots of the data were carried out in the same way as Fig.",
"3 in the main text.",
"These results indicate that the LP scheme can safely reduce the computational cost with respect to orbital numbers required for the description also for charge migration dynamics under a light field with using standard RT-TDDFT equivalent to LP-GDF-electron dynamics scheme with $N_g$ =1.",
"Figure: Light induced charge migration dynamics with N g N_g=1 for NPTL–TCNE.Panels (a), (b), (c) and (d) in this figurerespectively correspond to (c), (d), (g) and (h) in Fig.",
"2 in the main text.Figure: N proj N_{\\textrm {proj}} dependency of max deviations for donormolecule during dynamics from the results by full orbital calculationin Fig.",
"S.RT and FF in panel (A) were evaluated from the resultsin (a) and (c) of Fig.",
"S.while RT and FF in panel (B) were evaluated from the results in (b) and (d)."
],
[
"Group diabatic and canonical orbital energies of NPTL–TCNE dimer",
"Tab.",
"REF summarizes the energy differences of LUMO+1, LUMO and HOMO-1 measured from HOMO, which are presented for group diabatic and canonical representations associated with monomers and dimer system, respectively.",
"Table: (Upper table)Energy differences of group diabatized HOMO-1-1, LUMO, and LUMO+1+1 measuredfrom group diabatized HOMO of the electron donor NPTL and acceptor TCNE moietiesin a molecular complex, which are labeled with the symbols ofΔϵ H-1,H G \\Delta \\epsilon ^{\\rm G}_{{\\rm H}-1,{\\rm H}},Δϵ L,H G \\Delta \\epsilon ^{\\rm G}_{{\\rm L},{\\rm H}} , andΔϵ L+1,H G \\Delta \\epsilon ^{\\rm G}_{{\\rm L}+1,{\\rm H}} , respectively.Note that the monomer interactions are includedin the evaluations of the local orbital energies of monomers.",
"(Lower table)Energy differences of canonical HOMO-1-1, LUMO and LUMO+1+1 measured from HOMOfor the whole NPTL–TCNE dimer system, which are denoted by the symbols ofΔϵ H-1,H W \\Delta \\epsilon ^{\\rm W}_{{\\rm H}-1,{\\rm H}},Δϵ L,H W \\Delta \\epsilon ^{\\rm W}_{{\\rm L},{\\rm H}} , andΔϵ L+1,H W \\Delta \\epsilon ^{\\rm W}_{{\\rm L}+1,{\\rm H}} , respectively.In both tables, the unit is atomic unit.GD and CA mean 'group diabatic' and 'canonical', respectively."
],
[
"Excitation energies, transition dipoles and oscillator strength of 5UT",
"Tab.SREF summarizes the information on lower excitations of 5UT at the geometry given in the later section in this supplementary material.",
"We showed the excitation energies, transition dipole moment vectors, and oscillator strength.",
"Here, $(i \\rightarrow j)$ , $\\Delta E_{ij}\\equiv |E_i-E_j|$ , $\\vec{L}_{ij}$ and $f_{ij} \\equiv \\frac{2}{3}(\\Delta E_{ij}) |\\vec{L}_{ij}|^2 $ denote a state pairs of transition, energy difference of them, transition dipole moment vector for them, and its corresponding oscillator strength, respectively.",
"$E_i$ means $i$ -th adiabatic state with 0 being ground electronic state.",
"The geometry of this molecular system is given later in this supplementary material.",
"According to magnitudes of oscillator strength, we considered that 0$\\rightarrow $ 1, 0$\\rightarrow $ 3 and 0$\\rightarrow $ 5 are optically allowed transitions, and used as reference excitation energies in the presentation of Fig.",
"4 and 5 in the main text.",
"Though it is not the aim to reproduce the experimental data in the present article since the main focus is the projection scheme in electron dynamics calculation, we provide information near to the situation treated here with respect to the target system.",
"The experimental data of the position of the first peak in absorption spectrum of optical electronic transition for this 5UT system, however, in the CHCl$_3$ solvent but not in a gas phase, is reported in the former works, [34] and its value is 2.98 eV.",
"Though the absorption spectrum for electronic transition in a molecular system can be changed by an existence of a polar solvent associated with a complex modification of electric transition and structure relaxation, this is not significant in this system.",
"On the other hand, as seen in Fig.",
"4 of the main article as the comparison part of RT and FF results, the value given by RT-TDDFT, 2.2 eV, being consistent with LR-TDDFT result here within the present limited ab initio level is close to the experimental value mentioned above, compared to the peak value, 1.6 eV, given by FF approximation as found in Fig.",
"4.",
"This observation also supports the basic assertion of the importance of the self-consistent treatment for the Fock matrix and time-dependent density matrix in a calculation of absorption spectrum.",
"The remaining discrepancy in the LR-TDDFT result here is remedied by using more appropriate higher level basis set[34] and exchange-correlation functionals [32].",
"In fact, at the same structure, the use of the representative hybrid functionals with a long-range correction, namely, CAM-B3LYP[36] and LC-BOP[37], provide the optically allowed first electronic transition energies in gas phase, 2.80 eV and 2.99, respectively, while their corresponding values for the polarizable continuum model(PCM)[38] with CHCl$_3$ solvent at the same structure are 2.68 and 2.88.",
"In the calculations using the PCM combined with LR-TDDFT here, we used the ab initio program package for electronic structure calculation, GAMESS.",
"[39] Table: Transition properties of 5UT obtained by LR-TDDFT calculation with PBE/6-31G(d).See the main text in this supplementary material for the details."
],
[
"Geometry data of molecular systems",
"We summarize the data of Cartesian coordinates in Angstrom for the molecular systems used in the present article.",
"The information of atoms are also included below."
],
[
"NPTL–TCNE",
"C 1.824214 0.710474 -1.999379 C 1.824214 -0.710473 -1.999379 C 0.628132 -1.408746 -1.999379 C -0.619806 -0.721697 -1.999379 C -0.619809 0.721696 -1.999379 C 0.628130 1.408742 -1.999379 C -1.867746 1.408746 -1.999379 C -3.063829 0.710474 -1.999379 C -3.063829 -0.710473 -1.999379 C -1.867746 -1.408742 -1.999379 H 2.775875 1.251891 -1.999379 H 2.775876 -1.251890 -1.999379 H 0.624772 -2.504937 -1.999379 H 0.624771 2.504934 -1.999379 H -1.864388 2.504937 -1.999379 H -4.015491 1.251889 -1.999379 H -4.015490 -1.251891 -1.999379 H -1.864388 -2.504934 -1.999379 N 2.832029 2.070410 2.000621 N -1.591624 2.070452 2.000621 C 1.839916 1.437817 2.000621 C -0.599565 1.437776 2.000621 C 0.620188 0.694253 2.000621 C 0.620194 -0.694285 2.000621 C 1.839957 -1.437800 2.000621 C -0.599550 -1.437827 2.000621 N 2.832015 -2.070408 2.000621 N -1.591638 -2.070398 2.000621"
],
[
"5UT",
"C -8.572000 1.432812 0.000000 C -9.102000 0.159612 0.000000 S -7.858600 -1.051488 0.000000 C -6.587100 0.165912 0.000000 C -7.149500 1.439512 0.000000 H -9.185200 2.337012 0.000000 H -10.148900 -0.142688 0.000000 H -6.546000 2.351012 0.000000 C -4.632800 -1.495088 0.000000 C -5.196000 -0.220788 0.000000 S -3.927900 0.998512 0.000000 C -2.656500 -0.219888 0.000000 C -3.219600 -1.495488 0.000000 H -5.236800 -2.406288 0.000000 H -2.615500 -2.406588 0.000000 C -0.706100 1.441412 0.000000 C -1.270100 0.165512 0.000000 S 0.000000 -1.054088 0.000000 C 1.270100 0.165512 0.000000 C 0.706100 1.441412 0.000000 H -1.310300 2.352412 0.000000 H 1.310300 2.352412 0.000000 C 3.219600 -1.495488 0.000000 C 2.656500 -0.219888 0.000000 S 3.927900 0.998512 0.000000 C 5.196000 -0.220788 0.000000 C 4.632800 -1.495088 0.000000 H 2.615500 -2.406588 0.000000 H 5.236800 -2.406288 0.000000 C 7.149500 1.439512 0.000000 C 6.587100 0.165912 0.000000 S 7.858600 -1.051488 0.000000 C 9.102000 0.159612 0.000000 C 8.572000 1.432812 0.000000 H 6.546000 2.351012 0.000000 H 10.148900 -0.142688 0.000000 H 9.185200 2.337012 0.000000"
]
] | 1808.08362 |
[
[
"Collective Mode Interferences in Light--Matter Interactions"
],
[
"Abstract We present a theoretical and experimental analysis of transient optical properties of a dense cold atomic gas.",
"After the rapid extinction of a weak coherent driving field (mean photon number $\\sim 1.5$), a transient `flash' is observed.",
"Surprisingly the decay of the `flash' is faster than the decay of the fastest superradiant mode of the system.",
"We show that this `faster than superradiance decay' is expected due to the interference between collective eigenmodes that exhibit a range of frequency shifts away from the bare atomic transition.",
"Experimental results confirm that the initial decay rate of the superradiant flash increases with optical depth, in agreement with the numerical simulations for the experimental conditions."
],
[
"Eigenmode distribution",
"In Fig.",
"REF we plot the eigenmodes for an ensemble of $N=400$ atoms as in Fig.",
"1b in the main text, except now showing 100 random realizations rather than a single realization.",
"This gives a more complete representation of the overall eigenmode distribution.",
"Importantly, it shows that barely any eigenmode decay rates exceed $2\\Gamma _0$ even for 100 repetitions while the decay rate measured for the equivalent optical depth in Fig.",
"4a in the main text exceeds $\\sim 2.5\\Gamma _0$ .",
"This excludes the possibility that the enhanced decay rate could be due to spurious large eigenmode decay rates.",
"Figure: Eigenvalue distribution.a Eigenmode shifts Δ p \\Delta _p and widths Γ p \\Gamma _p for an ensemble of N=400N=400 interacting atoms using the same parameters as Fig.",
"1b from the main text, repeated 100 times.",
"b,c Eigenvalue probability density PP."
]
] | 1808.08415 |
[
[
"Tree-based Particle Smoothing Algorithms in a Hidden Markov Model"
],
[
"Abstract We provide a new strategy built on the divide-and-conquer approach by Lindsten et al.",
"(2017) to investigate the smoothing problem in a hidden Markov model.",
"We employ this approach to decompose a hidden Markov model into sub-models with intermediate target distributions based on an auxiliary tree structure and produce independent samples from the sub-models at the leaf nodes towards the original model of interest at the root.",
"We review the target distribution in the sub-models suggested by Lindsten et al.",
"and propose two new classes of target distributions, which are the estimates of the (joint) filtering distributions and the (joint) smoothing distributions.",
"The first proposed type is straightforwardly constructible by running a filtering algorithm in advance.",
"The algorithm using the second type of target distributions has an advantage of roughly retaining the marginals of all random variables invariant at all levels of the tree at the cost of approximating the marginal smoothing distributions in advance.",
"We further propose the constructions of these target distributions using pre-generated Monte Carlo samples.",
"We show empirically the algorithms with the proposed intermediate target distributions give stable and comparable results as the conventional smoothing methods in a linear Gaussian model and a non-linear model."
],
[
"Introduction",
"A hidden Markov model (HMM) is a discrete-time stochastic process $\\lbrace X_{t}, Y_{t}\\rbrace _{t \\ge 0}$ where $\\lbrace X_{t}\\rbrace _{t \\ge 0}$ is an unobserved Markov chain.",
"We only have access to $\\lbrace Y_{t}\\rbrace $ whose distribution depends on $\\lbrace X_{t}\\rbrace $ .",
"We make the following assumptions in the entire article: The densities of the initial state $X_{0}$ , the transition density $X_{t+1}$ given $X_{t} = x_{t}$ and the emission density $Y_{t}$ given $X_{t} = x_{t}$ taken with respect to some dominating measure exist and are denoted as follows: $X_{0} &\\sim p_{0}(x_{0}) &&\\\\X_{t+1}| \\lbrace X_{t} = x_{t} \\rbrace &\\sim p(x_{t+1} | x_{t}) && \\text{~~for } t = 0, \\ldots , T-1,\\\\Y_{t} | \\lbrace X_{t} = x_{t}\\rbrace &\\sim p(y_{t}|x_{t}) && \\text{~~for } t = 0, \\ldots , T,$ where $T$ is the final time step of the process.",
"We are interested in the (marginal) smoothing distributions $\\lbrace p(x_{t}|y_{0:T})\\rbrace _{t = 0, \\ldots , T}$ or the joint smoothing distribution $p(x_{0:T}|y_{0:T})$ where $x_{0:T}$ and $y_{0:T}$ are abbreviations of $(x_{0}, \\ldots , x_{T})$ and $(y_{0}, \\ldots , y_{T})$ , respectively.",
"Exact solutions are available for linear Gaussian HMM using a Rauch–Tung–Striebel smoother (RTSs) [21] and in a HMM with finite-space Markov chains [3].",
"In most other cases, the smoothing distributions are not analytically tractable.",
"A large body of work uses Monte Carlo methods to approximate the smoothing distributions $\\lbrace p(x_{t}|y_{0:T})\\rbrace _{t = 0, \\ldots , T}$ or the joint smoothing distribution $p(x_{0:T}|y_{0:T})$ .",
"Sequential Monte Carlo (SMC) methods [7] are commonly used to sequentially update the filtering distributions $\\lbrace p(x_{t}|y_{0:t})\\rbrace _{t = 0, \\ldots , T}$ .",
"SMC can in principle be used to estimate the joint smoothing density $p(x_{0:T}|y_{0:T})$ by updating the entire history of the random samples in each resampling step.",
"However, the performance can be poor, as path degeneracy will occur in many settings [2].",
"Advanced sequential Monte Carlo methods with desirable theoretical and practical results have been developed in recent years including sequential Quasi-Monte Carlo (SQMC) [11], divide-and-conquer sequential Monte Carlo (D&C SMC) [17], multilevel sequential Monte Carlo (MSMC) [4] and variational sequential Monte Carlo (VSMC) [20].",
"Other smoothing algorithms have been suggested previously.",
"[8] develop the forward filtering backward smoothing algorithm (FFBSm) for sampling from $\\lbrace p(x_{t}|y_{0:T})\\rbrace _{t = 0, \\ldots , T}$ based on the formula proposed by [14].",
"[12] propose the forward filtering backward simulation algorithm (FFBSi) which generates samples from the joint smoothing distribution $p(x_{0:T}|y_{0:T})$ .",
"[5] propose a two-filter smoother (TFS) which employs a standard forward particle filter and a backward information filter to sample from $\\lbrace p(x_{t}|y_{0:T})\\rbrace _{t = 0, \\ldots , T}$ .",
"Typically, these algorithms have quadratic complexities in $N$ for generating $N$ samples.",
"[9] and [16] propose two smoothing algorithms with lower computational complexity, but their methods do not provide unbiased estimates.",
"In this article, we suggest using the divide-and-conquer sequential Monte Carlo (D&C SMC) [17] approach to address the smoothing problem.",
"The D&C SMC algorithm performs statistical inferences in probabilistic graphical models.",
"It splits the random variables of the target distribution into multiple levels of disjoint sets based upon an auxiliary tree $\\mathcal {T}$ .",
"An intermediate target distribution needs to be assigned to each set of random variables yielding sub-models for each non-leaf node.",
"The choice of these intermediate target distributions is key for a good overall performance of the algorithm.",
"By sampling independently from the leaf nodes and gradually propagating, merging and resampling from the leaf nodes to the root, the D&C SMC algorithm eventually produces samples from the target distribution.",
"The merging step involves importance sampling.",
"Using the idea of D&C SMC, we aim to estimate the joint smoothing distribution $p(x_{0:T}|y_{0:T})$ and thus call the algorithm: `tree-based particle smoothing algorithm' (TPS).",
"The key differences between TPS and other smoothing algorithms lie in its non-sequential and more adaptive merging step of the samples.",
"Our main contribution is the proposition and investigation of three classes of intermediate target distributions to be used in the algorithm.",
"We denote a leaf node corresponding to a single random variable $X_{j}$ by $\\mathcal {T}_{j} \\in \\mathcal {T}$ and a non-leaf node corresponding to the random variables $X_{j:l}$ by $\\mathcal {T}_{j:l} \\in \\mathcal {T} (j < l)$ .",
"The first class advised by [17] has the density proportional to the product of all transition and emission densities associated to the target variable $X_{j}$ (resp.",
"$X_{j:l}$ ) in the sub-model.",
"This is equivalent to the unnormalised likelihood of a new HMM starting at time $j$ (resp.",
"from time $j$ to $l$ ) given the observations of the same time interval with an uninformative prior of $X_{j}$ if $j \\ne 0$ .",
"The second class uses an estimate of the filtering distribution $p(x_{j} | y_{0:j})$ at $\\mathcal {T}_{j} \\in \\mathcal {T}$ and an estimate of the joint filtering distribution $p(x_{j:l} | y_{0:l})$ at $\\mathcal {T}_{j:l} \\in \\mathcal {T}$ .",
"Working with this estimate involves tuning a preliminary particle filter.",
"The third class uses estimates of the marginal smoothing distribution $p(x_{j} | y_{0:T})$ at $\\mathcal {T}_{j} \\in \\mathcal {T}$ and of the joint smoothing distribution $p(x_{j:l} | y_{0:T})$ at $\\mathcal {T}_{j:l} \\in \\mathcal {T}$ .",
"We will see that this class of immediate distributions is optimal in a certain sense.",
"Furthermore, under this construction, we approximately retain the marginal distribution of all single random variables $\\lbrace X_{j}\\rbrace _{j = 0}^{T}$ invariant as the marginal smoothing distributions $\\lbrace p(x_{j} | y_{0:T}) \\rbrace _{j = 0}^{T}$ at every level of the tree.",
"The price of implementing TPS using the second class of intermediate target distributions relies on both the estimates of the filtering and the (marginal) smoothing distributions, but not necessarily the joint smoothing distribution.",
"We then propose some parametric and non-parametric approaches to construct these intermediate distributions based on the pre-generated Monte Carlo samples considering both efficiency and accuracy.",
"The article is structured as follows.",
"We first describe the divide-and-conquer approach for particle smoothing in Section .",
"We discuss the intermediate target distributions and the constructions of the initial sampling distributions at the leaf nodes in Section .",
"In Section , we conduct simulation studies in a linear Gaussian and non-linear non-Gaussian HMM to compare TPS with other smoothing algorithms.",
"The article finishes with a discussion in Section ."
],
[
"Tree-based Particle Smoothing Algorithm (TPS)",
"This section outlines an algorithm we call `tree-based particle smoothing algorithm' (TPS).",
"[17] describe the construction of an auxiliary tree for general probabilistic graphical models.",
"We demonstrate a unique construction of an auxiliary binary tree from a HMM bearing intermediate target distributions specified at each node.",
"We then illustrate the sampling procedure for the target distributions at the nodes.",
"We present an algorithm which can be applied recursively from the leaf nodes towards the root and yet generate the target samples."
],
[
"Construction of an auxiliary tree",
"TPS splits a HMM into sub-models based upon a binary tree decomposition.",
"It first divides the random variables $X_{0:T}$ into two disjoint subsets and recursively apply binary splits to the resulting two subsets until the resulting subset consists of only a single random variable.",
"Each generated subset corresponds to a tree node and is assigned an intermediate target distribution.",
"The root characterises the complete model with the target distribution $p(x_{0:T}|y_{0:T})$ .",
"Initial samples are generated at the leaf nodes, independent between leaves.",
"Theses samples are recursively merged using importance sampling until the root of the tree is reached.",
"We propose one intuitive way of implementing the binary splits which ensures that the left subtree is always a complete binary tree and contains at least as many nodes as the right subtree.",
"We split a non-leaf node with the variables $X_{j:l}$ where $0 \\le j < l \\le T$ , into two children $\\mathcal {T}_{j:k-1}$ and $\\mathcal {T}_{k:l}$ with the random variables $X_{j:k-1}$ and $X_{k:l}$ , where $k = j + 2^{p},$ and $p = \\lceil \\frac{\\log (l-j+1)}{\\log 2}\\rceil - 1.$ The auxiliary tree when $T = 5$ is shown in Figure REF .",
"This construction has several advantages: The random variables within each node have consecutive time indices.",
"The left subtree is also a complete binary tree of $2^{\\lceil \\frac{\\log (T+1)}{\\log 2}\\rceil }$ leave nodes.",
"$\\lbrace y_{T+1}, y_{T+2}, \\ldots \\rbrace $ become available, as samples from the complete subtree would not need to be updated.",
"Moreover, the tree has a height of $\\big (\\lceil \\frac{\\log (T+1)}{\\log 2}\\rceil + 1\\big )$ levels, which implies a maximum number of $\\lceil \\frac{\\log (T+1)}{\\log 2}\\rceil $ updates of the samples corresponding to a single random variable with different target distributions at different levels of the tree.",
"Usually, more updates potentially indicate more resampling steps, which may cause more serious degeneracy problems.",
"In Figure REF , the samples corresponding to $X_{0}, \\ldots , X_{3}$ need to be updated three times from the leave nodes and those of $X_{4}, X_{5}$ need to be updated twice.",
"When running a bootstrap particle filter to solve the smoothing problem, the samples at time step $t = 0$ need to be updated $T$ times and thus the maximum number of the updates become $T$ , which is no less than $ \\lceil \\frac{\\log (T+1)}{\\log 2}\\rceil $ .",
"[17] also propose a general way of constructing the auxiliary tree in a self-similar model family, where a HMM belongs to.",
"Their construction in the context of a HMM may not be identical to ours with no restriction on the choice of the cutting point.",
"Figure: An auxiliary binary tree consisting of random variables when T=5T = 5"
],
[
"Sampling procedure in the sub-models of tree",
"We describe the sampling approach from the target distribution at a leaf and non-leaf node of the constructed binary tree $\\mathcal {T}$ described in Section REF .",
"We denote a target density by $f_{j}$ which can be straightforwardly sampled from at a leaf node $\\mathcal {T}_{j} \\in \\mathcal {T}$ , a proper importance density by $h_{j:l}$ and a target density by $f_{j:l}$ respectively at a non-root tree node $\\mathcal {T}_{j:l} \\in \\mathcal {T}$ where $0 < l - j <T$ .",
"At the root, the target density is always $f_{0:T} = p(x_{0:T}|y_{0:T})$ .",
"At a leaf node $\\mathcal {T}_{j}$ , we sample from $f_{j}$ directly.",
"At a non-root node $\\mathcal {T}_{j:l}$ , we employ an importance sampling step with the proposal $h_{j:l} = f_{j:k-1} f_{k:l}$ being the product of the target densities from the two children of $\\mathcal {T}_{j:l}$ .",
"Practically, we merge the samples from $\\mathcal {T}_{j:k-1}$ and $\\mathcal {T}_{k:l}$ respectively and reweigh them.",
"[tbp] j = l Simulate $x_{j}^{(i)} \\sim f_{j}(x_{j})$ for $i = 1,2, \\ldots , N$ .",
"Return $\\lbrace x_{l}^{(i)}, w_{l}^{(i)} = \\frac{1}{N} \\rbrace _{i = 1}^{N}.$ Let $p = \\lceil \\frac{\\log (l-j+1)}{\\log 2}\\rceil - 1$ and $k = j+ 2^{p}$ .",
"$\\lbrace \\tilde{x}_{j:k-1}^{(i)}, \\tilde{w}_{j:k-1}^{(i)} \\rbrace _{i = 1}^{N} \\leftarrow \\texttt {TS}(j,k-1)$ from $\\mathcal {T}_{j:k-1}$ and $\\lbrace \\tilde{x}_{k:l}^{(i)}, \\tilde{w}_{k:l}^{(i)} \\rbrace _{i = 1}^{N} \\leftarrow \\texttt {TS}(k,l)$ from $\\mathcal {T}_{k:l}$ .",
"Denote the combined particles by $\\lbrace \\tilde{x}_{j:l}^{(i)} = (\\tilde{x}_{j:k-1}^{({i})}, \\tilde{x}_{k:l}^{({i})}), \\tilde{w}_{j:l}^{(i)} = \\tilde{w}_{j:k-1}^{({i})} \\tilde{w}_{k:l}^{({i})} \\rbrace _{i = 1}^{N}$ .",
"Update the unnormalised weights for $i = 1, \\ldots , N$ : $\\hat{w}^{(i)}_{j:l} = \\tilde{w}_{j:l}^{(i)} \\frac{f_{j:l} (\\tilde{x}_{j:l}^{(i)})}{ f_{j:k-1} ( \\tilde{x}^{({i})}_{j:k-1}) f_{k:l}(\\tilde{x}^{({i})}_{k:l})}.$ Resample $\\big \\lbrace \\tilde{x}_{j:l}^{(i)}, \\hat{w}^{(i)}_{j:l} \\big \\rbrace _{i = 1}^{N}$ to obtain the normalised weighted particles $\\big \\lbrace x_{j:l}^{(i)}, w^{(i)}_{j:l} \\big \\rbrace _{i = 1}^{N}$ .",
"Return $\\big \\lbrace x_{j:l}^{(i)}, w^{(i)}_{j:l} \\big \\rbrace _{i = 1}^{N}.$ Algorithm TS($j, l$ ) which generates weighted samples from the target $f_{j:l}$ Figure: Computational flow of 𝚃𝚂\\texttt {TS} (see Algorithm ) in a HMM for T=5T = 5.",
"Each non-root node contains the weighted samples from the intermediate target distributions.",
"The generation of the samples starts from the leaves following the branches towards the root of the auxiliary binary tree.Algorithm REF demonstrates the generation of $N$ weighted samples $\\big \\lbrace x_{j:l}^{(i)}, w^{(i)}_{j:l} \\big \\rbrace $ from the target $f_{j:l}$ at $\\mathcal {T}_{j:l}$ .",
"It adopts the pre-stored weighted particles $\\big \\lbrace \\tilde{x}_{j:k-1}^{(i)}, \\tilde{w}^{(i)}_{j:k-1} \\big \\rbrace _{i = 1}^{N}$ from $\\mathcal {T}_{j:k-1}$ and $\\big \\lbrace \\tilde{x}_{k:l}^{(i)}, \\tilde{w}^{(i)}_{k:l} \\big \\rbrace _{i = 1}^{N}$ from $\\mathcal {T}_{k:l}$ where $k$ is the cutting point defined in Equation (REF ).",
"The algorithm first merges the weighted particles $\\big \\lbrace \\tilde{x}_{j:l}^{(i)} = \\big ( \\tilde{x}_{j:k-1}^{({i})}, \\tilde{x}_{k:l}^{({i})} \\big ) \\big \\rbrace _{i = 1}^{N}$ from the children which forms an approximation of the distribution with density $f_{j:k-1} f_{k:l}$ .",
"The algorithm reweighs the combined samples using importance sampling to target the new distribution $f_{j:l}$ .",
"We retain the notation of the weights in the algorithm since some return unequal weights including Chopthin algorithm [10] while others including multinomial resampling, residual resampling [18] and systematic resampling [15] return equal weights.",
"We apply the algorithm recursively from the leaf nodes to the root of the auxiliary binary tree which yields the samples from the final target $f_{0:T} = p(x_{0:T} | y_{0:T})$ .",
"The computational flow is shown in Figure REF when $T = 5$ .",
"The setting of the algorithms are the same as the paper by [17] with additional attentions to the form of the proposals and intermediate target distributions associated to the tree nodes.",
"According to Proposition 1 and 2 in [17], the unbiasedness of the normalising constant and the consistency can be verified under some regularity conditions given valid proposals and an exchangeable resampling procedure."
],
[
"Intermediate target distributions in TPS",
"Given an auxiliary tree $\\mathcal {T}$ constructed in a way described in Section REF , we define the intermediate target distributions of the sub-models associated to the nodes in the tree.",
"We apply [17]'s method to build one class of intermediate target distribution $\\lbrace f_{j:l}\\rbrace _{\\mathcal {T}_{j:l} \\in \\mathcal {T}}$ and develop two new classes, based on the filtering and the smoothing distribution, respectively."
],
[
"Target suggested by {{cite:9a9318e0c7a094262eab626a214e9f7c1134677c}}",
"[17] recommends a class of intermediate target distributions with densities proportional to the product of the factors within the probabilistic graphical model.",
"We apply the method to a HMM which bears binary and unary factors.",
"A binary factor refers to a transition density of two consecutive hidden states.",
"An unary factor refers to a prior density of a hidden state or the emission density between a hidden state and its observation.",
"We call the tree-based particle smoothing algorithm with the above idea TPS-L as suggested by [17].",
"At a leaf node $\\mathcal {T}_{j}$ where the sub-model only contains a single random variable $X_{j}$ given the observation $Y_{j} = y_{j}$ , the target distribution contains no binary factor and is defined as $f_{0}(x_{0}) \\propto p_{0}(x_{0})p(y_{0} | x_{0})$ when $j = 0$ and $f_{j}(x_{j}) \\propto p(y_{j} | x_{j})$ when $j \\ne 0$ .",
"At a non-leaf node $\\mathcal {T}_{j:l}$ , the target density is proportional to the product of all transition and emission densities containing the hidden states in the sub-model: $f_{j:l}(x_{j:l}) &\\propto & p(y_{j} | x_{j}) \\prod ^{l-1}_{i = j} \\bigg \\lbrace p(x_{i+1} | x_{i}) p(y_{i+1} | x_{i+1}) \\bigg \\rbrace .$ When $j = 0$ , the prior density of $X_{0}$ is additionally multiplied.",
"Assume $\\mathcal {T}_{j:l}$ connects two children $\\mathcal {T}_{j:k-1}, \\mathcal {T}_{k:l} \\in \\mathcal {T}$ carrying the pre-generated particles: $\\lbrace \\tilde{x}^{(i)}_{j:k-1}, \\tilde{w}^{(i)}_{j:k-1} \\rbrace _{i = 1}^{N} \\sim f_{j:k-1}$ at $\\mathcal {T}_{j:k-1} \\in \\mathcal {T}$ and $\\lbrace \\tilde{x}^{(i)}_{k:l}, \\tilde{w}^{(i)}_{k:l} \\rbrace _{i = 1}^{N} \\sim f_{k:l}$ at $\\mathcal {T}_{k:l} \\in \\mathcal {T}$ .",
"The unnormalised importance weight $\\hat{w}^{(i)}_{j:l}$ of the combined particle $\\tilde{x}^{(i)}_{j:l} = (\\tilde{x}_{j:k-1}^{({i})}, \\tilde{x}_{k:l}^{({i})})$ in Equation (REF ) becomes: $\\hat{w}^{(i)}_{j:l} = \\tilde{w}_{j:l}^{(i)} p(\\tilde{x}^{(i)}_{k} | \\tilde{x}^{(i)}_{k-1}),$ where $\\tilde{x}^{(i)}_{k-1}$ is the last element in $\\tilde{x}_{j:k-1}^{({i})}$ and $\\tilde{x}^{(i)}_{k}$ is the first element in $\\tilde{x}_{k:l}^{({i})}$ .",
"The tree-based sampling algorithm employing this type of intermediate target distributions is simple to implement, which does not involve any estimation techniques in the algorithms discussed in Section REF and REF .",
"TPS-L only requires the initial sampling of the particles from $f_{j}$ and applies importance sampling with a straightforward weight formula to merge them towards the root of the tree.",
"The initial sampling distribution $f_{j}$ for $j \\ne 0$ is equivalent to the posterior given a single observation $y_{j}$ from an uninformative prior.",
"Correspondingly, the target distribution $\\mathcal {T}_{j:l}$ only incorporates the observations from time $j$ to $l$ with no information beforehand or afterward.",
"We will see in the simulation section that with only one observation conditioned on, the initial sampling distribution may be vastly different from the marginal smoothing distribution, thus resulting in poor estimation results."
],
[
"Estimates of filtering distributions as target",
"The second class of target distributions is based on estimates of filtering distributions and thus we name the algorithm TPS-EF.",
"At the root, the target distribution is $f_{0:T}(x_{0:T}) = p(x_{0:T}|y_{0:T}) = p_{0}(x_{0}) p(y_{0}|x_{0}) \\prod ^{T-1}_{i = 0} \\bigg \\lbrace p(x_{i+1} | x_{i}) p(y_{i+1} | x_{i+1}) \\bigg \\rbrace .$ At a leaf node $\\mathcal {T}_{j} \\in \\mathcal {T}$ , we use an estimate of the filtering distribution $f_{j}(x_{j}) = \\hat{p}(x_{j} | y_{0:j}) \\approx p(x_{j} | y_{0:j})$ whose exact form and sampling process will be discussed in Section REF .",
"At a non-leaf and non-root node $\\mathcal {T}_{j:l} \\in \\mathcal {T}$ , we define the intermediate target distribution: $f_{j:l}(x_{j:l}) &\\propto & \\hat{p}(x_{j} | y_{0:j}) \\prod ^{l-1}_{i = j} \\bigg \\lbrace p(x_{i+1} | x_{i}) p(y_{i+1} | x_{i+1}) \\bigg \\rbrace \\approx p(x_{j:l} | y_{0:l}).$ The weight of the merged sample $\\tilde{x}^{(i)}_{j:l} = (\\tilde{x}_{j:k-1}^{({i})}, \\tilde{x}_{k:l}^{({i})})$ in Equation (REF ) becomes: $\\hat{w}^{(i)}_{j:l} = \\tilde{w}^{(i)}_{j:l} \\frac{ p( \\tilde{x}_{k}^{({i})} | \\tilde{x}_{k-1}^{({i})} ) p(y_{k} | \\tilde{x}_{k}^{({i})}) }{ \\hat{p}_{k} (\\tilde{x}^{({i})}_{k} | y_{0:k}) }.",
"$ Under such constructions of the intermediate target distributions, the particles at the leaf nodes are initially generated from (an estimate of) the filtering distribution.",
"Whilst moving up the tree, their empirical marginal distributions gradually shifts towards the smoothing distributions.",
"One downside of this is that this may eliminate a large population of particles, as the transition is accomplished via importance sampling, particularly if the discrepancy between the filtering and smoothing distribution is large."
],
[
"Kullback–Leibler divergence between the target and proposal distribution",
"Before proposing the second type of intermediate target distributions, we present an optimal type of proposal attaining the minimum Kullback–Leibler (KL) divergence [6] by assuming the random variables $X_{j:k-1} \\in \\mathcal {T}$ and $X_{k:l} \\in \\mathcal {T}$ from the sibling nodes being independent.",
"Given the proposal $h_{j:l} = f_{j:k-1} f_{k:l}$ being the product of the densities of two independent random variables, the minimum KL divergence is met when the two densities are the marginals of the target densities with respect to the corresponding random variables.",
"For simplicity of the notations, we denote the target density at a non-leaf node to be $f(\\mathbf {x_{1}}, \\mathbf {x_{2}})$ where $\\mathbf {X_{1}}, \\mathbf {X_{2}}$ are the random variables with the same time indices from the children but not necessarily the same probability measure.",
"A valid proposal density $h_{1}( {\\mathbf {x_{1}}}) h_{2}({ \\mathbf {x_{2}}})$ satisfies $h_{1}( {\\mathbf {x_{1}}}) h_{2}({ \\mathbf {x_{2}}}) > 0$ whenever $f(\\mathbf {x_{1}}, \\mathbf {x_{2}}) > 0$ , where we assume $h_{1}$ and $h_{2}$ are the probability densities of two independent (joint) random variables $\\mathbf {X_{1}}$ and $\\mathbf {X_{2}}$ .",
"We claim that proposal $f_{1}({\\mathbf {x_{1}}}) f_{2}({ \\mathbf {x_{2}}})$ has the smallest KL divergence among all proposals of the form $h_{1}(\\mathbf {x_{1}})h_{2}({\\mathbf {x_{2}}})$ where $f_{1}({\\mathbf {x_{1}}})$ and $f_{2}({ \\mathbf {x_{2}}})$ are the marginal densities of $f(\\mathbf {x_{1}, x_{2}})$ with respect to $\\mathbf {X_{1}}$ and $\\mathbf {X_{2}}$ , respectively.",
"Theorem 1 Let $f$ be a probability density function defined on $\\mathbb {R}^{n_{1} + n_{2}}$ , let $h_{1}$ and $h_{2}$ be probability density functions on $\\mathbb {R}^{n_{1}}$ and $\\mathbb {R}^{n_{2}}$ , respectively.",
"If $h_{1}(\\mathbf {x_{1}}) h_{2} (\\mathbf {x_{2}}) > 0$ whenever $f(\\mathbf {x_{1}, x_{2}}) >0,$ then $ \\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_{1}}} f( \\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\bigg (\\frac{f(\\mathbf {x_{1}}, \\mathbf {x_{2}})}{h_{1}(\\mathbf {x_{1}}) h_{2}(\\mathbf {x_{2}}) } \\bigg ) \\mathrm {d} \\mathbf {x_{1}} \\mathrm {d} \\mathbf {x_{2}} \\ge \\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_{1}}} f(\\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\bigg (\\frac{f(\\mathbf {x_{1}}, \\mathbf {x_{2}})}{f_{1}(\\mathbf {x_{1}}) f_{2}(\\mathbf {x_{2}}) } \\bigg ) \\mathrm {d} \\mathbf {x_{1}} \\mathrm {d} \\mathbf {x_{2}},$ where $f_{1}({ \\mathbf {x_{1}}}) = \\int _{\\mathbb {R}^{n_{2}}} f({\\mathbf {x_{1}}}, \\mathbf {{x_{2}}}) \\mathrm {d} { \\mathbf {x_{2}}}$ and $f_{2}({ \\mathbf {x_{2}}}) = \\int _{\\mathbb {R}^{n_{1}}} f(\\mathbf {{x_{1}}}, \\mathbf {{x_{2}}}) \\mathrm {d} {\\mathbf {x_{1}}}$ are the densities of the marginal distributions of $f(\\mathbf {x_{1}}, \\mathbf {x_{2}})$ .",
"The proof of Theorem REF is in the Appendix."
],
[
"Estimates of smoothing distributions as target",
"We provide an alternative way of constructing the intermediate target distributions using the marginal smoothing distributions motivated by Theorem REF .",
"Since the closed-form solutions to the marginal smoothing distributions are not available in general, we employ the estimates of the distributions at the nodes.",
"At the root, we still use $f_{0:T} = p(x_{0:T}|y_{0:T})$ .",
"At a leaf node $\\mathcal {T}_{j} \\in \\mathcal {T}$ , we define $f_{j}(x_{j}) = \\hat{p}(x_{j} | y_{0:T}) \\approx p(x_{j} | y_{0:T})$ , which requires estimating the marginal smoothing distribution.",
"We thus name the algorithm TPS-ES.",
"At a non-leaf and non-root node $\\mathcal {T}_{j:l}$ , we define the target distribution $f_{j:l}$ : $f_{j:l}(x_{j:l}) &\\propto & \\hat{p}(x_{j} | y_{0:j}) \\frac{ \\hat{p}(x_{l} | y_{0:T})}{ \\hat{p}(x_{l} | y_{0:l})} \\prod ^{l-1}_{i = j} \\bigg \\lbrace p(x_{i+1} | x_{i}) p(y_{i+1} | x_{i+1}) \\bigg \\rbrace \\\\ &\\approx & p(x_{j} | y_{0:j}) \\frac{p(x_{l} | y_{0:T}) }{p(x_{l} | y_{0:l}) } \\prod ^{l-1}_{i = j} \\bigg \\lbrace p(x_{i+1} | x_{i}) p(y_{i+1} | x_{i+1}) \\bigg \\rbrace \\\\& = & p(x_{j:l} | y_{0:T}),$ where $\\hat{p}(x_{j} | y_{0:j})$ denotes a probability density approximating the filtering density at the $j$ th time step.",
"Hence, given the estimate smoothing densities $\\lbrace \\hat{p}(x_{j} | y_{0:T}) \\rbrace _{j = 0, \\ldots , T}$ and the estimating filtering densities $\\lbrace \\hat{p}(x_{j} | y_{0:j}) \\rbrace _{j = 0, \\ldots , T}$ , we build an estimator of the distribution $p(x_{j:l} | y_{0:T})$ at $\\mathcal {T}_{j:l} \\in \\mathcal {T}$ .",
"Merging the particles at $\\mathcal {T}_{j:l}$ from its children at $\\mathcal {T}_{j:k-1} \\in \\mathcal {T}$ and $\\mathcal {T}_{k:l} \\in \\mathcal {T}$ amounts to correlating the two sets of samples while roughly preserving their marginal distributions.",
"The weight of the merged sample $\\tilde{x}^{(i)}_{j:l} = (\\tilde{x}_{j:k-1}^{({i})}, \\tilde{x}_{k:l}^{({i})})$ in Equation (REF ) becomes: $\\hat{w}^{(i)}_{j:l} = \\tilde{w}^{(i)}_{j:l} \\frac{\\hat{p}(\\tilde{x}^{({i})}_{k-1}|y_{0:k-1})}{ \\hat{p}(\\tilde{x}^{({i})}_{k-1}|y_{0:T}) \\hat{p}(\\tilde{x}^{({i})}_{k}|y_{0:k})} p(\\tilde{x}^{({i})}_{k} | \\tilde{x}^{({i})}_{k-1}) p(y_{k}|\\tilde{x}^{({i})}_{k}).$ Applying TPS-ES demands the constructions of $\\lbrace \\hat{p}(x_{j} | y_{0:j}) \\rbrace _{j = 0, \\ldots , T}$ and $\\lbrace \\hat{p}(x_{j} | y_{0:T}) \\rbrace _{j = 0, \\ldots , T}$ in advance.",
"The new weight formula in Equation (REF ) additionally incorporates the ratio between the estimated filtering and smoothing densities of $x_{k-1}$ compared with Equation (REF ).",
"TPS-ES exhibits a sound property regarding the Kullback–Leibler divergence discussed in Section REF .",
"Given the target distribution $f_{j:l}(x_{j:l}) = \\hat{p}(x_{j:l}|y_{0:T}) $ estimating $p(x_{j:l}|y_{0:T}) $ at $\\mathcal {T}_{j:l}$ , the proposal $h_{j:l}(x_{j:l}) = f_{j:k-1}(x_{j:k-1}) f_{k:l}(x_{k:l})$ estimates $p(x_{j:k-1} | y_{0:T}) p(x_{k:l} | y_{0:T})$ .",
"We notice $p(x_{j:k-1} | y_{0:T})$ and $p(x_{k:l} | y_{0:T})$ are the marginal distributions, and their product forms a proposal attaining the minimum KL divergence from $p(x_{j:l}|y_{0:T})$ .",
"Hence, what the proposal density $h_{j:l}(x_{j:l})$ estimates has a minimum KL divergence from the smoothing density that our target distribution $f_{j:l}(x_{j:l})$ estimates.",
"Moreover, TPS-ES can be practically useful in some extreme models whereas the empirical marginal densities from other Monte Carlo smoothing algorithms may miss modes caused by the poor proposals.",
"Since TPS-ES leaves the marginal distributions of all random variables roughly invariant at all levels of the tree, we can diagnose each importance sampling step by inspecting the empirical marginals of the corresponding variables.",
"If there is a substantial difference between the empirical marginal distributions, we need to examine the combination step."
],
[
"Initial sampling distribution at leaf nodes",
"We illustrate the constructions of the univariate distributions $\\lbrace \\hat{p}(x_{j} | y_{1:j})\\rbrace _{j = 0, \\ldots , T}$ and $\\lbrace \\hat{p}(x_{j} | y_{0:T})_{j = 0, \\ldots ,T}\\rbrace $ mentioned in Section REF and Section REF , which are used in the initial sampling distributions at the leaf nodes.",
"In general, the solutions of the filtering and smoothing distribution of a HMM are analytically intractable and need to be estimated from Monte Carlo samples with some exceptions including linear Gaussian and discrete HMMs.",
"We aim to generate a probability density $\\hat{f}$ estimating a target density $f$ given the weighted samples $\\lbrace x_{i}, w_{i}\\rbrace _{i = 1}^{n}$ from $f$ .",
"In the context of $f$ being a filtering or smoothing distribution, we can obtain the weighted samples by running a filtering algorithm or a smoothing algorithm.",
"We are not interested in the empirical distribution since it is discrete and generally does not cover the full support of the random variable of interest.",
"We first consider some parametric approaches.",
"We can fit the data with some common probability distributions including a normal distribution and Student's $t$ -distribution.",
"We can also accommodate a mixture model to fit multiple modes of the target densities.",
"The parameters of the distributions can be estimated in various ways including moment matching, maximum likelihood method and EM algorithm.",
"The parametric approaches are reasonably quick and simple.",
"For instance, assuming a Gaussian distribution requires the evaluation of the mean and variance and can be easily obtained from the samples using moment matching.",
"The generation and evaluation of densities of the new particles are straightforward and fast to implement.",
"Nevertheless, the target distribution may not be well approximated under the parametric assumption.",
"Alternatively, we can employ some non-parametric approaches for instance, a kernel density estimator (KDS).",
"We need to select the type of kernels and bandwidth in advance.",
"The complexity of generating $N$ new samples is $O\\big (\\log (n)N\\big )$ and the evaluation of the densities is more computationally expensive with complexity $O(nN)$ .",
"We propose another non-parametric approximation method using piecewise constant functions with a lower computational effort than a KDS.",
"We first build a uniform grid consisting of the points $x_{1} < x_{2} < \\ldots < x_{n}$ with densities $d_{1}, \\ldots , d_{n}$ estimated by a KDS such that $x_{i+1} - x_{i} = \\Delta > 0$ for $i = 1, \\ldots , n$ .",
"The resulting probability density function formed by these grid points using piecewise constant functions is: $f(x) = \\sum _{i = 1}^{n} \\mathbb {1}_{x \\in [x_{i} - \\Delta /2, x_{i} + \\Delta /2)} d_{i}.$ The evaluation of the sample densities reduces significantly from $O(nN)$ to $O\\big (N \\big )$ compared to a KDS.",
"Such probability density functions using piecewise constant functions have several disadvantages though enjoy a fast computation of estimated densities.",
"Firstly, the estimator is biased since the proposal density generally does not cover the full support of the target density.",
"Moreover, in TPS-ES, if the estimated filtering and smoothing distributions are both generated using the piecewise constant functions with different samples, there is no guarantee their densities have the same support, which may cause zero or infinite weight in Equation (REF ).",
"To avoid this, we consider the mixture probability distributions using the piecewise constant functions accommodating the samples from both the filtering and smoothing distributions.",
"Assume at time step $j$ , the first uniform grid consists of the points $x^{f}_{1} < x^{f}_{2} < \\ldots < x^{f}_{n^{f}}$ such that $x^{f}_{i+1} - x^{f}_{i} = \\Delta ^{f}$ for $i = 1, \\ldots , n^{f}$ with estimated filtering densities $d^{f}_{1}, \\ldots , d^{f}_{n}$ from a KDS and assume the second uniform grid consists of the points $x^{s}_{1} < x^{s}_{2} < \\ldots < x^{s}_{n^{s}}$ such that $x^{s}_{i+1} - x^{s}_{i} = \\Delta ^{s}$ for $i = 1, \\ldots , n^{s}$ with estimated smoothing densities $d^{s}_{1}, \\ldots , d^{s}_{n}$ from another KDS.",
"Then the resulting estimated filtering density $\\hat{p}(x | y_{0:j})$ is given by $\\hat{p}(x | y_{0:j}) = \\alpha ^{f} \\sum _{i = 1}^{n^{f}} \\mathbb {1}_{x \\in [x^{f}_{i} - \\Delta ^{f}/2, x^{f}_{i} + \\Delta ^{f}/2)} d^{f}_{i} + (1-\\alpha ^{f}) \\sum _{i = 1}^{n^{s}} \\mathbb {1}_{x \\in [x^{s}_{i} - \\Delta ^{s}/2, x^{s}_{i} + \\Delta ^{s}/2)} d^{s}_{i},$ where $0 < \\alpha ^{f} < 1$ .",
"Similarly, the estimated smoothing density $\\hat{p}(x | y_{0:T})$ is given by $\\hat{p}(x | y_{0:T}) = \\alpha ^{s} \\sum _{i = 1}^{n^{s}} \\mathbb {1}_{x \\in [x^{s}_{i} - \\Delta ^{s}/2, x^{s}_{i} + \\Delta ^{s}/2)} d^{s}_{i} + (1-\\alpha ^{s}) \\sum _{i = 1}^{n^{f}} \\mathbb {1}_{x \\in [x^{f}_{i} - \\Delta ^{f}/2, x^{f}_{i} + \\Delta ^{f}/2)} d^{f}_{i},$ where $0 < \\alpha ^{s} < 1$ .",
"We have no conclusion of the values of $\\alpha ^{f}$ and $\\alpha ^{s}$ so far and choose them with values close to 1.",
"The resulting grid with the set of points $\\lbrace x^{f}_{1}, x^{f}_{2}, \\ldots , x^{f}_{n^{f}}, x^{s}_{1}, x^{s}_{2}, \\ldots , x^{s}_{n^{s}}\\rbrace $ is generally not uniform, but we ensure the estimated filtering and smoothing densities have the same support, though still finite."
],
[
"Simulations",
"We conduct simulations in a linear Gaussian HMM and a non-linear non-Gaussian HMM in this section.",
"We implement TPS-EF and other smoothing algorithms with roughly the same computational effort.",
"In the second example, we further compare TPS-EF and TPS-ES."
],
[
"Gaussian Linear Model",
"We consider a simple linear Gaussian HMM similar to [8].",
"$X_{t} &= 0.8 X_{t-1} + V_{t} ~~~&& t = 1, \\ldots , T,\\\\Y_{t} &= X_{t} + W_{t} ~~~&& t = 0, \\ldots , T.$ where $T = 127$ , where $X_0,V_1,\\dots ,V_T,W_0,\\dots ,W_T$ are independent with $X_{0} \\sim \\mathcal {N}(0,1)$ , $V_{t}\\sim N(0,1)$ , $W_{t}\\sim N(0,1)$ .",
"We implement the following smoothing algorithms.",
"We run TPS using normal distributions as the initial sampling distributions (TPS-N) whose means and variances are estimated using moment matching from the samples of a bootstrap particle filter.",
"The choice of a normal distribution is motivated by the fact that in this case the true smoothing distribution is a normal distribution.",
"We also implement the tree-based particle smoothing algorithm suggest by [17] (TPS-L), the Rauch–Tung–Striebel smoother (RTSs) [21] yielding the closed-form solutions, the bootstrap particle filter (BPF) which updates the entire history of the particles in each step, the forward filtering backward smoothing algorithm (FFBSm) [8] and the forward filtering backward simulation (FFBSi) [12].",
"We have implemented the above methods in R ourselves.",
"We set the required sample size $N = 10000$ in TPS-N as a benchmark and denote $n = 10000$ the number of samples pre-generated from a bootstrap particle filter in FFBSm, FFBSi, TPS-N and TPS-L. We adjust the number of particles in other algorithms to roughly keep the same running time.",
"As the implementations are not deterministic, we allow a 10% error regarding the running time for the rest of the algorithms compared to TPS-N. We run each algorithm $M = 500$ times with the same set of observations $\\lbrace y_{t}\\rbrace _{t = 0}^{127}$ .",
"As a criterion for comparison, we define the mean square error of means (MSEm) and variances (MSEv) in the $m$ th simulation: $\\text{MSEm}_{m} &=& \\frac{1}{T+1} \\sum ^{T}_{t = 0} \\big ( \\widehat{\\mathbb {E}}^{m}[X_{t}| Y_{0:T}] - \\mathbb {E}[X_{t}| Y_{0:T}]\\big )^{2}, \\\\\\text{MSEv}_{m} &=& \\frac{1}{T+1} \\sum ^{T}_{t = 0} \\big ( \\widehat{\\text{Var}}^{m}[X_{t}| Y_{0:T}] - \\text{Var}[X_{t}| Y_{0:T}]\\big )^{2}, \\\\$ where $ \\widehat{\\mathbb {E}}[X^{m}_{t}| Y_{0:T}]$ and $\\widehat{\\text{Var}}[x^{m}| y_{0:T}] $ are the Monte Carlo estimates of the mean and variance of the smoothing distribution at time step $t$ in the $m$ th simulation.",
"$\\mathbb {E}[X_{t}| Y_{0:T}]$ and $\\text{Var}[X_{t}| Y_{0:T}]$ are the true smoothing means and variances from a Rauch–Tung–Striebel smoother [21].",
"The simulation results are shown in Table REF .",
"When $N = n$ , the two tree-based sampling algorithms: TPS-L and TPS-N enjoy the same complexity $O(N)$ as BPF, and generate far more particles than FFBSm and FFBSi with quadratic complexities.",
"TPS-L has the smallest mean of MSEm and MSEv followed by TPS-N, which outperform FFBSm and FFBSi significantly in terms of MSEm.",
"Table: Simulation errors in the linear model"
],
[
"Non-linear Model",
"We consider a well-known non-linear model [13], [1]: $X_{t} &= \\frac{1}{2} X_{t-1} + 25 \\frac{X_{t-1}}{1 + X_{t-1}^2} + 8 \\cos (1.2 t) + V_{t},~~&&t = 1,2, \\ldots , T,\\\\Y_{t} &= \\frac{X^{2}_{t}}{20} + W_{t},~~&&t = 0,2, \\ldots , T,$ where $T = 511$ , where $X_0, V_1,...,V_T, W_0,...W_T$ are independent with $X_{0} \\sim \\mathcal {N}(0, 1)$ , $V_{t} \\sim \\mathcal {N}(0, \\tau ^2)$ and $W_{t} \\sim \\mathcal {N}(0,\\sigma ^2).$ We run the same algorithms BPF, FFBSm, FFBSi and TPS-L as in Section REF .",
"In TPS-EF, we use piecewise constant functions defined in Equation (REF ) for the approximation of the initial sampling distributions.",
"We call the algorithm TPS-EFP and set $N = n = 10000$ as a benchmark.",
"As before, we correspondingly adjust the sample sizes in other algorithms to achieve roughly the same computational effort.",
"We calculate the mean and standard deviation of the MSE of means (MSEm) in $M = 500$ simulations with the same set of observations.",
"Given no closed-form solutions to the true smoothing distributions, we apply a discrete analogue to the distributions of the initial hidden state $p_{0}(x_{0})$ and the transition distributions $\\lbrace p(x_{t+1}|x_{t})\\rbrace _{t = 0, \\ldots , 126}$ .",
"We then approximate the smoothing distributions of the original HMM using the solutions of the discrete-space HMM.",
"The MSEm of the $m$ th simulation in the non-linear model is defined as: $ \\text{MSEm}_{m} = \\frac{1}{T+1} \\sum ^{T}_{t = 0} \\big (\\widehat{\\mathbb {E}}^{m}[X_{t}| Y_{0:T}] - \\mathbb {E} (\\hat{X}_{t}\\big | y_{0:T}) )^{2}, $ where $\\mathbb {E} (\\hat{X}_{t}\\big | y_{0:T})$ is the mean of the smoothing distribution at time step $t$ of the discrete-space HMM.",
"We additionally perform Kolmogorov–Smirnov test [19] which measures a distance between the empirical distribution and the target probability distribution.",
"In the context of the smoothing problem in a non-linear hidden Markov model, the Kolmogorov–Smirnov statistic can be defined as $ D = \\sup _{x} | F^{(t)}_{1,N}(x) - F^{(t)}_{2}(x) |, $ where $F^{(t)}_{1,N}$ is the empirical cumulative function generated by $N$ samples at the time step $t$ from a smoothing algorithm and $F^{(t)}_{2}$ is the cumulative distribution function at time step $t$ of the smoothing distribution from a discrete-space HMM derived from the true model.",
"We denote KS$_{m}$ to be the sum of the KS statistic of all time steps in the $m$ th simulation.",
"Table: Simulation errors in the non-linear modelThe simulation results with different values of $\\tau $ and $\\sigma $ are shown in Table REF .",
"In the first two situations, TPS-L shows the largest error and KS statistic, especially when $\\tau = 1$ and $\\sigma = 5$ .",
"This can be explained by the poor proposal from the initial sampling distribution constructed by the algorithm.",
"We examine this by plotting the cumulative distribution function (CDF) of the initial sampling distribution $f_{j}$ in TPS-L, the filtering distribution $p(x_{j}| y_{0:j})$ and the marginal smoothing distribution $p(x_{j}| y_{0:T})$ at a particular time step when $j = 271$ .",
"In Figure REF , the CDF of the initial sampling are far more dissimilar to the marginal smoothing distribution than the filtering one, which contributes to very ineffective importance sampling steps during the built-up of the tree.",
"Figure: CDF of the smoothing, filtering and initial sampling distribution at time step j=271j = 271 of TPS-L in the non-linear model when τ=1,σ=5\\tau = 1, \\sigma = 5.Other algorithms provide different results in the three parameter settings.",
"When $\\tau = 1, \\sigma = 1$ , TPS-EFP shows much smaller MSEm and KS followed by BPF.",
"BPF however has the largest mean of KS.",
"When $\\tau = 1, \\sigma = 5$ , TPS-EFP has a larger mean of MSEm than BPF.",
"In terms of the KS statistic, TPS-EFP outperforms other smoothing algorithms.",
"When $\\tau = 5, \\sigma = 1$ , TPS-EFP and TPS-L produce dominant results with vastly smaller MSEm.",
"They also exhibit the smallest mean of KS among the smoothing algorithms whereas the BPF gives the largest result though generating the most samples.",
"To conclude, TPS-EFP and TPS-L perform well when the ratio between the standard deviation in the transition and emission density, i.e.",
"when $\\tau / \\sigma $ is large.",
"TPS-EFP has a more stable and appreciable performance, which provides low MSEm and consistently the smallest KS among the five smoothing algorithms.",
"In contrast, the result of TPS-L may be misleading due to its instability.",
"BPF works well regarding MSEm in some situations, but poorly in terms of Kolmogorov–Smirnov statistic.",
"FFBSm and FFBSi produces less accurate results due to higher computational complexity."
],
[
"Comparing TPS-EF and TPS-ES in the non-linear model",
"In this section, we conduct simulations in the same non-linear model using tree-based particle smoothing algorithm with estimated filtering (TPS-EF) and smoothing (TPS-ES) distributions as the intermediate target distributions.",
"As TPS-ES is not a good competitor given a relatively small sample size in Section REF , we compare its performance with TPS-EF with more computational budget.",
"We demonstrate the implementations of the two algorithms.",
"We apply the same smoothing algorithm TPS-EFP as described in Section REF which utilises the piecewise constant functions to estimate the filtering distributions.",
"As TPS-ES requires the estimated smoothing distributions as the initial sampling distributions, we achieve this by using piecewise constant functions for the estimation based on the samples from an initial run of TPS-EFP and thus call the algorithm TPS-ESP.",
"We specify the parameters in the simulations of TPS-EFP and TPS-ESP.",
"We denote the sample size by $N$ .",
"We set the parameters $\\alpha ^{s} = \\alpha ^{f} = 0.95$ appeared in Equation (REF ) and (REF ).",
"In TPS-EFP and TPS-ESP, the estimated filtering distributions are both constructed from $n$ samples from the particle filters.",
"Additionally, in TPS-ESP, the estimated smoothing distributions are constructed from TPS-EFP with $n^{\\prime }$ samples.",
"We run TPS-ESP in two different situations: The first one has the same sample size $N$ as TPS-EFP and requires more computational effort to estimate the initial sampling distributions based on $n^{\\prime }$ Monte Carlo samples.",
"The second one has roughly the same computational effort as TPS-EFP which generates fewer Monte Carlo samples for the estimation of the initial sampling distributions and the target samples.",
"We compare TPS-EFP and TPS-ESP with respect to the mean square error and Kolmogorov–Smirnov statistic defined in Section REF .",
"We run TPS-EFP and TPS-ESP for $M = 200$ times with different values of $\\tau $ and $\\sigma $ whose results are shown in Table REF .",
"TPS-ESP has an evident improvement of the Kolmogorov–Smirnov (KS) statistic in most situations and the comparisons between MSEm vary.",
"The MSEm of TPS-ESP always decreases when generating the same number of samples as TPS-EFP.",
"However, TPS-ESP does not provide convinced results under roughly the same computational effort.",
"Overall, the performance of TPS-ESP depends on the computational budget.",
"Given the same sample size as in TPS-EFP, TPS-ESP can potentially decrease both MSEm and KS statistic.",
"This may not be true when the algorithm is kept the same overall effort as TPS-EFP.",
"Table: Simulation errors between TPS-EF and TPS-ES in the non-linear model"
],
[
"Conclusion",
"This article introduces a Monte Carlo sampling method we call TPS built on the D&C SMC [17] to estimate the joint smoothing distribution $p(x_{0:T}|y_{0:T})$ in a hidden Markov model.",
"The method decomposes the model into sub-models with intermediate target distributions using a binary tree structure.",
"TPS samples independently from the leaves of the tree and gradually merges and resamples to target the new distributions upon the auxiliary tree.",
"We propose one generic way of constructing a binary tree which sequentially splits the joint random variables $X_{0:T}$ .",
"Furthermore, we discuss the sampling procedure of the target samples at a non-leave node by combining the samples from its children using importance sampling.",
"The computational effort is adjustable with a possible reduction to a linear effort with respect to the sample size.",
"Using the above settings, we investigate three algorithms with different types of intermediate target distributions at the non-root nodes.",
"TPS-L [17] constructs intermediate target distributions conditional on the observations from the same time interval as the target variables and imposes an uninformative prior.",
"TPS-L is very simple to implement with no additional tuning algorithms.",
"The algorithm is at the risk of providing very poor initial sampling distribution based on little information from the observations.",
"TPS-EF employs intermediate target distributions estimating of the (joint) filtering distributions which conditions on the observations up to the last time step in the target variable.",
"It is straightforward for implementation with an initial run of a filtering algorithm.",
"Nevertheless, the proposal in the importance sampling step may still not be satisfactory when the marginal filtering and smoothing distributions are vastly different.",
"TPS-ES builds the distributions estimating of the (joint) smoothing distributions which conditions on all the observations.",
"It roughly retains the marginal smoothing distributions from the intermediate target distributions at all levels of the auxiliary tree despite its more intensive computations.",
"We further propose the constructions of the estimated filtering and smoothing distributions based on the Monte Carlo samples.",
"Considering both accuracy and computational effort, we recommend parametric approaches such as normal assumptions in a linear Gaussian model and non-parametric approaches such as using piecewise constant functions in a non-linear model.",
"In the simulation studies, TPS-L has the smallest error in the linear model, but very unstable results in the different settings of the non-linear model.",
"TPS-EF exhibits more desirable simulation outcomes.",
"It is computationally less expensive than the most smoothing algorithms with quadratic complexity.",
"It also produces the smallest mean square errors in the linear Gaussian model and consistently the smallest average Kolmogorov–Smirnov statistic in different situations under the non-linear model.",
"In particular, it outperforms other algorithms substantially when the variance of the transition density is much larger than the emission density.",
"TPS-ES, however, has a better approximation of the smoothing distribution with respect to the Kolmogorov–Smirnov statistic compared with TPS-EF at the cost of an additional run of a smoothing algorithm.",
"To conclude, TPS with two proposed choices of the intermediate target distribution presents a new approach of addressing the smoothing problem which shows the following advantages: We have flexibilities of choosing and constructing the intermediate target distributions, which can potentially produce better proposals in the importance sampling steps.",
"TPS can escape from the quadratic complexity with respect to the sample size computationally, and produce more particles and accurate simulation results than some smoothing algorithms.",
"Nevertheless, its performance depends on the implementation of other filtering or smoothing algorithms and the estimation of the target distributions.",
"Due to its flexible and relatively fast implementations with stable and comparable simulation results, we regard it as a competitor with other smoothing algorithms."
],
[
"Proof of Theorem ",
"By Jensen's inequality, $&& \\int _{\\mathbb {R}^{n_{1}}} f_{1}(\\mathbf {x_{1}}) \\log \\big ( f_{1}(\\mathbf {x_{1}}) \\big ) \\mathrm {d} \\mathbf {x_{1}} - \\int _{\\mathbb {R}^{n_{1}}} f_{1}(\\mathbf {x_{1}}) \\log \\big ( h_{1}(\\mathbf {x_{1}}) \\big ) \\mathrm {d}\\mathbf {x_{1}}\\\\ &=& \\int _{\\mathbb {R}^{n_{1}}} f_{1} (\\mathbf {x_{1}}) \\log \\bigg ( \\frac{f_{1}(\\mathbf {x_{1}})}{h_{1} (\\mathbf {x_{1}})} \\bigg ) \\mathrm {d} \\mathbf {x_{1}}= \\mathbb {E} \\bigg [ \\log \\bigg ( \\frac{f_{1}(\\mathbf {X_{1}})}{ h_{1}(\\mathbf {X_{1}}) } \\bigg ) \\bigg ] = \\mathbb {E} \\bigg [ - \\log \\bigg ( \\frac{h_{1}(\\mathbf {X_{1}})}{ f_{1}(\\mathbf {X_{1}}) } \\bigg ) \\bigg ] \\\\&\\ge & - \\log \\bigg \\lbrace \\mathbb {E} \\bigg [ \\frac{h_{1}(\\mathbf {X_{1}})}{f_{1}(\\mathbf {X_{1}})} \\bigg ] \\bigg \\rbrace = 0.",
"$ Using this and the definition of marginal distribution, $&& \\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_1}} f(\\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\big ( f_{1}(\\mathbf {x_{1}}) \\big ) \\mathrm {d} \\mathbf {x_{1}} \\mathrm {d} \\mathbf {x_{2}}= \\int _{\\mathbb {R}^{n_{1}}} f_{1}(\\mathbf {x_{1}}) \\log \\big ( f_{1}(\\mathbf {x_{1}}) \\big ) \\mathrm {d} \\mathbf {x_{1}} \\nonumber \\\\&\\ge & \\int _{\\mathbb {R}^{n_{1}}} f_{1}(\\mathbf {x_{1}}) \\log \\big ( h_{1}(\\mathbf {x_{1}}) \\big ) \\mathrm {d} \\mathbf {x_{1}} = \\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_1}} f(\\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\big ( h_{1}(\\mathbf {x_{1}}) \\big ) \\mathrm {d} \\mathbf {x_{1}}\\mathrm {d} \\mathbf {x_{2}}.",
"\\nonumber \\\\$ Similarly, $\\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_{1}}} f(\\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\big ( f_{2}(\\mathbf {x_{2}}) \\big ) \\mathrm {d} \\mathbf {x_{1}} \\mathrm {d} \\mathbf {x_{2}} \\ge \\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_{1}}} f(\\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\big ( h_{2}(\\mathbf {x_{2}}) \\big ) \\mathrm {d} \\mathbf {x_{1}}\\mathrm {d} \\mathbf {x_{2}}.$ Multiplying (REF ) and (REF ) by -1 and adding them, we have $\\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_{1}}} f( \\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\bigg ( \\frac{1}{f_{1}(\\mathbf {x_{1}}) f_{2}(\\mathbf {x_{2}})} \\bigg ) \\mathrm {d} \\mathbf {x_{1}} \\mathrm {d}\\mathbf {x_{2}} \\le \\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_{1}}} f( \\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\bigg (\\frac{1}{h_{1}(\\mathbf {x_{1}}) h_{2}(\\mathbf {x_{2}})} \\bigg ) \\mathrm {d} \\mathbf {x_{1}} \\mathrm {d} \\mathbf {x_{2}}.$ Adding $ \\displaystyle \\int _{\\mathbb {R}^{n_{2}}} \\int _{\\mathbb {R}^{n_{1}}} f( \\mathbf {x_{1}}, \\mathbf {x_{2}}) \\log \\big ( f(\\mathbf {x_{1}}, \\mathbf {x_{2}}) \\big ) \\mathrm {d} \\mathbf {x_{1}} \\mathrm {d} \\mathbf {x_{2}} $ to both sides yields the result."
]
] | 1808.08400 |
[
[
"Integrated Server for Measurement-Device-Independent Quantum Key\n Distribution Network"
],
[
"Abstract Quantum key distribution (QKD), harnessing quantum physics and optoelectronics, may promise unconditionally secure information exchange in theory.",
"Recently, theoretical and experimental advances in measurement-device-independent (MDI-) QKD have successfully closed the physical backdoor in detection terminals.",
"However, the issues of scalability, stability, cost and loss prevent QKD systems from widespread application in practice.",
"Here, we propose and experimentally demonstrate a solution to build a star-topology quantum access network with an integrated server.",
"By using femtosecond laser direct writing technique, we construct integrated circuits for all the elements of Bell state analyzer together and are able to integrate several such analyzers on single photonic chip.",
"The measured high-visibility Bell state analysis suggests integrated server a promising platform for the practical application of MDI-QKD network."
],
[
"Integrated Server for Measurement-Device-Independent Quantum Key Distribution Network Ci-Yu Wang State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Jun Gao State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Zhi-Qiang Jiao State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Lu-Feng Qiao State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Ruo-Jing Ren State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Zhen Feng State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Yuan Chen State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Zeng-Quan Yan State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Yao Wang State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Hao Tang State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Xian-Min Jin xianmin.jin@sjtu.edu.cn State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Quantum key distribution (QKD), harnessing quantum physics and optoelectronics, may promise unconditionally secure information exchange in theory.",
"Recently, theoretical and experimental advances in measurement-device-independent (MDI-) QKD have successfully closed the physical backdoor in detection terminals.",
"However, the issues of scalability, stability, cost and loss prevent QKD systems from widespread application in practice.",
"Here, we propose and experimentally demonstrate a solution to build a star-topology quantum access network with an integrated server.",
"By using femtosecond laser direct writing technique, we construct integrated circuits for all the elements of Bell state analyzer together and are able to integrate several such analyzers on single photonic chip.",
"The measured high-visibility Bell state analysis suggests integrated server a promising platform for the practical application of MDI-QKD network.",
"Living in a new era of `Internet Of Everything', the ability to build a secure communication network is essential more than ever.",
"Since the emergence of QKD[1] – an indispensable part of quantum communication, many proof-of-principle demonstrations of quantum internet[2], [3], [4], [5], [6], [7], [8] have been done in increasingly complex systems.",
"Experiments have boomed in bulk components implementations[9], [10], [11], [12], leaving connecting multiple users an expensive as well as a massive construction project.",
"Integrated quantum device, featuring its cheap, compact, stable and flexible performance, becomes a revolutionary solution to establish quantum communication links.",
"Much efforts have been put forward on versatile integrated QKD platforms[13], [14], [15].",
"Chip-based client[16], [17], [18], [19], [20], [21] has been developed to investigate the practical possibilities for miniaturization of the quantum communication terminal.",
"Recently, Bunandar et al.",
"[22] achieved a 43-km-long link with silicon integrated devices, paving the way for one-way long-distance and high-speed chip-based QKD.",
"However, these point-to-point schemes have been threatened by many attacks through physical loopholes in detection terminals.",
"MDI-QKD[23], [24], [25], [26] introduces an independent untrusted third party (Charlie) to announce the measurement results, and the end users (Alice and Bob) only need to encode and send their photons (FIG.",
"1a).",
"It is immune to all attacks on detection and has a high tolerance of finite basis-dependent flaws.",
"In this letter, we propose a solution to build a star-topology[27], [28], [29], [30] quantum access network with an integrated server.",
"The star topology is found perfectly matching the features of MDI-QKD scheme.",
"All clients only need to build one quantum channel with the server, where an all-optical router can switch and connect any two clients on an integrated Bell state analyzer (FIG.",
"1b).",
"The keys are not touched in the routing process.",
"Furthermore, this solution is very resource-efficient for multi-user networks.",
"The proposed star network needs $N$ quantum channels to fully connect any two clients, which is far below the requirement of mesh network (FIG.",
"1c), $C^2_N$ quantum channels, constructed with point-to-point schemes.",
"Meanwhile, this solution needs plenty of Bell state analyzers for simultaneously supporting QKD among many pairs of clients, which, methodologically, can be well solved by constructing integrated circuits for all the elements of Bell state analyzer and assembling an array of such integrated analyzers on single photonic chip in server, as is shown in FIG.",
"1d.",
"Figure: Schematic of integrated server for MDI-QKD a. Overview of MDI-QKD protocol.",
"Two clients (Alice and Bob) encode their photons in BB84 bases by polarization modulator (Pol-M) and send them to an untrusted relay Charlie to conduct Bell state analysis.",
"Intensity modulators (IM) are used to generate decoy states.",
"The Bell state analyzer is composed of one 50/50 beam splitter (BS) and a polarization beam splitter(PBS) on each of its output ports for projection.",
"b. Star-topology network.",
"c. Mesh-topology network.",
"d. Integrated server for proof-of-principle test.",
"Two single photons from Alice and Bob are employed to qualify on-chip Bell state analyzer.",
"Signals from off-chip avalanche photodiode detectors (APD) are processed by a field-programmable gate array (FPGA).Figure: Design of integrated elements of on-chip Bell state analyzer.",
"a.",
"Top view geometry of integrated analyzer consisting of one polarization insensitive coupler (PIC) and two identical polarization directional couplers (PDC).",
"r: radius of fabrication circular arc.",
"CL: coupling length.",
"CW: coupling width.",
"b. Contour map of index simulation of PIC with designed two crossings.",
"c. Power distribution in waveguides with a Gaussian beam injected from In1.",
"d. Measured transmission of couplers by scanning coupling lengths when coupling width is fixed at 8m, for horizontal-(red square) and vertical-(blue circle) polarization input light.",
"Solid lines are the best-fitting curves according to coupling mode theory.",
"Dashed line marks the right coupling length for realizing a PDC.We employ femtosecond laser direct writing[31] to realize all the elements of Bell state analyzer, since the written chip can uniquely be single-mode-fiber compatible for low insertion loss, and can be fabricated in a single-step, mask-free and cost-effective way.",
"By controlling the nonlinear absorption, we are able to tune the mode profile and birefringence of the waveguide for polarization manipulating integrated devices[32], [33], [34], [35], [36], [37].",
"The waveguides used in this work are fabricated in boro-silicate glass via femtosecond laser direct writing.",
"Waveguides are formed by laser-induced permanent refractive index change, locating at 170 $$ m under the substrate surface by focusing femtosecond laser (wavelength 513nm, repetition rate 1 MHz, pulse energy 210 nJ, writing speed 20 mm/s) through a 50$\\times $ objective (numerical aperture 0.55).",
"Profiting from the mode control capacity of femtosecond laser direct writing, the written chip can be linked to a standard V-groove fiber array packaging with facet coupling as low as 1dB.",
"FIG.",
"2a shows the detailed design of the Bell state analyzer.",
"The special configuration of polarization insensitve coupler (PIC) is found not very sensitive to fabrication parameters in terms of splitting ratio.",
"The simulation results with commercially available softwares (RSoft BeamPROP and CAD suite) are shown in FIG.",
"2b and 2c.",
"Polarization insensitiveness is made possible by geometrically controlling interaction length with different coupling width to compensate the birefringence-induced difference of coupling coefficients between horizontal (H) and vertical (V) polarized light[38], [39].",
"The characterized performance shown in TABLE 1 confirms the high repeatability and uniformity of our PIC.",
"According to coupling mode theory, photons transferring from one waveguide to the other follows a sinusoidal law as shown in FIG.",
"2d.",
"Slight difference in coupling coefficients for H and V polarized light leads to different oscillation periods, and finally an opposite output behaves as a polarization directional coupler (PDC).",
"Two waveguides are prototyped with a fixed coupling width 8 $$ m, small enough but not overlapped, to ensure that the two propagating modes become coupled during evanescent field overlap.",
"Figure: Characterized polarization performance of on-chip Bell state analyzer.",
"a.",
"Measured polarization projection with a probe laser at 780 nm.",
"b.",
"Measured polarization projection with a probe laser at 786 nm.",
"c. and d. Measured projection coincidence counts per 100 seconds with heralded single photon at 780 nm injected from In1 and In2.Table: 50/50 PIC performance.",
"Output splitting ratios under different input conditions.Figure: Experimental results of Bell state analysis with integrated server.",
"a. Hong-Ou-Mandel Interference for all the coincidence counts.",
"An average visibility over 90%\\% is reached.",
"b.-e.",
"Relative conditional probability distribution with different input states in rectilinear and diagonal bases.",
"b. and d. show the theoretical distributions and c. and e. show the experimental coincidence distributions.The main task of MDI-QKD is to conduct Bell state analysis, thus we first use a probe laser to characterize our integrated analyzer.",
"The results obtained by injecting photons from In1 and In2 are presented with hatched and colored histograms respectively, and are piled together to clearly reveal their uniformity for different inputs and for different wavelengths of light at 780 nm (FIG.",
"3a) and 786 nm (FIG.",
"3b).",
"Our probe laser can only be tuned in a limited range, and however, the good performance at two different wavelengths implies the ability to fabricate our integrated analyzer in telecom band through similar parameter optimization method.",
"To test the performance of polarization projection on two PDCs, we prepare four typical input states for BB84 protocol bases, polarized at 0$^\\circ $ (H), 90$^\\circ $ (V), 45$^\\circ $ (D) and 135$^\\circ $ (A).",
"The normalized projection results agree with the expected relative ratio for H/V and A/D bases (red dashed lines).",
"More importantly, the average polarization extinction ratio for H/V bases is about 20:1, which can well identify $|\\psi ^+\\rangle $ out of $|\\phi ^\\pm \\rangle $ for Bell state analysis.",
"However, the response and noise level of detection system for laser and single photons are different.",
"Thus, we repeat the experiment with input states of heralded single photons.",
"We generate photon pairs via Type-II spontaneous parametric down-conversion by pumping a beta-barium borate (BBO) crystal with a Ti:sapphire laser.",
"Heralding efficiency of conditionally detecting one single photon by the other one is over 25$\\%$ .",
"Apparently, the detection and coupling efficiencies for different outputs can not be identical in practice, which may modify the projection results.",
"As we can see, the projection coincidence counts per 100 seconds shown in FIG.",
"3c and 3d slightly deviate from the results shown in FIG.",
"3a, but, are all around the expected relative ratio (red dashed lines).",
"Interestingly, the average polarization extinction ratio for H/V bases is as high as 59:1, which of course reflects the real performance of the device at single-photon level.",
"In order to qualify the integrated server suitable for MDI-QKD, we couple the pair of photons into one of on-chip Bell state analyzers, as is shown in FIG.",
"1d, both prepared in D, and scan their relative delay to find the zone of Hong-Ou-Mandel Interference[40].",
"Photons from Alice and Bob are encoded independently in BB84 protocol bases by rotating half wave plates.",
"Since our two PDCs can project incoming states H and V at the output ports 1, 4 and 2, 3 respectively, coincidence measurements could deterministically discriminate the states $|\\psi ^+\\rangle $ and $|\\psi ^-\\rangle $ by two-fold coincidences C12, C34 and C13, C24 via bunching or anti-bunching effect[41].",
"Meanwhile, two-fold coincidences C14 and C23 can not distinguish the states $|\\phi ^+\\rangle $ and $|\\phi ^-\\rangle $ , and therefore are not used as successful events in MDI-QKD protocols[42].",
"The Visibilities for all combinations are $89.8\\%, 92.4\\%, 93.0\\%, 92.3\\%, 92.2\\%$ and $83.7\\%$ for C12, C13, C14, C23, C24 and C34 (FIG.",
"4a).",
"Two-photon data are collected for eight linearly independent input conditions FIG.",
"4b and 4c (HH, HV, VH, VV), and FIG.",
"4d and 4e (DD, DA, AD, AA), with half wave plates and polarization compensation.",
"The gain values at each rectilinear ($Q_{i,j}^{r}$ ) and diagonal ($Q_{i,j}^{d}$ ) basis can be caculated from the coincidence measurements by[24]: $Q_{i,j}^{r}=\\frac{1}{4}(C_{Sum}^{HH}+C_{Sum}^{VV}+C_{Sum}^{HV}+C_{Sum}^{VH})$ $Q_{i,j}^{d}=\\frac{1}{4}(C_{Sum}^{++}+C_{Sum}^{--}+C_{Sum}^{+-}+C_{Sum}^{-+})$ i and j represent the different average coincidence counts used by Alice and Bob while we adopt the successful events by: $C_{Sum}^{AB}=C_{12}^{AB}+C_{34}^{AB}+C_{13}^{AB}+C_{24}^{AB}$ We get the final gain values: $Q_{i,j}^{r}=1.37\\times 10^{-6}$ bits/pulse, $Q_{i,j}^{d}=1.75\\times 10^{-6}$ bits/pulse.",
"We estimate the quantum bit error rate (QBER) by: $E_{i,j}^{r}=\\frac{C_{Sum}^{HH}+C_{Sum}^{VV}}{C_{Sum}^{HH}+C_{Sum}^{VV}+C_{Sum}^{HV}+C_{Sum}^{VH}}$ $\\begin{aligned}E_{i,j}^{d}=&\\frac{C_{13}^{++}+C_{24}^{++}+C_{13}^{--}+C_{24}^{--}}{C_{Sum}^{++}+C_{Sum}^{--}+C_{Sum}^{+-}+C_{Sum}^{-+}}\\\\+&\\frac{C_{12}^{+-}+C_{34}^{+-}+C_{12}^{-+}+C_{34}^{-+}}{C_{Sum}^{++}+C_{Sum}^{--}+C_{Sum}^{+-}+C_{Sum}^{-+}}\\end{aligned}$ From our experimental data, we estimate $E_{i,j}^{r}=0.058, E_{i,j}^{d}=0.081$ .",
"The nonideal QBER can be attributed to multi-photon events of the testing photon source, residual birefringence in PIC and polarization-dependent loss.",
"The last two points are being improved to meet the stringent requirement for low QBER and therefore for high final bit rate.",
"In summary, we propose and experimentally demonstrate an integrated server for MDI-QKD network and its compatibility to a star-topology quantum access network.",
"Every client sends their encoded photons to server through their sole channel.",
"Server can link any two clients simply by switching their photons to one of Bell state analyzers on a photonic chip.",
"All the combination between two clients is the same as a standard MDI-QKD and therefore is unconditionally secure.",
"The number of required channels is much less than mesh-topology with point-to-point schemes for realizing fully connected MDI-QKD networks, which is particularly significant when the networks go to large scales.",
"Acknowledgements: The authors thank Jian-Wei Pan, Hang Li, Xiao-Ling Pang, and Jian-Peng Dou for helpful discussions.",
"This work was supported by National Key R&D Program of China (2017YFA0303700); National Natural Science Foundation of China (NSFC) (11374211, 61734005, 11690033); Shanghai Municipal Education Commission (SMEC)(16SG09, 2017-01-07-00-02-E00049); Science and Technology Commission of Shanghai Municipality (STCSM) (15QA1402200, 16JC1400405).",
"X.-M.J. acknowledges support from the National Young 1000 Talents Plan."
]
] | 1808.08586 |
[
[
"An extension of a theorem of Zermelo"
],
[
"Abstract We show that if (M,E,E') satisfies the first order Zermelo-Fraenkel axioms of set theory when the membership relation is E and also when the membership relation is E', and in both cases the formulas are allowed to contain both E and E', then (M,E) and (M,E') are isomorphic, and the isomorphism is definable in (M,E,E').",
"This extends Zermelo's 1930 theorem about second order ZFC."
],
[
"We show that if $(M,\\in _1,\\in _2)$ satisfies the first order Zermelo-Fraenkel axioms of set theory when the membership relation is $\\in _1$ and also when the membership relation is $\\in _2$ , and in both cases the formulas are allowed to contain both $\\in _1$ and $\\in _2$ , then $(M,\\in _1)\\cong (M,\\in _2)$ , and the isomorphism is definable in $(M,\\in _1,\\in _2)$ .",
"This extends Zermelo's 1930 theorem in [6].",
"Zermelo [6] proved that if $(M,\\in _1)$ and $(M,\\in _2)$ both satisfy the second order Zermelo-Fraenkel axioms in which the Separation Schema and the Replacement Schema of $ZFC$ are replaced by single second order axioms, then $(M,\\in _1)\\cong (M,\\in _2)$ .",
"We extend this as follows: Let us consider the vocabulary $\\lbrace \\in _1,\\in _2\\rbrace $ , where both $\\in _1$ and $\\in _2$ are binary predicate symbols.",
"Let $ZFC(\\in _1)$ denote the first order Zermelo-Fraenkel axioms of set theory when $\\in _1$ is the membership relation but formulas are allowed to contain $\\in _2$ too.",
"Similarly, in $ZFC(\\in _2)$ the membership relation is $\\in _2$ but formulas are allowed to contain $\\in _1$ too.",
"We prove the following theorem: Theorem 1 If $(M,\\in _1,\\in _2)\\models ZFC(\\in _1)\\cup ZFC(\\in _2)$ , then $(M,\\in _1)\\cong (M,\\in _2)$ via a definable class function.",
"The result of Zermelo readily follows from our theorem.",
"The important difference between our result and Zermelo's result is that our theories $ZFC(\\in _1)$ and $ZFC(\\in _2)$ are first order theories.",
"It is important that we allow in these axiom systems formulas from the extended vocabulary $\\lbrace \\in _1,\\in _2\\rbrace $ .",
"Without this the result would be blatantly false as there are countable non-isomorphic models of $ZFC$ , assuming there are models of $ZFC$ at all.",
"Since the isomorphism in Theorem REF is definable, the result can be seen as a provable theorem of the first order theory $ZFC(\\in _1)\\cup ZFC(\\in _2)$ .",
"Theorem REF resembles the categoricity conclusion for set theory in [2].",
"There are two main differences: First, the axiomatization of set theory in [2] is informal, based on the Axiom of Extensionality and an informal full Comprehension Axiom, while our result is completely formal and in the context of ZFC.",
"Secondly, it is assumed in [2] that the two $\\in $ -relations give rise to the same (informal) structure of the ordinals, owing to the uniqueness of the ordinal concept.",
"We do not make this assumption but rather prove that the two $\\in $ -relations have isomorphic ordinals.",
"Martin's work has been extended to class theory in [5].",
"Theorem REF was stated without proof in [3].",
"We call our theorem an internal categoricity result because it shows that one cannot have in one and the same domain two non-isomorphic membership-relations $\\in _1$ and $\\in _2$ if these relations can “talk” about each other.",
"Our theorem is a strong robustness result for set theory.",
"Essentially, the model cannot be changed “internally”.",
"To get a non-isomorphic model one has to go “outside” the model.",
"Such robustness is important for set theory because set theory is already the “outside” of mathematics, the framework where mathematics is (or can be) built.",
"How are the numerous independence results in harmony with this internal categoricity?",
"Let us take the Continuum Hypothesis CH as an example.",
"CH is independent of $ZFC$ in the sense that both $ZFC\\cup \\lbrace CH\\rbrace $ and $ZFC\\cup \\lbrace \\lnot CH\\rbrace $ are consistent, if $ZFC$ itself is.",
"Internal categoricity means simply that if $(M,\\in _1)$ satisfies $CH$ and $(M,\\in _2)$ satisfies $\\lnot CH$ , then either $(M,\\in _1)$ or $(M,\\in _2)$ does not satisfy the Separation Schema or the Replacement Schema if formulas are allowed to mention the other membership-relation.",
"Such models cannot be internal to each other in the sense discussed.",
"In the below proof we will work in $ZFC(\\in _1)\\cup ZFC(\\in _2)$ but in fact operate all the time in either $\\in _1$ -set theory or in $\\in _2$ -set theory.",
"We have to keep the two set theories separate even though they also interact via the Separation and Replacement Schemas in the joint vocabulary $\\lbrace \\in _1,\\in _2\\rbrace $ .",
"Let $\\mathrm {tr}_i(x)$ be the formula $\\forall t\\in _i x\\forall w\\in _i t(w\\in _1 x)$ .",
"Let $\\mathrm {TC}_i(x)$ be the unique $u$ such that $\\mathrm {tr}_i(u)\\wedge \\forall v\\in _ix(v\\in _iu)\\wedge \\forall v((\\mathrm {tr}_i(v)\\wedge \\forall w\\in _ix(w\\in _iv))\\rightarrow \\forall w\\in _iu(w\\in _i v))$ (“$u$ is the $\\in _i$ -transitive closure of $x$ ”).",
"When we write $\\mathrm {TC}_i(\\lbrace x\\rbrace )$ , we mean by $\\lbrace x\\rbrace $ the singleton $\\lbrace x\\rbrace $ in the sense of $\\in _i$ .",
"Let $\\phi (x,y)$ be the formula $\\exists f\\psi (x,y,f)$ , where $\\psi (x,y,f)$ is the conjunction of the following formulas (where $f(t)$ , $f(w)$ and $f(x)$ are in the sense of $\\in _1$ ): (i) In the sense of $\\in _1$ , the set $f$ is a function with $\\mathrm {TC}_1(\\lbrace x\\rbrace )$ as its domain.",
"(ii) $\\forall t\\in _1 \\mathrm {TC}_1(x)(f(t)\\in _2 \\mathrm {TC}_2(y))$ (iii) $\\forall t\\in _2 \\mathrm {TC}_2(y)\\exists w\\in _1 \\mathrm {TC}_1(x)(t=f(w))$ (iv) $\\forall t\\in _1 \\mathrm {TC}_1(x)\\forall w\\in _1 \\mathrm {TC}_1(\\lbrace x\\rbrace )(t\\in _1w\\leftrightarrow f(t)\\in _2 f(w))$ (v) $f(x)=y$ In the sense of $\\in _1$ , the set $f$ is a function with $\\mathrm {TC}_1(\\lbrace x\\rbrace )$ as its domain.",
"$\\forall t\\in _1 \\mathrm {TC}_1(x)(f(t)\\in _2 \\mathrm {TC}_2(y))$ $\\forall t\\in _2 \\mathrm {TC}_2(y)\\exists w\\in _1 \\mathrm {TC}_1(x)(t=f(w))$ $\\forall t\\in _1 \\mathrm {TC}_1(x)\\forall w\\in _1 \\mathrm {TC}_1(\\lbrace x\\rbrace )(t\\in _1w\\leftrightarrow f(t)\\in _2 f(w))$ $f(x)=y$ We prove a sequence of lemmas about the formulas $\\phi (x,y)$ and $\\psi (x,y,f)$ : Lemma 2 If $\\psi (x,y,f)$ and $\\psi (x,y,f^{\\prime })$ , then $f=f^{\\prime }$ .",
"To prove $f=f^{\\prime }$ assume $w\\in _1 \\mathrm {TC}_1(\\lbrace x\\rbrace )$ .",
"We show $f(w)=f^{\\prime }(w)$ .",
"W.l.o.g.",
"$f(s)=f^{\\prime }(s)$ for $s\\in _1 w$ .",
"Suppose $t\\in _2 f(w)$ .",
"Clearly, $t\\in _2 \\mathrm {TC}_2(\\lbrace y\\rbrace )$ .",
"By (iii), $t=f(s)$ for some $s\\in _1 \\mathrm {TC}_1(\\lbrace x\\rbrace )$ .",
"By (iv), $s\\in _1 w$ .",
"By (iv) again, $f^{\\prime }(s)\\in _2 f^{\\prime }(w)$ .",
"By assumption, $f(s)=f^{\\prime }(s)$ .",
"Hence $t\\in _2 f^{\\prime }(w)$ .",
"Thus $\\forall t(t\\in _2 f(w)\\rightarrow t\\in _2 f^{\\prime }(w))$ .",
"By symmetry, $f(w)=f^{\\prime }(w)$ .",
"Lemma 3 Suppose $\\psi (x,y,f)$ .",
"If $x^{\\prime }\\in _1x$ , then $\\phi (x^{\\prime },f(x^{\\prime }))$ .",
"If $y^{\\prime }\\in _2y$ , then there is $x^{\\prime }\\in _1x$ such that $f(x^{\\prime })=y^{\\prime }$ and $\\phi (x^{\\prime },y^{\\prime })$ .",
"Let $y^{\\prime }=f(x^{\\prime })$ and $f^{\\prime }=f\\upharpoonright \\mathrm {TC}_1(\\lbrace x^{\\prime }\\rbrace )$ .",
"Clearly now $\\psi (x^{\\prime },y^{\\prime },f^{\\prime })$ .",
"Hence $\\phi (x^{\\prime },f(x^{\\prime }))$ .",
"The other claim is proved similarly.",
"Lemma 4 If $\\phi (x,y)$ and $\\phi (x,y^{\\prime })$ , then $y=y^{\\prime }$ .",
"If $\\phi (x,y)$ and $\\phi (x^{\\prime },y)$ , then $x=x^{\\prime }$ .",
"We may assume the claim holds for all $\\in _1$ -elements of $x$ .",
"Suppose $\\psi (x,y,f)$ and $\\psi (x,y^{\\prime },f^{\\prime })$ .",
"We prove $y=y^{\\prime }$ .",
"Let $s\\in _2 y$ .",
"By Lemma REF there is $t\\in _1 x$ such that $f(t)=s$ and $\\phi (t,s)$ .",
"By (iv), $s\\in _1 x$ .",
"Let $s^{\\prime }=f^{\\prime }(t)$ .",
"By (iv), $s^{\\prime }\\in _2 y^{\\prime }$ .",
"By Lemma REF again, $\\phi (t,s^{\\prime })$ .",
"By the Induction Hypothesis, $s=s^{\\prime }$ .",
"We have proved $\\forall s(s\\in _2 y\\rightarrow s\\in _2 y^{\\prime })$ .",
"The converse follows from symmetry.",
"Now to the second claim.",
"We may assume the claim holds for all $\\in _2$ -elements of $y$ .",
"Suppose $\\psi (x,y,f)$ and $\\psi (x^{\\prime },y,f^{\\prime })$ .",
"We prove $x=x^{\\prime }$ .",
"Let $s\\in _1 x$ .",
"Thus $f(s)\\in _2 y$ .",
"There is $s^{\\prime }\\in _1 \\mathrm {TC}_1(\\lbrace x^{\\prime }\\rbrace )$ such that $f^{\\prime }(s^{\\prime })=f(s)$ .",
"Now $\\phi (s,f(s))$ and $\\phi (s^{\\prime },f(s))$ by Lemma REF .",
"Since $f(s)\\in _2 y$ , $s=s^{\\prime }$ .",
"Hence $s\\in _1 x^{\\prime }$ .",
"We have proved $\\forall s(s\\in _1 x\\rightarrow s\\in _1 x^{\\prime })$ .",
"The converse follows from symmetry.",
"Lemma 5 If $\\phi (x,y)$ and $\\phi (x^{\\prime },y^{\\prime })$ , then $x\\in _1x^{\\prime }\\leftrightarrow y\\in _2y^{\\prime }$ .",
"Suppose $\\psi (x,y,f)$ and $\\psi (x^{\\prime },y^{\\prime },f^{\\prime })$ .",
"Suppose $x\\in _1 x^{\\prime }$ .",
"Then $z=f^{\\prime }(x)\\in _2 y^{\\prime }$ .",
"By Lemma REF , $\\phi (x,z)$ .",
"We have $\\phi (x,y)$ and $\\phi (x,z)$ .",
"By Lemma REF , $y=z$ .",
"Hence $y\\in _2 y^{\\prime }$ .",
"The converse is similar.",
"Let $\\mbox{On}_1(x)$ be the $\\in _1$ -formula saying that $x$ is an ordinal i.e.",
"a transitive set of transitive sets, and similarly $\\mbox{On}_2(x)$ .",
"For $\\mbox{On}_1(\\alpha )$ let $V^1_\\alpha $ be the $\\alpha ^{th}$ level of the cumulative hierarchy in the sense of $\\in _1$ , and similarly $V^2_y$ when $\\mbox{On}_2(y)$ .",
"Lemma 6 If $\\phi (\\alpha ,y)$ , then $\\mbox{On}_1(\\alpha )$ if and only if $\\mbox{On}_2(y)$ .",
"If $\\alpha $ is a limit ordinal then so is $y$ i.e.",
"if $\\forall u\\in _1 \\alpha \\exists v\\in _1\\alpha (u\\in _1 v)$ , then $\\forall u\\in _2 y\\exists v\\in _2 y(u\\in _2 v)$ , and vice versa.",
"Let us fix $y$ .",
"Suppose $\\psi (\\alpha ,y,f)$ .",
"We prove that $y$ is a transitive set of transitive sets.",
"Suppose $w\\in _2s\\in _2y$ .",
"There are $t\\in _1 \\alpha $ and $u\\in _1 t$ such that $f(t)=s$ and $f(u)=w$ .",
"Now $w\\in _2 y$ follows from $u\\in _1\\alpha $ .",
"This shows that $\\mathrm {tr}_2(y)$ .",
"Similarly one proves that all $s\\in _2y$ satisfy $\\mathrm {tr}_2(y)$ .",
"This ends the proof of the first claim.",
"The second claim is proved similarly.",
"Lemma 7 Suppose $\\psi (\\alpha ,y,f)$ .",
"If $\\mbox{On}_1(\\alpha )$ (or equivalently $\\mbox{On}_2(y)$ ), then there is $\\bar{f}\\supseteq f$ such that $\\psi (V^1_\\alpha ,V^2_y,\\bar{f})$ .",
"We use induction on $\\alpha $ .",
"Suppose the claim holds for $\\alpha $ .",
"We prove the claim for $\\alpha +1$ .",
"Suppose to this end $\\psi (\\alpha +1,y+1,f)$ .",
"We construct $\\bar{f}$ such that $\\psi (V^1_{\\alpha +1},V^2_{y+1},\\bar{f})$ .",
"From $\\psi (\\alpha +1,y+1,f)$ we obtain $\\psi (\\alpha ,y,f\\upharpoonright \\alpha )$ .",
"By assumption there is $g\\supseteq f\\upharpoonright \\alpha $ such that $\\psi (V^1_\\alpha ,V^2_y,g)$ .",
"Let $\\theta (u,v)$ be the formula $\\forall w(w\\in _2 v\\leftrightarrow (w\\in _2 V^2_y\\wedge \\exists t\\in _1 u(g(t)=w))).$ It follows from the Separation Schema of $ZFC(\\in _2)$ that for all $u\\in _1V^1_{\\alpha +1}$ there is $v$ such that $\\theta (u,v)$ .",
"By the Replacement Schema of $ZFC(\\in _1)$ , we can let $\\bar{f}$ be a function (in the sense of $\\lbrace \\in _1\\rbrace $ ) such that for all $u\\in _1V^1_{\\alpha +1}$ we have $\\theta (u,\\bar{f}(u))$ .",
"It is easy to see, using the Separation Schema of $ZFC(\\in _1)$ , that $\\psi (V^1_{\\alpha +1},V^2_{y+1},\\bar{f})$ .",
"Suppose then the claim holds for all $\\beta <\\alpha =\\bigcup \\alpha $ .",
"For each $\\beta <\\alpha $ there is thus some $g_\\beta $ such that $\\psi (V^1_\\beta ,V^2_{f(\\beta )},g_\\beta )$ .",
"By the Replacement Schema of $ZFC(\\in _1)$ we can form the $\\in _1$ -set $\\bar{f}=\\bigcup _{\\beta <\\alpha }g_\\beta $ .",
"It is easy to see that $\\psi (V^1_\\alpha ,V^2_y,\\bar{f})$ .",
"Lemma 8 $\\forall x\\exists y\\phi (x,y)$ and $\\forall y\\exists x\\phi (x,y)$ .",
"Let us first assume that both $\\forall \\alpha (\\mbox{On}_1(\\alpha )\\rightarrow \\exists y\\phi (\\alpha ,y))$ and $\\forall y(\\mbox{On}_2(y)\\rightarrow \\exists \\alpha \\phi (\\alpha ,y)).$ hold.",
"In order to prove $\\forall x\\exists y\\phi (x,y)$ , suppose $x$ is given.",
"There is $\\alpha $ such that $\\mbox{On}_1(\\alpha )$ and $x\\in _1 V^1_\\alpha $ .",
"By (REF ) there are $v$ and $f$ such that $\\psi (\\alpha ,v,f)$ .",
"By Lemma REF there is $\\bar{f}\\supseteq f$ such that $\\psi (V^1_\\alpha , V^2_v,\\bar{f})$ .",
"By Lemma REF , $\\phi (x,\\bar{f}(x))$ .",
"Thus $\\exists y\\phi (x,y)$ .",
"In order to prove $\\forall y\\exists x\\phi (x,y)$ , suppose $y$ is given.",
"There is $v$ such that $\\mbox{On}_2(v)$ and $y\\in _2 V^2_v$ .",
"By (REF ) there are $\\alpha $ and $f$ such that $\\psi (\\alpha ,v,f)$ .",
"By Lemma REF there is $\\bar{f}\\supseteq f$ such that $\\psi (V^1_\\alpha , V^2_v)$ .",
"By condition (iii) of the definition of $\\psi $ there is $w\\in _1 V^1_\\alpha $ such that $\\bar{f}(w)=y$ .",
"By Lemma REF , $\\phi (w,\\bar{f}(w))$ .",
"Thus $\\exists x\\phi (x,y)$ .",
"Thus it suffices to show that the failure of (REF ) or (REF ) to hold leads to a contradiction.",
"Case 1: $\\lnot $ (REF )$\\wedge \\lnot $ (REF ).",
"Let $\\alpha $ be the $\\in _1$ -least $\\alpha $ such that $\\mbox{On}_1(\\alpha )\\wedge \\lnot \\exists y\\phi (\\alpha ,y)$ .",
"Let $y$ be the $\\in _2$ -least $y$ such that $\\mbox{On}_2(y)\\wedge \\lnot \\exists \\beta \\phi (\\beta ,y)$ .",
"It is easy to see that $\\phi (\\alpha ,y)$ , a contradiction.",
"Case 2: (REF )$\\wedge \\lnot $ (REF ).",
"Let $y$ be the $\\in _2$ -least $y$ such that $\\mbox{On}_2(y)\\wedge \\lnot \\exists \\alpha \\phi (\\alpha ,y)$ .",
"Now, $\\forall t\\in _2 y\\exists \\alpha (\\mbox{On}_1(\\alpha )\\wedge \\phi (\\alpha ,t))$ .",
"Clearly, $y$ is an $\\in _2$ -limit ordinal.",
"Suppose $z\\in _2 V^2_t$ , where $t\\in _2 y$ .",
"Let $\\alpha $ and $f$ be such that $\\mbox{On}_1(\\alpha )\\wedge \\psi (\\alpha ,t,f)$ .",
"By Lemma REF there is $\\bar{f}\\supseteq f$ such that $\\psi (V^1_\\alpha , V^2_t,\\bar{f})$ .",
"There is $x\\in _1 V^1_\\alpha $ such that $\\bar{f}(x)=z$ .",
"Thus $\\phi (x,z)$ and hence $\\forall z\\in _2 V^2_y\\exists x\\ \\phi (x,z).$ By the Replacement Schema in $ZFC(\\in _2)$ there is $c$ such that $\\forall z\\in _2 V^2_y\\exists x\\in _2 c\\ \\phi (x,z).$ Let $\\alpha $ be such that $c\\in _1 V^1_\\alpha $ .",
"By (REF ) there are $t$ and $f$ such that $\\phi (\\alpha ,t,f)$ .",
"Necessarily, $t\\in _2 y$ .",
"By Lemma REF there is $\\bar{f}\\supseteq f$ such that $\\psi (V^1_\\alpha , V^2_t,\\bar{f})$ .",
"In particular, $\\bar{f}(c)\\in _2V^2_y$ .",
"By (REF ) there is $b\\in _2 c$ such that $\\phi (b,\\bar{f}(c))$ .",
"Since also $\\phi (c,\\bar{f}(c))$ , Lemma REF gives $c=b$ .",
"Thus $c\\in _2 c$ , a contradiction.",
"Case 3: $\\lnot $ (REF )$\\wedge $ (REF ).",
"This case is analogous to Case 2.",
"Proposition 9 The class defined by $\\phi (x,y)$ is an isomorphism between the $\\in _1$ -reduct and the $\\in _2$ -reduct.",
"By Lemmas REF , REF and REF .",
"A similar result holds for first order Peano arithmetic, extending the categoricity result of Dedekind [1] of second order Peano arithmetic.",
"The proof (see [4]) of this is similar, but somewhat easier."
]
] | 1808.08621 |
[
[
"Embedded Pilot-Aided Channel Estimation for OTFS in Delay-Doppler\n Channels"
],
[
"Abstract Orthogonal time frequency space (OTFS) modulation was shown to provide significant error performance advantages over orthogonal frequency division multiplexing (OFDM) in delay--Doppler channels.",
"In order to detect OTFS modulated data, the channel impulse response needs to be known at the receiver.",
"In this paper, we propose embedded pilot-aided channel estimation schemes for OTFS.",
"In each OTFS frame, we arrange pilot, guard, and data symbols in the delay--Doppler plane to suitably avoid interference between pilot and data symbols at the receiver.",
"We develop such symbol arrangements for OTFS over multipath channels with integer and fractional Doppler shifts, respectively.",
"At the receiver, channel estimation is performed based on a threshold method and the estimated channel information is used for data detection via a message passing (MP) algorithm.",
"Thanks to our specific embedded symbol arrangements, both channel estimation and data detection are performed within the same OTFS frame with a minimum overhead.",
"We compare by simulations the error performance of OTFS using the proposed channel estimation and OTFS with ideally known channel information and observe only a marginal performance loss.",
"We also demonstrate that the proposed channel estimation in OTFS significantly outperforms OFDM with known channel information.",
"Finally, we present extensions of the proposed schemes to MIMO and multi-user uplink/downlink."
],
[
"Introduction",
"Orthogonal frequency division multiplexing (OFDM) is a popular modulation scheme that are currently deployed in 4G long term evolution (LTE) mobile systems.",
"OFDM is known to achieve good robustness and high spectral efficiency for time-invariant frequency selective channels.",
"However, for high-mobility environments such as high-speed railway mobile communications, the channels can be typically time-varying with high Doppler spreads.",
"Under such high Doppler conditions, OFDM is no longer robust and suffers heavy performance degradations.",
"Hence, new modulation schemes that are robust to channel time-variations are being extensively explored.",
"Recently, orthogonal time frequency space (OTFS) modulation was proposed in [1], [2].",
"OTFS exhibits significant advantages over OFDM in multipath delay–Doppler channels where each path exhibits a different delay and Doppler shift.",
"In particular, the idea of transmission in the delay-Doppler domain was introduced in [1], [2].",
"The delay–Doppler domain provides as an alternative representation of a time-varying channel geometry due to moving objects (e.g.",
"transmitters, receivers, or reflectors) in the scene.",
"Leveraging on this representation, OTFS multiplexes each information symbol over a two dimensional (2D) orthogonal basis functions, specifically designed to combat the dynamics of time-varying multipath channels.",
"Then the information symbols placed in the delay-Doppler coordinate system can be converted to the standard time-frequency domain used by traditional modulation schemes such as OFDM.",
"More recently, in [12], a simplified OTFS structure was proposed by including OFDM for time-frequency signal modulation.",
"Its extension to the multiple-input multiple-output (MIMO) case was presented in [13].",
"In general, OTFS uses the delay-Doppler channel response [1], [2], [3] to parameterize the effects of a time-varying channel on any transmitted waveform.",
"In the delay-Doppler domain, the response captures the dominant scatterers in the channel, with their specific delay and Doppler parameters.",
"In the time-frequency domain, this corresponds to a standard time-varying impulse response.",
"Estimating delay-Doppler channel response at the receiver is necessary to perform OTFS detection [4]-[11].",
"Hence, in [10], [11], [14], [15], pilot-aided channel estimation techniques were investigated.",
"In [11], an entire OTFS frame was used for pilot transmission and the estimated channel information was used for data detection in next frame.",
"This method may not be effective if the channel estimation becomes outdated in the following frame.",
"In [10], [15], OTFS channel estimation was conducted in the time–frequency domain, resulting in higher implementation complexity than that of [11], [14], where the channel estimation was conducted in delay–Doppler domain.",
"In [14], channel estimation was considered for OTFS with ideal pulse-shaping waveform over channels with integer Doppler shifts only, i.e., when the channel Doppler taps are aligned to integer delay–Doppler grid.",
"Motivated by [14], in this paper, we consider multipath channels with integer and fractional Doppler shifts, respectivelyFractional Doppler shifts usually occur with a low Doppler resolution..",
"Under such setting, we propose an embedded OTFS channel estimation scheme for point-to-point single-input single-output (SISO) system with ideal and rectangular pulse-shaping waveforms, respectively.",
"Specifically, for each OTFS frame, we arrange a single pilot symbol, guard symbols, and data symbols in the delay–Doppler grid to suitably avoid the interferences between pilot and data symbols.",
"At the receiver, channel estimation is performed based on a threshold method and the estimated channel information is used for data detection via a message passing (MP) algorithm in [4].",
"Depending on the channel and symbol arrangement, the threshold is chosen to optimize the estimation accuracy.",
"Thanks to our specific embedded symbol arrangements, both channel estimation and data detection are performed within the same OTFS frame with a minimum overhead (1% for integer Doppler case and 8% for fractional Doppler case).",
"We compare by simulations the performance of OTFS using the proposed channel estimation schemes and OTFS with perfectly known channel information and observe only a marginal performance degradation.",
"Further, we show that OTFS with our channel estimation significantly outperforms OFDM, with known channel information.",
"Finally, we present the extensions of the proposed channel estimation schemes to MIMO and multi-user uplink/downlink.",
"The rest of the paper is organized as follows.",
"Section II reviews basic OTFS concepts and results, which lay the foundations for the development of OTFS-based channel estimation schemes in Section III.",
"Numerical results are presented in Section IV.",
"Extensions of the proposed channel estimation schemes to other different OTFS systems are presented in Section V followed by the conclusions in Section VI."
],
[
"OTFS: Basic concepts and results",
"In this section, we first review the basic concepts and results of OTFS from [1], [2], [4]."
],
[
"Basic OTFS concepts/notations",
"– The time–frequency signal plane is discretized to a $M\\times N$ grid (for some integers $N, M >0$ ) by sampling time and frequency axes at intervals $T$ (seconds) and $\\Delta f$ (Hz), respectively, i.e., $\\Lambda = \\bigl \\lbrace (nT,m\\Delta f),\\; n=0,\\hdots ,N-1, m=0,\\hdots ,M-1\\bigr \\rbrace \\nonumber $ – The modulated time–frequency samples $X[n,m], n=0,\\hdots ,N-1, m=0,\\hdots ,M-1$ , are transmitted over an OTFS frame with duration $T_f = NT$ and bandwidth $B = M\\Delta f$ .",
"– The delay–Doppler plane is discretized to a $M\\times N$ information grid $\\Gamma = \\Biggl \\lbrace \\left(\\frac{k}{NT},\\frac{l}{M\\Delta f}\\right),\\; k=0,\\hdots ,N-1, l=0,\\hdots ,M-1\\Biggr \\rbrace , \\nonumber $ where $1/M\\Delta f$ and $1/NT$ represent the quantization steps of the delay and Doppler frequency axes, respectively."
],
[
"OTFS mod/demod",
"The modulator first maps a set of $NM$ information symbols $\\lbrace x[k,l], k=0,\\ldots ,N-1, l=0,\\ldots , M-1\\rbrace $ from a modulation alphabet $\\mathbb {A} = \\lbrace a_1, \\cdots , a_{Q} \\rbrace $ (e.g.",
"QAM symbols) of size $Q$ , arranged on the delay–Doppler information grid $\\Gamma $ , to $X[n,m]$ in the time–frequency domain grid using the inverse symplectic finite Fourier transform (ISFFT).",
"Next, the Heisenberg transform is applied to $X[n,m]$ using transmit pulse $g_{\\rm tx}(t)$ to create the time-domain signal $s(t)$ .",
"The signal $s(t)$ is then transmitted over the wireless channel with complex baseband channel impulse response $h(\\tau ,\\nu )$ , which characterizes the channel response to an impulse with delay $\\tau $ and Doppler $\\nu $ [17].",
"The received signal $r(t)$ is processed with the Wigner transform (implementing a receiver filter with an impulse response $g_{\\rm rx}(t)$ ) followed by a sampler, yielding $Y[n,m]$ in the time–frequency domain.",
"We then apply SFFT on $Y[n,m]$ to obtain received symbols $y[k,l]$ in the delay–Doppler domain for symbol detection [1]."
],
[
"OTFS input–output analysis",
"We now look at the relations between received symbols $y[k,l]$ and transmitted symbols $x[k,l]$ .",
"We assume that $h(\\tau ,\\nu )$ has finite support bounded by $[0,\\tau _{\\rm max}]$ on the delay axis and $[-\\nu _{\\rm max},\\nu _{\\rm max}]$ on the Doppler axis, where $\\tau _{\\rm max}$ and $\\nu _{\\rm max}$ are the maximum delay and the maximum Doppler shift among all channel paths.",
"Since typically there are only a small number of reflectors in the channel with associated delays and Dopplers, very few parameters are needed to model the channel in the delay-Doppler domain.",
"The sparse representation of the channel is $h(\\tau ,\\nu ) = \\sum _{i=1}^{P} h_i \\delta (\\tau -\\tau _i) \\delta (\\nu -\\nu _i)$ where $P$ is the number of propagation paths, $h_i$ , $\\tau _i$ , and $\\nu _i$ represent the complex gain, delay, and Doppler shift associated with the $i$ -th path, and $\\delta (\\cdot )$ denotes the Dirac delta function.",
"We denote by $l_{\\tau _i}, k_{\\nu _i}$ the delay and Doppler taps for the $i$ -th path (relatively to the delay–Doppler grid $\\Gamma $ ) defined as $\\tau _i = \\frac{l_{\\tau _i}}{M\\Delta f},\\;\\;\\nu _i = \\frac{k_{\\nu _i} + \\kappa _{\\nu _i}}{NT}$ where $-\\frac{1}{2}< \\kappa _{\\nu _i} \\le \\frac{1}{2}$ represents the fractional Doppler, i.e., the fractional shift from the nearest Doppler tap $k_{\\nu _i}$ .",
"We do not need to consider fractional delays, since the resolution $1/M\\Delta f$ of the time axis is sufficient to approximate the path delays to the nearest sampling points in typical wide-band systems [19].",
"Let us denote $l_{\\tau }$ and $k_{\\nu }$ the delay and Doppler taps corresponding to the largest delay $\\tau _{\\rm max}$ and Doppler $\\nu _{\\rm max}$ .",
"We also assume that the pulses $g_{\\text{tx}}(t)$ and $g_{\\text{rx}}(t)$ are ideal, meaning that they satisfy the bi-orthogonal property condition [1], i.e., the cross-ambiguity function $A_{g_{\\text{rx}}, g_{\\text{tx}}}(t,f) = 0$ for $t \\in (nT-\\tau _{\\rm max},nT+\\tau _{\\rm max})$ , $f \\in (m\\Delta f-\\nu _{\\rm max},m\\Delta f+\\nu _{\\rm max})$ , $\\forall n,m$ , except for $n=0, m=0$ , where $A_{g_{\\text{rx}}, g_{\\text{tx}}}(t,f) = 1$ with $t \\in (-\\tau _{\\rm max},\\tau _{\\rm max})$ and $f \\in (-\\nu _{\\rm max}, +\\nu _{\\rm max})$ .",
"The case of non-ideal yet practical rectangular pulses is discussed in Section III."
],
[
"Integer Doppler shifts",
"The relation between $y[k,l]$ and $x[k,l]$ was derived in [4] as $y[k,l]=\\!\\!\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\sum _{l^{\\prime }=0}^{l_\\tau } b[k^{\\prime },l^{\\prime }] \\hat{h}[k^{\\prime },l^{\\prime }] x[[k-k^{\\prime }]_N,&[l-l^{\\prime }]_M] \\nonumber \\\\& + v[k,l] $ where $\\hat{h}[k^{\\prime },l^{\\prime }] = h[k^{\\prime },l^{\\prime }] e^{-j2\\pi \\frac{k^{\\prime }}{N T}\\frac{l^{\\prime }}{M\\Delta f}} $ , $b[k^{\\prime },l^{\\prime }] \\in \\lbrace 0,1\\rbrace $ is the path indicator, i.e., $b[k^{\\prime },l^{\\prime }]=1$ indicates that there is a path with Doppler tap $k^{\\prime }$ and delay tap $l^{\\prime }$ with corresponding path magnitude $\\hat{h}[k^{\\prime },l^{\\prime }]$ , otherwise, there is no such path, i.e., $b[k^{\\prime },l^{\\prime }]=0$ and $\\hat{h}[k^{\\prime },l^{\\prime }]=0$ .",
"Finally, the term $v[k,l] \\sim \\mathcal {CN} (0, \\sigma ^2)$ is an additive white noise with variance $\\sigma ^2$ , and $[\\cdot ]_N$ , $[\\cdot ]_M$ denote modulo $N$ and $M$ operations, respectively.",
"We have the total number of paths: $\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\sum _{l^{\\prime }=0}^{l_\\tau } b[k^{\\prime },l^{\\prime }]= P.$ Each path circularly shifts the transmitted symbols by the delay and Doppler taps.",
"Figure: The integer Doppler case"
],
[
"Fractional Doppler shifts",
"Similarly, the following result was derived in [4] for the fractional Doppler case $\\!\\!\\!\\!\\!\\!\\!\\!",
"\\!\\!\\!\\!",
"y[k,l]\\!\\!\\!\\!&=&\\!\\!\\!\\!",
"\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\sum _{l^{\\prime }=0}^{l_\\tau } b[k^{\\prime },l^{\\prime }] \\sum _{q=0}^{N-1} \\bar{h}[k^{\\prime },l^{\\prime },\\kappa ^{\\prime },q] \\nonumber \\\\&&\\quad x\\left[[k-k^{\\prime }+q]_N, [l - l^{\\prime }]_M\\right] + v[k,l]$ where $\\kappa ^{\\prime }$ denotes the fractional Doppler associated with the $(k^{\\prime },l^{\\prime })$ path, with the path gain $\\bar{h}[k^{\\prime },l^{\\prime },\\kappa ^{\\prime },q] = \\left(\\frac{e^{j {2\\pi } (-q - \\kappa ^{\\prime }) }-1}{N e^{j \\frac{2\\pi }{N} (- q - \\kappa ^{\\prime })}-N}\\right) h[k^{\\prime },l^{\\prime }] e^{-j2\\pi \\frac{k^{\\prime }+\\kappa ^{\\prime }}{N T}\\frac{l^{\\prime }}{M\\Delta f}}.$ It can be seen that with fractional Doppler shifts, each received symbol is affected by more transmitted symbols than in the case of integer Doppler in (REF ).",
"We can see from (REF ) that when $\\kappa ^{\\prime }=0$ , (REF ) simplifies to (REF ) as expected."
],
[
"OTFS data detection via message passing (MP)",
"From the received symbols $y[k,l]$ , if the channel parameters are known, we can employ the message passing (MP) algorithm in [4] to detect the data symbols $x[k,l]$ using the set of $MN$ linear equations (REF ) or (REF )."
],
[
"Embedded channel estimation for point-to-point SISO Case",
"We first assume that OTFS with ideal waveforms for multipath channel with integer and fractional Doppler cases.",
"Then we consider the extension to OTFS with practical rectangular waveforms."
],
[
"Integer Doppler Case",
"Let $x_p$ denote the pilot symbol with pilot SNR of ${\\rm SNR}_{p}$ , $x_d[k,l]$ denote the data symbols with data SNR of ${\\rm SNR}_{d}$ located at location $[k,l]$ in the delay–Doppler information grid, and 0 denotes the guard symbol.",
"Motivated by [14], we place one pilot symbol $x_p$ , $N_n$ of the guard symbols, and $MN-N_n-1$ information symbols in the delay–Doppler grid $\\Gamma $ for each OTFS frame transmission.",
"The symbols are located in such a way so that at the receiver, we can separate two distinct groups of received symbols: the first group that involves pilot and guard symbols is used for channel estimation, and the second group for data detection.",
"Moreover, the guard symbols guarantee that the received symbols for channel estimation and data detection are not interfered with each other.",
"This helps to provide a more accurate channel estimation to be used for data detection within the same frame.",
"Figure: The fractional Doppler case: Full guard symbolsFor a pilot, we first choose arbitrary grid location $[k_p,l_p]$ such that $0 \\le k_p \\le N-1 $ , and $0 \\le l_p \\le M-1$ .",
"For ease of representation, we choose $0 \\le l_p-l_{\\tau } \\le l_p \\le l_p+l_{\\tau } \\le M-1$ , and $0 \\le k_p-2k_{\\nu } \\le k_p \\le k_p+2k_{\\nu } \\le N-1$ .",
"Recall that $l_\\tau $ and $k_\\nu $ denote the taps corresponding to the maximum delay and Doppler values.",
"We arrange the pilot, guard, and data symbols in the delay–Doppler grid for an OTFS frame transmission as in Fig.",
"REF : $x[k,l] ={\\left\\lbrace \\begin{array}{ll}x_p & ~~k=k_p, l =l_p,\\\\0 & ~~k_p-2k_{\\nu } \\le k \\le k_p+2k_{\\nu }, \\\\&~~\\quad l_p-l_{\\tau } \\le l \\le l_p+l_{\\tau }, \\\\x_d[k,l] & ~~\\text{otherwise.}\\end{array}\\right.}",
"$ In this case, we have $N_n=(2l_{\\tau }+1)(4k_{\\nu }+1)-1$ guard symbols.",
"For example, in Long-Term Evolution (LTE) channels, the overhead for pilot and guard symbols is less than 1% of the data frame [16].",
"At the receiver, we use the received symbols $y[k,l]$ , $k_p - k_{\\nu }\\le k \\le k_p + k_{\\nu }, l_p \\le l \\le l_p+l_{\\tau }$ for channel estimation.",
"Then the remaining received symbols $y[k,l]$ on the grid are used for data detection, as shown in Fig.",
"REF .",
"Due to the transmit symbol arrangement in (REF ), using (REF ), we can express the received symbols for channel estimation as $y[k,l] = b[k-k_p,l-l_p] \\hat{h}[k-k_p,l-l_p] x_p + v[k,l].",
"$ for $k \\in [k_p - k_{\\nu }, k_p + k_{\\nu }], l \\in [l_p, l_p+l_{\\tau }]$ .",
"We can see that if there is a path with Doppler tap $k-k_p$ and delay tap $l-l_p$ , i.e., $b[k-k_p,l-l_p]=1$ , we have $y[k,l] = \\hat{h}[k-k_p,l-l_p] x_p + v[k,l]$ .",
"Otherwise, $y[k,l] = v[k,l]$ .",
"Similarly, we can express the received symbols for data detection as in (REF ), demonstrating no interference between the received symbols for channel estimation and data detection.",
"We propose a simple channel estimation algorithm as follows.",
"For $k \\in [k_p - k_{\\nu }, k_p + k_{\\nu }], l \\in [l_p, l_p+l_{\\tau }]$ , if the magnitude $|y[k,l]| \\ge \\mathcal {T}$ , where $\\mathcal {T}$ is some positive detection threshold, then we estimate $b[k-k_p,l-l_p]=1$ and $\\hat{h}[k-k_p,l-l_p]=y[k,l]/x_p$ .",
"Otherwise, we set $b[k-k_p,l-l_p]=\\hat{h}[k-k_p,l-l_p]=0$ .",
"The proposed threshold-based scheme relies on the fact that if a path exists, the received symbol is the scaled pilot signal with additive white Gaussian noise (see (REF )).",
"Otherwise, it is only noise.",
"By varying the threshold $\\mathcal {T}$ , we can alter the miss detection or false alarm probabilities on path detection.",
"As a result, the error performance of data detection is affected by $\\mathcal {T}$ , as will be shown in Section .",
"We then use the estimated information for data detection, i.e., the received symbols $y[k,l]$ for data detection are $y[k,l] =\\!\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\sum _{l^{\\prime }=0}^{l_\\tau } b[k^{\\prime },l^{\\prime }] \\hat{h}[k^{\\prime },l^{\\prime }] x_d[[k-k^{\\prime }]_N,&[l-l^{\\prime }]_M] \\nonumber \\\\& + v[k,l] $ for $k \\notin [k_p - k_{\\nu }, k_p + k_{\\nu }]$ or $ l \\notin [l_p,l_p+l_{\\tau }]$ .",
"Note that we have a total of $MN-(2k_\\nu +1)(l_\\tau +1)$ received symbols to detect a smaller number of $MN-(2l_{\\tau }+1)(4k_{\\nu }+1)$ data symbols via the MP algorithm in [4]."
],
[
"The fractional Doppler case",
"We consider two cases using full guard symbols and reduced guard symbols, respectively.",
"The former case offers better channel estimation at the expense of the lower spectral efficiency by using more guard symbols and less data symbols, in contrast to the latter case."
],
[
"The case with full guard symbols",
"We arrange the pilot, guard, and data symbols in the delay–Doppler grid, as depicted in Fig.",
"REF : $x[k,l] ={\\left\\lbrace \\begin{array}{ll}x_p, & k=k_p, l =l_p\\\\0, & 0 \\le k \\le N-1, l_p \\!-\\!l_\\tau \\le l \\le l_p \\!+\\!",
"l_\\tau \\\\x_d[k,l], & \\text{otherwise.}",
"\\end{array}\\right.",
"}$ For simplicity of notation, we choose $0 \\le l_p-l_{\\tau } \\le l_p \\le l_p+l_{\\tau } \\le M-1$ .",
"We have the number of guard symbols $N_n=(2l_{\\tau }+1)N-1$ , and the overhead for pilot and guard symbols is about 8% in LTE channels [16].",
"Figure: The fractional Doppler case: Reduced guard symbolsAt the receiver, we use the received symbols $y[k,l], 0 \\le k \\le N-1, l_p \\le l \\le l_p + l_\\tau $ for channel estimation, and the remaining received symbols $y[k,l]$ for data detection (see Fig.",
"REF ).",
"Using (REF ), the received symbols $y[k,l]$ for channel estimation are $y[k,l] \\!=\\!\\!\\!\\!\\!\\!\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\!\\!b[k^{\\prime },l\\!-\\!l_p] \\bar{h}[k^{\\prime },l\\!-\\!l_p,\\kappa ^{\\prime },[k_p\\!+\\!k^{\\prime }\\!-\\!k]_N\\!",
"]\\,x_p \\!\\!+\\!\\!",
"v[k,l] \\nonumber $ for $k \\in [0, N-1], l \\in [l_p, l_p + l_\\tau ]$ .",
"We can rewrite $y[k,l]$ as $y[k,l] = \\tilde{b}[l-l_p] \\tilde{h}[[k-k_p]_N,l-l_p]x_p + v[k,l] $ where $\\tilde{b}[l-l_p] ={\\left\\lbrace \\begin{array}{ll}1, & \\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\!\\!\\!b[k^{\\prime },l-l_p] \\ge 1\\\\0, & \\text{otherwise} \\nonumber \\end{array}\\right.",
"}$ is the path indicator, and $\\tilde{h}[[k-k_p\\!",
"]_N,l-l_p] = \\!\\!\\!\\!\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\!\\!\\!b[k^{\\prime },l-l_p] \\bar{h}[k^{\\prime },l-l_p,\\kappa ^{\\prime },[k_p\\!+\\!k^{\\prime }-k]_N\\!",
"]\\nonumber $ is the effective path gain from the pilot symbol $x_p$ at location $[k_p,l_p]$ to the received symbol $y[k,l]$ .",
"Then $\\tilde{b}[l-l_p]=1$ indicates that there is at least one path with delay tap $l-l_p$ , otherwise, $\\tilde{b}[l-l_p]=0$ .",
"Based on (REF ), we propose the following threshold-based channel estimation algorithm.",
"For $k \\in [0, N-1], l \\in [l_p, l_p + l_\\tau ]$ , if $|y[k,l]| \\ge \\mathcal {T}$ , then we have $\\tilde{b}[l-l_p]=1$ , and $\\tilde{h}[[k-k_p]_N,l-l_p] = y[k,l]/x_p.$ Otherwise, we set $ \\tilde{b}[l-l_p]=\\tilde{h}[[k-k_p]_N,l-l_p] = 0$ .",
"Unlike the integer Doppler case, where we estimate whether an individual path with given delay and Doppler taps exists, in this case, we estimate whether there exists at least one path with a given delay tap.",
"For data detection, similar to (REF ), we rewrite (REF ) as $y[k,l]= \\sum _{l^{\\prime }=0}^{l_\\tau }\\tilde{b}[l^{\\prime }] \\sum _{k^{\\prime }=0}^{N-1} \\tilde{h}[k^{\\prime },l^{\\prime }] x_d[[k\\!-\\!k_{i^{\\prime }}]_N,&[l-l^{\\prime }]_M] \\nonumber \\\\&+v[k,l] $ for $k \\in [0,N-1]$ and $l \\notin [l_p, l_p + l_\\tau ]$ .",
"Now we can adapt the MP algorithm in [4] for data detection in (REF ).",
"Note that, to guarantee no interference between the received symbols for channel estimation and data detection, the guard symbols need to expand over a wider range over the Doppler axis, when compared to the integer Doppler case."
],
[
"The case of reduced guard symbols",
"Employing full guard symbols to avoid interferences provide more accurate channel estimation but with reduced spectral efficiency.",
"To improve the spectral efficiency, we can reduce the number of guard symbols and thus increase the number of data symbols, as discussed below.",
"We arrange the symbols as in Fig.",
"REF $x[k,l] ={\\left\\lbrace \\begin{array}{ll}x_p & ~~k=k_p, l =l_p,\\\\0 & ~~k_p-2k_{\\nu }-2 \\hat{k} \\le k \\le k_p+2k_{\\nu } + 2 \\hat{k}, \\\\&~~\\quad l_p-l_{\\tau } \\le l \\le l_p+l_{\\tau }, \\\\x_d[k,l] & ~~\\text{otherwise}\\end{array}\\right.}",
"$ for some integer $\\hat{k}$ .",
"For smaller $\\hat{k}$ , less guard and more data symbols are used, resulting in an increased spectral efficiency.",
"The received symbols $y[k,l], k_p-k_{\\nu }- \\hat{k} \\le k \\le k_p+k_{\\nu } + \\hat{k}, l_p \\le l \\le l_p + l_\\tau $ are used for channel estimation, while the remaining $y[k,l]$ are used for data detection (see Fig.",
"REF ) From (REF ), for channel estimation, we have $y[k,l] = \\tilde{b}[l\\!-\\!l_p] \\tilde{h}[[k\\!-\\!k_p]_N,l\\!-\\!l_p]x_p \\!+\\!",
"\\mathcal {I}[k,l] \\!+\\!",
"v[k,l] $ for $k_p-k_{\\nu }- \\hat{k} \\le k \\le k_p+k_{\\nu } + \\hat{k}, l_p \\le l \\le l_p + l_\\tau $ .",
"The second term $\\mathcal {I}[k,l]$ is the interferences from all neighboring data symbols $x_d[k,l]$ , i.e., $\\mathcal {I}[k,l]&=&\\!\\!\\!\\!",
"\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\sum _{l^{\\prime }=0}^{l_\\tau } b[k^{\\prime },l^{\\prime }]\\hspace{-19.91692pt} \\sum _{q \\notin [k_p-2k_{\\nu }- 2\\hat{k}, k_p+2k_{\\nu } + 2\\hat{k}]} \\hspace{-19.91692pt} \\bar{h}[k^{\\prime },l^{\\prime },\\kappa ^{\\prime },q] \\nonumber \\\\&&\\quad x_d\\left[[k-k^{\\prime }+q]_N, [l - l^{\\prime }]_M\\right]$ We observe that the interference $\\mathcal {I}[k,l]$ gets larger for smaller $\\hat{k}$ , and similarly for the interference from pilot symbols to the received symbols for data detection.",
"Similar to the case of full guard symbols, we develop a threshold-based algorithm to estimate $\\tilde{b}[l-l_p]$ and $ \\tilde{h}[[k-k_p]_N,l-l_p]$ based on (REF ) by treating $\\mathcal {I}[k,l]$ as additive noise.",
"Based on the simulation results (see next section), we demonstrate that the performance gap of the full guard symbols case (8% overhead) and reduced guard symbols case (2% overhead) is indeed marginal."
],
[
"OTFS with rectangular waveforms",
"So far, we have assumed ideal transmit $g_{\\text{tx}}(t)$ and receive $g_{\\text{rx}}(t)$ pulses.",
"Since the ideal pulses cannot be realized in practice, we now investigate OTFS with the more practical rectangular pulses at both transmitter and receiver.",
"Although these pulses do not satisfy the bi-orthogonality conditions [5], we show that the proposed embedded channel estimation schemes can also be employed for this case.",
"Consider the integer Doppler case for simplicity.",
"With rectangular pulses, the input-output symbol relationship in [5] can be rewritten as $y[k,l]=\\!\\!\\!\\sum _{k^{\\prime }=-k_\\nu }^{k_\\nu } \\sum _{l^{\\prime }=0}^{l_\\tau } b[k^{\\prime },l^{\\prime }] \\hat{h}[k^{\\prime },l^{\\prime }]\\alpha [k,l] x[[k\\!-\\!k^{\\prime }]_N,&[l\\!-\\!l^{\\prime }]_M\\!]",
"\\nonumber \\\\& + v[k,l] \\nonumber $ where $\\alpha [k,l] & ={\\left\\lbrace \\begin{array}{ll}e^{j2\\pi \\left(\\frac{l-l^{\\prime }}{M}\\right) \\frac{k^{\\prime }}{N} } & l^{\\prime } \\le l < M\\\\\\frac{N-1}{N} e^{j2\\pi \\left(\\frac{l-l^{\\prime }}{M}\\right) \\frac{k^{\\prime }}{N} } e^{-j 2 \\pi \\left(\\frac{[k-k^{\\prime }]_N}{N} \\right)} & 0\\le l<l^{\\prime }.\\end{array}\\right.",
"}$ Hence, the threshold-based channel estimation technique can be straightforwardly employed by introducing a known phase $\\alpha [k,l]$ in the detection process.",
"The thresholds for the rectangular waveforms remains the same as the ideal waveforms, since the channel differs only by a phase."
],
[
"Numerical results",
"We illustrate the performance in term of bit-error-rate (BER) of the uncoded OTFS using the proposed channel estimation schemes for integer and fractional Doppler cases.",
"We adopt the following system parameters: Carrier frequency of 4 GHz, sub-carrier spacing of 15 KHz, $M$ = 512, $N = 128$ , and $4-$ QAM signaling.",
"For both OTFS and OFDM systems, Extended Vehicular A model [18] is used, and each delay tap has a single Doppler shift generated by using Jakes' formula, i.e., $\\nu _i = \\nu _{\\rm {max}}\\cos (\\theta _i)$ , where $\\nu _{\\rm max}$ is the maximum Doppler shift determined by the UE speed and $\\theta _i$ is uniformly distributed over $[-\\pi ,\\pi ]$ ."
],
[
"The integer Doppler case",
"Fig.",
"REF compares BER versus data SNRs (${\\rm SNR}_d$ ) for OTFS with known channel information (ideal case) and OTFS using the proposed channel estimation for the integer Doppler case with ${\\rm SNR}_p= 30,$ $35,$ and 40 dB and $\\mathcal {T}=3\\sigma $ .",
"We assume a delay–Doppler channel with maximum delay tap $l_{\\tau }=20$ and Doppler tap $k_{\\nu }=4$ , which corresponds to maximum Doppler speed of 120 Kmph.",
"The overhead for pilot and guard symbols is approximately $1\\%$ of an OTFS frame.",
"We observe that the BER reduces as ${\\rm SNR}_p$ increases, providing more accurate channel estimation and better data detection.",
"Moreover, the performance of OTFS with channel estimation is very close to the ideal case, when ${\\rm SNR}_p=40$ dB (at least 20dB higher than the data ${\\rm SNR}_d$ ).",
"Note that a large pilot power does not affect the peak transmit power as OTFS spreads each delay–Doppler symbol in the entire time–frequency plane thanks to the ISFFT operation.",
"In Fig.",
"REF , we perform comparisons of BER versus ${\\rm SNR}_d$ for different Doppler frequencies with ${\\rm SNR}_p=40$ dB, $l_{\\tau }=20$ , $\\mathcal {T}=3\\sigma $ , and 4-QAM.",
"Consider UE speeds of 30, 120, and 500 Kmph corresponding to maximum Doppler tap $k_{\\nu } = 1,4$ , and 16, respectively.",
"We observe that the proposed estimation scheme exhibits highly similar performance under different Doppler frequencies except a slight performance improvement under higher Doppler frequencies (i.e., $k_{\\nu } = 16$ ).",
"This is due to the fact that more guard symbols and less data symbols are transmitted, leading to better data detection capability at higher ${\\rm SNR}_d$ .",
"Since OTFS performs similarly at different frequencies, in the following, we consider only the UE speed of 120 kmph.",
"We next investigate the effect of the channel estimation threshold $\\mathcal {T}$ on the system performance.",
"Fix ${\\rm SNR}_p=40$ dB.",
"Fig.",
"REF displays BER versus ${\\rm SNR}_d$ with different $\\mathcal {T}$ .",
"We observe that the BER performance improves as $\\mathcal {T}$ increases.",
"For small threshold values, the path false detection probability is higher (i.e., it is more likely to detect non-existent paths), which degrades the BER performance.",
"However, at the same time, increasing the threshold beyond a certain value may cause the likely miss detection of paths with small path-gains, resulting in performance loss.",
"Hence, there is an optimal threshold to balance the false detection and miss detection probabilities.",
"For the given system parameters, we observe that the optimal threshold is approximately $3\\sigma $ .",
"Figure: BER versus SNR d {\\rm SNR}_d: Fractional Doppler with full guard symbols.Figure: BER versus SNR d {\\rm SNR}_d: Fractional Doppler with reduced guard symbols.Figure: BER versus SNR d {\\rm SNR}_d: Fractional Doppler with reduced guard symbols for 16-QAM.Figure: BER versus SNR d {\\rm SNR}_d: low latency communication"
],
[
"The fractional Doppler case",
"Fig.",
"REF shows the BER for different ${\\rm SNR}_p$ with a threshold of $\\mathcal {T}=3\\sigma $ .",
"In this case, the pilot and guard symbols occupy approximately 8% of an OTFS frame.",
"Similar to the integer Doppler case, as more pilot power is used, the error performance is improved.",
"As ${\\rm SNR}_p=50$ dB, OTFS with our proposed embedded channel estimation attains similar performance as OTFS with known channel information.",
"We can see that larger pilot power is required for channels with fractional Doppler shifts than integer Doppler shifts.",
"Last, we compare the BERs of OTFS with channel estimation and OFDM with known channel information and find that OTFS significantly outperforms OFDM, demonstrating the effectiveness of OTFS over delay–Doppler channels.",
"In Fig.",
"REF , we compare the BER performance of OTFS using the proposed channel estimation scheme with reduced guard symbols for $\\hat{k}=2$ and 5.",
"Fix ${\\rm SNR}_p=50$ dB, $\\mathcal {T}=3\\sigma $ , and 4-QAM.",
"With $\\hat{k}=2,$ and 5, the overheads for pilot and guard symbols are roughly 1.5% and 2.3%, respectively, which are much less than the full guard symbols case (roughly 8%).",
"We observe that, as $\\hat{k}$ becomes larger, the performance improves.",
"In particular, with $\\hat{k}=5$ , the performance is very close to that with full guard symbols.",
"For larger $\\hat{k}$ , smaller interference from neighboring data symbols improves the channel estimation accuracy.",
"Hence, there is a tradeoff between spectral efficiency and error performance.",
"In Fig.",
"REF , we illustrate the effectiveness of the proposed channel estimation schemes with full and reduced guard symbols, respectively, using 16-QAM, ${\\rm SNR}_p=60$ dB, and $\\mathcal {T}=3\\sigma $ .",
"We see that with the higher pilot power (i.e., 60 dB), the performance of our channel estimation scheme with full guard symbols is the same as that of the ideal case.",
"Moreover, with 16-QAM, more guard symbols are required (i.e., $\\hat{k}=10$ , about 3.6% guard symbols overhead) to achieve a performance close to the full guard symbols case, when compared to the 4-QAM case that adopts $\\hat{k}=5$ , about 2.3% guard symbols overhead.",
"This is due to the fact that the data detection of 16-QAM case is more sensitive to the channel estimation and hence requires more guard symbols.",
"Figure: Tx pilot, guard, and data symbols for MIMO OTFS system (□\\square : pilot; ∘\\circ : guard symbols)Figure: Rx symbol pattern at one antenna of MIMO OTFS system (▿\\triangledown : data detection, ⊞,⊠,⊗\\boxplus ,\\boxtimes ,\\otimes : channel estimation for Tx antenna 1, 2, and 3, respectively)Figure: Tx pilot, guard, and data symbols for multiuser uplink OTFS system (□\\square : pilot; ∘\\circ : guard symbols)Figure: Tx pilot and data arrangement for multiuser downlink OTFS system (□\\square : pilot; ∘\\circ : guard symbols; ×,⋄,⊕\\times , \\Diamond ,\\oplus : data symbols for users 1, 2, and 3, respectively)"
],
[
"OTFS under low latency communications",
"As next-generation wireless communications mostly require low latency communications, we next simulate the proposed OTFS channel estimation schemes under such scenario.",
"Fig.",
"REF shows the OTFS performance for low latency application with $N=16$ and $M=128$ , corresponding to frame duration of $1.1$ ms. We consider the channel estimation scheme with full guard symbols as the reduced guard symbols case will not improve significantly the spectral efficiency with small $N$ .",
"We observe that the OTFS performance with channel estimation is very close to the ideal case with ${\\rm SNR}_p=60$ dB.",
"Hence, we can conclude that the proposed channel estimation schemes are very efficient under low latency communications."
],
[
"Extensions to MIMO and Multiuser Uplink/Downlink",
"In this section, we extend our embedded channel estimation for point-to-point SISO OTFS systems to MIMO and multi-user uplink/downlink, respectively."
],
[
"Point-to-point MIMO",
"In a MIMO system, each transmit (Tx) antenna arranges its own pilot, guard, and information symbols on the delay–Doppler grid for transmission (see Fig.",
"REF ).",
"The pilot symbol is used to estimate the channels from that Tx antenna to each receive (Rx) antenna.",
"At each Rx antenna, different groups of received symbols are used for channel estimation from that Rx antenna to the Tx antennas, and for data detection from the Tx antennas.",
"Moreover, the received symbols for data detection of the Rx antennas are jointly decoded using MP algorithm.",
"The symbol arrangements from the Tx antennas have to be carefully designed to facilitate the channel estimation and data detection at the Rx antennas.",
"In the following, we describe one such arrangement.",
"Consider a MIMO system with arbitrary $N_t \\ge 1$ and $N_r \\ge 1$ .",
"For ease of presentation, we consider channels with integer Doppler shifts and the case of fractional Doppler shifts is a straightforward extension.",
"Inspired by our previous study in Section III, we propose the following symbol arrangement $x^{n_t}[k,l]$ for the $n_t$ -th Tx antenna ($n_t=1,\\hdots ,N_t$ ) $x^{n_t}[k,l] ={\\left\\lbrace \\begin{array}{ll}x_p & ~~k=k_p, l =l_p+(n_t-1)(l_{\\tau }+1),\\\\0 & ~~k_p-2k_{\\nu } \\le k \\le k_p+2k_{\\nu }, \\\\&~~\\quad l_p-l_{\\tau } \\le l \\le l_p+N_tl_{\\tau } + N_t-1, \\\\x_d^{n_t}[k,l] & ~~\\text{otherwise} \\nonumber \\end{array}\\right.}",
"$ where $x_d^{n_t}[k,l]$ denotes the data symbol at location $[k,l]$ of $n_t$ -th Tx antenna.",
"We can see that the pilot symbols of the Tx antennas are sufficiently separated (by the maximum delay tap $l_{\\tau }$ along the delay axis) so that they do not interfere with each other at the Rx antennas, as demonstrated in Fig.",
"REF for an exemplary MIMO system with three Tx antennas.",
"At the $n_r$ -th Rx antenna ($n_r=1,\\hdots ,N_r$ ), the received symbols $y^{n_r}[k,l], k_p-k_{\\nu } \\le k \\le k_p+k_{\\nu }, l_p+(n_t-1)(l_{\\tau }+1) \\le l \\le l_p+n_tl_{\\tau } + n_t-1$ , are used for channel estimation to the $n_t$ -th Tx antenna.",
"These received symbols are affected by the pilot signal of the $n_t$ -th Tx antenna and by the channel between the $n_t$ -th Tx and $n_r$ -th Rx antennas only, as shown in Fig.",
"REF .",
"Hence, the channel estimation technique in Section III can be applied straightforwardly.",
"The remaining received symbols of the $n_r$ -th Rx antenna are functions of the data symbols from all the Tx antennas and thus a joint detection in [11] can be applied.",
"We omit the details for brevity."
],
[
"Multiuser",
"Consider a multiuser system, where single-antenna users communicate with base station in uplink or downlnk.",
"The base station has either single or multiple antennas.",
"In the following, we present embedded channel estimation schemes using Tx symbol arrangement for the users and base station."
],
[
"Uplink",
"Consider single-antenna base station.",
"We assume orthogonal resource allocation among the users.",
"One example of the Tx symbol arrangements for three-user case is shown in Fig.",
"REF .",
"For each user, in each OTFS frame, the grid locations $[k,l], k_p-2k_{\\nu } \\le k \\le k_p+2k_{\\nu }, l_p-l_{\\tau } \\le l \\le l_p+N_ul_{\\tau } + N_u-1$ are used for pilot and guard symbols, where $N_u$ is the number of users.",
"The pilot symbols of the users are located sufficiently apart at suitable locations as in the MIMO case.",
"Moreover, each user occupies only a non-overlapping portion of the rest of the grid locations for its data transmissions with the remaining grid locations being used for zero symbols since orthogonal resource allocations is required, as shown in Fig.",
"REF , where green, blue, and yellow grids contains data for Users 1, 2, and 3, respectively.",
"The data portion for each user depends on the resource requirement/allocation.",
"Based on the Tx symbol arrangements, the base station exploits suitable received symbols for channel estimation and data detection for the users.",
"Remark 1 When the base station has multiple antennas, the grid locations for pilot and guard symbols for the users remain intact.",
"However, each user can exploit a larger portion, even full remaining grids for data transmissions, similar to the MIMO case."
],
[
"Downlink",
"Consider single-antenna base station, transmitting a pilot symbol being enclosed with guard symbols, similar to the point-to-point SISO case.",
"This pilot signal is used by all the users to estimate the channel from itself to the base station.",
"The rest of delay–Doppler grid locations is used for data transmissions to the users.",
"Since orthogonal resource allocation is required, data symbols for users should be sufficiently separated using guard symbols to avoid inter-user interferences, as shown in Fig.",
"REF , where yellow grids represent the guard symbols between users.",
"Each user exploits appropriate groups of received symbols for channel estimation and detection of its own data.",
"Table: Total number of pilot and guard symbols required for different embedded channel estimation schemesTable REF summarizes the total number of pilot and guard symbols required for the different channel estimation methods in our paper."
],
[
"Conclusion",
"In this work, we have developed embedded pilot-aided OTFS channel estimation schemes.",
"In particular, we arrange pilot, guard, and information symbols in the delay–Doppler grids to suitably avoid interference between pilot and data symbols.",
"We design such arrangements for OTFS with ideal and rectangular pulses over channels with integer or fractional Doppler paths, respectively.",
"At the receiver, channel estimation is performed based on a threshold method and the estimated channel information is used for data detection via a MP algorithm.",
"We compare by simulations the error performance of OTFS using the proposed channel estimation schemes and OTFS with perfectly known channel information and observe only a marginal performance loss.",
"Further, we show that OTFS with our channel estimation significantly outperforms OFDM with ideal channel information.",
"Extensions of the proposed schemes to MIMO and multi-user uplink/downlink have been presented."
],
[
"Acknowledgement",
"This research work is supported by the Australian Research Council under Discovery Project ARC DP160101077.",
"Simulations were undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government."
]
] | 1808.08360 |